libpari-dev 2.9.4-1 / usr / share / pari / pari.desc

Function: !_
Class: basic
Section: symbolic_operators
C-Name: gnot
Prototype: G
Help: !_
Description: 
 (negbool):bool:parens                $1
 (bool):negbool:parens                $1

Function: #_
Class: basic
Section: symbolic_operators
C-Name: glength
Prototype: lG
Help: #x: number of non code words in x, number of characters for a string.
Description: 
 (vecsmall):lg      lg($1)
 (vec):lg           lg($1)
 (pol):small        lgpol($1)
 (gen):small        glength($1)

Function: %
Class: basic
Section: symbolic_operators
C-Name: pari_get_hist
Prototype: D0,L,
Help: last history item.

Function: %#
Class: basic
Section: symbolic_operators
C-Name: pari_get_histtime
Prototype: lD0,L,
Help: time to compute last history item.

Function: +_
Class: basic
Section: symbolic_operators
Help: +_
Description: 
 (small):small:parens                      $1
 (int):int:parens:copy                     $1
 (real):real:parens:copy                   $1
 (mp):mp:parens:copy                       $1
 (gen):gen:parens:copy                     $1

Function: -_
Class: basic
Section: symbolic_operators
C-Name: gneg
Prototype: G
Help: -_
Description: 
 (small):small:parens           -$(1)
 (int):int                      negi($1)
 (real):real                    negr($1)
 (mp):mp                        mpneg($1)
 (gen):gen                      gneg($1)
 
 (Fp):Fp     Fp_neg($1, p)
 (FpX):FpX   FpX_neg($1, p)
 (Fq):Fq     Fq_neg($1, T, p)
 (FqX):FqX   FqX_neg($1, T, p)

Function: Catalan
Class: basic
Section: transcendental
C-Name: mpcatalan
Prototype: p
Help: Catalan=Catalan(): Catalan's number with current precision.
Description: 
 ():real:prec        mpcatalan($prec)
Doc: Catalan's constant $G = \sum_{n>=0}\dfrac{(-1)^n}{(2n+1)^2}=0.91596\cdots$.
 Note that \kbd{Catalan} is one of the few reserved names which cannot be
 used for user variables.

Function: Col
Class: basic
Section: conversions
C-Name: gtocol0
Prototype: GD0,L,
Help: Col(x, {n}): transforms the object x into a column vector of dimension n.
Description: 
 (gen):vec     gtocol($1)
Doc: 
 transforms the object $x$ into a column vector. The dimension of the
 resulting vector can be optionally specified via the extra parameter $n$.
 
 If $n$ is omitted or $0$, the dimension depends on the type of $x$; the
 vector has a single component, except when $x$ is
 
 \item a vector or a quadratic form (in which case the resulting vector
 is simply the initial object considered as a row vector),
 
 \item a polynomial or a power series. In the case of a polynomial, the
 coefficients of the vector start with the leading coefficient of the
 polynomial, while for power series only the significant coefficients are
 taken into account, but this time by increasing order of degree.
 In this last case, \kbd{Vec} is the reciprocal function of \kbd{Pol} and
 \kbd{Ser} respectively,
 
 \item a matrix (the column of row vector comprising the matrix is returned),
 
 \item a character string (a vector of individual characters is returned).
 
 In the last two cases (matrix and character string), $n$ is meaningless and
 must be omitted or an error is raised. Otherwise, if $n$ is given, $0$
 entries are appended at the end of the vector if $n > 0$, and prepended at
 the beginning if $n < 0$. The dimension of the resulting vector is $|n|$.
Variant: \fun{GEN}{gtocol}{GEN x} is also available.

Function: Colrev
Class: basic
Section: conversions
C-Name: gtocolrev0
Prototype: GD0,L,
Help: Colrev(x, {n}): transforms the object x into a column vector of
 dimension n in reverse order with respect to Col(x, {n}). Empty vector if x
 is omitted.
Description: 
 (gen):vec     gtocolrev($1)
Doc: 
 as $\kbd{Col}(x, -n)$, then reverse the result. In particular,
 \kbd{Colrev} is the reciprocal function of \kbd{Polrev}: the
 coefficients of the vector start with the constant coefficient of the
 polynomial and the others follow by increasing degree.
Variant: \fun{GEN}{gtocolrev}{GEN x} is also available.

Function: DEBUGLEVEL
Class: gp2c
C-Name: DEBUGLEVEL
Prototype: v
Description: 
 ():small                         DEBUGLEVEL

Function: Euler
Class: basic
Section: transcendental
C-Name: mpeuler
Prototype: p
Help: Euler=Euler(): Euler's constant with current precision.
Description: 
 ():real:prec        mpeuler($prec)
Doc: Euler's constant $\gamma=0.57721\cdots$. Note that
 \kbd{Euler} is one of the few reserved names which cannot be used for
 user variables.

Function: I
Class: basic
Section: transcendental
C-Name: gen_I
Prototype: 
Help: I=I(): square root of -1.
Description: 
Doc: the complex number $\sqrt{-1}$.

Function: List
Class: basic
Section: conversions
C-Name: gtolist
Prototype: DG
Help: List({x=[]}): transforms the vector or list x into a list. Empty list
 if x is omitted.
Description: 
 ():list           mklist()
 (gen):list        gtolist($1)
Doc: 
 transforms a (row or column) vector $x$ into a list, whose components are
 the entries of $x$. Similarly for a list, but rather useless in this case.
 For other types, creates a list with the single element $x$. Note that,
 except when $x$ is omitted, this function creates a small memory leak; so,
 either initialize all lists to the empty list, or use them sparingly.
Variant: The variant \fun{GEN}{mklist}{void} creates an empty list.

Function: Map
Class: basic
Section: conversions
C-Name: gtomap
Prototype: DG
Help: Map({x}): converts the matrix [a_1,b_1;a_2,b_2;...;a_n,b_n] to the map a_i->b_i.
Doc: A ``Map'' is an associative array, or dictionary: a data
 type composed of a collection of (\emph{key}, \emph{value}) pairs, such that
 each key appears just once in the collection. This function
 converts the matrix $[a_1,b_1;a_2,b_2;\dots;a_n,b_n]$ to the map $a_i\mapsto
 b_i$.
 \bprog
 ? M = Map(factor(13!));
 ? mapget(M,3)
 %2 = 5
 @eprog\noindent If the argument $x$ is omitted, creates an empty map, which
 may be filled later via \tet{mapput}.

Function: Mat
Class: basic
Section: conversions
C-Name: gtomat
Prototype: DG
Help: Mat({x=[]}): transforms any GEN x into a matrix. Empty matrix if x is
 omitted.
Description: 
 ():vec        cgetg(1, t_MAT)
 (gen):vec     gtomat($1)
Doc: 
 transforms the object $x$ into a matrix.
 If $x$ is already a matrix, a copy of $x$ is created.
 If $x$ is a row (resp. column) vector, this creates a 1-row (resp.
 1-column) matrix, \emph{unless} all elements are column (resp.~row) vectors
 of the same length, in which case the vectors are concatenated sideways
 and the attached big matrix is returned.
 If $x$ is a binary quadratic form, creates the attached $2\times 2$
 matrix. Otherwise, this creates a $1\times 1$ matrix containing $x$.
 
 \bprog
 ? Mat(x + 1)
 %1 =
 [x + 1]
 ? Vec( matid(3) )
 %2 = [[1, 0, 0]~, [0, 1, 0]~, [0, 0, 1]~]
 ? Mat(%)
 %3 =
 [1 0 0]
 
 [0 1 0]
 
 [0 0 1]
 ? Col( [1,2; 3,4] )
 %4 = [[1, 2], [3, 4]]~
 ? Mat(%)
 %5 =
 [1 2]
 
 [3 4]
 ? Mat(Qfb(1,2,3))
 %6 =
 [1 1]
 
 [1 3]
 @eprog

Function: Mod
Class: basic
Section: conversions
C-Name: gmodulo
Prototype: GG
Help: Mod(a,b): creates 'a modulo b'.
Description: 
 (small, small):gen         gmodulss($1, $2)
 (small, gen):gen           gmodulsg($1, $2)
 (gen, gen):gen             gmodulo($1, $2)
Doc: in its basic form, creates an intmod or a polmod $(a \mod b)$; $b$ must
 be an integer or a polynomial. We then obtain a \typ{INTMOD} and a
 \typ{POLMOD} respectively:
 \bprog
 ? t = Mod(2,17); t^8
 %1 = Mod(1, 17)
 ? t = Mod(x,x^2+1); t^2
 %2 = Mod(-1, x^2+1)
 @eprog\noindent If $a \% b$ makes sense and yields a result of the
 appropriate type (\typ{INT} or scalar/\typ{POL}), the operation succeeds as
 well:
 \bprog
 ? Mod(1/2, 5)
 %3 = Mod(3, 5)
 ? Mod(7 + O(3^6), 3)
 %4 = Mod(1, 3)
 ? Mod(Mod(1,12), 9)
 %5 = Mod(1, 3)
 ? Mod(1/x, x^2+1)
 %6 = Mod(-1, x^2+1)
 ? Mod(exp(x), x^4)
 %7 = Mod(1/6*x^3 + 1/2*x^2 + x + 1, x^4)
 @eprog
 If $a$ is a complex object, ``base change'' it to $\Z/b\Z$ or $K[x]/(b)$,
 which is equivalent to, but faster than, multiplying it by \kbd{Mod(1,b)}:
 \bprog
 ? Mod([1,2;3,4], 2)
 %8 =
 [Mod(1, 2) Mod(0, 2)]
 
 [Mod(1, 2) Mod(0, 2)]
 ? Mod(3*x+5, 2)
 %9 = Mod(1, 2)*x + Mod(1, 2)
 ? Mod(x^2 + y*x + y^3, y^2+1)
 %10 = Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1)*x + Mod(-y, y^2 + 1)
 @eprog
 
 This function is not the same as $x$ \kbd{\%} $y$, the result of which
 has no knowledge of the intended modulus $y$. Compare
 \bprog
 ? x = 4 % 5; x + 1
 %1 = 5
 ? x = Mod(4,5); x + 1
 %2 = Mod(0,5)
 @eprog Note that such ``modular'' objects can be lifted via \tet{lift} or
 \tet{centerlift}. The modulus of a \typ{INTMOD} or \typ{POLMOD} $z$ can
 be recovered via \kbd{$z$.mod}.

Function: O
Class: basic
Section: polynomials
C-Name: ggrando
Prototype: 
Help: O(p^e): p-adic or power series zero with precision given by e.
Doc: if $p$ is an integer
 greater than $2$, returns a $p$-adic $0$ of precision $e$. In all other
 cases, returns a power series zero with precision given by $e v$, where $v$
 is the $X$-adic valuation of $p$ with respect to its main variable.
Variant: \fun{GEN}{zeropadic}{GEN p, long e} for a $p$-adic and
 \fun{GEN}{zeroser}{long v, long e} for a power series zero in variable $v$.

Function: O(_^_)
Class: basic
Section: programming/internals
C-Name: ggrando
Prototype: GD1,L,
Help: O(p^e): p-adic or power series zero with precision given by e.
Description: 
 (gen):gen          ggrando($1, 1)
 (1,small):gen      ggrando(gen_1, $2)
 (int,small):gen    zeropadic($1, $2)
 (gen,small):gen    ggrando($1, $2)
 (var,small):gen    zeroser($1, $2)

Function: Pi
Class: basic
Section: transcendental
C-Name: mppi
Prototype: p
Help: Pi=Pi(): the constant pi, with current precision.
Description: 
 ():real:prec        mppi($prec)
Doc: the constant $\pi$ ($3.14159\cdots$). Note that \kbd{Pi} is one of the few
 reserved names which cannot be used for user variables.

Function: Pol
Class: basic
Section: conversions
C-Name: gtopoly
Prototype: GDn
Help: Pol(t,{v='x}): convert t (usually a vector or a power series) into a
 polynomial with variable v, starting with the leading coefficient.
Description: 
 (gen,?var):pol  gtopoly($1, $2)
Doc: 
 transforms the object $t$ into a polynomial with main variable $v$. If $t$
 is a scalar, this gives a constant polynomial. If $t$ is a power series with
 non-negative valuation or a rational function, the effect is similar to
 \kbd{truncate}, i.e.~we chop off the $O(X^k)$ or compute the Euclidean
 quotient of the numerator by the denominator, then change the main variable
 of the result to $v$.
 
 The main use of this function is when $t$ is a vector: it creates the
 polynomial whose coefficients are given by $t$, with $t[1]$ being the leading
 coefficient (which can be zero). It is much faster to evaluate
 \kbd{Pol} on a vector of coefficients in this way, than the corresponding
 formal expression $a_n X^n + \dots + a_0$, which is evaluated naively exactly
 as written (linear versus quadratic time in $n$). \tet{Polrev} can be used if
 one wants $x[1]$ to be the constant coefficient:
 \bprog
 ? Pol([1,2,3])
 %1 = x^2 + 2*x + 3
 ? Polrev([1,2,3])
 %2 = 3*x^2 + 2*x + 1
 @eprog\noindent
 The reciprocal function of \kbd{Pol} (resp.~\kbd{Polrev}) is \kbd{Vec} (resp.~
 \kbd{Vecrev}).
 \bprog
 ? Vec(Pol([1,2,3]))
 %1 = [1, 2, 3]
 ? Vecrev( Polrev([1,2,3]) )
 %2 = [1, 2, 3]
 @eprog\noindent
 
 \misctitle{Warning} This is \emph{not} a substitution function. It will not
 transform an object containing variables of higher priority than~$v$.
 \bprog
 ? Pol(x + y, y)
   ***   at top-level: Pol(x+y,y)
   ***                 ^----------
   *** Pol: variable must have higher priority in gtopoly.
 @eprog

Function: Polrev
Class: basic
Section: conversions
C-Name: gtopolyrev
Prototype: GDn
Help: Polrev(t,{v='x}): convert t (usually a vector or a power series) into a
 polynomial with variable v, starting with the constant term.
Description: 
 (gen,?var):pol  gtopolyrev($1, $2)
Doc: 
 transform the object $t$ into a polynomial
 with main variable $v$. If $t$ is a scalar, this gives a constant polynomial.
 If $t$ is a power series, the effect is identical to \kbd{truncate}, i.e.~it
 chops off the $O(X^k)$.
 
 The main use of this function is when $t$ is a vector: it creates the
 polynomial whose coefficients are given by $t$, with $t[1]$ being the
 constant term. \tet{Pol} can be used if one wants $t[1]$ to be the leading
 coefficient:
 \bprog
 ? Polrev([1,2,3])
 %1 = 3*x^2 + 2*x + 1
 ? Pol([1,2,3])
 %2 = x^2 + 2*x + 3
 @eprog
 The reciprocal function of \kbd{Pol} (resp.~\kbd{Polrev}) is \kbd{Vec} (resp.~
 \kbd{Vecrev}).

Function: Qfb
Class: basic
Section: conversions
C-Name: Qfb0
Prototype: GGGDGp
Help: Qfb(a,b,c,{D=0.}): binary quadratic form a*x^2+b*x*y+c*y^2. D is
 optional (0.0 by default) and initializes Shanks's distance if b^2-4*a*c>0.
Doc: creates the binary quadratic form\sidx{binary quadratic form}
 $ax^2+bxy+cy^2$. If $b^2-4ac>0$, initialize \idx{Shanks}' distance
 function to $D$. Negative definite forms are not implemented,
 use their positive definite counterpart instead.
Variant: Also available are
 \fun{GEN}{qfi}{GEN a, GEN b, GEN c} (assumes $b^2-4ac<0$) and
 \fun{GEN}{qfr}{GEN a, GEN b, GEN c, GEN D} (assumes $b^2-4ac>0$).

Function: Ser
Class: basic
Section: conversions
C-Name: gtoser
Prototype: GDnDP
Help: Ser(s,{v='x},{d=seriesprecision}): convert s into a power series with
 variable v and precision d, starting with the constant coefficient.
Doc: transforms the object $s$ into a power series with main variable $v$
 ($x$ by default) and precision (number of significant terms) equal to
 $d \geq 0$ ($d = \kbd{seriesprecision}$ by default). If $s$ is a
 scalar, this gives a constant power series in $v$ with precision \kbd{d}.
 If $s$ is a polynomial, the polynomial is truncated to $d$ terms if needed
 \bprog
 ? Ser(1, 'y, 5)
 %1 = 1 + O(y^5)
 ? Ser(x^2,, 5)
 %2 = x^2 + O(x^7)
 ? T = polcyclo(100)
 %3 = x^40 - x^30 + x^20 - x^10 + 1
 ? Ser(T, 'x, 11)
 %4 = 1 - x^10 + O(x^11)
 @eprog\noindent The function is more or less equivalent with multiplication by
 $1 + O(v^d)$ in theses cases, only faster.
 
 If $s$ is a vector, on the other hand, the coefficients of the vector are
 understood to be the coefficients of the power series starting from the
 constant term (as in \tet{Polrev}$(x)$), and the precision $d$ is ignored:
 in other words, in this case, we convert \typ{VEC} / \typ{COL} to the power
 series whose significant terms are exactly given by the vector entries.
 Finally, if $s$ is already a power series in $v$, we return it verbatim,
 ignoring $d$ again. If $d$ significant terms are desired in the last two
 cases, convert/truncate to \typ{POL} first.
 \bprog
 ? v = [1,2,3]; Ser(v, t, 7)
 %5 = 1 + 2*t + 3*t^2 + O(t^3)  \\ 3 terms: 7 is ignored!
 ? Ser(Polrev(v,t), t, 7)
 %6 = 1 + 2*t + 3*t^2 + O(t^7)
 ? s = 1+x+O(x^2); Ser(s, x, 7)
 %7 = 1 + x + O(x^2)  \\ 2 terms: 7 ignored
 ? Ser(truncate(s), x, 7)
 %8 = 1 + x + O(x^7)
 @eprog\noindent
 The warning given for \kbd{Pol} also applies here: this is not a substitution
 function.

Function: Set
Class: basic
Section: conversions
C-Name: gtoset
Prototype: DG
Help: Set({x=[]}): convert x into a set, i.e. a row vector with strictly
 increasing coefficients. Empty set if x is omitted.
Description: 
 ():vec           cgetg(1,t_VEC)
 (gen):vec        gtoset($1)
Doc: 
 converts $x$ into a set, i.e.~into a row vector, with strictly increasing
 entries with respect to the (somewhat arbitrary) universal comparison function
 \tet{cmp}. Standard container types \typ{VEC}, \typ{COL}, \typ{LIST} and
 \typ{VECSMALL} are converted to the set with corresponding elements. All
 others are converted to a set with one element.
 \bprog
 ? Set([1,2,4,2,1,3])
 %1 = [1, 2, 3, 4]
 ? Set(x)
 %2 = [x]
 ? Set(Vecsmall([1,3,2,1,3]))
 %3 = [1, 2, 3]
 @eprog

Function: Str
Class: basic
Section: conversions
C-Name: Str
Prototype: s*
Help: Str({x}*): concatenates its (string) argument into a single string.
Description: 
 (gen):genstr:copy:parens      $genstr:1
 (gen,gen):genstr              Str(mkvec2($1, $2))
 (gen,gen,gen):genstr          Str(mkvec3($1, $2, $3))
 (gen,gen,gen,gen):genstr      Str(mkvec4($1, $2, $3, $4))
 (gen,...):genstr              Str(mkvecn($#, $2))
Doc: 
 converts its argument list into a
 single character string (type \typ{STR}, the empty string if $x$ is omitted).
 To recover an ordinary \kbd{GEN} from a string, apply \kbd{eval} to it. The
 arguments of \kbd{Str} are evaluated in string context, see \secref{se:strings}.
 
 \bprog
 ? x2 = 0; i = 2; Str(x, i)
 %1 = "x2"
 ? eval(%)
 %2 = 0
 @eprog\noindent
 This function is mostly useless in library mode. Use the pair
 \tet{strtoGEN}/\tet{GENtostr} to convert between \kbd{GEN} and \kbd{char*}.
 The latter returns a malloced string, which should be freed after usage.
 %\syn{NO}

Function: Strchr
Class: basic
Section: conversions
C-Name: Strchr
Prototype: G
Help: Strchr(x): converts x to a string, translating each integer into a
 character.
Doc: 
 converts $x$ to a string, translating each integer
 into a character.
 \bprog
 ? Strchr(97)
 %1 = "a"
 ? Vecsmall("hello world")
 %2 = Vecsmall([104, 101, 108, 108, 111, 32, 119, 111, 114, 108, 100])
 ? Strchr(%)
 %3 = "hello world"
 @eprog

Function: Strexpand
Class: basic
Section: conversions
C-Name: Strexpand
Prototype: s*
Help: Strexpand({x}*): concatenates its (string) argument into a single
 string, performing tilde expansion.
Doc: 
 converts its argument list into a
 single character string (type \typ{STR}, the empty string if $x$ is omitted).
 Then perform \idx{environment expansion}, see \secref{se:envir}.
 This feature can be used to read \idx{environment variable} values.
 \bprog
 ? Strexpand("$HOME/doc")
 %1 = "/home/pari/doc"
 @eprog
 
 The individual arguments are read in string context, see \secref{se:strings}.
 %\syn{NO}

Function: Strprintf
Class: basic
Section: programming/specific
C-Name: Strprintf
Prototype: ss*
Help: Strprintf(fmt,{x}*): returns a string built from the remaining
 arguments according to the format fmt.
Doc: returns a string built from the remaining arguments according to the
 format fmt. The format consists of ordinary characters (not \%), printed
 unchanged, and conversions specifications. See \kbd{printf}.
 %\syn{NO}

Function: Strtex
Class: basic
Section: conversions
C-Name: Strtex
Prototype: s*
Help: Strtex({x}*): translates its (string) arguments to TeX format and
 returns the resulting string.
Doc: 
 translates its arguments to TeX
 format, and concatenates the results into a single character string (type
 \typ{STR}, the empty string if $x$ is omitted).
 
 The individual arguments are read in string context, see \secref{se:strings}.
 %\syn{NO}

Function: Vec
Class: basic
Section: conversions
C-Name: gtovec0
Prototype: GD0,L,
Help: Vec(x, {n}): transforms the object x into a vector of dimension n.
Description: 
 (gen):vec        gtovec($1)
Doc: 
 transforms the object $x$ into a row vector. The dimension of the
 resulting vector can be optionally specified via the extra parameter $n$.
 
 If $n$ is omitted or $0$, the dimension depends on the type of $x$; the
 vector has a single component, except when $x$ is
 
 \item a vector or a quadratic form: returns the initial object considered as a
 row vector,
 
 \item a polynomial or a power series: returns a vector consisting of the coefficients.
 In the case of a polynomial, the coefficients of the vector start with the leading
 coefficient of the polynomial, while for power series only the significant coefficients
 are taken into account, but this time by increasing order of degree.
 \kbd{Vec} is the reciprocal function of \kbd{Pol} for a polynomial and of
 \kbd{Ser} for a power series,
 
 \item a matrix: returns the vector of columns comprising the matrix,
 
 \item a character string: returns the vector of individual characters,
 
 \item a map: returns the vector of the domain of the map,
 
 \item an error context (\typ{ERROR}): returns the error components, see
 \tet{iferr}.
 
 In the last four cases (matrix, character string, map, error), $n$ is
 meaningless and must be omitted or an error is raised. Otherwise, if $n$ is
 given, $0$ entries are appended at the end of the vector if $n > 0$, and
 prepended at the beginning if $n < 0$. The dimension of the resulting vector
 is $|n|$. Variant: \fun{GEN}{gtovec}{GEN x} is also available.

Function: Vecrev
Class: basic
Section: conversions
C-Name: gtovecrev0
Prototype: GD0,L,
Help: Vecrev(x, {n}): transforms the object x into a vector of dimension n
 in reverse order with respect to Vec(x, {n}). Empty vector if x is omitted.
Description: 
 (gen):vec     gtovecrev($1)
Doc: 
 as $\kbd{Vec}(x, -n)$, then reverse the result. In particular,
 \kbd{Vecrev} is the reciprocal function of \kbd{Polrev}: the
 coefficients of the vector start with the constant coefficient of the
 polynomial and the others follow by increasing degree.
Variant: \fun{GEN}{gtovecrev}{GEN x} is also available.

Function: Vecsmall
Class: basic
Section: conversions
C-Name: gtovecsmall0
Prototype: GD0,L,
Help: Vecsmall(x, {n}): transforms the object x into a VECSMALL of dimension n.
Description: 
 (gen):vecsmall                gtovecsmall($1)
Doc: 
 transforms the object $x$ into a row vector of type \typ{VECSMALL}. The
 dimension of the resulting vector can be optionally specified via the extra
 parameter $n$.
 
 This acts as \kbd{Vec}$(x,n)$, but only on a limited set of objects:
 the result must be representable as a vector of small integers.
 If $x$ is a character string, a vector of individual characters in ASCII
 encoding is returned (\tet{Strchr} yields back the character string).
Variant: \fun{GEN}{gtovecsmall}{GEN x} is also available.

Function: [_.._]
Class: basic
Section: programming/internals
C-Name: vecrange
Prototype: GG
Help: [a..b] = [a,a+1,...,b]
Description: 
 (gen,gen):vec     vecrange($1, $2)
 (small,small):vec vecrangess($1, $2)

Function: [_|_<-_,_;_]
Class: basic
Section: programming/internals
C-Name: vecexpr1
Prototype: mGVDEDE
Help: [a(x)|x<-b,c(x);...]
Wrapper: (,,G,bG)
Description: 
 (gen,,closure):gen         veccatapply(${3 cookie}, ${3 wrapper}, $1)
 (gen,,closure,closure):gen veccatselapply(${4 cookie}, ${4 wrapper}, ${3 cookie}, ${3 wrapper}, $1)

Function: [_|_<-_,_]
Class: basic
Section: programming/internals
C-Name: vecexpr0
Prototype: GVDEDE
Help: [a(x)|x<-b,c(x)] = apply(a,select(c,b))
Wrapper: (,,G,bG)
Description: 
 (gen,,closure):gen         vecapply(${3 cookie}, ${3 wrapper}, $1)
 (gen,,,closure):gen        vecselect(${4 cookie}, ${4 wrapper}, $1)
 (gen,,closure,closure):gen vecselapply(${4 cookie}, ${4 wrapper}, ${3 cookie}, ${3 wrapper}, $1)

Function: _!
Class: basic
Section: symbolic_operators
C-Name: mpfact
Prototype: L
Help: n!: factorial of n.
Description: 
 (small):int                        mpfact($1)

Function: _!=_
Class: basic
Section: symbolic_operators
C-Name: gne
Prototype: GG
Help: _!=_
Description: 
 (small, small):bool:parens             $(1) != $(2)
 (lg, lg):bool:parens                   $(1) != $(2)
 (small, int):bool:parens               cmpsi($1, $2) != 0
 (int, small):bool:parens               cmpis($1, $2) != 0
 (int, 1):negbool                       equali1($1)
 (int, -1):negbool                      equalim1($1)
 (int, int):negbool                     equalii($1, $2)
 (real,real):bool                       cmprr($1, $2) != 0
 (mp, mp):bool:parens                   mpcmp($1, $2) != 0
 (errtyp, errtyp):bool:parens           $(1) != $(2)
 (errtyp, #str):bool:parens             $(1) != $(errtyp:2)
 (#str, errtyp):bool:parens             $(errtyp:1) != $(2)
 (typ, typ):bool:parens                 $(1) != $(2)
 (typ, #str):bool:parens                $(1) != $(typ:2)
 (#str, typ):bool:parens                $(typ:1) != $(2)
 (str, str):bool                        strcmp($1, $2)
 (small, gen):negbool                   gequalsg($1, $2)
 (gen, small):negbool                   gequalgs($1, $2)
 (gen, gen):negbool                     gequal($1, $2)

Function: _%=_
Class: basic
Section: symbolic_operators
C-Name: gmode
Prototype: &G
Help: x%=y: shortcut for x=x%y.
Description: 
 (*small, small):small:parens            $1 = smodss($1, $2)
 (*int, small):int:parens                $1 = modis($1, $2)
 (*int, int):int:parens                  $1 = modii($1, $2)
 (*pol, gen):gen:parens                  $1 = gmod($1, $2)
 (*gen, small):gen:parens                $1 = gmodgs($1, $2)
 (*gen, gen):gen:parens                  $1 = gmod($1, $2)

Function: _%_
Class: basic
Section: symbolic_operators
C-Name: gmod
Prototype: GG
Help: x%y: Euclidean remainder of x and y.
Description: 
 (small, small):small            smodss($1, $2)
 (small, int):int                modsi($1, $2)
 (int, small):small              smodis($1, $2)
 (int, int):int                  modii($1, $2)
 (gen, small):gen                gmodgs($1, $2)
 (small, gen):gen                gmodsg($1, $2)
 (gen, gen):gen                  gmod($1, $2)
 
 (FpX,FpX):FpX                   FpX_rem($1, $2, p)
 (FqX,FqX):FqX                   FqX_rem($1, $2, T, p)

Function: _&&_
Class: basic
Section: symbolic_operators
C-Name: andpari
Prototype: GE
Help: _&&_
Description: 
 (bool, bool):bool:parens               $(1) && $(2)

Function: _'
Class: basic
Section: symbolic_operators
C-Name: deriv
Prototype: GDn
Help: x': derivative of x with respect to the main variable.
Description: 
 (gen):gen                      deriv($1,-1)
 
 (FpX):FpX                      FpX_deriv($1, p)
 (FqX):FqX                      FqX_deriv($1, T, p)

Function: _(_)
Class: basic
Section: symbolic_operators
Help: f(a,b,...): evaluates the function f on a,b,...
Description: 
 (gen):gen          closure_callgenall($1, 0)
 (gen,gen):gen      closure_callgen1($1, $2)
 (gen,gen,gen):gen  closure_callgen2($1, $2, $3)
 (gen,gen,...):gen  closure_callgenall($1, ${nbarg 1 sub}, $3)

Function: _*=_
Class: basic
Section: symbolic_operators
C-Name: gmule
Prototype: &G
Help: x*=y: shortcut for x=x*y.
Description: 
 (*small, small):small:parens             $1 *= $(2)
 (*int, small):int:parens                 $1 = mulis($1, $2)
 (*int, int):int:parens                   $1 = mulii($1, $2)
 (*real, small):real:parens               $1 = mulrs($1, $2)
 (*real, int):real:parens                 $1 = mulri($1, $2)
 (*real, real):real:parens                $1 = mulrr($1, $2)
 (*mp, mp):mp:parens                      $1 = mpmul($1, $2)
 (*pol, small):gen:parens                 $1 = gmulgs($1, $2)
 (*pol, gen):gen:parens                   $1 = gmul($1, $2)
 (*vec, gen):gen:parens                   $1 = gmul($1, $2)
 (*gen, small):gen:parens                 $1 = gmulgs($1, $2)
 (*gen, gen):gen:parens                   $1 = gmul($1, $2)

Function: _*_
Class: basic
Section: symbolic_operators
C-Name: gmul
Prototype: GG
Help: x*y: product of x and y.
Description: 
 (small, small):small:parens     $(1)*$(2)
 (int, small):int                mulis($1, $2)
 (small, int):int                mulsi($1, $2)
 (int, int):int                  mulii($1, $2)
 (0, mp):small                   ($2, 0)/*for side effect*/
 (#small, real):real             mulsr($1, $2)
 (small, real):mp                mulsr($1, $2)
 (real, small):mp                mulrs($1, $2)
 (real, real):real               mulrr($1, $2)
 (mp, mp):mp                     mpmul($1, $2)
 (gen, small):gen                gmulgs($1, $2)
 (small, gen):gen                gmulsg($1, $2)
 (vecsmall, vecsmall):vecsmall   perm_mul($1, $2)
 (gen, gen):gen                  gmul($1, $2)
 
 (usmall,Fp):Fp                  Fp_mulu($2, $1, p)
 (small,Fp):Fp                   Fp_muls($2, $1, p)
 (Fp, usmall):Fp                 Fp_mulu($1, $2, p)
 (Fp, small):Fp                  Fp_muls($1, $2, p)
 (usmall,FpX):FpX                FpX_mulu($2, $1, p)
 (FpX, usmall):FpX               FpX_mulu($1, $2, p)
 (Fp, FpX):FpX                   FpX_Fp_mul($2, $1, p)
 (FpX, Fp):FpX                   FpX_Fp_mul($1, $2, p)
 (FpX, FpX):FpX                  FpX_mul($1, $2, p)
 
 (usmall,Fq):Fq                  Fq_mulu($2, $1, T, p)
 (Fq, usmall):Fq                 Fq_mulu($1, $2, T, p)
 (Fq,Fp):Fq                      Fq_Fp_mul($1, $2, T, p)
 (Fp,Fq):Fq                      Fq_Fp_mul($2, $1, T, p)
 (usmall,FqX):FqX                FqX_mulu($2, $1, T, p)
 (FqX, usmall):FqX               FqX_mulu($1, $2, T, p)
 (FqX,Fp):FqX                    FqX_Fp_mul($1, $2, T, p)
 (Fp,FqX):FqX                    FqX_Fp_mul($2, $1, T, p)
 (Fq, FqX):FqX                   FqX_Fq_mul($2, $1, T, p)
 (FqX, Fq):FqX                   FqX_Fq_mul($1, $2, T, p)
 (FqX, FqX):FqX                  FqX_mul($1, $2, T, p)

Function: _++
Class: basic
Section: symbolic_operators
C-Name: gadd1e
Prototype: &
Help: x++
Description: 
 (*bptr):bptr                            ++$1
 (*small):small                          ++$1
 (*lg):lg                                ++$1
 (*int):int:parens                       $1 = addis($1, 1)
 (*real):real:parens                     $1 = addrs($1, 1)
 (*mp):mp:parens                         $1 = mpadd($1, gen_1)
 (*pol):pol:parens                       $1 = gaddgs($1, 1)
 (*gen):gen:parens                       $1 = gaddgs($1, 1)

Function: _+=_
Class: basic
Section: symbolic_operators
C-Name: gadde
Prototype: &G
Help: x+=y: shortcut for x=x+y.
Description: 
 (*small, small):small:parens             $1 += $(2)
 (*lg, small):lg:parens                   $1 += $(2)
 (*int, small):int:parens                 $1 = addis($1, $2)
 (*int, int):int:parens                   $1 = addii($1, $2)
 (*real, small):real:parens               $1 = addrs($1, $2)
 (*real, int):real:parens                 $1 = addir($2, $1)
 (*real, real):real:parens                $1 = addrr($1, $2)
 (*mp, mp):mp:parens                      $1 = mpadd($1, $2)
 (*pol, small):gen:parens                 $1 = gaddgs($1, $2)
 (*pol, gen):gen:parens                   $1 = gadd($1, $2)
 (*vec, gen):gen:parens                   $1 = gadd($1, $2)
 (*gen, small):gen:parens                 $1 = gaddgs($1, $2)
 (*gen, gen):gen:parens                   $1 = gadd($1, $2)

Function: _+_
Class: basic
Section: symbolic_operators
C-Name: gadd
Prototype: GG
Help: x+y: sum of x and y.
Description: 
 (lg, 1):small:parens            $(1)
 (small, small):small:parens     $(1) + $(2)
 (lg, small):lg:parens           $(1) + $(2)
 (small, lg):lg:parens           $(1) + $(2)
 (int, small):int                addis($1, $2)
 (small, int):int                addsi($1, $2)
 (int, int):int                  addii($1, $2)
 (real, small):real              addrs($1, $2)
 (small, real):real              addsr($1, $2)
 (real, real):real               addrr($1, $2)
 (mp, real):real                 mpadd($1, $2)
 (real, mp):real                 mpadd($1, $2)
 (mp, mp):mp                     mpadd($1, $2)
 (gen, small):gen                gaddgs($1, $2)
 (small, gen):gen                gaddsg($1, $2)
 (gen, gen):gen                  gadd($1, $2)
 
 (Fp, Fp):Fp                     Fp_add($1, $2, p)
 (FpX, Fp):FpX                   FpX_Fp_add($1, $2, p)
 (Fp, FpX):FpX                   FpX_Fp_add($2, $1, p)
 (FpX, FpX):FpX                  FpX_add($1, $2, p)
 (Fq, Fq):Fq                     Fq_add($1, $2, T, p)
 (FqX, Fq):FqX                   FqX_Fq_add($1, $2, T, p)
 (Fq, FqX):FqX                   FqX_Fq_add($2, $1, T, p)
 (FqX, FqX):FqX                  FqX_add($1, $2, T, p)

Function: _--
Class: basic
Section: symbolic_operators
C-Name: gsub1e
Prototype: &
Help: x--
Description: 
 (*bptr):bptr                          --$1
 (*small):small                        --$1
 (*lg):lg                              --$1
 (*int):int:parens                     $1 = subis($1, 1)
 (*real):real:parens                   $1 = subrs($1, 1)
 (*mp):mp:parens                       $1 = mpsub($1, gen_1)
 (*pol):pol:parens                     $1 = gsubgs($1, 1)
 (*gen):gen:parens                     $1 = gsubgs($1, 1)

Function: _-=_
Class: basic
Section: symbolic_operators
C-Name: gsube
Prototype: &G
Help: x-=y: shortcut for x=x-y.
Description: 
 (*small, small):small:parens             $1 -= $(2)
 (*lg, small):lg:parens                   $1 -= $(2)
 (*int, small):int:parens                 $1 = subis($1, $2)
 (*int, int):int:parens                   $1 = subii($1, $2)
 (*real, small):real:parens               $1 = subrs($1, $2)
 (*real, int):real:parens                 $1 = subri($1, $2)
 (*real, real):real:parens                $1 = subrr($1, $2)
 (*mp, mp):mp:parens                      $1 = mpsub($1, $2)
 (*pol, small):gen:parens                 $1 = gsubgs($1, $2)
 (*pol, gen):gen:parens                   $1 = gsub($1, $2)
 (*vec, gen):gen:parens                   $1 = gsub($1, $2)
 (*gen, small):gen:parens                 $1 = gsubgs($1, $2)
 (*gen, gen):gen:parens                   $1 = gsub($1, $2)

Function: _-_
Class: basic
Section: symbolic_operators
C-Name: gsub
Prototype: GG
Help: x-y: difference of x and y.
Description: 
 (small, small):small:parens     $(1) - $(2)
 (lg, small):lg:parens           $(1) - $(2)
 (int, small):int                subis($1, $2)
 (small, int):int                subsi($1, $2)
 (int, int):int                  subii($1, $2)
 (real, small):real              subrs($1, $2)
 (small, real):real              subsr($1, $2)
 (real, real):real               subrr($1, $2)
 (mp, real):real                 mpsub($1, $2)
 (real, mp):real                 mpsub($1, $2)
 (mp, mp):mp                     mpsub($1, $2)
 (gen, small):gen                gsubgs($1, $2)
 (small, gen):gen                gsubsg($1, $2)
 (gen, gen):gen                  gsub($1, $2)
 
 (Fp, Fp):Fp                     Fp_sub($1, $2, p)
 (Fp, FpX):FpX                   Fp_FpX_sub($1, $2, p)
 (FpX, Fp):FpX                   FpX_Fp_sub($1, $2, p)
 (FpX, FpX):FpX                  FpX_sub($1, $2, p)
 (Fq, Fq):Fq                     Fq_sub($1, $2, T, p)
 (FqX, FqX):FqX                  FqX_sub($1, $2, T, p)

Function: _.a1
Class: basic
Section: member_functions
C-Name: member_a1
Prototype: mG
Help: _.a1
Description: 
 (ell):gen:copy        ell_get_a1($1)

Function: _.a2
Class: basic
Section: member_functions
C-Name: member_a2
Prototype: mG
Help: _.a2
Description: 
 (ell):gen:copy        ell_get_a2($1)

Function: _.a3
Class: basic
Section: member_functions
C-Name: member_a3
Prototype: mG
Help: _.a3
Description: 
 (ell):gen:copy        ell_get_a3($1)

Function: _.a4
Class: basic
Section: member_functions
C-Name: member_a4
Prototype: mG
Help: _.a4
Description: 
 (ell):gen:copy        ell_get_a4($1)

Function: _.a6
Class: basic
Section: member_functions
C-Name: member_a6
Prototype: mG
Help: _.a6
Description: 
 (ell):gen:copy         ell_get_a6($1)

Function: _.area
Class: basic
Section: member_functions
C-Name: member_area
Prototype: mG
Help: _.area

Function: _.b2
Class: basic
Section: member_functions
C-Name: member_b2
Prototype: mG
Help: _.b2
Description: 
 (ell):gen:copy         ell_get_b2($1)

Function: _.b4
Class: basic
Section: member_functions
C-Name: member_b4
Prototype: mG
Help: _.b4
Description: 
 (ell):gen:copy        ell_get_b4($1)

Function: _.b6
Class: basic
Section: member_functions
C-Name: member_b6
Prototype: mG
Help: _.b6
Description: 
 (ell):gen:copy               ell_get_b6($1)

Function: _.b8
Class: basic
Section: member_functions
C-Name: member_b8
Prototype: mG
Help: _.b8
Description: 
 (ell):gen:copy        ell_get_b8($1)

Function: _.bid
Class: basic
Section: member_functions
C-Name: member_bid
Prototype: mG
Help: _.bid
Description: 
 (bnr):gen:copy                 bnr_get_bid($1)
 (gen):gen:copy                 member_bid($1)

Function: _.bnf
Class: basic
Section: member_functions
C-Name: member_bnf
Prototype: mG
Help: _.bnf
Description: 
 (bnf):bnf:parens               $1
 (bnr):bnf:copy:parens          $bnf:1
 (gen):bnf:copy                 member_bnf($1)

Function: _.c4
Class: basic
Section: member_functions
C-Name: member_c4
Prototype: mG
Help: _.c4
Description: 
 (ell):gen:copy        ell_get_c4($1)

Function: _.c6
Class: basic
Section: member_functions
C-Name: member_c6
Prototype: mG
Help: _.c6
Description: 
 (ell):gen:copy        ell_get_c6($1)

Function: _.clgp
Class: basic
Section: member_functions
C-Name: member_clgp
Prototype: mG
Help: _.clgp
Description: 
 (bnf):clgp:copy:parens         $clgp:1
 (bnr):clgp:copy:parens         $clgp:1
 (clgp):clgp:parens             $1
 (gen):clgp:copy                member_clgp($1)

Function: _.codiff
Class: basic
Section: member_functions
C-Name: member_codiff
Prototype: mG
Help: _.codiff

Function: _.cyc
Class: basic
Section: member_functions
C-Name: member_cyc
Prototype: mG
Help: _.cyc
Description: 
 (bnr):vec:copy                 bnr_get_cyc($1)
 (bnf):vec:copy                 bnf_get_cyc($1)
 (clgp):vec:copy                gel($1, 2)
 (gen):vec:copy                 member_cyc($1)

Function: _.diff
Class: basic
Section: member_functions
C-Name: member_diff
Prototype: mG
Help: _.diff
Description: 
 (nf):gen:copy                  nf_get_diff($1)
 (gen):gen:copy                 member_diff($1)

Function: _.disc
Class: basic
Section: member_functions
C-Name: member_disc
Prototype: mG
Help: _.disc
Description: 
 (nf):int:copy                  nf_get_disc($1)
 (ell):gen:copy                 ell_get_disc($1)
 (gen):gen:copy                 member_disc($1)

Function: _.e
Class: basic
Section: member_functions
C-Name: member_e
Prototype: mG
Help: _.e
Description: 
 (prid):small        pr_get_e($1)

Function: _.eta
Class: basic
Section: member_functions
C-Name: member_eta
Prototype: mG
Help: _.eta

Function: _.f
Class: basic
Section: member_functions
C-Name: member_f
Prototype: mG
Help: _.f
Description: 
 (prid):small       pr_get_f($1)

Function: _.fu
Class: basic
Section: member_functions
C-Name: member_fu
Prototype: G
Help: _.fu
Description: 
 (bnr):void                $"ray units not implemented"
 (bnf):gen:copy         bnf_get_fu($1)
 (gen):gen              member_fu($1)

Function: _.futu
Class: basic
Section: member_functions
C-Name: member_futu
Prototype: mG
Help: _.futu

Function: _.gen
Class: basic
Section: member_functions
C-Name: member_gen
Prototype: mG
Help: _.gen
Description: 
 (bnr):vec:copy        bnr_get_gen($1)
 (bnf):vec:copy        bnf_get_gen($1)
 (gal):vec:copy        gal_get_gen($1)
 (clgp):vec:copy       gel($1, 3)
 (prid):gen:copy       pr_get_gen($1)
 (gen):gen:copy        member_gen($1)

Function: _.group
Class: basic
Section: member_functions
C-Name: member_group
Prototype: mG
Help: _.group
Description: 
 (gal):vec:copy        gal_get_group($1)
 (gen):vec:copy        member_group($1)

Function: _.index
Class: basic
Section: member_functions
C-Name: member_index
Prototype: mG
Help: _.index
Description: 
 (nf):int:copy                  nf_get_index($1)
 (gen):int:copy                 member_index($1)

Function: _.j
Class: basic
Section: member_functions
C-Name: member_j
Prototype: mG
Help: _.j
Description: 
 (ell):gen:copy        ell_get_j($1)

Function: _.mod
Class: basic
Section: member_functions
C-Name: member_mod
Prototype: mG
Help: _.mod

Function: _.nf
Class: basic
Section: member_functions
C-Name: member_nf
Prototype: mG
Help: _.nf
Description: 
 (nf):nf:parens                $1
 (gen):nf:copy                 member_nf($1)

Function: _.no
Class: basic
Section: member_functions
C-Name: member_no
Prototype: mG
Help: _.no
Description: 
 (bnr):int:copy                 bnr_get_no($1)
 (bnf):int:copy                 bnf_get_no($1)
 (clgp):int:copy                gel($1, 1)
 (gen):int:copy                 member_no($1)

Function: _.omega
Class: basic
Section: member_functions
C-Name: member_omega
Prototype: mG
Help: _.omega

Function: _.orders
Class: basic
Section: member_functions
C-Name: member_orders
Prototype: mG
Help: _.orders
Description: 
 (gal):vecsmall:copy   gal_get_orders($1)

Function: _.p
Class: basic
Section: member_functions
C-Name: member_p
Prototype: mG
Help: _.p
Description: 
 (gal):int:copy                 gal_get_p($1)
 (prid):int:copy                pr_get_p($1)
 (gen):int:copy                 member_p($1)

Function: _.pol
Class: basic
Section: member_functions
C-Name: member_pol
Prototype: mG
Help: _.pol
Description: 
 (gal):gen:copy                 gal_get_pol($1)
 (nf):gen:copy                  nf_get_pol($1)
 (gen):gen:copy                 member_pol($1)

Function: _.polabs
Class: basic
Section: member_functions
C-Name: member_polabs
Prototype: mG
Help: _.polabs

Function: _.r1
Class: basic
Section: member_functions
C-Name: member_r1
Prototype: mG
Help: _.r1
Description: 
 (nf):small                     nf_get_r1($1)
 (gen):int:copy                 member_r1($1)

Function: _.r2
Class: basic
Section: member_functions
C-Name: member_r2
Prototype: mG
Help: _.r2
Description: 
 (nf):small                     nf_get_r2($1)
 (gen):int:copy                 member_r2($1)

Function: _.reg
Class: basic
Section: member_functions
C-Name: member_reg
Prototype: mG
Help: _.reg
Description: 
 (bnr):real             $"ray regulator not implemented"
 (bnf):real:copy        bnf_get_reg($1)
 (gen):real:copy        member_reg($1)

Function: _.roots
Class: basic
Section: member_functions
C-Name: member_roots
Prototype: mG
Help: _.roots
Description: 
 (gal):vec:copy                 gal_get_roots($1)
 (nf):vec:copy                  nf_get_roots($1)
 (gen):vec:copy                 member_roots($1)

Function: _.sign
Class: basic
Section: member_functions
C-Name: member_sign
Prototype: mG
Help: _.sign
Description: 
 (nf):vec:copy                  gel($1, 2)
 (gen):vec:copy                 member_sign($1)

Function: _.t2
Class: basic
Section: member_functions
C-Name: member_t2
Prototype: G
Help: _.t2
Description: 
 (gen):vec                      member_t2($1)

Function: _.tate
Class: basic
Section: member_functions
C-Name: member_tate
Prototype: mG
Help: _.tate

Function: _.tu
Class: basic
Section: member_functions
C-Name: member_tu
Prototype: G
Help: _.tu
Description: 
 (gen):gen:copy        member_tu($1)

Function: _.tufu
Class: basic
Section: member_functions
C-Name: member_tufu
Prototype: mG
Help: _.tufu

Function: _.zk
Class: basic
Section: member_functions
C-Name: member_zk
Prototype: mG
Help: _.zk
Description: 
 (nf):vec:copy         nf_get_zk($1)
 (gen):vec:copy        member_zk($1)

Function: _.zkst
Class: basic
Section: member_functions
C-Name: member_zkst
Prototype: mG
Help: _.zkst
Description: 
 (bnr):gen:copy        bnr_get_bid($1)

Function: _/=_
Class: basic
Section: symbolic_operators
C-Name: gdive
Prototype: &G
Help: x/=y: shortcut for x=x/y.
Description: 
 (*small, gen):void                $"cannot divide small: use \= instead."
 (*int, gen):void                  $"cannot divide int: use \= instead."
 (*real, real):real:parens               $1 = divrr($1, $2)
 (*real, small):real:parens              $1 = divrs($1, $2)
 (*real, mp):real:parens                 $1 = mpdiv($1, $2)
 (*mp, real):mp:parens                   $1 = mpdiv($1, $2)
 (*pol, gen):gen:parens                  $1 = gdiv($1, $2)
 (*vec, gen):gen:parens                  $1 = gdiv($1, $2)
 (*gen, small):gen:parens                $1 = gdivgs($1, $2)
 (*gen, gen):gen:parens                  $1 = gdiv($1, $2)

Function: _/_
Class: basic
Section: symbolic_operators
C-Name: gdiv
Prototype: GG
Help: x/y: quotient of x and y.
Description: 
 (0, mp):small                   ($2, 0)/*for side effect*/
 (1, real):real                  invr($2)
 (#small, real):real             divsr($1, $2)
 (small, real):mp                divsr($1, $2)
 (real, small):real              divrs($1, $2)
 (real, real):real               divrr($1, $2)
 (real, mp):real                 mpdiv($1, $2)
 (mp, real):mp                   mpdiv($1, $2)
 (1, gen):gen                    ginv($2)
 (gen, small):gen                gdivgs($1, $2)
 (small, gen):gen                gdivsg($1, $2)
 (gen, gen):gen                  gdiv($1, $2)
 
 (Fp, 2):Fp                       Fp_halve($1, p)
 (Fp, Fp):Fp                     Fp_div($1, $2, p)
 (Fq, 2):Fq                       Fq_halve($1, T, p)
 (Fq, Fq):Fq                     Fq_div($1, $2, T, p)

Function: _<<=_
Class: basic
Section: symbolic_operators
C-Name: gshiftle
Prototype: &L
Help: x<<=y: shortcut for x=x<<y.
Description: 
 (*small, small):small:parens             $1 <<= $(2)
 (*int, small):int:parens                 $1 = shifti($1, $2)
 (*mp, small):mp:parens                   $1 = mpshift($1, $2)
 (*gen, small):mp:parens                  $1 = gshift($1, $2)

Function: _<<_
Class: basic
Section: symbolic_operators
C-Name: gshift
Prototype: GL
Help: x<<y
Description: 
 (int, small):int               shifti($1, $2)
 (mp, small):mp                 mpshift($1, $2)
 (gen, small):mp                gshift($1, $2)

Function: _<=_
Class: basic
Section: symbolic_operators
C-Name: gle
Prototype: GG
Help: x<=y: return 1 if x is less or equal to y, 0 otherwise.
Description: 
 (small, small):bool:parens              $(1) <= $(2)
 (small, lg):bool:parens                 $(1) < $(2)
 (lg, lg):bool:parens                    $(1) <= $(2)
 (small, int):bool:parens                cmpsi($1, $2) <= 0
 (int, lg):bool:parens                   cmpis($1, $2) < 0
 (int, small):bool:parens                cmpis($1, $2) <= 0
 (int, int):bool:parens                  cmpii($1, $2) <= 0
 (mp, mp):bool:parens                    mpcmp($1, $2) <= 0
 (str, str):bool:parens                  strcmp($1, $2) <= 0
 (small, gen):bool:parens                gcmpsg($1, $2) <= 0
 (gen, small):bool:parens                gcmpgs($1, $2) <= 0
 (gen, gen):bool:parens                  gcmp($1, $2) <= 0

Function: _<_
Class: basic
Section: symbolic_operators
C-Name: glt
Prototype: GG
Help: x<y: return 1 if x is strictly less than y, 0 otherwise.
Description: 
 (small, small):bool:parens              $(1) < $(2)
 (lg, lg):bool:parens                    $(1) < $(2)
 (lg, small):bool:parens                 $(1) <= $(2)
 (small, int):bool:parens                cmpsi($1, $2) < 0
 (int, small):bool:parens                cmpis($1, $2) < 0
 (int, int):bool:parens                  cmpii($1, $2) < 0
 (mp, mp):bool:parens                    mpcmp($1, $2) < 0
 (str, str):bool:parens                  strcmp($1, $2) < 0
 (small, gen):bool:parens                gcmpsg($1, $2) < 0
 (gen, small):bool:parens                gcmpgs($1, $2) < 0
 (gen, gen):bool:parens                  gcmp($1, $2) < 0

Function: _===_
Class: basic
Section: symbolic_operators
C-Name: gidentical
Prototype: iGG
Help: a === b : true if a and b are identical

Function: _==_
Class: basic
Section: symbolic_operators
C-Name: geq
Prototype: GG
Help: _==_
Description: 
 (small, small):bool:parens             $(1) == $(2)
 (lg, lg):bool:parens                   $(1) == $(2)
 (small, int):bool:parens               cmpsi($1, $2) == 0
 (mp, 0):bool                           !signe($1)
 (int, 1):bool                          equali1($1)
 (int, -1):bool                         equalim1($1)
 (int, small):bool:parens               cmpis($1, $2) == 0
 (int, int):bool                        equalii($1, $2)
 (gen, 0):bool                          gequal0($1)
 (gen, 1):bool                          gequal1($1)
 (gen, -1):bool                         gequalm1($1)
 (real,real):bool                       cmprr($1, $2) == 0
 (mp, mp):bool:parens                   mpcmp($1, $2) == 0
 (errtyp, errtyp):bool:parens           $(1) == $(2)
 (errtyp, #str):bool:parens             $(1) == $(errtyp:2)
 (#str, errtyp):bool:parens             $(errtyp:1) == $(2)
 (typ, typ):bool:parens                 $(1) == $(2)
 (typ, #str):bool:parens                $(1) == $(typ:2)
 (#str, typ):bool:parens                $(typ:1) == $(2)
 (str, str):negbool                     strcmp($1, $2)
 (small, gen):bool                      gequalsg($1, $2)
 (gen, small):bool                      gequalgs($1, $2)
 (gen, gen):bool                        gequal($1, $2)

Function: _>=_
Class: basic
Section: symbolic_operators
C-Name: gge
Prototype: GG
Help: x>=y: return 1 if x is greater or equal to y, 0 otherwise.
Description: 
 (small, small):bool:parens              $(1) >= $(2)
 (lg, lg):bool:parens                    $(1) >= $(2)
 (lg, small):bool:parens                 $(1) > $(2)
 (small, int):bool:parens                cmpsi($1, $2) >= 0
 (int, small):bool:parens                cmpis($1, $2) >= 0
 (int, int):bool:parens                  cmpii($1, $2) >= 0
 (mp, mp):bool:parens                    mpcmp($1, $2) >= 0
 (str, str):bool:parens                  strcmp($1, $2) >= 0
 (small, gen):bool:parens                gcmpsg($1, $2) >= 0
 (gen, small):bool:parens                gcmpgs($1, $2) >= 0
 (gen, gen):bool:parens                  gcmp($1, $2) >= 0

Function: _>>=_
Class: basic
Section: symbolic_operators
C-Name: gshiftre
Prototype: &L
Help: x>>=y: shortcut for x=x>>y.
Description: 
 (*small, small):small:parens             $1 >>= $(2)
 (*int, small):int:parens                 $1 = shifti($1, -$(2))
 (*mp, small):mp:parens                   $1 = mpshift($1, -$(2))
 (*gen, small):mp:parens                  $1 = gshift($1, -$(2))

Function: _>>_
Class: basic
Section: symbolic_operators
C-Name: gshift_right
Prototype: GL
Help: x>>y
Description: 
 (small, small):small:parens     $(1)>>$(2)
 (int, small):int                shifti($1, -$(2))
 (mp, small):mp                  mpshift($1, -$(2))
 (gen, small):mp                 gshift($1, -$(2))

Function: _>_
Class: basic
Section: symbolic_operators
C-Name: ggt
Prototype: GG
Help: x>y: return 1 if x is strictly greater than y, 0 otherwise.
Description: 
 (small, small):bool:parens              $(1) > $(2)
 (lg, lg):bool:parens                    $(1) > $(2)
 (small, lg):bool:parens                 $(1) >= $(2)
 (small, int):bool:parens                cmpsi($1, $2) > 0
 (int, small):bool:parens                cmpis($1, $2) > 0
 (int, int):bool:parens                  cmpii($1, $2) > 0
 (mp, mp):bool:parens                    mpcmp($1, $2) > 0
 (str, str):bool:parens                  strcmp($1, $2) > 0
 (small, gen):bool:parens                gcmpsg($1, $2) > 0
 (gen, small):bool:parens                gcmpgs($1, $2) > 0
 (gen, gen):bool:parens                  gcmp($1, $2) > 0

Function: _ZX_resultant_worker
Class: basic
Section: programming/internals
C-Name: ZX_resultant_worker
Prototype: GGGG
Help: worker for ZX_resultant

Function: _[_,]
Class: basic
Section: symbolic_operators
Help: x[y,]: y-th row of x.
Description: 
 (mp,small):gen                 $"Scalar has no rows"
 (vec,small):vec                rowcopy($1, $2)
 (gen,small):vec                rowcopy($1, $2)

Function: _[_,_]
Class: basic
Section: symbolic_operators
Help: x[i{,j}]: i coefficient of a vector, i,j coefficient of a matrix
Description: 
 (mp,small):gen                 $"Scalar has no components"
 (mp,small,small):gen           $"Scalar has no components"
 (vecsmall,small):small         $(1)[$2]
 (vecsmall,small,small):gen     $"Vecsmall are single-dimensional"
 (list,small):gen:copy          gel(list_data($1), $2)
 (vec,small):gen:copy           gel($1, $2)
 (vec,small,small):gen:copy     gcoeff($1, $2, $3)
 (gen,small):gen:copy           gel($1, $2)
 (gen,small,small):gen:copy     gcoeff($1, $2, $3)

Function: _[_.._,_.._]
Class: basic
Section: symbolic_operators
C-Name: matslice0
Prototype: GD0,L,D0,L,D0,L,D0,L,
Help: x[a..b,c..d] = [x[a,c],  x[a+1,c],  ...,x[b,c];
                      x[a,c+1],x[a+1,c+1],...,x[b,c+1];
                        ...       ...          ...
                      x[a,d],  x[a+1,d]  ,...,x[b,d]]

Function: _[_.._]
Class: basic
Section: symbolic_operators
C-Name: vecslice0
Prototype: GD0,L,L
Help: x[a..b] = [x[a],x[a+1],...,x[b]]

Function: _\/=_
Class: basic
Section: symbolic_operators
C-Name: gdivrounde
Prototype: &G
Help: x\/=y: shortcut for x=x\/y.
Description: 
 (*int, int):int:parens                         $1 = gdivround($1, $2)
 (*pol, gen):gen:parens                         $1 = gdivround($1, $2)
 (*gen, gen):gen:parens                         $1 = gdivround($1, $2)

Function: _\/_
Class: basic
Section: symbolic_operators
C-Name: gdivround
Prototype: GG
Help: x\/y: rounded Euclidean quotient of x and y.
Description: 
 (int, int):int                        gdivround($1, $2)
 (gen, gen):gen                        gdivround($1, $2)

Function: _\=_
Class: basic
Section: symbolic_operators
C-Name: gdivente
Prototype: &G
Help: x\=y: shortcut for x=x\y.
Description: 
 (*small, small):small:parens                   $1 /= $(2)
 (*int, int):int:parens                         $1 = gdivent($1, $2)
 (*pol, gen):gen:parens                         $1 = gdivent($1, $2)
 (*gen, gen):gen:parens                         $1 = gdivent($1, $2)

Function: _\_
Class: basic
Section: symbolic_operators
C-Name: gdivent
Prototype: GG
Help: x\y: Euclidean quotient of x and y.
Description: 
 (small, small):small:parens             $(1)/$(2)
 (int, small):int                        truedivis($1, $2)
 (small, int):int                        gdiventsg($1, $2)
 (int, int):int                          truedivii($1, $2)
 (gen, small):gen                        gdiventgs($1, $2)
 (small, gen):gen                        gdiventsg($1, $2)
 (gen, gen):gen                          gdivent($1, $2)

Function: _^_
Class: basic
Section: symbolic_operators
C-Name: gpow
Prototype: GGp
Help: x^y: compute x to the power y.
Description: 
 (int, 2):int                sqri($1)
 (int, 3):int                powiu($1, 3)
 (int, 4):int                powiu($1, 4)
 (int, 5):int                powiu($1, 5)
 (real, -1):real             invr($1)
 (mp, -1):mp                 ginv($1)
 (gen, -1):gen               ginv($1)
 (real, 2):real              sqrr($1)
 (mp, 2):mp                  mpsqr($1)
 (gen, 2):gen                gsqr($1)
 (int, small):gen            powis($1, $2)
 (real, small):real          gpowgs($1, $2)
 (gen, small):gen            gpowgs($1, $2)
 (real, int):real            powgi($1, $2)
 (gen, int):gen              powgi($1, $2)
 (gen, gen):gen:prec         gpow($1, $2, $prec)
 
 (Fp, 2):Fp                  Fp_sqr($1, p)
 (Fp, usmall):Fp             Fp_powu($1, $2, p)
 (Fp, small):Fp              Fp_pows($1, $2, p)
 (Fp, int):Fp                Fp_pow($1, $2, p)
 (FpX, 2):FpX                FpX_sqr($1, p)
 (FpX, usmall):FpX           FpX_powu($1, $2, p)
 (Fq, 2):Fq                  Fq_sqr($1, T, p)
 (Fq, usmall):Fq             Fq_powu($1, $2, T, p)
 (Fq, int):Fq                Fq_pow($1, $2, T, p)
 (Fq, 2):Fq                  Fq_sqr($1, T, p)
 (Fq, usmall):Fq             Fq_powu($1, $2, T, p)
 (Fq, int):Fq                Fq_pow($1, $2, T, p)
 (FqX, 2):FqX                FqX_sqr($1, T, p)
 (FqX, usmall):FqX           FqX_powu($1, $2, T, p)

Function: _^s
Class: basic
Section: programming/internals
C-Name: gpowgs
Prototype: GL
Help: return x^n where n is a small integer

Function: __
Class: basic
Section: symbolic_operators
Help: __
Description: 
 (genstr, genstr):genstr                gconcat($1, $2)
 (genstr, gen):genstr                   gconcat($1, $2)
 (gen, genstr):genstr                   gconcat($1, $2)
 (gen, gen):genstr                      gconcat($genstr:1, $2)

Function: _avma
Class: gp2c_internal
Description: 
 ():pari_sp                avma

Function: _badtype
Class: gp2c_internal
Help: Code to check types. If not void, will be used as if(...).
Description: 
 (int):bool:parens              typ($1) != t_INT
 (real):bool:parens             typ($1) != t_REAL
 (mp):negbool                   is_intreal_t(typ($1))
 (vec):negbool                  is_matvec_t(typ($1))
 (vecsmall):bool:parens         typ($1) != t_VECSMALL
 (pol):bool:parens              typ($1) != t_POL
 (*nf):void:parens              $1 = checknf($1)
 (*bnf):void:parens             $1 = checkbnf($1)
 (bnr):void                     checkbnr($1)
 (prid):void                    checkprid($1)
 (clgp):void                    checkabgrp($1)
 (ell):void                     checkell($1)
 (*gal):gal:parens              $1 = checkgal($1)

Function: _cast
Class: gp2c_internal
Help: (type1):type2 : cast expression of type1 to type2
Description: 
 (void):bool           0
 (#negbool):bool       ${1 value not}
 (negbool):bool        !$(1)
 (small_int):bool
 (usmall):bool
 (small):bool
 (lg):bool:parens      $(1)!=1
 (bptr):bool           *$(1)
 (gen):bool            !gequal0($1)
 (real):bool           signe($1)
 (int):bool            signe($1)
 (mp):bool             signe($1)
 (pol):bool            signe($1)
 
 (void):negbool        1
 (#bool):negbool       ${1 value not}
 (bool):negbool        !$(1)
 (lg):negbool:parens   $(1)==1
 (bptr):negbool        !*$(1)
 (gen):negbool         gequal0($1)
 (int):negbool         !signe($1)
 (real):negbool        !signe($1)
 (mp):negbool          !signe($1)
 (pol):negbool         !signe($1)
 
 (bool):small_int
 (typ):small_int
 (small):small_int
 
 (bool):usmall
 (typ):usmall
 (small):usmall
 
 (bool):small
 (typ):small
 (small_int):small
 (usmall):small
 (bptr):small           *$(1)
 (int):small            itos($1)
 (int):usmall           itou($1)
 (#lg):small:parens     ${1 value 1 sub}
 (lg):small:parens      $(1)-1
 (gen):small            gtos($1)
 (gen):usmall           gtou($1)
 
 (void):int             gen_0
 (-2):int               gen_m2
 (-1):int               gen_m1
 (0):int                gen_0
 (1):int                gen_1
 (2):int                gen_2
 (bool):int             stoi($1)
 (small):int            stoi($1)
 (usmall):int           utoi($1)
 (mp):int
 (gen):int
 
 (mp):real
 (gen):real
 
 (int):mp
 (real):mp
 (gen):mp
 
 (#bool):lg:parens             ${1 1 value add}
 (bool):lg:parens              $(1)+1
 (#small):lg:parens            ${1 1 value add}
 (small):lg:parens             $(1)+1
 
 (gen):error
 (gen):closure
 (gen):vecsmall
 
 (nf):vec
 (bnf):vec
 (bnr):vec
 (ell):vec
 (clgp):vec
 (prid):vec
 (gal):vec
 (gen):vec
 
 (gen):list
 
 (pol):var      varn($1)
 (gen):var      gvar($1)
 
 (var):pol      pol_x($1)
 (gen):pol
 
 (int):gen
 (mp):gen
 (vecsmall):gen
 (vec):gen
 (list):gen
 (pol):gen
 (genstr):gen
 (error):gen
 (closure):gen
 (Fp):gen
 (FpX):gen
 (Fq):gen
 (FqX):gen
 
 (gen):genstr GENtoGENstr($1)
 (str):genstr strtoGENstr($1)
 
 (gen):str GENtostr_unquoted($1)
 (genstr):str GSTR($1)
 (typ):str type_name($1)
 (errtyp):str numerr_name($1)
 
 (#str):typ  ${1 str_format}
 (#str):errtyp  ${1 str_format}
 
 (bnf):nf              bnf_get_nf($1)
 (gen):nf
 (bnr):bnf             bnr_get_bnf($1)
 (gen):bnf
 (gen):bnr
 (bnf):clgp            bnf_get_clgp($1)
 (bnr):clgp            bnr_get_clgp($1)
 (gen):clgp
 (gen):ell
 (gen):gal
 (gen):prid
 
 (Fp):Fq

Function: _cgetg
Class: gp2c_internal
Description: 
 (lg,#str):gen              cgetg($1, ${2 str_raw})
 (gen,lg,#str):gen          $1 = cgetg($2, ${3 str_raw})

Function: _const_expr
Class: gp2c_internal
Description: 
 (str):gen       readseq($1)

Function: _const_quote
Class: gp2c_internal
Description: 
 (str):var       fetch_user_var($1)

Function: _const_real
Class: gp2c_internal
Description: 
 (str):real:prec       strtor($1, $prec)

Function: _const_smallreal
Class: gp2c_internal
Description: 
 (0):real:prec       real_0($prec)
 (1):real:prec       real_1($prec)
 (-1):real:prec      real_m1($prec)
 (small):real:prec   stor($1, $prec)

Function: _decl_base
Class: gp2c_internal
Description: 
 (C!void)            void
 (C!long)            long
 (C!ulong)           ulong
 (C!int)             int
 (C!GEN)             GEN
 (C!char*)           char
 (C!byteptr)         byteptr
 (C!pari_sp)         pari_sp
 (C!func_GG)         GEN
 (C!forprime_t)      forprime_t
 (C!forcomposite_t)  forcomposite_t
 (C!forpart_t)       forpart_t
 (C!forvec_t)        forvec_t

Function: _decl_ext
Class: gp2c_internal
Description: 
 (C!char*)         *$1
 (C!func_GG)       (*$1)(GEN, GEN)

Function: _def_TeXstyle
Class: default
Section: default
C-Name: sd_TeXstyle
Prototype: 
Help: 
Doc: the bits of this default allow
 \kbd{gp} to use less rigid TeX formatting commands in the logfile. This
 default is only taken into account when $\kbd{log} = 3$. The bits of
 \kbd{TeXstyle} have the following meaning
 
 2: insert \kbd{\bs right} / \kbd{\bs left} pairs where appropriate.
 
 4: insert discretionary breaks in polynomials, to enhance the probability of
 a good line break.
 
 The default value is \kbd{0}.

Function: _def_breakloop
Class: default
Section: default
C-Name: sd_breakloop
Prototype: 
Help: 
Doc: if true, enables the ``break loop'' debugging mode, see
 \secref{se:break_loop}.
 
 The default value is \kbd{1} if we are running an interactive \kbd{gp}
 session, and \kbd{0} otherwise.

Function: _def_colors
Class: default
Section: default
C-Name: sd_colors
Prototype: 
Help: 
Doc: this default is only usable if \kbd{gp}
 is running within certain color-capable terminals. For instance \kbd{rxvt},
 \kbd{color\_xterm} and modern versions of \kbd{xterm} under X Windows, or
 standard Linux/DOS text consoles. It causes \kbd{gp} to use a small palette of
 colors for its output. With xterms, the colormap used corresponds to the
 resources \kbd{Xterm*color$n$} where $n$ ranges from $0$ to $15$ (see the
 file \kbd{misc/color.dft} for an example). Accepted values for this
 default are strings \kbd{"$a_1$,\dots,$a_k$"} where $k\le7$ and each
 $a_i$ is either
 
 \noindent\item the keyword \kbd{no} (use the default color, usually
 black on transparent background)
 
 \noindent\item an integer between 0 and 15 corresponding to the
 aforementioned colormap
 
 \noindent\item a triple $[c_0,c_1,c_2]$ where $c_0$ stands for foreground
 color, $c_1$ for background color, and $c_2$ for attributes (0 is default, 1
 is bold, 4 is underline).
 
 The output objects thus affected are respectively error messages,
 history numbers, prompt, input line, output, help messages, timer (that's
 seven of them). If $k < 7$, the remaining $a_i$ are assumed to be $no$. For
 instance
 %
 \bprog
 default(colors, "9, 5, no, no, 4")
 @eprog
 \noindent
 typesets error messages in color $9$, history numbers in color $5$, output in
 color $4$, and does not affect the rest.
 
 A set of default colors for dark (reverse video or PC console) and light
 backgrounds respectively is activated when \kbd{colors} is set to
 \kbd{darkbg}, resp.~\kbd{lightbg} (or any proper prefix: \kbd{d} is
 recognized as an abbreviation for \kbd{darkbg}). A bold variant of
 \kbd{darkbg}, called \kbd{boldfg}, is provided if you find the former too
 pale.
 
 \emacs In the present version, this default is incompatible with PariEmacs.
 Changing it will just fail silently (the alternative would be to display
 escape sequences as is, since Emacs will refuse to interpret them).
 You must customize color highlighting from the PariEmacs side, see its
 documentation.
 
 The default value is \kbd{""} (no colors).

Function: _def_compatible
Class: default
Section: default
C-Name: sd_compatible
Prototype: 
Help: 
Doc: Obsolete. This default is now a no-op.
Obsolete: 2014-10-11

Function: _def_datadir
Class: default
Section: default
C-Name: sd_datadir
Prototype: 
Help: 
Doc: the name of directory containing the optional data files. For now,
 this includes the \kbd{elldata}, \kbd{galdata}, \kbd{galpol}, \kbd{seadata}
 packages.
 
 The default value is \kbd{/usr/local/share/pari}, or the override specified
 via \kbd{Configure --datadir=}.

Function: _def_debug
Class: default
Section: default
C-Name: sd_debug
Prototype: 
Help: 
Doc: debugging level. If it is non-zero, some extra messages may be printed,
 according to what is going on (see~\b{g}).
 
 The default value is \kbd{0} (no debugging messages).

Function: _def_debugfiles
Class: default
Section: default
C-Name: sd_debugfiles
Prototype: 
Help: 
Doc: file usage debugging level. If it is non-zero, \kbd{gp} will print
 information on file descriptors in use, from PARI's point of view
 (see~\b{gf}).
 
 The default value is \kbd{0} (no debugging messages).

Function: _def_debugmem
Class: default
Section: default
C-Name: sd_debugmem
Prototype: 
Help: 
Doc: memory debugging level. If it is non-zero, \kbd{gp} will regularly print
 information on memory usage. If it's greater than 2, it will indicate any
 important garbage collecting and the function it is taking place in
 (see~\b{gm}).
 
 \noindent {\bf Important Note:} As it noticeably slows down the performance,
 the first functionality (memory usage) is disabled if you're not running a
 version compiled for debugging (see Appendix~A).
 
 The default value is \kbd{0} (no debugging messages).

Function: _def_echo
Class: default
Section: default
C-Name: sd_echo
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). When \kbd{echo}
 mode is on, each command is reprinted before being executed. This can be
 useful when reading a file with the \b{r} or \kbd{read} commands. For
 example, it is turned on at the beginning of the test files used to check
 whether \kbd{gp} has been built correctly (see \b{e}).
 
 The default value is \kbd{0} (no echo).

Function: _def_factor_add_primes
Class: default
Section: default
C-Name: sd_factor_add_primes
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). If on,
 the integer factorization machinery calls \tet{addprimes} on prime
 factors that were difficult to find (larger than $2^{24}$), so they are
 automatically tried first in other factorizations. If a routine is performing
 (or has performed) a factorization and is interrupted by an error or via
 Control-C, this lets you recover the prime factors already found. The
 downside is that a huge \kbd{addprimes} table unrelated to the current
 computations will slow down arithmetic functions relying on integer
 factorization; one should then empty the table using \tet{removeprimes}.
 
 The default value is \kbd{0}.

Function: _def_factor_proven
Class: default
Section: default
C-Name: sd_factor_proven
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). By
 default, the factors output by the integer factorization machinery are
 only pseudo-primes, not proven primes. If this toggle is
 set, a primality proof is done for each factor and all results depending on
 integer factorization are fully proven. This flag does not affect partial
 factorization when it is explicitly requested. It also does not affect the
 private table managed by \tet{addprimes}: its entries are included as is in
 factorizations, without being tested for primality.
 
 The default value is \kbd{0}.

Function: _def_format
Class: default
Section: default
C-Name: sd_format
Prototype: 
Help: 
Doc: of the form x$.n$, where x (conversion style)
 is a letter in $\{\kbd{e},\kbd{f},\kbd{g}\}$, and $n$ (precision) is an
 integer; this affects the way real numbers are printed:
 
 \item If the conversion style is \kbd{e}, real numbers are printed in
 \idx{scientific format}, always with an explicit exponent,
 e.g.~\kbd{3.3 E-5}.
 
 \item In style \kbd{f}, real numbers are generally printed in
 \idx{fixed floating point format} without exponent, e.g.~\kbd{0.000033}. A
 large real number, whose integer part is not well defined (not enough
 significant digits), is printed in style~\kbd{e}. For instance
 \kbd{10.\pow 100} known to ten significant digits is always printed in style
 \kbd{e}.
 
 \item In style \kbd{g}, non-zero real numbers are printed in \kbd{f} format,
 except when their decimal exponent is $< -4$, in which case they are printed
 in \kbd{e} format. Real zeroes (of arbitrary exponent) are printed in \kbd{e}
 format.
 
 The precision $n$ is the number of significant digits printed for real
 numbers, except if $n<0$ where all the significant digits will be printed
 (initial default 28, or 38 for 64-bit machines). For more powerful formatting
 possibilities, see \tet{printf} and \tet{Strprintf}.
 
 The default value is \kbd{"g.28"} and \kbd{"g.38"} on 32-bit and
 64-bit machines, respectively.

Function: _def_graphcolormap
Class: default
Section: default
C-Name: sd_graphcolormap
Prototype: 
Help: 
Doc: a vector of colors, to be
 used by hi-res graphing routines. Its length is arbitrary, but it must
 contain at least 3 entries: the first 3 colors are used for background,
 frame/ticks and axes respectively. All colors in the colormap may be freely
 used in \tet{plotcolor} calls.
 
 A color is either given as in the default by character strings or by an RGB
 code. For valid character strings, see the standard \kbd{rgb.txt} file in X11
 distributions, where we restrict to lowercase letters and remove all
 whitespace from color names. An RGB code is a vector with 3 integer entries
 between 0 and 255. For instance \kbd{[250, 235, 215]} and
 \kbd{"antiquewhite"} represent the same color. RGB codes are cryptic but
 often easier to generate.
 
 The default value is [\kbd{"white"}, \kbd{"black"}, \kbd{"blue"},
 \kbd{"violetred"}, \kbd{"red"}, \kbd{"green"}, \kbd{"grey"},
 \kbd{"gainsboro"}].

Function: _def_graphcolors
Class: default
Section: default
C-Name: sd_graphcolors
Prototype: 
Help: 
Doc: entries in the
 \tet{graphcolormap} that will be used to plot multi-curves. The successive
 curves are drawn in colors
 
 \kbd{graphcolormap[graphcolors[1]]}, \kbd{graphcolormap[graphcolors[2]]},
   \dots
 
 cycling when the \kbd{graphcolors} list is exhausted.
 
 The default value is \kbd{[4,5]}.

Function: _def_help
Class: default
Section: default
C-Name: sd_help
Prototype: 
Help: 
Doc: name of the external help program to use from within \kbd{gp} when
 extended help is invoked, usually through a \kbd{??} or \kbd{???} request
 (see \secref{se:exthelp}), or \kbd{M-H} under readline (see
 \secref{se:readline}).
 
 The default value is the path to the \kbd{gphelp} script we install.

Function: _def_histfile
Class: default
Section: default
C-Name: sd_histfile
Prototype: 
Help: 
Doc: name of a file where
 \kbd{gp} will keep a history of all \emph{input} commands (results are
 omitted). If this file exists when the value of \kbd{histfile} changes,
 it is read in and becomes part of the session history. Thus, setting this
 default in your gprc saves your readline history between sessions. Setting
 this default to the empty string \kbd{""} changes it to
 \kbd{$<$undefined$>$}
 
 The default value is \kbd{$<$undefined$>$} (no history file).

Function: _def_histsize
Class: default
Section: default
C-Name: sd_histsize
Prototype: 
Help: 
Doc: \kbd{gp} keeps a history of the last
 \kbd{histsize} results computed so far, which you can recover using the
 \kbd{\%} notation (see \secref{se:history}). When this number is exceeded,
 the oldest values are erased. Tampering with this default is the only way to
 get rid of the ones you do not need anymore.
 
 The default value is \kbd{5000}.

Function: _def_lines
Class: default
Section: default
C-Name: sd_lines
Prototype: 
Help: 
Doc: if set to a positive value, \kbd{gp} prints at
 most that many lines from each result, terminating the last line shown with
 \kbd{[+++]} if further material has been suppressed. The various \kbd{print}
 commands (see \secref{se:gp_program}) are unaffected, so you can always type
 \kbd{print(\%)} or \b{a} to view the full result. If the actual screen width
 cannot be determined, a ``line'' is assumed to be 80 characters long.
 
 The default value is \kbd{0}.

Function: _def_linewrap
Class: default
Section: default
C-Name: sd_linewrap
Prototype: 
Help: 
Doc: if set to a positive value, \kbd{gp} wraps every single line after
 printing that many characters.
 
 The default value is \kbd{0} (unset).

Function: _def_log
Class: default
Section: default
C-Name: sd_log
Prototype: 
Help: 
Doc: this can be either 0 (off) or 1, 2, 3
 (on, see below for the various modes). When logging mode is turned on, \kbd{gp}
 opens a log file, whose exact name is determined by the \kbd{logfile}
 default. Subsequently, all the commands and results will be written to that
 file (see \b{l}). In case a file with this precise name already existed, it
 will not be erased: your data will be \emph{appended} at the end.
 
 The specific positive values of \kbd{log} have the following meaning
 
 1: plain logfile
 
 2: emit color codes to the logfile (if \kbd{colors} is set).
 
 3: write LaTeX output to the logfile (can be further customized using
 \tet{TeXstyle}).
 
 The default value is \kbd{0}.

Function: _def_logfile
Class: default
Section: default
C-Name: sd_logfile
Prototype: 
Help: 
Doc: name of the log file to be used when the \kbd{log} toggle is on.
 Environment and time expansion are performed.
 
 The default value is \kbd{"pari.log"}.

Function: _def_nbthreads
Class: default
Section: default
C-Name: sd_nbthreads
Prototype: 
Help: 
Doc: Number of threads to use for parallel computing.
 The exact meaning an default depend on the \kbd{mt} engine used:
 
 \item \kbd{single}: not used (always one thread).
 
 \item \kbd{pthread}: number of threads (unlimited, default: number of core)
 
 \item \kbd{mpi}: number of MPI process to use (limited to the number allocated by \kbd{mpirun},
 default: use all allocated process).

Function: _def_new_galois_format
Class: default
Section: default
C-Name: sd_new_galois_format
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). If on,
 the \tet{polgalois} command will use a different, more
 consistent, naming scheme for Galois groups. This default is provided to
 ensure that scripts can control this behavior and do not break unexpectedly.
 
 The default value is \kbd{0}. This value will change to $1$ (set) in the next
 major version.

Function: _def_output
Class: default
Section: default
C-Name: sd_output
Prototype: 
Help: 
Doc: there are three possible values: 0
 (=~\var{raw}), 1 (=~\var{prettymatrix}), or 3
 (=~\var{external} \var{prettyprint}). This
 means that, independently of the default \kbd{format} for reals which we
 explained above, you can print results in three ways:
 
 \item \tev{raw format}, i.e.~a format which is equivalent to what you
 input, including explicit multiplication signs, and everything typed on a
 line instead of two dimensional boxes. This can have several advantages, for
 instance it allows you to pick the result with a mouse or an editor, and to
 paste it somewhere else.
 
 \item \tev{prettymatrix format}: this is identical to raw format, except
 that matrices are printed as boxes instead of horizontally. This is
 prettier, but takes more space and cannot be used for input. Column vectors
 are still printed horizontally.
 
 \item \tev{external prettyprint}: pipes all \kbd{gp}
 output in TeX format to an external prettyprinter, according to the value of
 \tet{prettyprinter}. The default script (\tet{tex2mail}) converts its input
 to readable two-dimensional text.
 
 Independently of the setting of this default, an object can be printed
 in any of the three formats at any time using the commands \b{a} and \b{m}
 and \b{B} respectively.
 
 The default value is \kbd{1} (\var{prettymatrix}).

Function: _def_parisize
Class: default
Section: default
C-Name: sd_parisize
Prototype: 
Help: 
Doc: \kbd{gp}, and in fact any program using the PARI
 library, needs a \tev{stack} in which to do its computations; \kbd{parisize}
 is the stack size, in bytes. It is recommended to increase this
 default using a \tet{gprc}, to the value you believe PARI should be happy
 with, given your typical computation. We strongly recommend to also
 set \tet{parisizemax} to a much larger value, about what you believe your
 machine can stand: PARI will then try to fit its computations within about
 \kbd{parisize} bytes, but will increase the stack size if needed (up to
 \kbd{parisizemax}). Once the memory intensive computation is over, PARI
 will restore the stack size to the originally requested \kbd{parisize}.
 
 The default value is 4M, resp.~8M on a 32-bit, resp.~64-bit machine.

Function: _def_parisizemax
Class: default
Section: default
C-Name: sd_parisizemax
Prototype: 
Help: 
Doc: \kbd{gp}, and in fact any program using the PARI library, needs a
 \tev{stack} in which to do its computations.  If non-zero,  \kbd{parisizemax}
 is the maximum size the stack can grow to, in bytes.  If zero, the stack will
 not automatically grow, and will be limited to the value of \kbd{parisize}.
 
 We strongly recommend to set \tet{parisizemax} to a non-zero value, about
 what you believe your machine can stand: PARI will then try to fit its
 computations within about \kbd{parisize} bytes, but will increase the stack
 size if needed (up to \kbd{parisizemax}). Once the memory intensive
 computation is over, PARI will restore the stack size to the originally
 requested \kbd{parisize}.
 
 The default value is $0$.

Function: _def_path
Class: default
Section: default
C-Name: sd_path
Prototype: 
Help: 
Doc: this is a list of directories, separated by colons ':'
 (semicolons ';' in the DOS world, since colons are preempted for drive names).
 When asked to read a file whose name is not given by an absolute path
 (does not start with \kbd{/}, \kbd{./} or \kbd{../}), \kbd{gp} will look for
 it in these directories, in the order they were written in \kbd{path}. Here,
 as usual, \kbd{.} means the current directory, and \kbd{..} its immediate
 parent. Environment expansion is performed.
 
 The default value is \kbd{".:\til:\til/gp"} on UNIX systems,
 \kbd{".;C:\bs;C:\bs GP"} on DOS, OS/2 and Windows, and \kbd{"."} otherwise.

Function: _def_prettyprinter
Class: default
Section: default
C-Name: sd_prettyprinter
Prototype: 
Help: 
Doc: the name of an external prettyprinter to use when
 \kbd{output} is~3 (alternate prettyprinter). Note that the default
 \tet{tex2mail} looks much nicer than the built-in ``beautified
 format'' ($\kbd{output} = 2$).
 
 The default value is \kbd{"tex2mail -TeX -noindent -ragged -by\_par"}.

Function: _def_primelimit
Class: default
Section: default
C-Name: sd_primelimit
Prototype: 
Help: 
Doc: \kbd{gp} precomputes a list of
 all primes less than \kbd{primelimit} at initialization time, and can build
 fast sieves on demand to quickly iterate over primes up to the \emph{square}
 of \kbd{primelimit}. These are used by many arithmetic functions, usually for
 trial division purposes. The maximal value is $2^{32} - 2049$ (resp $2^{64} -
 2049$) on a 32-bit (resp.~64-bit) machine, but values beyond $10^8$,
 allowing to iterate over primes up to $10^{16}$, do not seem useful.
 
 Since almost all arithmetic functions eventually require some table of prime
 numbers, PARI guarantees that the first 6547 primes, up to and
 including 65557, are precomputed, even if \kbd{primelimit} is $1$.
 
 This default is only used on startup: changing it will not recompute a new
 table.
 
 \misctitle{Deprecated feature} \kbd{primelimit} was used in some
 situations by algebraic number theory functions using the
 \tet{nf_PARTIALFACT} flag (\tet{nfbasis}, \tet{nfdisc}, \tet{nfinit}, \dots):
 this assumes that all primes $p > \kbd{primelimit}$ have a certain
 property (the equation order is $p$-maximal). This is never done by default,
 and must be explicitly set by the user of such functions. Nevertheless,
 these functions now provide a more flexible interface, and their use
 of the global default \kbd{primelimit} is deprecated.
 
 \misctitle{Deprecated feature} \kbd{factor(N, 0)} was used to partially
 factor integers by removing all prime factors $\leq$ \kbd{primelimit}.
 Don't use this, supply an explicit bound: \kbd{factor(N, bound)},
 which avoids relying on an unpredictable global variable.
 
 The default value is \kbd{500k}.

Function: _def_prompt
Class: default
Section: default
C-Name: sd_prompt
Prototype: 
Help: 
Doc: a string that will be printed as
 prompt. Note that most usual escape sequences are available there: \b{e} for
 Esc, \b{n} for Newline, \dots, \kbd{\bs\bs} for \kbd{\bs}. Time expansion is
 performed.
 
 This string is sent through the library function \tet{strftime} (on a
 Unix system, you can try \kbd{man strftime} at your shell prompt). This means
 that \kbd{\%} constructs have a special meaning, usually related to the time
 and date. For instance, \kbd{\%H} = hour (24-hour clock) and \kbd{\%M} =
 minute [00,59] (use \kbd{\%\%} to get a real \kbd{\%}).
 
 If you use \kbd{readline}, escape sequences in your prompt will result in
 display bugs. If you have a relatively recent \kbd{readline} (see the comment
 at the end of \secref{se:def,colors}), you can brace them with special sequences
 (\kbd{\bs[} and \kbd{\bs]}), and you will be safe. If these just result in
 extra spaces in your prompt, then you'll have to get a more recent
 \kbd{readline}. See the file \kbd{misc/gprc.dft} for an example.
 
 \emacs {\bf Caution}: PariEmacs needs to know about the prompt pattern to
 separate your input from previous \kbd{gp} results, without ambiguity. It is
 not a trivial problem to adapt automatically this regular expression to an
 arbitrary prompt (which can be self-modifying!). See PariEmacs's
 documentation.
 
 The default value is \kbd{"? "}.

Function: _def_prompt_cont
Class: default
Section: default
C-Name: sd_prompt_cont
Prototype: 
Help: 
Doc: a string that will be printed
 to prompt for continuation lines (e.g. in between braces, or after a
 line-terminating backslash). Everything that applies to \kbd{prompt}
 applies to \kbd{prompt\_cont} as well.
 
 The default value is \kbd{""}.

Function: _def_psfile
Class: default
Section: default
C-Name: sd_psfile
Prototype: 
Help: 
Doc: name of the default file where
 \kbd{gp} is to dump its PostScript drawings (these are appended, so that no
 previous data are lost). Environment and time expansion are performed.
 
 The default value is \kbd{"pari.ps"}.

Function: _def_readline
Class: default
Section: default
C-Name: sd_readline
Prototype: 
Help: 
Doc: switches readline line-editing
 facilities on and off. This may be useful if you are running \kbd{gp} in a Sun
 \tet{cmdtool}, which interacts badly with readline. Of course, until readline
 is switched on again, advanced editing features like automatic completion
 and editing history are not available.
 
 The default value is \kbd{1}.

Function: _def_realbitprecision
Class: default
Section: default
C-Name: sd_realbitprecision
Prototype: 
Help: 
Doc: the number of significant bits used to convert exact inputs given to
 transcendental functions (see \secref{se:trans}), or to create
 absolute floating point constants (input as \kbd{1.0} or \kbd{Pi} for
 instance). Unless you tamper with the \tet{format} default, this is also
 the number of significant bits used to print a \typ{REAL} number;
 \kbd{format} will override this latter behaviour, and allow you to have a
 large internal precision while outputting few digits for instance.
 
 Note that most PARI's functions currently handle precision on a word basis (by
 increments of 32 or 64 bits), hence bit precision may be a little larger
 than the number of bits you expected. For instance to get 10 bits of
 precision, you need one word of precision which, on a 64-bit machine,
 correspond to 64 bits. To make things even more confusing, this internal bit
 accuracy is converted to decimal digits when printing floating point numbers:
 now 64 bits correspond to 19 printed decimal digits
 ($19 <  \log_{10}(2^{64}) < 20$).
 
 The value returned when typing \kbd{default(realbitprecision)} is the internal
 number of significant bits, not the number of printed decimal digits:
 \bprog
 ? default(realbitprecision, 10)
 ? \pb
       realbitprecision = 64 significant bits
 ? default(realbitprecision)
 %1 = 64
 ? \p
       realprecision = 3 significant digits
 ? default(realprecision)
 %2 = 19
 @eprog\noindent Note that \tet{realprecision} and \kbd{\bs p} allow
 to view and manipulate the internal precision in decimal digits.
 
 The default value is \kbd{128}, resp.~\kbd{96}, on a 64-bit, resp~.32-bit,
 machine.

Function: _def_realprecision
Class: default
Section: default
C-Name: sd_realprecision
Prototype: 
Help: 
Doc: the number of significant digits used to convert exact inputs given to
 transcendental functions (see \secref{se:trans}), or to create
 absolute floating point constants (input as \kbd{1.0} or \kbd{Pi} for
 instance). Unless you tamper with the \tet{format} default, this is also
 the number of significant digits used to print a \typ{REAL} number;
 \kbd{format} will override this latter behaviour, and allow you to have a
 large internal precision while outputting few digits for instance.
 
 Note that PARI's internal precision works on a word basis (by increments of
 32 or 64 bits), hence may be a little larger than the number of decimal
 digits you expected. For instance to get 2 decimal digits you need one word
 of precision which, on a 64-bit machine, actually gives you 19 digits ($19 <
 \log_{10}(2^{64}) < 20$). The value returned when typing
 \kbd{default(realprecision)} is the internal number of significant digits,
 not the number of printed digits:
 \bprog
 ? default(realprecision, 2)
       realprecision = 19 significant digits (2 digits displayed)
 ? default(realprecision)
 %1 = 19
 @eprog
 The default value is \kbd{38}, resp.~\kbd{28}, on a 64-bit, resp.~32-bit,
 machine.

Function: _def_recover
Class: default
Section: default
C-Name: sd_recover
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). If you change this to $0$, any
 error becomes fatal and causes the gp interpreter to exit immediately. Can be
 useful in batch job scripts.
 
 The default value is \kbd{1}.

Function: _def_secure
Class: default
Section: default
C-Name: sd_secure
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). If on, the \tet{system} and
 \tet{extern} command are disabled. These two commands are potentially
 dangerous when you execute foreign scripts since they let \kbd{gp} execute
 arbitrary UNIX commands. \kbd{gp} will ask for confirmation before letting
 you (or a script) unset this toggle.
 
 The default value is \kbd{0}.

Function: _def_seriesprecision
Class: default
Section: default
C-Name: sd_seriesprecision
Prototype: 
Help: 
Doc: number of significant terms
 when converting a polynomial or rational function to a power series
 (see~\b{ps}).
 
 The default value is \kbd{16}.

Function: _def_simplify
Class: default
Section: default
C-Name: sd_simplify
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). When the PARI library computes
 something, the type of the
 result is not always the simplest possible. The only type conversions which
 the PARI library does automatically are rational numbers to integers (when
 they are of type \typ{FRAC} and equal to integers), and similarly rational
 functions to polynomials (when they are of type \typ{RFRAC} and equal to
 polynomials). This feature is useful in many cases, and saves time, but can
 be annoying at times. Hence you can disable this and, whenever you feel like
 it, use the function \kbd{simplify} (see Chapter 3) which allows you to
 simplify objects to the simplest possible types recursively (see~\b{y}).
 \sidx{automatic simplification}
 
 The default value is \kbd{1}.

Function: _def_sopath
Class: default
Section: default
C-Name: sd_sopath
Prototype: 
Help: 
Doc: this is a list of directories, separated by colons ':'
 (semicolons ';' in the DOS world, since colons are preempted for drive names).
 When asked to \tet{install} an external symbol from a shared library whose
 name is not given by an absolute path (does not start with \kbd{/}, \kbd{./}
 or \kbd{../}), \kbd{gp} will look for it in these directories, in the order
 they were written in \kbd{sopath}. Here, as usual, \kbd{.} means the current
 directory, and \kbd{..} its immediate parent. Environment expansion is
 performed.
 
 The default value is \kbd{""}, corresponding to an empty list of
 directories: \tet{install} will use the library name as input (and look in
 the current directory if the name is not an absolute path).

Function: _def_strictargs
Class: default
Section: default
C-Name: sd_strictargs
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). If on, all arguments to \emph{new}
 user functions are mandatory unless the function supplies an explicit default
 value.
 Otherwise arguments have the default value $0$.
 
 In this example,
 \bprog
   fun(a,b=2)=a+b
 @eprog
 \kbd{a} is mandatory, while \kbd{b} is optional. If \kbd{strictargs} is on:
 \bprog
 ? fun()
  ***   at top-level: fun()
  ***                 ^-----
  ***   in function fun: a,b=2
  ***                    ^-----
  ***   missing mandatory argument 'a' in user function.
 @eprog
 This applies to functions defined while \kbd{strictargs} is on. Changing \kbd{strictargs}
 does not affect the behavior of previously defined functions.
 
 The default value is \kbd{0}.

Function: _def_strictmatch
Class: default
Section: default
C-Name: sd_strictmatch
Prototype: 
Help: 
Doc: Obsolete. This toggle is now a no-op.
Obsolete: 2014-10-11

Function: _def_threadsize
Class: default
Section: default
C-Name: sd_threadsize
Prototype: 
Help: 
Doc: In parallel mode, each thread needs its own private \tev{stack} in which
 to do its computations, see \kbd{parisize}. This value determines the size
 in bytes of the stacks of each thread, so the total memory allocated will be
 $\kbd{parisize}+\kbd{nbthreads}\times\kbd{threadsize}$.
 
 If set to $0$, the value used is the same as \kbd{parisize}.
 
 The default value is $0$.

Function: _def_threadsizemax
Class: default
Section: default
C-Name: sd_threadsizemax
Prototype: 
Help: 
Doc: In parallel mode, each threads needs its own private \tev{stack} in which
 to do its computations, see \kbd{parisize}. This value determines the maximal
 size in bytes of the stacks of each thread, so the total memory allocated will
 be between $\kbd{parisize}+\kbd{nbthreads}\times\kbd{threadsize}$. and
 $\kbd{parisize}+\kbd{nbthreads}\times\kbd{threadsizemax}$.
 
 If set to $0$, the value used is the same as \kbd{threadsize}.
 
 The default value is $0$.

Function: _def_timer
Class: default
Section: default
C-Name: sd_timer
Prototype: 
Help: 
Doc: this toggle is either 1 (on) or 0 (off). Every instruction sequence
 in the gp calculator (anything ended by a newline in your input) is timed,
 to some accuracy depending on the hardware and operating system. When
 \tet{timer} is on, each such timing is printed immediately before the
 output as follows:
 \bprog
 ? factor(2^2^7+1)
 time = 108 ms.     \\ this line omitted if 'timer' is 0
 %1 =
 [     59649589127497217 1]
 
 [5704689200685129054721 1]
 @eprog\noindent (See also \kbd{\#} and \kbd{\#\#}.)
 
 The time measured is the user \idx{CPU time}, \emph{not} including the time
 for printing the results. If the time is negligible ($< 1$ ms.), nothing is
 printed: in particular, no timing should be printed when defining a user
 function or an alias, or installing a symbol from the library.
 
 The default value is \kbd{0} (off).

Function: _default_check
Class: gp2c_internal
Help: Code to check for the default marker
Description: 
 (C!GEN):bool    !$(1)
 (var):bool      $(1) == -1

Function: _default_marker
Class: gp2c_internal
Help: Code for default value of GP function
Description: 
 (C!GEN)      NULL
 (var)        -1
 (small)      0
 (str)        ""

Function: _derivfun
Class: basic
Section: programming/internals
C-Name: derivfun0
Prototype: GGp
Help: _derivfun(closure,[args]) numerical derivation of closure with respect to
 the first variable at (args).

Function: _diffptr
Class: gp2c_internal
Help: Table of difference of primes.
Description: 
 ():bptr        diffptr

Function: _err_primes
Class: gp2c_internal
Description: 
 ():void  pari_err(e_MAXPRIME)

Function: _err_type
Class: gp2c_internal
Description: 
 (str,gen):void  pari_err_TYPE($1,$2)

Function: _eval_mnemonic
Class: basic
Section: programming/internals
C-Name: eval_mnemonic
Prototype: lGs
Help: Convert a mnemonic string to a flag.

Function: _factor_Aurifeuille
Class: basic
Section: programming/internals
C-Name: factor_Aurifeuille
Prototype: GL
Help: _factor_Aurifeuille(a,d): return an algebraic factor of Phi_d(a), a != 0

Function: _factor_Aurifeuille_prime
Class: basic
Section: programming/internals
C-Name: factor_Aurifeuille_prime
Prototype: GL
Help: _factor_Aurifeuille_prime(p,d): return an algebraic factor of Phi_d(p), p prime

Function: _forcomposite_init
Class: gp2c_internal
Help: Initialize forcomposite_t.
Description: 
 (forcomposite,int):void                  forcomposite_init(&$1, $2, NULL)
 (forcomposite,int,int):void              forcomposite_init(&$1, $2, $3)

Function: _forcomposite_next
Class: gp2c_internal
Help: Compute the next composite.
Description: 
 (forcomposite):int                       forcomposite_next(&$1)

Function: _formatcode
Class: gp2c_internal
Description: 
 (#small):void                    $1
 (small):small                    %ld
 (#str):void                      $%1
 (str):str                        %s
 (gen):gen                        %Ps

Function: _forpart_init
Class: gp2c_internal
Help: Initialize forpart_t
Description: 
 (forpart,small,?gen,?gen):void      forpart_init(&$1, $2, $3, $4)

Function: _forpart_next
Class: gp2c_internal
Help: Compute the next part
Description: 
 (forpart):vecsmall                  forpart_next(&$1)

Function: _forprime_init
Class: gp2c_internal
Help: Initialize forprime_t.
Description: 
 (forprime,int,?int):void             forprime_init(&$1, $2, $3);

Function: _forprime_next
Class: gp2c_internal
Help: Compute the next prime from the diffptr table.
Description: 
 (*small,*bptr):void  NEXT_PRIME_VIADIFF($1, $2)

Function: _forprime_next_
Class: gp2c_internal
Help: Compute the next prime.
Description: 
 (forprime):int                       forprime_next(&$1)

Function: _forvec_init
Class: gp2c_internal
Help: Initializes parameters for forvec.
Description: 
 (forvec, gen, ?small):void    forvec_init(&$1, $2, $3)

Function: _forvec_next
Class: gp2c_internal
Help: Initializes parameters for forvec.
Description: 
 (forvec):vec    forvec_next(&$1)

Function: _gc_needed
Class: gp2c_internal
Description: 
 (pari_sp):bool                gc_needed($1, 1)

Function: _gerepileall
Class: gp2c_internal
Description: 
 (pari_sp,gen):void:parens    $2 = gerepilecopy($1, $2)
 (pari_sp,gen,...):void       gerepileall($1, ${nbarg 1 sub}, ${stdref 3 code})

Function: _gerepileupto
Class: gp2c_internal
Description: 
 (pari_sp, int):int               gerepileuptoint($1, $2)
 (pari_sp, mp):mp                 gerepileuptoleaf($1, $2)
 (pari_sp, vecsmall):vecsmall     gerepileuptoleaf($1, $2)
 (pari_sp, vec):vec               gerepileupto($1, $2)
 (pari_sp, gen):gen               gerepileupto($1, $2)

Function: _iferr_CATCH
Class: gp2c_internal
Description: 
  (0)               pari_CATCH(CATCH_ALL)
  (small)           pari_CATCH2(__iferr_old$1, CATCH_ALL)

Function: _iferr_CATCH_reset
Class: gp2c_internal
Description: 
  (0):void      pari_CATCH_reset()
  (small):void  pari_CATCH2_reset(__iferr_old$1)

Function: _iferr_ENDCATCH
Class: gp2c_internal
Description: 
  (0)        pari_ENDCATCH
  (small)    pari_ENDCATCH2(__iferr_old$1)

Function: _iferr_error
Class: gp2c_internal
Description: 
  ():error pari_err_last()

Function: _iferr_rethrow
Class: gp2c_internal
Description: 
  (error):void    pari_err(0, $1)

Function: _low_stack_lim
Class: gp2c_internal
Description: 
 (pari_sp,pari_sp):bool        low_stack($1, stack_lim($2, 1))

Function: _maxprime
Class: gp2c_internal
Description: 
 ():small                maxprime()

Function: _multi_if
Class: basic
Section: programming/internals
C-Name: ifpari_multi
Prototype: GE*
Help: internal variant of if() that allows more than 3 arguments.

Function: _ndec2nbits
Class: gp2c_internal
Description: 
 (small):small      ndec2nbits($1)

Function: _ndec2prec
Class: gp2c_internal
Description: 
 (small):small      ndec2prec($1)

Function: _norange
Class: gp2c_internal
Description: 
 ():small    LONG_MAX

Function: _parapply_worker
Class: basic
Section: programming/internals
C-Name: parapply_worker
Prototype: GG
Help: _parapply_worker(d,C): evaluate the closure C on d.

Function: _pareval_worker
Class: basic
Section: programming/internals
C-Name: pareval_worker
Prototype: G
Help: _pareval_worker(C): evaluate the closure C.

Function: _parfor_worker
Class: basic
Section: programming/internals
C-Name: parfor_worker
Prototype: GG
Help: _parfor_worker(i,C): evaluate the closure C on i and return [i,C(i)]

Function: _parvector_worker
Class: basic
Section: programming/internals
C-Name: parvector_worker
Prototype: GG
Help: _parvector_worker(i,C): evaluate the closure C on i.

Function: _polint_worker
Class: basic
Section: programming/internals
C-Name: nmV_polint_center_tree_worker
Prototype: GGGGG
Help: used for parallel chinese
Doc: used for parallel chinese

Function: _polmodular_worker
Class: basic
Section: programming/internals
C-Name: polmodular_worker
Prototype: UUUGGGGLGG
Help: used by polmodular
Doc: used by polmodular

Function: _proto_code
Class: gp2c_internal
Help: Code for argument of a function
Description: 
 (var)          n
 (C!long)       L
 (C!ulong)      U
 (C!GEN)        G
 (C!char*)      s

Function: _proto_max_args
Class: gp2c_internal
Help: Max number of arguments supported by install.
Description: 
 (20)

Function: _proto_ret
Class: gp2c_internal
Help: Code for return value of functions
Description: 
 (C!void)       v
 (C!int)        i
 (C!long)       l
 (C!ulong)      u
 (C!GEN)

Function: _safecoeff
Class: basic
Section: symbolic_operators
Help: safe version of x[a], x[,a] and x[a,b]. Must be lvalues.
Description: 
 (vecsmall,small):small         *safeel($1, $2)
 (list,small):gen:copy          *safelistel($1, $2)
 (gen,small):gen:copy           *safegel($1, $2)
 (gen,small,small):gen:copy     *safegcoeff($1, $2, $3)

Function: _stack_lim
Class: gp2c_internal
Description: 
 (pari_sp,small):pari_sp       stack_lim($1, $2)

Function: _strtoclosure
Class: gp2c_internal
Description: 
 (str):closure               strtofunction($1)
 (str,gen,...):closure       strtoclosure($1, ${nbarg 1 sub}, $3)

Function: _tovec
Class: gp2c_internal
Help: Create a vector holding the arguments (shallow)
Description: 
 ():vec                      cgetg(1, t_VEC)
 (gen):vec                   mkvec($1)
 (gen,gen):vec               mkvec2($1, $2)
 (gen,gen,gen):vec           mkvec3($1, $2, $3)
 (gen,gen,gen,gen):vec       mkvec4($1, $2, $3, $4)
 (gen,gen,gen,gen,gen):vec   mkvec5($1, $2, $3, $4, $5)
 (gen,...):vec               mkvecn($#, $2)

Function: _tovecprec
Class: gp2c_internal
Help: Create a vector holding the arguments and prec (shallow)
Description: 
 ():vec:prec                mkvecs($prec)
 (gen):vec:prec             mkvec2($1, stoi($prec))
 (gen,gen):vec:prec         mkvec3($1, $2, stoi($prec))
 (gen,gen,gen):vec:prec     mkvec4($1, $2, $3, stoi($prec))
 (gen,gen,gen,gen):vec:prec mkvec5($1, $2, $3, $4, stoi($prec))
 (gen,...):vec:prec         mkvecn(${nbarg 1 add}, $2, stoi($prec))

Function: _type_preorder
Class: gp2c_internal
Help: List of chains of type preorder.
Description: 
 (empty, void, bool, small, int, mp, gen)
 (empty, real, mp)
 (empty, bptr, small)
 (empty, bool, lg, small)
 (empty, bool, small_int, small)
 (empty, bool, usmall, small)
 (empty, void, negbool, bool)
 (empty, typ, str, genstr,gen)
 (empty, errtyp, str)
 (empty, vecsmall, gen)
 (empty, vec, gen)
 (empty, list, gen)
 (empty, closure, gen)
 (empty, error, gen)
 (empty, bnr, bnf, nf, vec)
 (empty, bnr, bnf, clgp, vec)
 (empty, ell, vec)
 (empty, prid, vec)
 (empty, gal, vec)
 (empty, var, pol, gen)
 (empty, Fp, Fq, gen)
 (empty, FpX, FqX, gen)

Function: _typedef
Class: gp2c_internal
Description: 
 (empty)        void
 (void)         void
 (negbool)      long
 (bool)         long
 (small_int)    int
 (usmall)       ulong
 (small)        long
 (int)          GEN
 (real)         GEN
 (mp)           GEN
 (lg)           long
 (vecsmall)     GEN
 (vec)          GEN
 (list)         GEN
 (var)          long
 (pol)          GEN
 (gen)          GEN
 (closure)      GEN
 (error)        GEN
 (genstr)       GEN
 (str)          char*
 (bptr)         byteptr
 (forcomposite) forcomposite_t
 (forpart)      forpart_t
 (forprime)     forprime_t
 (forvec)       forvec_t
 (func_GG)      func_GG
 (pari_sp)      pari_sp
 (typ)          long
 (errtyp)       long
 (nf)           GEN
 (bnf)          GEN
 (bnr)          GEN
 (ell)          GEN
 (clgp)         GEN
 (prid)         GEN
 (gal)          GEN
 (Fp)           GEN
 (FpX)          GEN
 (Fq)           GEN
 (FqX)          GEN

Function: _u_forprime_init
Class: gp2c_internal
Help: Initialize forprime_t (ulong version).
Description: 
 (forprime,small,):void              u_forprime_init(&$1, $2, LONG_MAX);
 (forprime,small,small):void         u_forprime_init(&$1, $2, $3);

Function: _u_forprime_next
Class: gp2c_internal
Help: Compute the next prime (ulong version).
Description: 
 (forprime):small                   u_forprime_next(&$1)

Function: _void_if
Class: basic
Section: programming/internals
C-Name: ifpari_void
Prototype: vGDIDI
Help: internal variant of if() that does not return a value.

Function: _wrap_G
Class: gp2c_internal
C-Name: gp_call
Prototype: G
Description: 
  (gen):gen    $1

Function: _wrap_GG
Class: gp2c_internal
C-Name: gp_call2
Prototype: GG
Description: 
  (gen):gen    $1

Function: _wrap_Gp
Class: gp2c_internal
C-Name: gp_callprec
Prototype: Gp
Description: 
  (gen):gen    $1

Function: _wrap_bG
Class: gp2c_internal
C-Name: gp_callbool
Prototype: lG
Description: 
  (bool):bool   $1

Function: _wrap_vG
Class: gp2c_internal
C-Name: gp_callvoid
Prototype: lG
Description: 
  (void):small  0

Function: _||_
Class: basic
Section: symbolic_operators
C-Name: orpari
Prototype: GE
Help: x||y: inclusive OR.
Description: 
 (bool, bool):bool:parens               $(1) || $(2)

Function: _~
Class: basic
Section: symbolic_operators
C-Name: gtrans
Prototype: G
Help: x~: transpose of x.
Description: 
 (vec):vec                        gtrans($1)
 (gen):gen                        gtrans($1)

Function: abs
Class: basic
Section: transcendental
C-Name: gabs
Prototype: Gp
Help: abs(x): absolute value (or modulus) of x.
Description: 
 (small):small    labs($1)
 (int):int        mpabs($1)
 (real):real      mpabs($1)
 (mp):mp          mpabs($1)
 (gen):gen:prec        gabs($1, $prec)
Doc: absolute value of $x$ (modulus if $x$ is complex).
 Rational functions are not allowed. Contrary to most transcendental
 functions, an exact argument is \emph{not} converted to a real number before
 applying \kbd{abs} and an exact result is returned if possible.
 \bprog
 ? abs(-1)
 %1 = 1
 ? abs(3/7 + 4/7*I)
 %2 = 5/7
 ? abs(1 + I)
 %3 = 1.414213562373095048801688724
 @eprog\noindent
 If $x$ is a polynomial, returns $-x$ if the leading coefficient is
 real and negative else returns $x$. For a power series, the constant
 coefficient is considered instead.

Function: acos
Class: basic
Section: transcendental
C-Name: gacos
Prototype: Gp
Help: acos(x): arc cosine of x.
Doc: principal branch of $\cos^{-1}(x) = -i \log (x + i\sqrt{1-x^2})$.
 In particular, $\Re(\text{acos}(x))\in [0,\pi]$ and if $x\in \R$ and $|x|>1$,
 then $\text{acos}(x)$ is complex. The branch cut is in two pieces:
 $]-\infty,-1]$ , continuous with quadrant II, and $[1,+\infty[$, continuous
 with quadrant IV. We have $\text{acos}(x) = \pi/2 - \text{asin}(x)$ for all
 $x$.

Function: acosh
Class: basic
Section: transcendental
C-Name: gacosh
Prototype: Gp
Help: acosh(x): inverse hyperbolic cosine of x.
Doc: principal branch of $\cosh^{-1}(x) = 2
  \log(\sqrt{(x+1)/2} + \sqrt{(x-1)/2})$. In particular,
 $\Re(\text{acosh}(x))\geq 0$ and
 $\Im(\text{acosh}(x))\in ]-\pi,\pi]$; if $x\in \R$ and $x<1$, then
 $\text{acosh}(x)$ is complex.

Function: addhelp
Class: basic
Section: programming/specific
C-Name: addhelp
Prototype: vrs
Help: addhelp(sym,str): add/change help message for the symbol sym.
Doc: changes the help message for the symbol \kbd{sym}. The string \var{str}
 is expanded on the spot and stored as the online help for \kbd{sym}. It is
 recommended to document global variables and user functions in this way,
 although \kbd{gp} will not protest if you don't.
 
 You can attach a help text to an alias, but it will never be
 shown: aliases are expanded by the \kbd{?} help operator and we get the help
 of the symbol the alias points to. Nothing prevents you from modifying the
 help of built-in PARI functions. But if you do, we would like to hear why you
 needed it!
 
 Without \tet{addhelp}, the standard help for user functions consists of its
 name and definition.
 \bprog
 gp> f(x) = x^2;
 gp> ?f
 f =
   (x)->x^2
 
 @eprog\noindent Once addhelp is applied to $f$, the function code is no
 longer included. It can still be consulted by typing the function name:
 \bprog
 gp> addhelp(f, "Square")
 gp> ?f
 Square
 
 gp> f
 %2 = (x)->x^2
 @eprog

Function: addprimes
Class: basic
Section: number_theoretical
C-Name: addprimes
Prototype: DG
Help: addprimes({x=[]}): add primes in the vector x to the prime table to
 be used in trial division. x may also be a single integer. Composite
 "primes" are NOT allowed.
Doc: adds the integers contained in the
 vector $x$ (or the single integer $x$) to a special table of
 ``user-defined primes'', and returns that table. Whenever \kbd{factor} is
 subsequently called, it will trial divide by the elements in this table.
 If $x$ is empty or omitted, just returns the current list of extra
 primes.
 
 The entries in $x$ must be primes: there is no internal check, even if
 the \tet{factor_proven} default is set. To remove primes from the list use
 \kbd{removeprimes}.

Function: agm
Class: basic
Section: transcendental
C-Name: agm
Prototype: GGp
Help: agm(x,y): arithmetic-geometric mean of x and y.
Doc: arithmetic-geometric mean of $x$ and $y$. In the
 case of complex or negative numbers, the optimal AGM is returned
 (the largest in absolute value over all choices of the signs of the square
 roots).  $p$-adic or power series arguments are also allowed. Note that
 a $p$-adic agm exists only if $x/y$ is congruent to 1 modulo $p$ (modulo
 16 for $p=2$). $x$ and $y$ cannot both be vectors or matrices.

Function: alarm
Class: basic
Section: programming/specific
C-Name: gp_alarm
Prototype: D0,L,DE
Help: alarm({s = 0},{code}): if code is omitted, trigger an "e_ALARM"
 exception after s seconds, cancelling any previously set alarm; stop a pending
 alarm if s = 0 or is omitted. Otherwise, evaluate code, aborting after s
 seconds.
Doc: if \var{code} is omitted, trigger an \var{e\_ALARM} exception after $s$
 seconds, cancelling any previously set alarm; stop a pending alarm if $s =
 0$ or is omitted.
 
 Otherwise, if $s$ is positive, the function evaluates \var{code},
 aborting after $s$ seconds. The return value is the value of \var{code} if
 it ran to completion before the alarm timeout, and a \typ{ERROR} object
 otherwise.
 \bprog
   ? p = nextprime(10^25); q = nextprime(10^26); N = p*q;
   ? E = alarm(1, factor(N));
   ? type(E)
   %3 = "t_ERROR"
   ? print(E)
   %4 = error("alarm interrupt after 964 ms.")
   ? alarm(10, factor(N));   \\ enough time
   %5 =
   [ 10000000000000000000000013 1]
 
   [100000000000000000000000067 1]
 @eprog\noindent Here is a more involved example: the function
 \kbd{timefact(N,sec)} below tries to factor $N$ and gives up after \var{sec}
 seconds, returning a partial factorisation.
 \bprog
 \\ Time-bounded partial factorization
 default(factor_add_primes,1);
 timefact(N,sec)=
 {
   F = alarm(sec, factor(N));
   if (type(F) == "t_ERROR", factor(N, 2^24), F);
 }
 @eprog\noindent We either return the factorization directly, or replace the
 \typ{ERROR} result by a simple bounded factorization \kbd{factor(N, 2\pow 24)}.
 Note the \tet{factor_add_primes} trick: any prime larger than $2^{24}$
 discovered while attempting the initial factorization is stored and
 remembered. When the alarm rings, the subsequent bounded factorization finds
 it right away.
 
 \misctitle{Caveat} It is not possible to set a new alarm \emph{within}
 another \kbd{alarm} code: the new timer erases the parent one.

Function: algabsdim
Class: basic
Section: algebras
C-Name: algabsdim
Prototype: lG
Help: algabsdim(al): dimension of the algebra al over its prime subfield.
Doc: Given an algebra \var{al} output by \tet{alginit} or by
 \tet{algtableinit}, returns the dimension of \var{al} over its prime subfield
 ($\Q$ or $\F_p$).
 \bprog
 ? nf = nfinit(y^3-y+1);
 ? A = alginit(nf, [-1,-1]);
 ? algabsdim(A)
 %3 = 12
 @eprog

Function: algadd
Class: basic
Section: algebras
C-Name: algadd
Prototype: GGG
Help: algadd(al,x,y): element x+y in al.
Doc: Given two elements $x$ and $y$ in \var{al}, computes their sum $x+y$ in
 the algebra~\var{al}.
 \bprog
 ? A = alginit(nfinit(y),[-1,1]);
 ? algadd(A,[1,0]~,[1,2]~)
 %2 = [2, 2]~
 @eprog
 
 Also accepts matrices with coefficients in \var{al}.

Function: algalgtobasis
Class: basic
Section: algebras
C-Name: algalgtobasis
Prototype: GG
Help: algalgtobasis(al,x): transforms the element x of the algebra al into a
 column vector on the integral basis of al.
Doc: Given an element \var{x} in the central simple algebra \var{al} output
 by \tet{alginit}, transforms it to a column vector on the integral basis of
 \var{al}. This is the inverse function of \tet{algbasistoalg}.
 \bprog
 ? A = alginit(nfinit(y^2-5),[2,y]);
 ? algalgtobasis(A,[y,1]~)
 %2 = [0, 2, 0, -1, 2, 0, 0, 0]~
 ? algbasistoalg(A,algalgtobasis(A,[y,1]~))
 %3 = [Mod(Mod(y, y^2 - 5), x^2 - 2), 1]~
 @eprog

Function: algaut
Class: basic
Section: algebras
C-Name: algaut
Prototype: mG
Help: algaut(al): the stored automorphism of the splitting field of the
 cyclic algebra al.
Doc: Given a cyclic algebra $\var{al} = (L/K,\sigma,b)$ output by
 \tet{alginit}, returns the automorphism $\sigma$.
 \bprog
 ? nf = nfinit(y);
 ? p = idealprimedec(nf,7)[1];
 ? p2 = idealprimedec(nf,11)[1];
 ? A = alginit(nf,[3,[[p,p2],[1/3,2/3]],[0]]);
 ? algaut(A)
 %5 = -1/3*x^2 + 1/3*x + 26/3
 @eprog

Function: algb
Class: basic
Section: algebras
C-Name: algb
Prototype: mG
Help: algb(al): the element b of the center of the cyclic algebra al used
 to define it.
Doc: Given a cyclic algebra $\var{al} = (L/K,\sigma,b)$ output by
 \tet{alginit}, returns the element $b\in K$.
 \bprog
 nf = nfinit(y);
 ? p = idealprimedec(nf,7)[1];
 ? p2 = idealprimedec(nf,11)[1];
 ? A = alginit(nf,[3,[[p,p2],[1/3,2/3]],[0]]);
 ? algb(A)
 %5 = Mod(-77, y)
 @eprog

Function: algbasis
Class: basic
Section: algebras
C-Name: algbasis
Prototype: mG
Help: algbasis(al): basis of the stored order of the central simple algebra al.
Doc: Given an central simple algebra \var{al} output by \tet{alginit}, returns
 a $\Z$-basis of the order~${\cal O}_0$ stored in \var{al} with respect to the
 natural order in \var{al}. It is a maximal order if one has been computed.
 \bprog
 A = alginit(nfinit(y), [-1,-1]);
 ? algbasis(A)
 %2 =
 [1 0 0 1/2]
 
 [0 1 0 1/2]
 
 [0 0 1 1/2]
 
 [0 0 0 1/2]
 @eprog

Function: algbasistoalg
Class: basic
Section: algebras
C-Name: algbasistoalg
Prototype: GG
Help: algbasistoalg(al,x): transforms the column vector x on the integral
 basis of al into an element of al in algebraic form.
Doc: Given an element \var{x} in the central simple algebra \var{al} output
 by \tet{alginit}, transforms it to its algebraic representation in \var{al}.
 This is the inverse function of \tet{algalgtobasis}.
 \bprog
 ? A = alginit(nfinit(y^2-5),[2,y]);
 ? z = algbasistoalg(A,[0,1,0,0,2,-3,0,0]~);
 ? liftall(z)
 %3 = [(-1/2*y - 2)*x + (-1/4*y + 5/4), -3/4*y + 7/4]~
 ? algalgtobasis(A,z)
 %4 = [0, 1, 0, 0, 2, -3, 0, 0]~
 @eprog

Function: algcenter
Class: basic
Section: algebras
C-Name: algcenter
Prototype: mG
Help: algcenter(al): center of the algebra al.
Doc: If \var{al} is a table algebra output by \tet{algtableinit}, returns a
 basis of the center of the algebra~\var{al} over its prime field ($\Q$ or
 $\F_p$). If \var{al} is a central simple algebra output by \tet{alginit},
 returns the center of~\var{al}, which is stored in \var{al}.
 
 A simple example: the $2\times 2$ upper triangular matrices over $\Q$,
 generated by $I_2$, $a = \kbd{[0,1;0,0]}$ and $b = \kbd{[0,0;0,1]}$,
 such that $a^2 = 0$, $ab = a$, $ba = 0$, $b^2 = b$: the diagonal matrices
 form the center.
 \bprog
 ? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
 ? A = algtableinit(mt);
 ? algcenter(A) \\ = (I_2)
 %3 =
 [1]
 
 [0]
 
 [0]
 @eprog
 
 An example in the central simple case:
 
 \bprog
 ? nf = nfinit(y^3-y+1);
 ? A = alginit(nf, [-1,-1]);
 ? algcenter(A).pol
 %3 = y^3 - y + 1
 @eprog

Function: algcentralproj
Class: basic
Section: algebras
C-Name: alg_centralproj
Prototype: GGD0,L,
Help: algcentralproj(al,z,{maps=0}): projections of the algebra al on the
 orthogonal central idempotents z[i].
Doc: Given a table algebra \var{al} output by \tet{algtableinit} and a
 \typ{VEC} $\var{z}=[z_1,\dots,z_n]$ of orthogonal central idempotents,
 returns a \typ{VEC} $[al_1,\dots,al_n]$ of algebras such that
 $al_i = z_i\, al$. If $\var{maps}=1$, each $al_i$ is a \typ{VEC}
 $[quo,proj,lift]$ where \var{quo} is the quotient algebra, \var{proj} is a
 \typ{MAT} representing the projection onto this quotient and \var{lift} is a
 \typ{MAT} representing a lift.
 
 A simple example: $\F_2\oplus \F_4$, generated by~$1=(1,1)$, $e=(1,0)$
 and~$x$ such that~$x^2+x+1=0$. We have~$e^2=e$, $x^2=x+1$ and~$ex=0$.
 \bprog
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,2);
 ? e = [0,1,0]~;
 ? e2 = algsub(A,[1,0,0]~,e);
 ? [a,a2] = algcentralproj(A,[e,e2]);
 ? algdim(a)
 %6 = 1
 ? algdim(a2)
 %7 = 2
 @eprog

Function: algchar
Class: basic
Section: algebras
C-Name: algchar
Prototype: mG
Help: algchar(al): characteristic of the algebra al.
Doc: Given an algebra \var{al} output by \tet{alginit} or \tet{algtableinit},
 returns the characteristic of \var{al}.
 \bprog
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,13);
 ? algchar(A)
 %3 = 13
 @eprog

Function: algcharpoly
Class: basic
Section: algebras
C-Name: algcharpoly
Prototype: GGDn
Help: algcharpoly(al,b,{v='x}): (reduced) characteristic polynomial of b in
 \var{al}, with respect to the variable $v$.
Doc: Given an element $b$ in \var{al}, returns its characteristic polynomial
 as a polynomial in the variable $v$. If \var{al} is a table algebra output
 by \tet{algtableinit}, returns the absolute characteristic polynomial of
 \var{b}, which is an element of $\F_p[v]$ or~$\Q[v]$; if \var{al} is a
 central simple algebra output by \tet{alginit}, returns the reduced
 characteristic polynomial of \var{b}, which is an element of $K[v]$ where~$K$
 is the center of \var{al}.
 \bprog
 ? al = alginit(nfinit(y), [-1,-1]); \\ (-1,-1)_Q
 ? algcharpoly(al, [0,1]~)
 %2 = x^2 + 1
 @eprog
 
 Also accepts a square matrix with coefficients in \var{al}.

Function: algdecomposition
Class: basic
Section: algebras
C-Name: alg_decomposition
Prototype: G
Help: algdecomposition(al): semisimple decomposition of the algebra al.
Doc: \var{al} being a table algebra output by \tet{algtableinit}, returns
 $[J,[al_1,\dots,al_n]]$ where $J$ is a basis of the Jacobson radical of
 \var{al} and $al_1,\dots,al_n$ are the simple factors of the semisimple
 algebra $al/J$.

Function: algdegree
Class: basic
Section: algebras
C-Name: algdegree
Prototype: lG
Help: algdegree(al): degree of the central simple algebra al.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, returns
 the degree of \var{al}.
 \bprog
 ? nf = nfinit(y^3-y+1);
 ? A = alginit(nf, [-1,-1]);
 ? algdegree(A)
 %3 = 2
 @eprog

Function: algdep
Class: basic
Section: linear_algebra
C-Name: algdep0
Prototype: GLD0,L,
Help: algdep(z,k,{flag=0}): algebraic relations up to degree n of z, using
 lindep([1,z,...,z^(k-1)], flag).
Doc: \sidx{algebraic dependence}
 $z$ being real/complex, or $p$-adic, finds a polynomial (in the variable
 \kbd{'x}) of degree at most
 $k$, with integer coefficients, having $z$ as approximate root. Note that the
 polynomial which is obtained is not necessarily the ``correct'' one. In fact
 it is not even guaranteed to be irreducible. One can check the closeness
 either by a polynomial evaluation (use \tet{subst}), or by computing the
 roots of the polynomial given by \kbd{algdep} (use \tet{polroots} or
 \tet{polrootspadic}).
 
 Internally, \tet{lindep}$([1,z,\ldots,z^k], \fl)$ is used. A non-zero value of
 $\fl$ may improve on the default behavior if the input number is known to a
 \emph{huge} accuracy, and you suspect the last bits are incorrect: if $\fl > 0$
 the computation is done with an accuracy of $\fl$ decimal  digits; to get
 meaningful results,  the parameter $\fl$ should be smaller than the number of
 correct decimal digits in the input.
 But default values are usually sufficient, so try without $\fl$ first:
 \bprog
 ? \p200
 ? z = 2^(1/6)+3^(1/5);
 ? algdep(z, 30);      \\ right in 280ms
 ? algdep(z, 30, 100); \\ wrong in 169ms
 ? algdep(z, 30, 170); \\ right in 288ms
 ? algdep(z, 30, 200); \\ wrong in 320ms
 ? \p250
 ? z = 2^(1/6)+3^(1/5); \\ recompute to new, higher, accuracy !
 ? algdep(z, 30);      \\ right in 329ms
 ? algdep(z, 30, 200); \\ right in 324ms
 ? \p500
 ? algdep(2^(1/6)+3^(1/5), 30); \\ right in 677ms
 ? \p1000
 ? algdep(2^(1/6)+3^(1/5), 30); \\ right in 1.5s
 @eprog\noindent
 The changes in \kbd{realprecision} only affect the quality of the
 initial approximation to $2^{1/6} + 3^{1/5}$, \kbd{algdep} itself uses
 exact operations. The size of its operands depend on the accuracy of the
 input of course: more accurate input means slower operations.
 
 Proceeding by increments of 5 digits of accuracy, \kbd{algdep} with default
 flag produces its first correct result at 195 digits, and from then on a
 steady stream of correct results:
 \bprog
   \\ assume T contains the correct result, for comparison
   forstep(d=100, 250, 5, localprec(d);\
     print(d, " ", algdep(2^(1/6)+3^(1/5),30) == T))
 @eprog
 
 The above example is the test case studied in a 2000 paper by Borwein and
 Lisonek: Applications of integer relation algorithms, \emph{Discrete Math.},
 {\bf 217}, p.~65--82. The version of PARI tested there was 1.39, which
 succeeded reliably from precision 265 on, in about 200 as much time as the
 current version.
Variant: Also available is \fun{GEN}{algdep}{GEN z, long k} ($\fl=0$).

Function: algdim
Class: basic
Section: algebras
C-Name: algdim
Prototype: lG
Help: algdim(al): dimension of the algebra al.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, returns
 the dimension of \var{al} over its center. Given a table algebra \var{al}
 output by \tet{algtableinit}, returns the dimension of \var{al} over its prime
 subfield ($\Q$ or $\F_p$).
 \bprog
 ? nf = nfinit(y^3-y+1);
 ? A = alginit(nf, [-1,-1]);
 ? algdim(A)
 %3 = 4
 @eprog

Function: algdisc
Class: basic
Section: algebras
C-Name: algdisc
Prototype: G
Help: algdisc(al): discriminant of the stored order of the algebra al.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, computes
 the discriminant of the order ${\cal O}_0$ stored in \var{al}, that is the
 determinant of the trace form $\rm{Tr} : {\cal O}_0\times {\cal O}_0 \to \Z$.
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-3,1-y]);
 ? [PR,h] = alghassef(A);
 %3 = [[[2, [2, 0]~, 1, 2, 1], [3, [3, 0]~, 1, 2, 1]], Vecsmall([0, 1])]
 ? n = algdegree(A);
 ? D = algabsdim(A);
 ? h = vector(#h, i, n - gcd(n,h[i]));
 ? n^D * nf.disc^(n^2) * idealnorm(nf, idealfactorback(nf,PR,h))^n
 %4 = 12960000
 ? algdisc(A)
 %5 = 12960000
 @eprog

Function: algdivl
Class: basic
Section: algebras
C-Name: algdivl
Prototype: GGG
Help: algdivl(al,x,y): element x\y in al.
Doc: Given two elements $x$ and $y$ in \var{al}, computes their left quotient
 $x\backslash y$ in the algebra \var{al}: an element $z$ such that $xz=y$ (such
 an element is not unique when $x$ is a zerodivisor). If~$x$ is invertible, this
 is the same as $x^{-1}y$. Assumes that $y$ is left divisible by $x$ (i.e. that
 $z$ exists). Also accepts matrices with coefficients in~\var{al}.

Function: algdivr
Class: basic
Section: algebras
C-Name: algdivr
Prototype: GGG
Help: algdivr(al,x,y): element x/y in al.
Doc: Given two elements $x$ and $y$ in \var{al}, return $xy^{-1}$. Also accepts
 matrices with coefficients in \var{al}.

Function: alggroup
Class: basic
Section: algebras
C-Name: alggroup
Prototype: GDG
Help: alggroup(gal, {p=0}): constructs the group algebra of gal over Q (resp. Fp).
Doc: initialize the group algebra~$K[G]$ over~$K=\Q$ ($p$ omitted) or~$\F_p$
 where~$G$ is the underlying group of the \kbd{galoisinit} structure~\var{gal}.
 The input~\var{gal} is also allowed to be a \typ{VEC} of permutations that is
 closed under products.
 
 Example:
 \bprog
 ? K = nfsplitting(x^3-x+1);
 ? gal = galoisinit(K);
 ? al = alggroup(gal);
 ? algissemisimple(al)
 %4 = 1
 ? G = [Vecsmall([1,2,3]), Vecsmall([1,3,2])];
 ? al2 = alggroup(G, 2);
 ? algissemisimple(al2)
 %8 = 0
 @eprog

Function: alghasse
Class: basic
Section: algebras
C-Name: alghasse
Prototype: GG
Help: alghasse(al,pl): the hasse invariant of the central simple algebra al at
 the place pl.
Doc: Given a central simple algebra \var{al} output by \tet{alginit} and a prime
 ideal or an integer between $1$ and $r_1+r_2$, returns a \typ{FRAC} $h$ : the
 local Hasse invariant of \var{al} at the place specified by \var{pl}.
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-1,y]);
 ? alghasse(A, 1)
 %3 = 1/2
 ? alghasse(A, 2)
 %4 = 0
 ? alghasse(A, idealprimedec(nf,2)[1])
 %5 = 1/2
 ? alghasse(A, idealprimedec(nf,5)[1])
 %6 = 0
 @eprog

Function: alghassef
Class: basic
Section: algebras
C-Name: alghassef
Prototype: mG
Help: alghassef(al): the hasse invariant of the central simple algebra al at finite places.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, returns
 a \typ{VEC} $[\kbd{PR}, h_f]$ describing the local Hasse invariants at the
 finite places of the center: \kbd{PR} is a \typ{VEC} of primes and $h_f$ is a
 \typ{VECSMALL} of integers modulo the degree $d$ of \var{al}.
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-1,2*y-1]);
 ? [PR,hf] = alghassef(A);
 ? PR
 %4 = [[19, [10, 2]~, 1, 1, [-8, 2; 2, -10]], [2, [2, 0]~, 1, 2, 1]]
 ? hf
 %5 = Vecsmall([1, 0])
 @eprog

Function: alghassei
Class: basic
Section: algebras
C-Name: alghassei
Prototype: mG
Help: alghassei(al): the hasse invariant of the central simple algebra al
 at infinite places.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, returns
 a \typ{VECSMALL} $h_i$ of $r_1$ integers modulo the degree $d$ of \var{al},
 where $r_1$ is the number of real places of the center: the local Hasse
 invariants of \var{al} at infinite places.
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-1,y]);
 ? alghassei(A)
 %3 = Vecsmall([1, 0])
 @eprog

Function: algindex
Class: basic
Section: algebras
C-Name: algindex
Prototype: lGDG
Help: algindex(al,{pl}): the index of the central simple algebra al. If pl is
 set, it should be a prime ideal of the center or an integer between 1 and
 r1+r2, and in that case return the local index at the place pl instead.
Doc: Return the index of the central simple algebra~$A$ over~$K$ (as output by
 alginit), that is the degree~$e$ of the unique central division algebra~$D$
 over $K$ such that~$A$ is isomorphic to some matrix algebra~$M_d(D)$. If
 \var{pl} is set, it should be a prime ideal of~$K$ or an integer between~$1$
 and~$r_1+r_2$, and in that case return the local index at the place \var{pl}
 instead.
 
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-1,y]);
 ? algindex(A, 1)
 %3 = 2
 ? algindex(A, 2)
 %4 = 1
 ? algindex(A, idealprimedec(nf,2)[1])
 %5 = 2
 ? algindex(A, idealprimedec(nf,5)[1])
 %6 = 1
 ? algindex(A)
 %7 = 2
 @eprog

Function: alginit
Class: basic
Section: algebras
C-Name: alginit
Prototype: GGDnD1,L,
Help: alginit(B, C, {v}, {flag = 1}): initialize the central simple algebra
 defined by data B, C. If flag = 1, compute a maximal order.
Doc: initialize the central simple algebra defined by data $B$, $C$ and
 variable $v$, as follows.
 
 \item (multiplication table) $B$ is the base number field $K$ in \tet{nfinit}
 form, $C$ is a ``multiplication table'' over $K$.
 As a $K$-vector space, the algebra is generated by a basis
 $(e_1 = 1,\dots, e_n)$; the table is given as a \typ{VEC} of $n$ matrices in
 $M_n(K)$, giving the left multiplication by the basis elements $e_i$, in the
 given basis.
 Assumes that $e_1= 1$, that the multiplication table is integral, and that
 $K[e_1,\dots,e_n]$ describes a central simple algebra over $K$.
 \bprog
 { m_i = [0,-1,0, 0;
          1, 0,0, 0;
          0, 0,0,-1;
          0, 0,1, 0];
   m_j = [0, 0,-1,0;
          0, 0, 0,1;
          1, 0, 0,0;
          0,-1, 0,0];
   m_k = [0, 0, 0, 0;
          0, 0,-1, 0;
          0, 1, 0, 0;
          1, 0, 0,-1];
   A = alginit(nfinit(y), [matid(4), m_i,m_j,m_k],  0); }
 @eprog represents (in a complicated way) the quaternion algebra $(-1,-1)_\Q$.
 See below for a simpler solution.
 
 \item (cyclic algebra) $B$ is an \kbd{rnf} structure attached to a cyclic
 number field extension $L/K$ of degree $d$, $C$ is a \typ{VEC}
 \kbd{[sigma,b]} with 2 components: \kbd{sigma} is a \typ{POLMOD} representing
 an automorphism generating $\text{Gal}(L/K)$, $b$ is an element in $K^*$. This
 represents the cyclic algebra~$(L/K,\sigma,b)$. Currently the element $b$ has
 to be integral.
 \bprog
  ? Q = nfinit(y); T = polcyclo(5, 'x); F = rnfinit(Q, T);
  ? A = alginit(F, [Mod(x^2,T), 3]);
 @eprog defines the cyclic algebra $(L/\Q, \sigma, 3)$, where
 $L = \Q(\zeta_5)$ and $\sigma:\zeta\mapsto\zeta^2$ generates
 $\text{Gal}(L/\Q)$.
 
 \item (quaternion algebra, special case of the above) $B$ is an \kbd{nf}
 structure attached to a number field $K$, $C = [a,b]$ is a vector
 containing two elements of $K^*$ with $a$ not a square in $K$, returns the quaternion algebra $(a,b)_K$.
 The variable $v$ (\kbd{'x} by default) must have higher priority than the
 variable of $K$\kbd{.pol} and is used to represent elements in the splitting
 field $L = K[x]/(x^2-a)$.
 \bprog
  ? Q = nfinit(y); A = alginit(Q, [-1,-1]);  \\@com $(-1,-1)_\Q$
 @eprog
 
 \item (algebra/$K$ defined by local Hasse invariants)
 $B$ is an \kbd{nf} structure attached to a number field $K$,
 $C = [d, [\kbd{PR},h_f], h_i]$ is a triple
 containing an integer $d > 1$, a pair $[\kbd{PR}, h_f]$ describing the
 Hasse invariants at finite places, and $h_i$ the Hasse invariants
 at archimedean (real) places. A local Hasse invariant belongs to $(1/d)\Z/\Z
 \subset \Q/\Z$, and is given either as a \typ{FRAC} (lift to $(1/d)\Z$),
 a \typ{INT} or \typ{INTMOD} modulo $d$ (lift to $\Z/d\Z$); a whole vector
 of local invariants can also be given as a \typ{VECSMALL}, whose
 entries are handled as \typ{INT}s. \kbd{PR} is a list of prime ideals
 (\kbd{prid} structures), and $h_f$ is a vector of the same length giving the
 local invariants at those maximal ideals. The invariants at infinite real
 places are indexed by the real roots $K$\kbd{.roots}: if the Archimedean
 place $v$ is attached to the $j$-th root, the value of
 $h_v$ is given by $h_i[j]$, must be $0$ or $1/2$ (or~$d/2$ modulo~$d$), and
 can be nonzero only if~$d$ is even.
 
 By class field theory, provided the local invariants $h_v$ sum to $0$, up
 to Brauer equivalence, there is a unique central simple algebra over $K$
 with given local invariants and trivial invariant elsewhere. In particular,
 up to isomorphism, there is a unique such algebra $A$ of degree $d$.
 
 We realize $A$ as a cyclic algebra through class field theory. The variable $v$
 (\kbd{'x} by default) must have higher priority than the variable of
 $K$\kbd{.pol} and is used to represent elements in the (cyclic) splitting
 field extension $L/K$ for $A$.
 
 \bprog
  ? nf = nfinit(y^2+1);
  ? PR = idealprimedec(nf,5); #PR
  %2 = 2
  ? hi = [];
  ? hf = [PR, [1/3,-1/3]];
  ? A = alginit(nf, [3,hf,hi]);
  ? algsplittingfield(A).pol
  %6 = x^3 - 21*x + 7
 @eprog
 
 \item (matrix algebra, toy example) $B$ is an \kbd{nf} structure attached
 to a number field $K$, $C = d$ is a positive integer. Returns a cyclic
 algebra isomorphic to the matrix algebra $M_d(K)$.
 
 In all cases, this function computes a maximal order for the algebra by default,
 which may require a lot of time. Setting $\fl = 0$ prevents this computation.
 
 The pari object representing such an algebra $A$ is a \typ{VEC} with the
 following data:
 
  \item A splitting field $L$ of $A$ of the same degree over $K$ as $A$, in
 \kbd{rnfinit} format, accessed with \kbd{algsplittingfield}.
 
  \item The same splitting field $L$ in \kbd{nfinit} format.
 
  \item The Hasse invariants at the real places of $K$, accessed with
 \kbd{alghassei}.
 
  \item The Hasse invariants of $A$ at the finite primes of $K$ that ramify in
 the natural order of $A$, accessed with \kbd{alghassef}.
 
  \item A basis of an order ${\cal O}_0$ expressed on the basis of the natural
 order, accessed with \kbd{algbasis}.
 
  \item A basis of the natural order expressed on the basis of ${\cal O}_0$,
 accessed with \kbd{alginvbasis}.
 
  \item The left multiplication table of ${\cal O}_0$ on the previous basis,
 accessed with \kbd{algmultable}.
 
  \item The characteristic of $A$ (always $0$), accessed with \kbd{algchar}.
 
  \item The absolute traces of the elements of the basis of ${\cal O}_0$.
 
  \item If $A$ was constructed as a cyclic algebra~$(L/K,\sigma,b)$ of degree
 $d$, a \typ{VEC} $[\sigma,\sigma^2,\dots,\sigma^{d-1}]$. The function
 \kbd{algaut} returns $\sigma$.
 
  \item If $A$ was constructed as a cyclic algebra~$(L/K,\sigma,b)$, the
 element $b$, accessed with \kbd{algb}.
 
  \item If $A$ was constructed with its multiplication table $mt$ over $K$,
 the \typ{VEC} of \typ{MAT} $mt$, accessed with \kbd{algrelmultable}.
 
  \item If $A$ was constructed with its multiplication table $mt$ over $K$,
 a \typ{VEC} with three components: a \typ{COL} representing an element of $A$
 generating the splitting field $L$ as a maximal subfield of $A$, a \typ{MAT}
 representing an $L$-basis ${\cal B}$ of $A$ expressed on the $\Z$-basis of
 ${\cal O}_0$, and a \typ{MAT} representing the $\Z$-basis of ${\cal O}_0$
 expressed on ${\cal B}$. This data is accessed with \kbd{algsplittingdata}.

Function: alginv
Class: basic
Section: algebras
C-Name: alginv
Prototype: GG
Help: alginv(al,x): element 1/x in al.
Doc: Given an element $x$ in \var{al}, computes its inverse $x^{-1}$ in the
 algebra \var{al}. Assumes that $x$ is invertible.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? alginv(A,[1,1,0,0]~)
 %2 = [1/2, 1/2, 0, 0]~
 @eprog
 
 Also accepts matrices with coefficients in \var{al}.

Function: alginvbasis
Class: basic
Section: algebras
C-Name: alginvbasis
Prototype: mG
Help: alginvbasis(al): basis of the natural order of the central simple algebra
 al in terms of the stored order.
Doc: Given an central simple algebra \var{al} output by \tet{alginit}, returns
 a $\Z$-basis of the natural order in \var{al} with respect to the
 order~${\cal O}_0$ stored in \var{al}.
 \bprog
 A = alginit(nfinit(y), [-1,-1]);
 ? alginvbasis(A)
 %2 =
 [1 0 0 -1]
 
 [0 1 0 -1]
 
 [0 0 1 -1]
 
 [0 0 0  2]
 @eprog

Function: algisassociative
Class: basic
Section: algebras
C-Name: algisassociative
Prototype: iGD0,G,
Help: algisassociative(mt,p=0): true (1) if the multiplication table mt is
 suitable for algtableinit(mt,p), false (0) otherwise.
Doc: Returns 1 if the multiplication table \kbd{mt} is suitable for
 \kbd{algtableinit(mt,p)}, 0 otherwise. More precisely, \kbd{mt} should be
 a \typ{VEC} of $n$ matrices in $M_n(K)$, giving the left multiplications
 by the basis elements $e_1, \dots, e_n$ (structure constants).
 We check whether the first basis element $e_1$ is $1$ and $e_i(e_je_k) =
 (e_ie_j)e_k$ for all $i,j,k$.
 \bprog
  ? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
  ? algisassociative(mt)
  %2 = 1
 @eprog
 
 May be used to check a posteriori an algebra: we also allow \kbd{mt} as
 output by \tet{algtableinit} ($p$ is ignored in this case).

Function: algiscommutative
Class: basic
Section: algebras
C-Name: algiscommutative
Prototype: iG
Help: algiscommutative(al): test whether the algebra al is commutative.
Doc: \var{al} being a table algebra output by \tet{algtableinit} or a central
 simple algebra output by \tet{alginit}, tests whether the algebra \var{al} is
 commutative.
 \bprog
 ? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
 ? A = algtableinit(mt);
 ? algiscommutative(A)
 %3 = 0
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,2);
 ? algiscommutative(A)
 %6 = 1
 @eprog

Function: algisdivision
Class: basic
Section: algebras
C-Name: algisdivision
Prototype: iGDG
Help: algisdivision(al,{pl}): test whether the central simple algebra al is a
 division algebra. If pl is set, it should be a prime ideal of the center or an
 integer between 1 and r1+r2, and in that case test whether al is locally a
 division algebra at the place pl instead.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, test
 whether \var{al} is a division algebra. If \var{pl} is set, it should be a
 prime ideal of~$K$ or an integer between~$1$ and~$r_1+r_2$, and in that case
 test whether \var{al} is locally a division algebra at the place \var{pl}
 instead.
 
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-1,y]);
 ? algisdivision(A, 1)
 %3 = 1
 ? algisdivision(A, 2)
 %4 = 0
 ? algisdivision(A, idealprimedec(nf,2)[1])
 %5 = 1
 ? algisdivision(A, idealprimedec(nf,5)[1])
 %6 = 0
 ? algisdivision(A)
 %7 = 1
 @eprog

Function: algisdivl
Class: basic
Section: algebras
C-Name: algisdivl
Prototype: iGGGD&
Help: algisdivl(al,x,y,{&z}): tests whether y is left divisible by x and sets z
 to the left quotient x\y.
Doc: Given two elements $x$ and $y$ in \var{al}, tests whether $y$ is left
 divisible by $x$, that is whether there exists~$z$ in \var{al} such
 that~$xz=y$, and sets $z$ to this element if it exists.
 \bprog
 ? A = alginit(nfinit(y), [-1,1]);
 ? algisdivl(A,[x+2,-x-2]~,[x,1]~)
 %2 = 0
 ? algisdivl(A,[x+2,-x-2]~,[-x,x]~,&z)
 %3 = 1
 ? z
 %4 = [Mod(-2/5*x - 1/5, x^2 + 1), 0]~
 @eprog
 
 Also accepts matrices with coefficients in \var{al}.

Function: algisinv
Class: basic
Section: algebras
C-Name: algisinv
Prototype: iGGD&
Help: algisinv(al,x,{&ix}): tests whether x is invertible and sets ix to the
 inverse of x.
Doc: Given an element $x$ in \var{al}, tests whether $x$ is invertible, and sets
 $ix$ to the inverse of $x$.
 \bprog
 ? A = alginit(nfinit(y), [-1,1]);
 ? algisinv(A,[-1,1]~)
 %2 = 0
 ? algisinv(A,[1,2]~,&ix)
 %3 = 1
 ? ix
 %4 = [Mod(Mod(-1/3, y), x^2 + 1), Mod(Mod(2/3, y), x^2 + 1)]~
 @eprog
 
 Also accepts matrices with coefficients in \var{al}.

Function: algisramified
Class: basic
Section: algebras
C-Name: algisramified
Prototype: iGDG
Help: algisramified(al,{pl}): test whether the central simple algebra al is
 ramified, i.e. not isomorphic to a matrix ring over its center. If pl is set,
 it should be a prime ideal of the center or an integer between 1 and r1+r2, and
 in that case test whether al is locally ramified at the place pl instead.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, test
 whether \var{al} is ramified, i.e. not isomorphic to a matrix algebra over its
 center. If \var{pl} is set, it should be a prime ideal of~$K$ or an integer
 between~$1$ and~$r_1+r_2$, and in that case test whether \var{al} is locally
 ramified at the place \var{pl} instead.
 
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-1,y]);
 ? algisramified(A, 1)
 %3 = 1
 ? algisramified(A, 2)
 %4 = 0
 ? algisramified(A, idealprimedec(nf,2)[1])
 %5 = 1
 ? algisramified(A, idealprimedec(nf,5)[1])
 %6 = 0
 ? algisramified(A)
 %7 = 1
 @eprog

Function: algissemisimple
Class: basic
Section: algebras
C-Name: algissemisimple
Prototype: iG
Help: algissemisimple(al): test whether the algebra al is semisimple.
Doc: \var{al} being a table algebra output by \tet{algtableinit} or a central
 simple algebra output by \tet{alginit}, tests whether the algebra \var{al} is
 semisimple.
 \bprog
 ? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
 ? A = algtableinit(mt);
 ? algissemisimple(A)
 %3 = 0
 ? m_i=[0,-1,0,0;1,0,0,0;0,0,0,-1;0,0,1,0]; \\ quaternion algebra (-1,-1)
 ? m_j=[0,0,-1,0;0,0,0,1;1,0,0,0;0,-1,0,0];
 ? m_k=[0,0,0,-1;0,0,-1,0;0,1,0,0;1,0,0,0];
 ? mt = [matid(4), m_i, m_j, m_k];
 ? A = algtableinit(mt);
 ? algissemisimple(A)
 %9 = 1
 @eprog

Function: algissimple
Class: basic
Section: algebras
C-Name: algissimple
Prototype: iGD0,L,
Help: algissimple(al, {ss = 0}): test whether the algebra al is simple.
Doc: \var{al} being a table algebra output by \tet{algtableinit} or a central
 simple algebra output by \tet{alginit}, tests whether the algebra \var{al} is
 simple. If $\var{ss}=1$, assumes that the algebra~\var{al} is semisimple
 without testing it.
 \bprog
 ? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
 ? A = algtableinit(mt); \\ matrices [*,*; 0,*]
 ? algissimple(A)
 %3 = 0
 ? algissimple(A,1) \\ incorrectly assume that A is semisimple
 %4 = 1
 ? m_i=[0,-1,0,0;1,0,0,0;0,0,0,-1;0,0,1,0];
 ? m_j=[0,0,-1,0;0,0,0,1;1,0,0,0;0,-1,0,0];
 ? m_k=[0,0,0,-1;0,0,b,0;0,1,0,0;1,0,0,0];
 ? mt = [matid(4), m_i, m_j, m_k];
 ? A = algtableinit(mt); \\ quaternion algebra (-1,-1)
 ? algissimple(A)
 %10 = 1
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,2); \\ direct sum F_4+F_2
 ? algissimple(A)
 %13 = 0
 @eprog

Function: algissplit
Class: basic
Section: algebras
C-Name: algissplit
Prototype: iGDG
Help: algissplit(al,{pl}): test whether the central simple algebra al is
 split, i.e. isomorphic to a matrix ring over its center. If pl is set, it
 should be a prime ideal of the center or an integer between 1 and r1+r2, and in
 that case test whether al is locally split at the place pl instead.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, test
 whether \var{al} is split, i.e. isomorphic to a matrix algebra over its center.
 If \var{pl} is set, it should be a prime ideal of~$K$ or an integer between~$1$
 and~$r_1+r_2$, and in that case test whether \var{al} is locally split at the
 place \var{pl} instead.
 
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-1,y]);
 ? algissplit(A, 1)
 %3 = 0
 ? algissplit(A, 2)
 %4 = 1
 ? algissplit(A, idealprimedec(nf,2)[1])
 %5 = 0
 ? algissplit(A, idealprimedec(nf,5)[1])
 %6 = 1
 ? algissplit(A)
 %7 = 0
 @eprog

Function: alglathnf
Class: basic
Section: algebras
C-Name: alglathnf
Prototype: GG
Help: alglathnf(al,m): the lattice generated by the columns of m.
Doc: Given an algebra \var{al} and a square invertible matrix \var{m} with size
 the dimension of \var{al}, returns the lattice generated by the columns of
 \var{m}.
 \bprog
 ? al = alginit(nfinit(y^2+7), [-1,-1]);
 ? a = [1,1,-1/2,1,1/3,-1,1,1]~;
 ? mt = algleftmultable(al,a);
 ? lat = alglathnf(al,mt);
 ? lat[2]
 %5 = 1/6
 @eprog

Function: algleftmultable
Class: basic
Section: algebras
C-Name: algleftmultable
Prototype: GG
Help: algleftmultable(al,x): left multiplication table of x.
Doc: Given an element \var{x} in \var{al}, computes its left multiplication
 table. If \var{x} is given in basis form, returns its multiplication table on
 the integral basis; if \var{x} is given in algebraic form, returns its
 multiplication table on the basis corresponding to the algebraic form of
 elements of \var{al}. In every case, if \var{x} is a \typ{COL} of length $n$,
 then the output is a $n\times n$ \typ{MAT}.
 Also accepts a square matrix with coefficients in \var{al}.
 
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algleftmultable(A,[0,1,0,0]~)
 %2 =
 [0 -1  1  0]
 
 [1  0  1  1]
 
 [0  0  1  1]
 
 [0  0 -2 -1]
 @eprog

Function: algmul
Class: basic
Section: algebras
C-Name: algmul
Prototype: GGG
Help: algmul(al,x,y): element x*y in al.
Doc: Given two elements $x$ and $y$ in \var{al}, computes their product $x*y$
 in the algebra~\var{al}.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algmul(A,[1,1,0,0]~,[0,0,2,1]~)
 %2 = [2, 3, 5, -4]~
 @eprog
 
 Also accepts matrices with coefficients in \var{al}.

Function: algmultable
Class: basic
Section: algebras
C-Name: algmultable
Prototype: mG
Help: algmultable(al): multiplication table of al over its prime subfield.
Doc: 
 returns a multiplication table of \var{al} over its
 prime subfield ($\Q$ or $\F_p$), as a \typ{VEC} of \typ{MAT}: the left
 multiplication tables of basis elements. If \var{al} was output by
 \tet{algtableinit}, returns the multiplication table used to define \var{al}.
 If \var{al} was output by \tet{alginit}, returns the multiplication table of
 the order~${\cal O}_0$ stored in \var{al}.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? M = algmultable(A);
 ? #M
 %3 = 4
 ? M[1]  \\ multiplication by e_1 = 1
 %4 =
 [1 0 0 0]
 
 [0 1 0 0]
 
 [0 0 1 0]
 
 [0 0 0 1]
 
 ? M[2]
 %5 =
 [0 -1  1  0]
 
 [1  0  1  1]
 
 [0  0  1  1]
 
 [0  0 -2 -1]
 @eprog

Function: algneg
Class: basic
Section: algebras
C-Name: algneg
Prototype: GG
Help: algneg(al,x): element -x in al.
Doc: Given an element $x$ in \var{al}, computes its opposite $-x$ in the
 algebra \var{al}.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algneg(A,[1,1,0,0]~)
 %2 = [-1, -1, 0, 0]~
 @eprog
 
 Also accepts matrices with coefficients in \var{al}.

Function: algnorm
Class: basic
Section: algebras
C-Name: algnorm
Prototype: GG
Help: algnorm(al,x): (reduced) norm of x.
Doc: Given an element \var{x} in \var{al}, computes its norm. If \var{al} is
 a table algebra output by \tet{algtableinit}, returns the absolute norm of
 \var{x}, which is an element of $\F_p$ of~$\Q$; if \var{al} is a central
 simple algebra output by \tet{alginit}, returns the reduced norm of \var{x},
 which is an element of the center of \var{al}.
 \bprog
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,19);
 ? algnorm(A,[0,-2,3]~)
 %3 = 18
 @eprog
 
 Also accepts a square matrix with coefficients in \var{al}.

Function: algpoleval
Class: basic
Section: algebras
C-Name: algpoleval
Prototype: GGG
Help: algpoleval(al,T,b): T in K[X] evaluate T(b) in al.
Doc: Given an element $b$ in \var{al} and a polynomial $T$ in $K[X]$,
 computes $T(b)$ in \var{al}.

Function: algpow
Class: basic
Section: algebras
C-Name: algpow
Prototype: GGG
Help: algpow(al,x,n): element x^n in al.
Doc: Given an element $x$ in \var{al} and an integer $n$, computes the
 power $x^n$ in the algebra \var{al}.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algpow(A,[1,1,0,0]~,7)
 %2 = [8, -8, 0, 0]~
 @eprog
 
 Also accepts a square matrix with coefficients in \var{al}.

Function: algprimesubalg
Class: basic
Section: algebras
C-Name: algprimesubalg
Prototype: G
Help: algprimesubalg(al): prime subalgebra of the positive characteristic,
 semisimple algebra al.
Doc: \var{al} being the output of \tet{algtableinit} representing a semisimple
 algebra of positive characteristic, returns a basis of the prime subalgebra
 of~\var{al}. The prime subalgebra of~\var{al} is the subalgebra fixed by the
 Frobenius automorphism of the center of \var{al}. It is abstractly isomorphic
 to a product of copies of $\F_p$.
 \bprog
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,2);
 ? algprimesubalg(A)
 %3 =
 [1 0]
 
 [0 1]
 
 [0 0]
 @eprog

Function: algquotient
Class: basic
Section: algebras
C-Name: alg_quotient
Prototype: GGD0,L,
Help: algquotient(al,I,{flag=0}): quotient of the algebra al by the two-sided
 ideal I.
Doc: \var{al} being a table algebra output by \tet{algtableinit} and \var{I}
 being a basis of a two-sided ideal of \var{al} represented by a matrix,
 returns the quotient $\var{al}/\var{I}$. When $\var{flag}=1$, returns a
 \typ{VEC} $[\var{al}/\var{I},\var{proj},\var{lift}]$ where \var{proj} and
 \var{lift} are matrices respectively representing the projection map and a
 section of it.
 \bprog
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,2);
 ? AQ = algquotient(A,[0;1;0]);
 ? algdim(AQ)
 %4 = 2
 @eprog

Function: algradical
Class: basic
Section: algebras
C-Name: algradical
Prototype: G
Help: algradical(al): Jacobson radical of the algebra al.
Doc: \var{al} being a table algebra output by \tet{algtableinit}, returns a
 basis of the Jacobson radical of the algebra \var{al} over its prime field
 ($\Q$ or $\F_p$).
 
 Here is an example with $A = \Q[x]/(x^2)$, generated by $(1,x)$:
 \bprog
 ? mt = [matid(2),[0,0;1,0]];
 ? A = algtableinit(mt);
 ? algradical(A) \\ = (x)
 %3 =
 [0]
 
 [1]
 @eprog
 
 Another one with $2\times 2$ upper triangular matrices over $\Q$, generated
 by $I_2$, $a = \kbd{[0,1;0,0]}$ and $b = \kbd{[0,0;0,1]}$, such that $a^2 =
 0$, $ab = a$, $ba = 0$, $b^2 = b$:
 \bprog
 ? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
 ? A = algtableinit(mt);
 ? algradical(A) \\ = (a)
 %6 =
 [0]
 
 [1]
 
 [0]
 @eprog

Function: algramifiedplaces
Class: basic
Section: algebras
C-Name: algramifiedplaces
Prototype: G
Help: algramifiedplaces(al): vector of the places of the center of al that
 ramify in al. Each place is described as an integer between 1 and r1 or as a
 prime ideal.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, return a
 \typ{VEC} containing the list of places of the center of \var{al} that are
 ramified in \var{al}. Each place is described as an integer between~$1$
 and~$r_1$ or as a prime ideal.
 
 \bprog
 ? nf = nfinit(y^2-5);
 ? A = alginit(nf, [-1,y]);
 ? algramifiedplaces(A)
 %3 = [1, [2, [2, 0]~, 1, 2, 1]]
 @eprog

Function: algrandom
Class: basic
Section: algebras
C-Name: algrandom
Prototype: GG
Help: algrandom(al,b): random element in al with coefficients in [-b,b].
Doc: Given an algebra \var{al} and an integer \var{b}, returns a random
 element in \var{al} with coefficients in~$[-b,b]$.

Function: algrelmultable
Class: basic
Section: algebras
C-Name: algrelmultable
Prototype: mG
Help: algrelmultable(al): multiplication table of the central simple
 algebra al over its center.
Doc: Given a central simple algebra \var{al} output by \tet{alginit} defined by a multiplication table over its center (a number field), returns this multiplication table.
 \bprog
 ? nf = nfinit(y^3-5); a = y; b = y^2;
 ? {m_i = [0,a,0,0;
           1,0,0,0;
           0,0,0,a;
           0,0,1,0];}
 ? {m_j = [0, 0,b, 0;
           0, 0,0,-b;
           1, 0,0, 0;
           0,-1,0, 0];}
 ? {m_k = [0, 0,0,-a*b;
           0, 0,b,   0;
           0,-a,0,   0;
           1, 0,0,   0];}
 ? mt = [matid(4), m_i, m_j, m_k];
 ? A = alginit(nf,mt,'x);
 ? M = algrelmultable(A);
 ? M[2] == m_i
 %8 = 1
 ? M[3] == m_j
 %9 = 1
 ? M[4] == m_k
 %10 = 1
 @eprog

Function: algsimpledec
Class: basic
Section: algebras
C-Name: algsimpledec
Prototype: GD0,L,
Help: algsimpledec(al,{flag=0}): decomposition into simple algebras of the
 semisimple algebra al.
Doc: \var{al} being the output of \tet{algtableinit} representing a semisimple
 algebra, returns a \typ{VEC} $[\var{al}_1,\var{al}_2,\dots,\var{al}_n]$ such
 that~\var{al} is isomorphic to the direct sum of the simple algebras
 $\var{al}_i$. When $\var{flag}=1$, each component is instead a \typ{VEC}
 $[\var{al}_i,\var{proj}_i,\var{lift}_i]$ where $\var{proj}_i$
 and~$\var{lift}_i$ are matrices respectively representing the projection map
 on the $i$-th factor and a section of it. The factors are sorted by
 increasing dimension, then increasing dimension of the center. This ensures
 that the ordering of the isomorphism classes of the factors is deterministic
 over finite fields, but not necessarily over~$\Q$.
 
 \misctitle{Warning} The images of the $\var{lift}_i$ are not guaranteed to form a direct sum.

Function: algsplittingdata
Class: basic
Section: algebras
C-Name: algsplittingdata
Prototype: mG
Help: algsplittingdata(al): data stored in the central simple algebra al to
 compute a splitting of al over an extension.
Doc: Given a central simple algebra \var{al} output by \tet{alginit} defined
 by a multiplication table over its center~$K$ (a number field), returns data
 stored to compute a splitting of \var{al} over an extension. This data is a
 \typ{VEC} \kbd{[t,Lbas,Lbasinv]} with $3$ components:
 
  \item an element $t$ of \var{al} such that $L=K(t)$ is a maximal subfield
 of \var{al};
 
  \item a matrix \kbd{Lbas} expressing a $L$-basis of \var{al} (given an
 $L$-vector space structure by multiplication on the right) on the integral
 basis of \var{al};
 
  \item a matrix \kbd{Lbasinv} expressing the integral basis of \var{al} on
 the previous $L$-basis.
 
 \bprog
 ? nf = nfinit(y^3-5); a = y; b = y^2;
 ? {m_i = [0,a,0,0;
           1,0,0,0;
           0,0,0,a;
           0,0,1,0];}
 ? {m_j = [0, 0,b, 0;
           0, 0,0,-b;
           1, 0,0, 0;
           0,-1,0, 0];}
 ? {m_k = [0, 0,0,-a*b;
           0, 0,b,   0;
           0,-a,0,   0;
           1, 0,0,   0];}
 ? mt = [matid(4), m_i, m_j, m_k];
 ? A = alginit(nf,mt,'x);
 ? [t,Lb,Lbi] = algsplittingdata(A);
 ? t
 %8 = [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]~;
 ? matsize(Lb)
 %9 = [12, 2]
 ? matsize(Lbi)
 %10 = [2, 12]
 @eprog

Function: algsplittingfield
Class: basic
Section: algebras
C-Name: algsplittingfield
Prototype: mG
Help: algsplittingfield(al): the stored splitting field of the central simple
 algebra al.
Doc: Given a central simple algebra \var{al} output by \tet{alginit}, returns
 an \kbd{rnf} structure: the splitting field of \var{al} that is stored in
 \var{al}, as a relative extension of the center.
 \bprog
 nf = nfinit(y^3-5);
 a = y; b = y^2;
 {m_i = [0,a,0,0;
        1,0,0,0;
        0,0,0,a;
        0,0,1,0];}
 {m_j = [0, 0,b, 0;
        0, 0,0,-b;
        1, 0,0, 0;
        0,-1,0, 0];}
 {m_k = [0, 0,0,-a*b;
        0, 0,b,   0;
        0,-a,0,   0;
        1, 0,0,   0];}
 mt = [matid(4), m_i, m_j, m_k];
 A = alginit(nf,mt,'x);
 algsplittingfield(A).pol
 %8 = x^2 - y
 @eprog

Function: algsplittingmatrix
Class: basic
Section: algebras
C-Name: algsplittingmatrix
Prototype: GG
Help: algsplittingmatrix(al,x): image of x under a splitting of al.
Doc: A central simple algebra \var{al} output by \tet{alginit} contains data
 describing an isomorphism~$\phi : A\otimes_K L \to M_d(L)$, where $d$ is the
 degree of the algebra and $L$ is an extension of $L$ with~$[L:K]=d$. Returns
 the matrix $\phi(x)$.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algsplittingmatrix(A,[0,0,0,2]~)
 %2 =
 [Mod(x + 1, x^2 + 1) Mod(Mod(1, y)*x + Mod(-1, y), x^2 + 1)]
 
 [Mod(x + 1, x^2 + 1)                   Mod(-x + 1, x^2 + 1)]
 @eprog
 
 Also accepts matrices with coefficients in \var{al}.

Function: algsqr
Class: basic
Section: algebras
C-Name: algsqr
Prototype: GG
Help: algsqr(al,x): element x^2 in al.
Doc: Given an element $x$ in \var{al}, computes its square $x^2$ in the
 algebra \var{al}.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algsqr(A,[1,0,2,0]~)
 %2 = [-3, 0, 4, 0]~
 @eprog
 
 Also accepts a square matrix with coefficients in \var{al}.

Function: algsub
Class: basic
Section: algebras
C-Name: algsub
Prototype: GGG
Help: algsub(al,x,y): element x-y in al.
Doc: Given two elements $x$ and $y$ in \var{al}, computes their difference
 $x-y$ in the algebra \var{al}.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algsub(A,[1,1,0,0]~,[1,0,1,0]~)
 %2 = [0, 1, -1, 0]~
 @eprog
 
 Also accepts matrices with coefficients in \var{al}.

Function: algsubalg
Class: basic
Section: algebras
C-Name: algsubalg
Prototype: GG
Help: algsubalg(al,B): subalgebra of al with basis B.
Doc: \var{al} being a table algebra output by \tet{algtableinit} and \var{B}
 being a basis of a subalgebra of \var{al} represented by a matrix, returns an
 algebra isomorphic to \var{B}.
 \bprog
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,2);
 ? B = algsubalg(A,[1,0; 0,0; 0,1]);
 ? algdim(A)
 %4 = 3
 ? algdim(B)
 %5 = 2
 @eprog

Function: algtableinit
Class: basic
Section: algebras
C-Name: algtableinit
Prototype: GDG
Help: algtableinit(mt, {p=0}): initialize the associative algebra
 over Q (resp. Fp) defined by the multiplication table mt.
Doc: initialize the associative algebra over $K = \Q$ (p omitted) or $\F_p$
 defined by the multiplication table \var{mt}.
 As a $K$-vector space, the algebra is generated by a basis
 $(e_1 = 1, e_2, \dots, e_n)$; the table is given as a \typ{VEC} of $n$ matrices in
 $M_n(K)$, giving the left multiplication by the basis elements $e_i$, in the
 given basis.
 Assumes that $e_1=1$, that $K e_1\oplus \dots\oplus K e_n]$ describes an
 associative algebra over $K$, and in the case $K=\Q$ that the multiplication
 table is integral. If the algebra is already known to be central
 and simple, then the case $K = \F_p$ is useless, and one should use
 \tet{alginit} directly.
 
 The point of this function is to input a finite dimensional $K$-algebra, so
 as to later compute its radical, then to split the quotient algebra as a
 product of simple algebras over $K$.
 
 The pari object representing such an algebra $A$ is a \typ{VEC} with the
 following data:
 
  \item The characteristic of $A$, accessed with \kbd{algchar}.
 
  \item The multiplication table of $A$, accessed with \kbd{algmultable}.
 
  \item The traces of the elements of the basis.
 
 A simple example: the $2\times 2$ upper triangular matrices over $\Q$,
 generated by $I_2$, $a = \kbd{[0,1;0,0]}$ and $b = \kbd{[0,0;0,1]}$,
 such that $a^2 = 0$, $ab = a$, $ba = 0$, $b^2 = b$:
 \bprog
 ? mt = [matid(3),[0,0,0;1,0,1;0,0,0],[0,0,0;0,0,0;1,0,1]];
 ? A = algtableinit(mt);
 ? algradical(A) \\ = (a)
 %6 =
 [0]
 
 [1]
 
 [0]
 ? algcenter(A) \\ = (I_2)
 %7 =
 [1]
 
 [0]
 
 [0]
 @eprog

Function: algtensor
Class: basic
Section: algebras
C-Name: algtensor
Prototype: GGD1,L,
Help: algtensor(al1,al2,{maxord=1}): tensor product of al1 and al2.
Doc: Given two algebras \var{al1} and \var{al2}, computes their tensor
 product. For table algebras output by \tet{algtableinit}, the flag
 \var{maxord} is ignored. For central simple algebras output by \tet{alginit},
 computes a maximal order by default. Prevent this computation by setting
 $\var{maxord}=0$.
 
 Currently only implemented for cyclic algebras of coprime degree over the same
 center~$K$, and the tensor product is over~$K$.

Function: algtrace
Class: basic
Section: algebras
C-Name: algtrace
Prototype: GG
Help: algtrace(al,x): (reduced) trace of x.
Doc: Given an element \var{x} in \var{al}, computes its trace. If \var{al} is
 a table algebra output by \tet{algtableinit}, returns the absolute trace of
 \var{x}, which is an element of $\F_p$ or~$\Q$; if \var{al} is the output of
 \tet{alginit}, returns the reduced trace of \var{x}, which is an element of
 the center of \var{al}.
 \bprog
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algtrace(A,[5,0,0,1]~)
 %2 = 11
 @eprog
 
 Also accepts a square matrix with coefficients in \var{al}.

Function: algtype
Class: basic
Section: algebras
C-Name: algtype
Prototype: lG
Help: algtype(al): type of the algebra al.
Doc: Given an algebra \var{al} output by \tet{alginit} or by \tet{algtableinit}, returns an integer indicating the type of algebra:
 
 \item $0$: not a valid algebra.
 
 \item $1$: table algebra output by \tet{algtableinit}.
 
 \item $2$: central simple algebra output by \tet{alginit} and represented by
 a multiplication table over its center.
 
 \item $3$: central simple algebra output by \tet{alginit} and represented by
 a cyclic algebra.
 \bprog
 ? algtype([])
 %1 = 0
 ? mt = [matid(3), [0,0,0; 1,1,0; 0,0,0], [0,0,1; 0,0,0; 1,0,1]];
 ? A = algtableinit(mt,2);
 ? algtype(A)
 %4 = 1
 ? nf = nfinit(y^3-5);
 ?  a = y; b = y^2;
 ?  {m_i = [0,a,0,0;
            1,0,0,0;
            0,0,0,a;
            0,0,1,0];}
 ?  {m_j = [0, 0,b, 0;
            0, 0,0,-b;
            1, 0,0, 0;
            0,-1,0, 0];}
 ?  {m_k = [0, 0,0,-a*b;
            0, 0,b,   0;
            0,-a,0,   0;
            1, 0,0,   0];}
 ?  mt = [matid(4), m_i, m_j, m_k];
 ?  A = alginit(nf,mt,'x);
 ? algtype(A)
 %12 = 2
 ? A = alginit(nfinit(y), [-1,-1]);
 ? algtype(A)
 %14 = 3
 @eprog

Function: alias
Class: basic
Section: programming/specific
C-Name: alias0
Prototype: vrr
Help: alias(newsym,sym): defines the symbol newsym as an alias for the symbol
 sym.
Doc: defines the symbol \var{newsym} as an alias for the symbol \var{sym}:
 \bprog
 ? alias("det", "matdet");
 ? det([1,2;3,4])
 %1 = -2
 @eprog\noindent
 You are not restricted to ordinary functions, as in the above example:
 to alias (from/to) member functions, prefix them with `\kbd{\_.}';
 to alias operators, use their internal name, obtained by writing
 \kbd{\_} in lieu of the operators argument: for instance, \kbd{\_!} and
 \kbd{!\_} are the internal names of the factorial and the
 logical negation, respectively.
 \bprog
 ? alias("mod", "_.mod");
 ? alias("add", "_+_");
 ? alias("_.sin", "sin");
 ? mod(Mod(x,x^4+1))
 %2 = x^4 + 1
 ? add(4,6)
 %3 = 10
 ? Pi.sin
 %4 = 0.E-37
 @eprog
 Alias expansion is performed directly by the internal GP compiler.
 Note that since alias is performed at compilation-time, it does not
 require any run-time processing, however it only affects GP code
 compiled \emph{after} the alias command is evaluated. A slower but more
 flexible alternative is to use variables. Compare
 \bprog
 ? fun = sin;
 ? g(a,b) = intnum(t=a,b,fun(t));
 ? g(0, Pi)
 %3 = 2.0000000000000000000000000000000000000
 ? fun = cos;
 ? g(0, Pi)
 %5 = 1.8830410776607851098 E-39
 @eprog\noindent
 with
 \bprog
 ? alias(fun, sin);
 ? g(a,b) = intnum(t=a,b,fun(t));
 ? g(0,Pi)
 %2 = 2.0000000000000000000000000000000000000
 ? alias(fun, cos);  \\ Oops. Does not affect *previous* definition!
 ? g(0,Pi)
 %3 = 2.0000000000000000000000000000000000000
 ? g(a,b) = intnum(t=a,b,fun(t)); \\ Redefine, taking new alias into account
 ? g(0,Pi)
 %5 = 1.8830410776607851098 E-39
 @eprog
 
 A sample alias file \kbd{misc/gpalias} is provided with
 the standard distribution.

Function: allocatemem
Class: basic
Section: programming/specific
C-Name: gp_allocatemem
Prototype: vDG
Help: allocatemem({s=0}): allocates a new stack of s bytes. doubles the
 stack if s is omitted.
Doc: this special operation changes the stack size \emph{after}
 initialization. $x$ must be a non-negative integer. If $x > 0$, a new stack
 of at least $x$ bytes is allocated. We may allocate more than $x$ bytes if
 $x$ is way too small, or for alignment reasons: the current formula is
 $\max(16*\ceil{x/16}, 500032)$ bytes.
 
 If $x=0$, the size of the new stack is twice the size of the old one.
 
 This command is much more useful if \tet{parisizemax} is non-zero, and we
 describe this case first. With \kbd{parisizemax} enabled, there are three
 sizes of interest:
 
 \item a virtual stack size, \tet{parisizemax}, which is an absolute upper
 limit for the stack size; this is set by \kbd{default(parisizemax, ...)}.
 
 \item the desired typical stack size, \tet{parisize}, that will grow as
 needed, up to \tet{parisizemax}; this is set by \kbd{default(parisize, ...)}.
 
 \item the current stack size, which is less that \kbd{parisizemax},
 typically equal to \kbd{parisize} but possibly larger and increasing
 dynamically as needed; \kbd{allocatemem} allows to change that one
 explicitly.
 
 The \kbd{allocatemem} command forces stack
 usage to increase temporarily (up to \kbd{parisizemax} of course); for
 instance if you notice using \kbd{\bs gm2} that we seem to collect garbage a
 lot, e.g.
 \bprog
 ? \gm2
   debugmem = 2
 ? default(parisize,"32M")
  ***   Warning: new stack size = 32000000 (30.518 Mbytes).
 ? bnfinit('x^2+10^30-1)
  *** bnfinit: collecting garbage in hnffinal, i = 1.
  *** bnfinit: collecting garbage in hnffinal, i = 2.
  *** bnfinit: collecting garbage in hnffinal, i = 3.
 @eprog\noindent and so on for hundred of lines. Then, provided the
 \tet{breakloop} default is set, you can interrupt the computation, type
 \kbd{allocatemem(100*10\pow6)} at the break loop prompt, then let the
 computation go on by typing \kbd{<Enter>}. Back at the \kbd{gp} prompt,
 the desired stack size of \kbd{parisize} is restored. Note that changing either
 \kbd{parisize} or \kbd{parisizemax} at the break loop prompt would interrupt
 the computation, contrary to the above.
 
 In most cases, \kbd{parisize} will increase automatically (up to
 \kbd{parisizemax}) and there is no need to perform the above maneuvers.
 But that the garbage collector is sufficiently efficient that
 a given computation can still run without increasing the stack size,
 albeit very slowly due to the frequent garbage collections.
 
 \misctitle{Deprecated: when \kbd{parisizemax} is unset}
 This is currently still the default behavior in order not to break backward
 compatibility. The rest of this section documents the
 behavior of \kbd{allocatemem} in that (deprecated) situation: it becomes a
 synonym for \kbd{default(parisize,...)}. In that case, there is no
 notion of a virtual stack, and the stack size is always equal to
 \kbd{parisize}. If more memory is needed, the PARI stack overflows, aborting
 the computation.
 
 Thus, increasing \kbd{parisize} via \kbd{allocatemem} or
 \kbd{default(parisize,...)} before a big computation is important.
 Unfortunately, either must be typed at the \kbd{gp} prompt in
 interactive usage, or left by itself at the start of batch files.
 They cannot be used meaningfully in loop-like constructs, or as part of a
 larger expression sequence, e.g
 \bprog
    allocatemem(); x = 1;   \\@com This will not set \kbd{x}!
 @eprog\noindent
 In fact, all loops are immediately exited, user functions terminated, and
 the rest of the sequence following \kbd{allocatemem()} is silently
 discarded, as well as all pending sequences of instructions. We just go on
 reading the next instruction sequence from the file we are in (or from the
 user). In particular, we have the following possibly unexpected behavior: in
 \bprog
    read("file.gp"); x = 1
 @eprog\noindent were \kbd{file.gp} contains an \kbd{allocatemem} statement,
 the \kbd{x = 1} is never executed, since all pending instructions in the
 current sequence are discarded.
 
 The reason for these unfortunate side-effects is that, with
 \kbd{parisizemax} disabled, increasing the stack size physically
 moves the stack, so temporary objects created during the current expression
 evaluation are not correct anymore. (In particular byte-compiled expressions,
 which are allocated on the stack.) To avoid accessing obsolete pointers to
 the old stack, this routine ends by a \kbd{longjmp}.

Function: apply
Class: basic
Section: programming/specific
C-Name: apply0
Prototype: GG
Help: apply(f, A): apply function f to each entry in A.
Wrapper: (G)
Description: 
  (closure,gen):gen    genapply(${1 cookie}, ${1 wrapper}, $2)
Doc: Apply the \typ{CLOSURE} \kbd{f} to the entries of \kbd{A}. If \kbd{A}
 is a scalar, return \kbd{f(A)}. If \kbd{A} is a polynomial or power series,
 apply \kbd{f} on all coefficients. If \kbd{A} is a vector or list, return
 the elements $f(x)$ where $x$ runs through \kbd{A}. If \kbd{A} is a matrix,
 return the matrix whose entries are the $f(\kbd{A[i,j]})$.
 \bprog
 ? apply(x->x^2, [1,2,3,4])
 %1 = [1, 4, 9, 16]
 ? apply(x->x^2, [1,2;3,4])
 %2 =
 [1 4]
 
 [9 16]
 ? apply(x->x^2, 4*x^2 + 3*x+ 2)
 %3 = 16*x^2 + 9*x + 4
 @eprog\noindent Note that many functions already act componentwise on
 vectors or matrices, but they almost never act on lists; in this
 case, \kbd{apply} is a good solution:
 \bprog
 ? L = List([Mod(1,3), Mod(2,4)]);
 ? lift(L)
   ***   at top-level: lift(L)
   ***                 ^-------
   *** lift: incorrect type in lift.
 ? apply(lift, L);
 %2 = List([1, 2])
 @eprog
 \misctitle{Remark} For $v$ a \typ{VEC}, \typ{COL}, \typ{LIST} or \typ{MAT},
 the alternative set-notations
 \bprog
 [g(x) | x <- v, f(x)]
 [x | x <- v, f(x)]
 [g(x) | x <- v]
 @eprog\noindent
 are available as shortcuts for
 \bprog
 apply(g, select(f, Vec(v)))
 select(f, Vec(v))
 apply(g, Vec(v))
 @eprog\noindent respectively:
 \bprog
 ? L = List([Mod(1,3), Mod(2,4)]);
 ? [ lift(x) | x<-L ]
 %2 = [1, 2]
 @eprog
 
 \synt{genapply}{void *E, GEN (*fun)(void*,GEN), GEN a}.

Function: arg
Class: basic
Section: transcendental
C-Name: garg
Prototype: Gp
Help: arg(x): argument of x, such that -pi<arg(x)<=pi.
Doc: argument of the complex number $x$, such that $-\pi < \arg(x) \le \pi$.

Function: asin
Class: basic
Section: transcendental
C-Name: gasin
Prototype: Gp
Help: asin(x): arc sine of x.
Doc: principal branch of $\sin^{-1}(x) = -i \log(ix + \sqrt{1 - x^2})$.
 In particular, $\Re(\text{asin}(x))\in [-\pi/2,\pi/2]$ and if $x\in \R$ and
 $|x|>1$ then $\text{asin}(x)$ is complex. The branch cut is in two pieces:
 $]-\infty,-1]$, continuous with quadrant II, and $[1,+\infty[$ continuous
 with quadrant IV. The function satisfies $i \text{asin}(x) =
 \text{asinh}(ix)$.

Function: asinh
Class: basic
Section: transcendental
C-Name: gasinh
Prototype: Gp
Help: asinh(x): inverse hyperbolic sine of x.
Doc: principal branch of $\sinh^{-1}(x) = \log(x + \sqrt{1+x^2})$. In
 particular $\Im(\text{asinh}(x))\in [-\pi/2,\pi/2]$.
 The branch cut is in two pieces: $]-i \infty ,-i]$, continuous with quadrant
 III and $[+i,+i \infty[$, continuous with quadrant I.

Function: asympnum
Class: basic
Section: sums
C-Name: asympnum0
Prototype: GD0,L,DGp
Help: asympnum(expr,{k=20},{alpha = 1}): asymptotic expansion of expr assuming
 it has rational coefficients with reasonable height; k and alpha are as
 in limitnum.
Doc: Asymptotic expansion of \var{expr}, corresponding to a sequence $u(n)$,
 assuming it has the shape
 $$u(n) \approx \sum_{i \geq 0} a_i n^{-i\alpha}$$
 with rational coefficients $a_i$ with reasonable height; the algorithm
 is heuristic and performs repeated calls to limitnum, with
 \kbd{k} and \kbd{alpha} are as in \kbd{limitnum}
 \bprog
 ? f(n) = n! / (n^n*exp(-n)*sqrt(n));
 ? asympnum(f)
 %2 = []   \\ failure !
 ? l = limitnum(f)
 %3 = 2.5066282746310005024157652848110452530
 ? asympnum(n->f(n)/l) \\ normalize
 %4 = [1, 1/12, 1/288, -139/51840]
 @eprog\noindent and we indeed get a few terms of Stirling's expansion. Note
 that it helps to normalize with a limit computed to higher accuracy:
 \bprog
 ? \p100
 ? L = limitnum(f)
 ? \p38
 ? asympnum(n->f(n)/L) \\ we get more terms!
 %6 = [1, 1/12, 1/288, -139/51840, -571/2488320, 163879/209018880,\
       5246819/75246796800, -534703531/902961561600]
 @eprog\noindent If \kbd{alpha} is not an integer, loss of accuracy is
 expected, so it should be precomputed to double accuracy, say:
 \bprog
 ? \p38
 ? asympnum(n->-log(1-1/n^Pi),,Pi)
 %1 = [0, 1, 1/2, 1/3]
 ? asympnum(n->-log(1-1/sqrt(n)),,1/2)
 %2 = [0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12, \
   1/13, 1/14, 1/15, 1/16, 1/17, 1/18, 1/19, 1/20, 1/21, 1/22]
 
 ? localprec(100); a = Pi;
 ? asympnum(n->-log(1-1/n^a),,a) \\ better !
 %4 = [0, 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/12]
 @eprog
 
 \synt{asympnum}{void *E, GEN (*u)(void *,GEN,long), long muli, GEN alpha, long prec}, where \kbd{u(E, n, prec)} must return $u(n)$ in precision \kbd{prec}.
 Also available is
 \fun{GEN}{asympnum0}{GEN u, long muli, GEN alpha, long prec}, where $u$
 must be a vector of sufficient length as above.

Function: atan
Class: basic
Section: transcendental
C-Name: gatan
Prototype: Gp
Help: atan(x): arc tangent of x.
Doc: principal branch of $\text{tan}^{-1}(x) = \log ((1+ix)/(1-ix)) /
 2i$. In particular the real part of $\text{atan}(x)$ belongs to
 $]-\pi/2,\pi/2[$.
 The branch cut is in two pieces:
 $]-i\infty,-i[$, continuous with quadrant IV, and $]i,+i \infty[$ continuous
 with quadrant II. The function satisfies $\text{atan}(x) =
 -i\text{atanh}(ix)$ for all $x\neq \pm i$.

Function: atanh
Class: basic
Section: transcendental
C-Name: gatanh
Prototype: Gp
Help: atanh(x): inverse hyperbolic tangent of x.
Doc: principal branch of $\text{tanh}^{-1}(x) = \log ((1+x)/(1-x)) / 2$. In
 particular the imaginary part of $\text{atanh}(x)$ belongs to
 $[-\pi/2,\pi/2]$; if $x\in \R$ and $|x|>1$ then $\text{atanh}(x)$ is complex.

Function: bernfrac
Class: basic
Section: transcendental
C-Name: bernfrac
Prototype: L
Help: bernfrac(x): Bernoulli number B_x, as a rational number.
Doc: Bernoulli number\sidx{Bernoulli numbers} $B_x$,
 where $B_0=1$, $B_1=-1/2$, $B_2=1/6$,\dots, expressed as a rational number.
 The argument $x$ should be of type integer.

Function: bernpol
Class: basic
Section: transcendental
C-Name: bernpol
Prototype: LDn
Help: bernpol(n, {v = 'x}): Bernoulli polynomial B_n, in variable v.
Doc: \idx{Bernoulli polynomial} $B_n$ in variable $v$.
 \bprog
 ? bernpol(1)
 %1 = x - 1/2
 ? bernpol(3)
 %2 = x^3 - 3/2*x^2 + 1/2*x
 @eprog

Function: bernreal
Class: basic
Section: transcendental
C-Name: bernreal
Prototype: Lp
Help: bernreal(x): Bernoulli number B_x, as a real number with the current
 precision.
Doc: Bernoulli number\sidx{Bernoulli numbers}
 $B_x$, as \kbd{bernfrac}, but $B_x$ is returned as a real number
 (with the current precision).

Function: bernvec
Class: basic
Section: transcendental
C-Name: bernvec
Prototype: L
Help: bernvec(x): this routine is obsolete, use bernfrac repeatedly.
Doc: This routine is obsolete, kept for backward compatibility only.
Obsolete: 2007-03-30

Function: besselh1
Class: basic
Section: transcendental
C-Name: hbessel1
Prototype: GGp
Help: besselh1(nu,x): H^1-bessel function of index nu and argument x.
Doc: $H^1$-Bessel function of index \var{nu} and argument $x$.

Function: besselh2
Class: basic
Section: transcendental
C-Name: hbessel2
Prototype: GGp
Help: besselh2(nu,x): H^2-bessel function of index nu and argument x.
Doc: $H^2$-Bessel function of index \var{nu} and argument $x$.

Function: besseli
Class: basic
Section: transcendental
C-Name: ibessel
Prototype: GGp
Help: besseli(nu,x): I-bessel function of index nu and argument x.
Doc: $I$-Bessel function of index \var{nu} and
 argument $x$. If $x$ converts to a power series, the initial factor
 $(x/2)^\nu/\Gamma(\nu+1)$ is omitted (since it cannot be represented in PARI
 when $\nu$ is not integral).

Function: besselj
Class: basic
Section: transcendental
C-Name: jbessel
Prototype: GGp
Help: besselj(nu,x): J-bessel function of index nu and argument x.
Doc: $J$-Bessel function of index \var{nu} and
 argument $x$. If $x$ converts to a power series, the initial factor
 $(x/2)^\nu/\Gamma(\nu+1)$ is omitted (since it cannot be represented in PARI
 when $\nu$ is not integral).

Function: besseljh
Class: basic
Section: transcendental
C-Name: jbesselh
Prototype: GGp
Help: besseljh(n,x): J-bessel function of index n+1/2 and argument x, where
 n is a non-negative integer.
Doc: $J$-Bessel function of half integral index.
 More precisely, $\kbd{besseljh}(n,x)$ computes $J_{n+1/2}(x)$ where $n$
 must be of type integer, and $x$ is any element of $\C$. In the
 present version \vers, this function is not very accurate when $x$ is small.

Function: besselk
Class: basic
Section: transcendental
C-Name: kbessel
Prototype: GGp
Help: besselk(nu,x): K-bessel function of index nu and argument x.
Doc: $K$-Bessel function of index \var{nu} and argument $x$.

Function: besseln
Class: basic
Section: transcendental
C-Name: nbessel
Prototype: GGp
Help: besseln(nu,x): N-bessel function of index nu and argument x.
Doc: $N$-Bessel function of index \var{nu} and argument $x$.

Function: bestappr
Class: basic
Section: number_theoretical
C-Name: bestappr
Prototype: GDG
Help: bestappr(x, {B}): returns a rational approximation to x, whose
 denominator is limited by B, if present. This function applies to reals,
 intmods, p-adics, and rationals of course. Otherwise it applies recursively
 to all components.
Doc: using variants of the extended Euclidean algorithm, returns a rational
 approximation $a/b$ to $x$, whose denominator is limited
 by $B$, if present. If $B$ is omitted, return the best approximation
 affordable given the input accuracy; if you are looking for true rational
 numbers, presumably approximated to sufficient accuracy, you should first
 try that option. Otherwise, $B$ must be a positive real scalar (impose
 $0 < b \leq B$).
 
 \item If $x$ is a \typ{REAL} or a \typ{FRAC}, this function uses continued
 fractions.
 \bprog
 ? bestappr(Pi, 100)
 %1 = 22/7
 ? bestappr(0.1428571428571428571428571429)
 %2 = 1/7
 ? bestappr([Pi, sqrt(2) + 'x], 10^3)
 %3 = [355/113, x + 1393/985]
 @eprog
 By definition, $a/b$ is the best rational approximation to $x$ if
 $|b x - a| < |v x - u|$ for all integers $(u,v)$ with $0 < v \leq B$.
 (Which implies that $n/d$ is a convergent of the continued fraction of $x$.)
 
 \item If $x$ is a \typ{INTMOD} modulo $N$ or a \typ{PADIC} of precision $N =
 p^k$, this function performs rational modular reconstruction modulo $N$. The
 routine then returns the unique rational number $a/b$ in coprime integers
 $|a| < N/2B$ and $b\leq B$ which is congruent to $x$ modulo $N$. Omitting
 $B$ amounts to choosing it of the order of $\sqrt{N/2}$. If rational
 reconstruction is not possible (no suitable $a/b$ exists), returns $[]$.
 \bprog
 ? bestappr(Mod(18526731858, 11^10))
 %1 = 1/7
 ? bestappr(Mod(18526731858, 11^20))
 %2 = []
 ? bestappr(3 + 5 + 3*5^2 + 5^3 + 3*5^4 + 5^5 + 3*5^6 + O(5^7))
 %2 = -1/3
 @eprog\noindent In most concrete uses, $B$ is a prime power and we performed
 Hensel lifting to obtain $x$.
 
 The function applies recursively to components of complex objects
 (polynomials, vectors, \dots). If rational reconstruction fails for even a
 single entry, return $[]$.

Function: bestapprPade
Class: basic
Section: number_theoretical
C-Name: bestapprPade
Prototype: GD-1,L,
Help: bestapprPade(x, {B}): returns a rational function approximation to x.
 This function applies to series, polmods, and rational functions of course.
 Otherwise it applies recursively to all components.
Doc: using variants of the extended Euclidean algorithm, returns a rational
 function approximation $a/b$ to $x$, whose denominator is limited
 by $B$, if present. If $B$ is omitted, return the best approximation
 affordable given the input accuracy; if you are looking for true rational
 functions, presumably approximated to sufficient accuracy, you should first
 try that option. Otherwise, $B$ must be a non-negative real
 (impose $0 \leq \text{degree}(b) \leq B$).
 
 \item If $x$ is a \typ{POLMOD} modulo $N$ this function performs rational
 modular reconstruction modulo $N$. The routine then returns the unique
 rational function $a/b$ in coprime polynomials, with $\text{degree}(b)\leq B$
 and $\text{degree}(a)$ minimal, which is congruent to $x$ modulo $N$.
 Omitting $B$ amounts to choosing it equal to the floor of
 $\text{degree}(N) / 2$. If rational reconstruction is not possible (no
 suitable $a/b$ exists), returns $[]$.
 \bprog
 ? T = Mod(x^3 + x^2 + x + 3, x^4 - 2);
 ? bestapprPade(T)
 %2 = (2*x - 1)/(x - 1)
 ? U = Mod(1 + x + x^2 + x^3 + x^5, x^9);
 ? bestapprPade(U)  \\ internally chooses B = 4
 %3 = []
 ? bestapprPade(U, 5) \\ with B = 5, a solution exists
 %4 = (2*x^4 + x^3 - x - 1)/(-x^5 + x^3 + x^2 - 1)
 @eprog
 
 \item If $x$ is a \typ{RFRAC} or \typ{SER}, this function implicitly
 converts the input to \typ{POLMOD} modulo $N = t^k$
 fractions.
 \bprog
 ? T = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + O(x^7);
 ? bestapprPade(T)
 %1 = 1/(-x + 1)
 @eprog\noindent
 The function applies recursively to components of complex objects
 (polynomials, vectors, \dots). If rational reconstruction fails for even a
 single entry, return $[]$.

Function: bezout
Class: basic
Section: number_theoretical
C-Name: gcdext0
Prototype: GG
Help: bezout(x,y): deprecated alias for gcdext.
Doc: deprecated alias for \kbd{gcdext}
Obsolete: 2013-04-03

Function: bezoutres
Class: basic
Section: polynomials
C-Name: polresultantext0
Prototype: GGDn
Help: bezoutres(A,B,{v}): deprecated alias for polresultantext.
Doc: deprecated alias for \kbd{polresultantext}
Obsolete: 2015-01-13

Function: bigomega
Class: basic
Section: number_theoretical
C-Name: bigomega
Prototype: lG
Help: bigomega(x): number of prime divisors of x, counted with multiplicity.
Doc: number of prime divisors of the integer $|x|$ counted with
 multiplicity:
 \bprog
 ? factor(392)
 %1 =
 [2 3]
 
 [7 2]
 
 ? bigomega(392)
 %2 = 5;  \\ = 3+2
 ? omega(392)
 %3 = 2;  \\ without multiplicity
 @eprog

Function: binary
Class: basic
Section: conversions
C-Name: binaire
Prototype: G
Help: binary(x): gives the vector formed by the binary digits of x (x
 integer).
Doc: 
 outputs the vector of the binary digits of $|x|$. Here $x$ can be an
 integer, a real number (in which case the result has two components, one for
 the integer part, one for the fractional part) or a vector/matrix.
 \bprog
 ? binary(10)
 %1 = [1, 0, 1, 0]
 
 ? binary(3.14)
 %2 = [[1, 1], [0, 0, 1, 0, 0, 0, [...]]
 
 ? binary([1,2])
 %3 = [[1], [1, 0]]
 @eprog\noindent By convention, $0$ has no digits:
 \bprog
 ? binary(0)
 %4 = []
 @eprog

Function: binomial
Class: basic
Section: number_theoretical
C-Name: binomial
Prototype: GL
Help: binomial(x,y): binomial coefficient x*(x-1)...*(x-y+1)/y! defined for
 y in Z and any x.
Doc: \idx{binomial coefficient} $\binom{x}{y}$.
 Here $y$ must be an integer, but $x$ can be any PARI object.
Variant: The function
 \fun{GEN}{binomialuu}{ulong n, ulong k} is also available, and so is
 \fun{GEN}{vecbinome}{long n}, which returns a vector $v$
 with $n+1$ components such that $v[k+1] = \kbd{binomial}(n,k)$ for $k$ from
 $0$ up to $n$.

Function: bitand
Class: basic
Section: conversions
C-Name: gbitand
Prototype: GG
Help: bitand(x,y): bitwise "and" of two integers x and y. Negative numbers
 behave as if modulo big power of 2.
Description: 
 (small, small):small:parens        $(1)&$(2)
 (gen, gen):int        gbitand($1, $2)
Doc: 
 bitwise \tet{and}
 \sidx{bitwise and}of two integers $x$ and $y$, that is the integer
 $$\sum_i (x_i~\kbd{and}~y_i) 2^i$$
 
 Negative numbers behave $2$-adically, i.e.~the result is the $2$-adic limit
 of \kbd{bitand}$(x_n,y_n)$, where $x_n$ and $y_n$ are non-negative integers
 tending to $x$ and $y$ respectively. (The result is an ordinary integer,
 possibly negative.)
 
 \bprog
 ? bitand(5, 3)
 %1 = 1
 ? bitand(-5, 3)
 %2 = 3
 ? bitand(-5, -3)
 %3 = -7
 @eprog
Variant: Also available is
 \fun{GEN}{ibitand}{GEN x, GEN y}, which returns the bitwise \emph{and}
 of $|x|$ and $|y|$, two integers.

Function: bitneg
Class: basic
Section: conversions
C-Name: gbitneg
Prototype: GD-1,L,
Help: bitneg(x,{n=-1}): bitwise negation of an integers x truncated to n
 bits. n=-1 means represent infinite sequences of bit 1 as negative numbers.
 Negative numbers behave as if modulo big power of 2.
Doc: 
 \idx{bitwise negation} of an integer $x$,
 truncated to $n$ bits, $n\geq 0$, that is the integer
 $$\sum_{i=0}^{n-1} \kbd{not}(x_i) 2^i.$$
 The special case $n=-1$ means no truncation: an infinite sequence of
 leading $1$ is then represented as a negative number.
 
 See \secref{se:bitand} for the behavior for negative arguments.

Function: bitnegimply
Class: basic
Section: conversions
C-Name: gbitnegimply
Prototype: GG
Help: bitnegimply(x,y): bitwise "negated imply" of two integers x and y,
 in other words, x BITAND BITNEG(y). Negative numbers behave as if modulo big
 power of 2.
Description: 
 (small, small):small:parens        $(1)&~$(2)
 (gen, gen):int        gbitnegimply($1, $2)
Doc: 
 bitwise negated imply of two integers $x$ and
 $y$ (or \kbd{not} $(x \Rightarrow y)$), that is the integer $$\sum
 (x_i~\kbd{and not}(y_i)) 2^i$$
 
 See \secref{se:bitand} for the behavior for negative arguments.
Variant: Also available is
 \fun{GEN}{ibitnegimply}{GEN x, GEN y}, which returns the bitwise negated
 imply of $|x|$ and $|y|$, two integers.

Function: bitor
Class: basic
Section: conversions
C-Name: gbitor
Prototype: GG
Help: bitor(x,y): bitwise "or" of two integers x and y. Negative numbers
 behave as if modulo big power of 2.
Description: 
 (small, small):small:parens        $(1)|$(2)
 (gen, gen):int        gbitor($1, $2)
Doc: 
 \sidx{bitwise inclusive or}bitwise (inclusive)
 \tet{or} of two integers $x$ and $y$, that is the integer $$\sum
 (x_i~\kbd{or}~y_i) 2^i$$
 
 See \secref{se:bitand} for the behavior for negative arguments.
Variant: Also available is
 \fun{GEN}{ibitor}{GEN x, GEN y}, which returns the bitwise \emph{ir}
 of $|x|$ and $|y|$, two integers.

Function: bitprecision
Class: basic
Section: conversions
C-Name: bitprecision0
Prototype: GD0,L,
Help: bitprecision(x,{n}): if n is present and positive, return x at precision
 n bits. If n is omitted, return real precision of object x in bits.
Doc: the function behaves differently according to whether $n$ is
 present and positive or not. If $n$ is missing, the function returns the
 (floating point) precision in bits of the PARI object $x$. If $x$ is an
 exact object, the function returns \kbd{+oo}.
 \bprog
 ? bitprecision(exp(1e-100))
 %1 = 512                 \\ 512 bits
 ? bitprecision( [ exp(1e-100), 0.5 ] )
 %2 = 128                 \\ minimal accuracy among components
 ? bitprecision(2 + x)
 %3 = +oo                  \\ exact object
 @eprog
 
 If $n$ is present and positive, the function creates a new object equal to $x$
 with the new bit-precision roughly $n$. In fact, the smallest multiple of 64
 (resp.~32 on a 32-bit machine) larger than or equal to $n$.
 
 For $x$ a vector or a matrix, the operation is
 done componentwise; for series and polynomials, the operation is done
 coefficientwise. For real $x$, $n$ is the number of desired significant
 \emph{bits}. If $n$ is smaller than the precision of $x$, $x$ is truncated,
 otherwise $x$ is extended with zeros. For exact or non-floating point types,
 no change.
 \bprog
 ? bitprecision(Pi, 10)    \\ actually 64 bits ~ 19 decimal digits
 %1 = 3.141592653589793239
 ? bitprecision(1, 10)
 %2 = 1
 ? bitprecision(1 + O(x), 10)
 %3 = 1 + O(x)
 ? bitprecision(2 + O(3^5), 10)
 %4 = 2 + O(3^5)
 @eprog\noindent

Function: bittest
Class: basic
Section: conversions
C-Name: gbittest
Prototype: GL
Help: bittest(x,n): gives bit number n (coefficient of 2^n) of the integer x.
 Negative numbers behave as if modulo big power of 2.
Description: 
 (small, small):bool:parens     ($(1)>>$(2))&1
 (int, small):bool              bittest($1, $2)
 (gen, small):gen               gbittest($1, $2)
Doc: 
 outputs the $n^{\text{th}}$ bit of $x$ starting
 from the right (i.e.~the coefficient of $2^n$ in the binary expansion of $x$).
 The result is 0 or 1.
 \bprog
 ? bittest(7, 0)
 %1 = 1 \\ the bit 0 is 1
 ? bittest(7, 2)
 %2 = 1 \\ the bit 2 is 1
 ? bittest(7, 3)
 %3 = 0 \\ the bit 3 is 0
 @eprog\noindent
 See \secref{se:bitand} for the behavior at negative arguments.
Variant: For a \typ{INT} $x$, the variant \fun{long}{bittest}{GEN x, long n} is
 generally easier to use, and if furthermore $n\ge 0$ the low-level function
 \fun{ulong}{int_bit}{GEN x, long n} returns \kbd{bittest(abs(x),n)}.

Function: bitxor
Class: basic
Section: conversions
C-Name: gbitxor
Prototype: GG
Help: bitxor(x,y): bitwise "exclusive or" of two integers x and y.
 Negative numbers behave as if modulo big power of 2.
Description: 
 (small, small):small:parens        $(1)^$(2)
 (gen, gen):int        gbitxor($1, $2)
Doc: 
 bitwise (exclusive) \tet{or}
 \sidx{bitwise exclusive or}of two integers $x$ and $y$, that is the integer
 $$\sum (x_i~\kbd{xor}~y_i) 2^i$$
 
 See \secref{se:bitand} for the behavior for negative arguments.
Variant: Also available is
 \fun{GEN}{ibitxor}{GEN x, GEN y}, which returns the bitwise \emph{xor}
 of $|x|$ and $|y|$, two integers.

Function: bnfcertify
Class: basic
Section: number_fields
C-Name: bnfcertify0
Prototype: lGD0,L,
Help: bnfcertify(bnf,{flag = 0}): certify the correctness (i.e. remove the GRH) of the bnf data output by bnfinit. If flag is present, only certify that the class group is a quotient of the one computed in bnf (much simpler in general).
Doc: $\var{bnf}$ being as output by
 \kbd{bnfinit}, checks whether the result is correct, i.e.~whether it is
 possible to remove the assumption of the Generalized Riemann
 Hypothesis\sidx{GRH}. It is correct if and only if the answer is 1. If it is
 incorrect, the program may output some error message, or loop indefinitely.
 You can check its progress by increasing the debug level. The \var{bnf}
 structure must contain the fundamental units:
 \bprog
 ? K = bnfinit(x^3+2^2^3+1); bnfcertify(K)
   ***   at top-level: K=bnfinit(x^3+2^2^3+1);bnfcertify(K)
   ***                                        ^-------------
   *** bnfcertify: missing units in bnf.
 ? K = bnfinit(x^3+2^2^3+1, 1); \\ include units
 ? bnfcertify(K)
 %3 = 1
 @eprog
 
 If flag is present, only certify that the class group is a quotient of the
 one computed in bnf (much simpler in general); likewise, the computed units
 may form a subgroup of the full unit group. In this variant, the units are
 no longer needed:
 \bprog
 ? K = bnfinit(x^3+2^2^3+1); bnfcertify(K, 1)
 %4 = 1
 @eprog
Variant: Also available is  \fun{GEN}{bnfcertify}{GEN bnf} ($\fl=0$).

Function: bnfcompress
Class: basic
Section: number_fields
C-Name: bnfcompress
Prototype: G
Help: bnfcompress(bnf): converts bnf to a much smaller sbnf, containing the
 same information. Use bnfinit(sbnf) to recover a true bnf.
Doc: computes a compressed version of \var{bnf} (from \tet{bnfinit}), a
 ``small Buchmann's number field'' (or \var{sbnf} for short) which contains
 enough information to recover a full $\var{bnf}$ vector very rapidly, but
 which is much smaller and hence easy to store and print. Calling
 \kbd{bnfinit} on the result recovers a true \kbd{bnf}, in general different
 from the original. Note that an \tev{snbf} is useless for almost all
 purposes besides storage, and must be converted back to \tev{bnf} form
 before use; for instance, no \kbd{nf*}, \kbd{bnf*} or member function
 accepts them.
 
 An \var{sbnf} is a 12 component vector $v$, as follows. Let \kbd{bnf} be
 the result of a full \kbd{bnfinit}, complete with units. Then $v[1]$ is
 \kbd{bnf.pol}, $v[2]$ is the number of real embeddings \kbd{bnf.sign[1]},
 $v[3]$ is \kbd{bnf.disc}, $v[4]$ is \kbd{bnf.zk}, $v[5]$ is the list of roots
 \kbd{bnf.roots}, $v[7]$ is the matrix $\kbd{W} = \kbd{bnf[1]}$,
 $v[8]$ is the matrix $\kbd{matalpha}=\kbd{bnf[2]}$,
 $v[9]$ is the prime ideal factor base \kbd{bnf[5]} coded in a compact way,
 and ordered according to the permutation \kbd{bnf[6]}, $v[10]$ is the
 2-component vector giving the number of roots of unity and a generator,
 expressed on the integral basis, $v[11]$ is the list of fundamental units,
 expressed on the integral basis, $v[12]$ is a vector containing the algebraic
 numbers alpha corresponding to the columns of the matrix \kbd{matalpha},
 expressed on the integral basis.
 
 All the components are exact (integral or rational), except for the roots in
 $v[5]$.

Function: bnfdecodemodule
Class: basic
Section: number_fields
C-Name: decodemodule
Prototype: GG
Help: bnfdecodemodule(nf,m): given a coded module m as in bnrdisclist,
 gives the true module.
Doc: if $m$ is a module as output in the
 first component of an extension given by \kbd{bnrdisclist}, outputs the
 true module.
 \bprog
 ? K = bnfinit(x^2+23); L = bnrdisclist(K, 10); s = L[1][2]
 %1 = [[Mat([8, 1]), [[0, 0, 0]]], [Mat([9, 1]), [[0, 0, 0]]]]
 ? bnfdecodemodule(K, s[1][1])
 %2 =
 [2 0]
 
 [0 1]
 @eprog

Function: bnfinit
Class: basic
Section: number_fields
C-Name: bnfinit0
Prototype: GD0,L,DGp
Help: bnfinit(P,{flag=0},{tech=[]}): compute the necessary data for future
 use in ideal and unit group computations, including fundamental units if
 they are not too large. flag and tech are both optional. flag can be any of
 0: default, 1: insist on having fundamental units.
 See manual for details about tech.
Description: 
 (gen):bnf:prec           Buchall($1, 0, $prec)
 (gen, 0):bnf:prec        Buchall($1, 0, $prec)
 (gen, 1):bnf:prec        Buchall($1, nf_FORCE, $prec)
 (gen, ?small, ?gen):bnf:prec        bnfinit0($1, $2, $3, $prec)
Doc: initializes a
 \kbd{bnf} structure. Used in programs such as \kbd{bnfisprincipal},
 \kbd{bnfisunit} or \kbd{bnfnarrow}. By default, the results are conditional
 on the GRH, see \ref{se:GRHbnf}. The result is a
 10-component vector \var{bnf}.
 
 This implements \idx{Buchmann}'s sub-exponential algorithm for computing the
 class group, the regulator and a system of \idx{fundamental units} of the
 general algebraic number field $K$ defined by the irreducible polynomial $P$
 with integer coefficients.
 
 If the precision becomes insufficient, \kbd{gp} does not strive to compute
 the units by default ($\fl=0$).
 
 When $\fl=1$, we insist on finding the fundamental units exactly. Be
 warned that this can take a very long time when the coefficients of the
 fundamental units on the integral basis are very large. If the fundamental
 units are simply too large to be represented in this form, an error message
 is issued. They could be obtained using the so-called compact representation
 of algebraic numbers as a formal product of algebraic integers. The latter is
 implemented internally but not publicly accessible yet.
 
 $\var{tech}$ is a technical vector (empty by default, see \ref{se:GRHbnf}).
 Careful use of this parameter may speed up your computations,
 but it is mostly obsolete and you should leave it alone.
 
 \smallskip
 
 The components of a \var{bnf} or \var{sbnf} are technical and never used by
 the casual user. In fact: \emph{never access a component directly, always use
 a proper member function.} However, for the sake of completeness and internal
 documentation, their description is as follows. We use the notations
 explained in the book by H. Cohen, \emph{A Course in Computational Algebraic
 Number Theory}, Graduate Texts in Maths \key{138}, Springer-Verlag, 1993,
 Section 6.5, and subsection 6.5.5 in particular.
 
 $\var{bnf}[1]$ contains the matrix $W$, i.e.~the matrix in Hermite normal
 form giving relations for the class group on prime ideal generators
 $(\goth{p}_i)_{1\le i\le r}$.
 
 $\var{bnf}[2]$ contains the matrix $B$, i.e.~the matrix containing the
 expressions of the prime ideal factorbase in terms of the $\goth{p}_i$.
 It is an $r\times c$ matrix.
 
 $\var{bnf}[3]$ contains the complex logarithmic embeddings of the system of
 fundamental units which has been found. It is an $(r_1+r_2)\times(r_1+r_2-1)$
 matrix.
 
 $\var{bnf}[4]$ contains the matrix $M''_C$ of Archimedean components of the
 relations of the matrix $(W|B)$.
 
 $\var{bnf}[5]$ contains the prime factor base, i.e.~the list of prime
 ideals used in finding the relations.
 
 $\var{bnf}[6]$ used to contain a permutation of the prime factor base, but
 has been obsoleted. It contains a dummy $0$.
 
 $\var{bnf}[7]$ or \kbd{\var{bnf}.nf} is equal to the number field data
 $\var{nf}$ as would be given by \kbd{nfinit}.
 
 $\var{bnf}[8]$ is a vector containing the classgroup \kbd{\var{bnf}.clgp}
 as a finite abelian group, the regulator \kbd{\var{bnf}.reg}, a $1$ (used to
 contain an obsolete ``check number''), the number of roots of unity and a
 generator \kbd{\var{bnf}.tu}, the fundamental units \kbd{\var{bnf}.fu}.
 
 $\var{bnf}[9]$ is a 3-element row vector used in \tet{bnfisprincipal} only
 and obtained as follows. Let $D = U W V$ obtained by applying the
 \idx{Smith normal form} algorithm to the matrix $W$ (= $\var{bnf}[1]$) and
 let $U_r$ be the reduction of $U$ modulo $D$. The first elements of the
 factorbase are given (in terms of \kbd{bnf.gen}) by the columns of $U_r$,
 with Archimedean component $g_a$; let also $GD_a$ be the Archimedean
 components of the generators of the (principal) ideals defined by the
 \kbd{bnf.gen[i]\pow bnf.cyc[i]}. Then $\var{bnf}[9]=[U_r, g_a, GD_a]$.
 
 $\var{bnf}[10]$ is by default unused and set equal to 0. This field is used
 to store further information about the field as it becomes available, which
 is rarely needed, hence would be too expensive to compute during the initial
 \kbd{bnfinit} call. For instance, the generators of the principal ideals
 \kbd{bnf.gen[i]\pow bnf.cyc[i]} (during a call to \tet{bnrisprincipal}), or
 those corresponding to the relations in $W$ and $B$ (when the \kbd{bnf}
 internal precision needs to be increased).
Variant: 
 Also available is \fun{GEN}{Buchall}{GEN P, long flag, long prec},
 corresponding to \kbd{tech = NULL}, where
 \kbd{flag} is either $0$ (default) or \tet{nf_FORCE} (insist on finding
 fundamental units). The function
 \fun{GEN}{Buchall_param}{GEN P, double c1, double c2, long nrpid, long flag, long prec} gives direct access to the technical parameters.

Function: bnfisintnorm
Class: basic
Section: number_fields
C-Name: bnfisintnorm
Prototype: GG
Help: bnfisintnorm(bnf,x): compute a complete system of solutions (modulo
 units of positive norm) of the absolute norm equation N(a)=x, where a
 belongs to the maximal order of big number field bnf (if bnf is not
 certified, this depends on GRH).
Doc: computes a complete system of
 solutions (modulo units of positive norm) of the absolute norm equation
 $\Norm(a)=x$,
 where $a$ is an integer in $\var{bnf}$. If $\var{bnf}$ has not been certified,
 the correctness of the result depends on the validity of \idx{GRH}.
 
 See also \tet{bnfisnorm}.
Variant: The function \fun{GEN}{bnfisintnormabs}{GEN bnf, GEN a}
 returns a complete system of solutions modulo units of the absolute norm
 equation $|\Norm(x)| = |a|$. As fast as \kbd{bnfisintnorm}, but solves
 the two equations $\Norm(x) = \pm a$ simultaneously.

Function: bnfisnorm
Class: basic
Section: number_fields
C-Name: bnfisnorm
Prototype: GGD1,L,
Help: bnfisnorm(bnf,x,{flag=1}): tries to tell whether x (in Q) is the norm
 of some fractional y (in bnf). Returns a vector [a,b] where x=Norm(a)*b.
 Looks for a solution which is a S-unit, with S a certain list of primes (in
 bnf) containing (among others) all primes dividing x. If bnf is known to be
 Galois, set flag=0 (in this case, x is a norm iff b=1). If flag is non zero
 the program adds to S all the primes: dividing flag if flag<0, or less than
 flag if flag>0. The answer is guaranteed (i.e x norm iff b=1) under GRH, if
 S contains all primes less than 12.log(disc(Bnf))^2, where Bnf is the Galois
 closure of bnf.
Doc: tries to tell whether the
 rational number $x$ is the norm of some element y in $\var{bnf}$. Returns a
 vector $[a,b]$ where $x=Norm(a)*b$. Looks for a solution which is an $S$-unit,
 with $S$ a certain set of prime ideals containing (among others) all primes
 dividing $x$. If $\var{bnf}$ is known to be \idx{Galois}, set $\fl=0$ (in
 this case, $x$ is a norm iff $b=1$). If $\fl$ is non zero the program adds to
 $S$ the following prime ideals, depending on the sign of $\fl$. If $\fl>0$,
 the ideals of norm less than $\fl$. And if $\fl<0$ the ideals dividing $\fl$.
 
 Assuming \idx{GRH}, the answer is guaranteed (i.e.~$x$ is a norm iff $b=1$),
 if $S$ contains all primes less than $12\log(\disc(\var{Bnf}))^2$, where
 $\var{Bnf}$ is the Galois closure of $\var{bnf}$.
 
 See also \tet{bnfisintnorm}.

Function: bnfisprincipal
Class: basic
Section: number_fields
C-Name: bnfisprincipal0
Prototype: GGD1,L,
Help: bnfisprincipal(bnf,x,{flag=1}): bnf being output by bnfinit (with
 flag<=2), gives [v,alpha], where v is the vector of exponents on
 the class group generators and alpha is the generator of the resulting
 principal ideal. In particular x is principal if and only if v is the zero
 vector. flag is optional, whose binary digits mean 1: output [v,alpha] (only v
 if unset); 2: increase precision until alpha can be computed (do not insist
 if unset).
Doc: $\var{bnf}$ being the \sidx{principal ideal}
 number field data output by \kbd{bnfinit}, and $x$ being an ideal, this
 function tests whether the ideal is principal or not. The result is more
 complete than a simple true/false answer and solves general discrete
 logarithm problem. Assume the class group is $\oplus (\Z/d_i\Z)g_i$
 (where the generators $g_i$ and their orders $d_i$ are respectively given by
 \kbd{bnf.gen} and \kbd{bnf.cyc}). The routine returns a row vector $[e,t]$,
 where $e$ is a vector of exponents $0 \leq e_i < d_i$, and $t$ is a number
 field element such that
 $$ x = (t) \prod_i  g_i^{e_i}.$$
 For \emph{given} $g_i$ (i.e. for a given \kbd{bnf}), the $e_i$ are unique,
 and $t$ is unique modulo units.
 
 In particular, $x$ is principal if and only if $e$ is the zero vector. Note
 that the empty vector, which is returned when the class number is $1$, is
 considered to be a zero vector (of dimension $0$).
 \bprog
 ? K = bnfinit(y^2+23);
 ? K.cyc
 %2 = [3]
 ? K.gen
 %3 = [[2, 0; 0, 1]]          \\ a prime ideal above 2
 ? P = idealprimedec(K,3)[1]; \\ a prime ideal above 3
 ? v = bnfisprincipal(K, P)
 %5 = [[2]~, [3/4, 1/4]~]
 ? idealmul(K, v[2], idealfactorback(K, K.gen, v[1]))
 %6 =
 [3 0]
 
 [0 1]
 ? % == idealhnf(K, P)
 %7 = 1
 @eprog
 
 \noindent The binary digits of \fl mean:
 
 \item $1$: If set, outputs $[e,t]$ as explained above, otherwise returns
 only $e$, which is much easier to compute. The following idiom only tests
 whether an ideal is principal:
 \bprog
   is_principal(bnf, x) = !bnfisprincipal(bnf,x,0);
 @eprog
 
 \item $2$: It may not be possible to recover $t$, given the initial accuracy
 to which the \kbd{bnf} structure was computed. In that case, a warning is
 printed and $t$ is set equal to the empty vector \kbd{[]\til}. If this bit is
 set, increase the precision and recompute needed quantities until $t$ can be
 computed. Warning: setting this may induce \emph{lengthy} computations.
Variant: Instead of the above hardcoded numerical flags, one should
 rather use an or-ed combination of the symbolic flags \tet{nf_GEN} (include
 generators, possibly a place holder if too difficult) and \tet{nf_FORCE}
 (insist on finding the generators).

Function: bnfissunit
Class: basic
Section: number_fields
C-Name: bnfissunit
Prototype: GGG
Help: bnfissunit(bnf,sfu,x): bnf being output by bnfinit (with flag<=2), sfu
 by bnfsunit, gives the column vector of exponents of x on the fundamental
 S-units and the roots of unity if x is a unit, the empty vector otherwise.
Doc: $\var{bnf}$ being output by
 \kbd{bnfinit}, \var{sfu} by \kbd{bnfsunit}, gives the column vector of
 exponents of $x$ on the fundamental $S$-units and the roots of unity.
 If $x$ is not a unit, outputs an empty vector.

Function: bnfisunit
Class: basic
Section: number_fields
C-Name: bnfisunit
Prototype: GG
Help: bnfisunit(bnf,x): bnf being output by bnfinit, gives
 the column vector of exponents of x on the fundamental units and the roots
 of unity if x is a unit, the empty vector otherwise.
Doc: \var{bnf} being the number field data
 output by \kbd{bnfinit} and $x$ being an algebraic number (type integer,
 rational or polmod), this outputs the decomposition of $x$ on the fundamental
 units and the roots of unity if $x$ is a unit, the empty vector otherwise.
 More precisely, if $u_1$,\dots,$u_r$ are the fundamental units, and $\zeta$
 is the generator of the group of roots of unity (\kbd{bnf.tu}), the output is
 a vector $[x_1,\dots,x_r,x_{r+1}]$ such that $x=u_1^{x_1}\cdots
 u_r^{x_r}\cdot\zeta^{x_{r+1}}$. The $x_i$ are integers for $i\le r$ and is an
 integer modulo the order of $\zeta$ for $i=r+1$.
 
 Note that \var{bnf} need not contain the fundamental unit explicitly:
 \bprog
 ? setrand(1); bnf = bnfinit(x^2-x-100000);
 ? bnf.fu
   ***   at top-level: bnf.fu
   ***                     ^--
   *** _.fu: missing units in .fu.
 ? u = [119836165644250789990462835950022871665178127611316131167, \
        379554884019013781006303254896369154068336082609238336]~;
 ? bnfisunit(bnf, u)
 %3 = [-1, Mod(0, 2)]~
 @eprog\noindent The given $u$ is the inverse of the fundamental unit
 implicitly stored in \var{bnf}. In this case, the fundamental unit was not
 computed and stored in algebraic form since the default accuracy was too
 low. (Re-run the command at \bs g1 or higher to see such diagnostics.)

Function: bnflog
Class: basic
Section: number_fields
C-Name: bnflog
Prototype: GG
Help: bnflog(bnf, l): let bnf be attached to a number field F and let l be
 a prime number. Return the logarithmic l-class group Cl~_F.
Doc: let \var{bnf} be attached to a number field $F$ and let $l$ be
 a prime number (hereafter denoted $\ell$ for typographical reasons). Return
 the logarithmic $\ell$-class group $\widetilde{Cl}_F$
 of $F$. This is an abelian group, conjecturally finite (known to be finite
 if $F/\Q$ is abelian). The function returns if and only if
 the group is indeed finite (otherwise it would run into an infinite loop).
 Let $S = \{ \goth{p}_1,\dots, \goth{p}_k\}$ be the set of $\ell$-adic places
 (maximal ideals containing $\ell$).
 The function returns $[D, G(\ell), G']$, where
 
 \item $D$ is the vector of elementary divisors for $\widetilde{Cl}_F$;
 
 \item $G(\ell)$ is the vector of elementary divisors for
 the (conjecturally finite) abelian group
 $$\widetilde{\Cl}(\ell) =
 \{ \goth{a} = \sum_{i \leq k} a_i \goth{p}_i :~\deg_F \goth{a} = 0\},$$
 where the $\goth{p}_i$ are the $\ell$-adic places of $F$; this is a
 subgroup of $\widetilde{\Cl}$.
 
 \item $G'$ is the vector of elementary divisors for the $\ell$-Sylow $Cl'$
 of the $S$-class group of $F$; the group $\widetilde{\Cl}$ maps to $Cl'$
 with a simple co-kernel.

Function: bnflogdegree
Class: basic
Section: number_fields
C-Name: bnflogdegree
Prototype: GGG
Help: bnflogdegree(nf, A, l): let A be an ideal, return exp(deg_F A)
 the exponential of the l-adic logarithmic degree.
Doc: Let \var{nf} be the number field data output by \kbd{nfinit},
 attached to the field $F$, and let $l$ be a prime number (hereafter
 denoted $\ell$). The
 $\ell$-adified group of id\`{e}les of $F$ quotiented by
 the group of logarithmic units is identified to the $\ell$-group
 of logarithmic divisors $\oplus \Z_\ell [\goth{p}]$, generated by the
 maximal ideals of $F$.
 
 The \emph{degree} map $\deg_F$ is additive with values in $\Z_\ell$,
 defined by $\deg_F \goth{p} = \tilde{f}_{\goth{p}} \deg_\ell p$,
 where the integer $\tilde{f}$ is as in \tet{bnflogef} and $\deg_\ell p$
 is $\log_\ell p$ for $p\neq \ell$, $\log_\ell (1 + \ell)$ for
 $p = \ell\neq 2$ and $\log_\ell (1 + 2^2)$ for $p = \ell = 2$.
 
 Let $A = \prod \goth{p}^{n_{\goth{p}}}$ be an ideal and let $\tilde{A} =
 \sum n_\goth{p} [\goth{p}]$ be the attached logarithmic divisor. Return the
 exponential of the $\ell$-adic logarithmic degree $\deg_F A$, which is a
 natural number.

Function: bnflogef
Class: basic
Section: number_fields
C-Name: bnflogef
Prototype: GG
Help: bnflogef(nf,pr): return [e~, f~] the logarithmic ramification and
 residue degrees for the maximal ideal pr.
Doc: let $F$ be a number field represented by the \var{nf} structure,
 and let \var{pr} be a \kbd{prid} structure attached to the
 maximal ideal $\goth{p} / p$. Return
 $[\tilde{e}(F_\goth{p} / \Q_p), \tilde{f}(F_\goth{p} / \Q_p)]$
 the logarithmic ramification and residue degrees. Let $\Q_p^c/\Q_p$ be the
 cyclotomic $\Z_p$-extension, then
 $\tilde{e} = [F_\goth{p} \colon F_\goth{p} \cap \Q_p^c]$
 $\tilde{f} = [F_\goth{p} \cap \Q_p^c \colon \Q_p]$. Note that
 $\tilde{e}\tilde{f} = e(\goth{p}/p) f(\goth{p}/p)$, where $e,f$ denote the
 usual ramification and residue degrees.
 \bprog
 ? F = nfinit(y^6 - 3*y^5 + 5*y^3 - 3*y + 1);
 ? bnflogef(F, idealprimedec(F,2)[1])
 %2 = [6, 1]
 ? bnflogef(F, idealprimedec(F,5)[1])
 %3 = [1, 2]
 @eprog

Function: bnfnarrow
Class: basic
Section: number_fields
C-Name: buchnarrow
Prototype: G
Help: bnfnarrow(bnf): given a big number field as output by bnfinit, gives
 as a 3-component vector the structure of the narrow class group.
Doc: \var{bnf} being as output by
 \kbd{bnfinit}, computes the narrow class group of \var{bnf}. The output is
 a 3-component row vector $v$ analogous to the corresponding class group
 component \kbd{\var{bnf}.clgp}: the first component
 is the narrow class number \kbd{$v$.no}, the second component is a vector
 containing the SNF\sidx{Smith normal form} cyclic components \kbd{$v$.cyc} of
 the narrow class group, and the third is a vector giving the generators of
 the corresponding \kbd{$v$.gen} cyclic groups. Note that this function is a
 special case of \kbd{bnrinit}; the \var{bnf} need not contain fundamental
 units.

Function: bnfsignunit
Class: basic
Section: number_fields
C-Name: signunits
Prototype: G
Help: bnfsignunit(bnf): matrix of signs of the real embeddings of the system
 of fundamental units found by bnfinit.
Doc: $\var{bnf}$ being as output by
 \kbd{bnfinit}, this computes an $r_1\times(r_1+r_2-1)$ matrix having $\pm1$
 components, giving the signs of the real embeddings of the fundamental units.
 The following functions compute generators for the totally positive units:
 
 \bprog
 /* exponents of totally positive units generators on bnf.tufu */
 tpuexpo(bnf)=
 { my(K, S = bnfsignunit(bnf), [m,n] = matsize(S));
   \\ m = bnf.r1, n = r1+r2-1
   S = matrix(m,n, i,j, if (S[i,j] < 0, 1,0));
   S = concat(vectorv(m,i,1), S);   \\ add sign(-1)
   K = matker(S * Mod(1,2));
   if (K, mathnfmodid(lift(K), 2), 2*matid(n+1))
 }
 
 /* totally positive fundamental units */
 tpu(bnf)=
 { my(ex = tpuexpo(bnf)[,2..-1]); \\ remove ex[,1], corresponds to 1 or -1
   vector(#ex, i, nffactorback(bnf, bnf.tufu, ex[,i]));
 }
 @eprog

Function: bnfsunit
Class: basic
Section: number_fields
C-Name: bnfsunit
Prototype: GGp
Help: bnfsunit(bnf,S): compute the fundamental S-units of the number field
 bnf output by bnfinit, S being a list of prime ideals. res[1] contains the
 S-units, res[5] the S-classgroup. See manual for details.
Doc: computes the fundamental $S$-units of the
 number field $\var{bnf}$ (output by \kbd{bnfinit}), where $S$ is a list of
 prime ideals (output by \kbd{idealprimedec}). The output is a vector $v$ with
 6 components.
 
 $v[1]$ gives a minimal system of (integral) generators of the $S$-unit group
 modulo the unit group.
 
 $v[2]$ contains technical data needed by \kbd{bnfissunit}.
 
 $v[3]$ is an empty vector (used to give the logarithmic embeddings of the
 generators in $v[1]$ in version 2.0.16).
 
 $v[4]$ is the $S$-regulator (this is the product of the regulator, the
 determinant of $v[2]$ and the natural logarithms of the norms of the ideals
 in $S$).
 
 $v[5]$ gives the $S$-class group structure, in the usual format
 (a row vector whose three components give in order the $S$-class number,
 the cyclic components and the generators).
 
 $v[6]$ is a copy of $S$.

Function: bnrL1
Class: basic
Section: number_fields
C-Name: bnrL1
Prototype: GDGD0,L,p
Help: bnrL1(bnr, {H}, {flag=0}): bnr being output by bnrinit(,,1) and
 H being a square matrix defining a congruence subgroup of bnr (the
 trivial subgroup if omitted), for each character of bnr trivial on this
 subgroup, compute L(1, chi) (or equivalently the first non-zero term c(chi)
 of the expansion at s = 0). The binary digits of flag mean 1: if 0 then
 compute the term c(chi) and return [r(chi), c(chi)] where r(chi) is the
 order of L(s, chi) at s = 0, or if 1 then compute the value at s = 1 (and in
 this case, only for non-trivial characters), 2: if 0 then compute the value
 of the primitive L-function attached to chi, if 1 then compute the value
 of the L-function L_S(s, chi) where S is the set of places dividing the
 modulus of bnr (and the infinite places), 3: return also the characters.
Doc: let \var{bnr} be the number field data output by \kbd{bnrinit(,,1)} and
 \var{H} be a square matrix defining a congruence subgroup of the
 ray class group corresponding to \var{bnr} (the trivial congruence subgroup
 if omitted). This function returns, for each \idx{character} $\chi$ of the ray
 class group which is trivial on $H$, the value at $s = 1$ (or $s = 0$) of the
 abelian $L$-function attached to $\chi$. For the value at $s = 0$, the
 function returns in fact for each $\chi$ a vector $[r_\chi, c_\chi]$ where
 $$L(s, \chi) = c \cdot s^r + O(s^{r + 1})$$
 \noindent near $0$.
 
 The argument \fl\ is optional, its binary digits
 mean 1: compute at $s = 0$ if unset or $s = 1$ if set, 2: compute the
 primitive $L$-function attached to $\chi$ if unset or the $L$-function
 with Euler factors at prime ideals dividing the modulus of \var{bnr} removed
 if set (that is $L_S(s, \chi)$, where $S$ is the
 set of infinite places of the number field together with the finite prime
 ideals dividing the modulus of \var{bnr}), 3: return also the character if
 set.
 \bprog
 K = bnfinit(x^2-229);
 bnr = bnrinit(K,1,1);
 bnrL1(bnr)
 @eprog\noindent
 returns the order and the first non-zero term of $L(s, \chi)$ at $s = 0$
 where $\chi$ runs through the characters of the class group of
 $K = \Q(\sqrt{229})$. Then
 \bprog
 bnr2 = bnrinit(K,2,1);
 bnrL1(bnr2,,2)
 @eprog\noindent
 returns the order and the first non-zero terms of $L_S(s, \chi)$ at $s = 0$
 where $\chi$ runs through the characters of the class group of $K$ and $S$ is
 the set of infinite places of $K$ together with the finite prime $2$. Note
 that the ray class group modulo $2$ is in fact the class group, so
 \kbd{bnrL1(bnr2,0)} returns the same answer as \kbd{bnrL1(bnr,0)}.
 
 This function will fail with the message
 \bprog
  *** bnrL1: overflow in zeta_get_N0 [need too many primes].
 @eprog\noindent if the approximate functional equation requires us to sum
 too many terms (if the discriminant of $K$ is too large).

Function: bnrchar
Class: basic
Section: number_fields
C-Name: bnrchar
Prototype: GGDG
Help: bnrchar(bnr,g,{v}): returns all characters chi on bnr.clgp such that
 chi(g[i]) = e(v[i]); if v is omitted, returns all characters that are
 trivial on the g[i].
Doc: returns all characters $\chi$ on \kbd{bnr.clgp} such that
 $\chi(g_i) = e(v_i)$, where $e(x) = \exp(2i\pi x)$. If $v$ is omitted,
 returns all characters that are trivial on the $g_i$. Else the vectors $g$
 and $v$ must have the same length, the $g_i$ must be ideals in any form, and
 each $v_i$ is a rational number whose denominator must divide the order of
 $g_i$ in the ray class group. For convenience, the vector of the $g_i$
 can be replaced by a matrix whose columns give their discrete logarithm,
 as given by \kbd{bnrisprincipal}; this allows to specify abstractly a
 subgroup of the ray class group.
 
 \bprog
 ? bnr = bnrinit(bnfinit(x), [160,[1]], 1); /* (Z/160Z)^* */
 ? bnr.cyc
 %2 = [8, 4, 2]
 ? g = bnr.gen;
 ? bnrchar(bnr, g, [1/2,0,0])
 %4 = [[4, 0, 0]]  \\ a unique character
 ? bnrchar(bnr, [g[1],g[3]]) \\ all characters trivial on g[1] and g[3]
 %5 = [[0, 1, 0], [0, 2, 0], [0, 3, 0], [0, 0, 0]]
 ? bnrchar(bnr, [1,0,0;0,1,0;0,0,2])
 %6 = [[0, 0, 1], [0, 0, 0]]  \\ characters trivial on given subgroup
 @eprog

Function: bnrclassno
Class: basic
Section: number_fields
C-Name: bnrclassno0
Prototype: GDGDG
Help: bnrclassno(A,{B},{C}): relative degree of the class field defined by
 A,B,C. [A,{B},{C}] is of type [bnr], [bnr,subgroup], [bnf,modulus],
 or [bnf,modulus,subgroup].
 Faster than bnrinit if only the ray class number is wanted.
Doc: 
  let $A$, $B$, $C$ define a class field $L$ over a ground field $K$
 (of type \kbd{[\var{bnr}]},
 \kbd{[\var{bnr}, \var{subgroup}]},
 or \kbd{[\var{bnf}, \var{modulus}]},
 or \kbd{[\var{bnf}, \var{modulus},\var{subgroup}]},
 \secref{se:CFT}); this function returns the relative degree $[L:K]$.
 
 In particular if $A$ is a \var{bnf} (with units), and $B$ a modulus,
 this function returns the corresponding ray class number modulo $B$.
 One can input the attached \var{bid} (with generators if the subgroup
 $C$ is non trivial) for $B$ instead of the module itself, saving some time.
 
 This function is faster than \kbd{bnrinit} and should be used if only the
 ray class number is desired. See \tet{bnrclassnolist} if you need ray class
 numbers for all moduli less than some bound.
Variant: Also available is
 \fun{GEN}{bnrclassno}{GEN bnf,GEN f} to compute the ray class number
 modulo~$f$.

Function: bnrclassnolist
Class: basic
Section: number_fields
C-Name: bnrclassnolist
Prototype: GG
Help: bnrclassnolist(bnf,list): if list is as output by ideallist or
 similar, gives list of corresponding ray class numbers.
Doc: $\var{bnf}$ being as
 output by \kbd{bnfinit}, and \var{list} being a list of moduli (with units) as
 output by \kbd{ideallist} or \kbd{ideallistarch}, outputs the list of the
 class numbers of the corresponding ray class groups. To compute a single
 class number, \tet{bnrclassno} is more efficient.
 
 \bprog
 ? bnf = bnfinit(x^2 - 2);
 ? L = ideallist(bnf, 100, 2);
 ? H = bnrclassnolist(bnf, L);
 ? H[98]
 %4 = [1, 3, 1]
 ? l = L[1][98]; ids = vector(#l, i, l[i].mod[1])
 %5 = [[98, 88; 0, 1], [14, 0; 0, 7], [98, 10; 0, 1]]
 @eprog
 The weird \kbd{l[i].mod[1]}, is the first component of \kbd{l[i].mod}, i.e.
 the finite part of the conductor. (This is cosmetic: since by construction
 the Archimedean part is trivial, I do not want to see it). This tells us that
 the ray class groups modulo the ideals of norm 98 (printed as \kbd{\%5}) have
 respectively order $1$, $3$ and $1$. Indeed, we may check directly:
 \bprog
 ? bnrclassno(bnf, ids[2])
 %6 = 3
 @eprog

Function: bnrconductor
Class: basic
Section: number_fields
C-Name: bnrconductor0
Prototype: GDGDGD0,L,
Help: bnrconductor(A,{B},{C},{flag=0}): conductor f of the subfield of
 the ray class field given by A,B,C. flag is optional and
 can be 0: default, 1: returns [f, Cl_f, H], H subgroup of the ray class
 group modulo f defining the extension, 2: returns [f, bnr(f), H].
Doc: conductor $f$ of the subfield of a ray class field as defined by $[A,B,C]$
 (of type \kbd{[\var{bnr}]},
 \kbd{[\var{bnr}, \var{subgroup}]},
 \kbd{[\var{bnf}, \var{modulus}]} or
 \kbd{[\var{bnf}, \var{modulus}, \var{subgroup}]},
 \secref{se:CFT})
 
 If $\fl = 0$, returns $f$.
 
 If $\fl = 1$, returns $[f, Cl_f, H]$, where $Cl_f$ is the ray class group
 modulo $f$, as a finite abelian group; finally $H$ is the subgroup of $Cl_f$
 defining the extension.
 
 If $\fl = 2$, returns $[f, \var{bnr}(f), H]$, as above except $Cl_f$ is
 replaced by a \kbd{bnr} structure, as output by $\tet{bnrinit}(,f,1)$.
 
 In place of a subgroup $H$, this function also accepts a character
 \kbd{chi}  $=(a_j)$, expressed as usual in terms of the generators
 \kbd{bnr.gen}: $\chi(g_j) = \exp(2i\pi a_j / d_j)$, where $g_j$ has
 order $d_j = \kbd{bnr.cyc[j]}$. In which case, the function returns
 respectively
 
 If $\fl = 0$, the conductor $f$ of $\text{Ker} \chi$.
 
 If $\fl = 1$, $[f, Cl_f, \chi_f]$, where $\chi_f$ is $\chi$ expressed
 on the minimal ray class group, whose modulus is the conductor.
 
 If $\fl = 2$, $[f, \var{bnr}(f), \chi_f]$.
Variant: 
 Also available is \fun{GEN}{bnrconductor}{GEN bnr, GEN H, long flag}

Function: bnrconductorofchar
Class: basic
Section: number_fields
C-Name: bnrconductorofchar
Prototype: GG
Help: bnrconductorofchar(bnr,chi): this function is obsolete, use bnrconductor.
Doc: This function is obsolete, use \tev{bnrconductor}.
Obsolete: 2015-11-11

Function: bnrdisc
Class: basic
Section: number_fields
C-Name: bnrdisc0
Prototype: GDGDGD0,L,
Help: bnrdisc(A,{B},{C},{flag=0}): absolute or relative [N,R1,discf] of
 the field defined by A,B,C. [A,{B},{C}] is of type [bnr],
 [bnr,subgroup], [bnf, modulus] or [bnf,modulus,subgroup], where bnf is as
 output by bnfinit, bnr by bnrinit, and
 subgroup is the HNF matrix of a subgroup of the corresponding ray class
 group (if omitted, the trivial subgroup). flag is optional whose binary
 digits mean 1: give relative data; 2: return 0 if modulus is not the
 conductor.
Doc: $A$, $B$, $C$ defining a class field $L$ over a ground field $K$
 (of type \kbd{[\var{bnr}]},
 \kbd{[\var{bnr}, \var{subgroup}]},
 \kbd{[\var{bnr}, \var{character}]},
 \kbd{[\var{bnf}, \var{modulus}]} or
 \kbd{[\var{bnf}, \var{modulus}, \var{subgroup}]},
 \secref{se:CFT}), outputs data $[N,r_1,D]$ giving the discriminant and
 signature of $L$, depending on the binary digits of \fl:
 
 \item 1: if this bit is unset, output absolute data related to $L/\Q$:
 $N$ is the absolute degree $[L:\Q]$, $r_1$ the number of real places of $L$,
 and $D$ the discriminant of $L/\Q$. Otherwise, output relative data for $L/K$:
 $N$ is the relative degree $[L:K]$, $r_1$ is the number of real places of $K$
 unramified in $L$ (so that the number of real places of $L$ is equal to $r_1$
 times $N$), and $D$ is the relative discriminant ideal of $L/K$.
 
 \item 2: if this bit is set and if the modulus is not the conductor of $L$,
 only return 0.

Function: bnrdisclist
Class: basic
Section: number_fields
C-Name: bnrdisclist0
Prototype: GGDG
Help: bnrdisclist(bnf,bound,{arch}): gives list of discriminants of
 ray class fields of all conductors up to norm bound, in a long vector
 The ramified Archimedean places are given by arch; all possible values are
 taken if arch is omitted. Supports the alternative syntax
 bnrdisclist(bnf,list), where list is as output by ideallist or ideallistarch
 (with units).
Doc: $\var{bnf}$ being as output by \kbd{bnfinit} (with units), computes a
 list of discriminants of Abelian extensions of the number field by increasing
 modulus norm up to bound \var{bound}. The ramified Archimedean places are
 given by \var{arch}; all possible values are taken if \var{arch} is omitted.
 
 The alternative syntax $\kbd{bnrdisclist}(\var{bnf},\var{list})$ is
 supported, where \var{list} is as output by \kbd{ideallist} or
 \kbd{ideallistarch} (with units), in which case \var{arch} is disregarded.
 
 The output $v$ is a vector of vectors, where $v[i][j]$ is understood to be in
 fact $V[2^{15}(i-1)+j]$ of a unique big vector $V$. (This awkward scheme
 allows for larger vectors than could be otherwise represented.)
 
 $V[k]$ is itself a vector $W$, whose length is the number of ideals of norm
 $k$. We consider first the case where \var{arch} was specified. Each
 component of $W$ corresponds to an ideal $m$ of norm $k$, and
 gives invariants attached to the ray class field $L$ of $\var{bnf}$ of
 conductor $[m, \var{arch}]$. Namely, each contains a vector $[m,d,r,D]$ with
 the following meaning: $m$ is the prime ideal factorization of the modulus,
 $d = [L:\Q]$ is the absolute degree of $L$, $r$ is the number of real places
 of $L$, and $D$ is the factorization of its absolute discriminant. We set $d
 = r = D = 0$ if $m$ is not the finite part of a conductor.
 
 If \var{arch} was omitted, all $t = 2^{r_1}$ possible values are taken and a
 component of $W$ has the form $[m, [[d_1,r_1,D_1], \dots, [d_t,r_t,D_t]]]$,
 where $m$ is the finite part of the conductor as above, and
 $[d_i,r_i,D_i]$ are the invariants of the ray class field of conductor
 $[m,v_i]$, where $v_i$ is the $i$-th Archimedean component, ordered by
 inverse lexicographic order; so $v_1 = [0,\dots,0]$, $v_2 = [1,0\dots,0]$,
 etc. Again, we set $d_i = r_i = D_i = 0$ if $[m,v_i]$ is not a conductor.
 
 Finally, each prime ideal $pr = [p,\alpha,e,f,\beta]$ in the prime
 factorization $m$ is coded as the integer $p\cdot n^2+(f-1)\cdot n+(j-1)$,
 where $n$ is the degree of the base field and $j$ is such that
 
 \kbd{pr = idealprimedec(\var{nf},p)[j]}.
 
 \noindent $m$ can be decoded using \tet{bnfdecodemodule}.
 
 Note that to compute such data for a single field, either \tet{bnrclassno}
 or \tet{bnrdisc} is more efficient.

Function: bnrgaloisapply
Class: basic
Section: number_fields
C-Name: bnrgaloisapply
Prototype: GGG
Help: bnrgaloisapply(bnr, mat, H): apply the automorphism given by its matrix
 mat to the congruence subgroup H given as a HNF matrix. The matrix mat can be
 computed with bnrgaloismatrix.
Doc: apply the automorphism given by its matrix \var{mat} to the congruence
 subgroup $H$ given as a HNF matrix.
 The matrix \var{mat} can be computed with \tet{bnrgaloismatrix}.

Function: bnrgaloismatrix
Class: basic
Section: number_fields
C-Name: bnrgaloismatrix
Prototype: GG
Help: bnrgaloismatrix(bnr,aut): return the matrix of the action of the
 automorphism aut of the base field bnf.nf on the generators of the ray class
 field bnr.gen. aut can be given as a polynomial, or a vector of automorphisms
 or a galois group as output by galoisinit, in which case a vector of matrices
 is returned (in the later case, only for the generators aut.gen).
Doc: return the matrix of the action of the automorphism \var{aut} of the base
 field \kbd{bnf.nf} on the generators of the ray class field \kbd{bnr.gen}.
 \var{aut} can be given as a polynomial, an algebraic number, or a vector of
 automorphisms or a Galois group as output by \kbd{galoisinit}, in which case a
 vector of matrices is returned (in the later case, only for the generators
 \kbd{aut.gen}).
 
 See \kbd{bnrisgalois} for an example.
Variant: When $aut$ is a polynomial or an algebraic number,
 \fun{GEN}{bnrautmatrix}{GEN bnr, GEN aut} is available.

Function: bnrinit
Class: basic
Section: number_fields
C-Name: bnrinit0
Prototype: GGD0,L,
Help: bnrinit(bnf,f,{flag=0}): given a bnf as output by
 bnfinit and a modulus f, initializes data
 linked to the ray class group structure corresponding to this module. flag
 is optional, and can be 0: default, 1: compute also the generators.
Description: 
 (gen,gen,?small):bnr       bnrinit0($1, $2, $3)
Doc: $\var{bnf}$ is as
 output by \kbd{bnfinit} (including fundamental units), $f$ is a modulus,
 initializes data linked to the ray class group structure corresponding to
 this module, a so-called \kbd{bnr} structure. One can input the attached
 \var{bid} with generators for $f$ instead of the module itself, saving some
 time. (As in \tet{idealstar}, the finite part of the conductor may be given
 by a factorization into prime ideals, as produced by \tet{idealfactor}.)
 
 The following member functions are available
 on the result: \kbd{.bnf} is the underlying \var{bnf},
 \kbd{.mod} the modulus, \kbd{.bid} the \kbd{bid} structure attached to the
 modulus; finally, \kbd{.clgp}, \kbd{.no}, \kbd{.cyc}, \kbd{.gen} refer to the
 ray class group (as a finite abelian group), its cardinality, its elementary
 divisors, its generators (only computed if $\fl = 1$).
 
 The last group of functions are different from the members of the underlying
 \var{bnf}, which refer to the class group; use \kbd{\var{bnr}.bnf.\var{xxx}}
 to access these, e.g.~\kbd{\var{bnr}.bnf.cyc} to get the cyclic decomposition
 of the class group.
 
 They are also different from the members of the underlying \var{bid}, which
 refer to $(\Z_K/f)^*$; use \kbd{\var{bnr}.bid.\var{xxx}} to access these,
 e.g.~\kbd{\var{bnr}.bid.no} to get $\phi(f)$.
 
 If $\fl=0$ (default), the generators of the ray class group are not computed,
 which saves time. Hence \kbd{\var{bnr}.gen} would produce an error.
 
 If $\fl=1$, as the default, except that generators are computed.
Variant: Instead the above  hardcoded  numerical flags,  one should rather use
 \fun{GEN}{Buchray}{GEN bnf, GEN module, long flag}
 where flag is an or-ed combination of \kbd{nf\_GEN} (include generators)
 and \kbd{nf\_INIT} (if omitted, return just the cardinality of the ray class
 group and its structure), possibly 0.

Function: bnrisconductor
Class: basic
Section: number_fields
C-Name: bnrisconductor0
Prototype: lGDGDG
Help: bnrisconductor(A,{B},{C}): returns 1 if the modulus is the
 conductor of the subfield of the ray class field given by A,B,C (see
 bnrdisc), and 0 otherwise. Slightly faster than bnrconductor if this is the
 only desired result.
Doc: fast variant of \kbd{bnrconductor}$(A,B,C)$; $A$, $B$, $C$ represent
 an extension of the base field, given by class field theory
 (see~\secref{se:CFT}). Outputs 1 if this modulus is the conductor, and 0
 otherwise. This is slightly faster than \kbd{bnrconductor} when the
 character or subgroup is not primitive.

Function: bnrisgalois
Class: basic
Section: number_fields
C-Name: bnrisgalois
Prototype: lGGG
Help: bnrisgalois(bnr, gal, H): check whether the class field attached to
 the subgroup H is Galois over the subfield of bnr.nf fixed by the Galois
 group gal, which can be given as output by galoisinit, or as a matrix or a
 vector of matrices as output by bnrgaloismatrix. The ray class field
 attached to bnr need to be Galois, which is not checked.
Doc: check whether the class field attached to the subgroup $H$ is Galois
 over the subfield of \kbd{bnr.nf} fixed by the group \var{gal}, which can be
 given as output by \tet{galoisinit}, or as a matrix or a vector of matrices as
 output by \kbd{bnrgaloismatrix}, the second option being preferable, since it
 saves the recomputation of the matrices.  Note: The function assumes that the
 ray class field attached to bnr is Galois, which is not checked.
 
 In the following example, we lists the congruence subgroups of subextension of
 degree at most $3$ of the ray class field of conductor $9$ which are Galois
 over the rationals.
 
 \bprog
 K=bnfinit(a^4-3*a^2+253009);
 G=galoisinit(K);
 B=bnrinit(K,9,1);
 L1=[H|H<-subgrouplist(B,3), bnrisgalois(B,G,H)]
 ##
 M=bnrgaloismatrix(B,G)
 L2=[H|H<-subgrouplist(B,3), bnrisgalois(B,M,H)]
 ##
 @eprog
 The second computation is much faster since \kbd{bnrgaloismatrix(B,G)} is
 computed only once.

Function: bnrisprincipal
Class: basic
Section: number_fields
C-Name: bnrisprincipal
Prototype: GGD1,L,
Help: bnrisprincipal(bnr,x,{flag=1}): bnr being output by bnrinit, gives
 [v,alpha], where v is the vector of exponents on the class group
 generators and alpha is the generator of the resulting principal ideal. In
 particular x is principal if and only if v is the zero vector. If (optional)
 flag is set to 0, output only v.
Doc: \var{bnr} being the
 number field data which is output by \kbd{bnrinit}$(,,1)$ and $x$ being an
 ideal in any form, outputs the components of $x$ on the ray class group
 generators in a way similar to \kbd{bnfisprincipal}. That is a 2-component
 vector $v$ where $v[1]$ is the vector of components of $x$ on the ray class
 group generators, $v[2]$ gives on the integral basis an element $\alpha$ such
 that $x=\alpha\prod_ig_i^{x_i}$.
 
 If $\fl=0$, outputs only $v_1$. In that case, \var{bnr} need not contain the
 ray class group generators, i.e.~it may be created with \kbd{bnrinit}$(,,0)$
 If $x$ is not coprime to the modulus of \var{bnr} the result is undefined.
Variant: Instead of hardcoded  numerical flags,  one should rather
 use
 \fun{GEN}{isprincipalray}{GEN bnr, GEN x} for $\kbd{flag} = 0$, and if you
 want generators:
 \bprog
   bnrisprincipal(bnr, x, nf_GEN)
 @eprog

Function: bnrrootnumber
Class: basic
Section: number_fields
C-Name: bnrrootnumber
Prototype: GGD0,L,p
Help: bnrrootnumber(bnr,chi,{flag=0}): returns the so-called Artin Root
 Number, i.e. the constant W appearing in the functional equation of the
 Hecke L-function attached to chi. Set flag = 1 if the character is known
 to be primitive.
Doc: if $\chi=\var{chi}$ is a
 \idx{character} over \var{bnr}, not necessarily primitive, let
 $L(s,\chi) = \sum_{id} \chi(id) N(id)^{-s}$ be the attached
 \idx{Artin L-function}. Returns the so-called \idx{Artin root number}, i.e.~the
 complex number $W(\chi)$ of modulus 1 such that
 %
 $$\Lambda(1-s,\chi) = W(\chi) \Lambda(s,\overline{\chi})$$
 %
 \noindent where $\Lambda(s,\chi) = A(\chi)^{s/2}\gamma_\chi(s) L(s,\chi)$ is
 the enlarged L-function attached to $L$.
 
 The generators of the ray class group are needed, and you can set $\fl=1$ if
 the character is known to be primitive. Example:
 
 \bprog
 bnf = bnfinit(x^2 - x - 57);
 bnr = bnrinit(bnf, [7,[1,1]], 1);
 bnrrootnumber(bnr, [2,1])
 @eprog\noindent
 returns the root number of the character $\chi$ of
 $\Cl_{7\infty_1\infty_2}(\Q(\sqrt{229}))$ defined by $\chi(g_1^ag_2^b)
 = \zeta_1^{2a}\zeta_2^b$. Here $g_1, g_2$ are the generators of the
 ray-class group given by \kbd{bnr.gen} and $\zeta_1 = e^{2i\pi/N_1},
 \zeta_2 = e^{2i\pi/N_2}$ where $N_1, N_2$ are the orders of $g_1$ and
 $g_2$ respectively ($N_1=6$ and $N_2=3$ as \kbd{bnr.cyc} readily tells us).

Function: bnrstark
Class: basic
Section: number_fields
C-Name: bnrstark
Prototype: GDGp
Help: bnrstark(bnr,{subgroup}): bnr being as output by
 bnrinit(,,1), finds a relative equation for the class field corresponding to
 the module in bnr and the given congruence subgroup (the trivial subgroup if
 omitted) using Stark's units. The ground field and the class field must be
 totally real.
Doc: \var{bnr} being as output by \kbd{bnrinit(,,1)}, finds a relative equation
 for the class field corresponding to the modulus in \var{bnr} and the given
 congruence subgroup (as usual, omit $\var{subgroup}$ if you want the whole ray
 class group).
 
 The main variable of \var{bnr} must not be $x$, and the ground field and the
 class field must be totally real. When the base field is $\Q$, the vastly
 simpler \tet{galoissubcyclo} is used instead. Here is an example:
 \bprog
 bnf = bnfinit(y^2 - 3);
 bnr = bnrinit(bnf, 5, 1);
 bnrstark(bnr)
 @eprog\noindent
 returns the ray class field of $\Q(\sqrt{3})$ modulo $5$. Usually, one wants
 to apply to the result one of
 \bprog
 rnfpolredabs(bnf, pol, 16)     \\@com compute a reduced relative polynomial
 rnfpolredabs(bnf, pol, 16 + 2) \\@com compute a reduced absolute polynomial
 @eprog
 
 The routine uses \idx{Stark units} and needs to find a suitable auxiliary
 conductor, which may not exist when the class field is not cyclic over the
 base. In this case \kbd{bnrstark} is allowed to return a vector of
 polynomials defining \emph{independent} relative extensions, whose compositum
 is the requested class field. It was decided that it was more useful
 to keep the extra information thus made available, hence the user has to take
 the compositum herself.
 
 Even if it exists, the auxiliary conductor may be so large that later
 computations become unfeasible. (And of course, Stark's conjecture may simply
 be wrong.) In case of difficulties, try \tet{rnfkummer}:
 \bprog
 ? bnr = bnrinit(bnfinit(y^8-12*y^6+36*y^4-36*y^2+9,1), 2, 1);
 ? bnrstark(bnr)
   ***   at top-level: bnrstark(bnr)
   ***                 ^-------------
   *** bnrstark: need 3919350809720744 coefficients in initzeta.
   *** Computation impossible.
 ? lift( rnfkummer(bnr) )
 time = 24 ms.
 %2 = x^2 + (1/3*y^6 - 11/3*y^4 + 8*y^2 - 5)
 @eprog

Function: break
Class: basic
Section: programming/control
C-Name: break0
Prototype: D1,L,
Help: break({n=1}): interrupt execution of current instruction sequence, and
 exit from the n innermost enclosing loops.
Doc: interrupts execution of current \var{seq}, and
 immediately exits from the $n$ innermost enclosing loops, within the
 current function call (or the top level loop); the integer $n$ must be
 positive. If $n$ is greater than the number of enclosing loops, all
 enclosing loops are exited.

Function: breakpoint
Class: gp
Section: programming/control
C-Name: pari_breakpoint
Prototype: v
Help: breakpoint(): interrupt the program and enter the breakloop. The program
 continues when the breakloop is exited.
Doc: Interrupt the program and enter the breakloop. The program continues when
 the breakloop is exited.
 \bprog
 ? f(N,x)=my(z=x^2+1);breakpoint();gcd(N,z^2+1-z);
 ? f(221,3)
   ***   at top-level: f(221,3)
   ***                 ^--------
   ***   in function f: my(z=x^2+1);breakpoint();gcd(N,z
   ***                              ^--------------------
   ***   Break loop: type <Return> to continue; 'break' to go back to GP
 break> z
 10
 break>
 %2 = 13
 @eprog

Function: call
Class: basic
Section: programming/specific
C-Name: call0
Prototype: GG
Help: call(f, A): A being a vector, evaluates f(A[1],...,A[#A]).
Doc: $A=[a_1,\dots, a_n]$ being a vector and $f$ being a function, returns the
 evaluation of $f(a_1,\dots,a_n)$.
 $f$ can also be the name of a built-in GP function.
 If $\# A =1$, \tet{call}($f,A$) = \tet{apply}($f,A$)[1].
 If $f$ is variadic, the variadic arguments must grouped in a vector in
 the last component of $A$.
 
 This function is useful
 
 \item when writing a variadic function, to call another one:
 \bprog
 fprintf(file,format,args[..]) = write(file,call(Strprintf,[format,args]))
 @eprog
 
 \item when dealing with function arguments with unspecified arity
 
 The function below implements a global memoization interface:
 \bprog
 memo=Map();
 memoize(f,A[..])=
 {
   my(res);
   if(!mapisdefined(memo, [f,A], &res),
     res = call(f,A);
     mapput(memo,[f,A],res));
  res;
 }
 @eprog
 for example:
 \bprog
 ? memoize(factor,2^128+1)
 %3 = [59649589127497217,1;5704689200685129054721,1]
 ? ##
   ***   last result computed in 76 ms.
 ? memoize(factor,2^128+1)
 %4 = [59649589127497217,1;5704689200685129054721,1]
 ? ##
   ***   last result computed in 0 ms.
 ? memoize(ffinit,3,3)
 %5 = Mod(1,3)*x^3+Mod(1,3)*x^2+Mod(1,3)*x+Mod(2,3)
 ? fibo(n)=if(n==0,0,n==1,1,memoize(fibo,n-2)+memoize(fibo,n-1));
 ? fibo(100)
 %7 = 354224848179261915075
 @eprog
 
 \item to call operators through their internal names without using
 \kbd{alias}
 \bprog
 matnbelts(M) = call("_*_",matsize(M))
 @eprog

Function: ceil
Class: basic
Section: conversions
C-Name: gceil
Prototype: G
Help: ceil(x): ceiling of x = smallest integer >= x.
Description: 
 (small):small:parens   $1
 (int):int:copy:parens  $1
 (real):int             ceilr($1)
 (mp):int               mpceil($1)
 (gen):gen              gceil($1)
Doc: 
 ceiling of $x$. When $x$ is in $\R$, the result is the
 smallest integer greater than or equal to $x$. Applied to a rational
 function, $\kbd{ceil}(x)$ returns the Euclidean quotient of the numerator by
 the denominator.

Function: centerlift
Class: basic
Section: conversions
C-Name: centerlift0
Prototype: GDn
Help: centerlift(x,{v}): centered lift of x. Same as lift except for
 intmod and padic components.
Description: 
 (pol):pol        centerlift($1)
 (vec):vec        centerlift($1)
 (gen):gen        centerlift($1)
 (pol, var):pol        centerlift0($1, $2)
 (vec, var):vec        centerlift0($1, $2)
 (gen, var):gen        centerlift0($1, $2)
Doc: Same as \tet{lift}, except that \typ{INTMOD} and \typ{PADIC} components
 are lifted using centered residues:
 
 \item for a \typ{INTMOD} $x\in \Z/n\Z$, the lift $y$ is such that
 $-n/2<y\le n/2$.
 
 \item  a \typ{PADIC} $x$ is lifted in the same way as above (modulo
 $p^\kbd{padicprec(x)}$) if its valuation $v$ is non-negative; if not, returns
 the fraction $p^v$ \kbd{centerlift}$(x p^{-v})$; in particular, rational
 reconstruction is not attempted. Use \tet{bestappr} for this.
 
 For backward compatibility, \kbd{centerlift(x,'v)} is allowed as an alias
 for \kbd{lift(x,'v)}.
 
 \synt{centerlift}{GEN x}.

Function: characteristic
Class: basic
Section: conversions
C-Name: characteristic
Prototype: mG
Help: characteristic(x): characteristic of the base ring over which x is
 defined.
Doc: 
 returns the characteristic of the base ring over which $x$ is defined (as
 defined by \typ{INTMOD} and \typ{FFELT} components). The function raises an
 exception if incompatible primes arise from \typ{FFELT} and \typ{PADIC}
 components.
 \bprog
 ? characteristic(Mod(1,24)*x + Mod(1,18)*y)
 %1 = 6
 @eprog

Function: charconj
Class: basic
Section: number_theoretical
C-Name: charconj0
Prototype: GG
Help: charconj(cyc,chi): given a finite abelian group (by its elementary
 divisors cyc) and a character chi, return the conjugate character.
Doc: let \var{cyc} represent a finite abelian group by its elementary
 divisors, i.e. $(d_j)$ represents $\sum_{j \leq k} \Z/d_j\Z$ with $d_k
 \mid \dots \mid d_1$; any object which has a \kbd{.cyc} method is also
 allowed, e.g.~the output of \kbd{znstar} or \kbd{bnrinit}. A character
 on this group is given by a row vector $\chi = [a_1,\ldots,a_n]$ such that
 $\chi(\prod g_j^{n_j}) = \exp(2\pi i\sum a_j n_j / d_j)$, where $g_j$ denotes
 the generator (of order $d_j$) of the $j$-th cyclic component.
 
 This function returns the conjugate character.
 \bprog
 ? cyc = [15,5]; chi = [1,1];
 ? charconj(cyc, chi)
 %2 = [14, 4]
 ? bnf = bnfinit(x^2+23);
 ? bnf.cyc
 %4 = [3]
 ? charconj(bnf, [1])
 %5 = [2]
 @eprog\noindent For Dirichlet characters (when \kbd{cyc} is
 \kbd{idealstar(,q)}), characters in Conrey representation are available,
 see \secref{se:dirichletchar} or \kbd{??character}:
 \bprog
 ? G = idealstar(,8);  \\ (Z/8Z)^*
 ? charorder(G, 3)  \\ Conrey label
 %2 = 2
 ? chi = znconreylog(G, 3);
 ? charorder(G, chi)  \\ Conrey logarithm
 %4 = 2
 @eprog
Variant: Also available is
 \fun{GEN}{charconj}{GEN cyc, GEN chi}, when \kbd{cyc} is known to
 be a vector of elementary divisors and \kbd{chi} a compatible character
 (no checks).

Function: chardiv
Class: basic
Section: number_theoretical
C-Name: chardiv0
Prototype: GGG
Help: chardiv(cyc, a,b): given a finite abelian group (by its elementary
 divisors cyc) and two characters a and b, return the character a/b.
Doc: let \var{cyc} represent a finite abelian group by its elementary
 divisors, i.e. $(d_j)$ represents $\sum_{j \leq k} \Z/d_j\Z$ with $d_k
 \mid \dots \mid d_1$; any object which has a \kbd{.cyc} method is also
 allowed, e.g.~the output of \kbd{znstar} or \kbd{bnrinit}. A character
 on this group is given by a row vector $a = [a_1,\ldots,a_n]$ such that
 $\chi(\prod g_j^{n_j}) = \exp(2\pi i\sum a_j n_j / d_j)$, where $g_j$ denotes
 the generator (of order $d_j$) of the $j$-th cyclic component.
 
 Given two characters $a$ and $b$, return the character
 $a / b = a \overline{b}$.
 \bprog
 ? cyc = [15,5]; a = [1,1]; b =  [2,4];
 ? chardiv(cyc, a,b)
 %2 = [14, 2]
 ? bnf = bnfinit(x^2+23);
 ? bnf.cyc
 %4 = [3]
 ? chardiv(bnf, [1], [2])
 %5 = [2]
 @eprog\noindent For Dirichlet characters on  $(\Z/N\Z)^*$, additional
 representations are available (Conrey labels, Conrey logarithm),
 see \secref{se:dirichletchar} or \kbd{??character}.
 If the two characters are in the same format, the
 result is given in the same format, otherwise a Conrey logarithm is used.
 \bprog
 ? G = idealstar(,100);
 ? G.cyc
 %2 = [20, 2]
 ? a = [10, 1]; \\ usual representation for characters
 ? b = 7; \\ Conrey label;
 ? c = znconreylog(G, 11); \\ Conrey log
 ? chardiv(G, b,b)
 %6 = 1   \\ Conrey label
 ? chardiv(G, a,b)
 %7 = [0, 5]~  \\ Conrey log
 ? chardiv(G, a,c)
 %7 = [0, 14]~   \\ Conrey log
 @eprog
Variant: Also available is
 \fun{GEN}{chardiv}{GEN cyc, GEN a, GEN b}, when \kbd{cyc} is known to
 be a vector of elementary divisors and $a, b$ are compatible characters
 (no checks).

Function: chareval
Class: basic
Section: number_theoretical
C-Name: chareval
Prototype: GGGDG
Help: chareval(G, chi, x, {z})): given an abelian group structure affording
 a discrete logarithm method, e.g. G = idealstar(,N) or a bnr structure,
 let x be an element of G and let chi be a character of G. This function
 returns the value of chi at x, where the encoding depends on the optional
 argument z; if z is omitted, we fix a canonical o-th root of 1, zeta_o,
 where o is the character order and return the rational number c/o where
 chi(x) = (zeta_o)^c.
Doc: 
 Let $G$ be an abelian group structure affording a discrete logarithm
 method, e.g $G = \kbd{idealstar}(,N)$ for $(\Z/N\Z)^*$ or a \kbd{bnr}
 structure, let $x$ be an element of $G$ and let \var{chi} be a character of
 $G$ (see the note below for details). This function returns the value of
 \var{chi} at $x$.
 
 \misctitle{Note on characters}
 Let $K$ be some field. If $G$ is an abelian group,
 let $\chi: G \to K^*$ be a character of finite order and let $o$ be a
 multiple of the character order such that $\chi(n) = \zeta^{c(n)}$ for some
 fixed $\zeta\in K^*$ of multiplicative order $o$ and a unique morphism $c: G
 \to (\Z/o\Z,+)$. Our usual convention is to write
 $$G = (\Z/o_1\Z) g_1 \oplus \cdots \oplus (\Z/o_d\Z) g_d$$
 for some generators $(g_i)$ of respective order $d_i$, where the group has
 exponent $o := \text{lcm}_i o_i$. Since $\zeta^o = 1$, the vector $(c_i)$ in
 $\prod (\Z/o_i\Z)$ defines a character $\chi$ on $G$ via $\chi(g_i) =
 \zeta^{c_i (o/o_i)}$ for all $i$. Classical Dirichlet characters have values
 in $K = \C$ and we can take $\zeta = \exp(2i\pi/o)$.
 
 \misctitle{Note on Dirichlet characters}
 In the special case where \var{bid} is attached to $G = (\Z/q\Z)^*$
 (as per \kbd{bid = idealstar(,q)}), the Dirichlet
 character \var{chi} can be written in one of the usual 3 formats: a \typ{VEC}
 in terms of \kbd{bid.gen} as above, a \typ{COL} in terms of the Conrey
 generators, or a \typ{INT} (Conrey label);
 see \secref{se:dirichletchar} or \kbd{??character}.
 
 The character value is encoded as follows, depending on the optional
 argument $z$:
 
 \item If $z$ is omitted: return the rational number $c(x)/o$ for $x$ coprime
 to $q$, where we normalize $0\leq c(x) < o$. If $x$ can not be mapped to the
 group (e.g. $x$ is not coprime to the conductor of a Dirichlet or Hecke
 character) we return the sentinel value $-1$.
 
 \item If $z$ is an integer $o$, then we assume that $o$ is a multiple of the
 character order and we return the integer $c(x)$ when $x$ belongs
 to the group, and the sentinel value $-1$ otherwise.
 
 \item $z$ can be of the form $[\var{zeta}, o]$, where \var{zeta}
 is an $o$-th root of $1$ and $o$ is a multiple of the character order.
 We return $\zeta^{c(x)}$ if $x$ belongs to the group, and the sentinel
 value $0$ otherwise. (Note that this coincides  with the usual extension
 of Dirichlet characters to $\Z$, or of Hecke characters to general ideals.)
 
 \item Finally, $z$ can be of the form $[\var{vzeta}, o]$, where
 \var{vzeta} is a vector of powers $\zeta^0, \dots, \zeta^{o-1}$
 of some $o$-th root of $1$ and $o$ is a multiple of the character order.
 As above, we return $\zeta^{c(x)}$ after a table lookup. Or the sentinel
 value $0$.

Function: charker
Class: basic
Section: number_theoretical
C-Name: charker0
Prototype: GG
Help: charker(cyc,chi): given a finite abelian group (by its elementary
 divisors cyc) and a character chi, return its kernel.
Doc: let \var{cyc} represent a finite abelian group by its elementary
 divisors, i.e. $(d_j)$ represents $\sum_{j \leq k} \Z/d_j\Z$ with $d_k
 \mid \dots \mid d_1$; any object which has a \kbd{.cyc} method is also
 allowed, e.g.~the output of \kbd{znstar} or \kbd{bnrinit}. A character
 on this group is given by a row vector $\chi = [a_1,\ldots,a_n]$ such that
 $\chi(\prod g_j^{n_j}) = \exp(2\pi i\sum a_j n_j / d_j)$, where $g_j$ denotes
 the generator (of order $d_j$) of the $j$-th cyclic component.
 
 This function returns the kernel of $\chi$, as a matrix $K$ in HNF which is a
 left-divisor of \kbd{matdiagonal(d)}. Its columns express in terms of
 the $g_j$ the generators of the subgroup. The determinant of $K$ is the
 kernel index.
 \bprog
 ? cyc = [15,5]; chi = [1,1];
 ? charker(cyc, chi)
 %2 =
 [15 12]
 
 [ 0  1]
 
 ? bnf = bnfinit(x^2+23);
 ? bnf.cyc
 %4 = [3]
 ? charker(bnf, [1])
 %5 =
 [3]
 @eprog\noindent Note that for Dirichlet characters (when \kbd{cyc} is
 \kbd{idealstar(,q)}), characters in Conrey representation are available,
 see \secref{se:dirichletchar} or \kbd{??character}.
 \bprog
 ? G = idealstar(,8);  \\ (Z/8Z)^*
 ? charker(G, 1) \\ Conrey label for trivial character
 %2 =
 [1 0]
 
 [0 1]
 @eprog
Variant: Also available is
 \fun{GEN}{charker}{GEN cyc, GEN chi}, when \kbd{cyc} is known to
 be a vector of elementary divisors and \kbd{chi} a compatible character
 (no checks).

Function: charmul
Class: basic
Section: number_theoretical
C-Name: charmul0
Prototype: GGG
Help: charmul(cyc, a,b): given a finite abelian group (by its elementary
 divisors cyc) and two characters a and b, return the product character
 ab.
Doc: let \var{cyc} represent a finite abelian group by its elementary
 divisors, i.e. $(d_j)$ represents $\sum_{j \leq k} \Z/d_j\Z$ with $d_k
 \mid \dots \mid d_1$; any object which has a \kbd{.cyc} method is also
 allowed, e.g.~the output of \kbd{znstar} or \kbd{bnrinit}. A character
 on this group is given by a row vector $a = [a_1,\ldots,a_n]$ such that
 $\chi(\prod g_j^{n_j}) = \exp(2\pi i\sum a_j n_j / d_j)$, where $g_j$ denotes
 the generator (of order $d_j$) of the $j$-th cyclic component.
 
 Given two characters $a$ and $b$, return the product character $ab$.
 \bprog
 ? cyc = [15,5]; a = [1,1]; b =  [2,4];
 ? charmul(cyc, a,b)
 %2 = [3, 0]
 ? bnf = bnfinit(x^2+23);
 ? bnf.cyc
 %4 = [3]
 ? charmul(bnf, [1], [2])
 %5 = [0]
 @eprog\noindent For Dirichlet characters on  $(\Z/N\Z)^*$, additional
 representations are available (Conrey labels, Conrey logarithm), see
 \secref{se:dirichletchar} or \kbd{??character}. If the two characters are in
 the same format, their
 product is given in the same format, otherwise a Conrey logarithm is used.
 \bprog
 ? G = idealstar(,100);
 ? G.cyc
 %2 = [20, 2]
 ? a = [10, 1]; \\ usual representation for characters
 ? b = 7; \\ Conrey label;
 ? c = znconreylog(G, 11); \\ Conrey log
 ? charmul(G, b,b)
 %6 = 49   \\ Conrey label
 ? charmul(G, a,b)
 %7 = [0, 15]~  \\ Conrey log
 ? charmul(G, a,c)
 %7 = [0, 6]~   \\ Conrey log
 @eprog
Variant: Also available is
 \fun{GEN}{charmul}{GEN cyc, GEN a, GEN b}, when \kbd{cyc} is known to
 be a vector of elementary divisors and $a, b$ are compatible characters
 (no checks).

Function: charorder
Class: basic
Section: number_theoretical
C-Name: charorder0
Prototype: GG
Help: charorder(cyc,chi): given a finite abelian group (by its elementary
 divisors cyc) and a character chi, return the order of chi.
Doc: let \var{cyc} represent a finite abelian group by its elementary
 divisors, i.e. $(d_j)$ represents $\sum_{j \leq k} \Z/d_j\Z$ with $d_k
 \mid \dots \mid d_1$; any object which has a \kbd{.cyc} method is also
 allowed, e.g.~the output of \kbd{znstar} or \kbd{bnrinit}. A character
 on this group is given by a row vector $\chi = [a_1,\ldots,a_n]$ such that
 $\chi(\prod g_j^{n_j}) = \exp(2\pi i\sum a_j n_j / d_j)$, where $g_j$ denotes
 the generator (of order $d_j$) of the $j$-th cyclic component.
 
 This function returns the order of the character \kbd{chi}.
 \bprog
 ? cyc = [15,5]; chi = [1,1];
 ? charorder(cyc, chi)
 %2 = 15
 ? bnf = bnfinit(x^2+23);
 ? bnf.cyc
 %4 = [3]
 ? charorder(bnf, [1])
 %5 = 3
 @eprog\noindent For Dirichlet characters (when \kbd{cyc} is
 \kbd{idealstar(,q)}), characters in Conrey representation are available,
 see \secref{se:dirichletchar} or \kbd{??character}:
 \bprog
 ? G = idealstar(,100); \\ (Z/100Z)^*
 ? charorder(G, 7)   \\ Conrey label
 %2 = 4
 @eprog
Variant: Also available is
 \fun{GEN}{charorder}{GEN cyc, GEN chi}, when \kbd{cyc} is known to
 be a vector of elementary divisors and \kbd{chi} a compatible character
 (no checks).

Function: charpoly
Class: basic
Section: linear_algebra
C-Name: charpoly0
Prototype: GDnD5,L,
Help: charpoly(A,{v='x},{flag=5}): det(v*Id-A)=characteristic polynomial of
 the matrix or polmod A. flag is optional and ignored unless A is a matrix;
 it may be set to 0 (Le Verrier), 1 (Lagrange interpolation),
 2 (Hessenberg form), 3 (Berkowitz), 4 (modular) if A is integral,
 or 5 (default, choose best method).
 Algorithms 0 (Le Verrier) and 1 (Lagrange) assume that n! is invertible,
 where n is the dimension of the matrix.
Doc: 
 \idx{characteristic polynomial}
 of $A$ with respect to the variable $v$, i.e.~determinant of $v*I-A$ if $A$
 is a square matrix.
 \bprog
 ? charpoly([1,2;3,4]);
 %1 = x^2 - 5*x - 2
 ? charpoly([1,2;3,4],, 't)
 %2 = t^2 - 5*t - 2
 @eprog\noindent
 If $A$ is not a square matrix, the function returns the characteristic
 polynomial of the map ``multiplication by $A$'' if $A$ is a scalar:
 \bprog
 ? charpoly(Mod(x+2, x^3-2))
 %1 = x^3 - 6*x^2 + 12*x - 10
 ? charpoly(I)
 %2 = x^2 + 1
 ? charpoly(quadgen(5))
 %3 = x^2 - x - 1
 ? charpoly(ffgen(ffinit(2,4)))
 %4 = Mod(1, 2)*x^4 + Mod(1, 2)*x^3 + Mod(1, 2)*x^2 + Mod(1, 2)*x + Mod(1, 2)
 @eprog
 
 The value of $\fl$ is only significant for matrices, and we advise to stick
 to the default value. Let $n$ be the dimension of $A$.
 
 If $\fl=0$, same method (Le Verrier's) as for computing the adjoint matrix,
 i.e.~using the traces of the powers of $A$. Assumes that $n!$ is
 invertible; uses $O(n^4)$ scalar operations.
 
 If $\fl=1$, uses Lagrange interpolation which is usually the slowest method.
 Assumes that $n!$ is invertible; uses $O(n^4)$ scalar operations.
 
 If $\fl=2$, uses the Hessenberg form. Assumes that the base ring is a field.
 Uses $O(n^3)$ scalar operations, but suffers from coefficient explosion
 unless the base field is finite or $\R$.
 
 If $\fl=3$, uses Berkowitz's division free algorithm, valid over any
 ring (commutative, with unit). Uses $O(n^4)$ scalar operations.
 
 If $\fl=4$, $x$ must be integral. Uses a modular algorithm: Hessenberg form
 for various small primes, then Chinese remainders.
 
 If $\fl=5$ (default), uses the ``best'' method given $x$.
 This means we use Berkowitz unless the base ring is $\Z$ (use $\fl=4$)
 or a field where coefficient explosion does not occur,
 e.g.~a finite field or the reals (use $\fl=2$).
Variant: Also available are
 \fun{GEN}{charpoly}{GEN x, long v} ($\fl=5$),
 \fun{GEN}{caract}{GEN A, long v} ($\fl=1$),
 \fun{GEN}{carhess}{GEN A, long v} ($\fl=2$),
 \fun{GEN}{carberkowitz}{GEN A, long v} ($\fl=3$) and
 \fun{GEN}{caradj}{GEN A, long v, GEN *pt}. In this
 last case, if \var{pt} is not \kbd{NULL}, \kbd{*pt} receives the address of
 the adjoint matrix of $A$ (see \tet{matadjoint}), so both can be obtained at
 once.

Function: chinese
Class: basic
Section: number_theoretical
C-Name: chinese
Prototype: GDG
Help: chinese(x,{y}): x,y being both intmods (or polmods) computes z in the
 same residue classes as x and y.
Description: 
 (gen):gen      chinese1($1)
 (gen, gen):gen chinese($1, $2)
Doc: if $x$ and $y$ are both intmods or both polmods, creates (with the same
 type) a $z$ in the same residue class as $x$ and in the same residue class as
 $y$, if it is possible.
 \bprog
 ? chinese(Mod(1,2), Mod(2,3))
 %1 = Mod(5, 6)
 ? chinese(Mod(x,x^2-1), Mod(x+1,x^2+1))
 %2 = Mod(-1/2*x^2 + x + 1/2, x^4 - 1)
 @eprog\noindent
 This function also allows vector and matrix arguments, in which case the
 operation is recursively applied to each component of the vector or matrix.
 \bprog
 ? chinese([Mod(1,2),Mod(1,3)], [Mod(1,5),Mod(2,7)])
 %3 = [Mod(1, 10), Mod(16, 21)]
 @eprog\noindent
 For polynomial arguments in the same variable, the function is applied to each
 coefficient; if the polynomials have different degrees, the high degree terms
 are copied verbatim in the result, as if the missing high degree terms in the
 polynomial of lowest degree had been \kbd{Mod(0,1)}. Since the latter
 behavior is usually \emph{not} the desired one, we propose to convert the
 polynomials to vectors of the same length first:
 \bprog
  ? P = x+1; Q = x^2+2*x+1;
  ? chinese(P*Mod(1,2), Q*Mod(1,3))
  %4 = Mod(1, 3)*x^2 + Mod(5, 6)*x + Mod(3, 6)
  ? chinese(Vec(P,3)*Mod(1,2), Vec(Q,3)*Mod(1,3))
  %5 = [Mod(1, 6), Mod(5, 6), Mod(4, 6)]
  ? Pol(%)
  %6 = Mod(1, 6)*x^2 + Mod(5, 6)*x + Mod(4, 6)
 @eprog
 
 If $y$ is omitted, and $x$ is a vector, \kbd{chinese} is applied recursively
 to the components of $x$, yielding a residue belonging to the same class as all
 components of $x$.
 
 Finally $\kbd{chinese}(x,x) = x$ regardless of the type of $x$; this allows
 vector arguments to contain other data, so long as they are identical in both
 vectors.
Variant: \fun{GEN}{chinese1}{GEN x} is also available.

Function: clone
Class: gp2c
Description: 
 (small):small:parens             $1
 (int):int                        gclone($1)
 (real):real                      gclone($1)
 (mp):mp                          gclone($1)
 (vecsmall):vecsmall              gclone($1)
 (vec):vec                        gclone($1)
 (pol):pol                        gclone($1)
 (list):list                      gclone($1)
 (closure):closure                gclone($1)
 (genstr):genstr                  gclone($1)
 (gen):gen                        gclone($1)

Function: cmp
Class: basic
Section: operators
C-Name: cmp_universal
Prototype: iGG
Help: cmp(x,y): compare two arbitrary objects x and y (1 if x>y, 0 if x=y, -1
 if x<y). The function is used to implement sets, and has no useful
 mathematical meaning.
Doc: gives the result of a comparison between arbitrary objects $x$ and $y$
 (as $-1$, $0$ or $1$). The underlying order relation is transitive,
 the function returns $0$ if and only if $x~\kbd{===}~y$, and its
 restriction to integers coincides with the customary one. Besides that,
 it has no useful mathematical meaning.
 
 In case all components are equal up to the smallest length of the operands,
 the more complex is considered to be larger. More precisely, the longest is
 the largest; when lengths are equal, we have matrix $>$ vector $>$ scalar.
 For example:
 \bprog
 ? cmp(1, 2)
 %1 = -1
 ? cmp(2, 1)
 %2 = 1
 ? cmp(1, 1.0)   \\ note that 1 == 1.0, but (1===1.0) is false.
 %3 = -1
 ? cmp(x + Pi, [])
 %4 = -1
 @eprog\noindent This function is mostly useful to handle sorted lists or
 vectors of arbitrary objects. For instance, if $v$ is a vector, the
 construction \kbd{vecsort(v, cmp)} is equivalent to \kbd{Set(v)}.

Function: component
Class: basic
Section: conversions
C-Name: compo
Prototype: GL
Help: component(x,n): the n'th component of the internal representation of
 x. For vectors or matrices, it is simpler to use x[]. For list objects such
 as nf, bnf, bnr or ell, it is much easier to use member functions starting
 with ".".
Description: 
 (error,small):gen     err_get_compo($1, $2)
 (gen,small):gen       compo($1,$2)
Doc: extracts the $n^{\text{th}}$-component of $x$. This is to be understood
 as follows: every PARI type has one or two initial \idx{code words}. The
 components are counted, starting at 1, after these code words. In particular
 if $x$ is a vector, this is indeed the $n^{\text{th}}$-component of $x$, if
 $x$ is a matrix, the $n^{\text{th}}$ column, if $x$ is a polynomial, the
 $n^{\text{th}}$ coefficient (i.e.~of degree $n-1$), and for power series,
 the $n^{\text{th}}$ significant coefficient.
 
 For polynomials and power series, one should rather use \tet{polcoeff}, and
 for vectors and matrices, the \kbd{[$\,$]} operator. Namely, if $x$ is a
 vector, then \tet{x[n]} represents the $n^{\text{th}}$ component of $x$. If
 $x$ is a matrix, \tet{x[m,n]} represents the coefficient of row \kbd{m} and
 column \kbd{n} of the matrix, \tet{x[m,]} represents the $m^{\text{th}}$
 \emph{row} of $x$, and \tet{x[,n]} represents the $n^{\text{th}}$
 \emph{column} of $x$.
 
 Using of this function requires detailed knowledge of the structure of the
 different PARI types, and thus it should almost never be used directly.
 Some useful exceptions:
 \bprog
     ? x = 3 + O(3^5);
     ? component(x, 2)
     %2 = 81   \\ p^(p-adic accuracy)
     ? component(x, 1)
     %3 = 3    \\ p
     ? q = Qfb(1,2,3);
     ? component(q, 1)
     %5 = 1
 @eprog

Function: concat
Class: basic
Section: linear_algebra
C-Name: gconcat
Prototype: GDG
Help: concat(x,{y}): concatenation of x and y, which can be scalars, vectors
 or matrices, or lists (in this last case, both x and y have to be lists). If
 y is omitted, x has to be a list or row vector and its elements are
 concatenated.
Description: 
 (mp,mp):vec           gconcat($1, $2)
 (vec,mp):vec          gconcat($1, $2)
 (mp,vec):vec          gconcat($1, $2)
 (vec,vec):vec         gconcat($1, $2)
 (list,list):list      gconcat($1, $2)
 (genstr,gen):genstr   gconcat($1, $2)
 (gen,genstr):genstr   gconcat($1, $2)
 (gen):gen             gconcat1($1)
 (gen,):gen            gconcat1($1)
 (gen,gen):gen         gconcat($1, $2)
Doc: concatenation of $x$ and $y$. If $x$ or $y$ is
 not a vector or matrix, it is considered as a one-dimensional vector. All
 types are allowed for $x$ and $y$, but the sizes must be compatible. Note
 that matrices are concatenated horizontally, i.e.~the number of rows stays
 the same. Using transpositions, one can concatenate them vertically,
 but it is often simpler to use \tet{matconcat}.
 \bprog
 ? x = matid(2); y = 2*matid(2);
 ? concat(x,y)
 %2 =
 [1 0 2 0]
 
 [0 1 0 2]
 ? concat(x~,y~)~
 %3 =
 [1 0]
 
 [0 1]
 
 [2 0]
 
 [0 2]
 ? matconcat([x;y])
 %4 =
 [1 0]
 
 [0 1]
 
 [2 0]
 
 [0 2]
 @eprog\noindent
 To concatenate vectors sideways (i.e.~to obtain a two-row or two-column
 matrix), use \tet{Mat} instead, or \tet{matconcat}:
 \bprog
 ? x = [1,2];
 ? y = [3,4];
 ? concat(x,y)
 %3 = [1, 2, 3, 4]
 
 ? Mat([x,y]~)
 %4 =
 [1 2]
 
 [3 4]
 ? matconcat([x;y])
 %5 =
 [1 2]
 
 [3 4]
 @eprog
 Concatenating a row vector to a matrix having the same number of columns will
 add the row to the matrix (top row if the vector is $x$, i.e.~comes first, and
 bottom row otherwise).
 
 The empty matrix \kbd{[;]} is considered to have a number of rows compatible
 with any operation, in particular concatenation. (Note that this is
 \emph{not} the case for empty vectors \kbd{[~]} or \kbd{[~]\til}.)
 
 If $y$ is omitted, $x$ has to be a row vector or a list, in which case its
 elements are concatenated, from left to right, using the above rules.
 \bprog
 ? concat([1,2], [3,4])
 %1 = [1, 2, 3, 4]
 ? a = [[1,2]~, [3,4]~]; concat(a)
 %2 =
 [1 3]
 
 [2 4]
 
 ? concat([1,2; 3,4], [5,6]~)
 %3 =
 [1 2 5]
 
 [3 4 6]
 ? concat([%, [7,8]~, [1,2,3,4]])
 %5 =
 [1 2 5 7]
 
 [3 4 6 8]
 
 [1 2 3 4]
 @eprog
Variant: \fun{GEN}{gconcat1}{GEN x} is a shortcut for \kbd{gconcat(x,NULL)}.

Function: conj
Class: basic
Section: conversions
C-Name: gconj
Prototype: G
Help: conj(x): the algebraic conjugate of x.
Doc: 
 conjugate of $x$. The meaning of this
 is clear, except that for real quadratic numbers, it means conjugation in the
 real quadratic field. This function has no effect on integers, reals,
 intmods, fractions or $p$-adics. The only forbidden type is polmod
 (see \kbd{conjvec} for this).

Function: conjvec
Class: basic
Section: conversions
C-Name: conjvec
Prototype: Gp
Help: conjvec(z): conjugate vector of the algebraic number z.
Doc: 
 conjugate vector representation of $z$. If $z$ is a
 polmod, equal to \kbd{Mod}$(a,T)$, this gives a vector of length
 $\text{degree}(T)$ containing:
 
 \item the complex embeddings of $z$ if $T$ has rational coefficients,
 i.e.~the $a(r[i])$ where $r = \kbd{polroots}(T)$;
 
 \item the conjugates of $z$ if $T$ has some intmod coefficients;
 
 \noindent if $z$ is a finite field element, the result is the vector of
 conjugates $[z,z^p,z^{p^2},\ldots,z^{p^{n-1}}]$ where $n=\text{degree}(T)$.
 
 \noindent If $z$ is an integer or a rational number, the result is~$z$. If
 $z$ is a (row or column) vector, the result is a matrix whose columns are
 the conjugate vectors of the individual elements of $z$.

Function: content
Class: basic
Section: number_theoretical
C-Name: content
Prototype: G
Help: content(x): gcd of all the components of x, when this makes sense.
Doc: computes the gcd of all the coefficients of $x$,
 when this gcd makes sense. This is the natural definition
 if $x$ is a polynomial (and by extension a power series) or a
 vector/matrix. This is in general a weaker notion than the \emph{ideal}
 generated by the coefficients:
 \bprog
 ? content(2*x+y)
 %1 = 1            \\ = gcd(2,y) over Q[y]
 @eprog
 
 If $x$ is a scalar, this simply returns the absolute value of $x$ if $x$ is
 rational (\typ{INT} or \typ{FRAC}), and either $1$ (inexact input) or $x$
 (exact input) otherwise; the result should be identical to \kbd{gcd(x, 0)}.
 
 The content of a rational function is the ratio of the contents of the
 numerator and the denominator. In recursive structures, if a
 matrix or vector \emph{coefficient} $x$ appears, the gcd is taken
 not with $x$, but with its content:
 \bprog
 ? content([ [2], 4*matid(3) ])
 %1 = 2
 @eprog\noindent The content of a \typ{VECSMALL} is computed assuming the
 entries are signed integers.

Function: contfrac
Class: basic
Section: number_theoretical
C-Name: contfrac0
Prototype: GDGD0,L,
Help: contfrac(x,{b},{nmax}): continued fraction expansion of x (x
 rational,real or rational function). b and nmax are both optional, where b
 is the vector of numerators of the continued fraction, and nmax is a bound
 for the number of terms in the continued fraction expansion.
Doc: returns the row vector whose components are the partial quotients of the
 \idx{continued fraction} expansion of $x$. In other words, a result
 $[a_0,\dots,a_n]$ means that $x \approx a_0+1/(a_1+\dots+1/a_n)$. The
 output is normalized so that $a_n \neq 1$ (unless we also have $n = 0$).
 
 The number of partial quotients $n+1$ is limited by \kbd{nmax}. If
 \kbd{nmax} is omitted, the expansion stops at the last significant partial
 quotient.
 \bprog
 ? \p19
   realprecision = 19 significant digits
 ? contfrac(Pi)
 %1 = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2]
 ? contfrac(Pi,, 3)  \\ n = 2
 %2 = [3, 7, 15]
 @eprog\noindent
 $x$ can also be a rational function or a power series.
 
 If a vector $b$ is supplied, the numerators are equal to the coefficients
 of $b$, instead of all equal to $1$ as above; more precisely, $x \approx
 (1/b_0)(a_0+b_1/(a_1+\dots+b_n/a_n))$; for a numerical continued fraction
 ($x$ real), the $a_i$ are integers, as large as possible; if $x$ is a
 rational function, they are polynomials with $\deg a_i = \deg b_i + 1$.
 The length of the result is then equal to the length of $b$, unless the next
 partial quotient cannot be reliably computed, in which case the expansion
 stops. This happens when a partial remainder is equal to zero (or too small
 compared to the available significant digits for $x$ a \typ{REAL}).
 
 A direct implementation of the numerical continued fraction
 \kbd{contfrac(x,b)} described above would be
 \bprog
 \\ "greedy" generalized continued fraction
 cf(x, b) =
 { my( a= vector(#b), t );
 
   x *= b[1];
   for (i = 1, #b,
     a[i] = floor(x);
     t = x - a[i]; if (!t || i == #b, break);
     x = b[i+1] / t;
   ); a;
 }
 @eprog\noindent There is some degree of freedom when choosing the $a_i$; the
 program above can easily be modified to derive variants of the standard
 algorithm. In the same vein, although no builtin
 function implements the related \idx{Engel expansion} (a special kind of
 \idx{Egyptian fraction} decomposition: $x = 1/a_1 + 1/(a_1a_2) + \dots$ ),
 it can be obtained as follows:
 \bprog
 \\ n terms of the Engel expansion of x
 engel(x, n = 10) =
 { my( u = x, a = vector(n) );
   for (k = 1, n,
     a[k] = ceil(1/u);
     u = u*a[k] - 1;
     if (!u, break);
   ); a
 }
 @eprog
 
 \misctitle{Obsolete hack} (don't use this): if $b$ is an integer, \var{nmax}
 is ignored and the command is understood as \kbd{contfrac($x,, b$)}.
Variant: Also available are \fun{GEN}{gboundcf}{GEN x, long nmax},
 \fun{GEN}{gcf}{GEN x} and \fun{GEN}{gcf2}{GEN b, GEN x}.

Function: contfraceval
Class: basic
Section: sums
C-Name: contfraceval
Prototype: GGD-1,L,
Help: contfraceval(CF,t,{lim=-1}): given a continued fraction CF from
 contfracinit, evaluate the first lim terms of the continued fraction at t
 (all terms if lim is negative or omitted).
Doc: Given a continued fraction \kbd{CF} output by \kbd{contfracinit}, evaluate
 the first \kbd{lim} terms of the continued fraction at \kbd{t} (all
 terms if \kbd{lim} is negative or omitted; if positive, \kbd{lim} must be
 less than or equal to the length of \kbd{CF}.

Function: contfracinit
Class: basic
Section: sums
C-Name: contfracinit
Prototype: GD-1,L,
Help: contfracinit(M,{lim = -1}): given M representing the power
 series S = sum_{n>=0} M[n+1]z^n, transform it into a continued fraction
 suitable for evaluation.
Doc: Given $M$ representing the power series $S=\sum_{n\ge0} M[n+1]z^n$,
 transform it into a continued fraction; restrict to $n\leq \kbd{lim}$
 if latter is non-negative. $M$ can be a vector, a power
 series, a polynomial, or a rational function.
 The result is a 2-component vector $[A,B]$ such that
 $S = M[1] / (1+A[1]z+B[1]z^2/(1+A[2]z+B[2]z^2/(1+...1/(1+A[lim/2]z))))$.
 Does not work if any coefficient of $M$ vanishes, nor for series for
 which certain partial denominators vanish.

Function: contfracpnqn
Class: basic
Section: number_theoretical
C-Name: contfracpnqn
Prototype: GD-1,L,
Help: contfracpnqn(x, {n=-1}): [p_n,p_{n-1}; q_n,q_{n-1}] corresponding to the
 continued fraction x. If n >= 0 is present, returns all convergents from
 p_0/q_0 up to p_n/q_n.
Doc: when $x$ is a vector or a one-row matrix, $x$
 is considered as the list of partial quotients $[a_0,a_1,\dots,a_n]$ of a
 rational number, and the result is the 2 by 2 matrix
 $[p_n,p_{n-1};q_n,q_{n-1}]$ in the standard notation of continued fractions,
 so $p_n/q_n=a_0+1/(a_1+\dots+1/a_n)$. If $x$ is a matrix with two rows
 $[b_0,b_1,\dots,b_n]$ and $[a_0,a_1,\dots,a_n]$, this is then considered as a
 generalized continued fraction and we have similarly
 $p_n/q_n=(1/b_0)(a_0+b_1/(a_1+\dots+b_n/a_n))$. Note that in this case one
 usually has $b_0=1$.
 
 If $n \geq 0$ is present, returns all convergents from $p_0/q_0$ up to
 $p_n/q_n$. (All convergents if $x$ is too small to compute the $n+1$
 requested convergents.)
 \bprog
 ? a=contfrac(Pi,20)
 %1 = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2]
 ? contfracpnqn(a,3)
 %2 =
 [3 22 333 355]
 
 [1  7 106 113]
 
 ? contfracpnqn(a,7)
 %3 =
 [3 22 333 355 103993 104348 208341 312689]
 
 [1  7 106 113  33102  33215  66317  99532]
 @eprog
Variant: also available is \fun{GEN}{pnqn}{GEN x} for $n = -1$.

Function: copy
Class: gp2c
Description: 
 (small):small:parens             $1
 (int):int                        icopy($1)
 (real):real                      gcopy($1)
 (mp):mp                          gcopy($1)
 (vecsmall):vecsmall              gcopy($1)
 (vec):vec                        gcopy($1)
 (pol):pol                        gcopy($1)
 (gen):gen                        gcopy($1)

Function: core
Class: basic
Section: number_theoretical
C-Name: core0
Prototype: GD0,L,
Help: core(n,{flag=0}): unique squarefree integer d
 dividing n such that n/d is a square. If (optional) flag is non-null, output
 the two-component row vector [d,f], where d is the unique squarefree integer
 dividing n such that n/d=f^2 is a square.
Doc: if $n$ is an integer written as
 $n=df^2$ with $d$ squarefree, returns $d$. If $\fl$ is non-zero,
 returns the two-element row vector $[d,f]$. By convention, we write $0 = 0
 \times 1^2$, so \kbd{core(0, 1)} returns $[0,1]$.
Variant: Also available are \fun{GEN}{core}{GEN n} ($\fl = 0$) and
 \fun{GEN}{core2}{GEN n} ($\fl = 1$)

Function: coredisc
Class: basic
Section: number_theoretical
C-Name: coredisc0
Prototype: GD0,L,
Help: coredisc(n,{flag=0}): discriminant of the quadratic field Q(sqrt(n)).
 If (optional) flag is non-null, output a two-component row vector [d,f],
 where d is the discriminant of the quadratic field Q(sqrt(n)) and n=df^2. f
 may be a half integer.
Doc: a \emph{fundamental discriminant} is an integer of the form $t\equiv 1
 \mod 4$ or $4t \equiv 8,12 \mod 16$, with $t$ squarefree (i.e.~$1$ or the
 discriminant of a quadratic number field). Given a non-zero integer
 $n$, this routine returns the (unique) fundamental discriminant $d$
 such that $n=df^2$, $f$ a positive rational number. If $\fl$ is non-zero,
 returns the two-element row vector $[d,f]$. If $n$ is congruent to
 0 or 1 modulo 4, $f$ is an integer, and a half-integer otherwise.
 
 By convention, \kbd{coredisc(0, 1))} returns $[0,1]$.
 
 Note that \tet{quaddisc}$(n)$ returns the same value as \kbd{coredisc}$(n)$,
 and also works with rational inputs $n\in\Q^*$.
Variant: Also available are \fun{GEN}{coredisc}{GEN n} ($\fl = 0$) and
 \fun{GEN}{coredisc2}{GEN n} ($\fl = 1$)

Function: cos
Class: basic
Section: transcendental
C-Name: gcos
Prototype: Gp
Help: cos(x): cosine of x.
Doc: cosine of $x$.

Function: cosh
Class: basic
Section: transcendental
C-Name: gcosh
Prototype: Gp
Help: cosh(x): hyperbolic cosine of x.
Doc: hyperbolic cosine of $x$.

Function: cotan
Class: basic
Section: transcendental
C-Name: gcotan
Prototype: Gp
Help: cotan(x): cotangent of x.
Doc: cotangent of $x$.

Function: cotanh
Class: basic
Section: transcendental
C-Name: gcotanh
Prototype: Gp
Help: cotanh(x): hyperbolic cotangent of x.
Doc: hyperbolic cotangent of $x$.

Function: dbg_down
Class: gp
Section: programming/control
C-Name: dbg_down
Prototype: vD1,L,
Help: dbg_down({n=1}): (break loop) go down n frames. Cancel a previous dbg_up.
Doc: (In the break loop) go down n frames. This allows to cancel a previous call to
 \kbd{dbg\_up}.

Function: dbg_err
Class: gp
Section: programming/control
C-Name: dbg_err
Prototype: 
Help: dbg_err(): (break loop) return the error data of the current error, if any.
Doc: In the break loop, return the error data of the current error, if any.
 See \tet{iferr} for details about error data.  Compare:
 \bprog
 ? iferr(1/(Mod(2,12019)^(6!)-1),E,Vec(E))
 %1 = ["e_INV", "Fp_inv", Mod(119, 12019)]
 ? 1/(Mod(2,12019)^(6!)-1)
   ***   at top-level: 1/(Mod(2,12019)^(6!)-
   ***                  ^--------------------
   *** _/_: impossible inverse in Fp_inv: Mod(119, 12019).
   ***   Break loop: type 'break' to go back to GP prompt
 break> Vec(dbg_err())
 ["e_INV", "Fp_inv", Mod(119, 12019)]
 @eprog

Function: dbg_up
Class: gp
Section: programming/control
C-Name: dbg_up
Prototype: vD1,L,
Help: dbg_up({n=1}): (break loop) go up n frames. Allow to inspect data of the parent function.
Doc: (In the break loop) go up n frames. This allows to inspect data of the
 parent function. To cancel a \tet{dbg_up} call, use \tet{dbg_down}

Function: dbg_x
Class: basic
Section: programming/control
C-Name: dbgGEN
Prototype: vGD-1,L,
Help: dbg_x(A,{n}): print inner structure of A, complete if n is omitted, up to
 level n otherwise. Intended for debugging.
Doc: Print the inner structure of \kbd{A}, complete if \kbd{n} is omitted, up
 to level \kbd{n} otherwise. This is useful for debugging. This is similar to
 \b{x} but does not require \kbd{A} to be an history entry. In particular,
 it can be used in the break loop.

Function: default
Class: basic
Section: programming/specific
C-Name: default0
Prototype: DrDs
Help: default({key},{val}): returns the current value of the
 default key. If val is present, set opt to val first. If no argument is
 given, print a list of all defaults as well as their values.
Description: 
 ("realprecision"):small:prec              getrealprecision()
 ("realprecision",small):small:prec        setrealprecision($2, &$prec)
 ("seriesprecision"):small                 precdl
 ("seriesprecision",small):small:parens    precdl = $2
 ("debug"):small                           DEBUGLEVEL
 ("debug",small):small:parens              DEBUGLEVEL = $2
 ("debugmem"):small                        DEBUGMEM
 ("debugmem",small):small:parens           DEBUGMEM = $2
 ("debugfiles"):small                      DEBUGFILES
 ("debugfiles",small):small:parens         DEBUGFILES = $2
 ("factor_add_primes"):small               factor_add_primes
 ("factor_add_primes",small):small         factor_add_primes = $2
 ("factor_proven"):small                   factor_proven
 ("factor_proven",small):small             factor_proven = $2
 ("new_galois_format"):small               new_galois_format
 ("new_galois_format",small):small         new_galois_format = $2
Doc: returns the default corresponding to keyword \var{key}. If \var{val} is
 present, sets the default to \var{val} first (which is subject to string
 expansion first). Typing \kbd{default()} (or \b{d}) yields the complete
 default list as well as their current values. See \secref{se:defaults} for an
 introduction to GP defaults, \secref{se:gp_defaults} for a
 list of available defaults, and \secref{se:meta} for some shortcut
 alternatives. Note that the shortcuts are meant for interactive use and
 usually display more information than \kbd{default}.

Function: denominator
Class: basic
Section: conversions
C-Name: denom
Prototype: G
Help: denominator(x): denominator of x (or lowest common denominator in case
 of an array).
Doc: 
 denominator of $x$. The meaning of this
 is clear when $x$ is a rational number or function. If $x$ is an integer
 or a polynomial, it is treated as a rational number or function,
 respectively, and the result is equal to $1$. For polynomials, you
 probably want to use
 \bprog
 denominator( content(x) )
 @eprog\noindent
 instead. As for modular objects, \typ{INTMOD} and \typ{PADIC} have
 denominator $1$, and the denominator of a \typ{POLMOD} is the denominator
 of its (minimal degree) polynomial representative.
 
 If $x$ is a recursive structure, for instance a vector or matrix, the lcm
 of the denominators of its components (a common denominator) is computed.
 This also applies for \typ{COMPLEX}s and \typ{QUAD}s.
 
 \misctitle{Warning} Multivariate objects are created according to variable
 priorities, with possibly surprising side effects ($x/y$ is a polynomial, but
 $y/x$ is a rational function). See \secref{se:priority}.

Function: deriv
Class: basic
Section: polynomials
C-Name: deriv
Prototype: GDn
Help: deriv(x,{v}): derivative of x with respect to v, or to the main
 variable of x if v is omitted.
Doc: 
 derivative of $x$ with respect to the main
 variable if $v$ is omitted, and with respect to $v$ otherwise. The derivative
 of a scalar type is zero, and the derivative of a vector or matrix is done
 componentwise. One can use $x'$ as a shortcut if the derivative is with
 respect to the main variable of $x$.
 
 By definition, the main variable of a \typ{POLMOD} is the main variable among
 the coefficients from its two polynomial components (representative and
 modulus); in other words, assuming a polmod represents an element of
 $R[X]/(T(X))$, the variable $X$ is a mute variable and the derivative is
 taken with respect to the main variable used in the base ring $R$.

Function: derivnum
Class: basic
Section: sums
C-Name: derivnum0
Prototype: V=GEp
Help: derivnum(X=a,expr): numerical derivation of expr with respect to
 X at X = a.
Wrapper: (,Gp)
Description: 
  (gen,gen):gen:prec derivnum(${2 cookie}, ${2 wrapper}, $1, $prec)
Doc: numerical derivation of \var{expr} with respect to $X$ at $X=a$.
 
 \bprog
 ? derivnum(x=0,sin(exp(x))) - cos(1)
 %1 = -1.262177448 E-29
 @eprog
 A clumsier approach, which would not work in library mode, is
 \bprog
 ? f(x) = sin(exp(x))
 ? f'(0) - cos(1)
 %1 = -1.262177448 E-29
 @eprog
 When $a$ is a power series, compute \kbd{derivnum(t=a,f)} as $f'(a) =
 (f(a))'/a'$.
 
 \synt{derivnum}{void *E, GEN (*eval)(void*,GEN), GEN a, long prec}. Also
 available is \fun{GEN}{derivfun}{void *E, GEN (*eval)(void *, GEN), GEN a, long prec}, which also allows power series for $a$.

Function: diffop
Class: basic
Section: polynomials
C-Name: diffop0
Prototype: GGGD1,L,
Help: diffop(x,v,d,{n=1}): apply the differential operator D to x, where D is defined
 by D(v[i])=d[i], where v is a vector of variable names. D is 0 for variables
 outside of v unless they appear as modulus of a POLMOD. If the optional parameter n
 is given, return D^n(x) instead.
Description: 
 (gen,gen,gen,?1):gen    diffop($1, $2, $3)
 (gen,gen,gen,small):gen diffop0($1, $2, $3, $4)
Doc: 
 Let $v$ be a vector of variables, and $d$ a vector of the same length,
 return the image of $x$ by the $n$-power ($1$ if n is not given) of the differential
 operator $D$ that assumes the value \kbd{d[i]} on the variable \kbd{v[i]}.
 The value of $D$ on a scalar type is zero, and $D$ applies componentwise to a vector
 or matrix. When applied to a \typ{POLMOD}, if no value is provided for the variable
 of the modulus, such value is derived using the implicit function theorem.
 
 Some examples:
 This function can be used to differentiate formal expressions:
 If $E=\exp(X^2)$ then we have $E'=2*X*E$. We can derivate $X*exp(X^2)$ as follow:
 \bprog
 ? diffop(E*X,[X,E],[1,2*X*E])
 %1 = (2*X^2 + 1)*E
 @eprog
 Let \kbd{Sin} and \kbd{Cos} be two function such that $\kbd{Sin}^2+\kbd{Cos}^2=1$
 and $\kbd{Cos}'=-\kbd{Sin}$. We can differentiate $\kbd{Sin}/\kbd{Cos}$ as follow,
 PARI inferring the value of $\kbd{Sin}'$ from the equation:
 \bprog
 ? diffop(Mod('Sin/'Cos,'Sin^2+'Cos^2-1),['Cos],[-'Sin])
 %1 = Mod(1/Cos^2, Sin^2 + (Cos^2 - 1))
 
 @eprog
 Compute the Bell polynomials (both complete and partial) via the Faa di Bruno formula:
 \bprog
 Bell(k,n=-1)=
 {
   my(var(i)=eval(Str("X",i)));
   my(x,v,dv);
   v=vector(k,i,if(i==1,'E,var(i-1)));
   dv=vector(k,i,if(i==1,'X*var(1)*'E,var(i)));
   x=diffop('E,v,dv,k)/'E;
   if(n<0,subst(x,'X,1),polcoeff(x,n,'X))
 }
 @eprog
Variant: 
 For $n=1$, the function \fun{GEN}{diffop}{GEN x, GEN v, GEN d} is also available.

Function: digits
Class: basic
Section: conversions
C-Name: digits
Prototype: GDG
Help: digits(x,{b=10}): gives the vector formed by the digits of x in base b (x and b
 integers).
Doc: 
 outputs the vector of the digits of $|x|$ in base $b$, where $x$ and $b$ are
 integers ($b = 10$ by default). See \kbd{fromdigits} for the reverse
 operation.
 
 \bprog
 ? digits(123)
 %1 = [1, 2, 3, 0]
 
 ? digits(10, 2) \\ base 2
 %2 = [1, 0, 1, 0]
 @eprog\noindent By convention, $0$ has no digits:
 \bprog
 ? digits(0)
 %3 = []
 @eprog

Function: dilog
Class: basic
Section: transcendental
C-Name: dilog
Prototype: Gp
Help: dilog(x): dilogarithm of x.
Doc: principal branch of the dilogarithm of $x$,
 i.e.~analytic continuation of the power series $\log_2(x)=\sum_{n\ge1}x^n/n^2$.

Function: dirdiv
Class: basic
Section: number_theoretical
C-Name: dirdiv
Prototype: GG
Help: dirdiv(x,y): division of the Dirichlet series x by the Dirichlet
 series y.
Doc: $x$ and $y$ being vectors of perhaps different
 lengths but with $y[1]\neq 0$ considered as \idx{Dirichlet series}, computes
 the quotient of $x$ by $y$, again as a vector.

Function: direuler
Class: basic
Section: number_theoretical
C-Name: direuler0
Prototype: V=GGEDG
Help: direuler(p=a,b,expr,{c}): Dirichlet Euler product of expression expr
 from p=a to p=b, limited to b terms. Expr should be a polynomial or rational
 function in p and X, and X is understood to mean p^(-s). If c is present,
 output only the first c terms.
Wrapper: (,,G)
Description: 
  (gen,gen,closure,?gen):gen direuler(${3 cookie}, ${3 wrapper}, $1, $2, $4)
Doc: computes the \idx{Dirichlet series} attached to the
 \idx{Euler product} of expression \var{expr} as $p$ ranges through the primes
 from $a$
 to $b$. \var{expr} must be a polynomial or rational function in another
 variable than $p$ (say $X$) and $\var{expr}(X)$ is understood as the local
 factor $\var{expr}(p^{-s})$.
 
 The series is output as a vector of coefficients. If $c$ is omitted, output
 the first $b$ coefficients of the series; otherwise, output the first $c$
 coefficients. The following command computes the \teb{sigma} function,
 attached to $\zeta(s)\zeta(s-1)$:
 \bprog
 ? direuler(p=2, 10, 1/((1-X)*(1-p*X)))
 %1 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18]
 
 ? direuler(p=2, 10, 1/((1-X)*(1-p*X)), 5) \\ fewer terms
 %2 = [1, 3, 4, 7, 6]
 @eprog\noindent Setting $c < b$ is useless (the same effect would be
 achieved by setting $b = c)$. If $c > b$, the computed coefficients are
 ``missing'' Euler factors:
 \bprog
 ? direuler(p=2, 10, 1/((1-X)*(1-p*X)), 15) \\ more terms, no longer = sigma !
 %3 = [1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 0, 28, 0, 24, 24]
 @eprog
 
 \synt{direuler}{void *E, GEN (*eval)(void*,GEN), GEN a, GEN b}

Function: dirmul
Class: basic
Section: number_theoretical
C-Name: dirmul
Prototype: GG
Help: dirmul(x,y): multiplication of the Dirichlet series x by the Dirichlet
 series y.
Doc: $x$ and $y$ being vectors of perhaps different lengths representing
 the \idx{Dirichlet series} $\sum_n x_n n^{-s}$ and $\sum_n y_n n^{-s}$,
 computes the product of $x$ by $y$, again as a vector.
 \bprog
 ? dirmul(vector(10,n,1), vector(10,n,moebius(n)))
 %1 = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
 @eprog\noindent
 The product
 length is the minimum of $\kbd{\#}x\kbd{*}v(y)$ and $\kbd{\#}y\kbd{*}v(x)$,
 where $v(x)$ is the index of the first non-zero coefficient.
 \bprog
 ? dirmul([0,1], [0,1]);
 %2 = [0, 0, 0, 1]
 @eprog

Function: dirzetak
Class: basic
Section: number_fields
C-Name: dirzetak
Prototype: GG
Help: dirzetak(nf,b): Dirichlet series of the Dedekind zeta function of the
 number field nf up to the bound b-1.
Doc: gives as a vector the first $b$
 coefficients of the \idx{Dedekind} zeta function of the number field $\var{nf}$
 considered as a \idx{Dirichlet series}.

Function: divisors
Class: basic
Section: number_theoretical
C-Name: divisors
Prototype: G
Help: divisors(x): gives a vector formed by the divisors of x in increasing
 order.
Description: 
 (gen):vec        divisors($1)
Doc: creates a row vector whose components are the
 divisors of $x$. The factorization of $x$ (as output by \tet{factor}) can
 be used instead.
 
 By definition, these divisors are the products of the irreducible
 factors of $n$, as produced by \kbd{factor(n)}, raised to appropriate
 powers (no negative exponent may occur in the factorization). If $n$ is
 an integer, they are the positive divisors, in increasing order.

Function: divrem
Class: basic
Section: operators
C-Name: divrem
Prototype: GGDn
Help: divrem(x,y,{v}): euclidean division of x by y giving as a
 2-dimensional column vector the quotient and the remainder, with respect to
 v (to main variable if v is omitted).
Doc: creates a column vector with two components, the first being the Euclidean
 quotient (\kbd{$x$ \bs\ $y$}), the second the Euclidean remainder
 (\kbd{$x$ - ($x$\bs$y$)*$y$}), of the division of $x$ by $y$. This avoids the
 need to do two divisions if one needs both the quotient and the remainder.
 If $v$ is present, and $x$, $y$ are multivariate
 polynomials, divide with respect to the variable $v$.
 
 Beware that \kbd{divrem($x$,$y$)[2]} is in general not the same as
 \kbd{$x$ \% $y$}; no GP operator corresponds to it:
 \bprog
 ? divrem(1/2, 3)[2]
 %1 = 1/2
 ? (1/2) % 3
 %2 = 2
 ? divrem(Mod(2,9), 3)[2]
  ***   at top-level: divrem(Mod(2,9),3)[2
  ***                 ^--------------------
  ***   forbidden division t_INTMOD \ t_INT.
 ? Mod(2,9) % 6
 %3 = Mod(2,3)
 @eprog
Variant: Also available is \fun{GEN}{gdiventres}{GEN x, GEN y} when $v$ is
 not needed.

Function: eint1
Class: basic
Section: transcendental
C-Name: veceint1
Prototype: GDGp
Help: eint1(x,{n}): exponential integral E1(x). If n is present and x > 0,
 computes the vector of the first n values of the exponential integral E1(n x).
Doc: exponential integral $\int_x^\infty \dfrac{e^{-t}}{t}\,dt =
 \kbd{incgam}(0, x)$, where the latter expression extends the function
 definition from real $x > 0$ to all complex $x \neq 0$.
 
 If $n$ is present, we must have $x > 0$; the function returns the
 $n$-dimensional vector $[\kbd{eint1}(x),\dots,\kbd{eint1}(nx)]$. Contrary to
 other transcendental functions, and to the default case ($n$ omitted), the
 values are correct up to a bounded \emph{absolute}, rather than relative,
 error $10^{-n}$, where $n$ is \kbd{precision}$(x)$ if $x$ is a \typ{REAL}
 and defaults to \kbd{realprecision} otherwise. (In the most important
 application, to the computation of $L$-functions via approximate functional
 equations, those values appear as weights in long sums and small individual
 relative errors are less useful than controlling the absolute error.) This is
 faster than repeatedly calling \kbd{eint1($i$ * x)}, but less precise.
Variant: Also available is \fun{GEN}{eint1}{GEN x, long prec}.

Function: ellL1
Class: basic
Section: elliptic_curves
C-Name: ellL1_bitprec
Prototype: GD0,L,b
Help: ellL1(e, {r = 0}): returns the value at s=1 of the derivative of order r of the L-function of the elliptic curve e.
Doc: returns the value at $s=1$ of the derivative of order $r$ of the
 $L$-function of the elliptic curve $e$.
 \bprog
 ? e = ellinit("11a1"); \\ order of vanishing is 0
 ? ellL1(e)
 %2 = 0.2538418608559106843377589233
 ? e = ellinit("389a1");  \\ order of vanishing is 2
 ? ellL1(e)
 %4 = -5.384067311837218089235032414 E-29
 ? ellL1(e, 1)
 %5 = 0
 ? ellL1(e, 2)
 %6 = 1.518633000576853540460385214
 @eprog\noindent
 The main use of this function, after computing at \emph{low} accuracy the
 order of vanishing using \tet{ellanalyticrank}, is to compute the
 leading term at \emph{high} accuracy to check (or use) the Birch and
 Swinnerton-Dyer conjecture:
 \bprog
 ? \p18
   realprecision = 18 significant digits
 ? e = ellinit("5077a1"); ellanalyticrank(e)
 time = 8 ms.
 %1 = [3, 10.3910994007158041]
 ? \p200
   realprecision = 202 significant digits (200 digits displayed)
 ? ellL1(e, 3)
 time = 104 ms.
 %3 = 10.3910994007158041387518505103609170697263563756570092797@com$[\dots]$
 @eprog

Function: elladd
Class: basic
Section: elliptic_curves
C-Name: elladd
Prototype: GGG
Help: elladd(E,z1,z2): sum of the points z1 and z2 on elliptic curve E.
Doc: 
 sum of the points $z1$ and $z2$ on the
 elliptic curve corresponding to $E$.

Function: ellak
Class: basic
Section: elliptic_curves
C-Name: akell
Prototype: GG
Help: ellak(E,n): computes the n-th Fourier coefficient of the L-function of
 the elliptic curve E (assumed E is an integral model).
Doc: 
 computes the coefficient $a_n$ of the $L$-function of the elliptic curve
 $E/\Q$, i.e.~coefficients of a newform of weight 2 by the modularity theorem
 (\idx{Taniyama-Shimura-Weil conjecture}). $E$ must be an \kbd{ell} structure
 over $\Q$ as output by \kbd{ellinit}. $E$ must be given by an integral model,
 not necessarily minimal, although a minimal model will make the function
 faster.
 \bprog
 ? E = ellinit([0,1]);
 ? ellak(E, 10)
 %2 = 0
 ? e = ellinit([5^4,5^6]); \\ not minimal at 5
 ? ellak(e, 5) \\ wasteful but works
 %3 = -3
 ? E = ellminimalmodel(e); \\ now minimal
 ? ellak(E, 5)
 %5 = -3
 @eprog\noindent If the model is not minimal at a number of bad primes, then
 the function will be slower on those $n$ divisible by the bad primes.
 The speed should be comparable for other $n$:
 \bprog
 ? for(i=1,10^6, ellak(E,5))
 time = 820 ms.
 ? for(i=1,10^6, ellak(e,5)) \\ 5 is bad, markedly slower
 time = 1,249 ms.
 
 ? for(i=1,10^5,ellak(E,5*i))
 time = 977 ms.
 ? for(i=1,10^5,ellak(e,5*i)) \\ still slower but not so much on average
 time = 1,008 ms.
 @eprog

Function: ellan
Class: basic
Section: elliptic_curves
C-Name: ellan
Prototype: GL
Help: ellan(E,n): computes the first n Fourier coefficients of the
 L-function of the elliptic curve E defined over a number field
 (n<2^24 on a 32-bit machine).
Doc: computes the vector of the first $n$ Fourier coefficients $a_k$
 corresponding to the elliptic curve $E$ defined over a number field.
 If $E$ is defined over $\Q$, the curve may be given by an
 arbitrary model, not necessarily minimal,
 although a minimal model will make the function faster. Over a more general
 number field, the model must be locally minimal at all primes above $2$
 and $3$.
Variant: Also available is \fun{GEN}{ellanQ_zv}{GEN e, long n}, which
 returns a \typ{VECSMALL} instead of a \typ{VEC}, saving on memory.

Function: ellanalyticrank
Class: basic
Section: elliptic_curves
C-Name: ellanalyticrank_bitprec
Prototype: GDGb
Help: ellanalyticrank(e, {eps}): returns the order of vanishing at s=1
 of the L-function of the elliptic curve e and the value of the first
 non-zero derivative. To determine this order, it is assumed that any
 value less than eps is zero. If no value of eps is given, a value of
 half the current precision is used.
Doc: returns the order of vanishing at $s=1$ of the $L$-function of the
 elliptic curve $e$ and the value of the first non-zero derivative. To
 determine this order, it is assumed that any value less than \kbd{eps} is
 zero. If no value of \kbd{eps} is given, a value of half the current
 precision is used.
 \bprog
 ? e = ellinit("11a1"); \\ rank 0
 ? ellanalyticrank(e)
 %2 = [0, 0.2538418608559106843377589233]
 ? e = ellinit("37a1"); \\ rank 1
 ? ellanalyticrank(e)
 %4 = [1, 0.3059997738340523018204836835]
 ? e = ellinit("389a1"); \\ rank 2
 ? ellanalyticrank(e)
 %6 = [2, 1.518633000576853540460385214]
 ? e = ellinit("5077a1"); \\ rank 3
 ? ellanalyticrank(e)
 %8 = [3, 10.39109940071580413875185035]
 @eprog

Function: ellap
Class: basic
Section: elliptic_curves
C-Name: ellap
Prototype: GDG
Help: ellap(E,{p}): computes the trace of Frobenius a_p for the elliptic
 curve E, defined over Q or a finite field.
Doc: 
 Let $E$ be an \kbd{ell} structure as output by \kbd{ellinit}, defined over
 a number field or a finite field $\F_q$. The argument $p$ is best left
 omitted if the curve is defined over a finite field, and must be a prime
 number or a maximal ideal otherwise. This function computes the trace of
 Frobenius $t$ for the elliptic curve $E$, defined by the equation $\#E(\F_q)
 = q+1 - t$ (for primes of good reduction).
 
 When the characteristic of the finite field is large, the availability of
 the \kbd{seadata} package will speed the computation.
 
 If the curve is defined over $\Q$, $p$ must be explicitly given and the
 function computes the trace of the reduction over $\F_p$.
 The trace of Frobenius is also the $a_p$ coefficient in the curve $L$-series
 $L(E,s) = \sum_n a_n n^{-s}$, whence the function name. The equation must be
 integral at $p$ but need not be minimal at $p$; of course, a minimal model
 will be more efficient.
 \bprog
 ? E = ellinit([0,1]);  \\ y^2 = x^3 + 0.x + 1, defined over Q
 ? ellap(E, 7) \\ 7 necessary here
 %2 = -4       \\ #E(F_7) = 7+1-(-4) = 12
 ? ellcard(E, 7)
 %3 = 12       \\ OK
 
 ? E = ellinit([0,1], 11);  \\ defined over F_11
 ? ellap(E)       \\ no need to repeat 11
 %4 = 0
 ? ellap(E, 11)   \\ ... but it also works
 %5 = 0
 ? ellgroup(E, 13) \\ ouch, inconsistent input!
    ***   at top-level: ellap(E,13)
    ***                 ^-----------
    *** ellap: inconsistent moduli in Rg_to_Fp:
      11
      13
 
 ? Fq = ffgen(ffinit(11,3), 'a); \\ defines F_q := F_{11^3}
 ? E = ellinit([a+1,a], Fq);  \\ y^2 = x^3 + (a+1)x + a, defined over F_q
 ? ellap(E)
 %8 = -3
 @eprog
 
 If the curve is defined over a more general number field than $\Q$,
 the maximal ideal $p$ must be explicitly given in \kbd{idealprimedec}
 format. If $p$ is above $2$ or $3$, the function currently assumes (without
 checking) that the given model is locally minimal at $p$. There is no
 restriction at other primes.
 \bprog
 ? K = nfinit(a^2+1); E = ellinit([1+a,0,1,0,0], K);
 ? fa = idealfactor(K, E.disc)
 %2 =
 [  [5, [-2, 1]~, 1, 1, [2, -1; 1, 2]] 1]
 
 [[13, [5, 1]~, 1, 1, [-5, -1; 1, -5]] 2]
 ? ellap(E, fa[1,1])
 %3 = -1 \\ non-split multiplicative reduction
 ? ellap(E, fa[2,1])
 %4 = 1  \\ split multiplicative reduction
 ? P17 = idealprimedec(K,17)[1];
 ? ellap(E, P17)
 %6 = 6  \\ good reduction
 ? E2 = ellchangecurve(E, [17,0,0,0]);
 ? ellap(E2, P17)
 %8 = 6  \\ same, starting from a non-miminal model
 
 ? P3 = idealprimedec(K,3)[1];
 ? E3 = ellchangecurve(E, [3,0,0,0]);
 ? ellap(E, P3)  \\ OK: E is minimal at P3
 %11 = -2
 ? ellap(E3, P3) \\ junk: E3 is not minimal at P3 | 3
 %12 = 0
 @eprog
 
 \misctitle{Algorithms used} If $E/\F_q$ has CM by a principal imaginary
 quadratic order we use a fast explicit formula (involving essentially
 Kronecker symbols and Cornacchia's algorithm), in $O(\log q)^2$.
 Otherwise, we use Shanks-Mestre's baby-step/giant-step method, which runs in
 time $\tilde{O}(q^{1/4})$ using $\tilde{O}(q^{1/4})$ storage, hence becomes
 unreasonable when $q$ has about 30~digits. Above this range, the \tet{SEA}
 algorithm becomes available, heuristically in $\tilde{O}(\log q)^4$, and
 primes of the order of 200~digits become feasible.  In small
 characteristic we use Mestre's (p=2), Kohel's (p=3,5,7,13), Satoh-Harley
 (all in $\tilde{O}(p^{2}\*n^2)$) or Kedlaya's (in $\tilde{O}(p\*n^3)$)
 algorithms.

Function: ellbil
Class: basic
Section: elliptic_curves
C-Name: bilhell
Prototype: GGGp
Help: ellbil(E,z1,z2): deprecated alias for ellheight(E,P,Q).
Doc: deprecated alias for \kbd{ellheight(E,P,Q)}.
Obsolete: 2014-05-21

Function: ellcard
Class: basic
Section: elliptic_curves
C-Name: ellcard
Prototype: GDG
Help: ellcard(E,{p}): computes the order of the group E(Fq)
 for the elliptic curve E, defined over Q (for q=p) or a finite field.
Doc: Let $E$ be an \kbd{ell} structure as output by \kbd{ellinit}, defined over
 $\Q$ or a finite field $\F_q$. The argument $p$ is best left omitted if the
 curve is defined over a finite field, and must be a prime number otherwise.
 This function computes the order of the group $E(\F_q)$ (as would be
 computed by \tet{ellgroup}).
 
 When the characteristic of the finite field is large, the availability of
 the \kbd{seadata} package will speed the computation.
 
 If the curve is defined over $\Q$, $p$ must be explicitly given and the
 function computes the cardinality of the reduction over $\F_p$; the
 equation need not be minimal at $p$, but a minimal model will be more
 efficient. The reduction is allowed to be singular, and we return the order
 of the group of non-singular points in this case.
Variant: Also available is \fun{GEN}{ellcard}{GEN E, GEN p} where $p$ is not
 \kbd{NULL}.

Function: ellchangecurve
Class: basic
Section: elliptic_curves
C-Name: ellchangecurve
Prototype: GG
Help: ellchangecurve(E,v): change data on elliptic curve according to
 v=[u,r,s,t].
Description: 
 (gen, gen):ell        ellchangecurve($1, $2)
Doc: 
 changes the data for the elliptic curve $E$
 by changing the coordinates using the vector \kbd{v=[u,r,s,t]}, i.e.~if $x'$
 and $y'$ are the new coordinates, then $x=u^2x'+r$, $y=u^3y'+su^2x'+t$.
 $E$ must be an \kbd{ell} structure as output by \kbd{ellinit}. The special
 case $v = 1$ is also used instead of $[1,0,0,0]$ to denote the
 trivial coordinate change.

Function: ellchangepoint
Class: basic
Section: elliptic_curves
C-Name: ellchangepoint
Prototype: GG
Help: ellchangepoint(x,v): change data on point or vector of points x on an
 elliptic curve according to v=[u,r,s,t].
Doc: 
 changes the coordinates of the point or
 vector of points $x$ using the vector \kbd{v=[u,r,s,t]}, i.e.~if $x'$ and
 $y'$ are the new coordinates, then $x=u^2x'+r$, $y=u^3y'+su^2x'+t$ (see also
 \kbd{ellchangecurve}).
 \bprog
 ? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4];
 ? E = ellchangecurve(E0, v);
 ? P = ellchangepoint(P0,v)
 %3 = [-2, 3]
 ? ellisoncurve(E, P)
 %4 = 1
 ? ellchangepointinv(P,v)
 %5 = [0, 1]
 @eprog
Variant: The reciprocal function \fun{GEN}{ellchangepointinv}{GEN x, GEN ch}
 inverts the coordinate change.

Function: ellchangepointinv
Class: basic
Section: elliptic_curves
C-Name: ellchangepointinv
Prototype: GG
Help: ellchangepointinv(x,v): change data on point or vector of points x on an
 elliptic curve according to v=[u,r,s,t], inverse of ellchangepoint.
Doc: 
 changes the coordinates of the point or vector of points $x$ using
 the inverse of the isomorphism attached to \kbd{v=[u,r,s,t]},
 i.e.~if $x'$ and $y'$ are the old coordinates, then $x=u^2x'+r$,
 $y=u^3y'+su^2x'+t$ (inverse of \kbd{ellchangepoint}).
 \bprog
 ? E0 = ellinit([1,1]); P0 = [0,1]; v = [1,2,3,4];
 ? E = ellchangecurve(E0, v);
 ? P = ellchangepoint(P0,v)
 %3 = [-2, 3]
 ? ellisoncurve(E, P)
 %4 = 1
 ? ellchangepointinv(P,v)
 %5 = [0, 1]  \\ we get back P0
 @eprog

Function: ellconvertname
Class: basic
Section: elliptic_curves
C-Name: ellconvertname
Prototype: G
Help: ellconvertname(name): convert an elliptic curve name (as found in
 the elldata database) from a string to a triplet [conductor, isogeny class,
 index]. It will also convert a triplet back to a curve name.
Doc: 
 converts an elliptic curve name, as found in the \tet{elldata} database,
 from a string to a triplet $[\var{conductor}, \var{isogeny class},
 \var{index}]$. It will also convert a triplet back to a curve name.
 Examples:
 \bprog
 ? ellconvertname("123b1")
 %1 = [123, 1, 1]
 ? ellconvertname(%)
 %2 = "123b1"
 @eprog

Function: elldivpol
Class: basic
Section: elliptic_curves
C-Name: elldivpol
Prototype: GLDn
Help: elldivpol(E,n,{v='x}): n-division polynomial f_n for the curve E in the
 variable v.
Doc: $n$-division polynomial $f_n$ for the curve $E$ in the
 variable $v$. In standard notation, for any affine point $P = (X,Y)$ on the
 curve, we have
 $$[n]P = (\phi_n(P)\psi_n(P) : \omega_n(P) : \psi_n(P)^3)$$
 for some polynomials $\phi_n,\omega_n,\psi_n$ in
 $\Z[a_1,a_2,a_3,a_4,a_6][X,Y]$. We have $f_n(X) = \psi_n(X)$ for $n$ odd, and
 $f_n(X) = \psi_n(X,Y) (2Y + a_1X+a_3)$ for $n$ even. We have
 $$ f_1  = 1,\quad f_2 = 4X^3 + b_2X^2 + 2b_4 X + b_6, \quad f_3 = 3 X^4 + b_2 X^3 + 3b_4 X^2 + 3 b_6 X + b8, $$
 $$ f_4 = f_2(2X^6 + b_2 X^5 + 5b_4 X^4 + 10 b_6 X^3 + 10 b_8 X^2 +
 (b_2b_8-b_4b_6)X + (b_8b_4 - b_6^2)), \dots $$
 For $n \geq 2$, the roots of $f_n$ are the $X$-coordinates of points in $E[n]$.

Function: elleisnum
Class: basic
Section: elliptic_curves
C-Name: elleisnum
Prototype: GLD0,L,p
Help: elleisnum(w,k,{flag=0}): k being an even positive integer, computes the
 numerical value of the Eisenstein series of weight k at the lattice
 w, as given by ellperiods. When flag is non-zero and k=4 or 6, this gives the
 elliptic invariants g2 or g3 with the correct normalization.
Doc: $k$ being an even positive integer, computes the numerical value of the
 Eisenstein series of weight $k$ at the lattice $w$, as given by
 \tet{ellperiods}, namely
 $$
 (2i \pi/\omega_2)^k
 \Big(1 + 2/\zeta(1-k) \sum_{n\geq 1} n^{k-1}q^n / (1-q^n)\Big),
 $$
 where $q = \exp(2i\pi \tau)$ and $\tau:=\omega_1/\omega_2$ belongs to the
 complex upper half-plane. It is also possible to directly input $w =
 [\omega_1,\omega_2]$, or an elliptic curve $E$ as given by \kbd{ellinit}.
 \bprog
 ? w = ellperiods([1,I]);
 ? elleisnum(w, 4)
 %2 = 2268.8726415508062275167367584190557607
 ? elleisnum(w, 6)
 %3 = -3.977978632282564763 E-33
 ? E = ellinit([1, 0]);
 ? elleisnum(E, 4, 1)
 %5 = -47.999999999999999999999999999999999998
 @eprog
 
 When \fl\ is non-zero and $k=4$ or 6, returns the elliptic invariants $g_2$
 or $g_3$, such that
 $$y^2 = 4x^3 - g_2 x - g_3$$
 is a Weierstrass equation for $E$.

Function: elleta
Class: basic
Section: elliptic_curves
C-Name: elleta
Prototype: Gp
Help: elleta(w): w=[w1,w2], returns the vector [eta1,eta2] of quasi-periods
 attached to [w1,w2].
Doc: returns the quasi-periods $[\eta_1,\eta_2]$
 attached to the lattice basis $\var{w} = [\omega_1, \omega_2]$.
 Alternatively, \var{w} can be an elliptic curve $E$ as output by
 \kbd{ellinit}, in which case, the quasi periods attached to the period
 lattice basis \kbd{$E$.omega} (namely, \kbd{$E$.eta}) are returned.
 \bprog
 ? elleta([1, I])
 %1 = [3.141592653589793238462643383, 9.424777960769379715387930149*I]
 @eprog

Function: ellformaldifferential
Class: basic
Section: elliptic_curves
C-Name: ellformaldifferential
Prototype: GDPDn
Help: ellformaldifferential(E, {n=seriesprecision}, {t = 'x}) : E elliptic curve,
 n integer. Returns n terms of the power series [f, g] such that
 omega = dx/(2y+a_1x+a_3) = f(t) dt and eta = x(t) * omega = g(t) dt in the
 local parameter t=-x/y.
Doc: Let $\omega := dx / (2y+a_1x+a_3)$ be the invariant differential form
 attached to the model $E$ of some elliptic curve (\kbd{ellinit} form),
 and $\eta := x(t)\omega$. Return $n$ terms (\tet{seriesprecision} by default)
 of $f(t),g(t)$ two power series in the formal parameter $t=-x/y$ such that
 $\omega = f(t) dt$, $\eta = g(t) dt$:
  $$f(t) = 1+a_1 t + (a_1^2 + a_2) t^2 + \dots,\quad
    g(t) = t^{-2} +\dots $$
  \bprog
  ? E = ellinit([-1,1/4]); [f,g] = ellformaldifferential(E,7,'t);
  ? f
  %2 = 1 - 2*t^4 + 3/4*t^6 + O(t^7)
  ? g
  %3 = t^-2 - t^2 + 1/2*t^4 + O(t^5)
 @eprog

Function: ellformalexp
Class: basic
Section: elliptic_curves
C-Name: ellformalexp
Prototype: GDPDn
Help: ellformalexp(E, {n = seriesprecision}, {z = 'x}) : E elliptic curve,
 returns n terms of the formal elliptic exponential on E as a series in z.
Doc: The elliptic formal exponential \kbd{Exp} attached to $E$ is the
 isomorphism from the formal additive law to the formal group of $E$. It is
 normalized so as to be the inverse of the elliptic logarithm (see
 \tet{ellformallog}): $\kbd{Exp} \circ L = \Id$. Return $n$ terms of this
 power series:
 \bprog
 ? E=ellinit([-1,1/4]); Exp = ellformalexp(E,10,'z)
 %1 = z + 2/5*z^5 - 3/28*z^7 + 2/15*z^9 + O(z^11)
 ? L = ellformallog(E,10,'t);
 ? subst(Exp,z,L)
 %3 = t + O(t^11)
 @eprog

Function: ellformallog
Class: basic
Section: elliptic_curves
C-Name: ellformallog
Prototype: GDPDn
Help: ellformallog(E, {n = seriesprecision}, {v = 'x}): E elliptic curve,
 returns n terms of the elliptic logarithm as a series of t =-x/y.
Doc: The formal elliptic logarithm is a series $L$ in $t K[[t]]$
 such that $d L = \omega = dx / (2y + a_1x + a_3)$, the canonical invariant
 differential attached to the model $E$. It gives an isomorphism
 from the formal group of $E$ to the additive formal group.
 \bprog
 ? E = ellinit([-1,1/4]); L = ellformallog(E, 9, 't)
 %1 = t - 2/5*t^5 + 3/28*t^7 + 2/3*t^9 + O(t^10)
 ? [f,g] = ellformaldifferential(E,8,'t);
 ? L' - f
 %3 = O(t^8)
 @eprog

Function: ellformalpoint
Class: basic
Section: elliptic_curves
C-Name: ellformalpoint
Prototype: GDPDn
Help: ellformalpoint(E, {n = seriesprecision}, {v = 'x}): E elliptic curve,
 n integer; return the coordinates [x(t), y(t)] on the elliptic curve as a
 formal expansion in the formal parameter t = -x/y.
Doc: If $E$ is an elliptic curve, return the coordinates $x(t), y(t)$ in the
 formal group of the elliptic curve $E$ in the formal parameter $t = -x/y$
 at $\infty$:
 $$ x = t^{-2} -a_1 t^{-1} - a_2 - a_3 t + \dots $$
 $$ y = - t^{-3} -a_1 t^{-2} - a_2t^{-1} -a_3 + \dots $$
 Return $n$ terms (\tet{seriesprecision} by default) of these two power
 series, whose coefficients are in $\Z[a_1,a_2,a_3,a_4,a_6]$.
 \bprog
 ? E = ellinit([0,0,1,-1,0]); [x,y] = ellformalpoint(E,8,'t);
 ? x
 %2 = t^-2 - t + t^2 - t^4 + 2*t^5 + O(t^6)
 ? y
 %3 = -t^-3 + 1 - t + t^3 - 2*t^4 + O(t^5)
 ? E = ellinit([0,1/2]); ellformalpoint(E,7)
 %4 = [x^-2 - 1/2*x^4 + O(x^5), -x^-3 + 1/2*x^3 + O(x^4)]
 @eprog

Function: ellformalw
Class: basic
Section: elliptic_curves
C-Name: ellformalw
Prototype: GDPDn
Help: ellformalw(E, {n = seriesprecision}, {t = 'x}): E elliptic curve,
 n integer; returns n terms of the formal expansion of w = -1/y in the formal
 parameter t = -x/y.
Doc: Return the formal power series $w$ attached to the elliptic curve $E$,
 in the variable $t$:
 $$ w(t) = t^3 + a_1 t^4 + (a_2 + a_1^2) t^5 + \cdots + O(t^{n+3}),$$
 which is the formal expansion of $-1/y$ in the formal parameter $t := -x/y$
 at $\infty$ (take $n = \tet{seriesprecision}$ if $n$ is omitted). The
 coefficients of $w$ belong to $\Z[a_1,a_2,a_3,a_4,a_6]$.
 \bprog
 ? E=ellinit([3,2,-4,-2,5]); ellformalw(E, 5, 't)
 %1 = t^3 + 3*t^4 + 11*t^5 + 35*t^6 + 101*t^7 + O(t^8)
 @eprog

Function: ellfromeqn
Class: basic
Section: elliptic_curves
C-Name: ellfromeqn
Prototype: G
Help: ellfromeqn(P): given a genus 1 plane curve, defined by the affine
 equation f(x,y) = 0, return the coefficients [a1,a2,a3,a4,a6] of a
 Weierstrass equation for its Jacobian.
 This allows to recover a Weierstrass model for an elliptic curve given by a
 general plane cubic or by a binary quartic or biquadratic model.
Doc: 
 Given a genus $1$ plane curve, defined by the affine equation $f(x,y) = 0$,
 return the coefficients $[a_1,a_2,a_3,a_4,a_6]$ of a Weierstrass equation
 for its Jacobian. This allows to recover a Weierstrass model for an elliptic
 curve given by a general plane cubic or by a binary quartic or biquadratic
 model. The function implements the $f \mapsto f^*$ formulae of Artin, Tate
 and Villegas (Advances in Math. 198 (2005), pp. 366--382).
 
 In the example below, the function is used to convert between twisted Edwards
 coordinates and Weierstrass coordinates.
 \bprog
 ? e = ellfromeqn(a*x^2+y^2 - (1+d*x^2*y^2))
 %1 = [0, -a - d, 0, -4*d*a, 4*d*a^2 + 4*d^2*a]
 ? E = ellinit(ellfromeqn(y^2-x^2 - 1 +(121665/121666*x^2*y^2)),2^255-19);
 ? isprime(ellcard(E) / 8)
 %3 = 1
 @eprog
 
 The elliptic curve attached to the sum of two cubes is given by
 \bprog
 ? ellfromeqn(x^3+y^3 - a)
 %1 = [0, 0, -9*a, 0, -27*a^2]
 @eprog
 
 \misctitle{Congruent number problem:}
 Let $n$ be an integer, if $a^2+b^2=c^2$ and $a\*b=2\*n$,
 then by substituting $b$ by $2\*n/a$ in the first equation,
 we get $((a^2+(2\*n/a)^2)-c^2)\*a^2 = 0$.
 We set $x=a$, $y=a\*c$.
 \bprog
 ? En = ellfromeqn((x^2 + (2*n/x)^2 - (y/x)^2)*x^2)
 %1 = [0, 0, 0, -16*n^2, 0]
 @eprog
 For example $23$ is congruent since the curve has a point of infinite order,
 namely:
 \bprog
 ? ellheegner( ellinit(subst(En, n, 23)) )
 %2 = [168100/289, 68053440/4913]
 @eprog

Function: ellfromj
Class: basic
Section: elliptic_curves
C-Name: ellfromj
Prototype: G
Help: ellfromj(j): returns the coefficients [a1,a2,a3,a4,a6] of a fixed
 elliptic curve with j-invariant j.
Doc: returns the coefficients $[a_1,a_2,a_3,a_4,a_6]$ of a fixed elliptic curve
 with $j$-invariant $j$.

Function: ellgenerators
Class: basic
Section: elliptic_curves
C-Name: ellgenerators
Prototype: G
Help: ellgenerators(E): if E is an elliptic curve over the rationals,
 return the generators of the Mordell-Weil group attached to the curve.
 This relies on the curve being referenced in the elldata database.
 If E is an elliptic curve over a finite field Fq as output by ellinit(),
 return a minimal set of generators for the group E(Fq).
Doc: 
 If $E$ is an elliptic curve over the rationals, return a $\Z$-basis of the
 free part of the \idx{Mordell-Weil group} attached to $E$.  This relies on
 the \tet{elldata} database being installed and referencing the curve, and so
 is only available for curves over $\Z$ of small conductors.
 If $E$ is an elliptic curve over a finite field $\F_q$ as output by
 \tet{ellinit}, return a minimal set of generators for the group $E(\F_q)$.

Function: ellglobalred
Class: basic
Section: elliptic_curves
C-Name: ellglobalred
Prototype: G
Help: ellglobalred(E): E being an elliptic curve over a number field,
 returns [N, v, c, faN, L], where N is the conductor of E,
 c is the product of the local Tamagawa numbers c_p, faN is the
 factorization of N and L[i] is elllocalred(E, faN[i,1]); v is an obsolete
 field.
Description: 
 (gen):gen        ellglobalred($1)
Doc: let $E$ be an \kbd{ell} structure as output by \kbd{ellinit} attached
 to an elliptic curve defined over a number field. This function calculates
 the arithmetic conductor and the global \idx{Tamagawa number} $c$.
 The result $[N,v,c,F,L]$ is slightly different if $E$ is defined
 over $\Q$ (domain $D = 1$ in \kbd{ellinit}) or over a number field
 (domain $D$ is a number field structure, including \kbd{nfinit(x)}
 representing $\Q$ !):
 
 \item $N$ is the arithmetic conductor of the curve,
 
 \item $v$ is an obsolete field, left in place for backward compatibility.
 If $E$ is defined over $\Q$, $v$ gives the coordinate change for $E$ to the
 standard minimal integral model (\tet{ellminimalmodel} provides it in a
 cheaper way); if $E$ is defined over another number field, $v$ gives a
 coordinate change to an integral model (\tet{ellintegralmodel} provides it
 in a cheaper way).
 
 \item $c$ is the product of the local Tamagawa numbers $c_p$, a quantity
 which enters in the \idx{Birch and Swinnerton-Dyer conjecture},
 
 \item $F$ is the factorization of $N$,
 
 \item $L$ is a vector, whose $i$-th entry contains the local data
 at the $i$-th prime ideal divisor of $N$, i.e.
 \kbd{L[i] = elllocalred(E,F[i,1])}. If $E$ is defined over $\Q$, the local
 coordinate change has been deleted and replaced by a 0; if $E$ is defined
 over another number field the local coordinate change to a local minimal
 model is given relative to the integral model afforded by $v$ (so either
 start from an integral model so that $v$ be trivial, or apply $v$ first).

Function: ellgroup
Class: basic
Section: elliptic_curves
C-Name: ellgroup0
Prototype: GDGD0,L,
Help: ellgroup(E,{p},{flag}): computes the structure of the group E(Fp)
 If flag is 1, return also generators.
Doc: Let $E$ be an \kbd{ell} structure as output by \kbd{ellinit}, defined over
 $\Q$ or a finite field $\F_q$. The argument $p$ is best left omitted if the
 curve is defined over a finite field, and must be a prime number otherwise.
 This function computes the structure of the group $E(\F_q) \sim \Z/d_1\Z
 \times \Z/d_2\Z$, with $d_2\mid d_1$.
 
 If the curve is defined over $\Q$, $p$ must be explicitly given and the
 function computes the structure of the reduction over $\F_p$; the
 equation need not be minimal at $p$, but a minimal model will be more
 efficient. The reduction is allowed to be singular, and we return the
 structure of the (cyclic) group of non-singular points in this case.
 
 If the flag is $0$ (default), return $[d_1]$ or $[d_1, d_2]$, if $d_2>1$.
 If the flag is $1$, return a triple $[h,\var{cyc},\var{gen}]$, where
 $h$ is the curve cardinality, \var{cyc} gives the group structure as a
 product of cyclic groups (as per $\fl = 0$). More precisely, if $d_2 > 1$,
 the output is $[d_1d_2, [d_1,d_2],[P,Q]]$ where $P$ is
 of order $d_1$ and $[P,Q]$ generates the curve.
 \misctitle{Caution} It is not guaranteed that $Q$ has order $d_2$, which in
 the worst case requires an expensive discrete log computation. Only that
 \kbd{ellweilpairing(E, P, Q, d1)} has order $d_2$.
 \bprog
 ? E = ellinit([0,1]);  \\ y^2 = x^3 + 0.x + 1, defined over Q
 ? ellgroup(E, 7)
 %2 = [6, 2] \\ Z/6 x Z/2, non-cyclic
 ? E = ellinit([0,1] * Mod(1,11));  \\ defined over F_11
 ? ellgroup(E)   \\ no need to repeat 11
 %4 = [12]
 ? ellgroup(E, 11)   \\ ... but it also works
 %5 = [12]
 ? ellgroup(E, 13) \\ ouch, inconsistent input!
    ***   at top-level: ellgroup(E,13)
    ***                 ^--------------
    *** ellgroup: inconsistent moduli in Rg_to_Fp:
      11
      13
 ? ellgroup(E, 7, 1)
 %6 = [12, [6, 2], [[Mod(2, 7), Mod(4, 7)], [Mod(4, 7), Mod(4, 7)]]]
 @eprog\noindent
 If $E$ is defined over $\Q$, we allow singular reduction and in this case we
 return the structure of the group of non-singular points, satisfying
 $\#E_{ns}(\F_p) = p - a_p$.
 \bprog
 ? E = ellinit([0,5]);
 ? ellgroup(E, 5, 1)
 %2 = [5, [5], [[Mod(4, 5), Mod(2, 5)]]]
 ? ellap(E, 5)
 %3 = 0 \\ additive reduction at 5
 ? E = ellinit([0,-1,0,35,0]);
 ? ellgroup(E, 5, 1)
 %5 = [4, [4], [[Mod(2, 5), Mod(2, 5)]]]
 ? ellap(E, 5)
 %6 = 1 \\ split multiplicative reduction at 5
 ? ellgroup(E, 7, 1)
 %7 = [8, [8], [[Mod(3, 7), Mod(5, 7)]]]
 ? ellap(E, 7)
 %8 = -1 \\ non-split multiplicative reduction at 7
 @eprog
Variant: Also available is \fun{GEN}{ellgroup}{GEN E, GEN p}, corresponding
 to \fl = 0.

Function: ellheegner
Class: basic
Section: elliptic_curves
C-Name: ellheegner
Prototype: G
Help: ellheegner(E): return a rational non-torsion point on the elliptic curve E
 assumed to be of rank 1.
Doc: Let $E$ be an elliptic curve over the rationals, assumed to be of
 (analytic) rank $1$. This returns a non-torsion rational point on the curve,
 whose canonical height is equal to the product of the elliptic regulator by the
 analytic Sha.
 
 This uses the Heegner point method, described in Cohen GTM 239; the complexity
 is proportional to the product of the square root of the conductor and the
 height of the point (thus, it is preferable to apply it to strong Weil curves).
 \bprog
 ? E = ellinit([-157^2,0]);
 ? u = ellheegner(E); print(u[1], "\n", u[2])
 69648970982596494254458225/166136231668185267540804
 538962435089604615078004307258785218335/67716816556077455999228495435742408
 ? ellheegner(ellinit([0,1]))         \\ E has rank 0 !
  ***   at top-level: ellheegner(E=ellinit
  ***                 ^--------------------
  *** ellheegner: The curve has even analytic rank.
 @eprog

Function: ellheight
Class: basic
Section: elliptic_curves
C-Name: ellheight0
Prototype: GGDGp
Help: ellheight(E,P,{Q}): canonical height of point P on elliptic curve E,
 resp. the value of the attached bilinear form at (P,Q).
Doc: global N\'eron-Tate height $h(P)$ of the point $P$ on the elliptic curve
 $E/\Q$, using the normalization in Cremona's \emph{Algorithms for modular
 elliptic curves}. $E$ must be an \kbd{ell} as output by \kbd{ellinit}; it
 needs not be given by a minimal model although the computation will be faster
 if it is.
 
 If the argument $Q$ is present, computes the value of the bilinear
 form $(h(P+Q)-h(P-Q)) / 4$.
Variant: Also available is \fun{GEN}{ellheight}{GEN E, GEN P, long prec}
 ($Q$ omitted).

Function: ellheightmatrix
Class: basic
Section: elliptic_curves
C-Name: ellheightmatrix
Prototype: GGp
Help: ellheightmatrix(E,x): gives the height matrix for vector of points x
 on elliptic curve E.
Doc: $x$ being a vector of points, this
 function outputs the Gram matrix of $x$ with respect to the N\'eron-Tate
 height, in other words, the $(i,j)$ component of the matrix is equal to
 \kbd{ellbil($E$,x[$i$],x[$j$])}. The rank of this matrix, at least in some
 approximate sense, gives the rank of the set of points, and if $x$ is a
 basis of the \idx{Mordell-Weil group} of $E$, its determinant is equal to
 the regulator of $E$. Note our height normalization follows Cremona's
 \emph{Algorithms for modular elliptic curves}: this matrix should be divided
 by 2 to be in accordance with, e.g., Silverman's normalizations.

Function: ellidentify
Class: basic
Section: elliptic_curves
C-Name: ellidentify
Prototype: G
Help: ellidentify(E): look up the elliptic curve E in the elldata database and
 return [[N, M, ...], C] where N is the name of the curve in Cremona's
 database, M the minimal model and C the coordinates change (see
 ellchangecurve).
Doc: look up the elliptic curve $E$, defined by an arbitrary model over $\Q$,
 in the \tet{elldata} database.
 Return \kbd{[[N, M, G], C]}  where $N$ is the curve name in Cremona's
 elliptic curve database, $M$ is the minimal model, $G$ is a $\Z$-basis of
 the free part of the \idx{Mordell-Weil group} $E(\Q)$ and $C$ is the
 change of coordinates change, suitable for \kbd{ellchangecurve}.

Function: ellinit
Class: basic
Section: elliptic_curves
C-Name: ellinit
Prototype: GDGp
Help: ellinit(x,{D=1}): let x be a vector [a1,a2,a3,a4,a6], or [a4,a6] if
 a1=a2=a3=0, defining the curve Y^2 + a1.XY + a3.Y = X^3 + a2.X^2 + a4.X +
 a6; x can also be a string, in which case the curve with matching name is
 retrieved from the elldata database, if available. This function initializes
 an elliptic curve over the domain D (inferred from coefficients if omitted).
Description: 
 (gen, gen, small):ell:prec  ellinit($1, $2, $prec)
Doc: 
 initialize an \tet{ell} structure, attached to the elliptic curve $E$.
 $E$ is either
 
 \item a $5$-component vector $[a_1,a_2,a_3,a_4,a_6]$ defining the elliptic
 curve with Weierstrass equation
 $$ Y^2 + a_1 XY + a_3 Y = X^3 + a_2 X^2 + a_4 X + a_6, $$
 
 \item a $2$-component vector $[a_4,a_6]$ defining the elliptic
 curve with short Weierstrass equation
 $$ Y^2 = X^3 + a_4 X + a_6, $$
 
 \item a character string in Cremona's notation, e.g. \kbd{"11a1"}, in which
 case the curve is retrieved from the \tet{elldata} database if available.
 
 The optional argument $D$ describes the domain over which the curve is
 defined:
 
 \item the \typ{INT} $1$ (default): the field of rational numbers $\Q$.
 
 \item a \typ{INT} $p$, where $p$ is a prime number: the prime finite field
 $\F_p$.
 
 \item an \typ{INTMOD} \kbd{Mod(a, p)}, where $p$ is a prime number: the
 prime finite field $\F_p$.
 
 \item a \typ{FFELT}, as returned by \tet{ffgen}: the corresponding finite
 field $\F_q$.
 
 \item a \typ{PADIC}, $O(p^n)$: the field $\Q_p$, where $p$-adic quantities
 will be computed to a relative accuracy of $n$ digits. We advise to input a
 model defined over $\Q$ for such curves. In any case, if you input an
 approximate model with \typ{PADIC} coefficients, it will be replaced by a lift
 to $\Q$ (an exact model ``close'' to the one that was input) and all quantities
 will then be computed in terms of this lifted model, at the given accuracy.
 
 \item a \typ{REAL} $x$: the field $\C$ of complex numbers, where floating
 point quantities are by default computed to a relative accuracy of
 \kbd{precision}$(x)$. If no such argument is given, the value of
 \kbd{realprecision} at the time \kbd{ellinit} is called will be used.
 
 \item a number field $K$, given by a \kbd{nf} or \kbd{bnf} structure; a
 \kbd{bnf} is required for \kbd{ellminimalmodel}.
 
 \item a prime ideal $\goth{p}$, given by a \kbd{prid} structure; valid if
 $x$ is a curve defined over a number field $K$ and the equation is integral
 and minimal at $\goth{p}$.
 
 This argument $D$ is indicative: the curve coefficients are checked for
 compatibility, possibly changing $D$; for instance if $D = 1$ and
 an \typ{INTMOD} is found. If inconsistencies are detected, an error is
 raised:
 \bprog
 ? ellinit([1 + O(5), 1], O(7));
  ***   at top-level: ellinit([1+O(5),1],O
  ***                 ^--------------------
  *** ellinit: inconsistent moduli in ellinit: 7 != 5
 @eprog\noindent If the curve coefficients are too general to fit any of the
 above domain categories, only basic operations, such as point addition, will
 be supported later.
 
 If the curve (seen over the domain $D$) is singular, fail and return an
 empty vector $[]$.
 \bprog
 ? E = ellinit([0,0,0,0,1]); \\ y^2 = x^3 + 1, over Q
 ? E = ellinit([0,1]);       \\ the same curve, short form
 ? E = ellinit("36a1");      \\ sill the same curve, Cremona's notations
 ? E = ellinit([0,1], 2)     \\ over F2: singular curve
 %4 = []
 ? E = ellinit(['a4,'a6] * Mod(1,5));  \\ over F_5[a4,a6], basic support !
 @eprog\noindent
 
 The result of \tet{ellinit} is an \tev{ell} structure. It contains at least
 the following information in its components:
 %
 $$ a_1,a_2,a_3,a_4,a_6,b_2,b_4,b_6,b_8,c_4,c_6,\Delta,j.$$
 %
 All are accessible via member functions. In particular, the discriminant is
 \kbd{$E$.disc}, and the $j$-invariant is \kbd{$E$.j}.
 \bprog
 ? E = ellinit([a4, a6]);
 ? E.disc
 %2 = -64*a4^3 - 432*a6^2
 ? E.j
 %3 = -6912*a4^3/(-4*a4^3 - 27*a6^2)
 @eprog
 Further components contain domain-specific data, which are in general dynamic:
 only computed when needed, and then cached in the structure.
 \bprog
 ? E = ellinit([2,3], 10^60+7);  \\ E over F_p, p large
 ? ellap(E)
 time = 4,440 ms.
 %2 = -1376268269510579884904540406082
 ? ellcard(E);  \\ now instantaneous !
 time = 0 ms.
 ? ellgenerators(E);
 time = 5,965 ms.
 ? ellgenerators(E); \\ second time instantaneous
 time = 0 ms.
 @eprog
 See the description of member functions related to elliptic curves at the
 beginning of this section.

Function: ellintegralmodel
Class: basic
Section: elliptic_curves
C-Name: ellintegralmodel
Prototype: GD&
Help: ellintegralmodel(E,{&v}): given an elliptic curve E defined
 over a number field, returns an integral model. If v is present,
 sets the variable v to the corresponding change of variable.
Doc: Let $E$ be an \kbd{ell} structure over a number field $K$. This function
 returns an integral model. If $v$ is present, sets $v = [u,0,0,0]$ to the
 corresponding change of variable: the return value is identical to that of
 \kbd{ellchangecurve(E, v)}.

Function: ellisdivisible
Class: basic
Section: elliptic_curves
C-Name: ellisdivisible
Prototype: lGGGD&
Help: ellisdivisible(E,P,n,{&Q})): given E/K and P in E(K),
 checks whether P = [n]R for some R in E(K) and sets Q to one such R if so;
 the integer n >= 0 may be given as ellxn(E,n).
Doc: given $E/K$ a number field and $P$ in $E(K)$
 return $1$ if $P = [n]R$ for some $R$ in $E(K)$ and set $Q$ to one such $R$;
 and return $0$ otherwise. The integer $n \geq 0$ may be given as
 \kbd{ellxn(E,n)}, if many points need to be tested.
 \bprog
 ? K = nfinit(polcyclo(11,t));
 ? E = ellinit([0,-1,1,0,0], K);
 ? P = [0,0];
 ? ellorder(E,P)
 %4 = 5
 ? ellisdivisible(E,P,5, &Q)
 %5 = 1
 ? lift(Q)
 %6 = [-t^7-t^6-t^5-t^4+1, -t^9-2*t^8-2*t^7-3*t^6-3*t^5-2*t^4-2*t^3-t^2-1]
 ? ellorder(E, Q)
 %7 = 25
 @eprog\noindent The algebraic complexity of the underlying algorithm is in
 $O(n^4)$, so it is advisable to first factor $n$, then use a chain of checks
 attached to the prime divisors of $n$: the function will do it itself unless
 $n$ is given in \kbd{ellxn} form.

Function: ellisogeny
Class: basic
Section: elliptic_curves
C-Name: ellisogeny
Prototype: GGD0,L,DnDn
Help: ellisogeny(E, G, {only_image = 0}, {x = 'x}, {y = 'y}): compute the image
 and isogeny corresponding to the quotient of E by the subgroup G.
Doc: 
 Given an elliptic curve $E$, a finite subgroup $G$ of $E$ is given either
 as a generating point $P$ (for a cyclic $G$) or as a polynomial whose roots
 vanish on the $x$-coordinates of the non-zero elements of $G$ (general case
 and more efficient if available). This function returns the
 $[a_1,a_2,a_3,a_4,a_6]$ invariants of the quotient elliptic curve $E/G$ and
 (if \var{only\_image} is zero (the default)) a vector of rational
 functions $[f, g, h]$ such that the isogeny $E \to E/G$ is given by $(x,y)
 \mapsto (f(x)/h(x)^2, g(x,y)/h(x)^3)$.
 \bprog
 ? E = ellinit([0,1]);
 ? elltors(E)
 %2 = [6, [6], [[2, 3]]]
 ? ellisogeny(E, [2,3], 1)  \\ Weierstrass model for E/<P>
 %3 = [0, 0, 0, -135, -594]
 ? ellisogeny(E,[-1,0])
 %4 = [[0,0,0,-15,22], [x^3+2*x^2+4*x+3, y*x^3+3*y*x^2-2*y, x+1]]
 @eprog

Function: ellisogenyapply
Class: basic
Section: elliptic_curves
C-Name: ellisogenyapply
Prototype: GG
Help: ellisogenyapply(f, g): given an isogeny f and g either a point P (in the
 domain of f) or an isogeny, apply f to g: return the image of P under f or
 the composite isogeny f o g.
Doc: 
 Given an isogeny of elliptic curves $f:E'\to E$ (being the result of a call
 to \tet{ellisogeny}), apply $f$ to $g$:
 
 \item if $g$ is a point $P$ in the domain of $f$, return the image $f(P)$;
 
 \item if $g:E''\to E'$ is a compatible isogeny, return the composite
 isogeny $f \circ g:  E''\to E$.
 
 \bprog
 ? one = ffgen(101, 't)^0;
 ? E = ellinit([6, 53, 85, 32, 34] * one);
 ? P = [84, 71] * one;
 ? ellorder(E, P)
 %4 = 5
 ? [F, f] = ellisogeny(E, P);  \\ f: E->F = E/<P>
 ? ellisogenyapply(f, P)
 %6 = [0]
 ? F = ellinit(F);
 ? Q = [89, 44] * one;
 ? ellorder(F, Q)
 %9 = 2
 ? [G, g] = ellisogeny(F, Q); \\  g: F->G = F/<Q>
 ? gof = ellisogenyapply(g, f); \\ gof: E -> G
 @eprog

Function: ellisomat
Class: basic
Section: elliptic_curves
C-Name: ellisomat
Prototype: GD0,L,
Help: ellisomat(E, {fl=0}): E being an elliptic curve over Q, return a list of
 representatives of the isomorphism classes of elliptic curves isogenous to E,
 with the corresponding isogenies from E and their dual, and the matrix of the
 degrees of the isogenies between the curves. If the flag fl is 1, the
 isogenies are not computed, which saves time.
Doc: 
 Given an elliptic curve $E$ defined over $\Q$, compute representatives of the
 isomorphism classes of elliptic curves $\Q$-isogenous to $E$. The function
 returns a vector $[L,M]$ where $L$ is a list of triples $[E_i, f_i, g_i]$,
 where $E_i$ is an elliptic curve in $[a_4,a_6]$ form, $f_i: E \to E_i$
 is a rational isogeny, $g_i: E_i \to E$ is the dual isogeny of $f_i$,
 and $M$ is the matrix such that $M_{i,j}$ is the degree of the isogeny between
 $E_i$ and $E_j$. Furthermore the first curve $E_1$ is isomorphic to $E$
 by $f_1$. If the flag $\var{fl}=1$, the $f_i$ and $g_i$ are not computed,
 which saves time, and $L$ is the list of the curves $E_i$.
 \bprog
 ? E = ellinit("14a1");
 ? [L,M] = ellisomat(E);
 ? LE = apply(x->x[1], L)  \\ list of curves
 %3 = [[215/48,-5291/864],[-675/16,6831/32],[-8185/48,-742643/864],
      [-1705/48,-57707/864],[-13635/16,306207/32],[-131065/48,-47449331/864]]
 ? L[2][2]  \\ isogeny f_2
 %4 = [x^3+3/4*x^2+19/2*x-311/12,
       1/2*x^4+(y+1)*x^3+(y-4)*x^2+(-9*y+23)*x+(55*y+55/2),x+1/3]
 ? L[2][3]  \\ dual isogeny g_2
 %5 = [1/9*x^3-1/4*x^2-141/16*x+5613/64,
       -1/18*x^4+(1/27*y-1/3)*x^3+(-1/12*y+87/16)*x^2+(49/16*y-48)*x
       +(-3601/64*y+16947/512),x-3/4]
 ? apply(E->ellidentify(ellinit(E))[1][1], LE)
 %6 = ["14a1","14a4","14a3","14a2","14a6","14a5"]
 ? M
 %7 =
 [1  3  3 2  6  6]
 
 [3  1  9 6  2 18]
 
 [3  9  1 6 18  2]
 
 [2  6  6 1  3  3]
 
 [6  2 18 3  1  9]
 
 [6 18  2 3  9  1]
 @eprog

Function: ellisoncurve
Class: basic
Section: elliptic_curves
C-Name: ellisoncurve
Prototype: GG
Help: ellisoncurve(E,z): true(1) if z is on elliptic curve E, false(0) if not.
Doc: gives 1 (i.e.~true) if the point $z$ is on the elliptic curve $E$, 0
 otherwise. If $E$ or $z$ have imprecise coefficients, an attempt is made to
 take this into account, i.e.~an imprecise equality is checked, not a precise
 one. It is allowed for $z$ to be a vector of points in which case a vector
 (of the same type) is returned.
Variant: Also available is \fun{int}{oncurve}{GEN E, GEN z} which does not
 accept vectors of points.

Function: ellissupersingular
Class: basic
Section: elliptic_curves
C-Name: ellissupersingular
Prototype: iGDG
Help: ellissupersingular(E,{p}): decide whether the elliptic curve E, defined
 over a number field or a finite field, is supersingular at p or not.
Doc: 
 Return 1 if the elliptic curve $E$ defined over a number field
 or a finite field is supersingular at $p$, and $0$ otherwise.
 If the curve is defined over a number field, $p$ must be explicitly given,
 and must be a prime number, resp.~a maximal ideal, if the curve is defined
 over $\Q$, resp.~a general number field: we return $1$ if and only if $E$
 has supersingular good reduction at $p$.
 
 Alternatively, $E$ can be given by its $j$-invariant in a finite field. In
 this case $p$ must be omitted.
 \bprog
 ? g = ffprimroot(ffgen(7^5))
 %1 = x^3 + 2*x^2 + 3*x + 1
 ? [g^n | n <- [1 .. 7^5 - 1], ellissupersingular(g^n)]
 %2 = [6]
 
 ? K = nfinit(y^3-2); P = idealprimedec(K, 2)[1];
 ? E = ellinit([y,1], K);
 ? ellissupersingular(E, P)
 %5 = 1
 @eprog
Variant: Also available is
 \fun{int}{elljissupersingular}{GEN j} where $j$ is a $j$-invariant of a curve
 over a finite field.

Function: ellj
Class: basic
Section: elliptic_curves
C-Name: jell
Prototype: Gp
Help: ellj(x): elliptic j invariant of x.
Doc: 
 elliptic $j$-invariant. $x$ must be a complex number
 with positive imaginary part, or convertible into a power series or a
 $p$-adic number with positive valuation.

Function: elllocalred
Class: basic
Section: elliptic_curves
C-Name: elllocalred
Prototype: GG
Help: elllocalred(E,p): E being an elliptic curve, returns
 [f,kod,[u,r,s,t],c], where f is the conductor's exponent, kod is the Kodaira
 type for E at p, [u,r,s,t] is the change of variable needed to make E
 minimal at p, and c is the local Tamagawa number c_p.
Doc: 
 calculates the \idx{Kodaira} type of the local fiber of the elliptic curve
 $E$ at $p$. $E$ must be an \kbd{ell} structure as output by
 \kbd{ellinit}, over $\Q$ ($p$ a rational prime) or a number field $K$ ($p$
 a maximal ideal given by a \kbd{prid} structure), and is assumed to have all
 its coefficients $a_i$ integral.
 The result is a 4-component vector $[f,kod,v,c]$. Here $f$ is the exponent of
 $p$ in the arithmetic conductor of $E$, and $kod$ is the Kodaira type which
 is coded as follows:
 
 1 means good reduction (type I$_0$), 2, 3 and 4 mean types II, III and IV
 respectively, $4+\nu$ with $\nu>0$ means type I$_\nu$;
 finally the opposite values $-1$, $-2$, etc.~refer to the starred types
 I$_0^*$, II$^*$, etc. The third component $v$ is itself a vector $[u,r,s,t]$
 giving the coordinate changes done during the local reduction;
 $u = 1$ if and only if the given equation was already minimal at $p$.
 Finally, the last component $c$ is the local \idx{Tamagawa number} $c_p$.

Function: elllog
Class: basic
Section: elliptic_curves
C-Name: elllog
Prototype: GGGDG
Help: elllog(E,P,G,{o}): return the discrete logarithm of the point P of
 the elliptic curve E in base G. If present, o represents the order of G.
 If not present, assume that G generates the curve.
Doc: given two points $P$ and $G$ on the elliptic curve $E/\F_q$, returns the
 discrete logarithm of $P$ in base $G$, i.e. the smallest non-negative
 integer $n$ such that $P = [n]G$.
 See \tet{znlog} for the limitations of the underlying discrete log algorithms.
 If present, $o$ represents the order of $G$, see \secref{se:DLfun};
 the preferred format for this parameter is \kbd{[N, factor(N)]}, where $N$
 is  the order of $G$.
 
 If no $o$ is given, assume that $G$ generates the curve.
 The function also assumes that $P$ is a multiple of $G$.
 \bprog
 ? a = ffgen(ffinit(2,8),'a);
 ? E = ellinit([a,1,0,0,1]);  \\ over F_{2^8}
 ? x = a^3; y = ellordinate(E,x)[1];
 ? P = [x,y]; G = ellmul(E, P, 113);
 ? ord = [242, factor(242)]; \\ P generates a group of order 242. Initialize.
 ? ellorder(E, G, ord)
 %4 = 242
 ? e = elllog(E, P, G, ord)
 %5 = 15
 ? ellmul(E,G,e) == P
 %6 = 1
 @eprog

Function: elllseries
Class: basic
Section: elliptic_curves
C-Name: elllseries
Prototype: GGDGp
Help: elllseries(E,s,{A=1}): L-series at s of the elliptic curve E, where A
 a cut-off point close to 1.
Doc: 
 This function is deprecated, use \kbd{lfun(E,s)} instead.
 
 $E$ being an elliptic curve, given by an arbitrary model over $\Q$ as output
 by \kbd{ellinit}, this function computes the value of the $L$-series of $E$ at
 the (complex) point $s$. This function uses an $O(N^{1/2})$ algorithm, where
 $N$ is the conductor.
 
 The optional parameter $A$ fixes a cutoff point for the integral and is best
 left omitted; the result must be independent of $A$, up to
 \kbd{realprecision}, so this allows to check the function's accuracy.
Obsolete: 2016-08-08

Function: ellminimalmodel
Class: basic
Section: elliptic_curves
C-Name: ellminimalmodel
Prototype: GD&
Help: ellminimalmodel(E,{&v}): determines whether the elliptic curve E defined
 over a number field admits a global minimal model. If so return it
 and sets v to the corresponding change of variable. Else return the
 (non-principal) Weierstrass class of E.
Doc: Let $E$ be an \kbd{ell} structure over a number field $K$. This function
 determines whether $E$ admits a global minimal integral model. If so, it
 returns it and sets $v = [u,r,s,t]$ to the corresponding change of variable:
 the return value is identical to that of \kbd{ellchangecurve(E, v)}.
 
 Else return the (non-principal) Weierstrass class of $E$, i.e. the class of
 $\prod \goth{p}^{(v_{\goth{p}}{\Delta} - \delta_{\goth{p}}) / 12}$ where
 $\Delta = \kbd{E.disc}$ is the model's discriminant and
 $\goth{p} ^ \delta_{\goth{p}}$ is the local minimal discriminant.
 This function requires either that $E$ be defined
 over the rational field $\Q$ (with domain $D = 1$ in \kbd{ellinit}),
 in which case a global minimal model always exists, or over a number
 field given by a \var{bnf} structure. The Weierstrass class is given in
 \kbd{bnfisprincipal} format, i.e. in terms of the \kbd{K.gen} generators.
 
 The resulting model has integral coefficients and is everywhere minimal, the
 coefficients $a_1$ and $a_3$ are reduced modulo $2$ (in terms of the fixed
 integral basis \kbd{K.zk}) and $a_2$ is reduced modulo $3$. Over $\Q$, we
 further require that $a_1$ and $a_3$ be $0$ or $1$, that $a_2$ be $0$ or $\pm
 1$ and that $u > 0$ in the change of variable: both the model and the change
 of variable $v$ are then unique.\sidx{minimal model}
 
 \bprog
 ? e = ellinit([6,6,12,55,233]);  \\ over Q
 ? E = ellminimalmodel(e, &v);
 ? E[1..5]
 %3 = [0, 0, 0, 1, 1]
 ? v
 %4 = [2, -5, -3, 9]
 @eprog
 
 \bprog
 ? K = bnfinit(a^2-65);  \\ over a non-principal number field
 ? K.cyc
 %2 = [2]
 ? u = Mod(8+a, K.pol);
 ? E = ellinit([1,40*u+1,0,25*u^2,0], K);
 ? ellminimalmodel(E) \\ no global minimal model exists over Z_K
 %6 = [1]~
 @eprog

Function: ellminimaltwist
Class: basic
Section: elliptic_curves
C-Name: ellminimaltwist0
Prototype: GD0,L,
Help: ellminimaltwist(E, {flag=0}): E being an elliptic curve defined over Q, return
 a discriminant D such the twist of E by D is minimal among all possible quadratic
 twists, i.e.  if flag=0, its minimal model has minimal discriminant,
 or if flag=1, it has minimal conductor.
Doc: Let $E$ be an elliptic curve defined over $\Q$, return
 a discriminant $D$ such that the twist of $E$ by $D$ is minimal among all
 possible quadratic twists, i.e. if $\fl=0$, its minimal model has minimal
 discriminant, or if $\fl=1$, it has minimal conductor.
 
 In the example below, we find a curve with $j$-invariant $3$ and minimal
 conductor.
 \bprog
 ? E=ellminimalmodel(ellinit(ellfromj(3)));
 ? ellglobalred(E)[1]
 %2 = 357075
 ? D = ellminimaltwist(E,1)
 %3 = -15
 ? E2=ellminimalmodel(ellinit(elltwist(E,D)));
 ? ellglobalred(E2)[1]
 %5 = 14283
 @eprog
Variant: Also available are
 \fun{GEN}{ellminimaltwist}{E} for $\fl=0$, and
 \fun{GEN}{ellminimaltwistcond}{E} for $\fl=1$.

Function: ellmoddegree
Class: basic
Section: elliptic_curves
C-Name: ellmoddegree
Prototype: Gb
Help: ellmoddegree(e): e being an elliptic curve defined over Q output by
  ellinit, compute the modular degree of e divided by the square of the
  Manin constant. Return [D, err], where D is a rational number and
  err is the exponent of the truncation error.
Doc: $e$ being an elliptic curve defined over $\Q$ output by \kbd{ellinit},
  compute the modular degree of $e$ divided by the square of
  the Manin constant. Return $[D, err]$, where $D$ is a rational number and
  err is exponent of the truncation error.

Function: ellmodulareqn
Class: basic
Section: elliptic_curves
C-Name: ellmodulareqn
Prototype: LDnDn
Help: ellmodulareqn(N,{x},{y}): given a prime N < 500, return a vector [P, t]
 where P(x,y) is a modular equation of level N. This requires the package
 seadata. The equation is either of canonical type (t=0) or of Atkin type (t=1).
Doc: given a prime $N < 500$, return a vector $[P,t]$ where $P(x,y)$
 is a modular equation of level $N$, i.e.~a bivariate polynomial with integer
 coefficients; $t$ indicates the type of this equation: either
 \emph{canonical} ($t = 0$) or \emph{Atkin} ($t = 1$). This function requires
 the \kbd{seadata} package and its only use is to give access to the package
 contents. See \tet{polmodular} for a more general and more flexible function.
 
 Let $j$ be the $j$-invariant function. The polynomial $P$ satisfies
 the functional equation,
 $$ P(f,j) = P(f \mid W_N, j \mid W_N) = 0 $$
 for some modular function $f = f_N$ (hand-picked for each fixed $N$ to
 minimize its size, see below), where $W_N(\tau) = -1 / (N\*\tau)$ is the
 Atkin-Lehner involution. These two equations allow to compute the values of
 the classical modular polynomial $\Phi_N$, such that $\Phi_N(j(\tau),
 j(N\tau)) = 0$, while being much smaller than the latter. More precisely, we
 have $j(W_N(\tau)) = j(N\*\tau)$; the function $f$ is invariant under
 $\Gamma_0(N)$ and also satisfies
 
 \item for Atkin type: $f \mid W_N = f$;
 
 \item for canonical type: let $s = 12/\gcd(12,N-1)$, then
 $f \mid W_N = N^s / f$. In this case, $f$ has a simple definition:
 $f(\tau) = N^s \* \big(\eta(N\*\tau) / \eta(\tau) \big)^{2\*s}$,
 where $\eta$ is Dedekind's eta function.
 
 The following GP function returns values of the classical modular polynomial
 by eliminating $f_N(\tau)$ in the above functional equation,
 for $N\leq 31$ or $N\in\{41,47,59,71\}$.
 
 \bprog
 classicaleqn(N, X='X, Y='Y)=
 {
   my([P,t] = ellmodulareqn(N), Q, d);
   if (poldegree(P,'y) > 2, error("level unavailable in classicaleqn"));
   if (t == 0, \\ Canonical
     my(s = 12/gcd(12,N-1));
     Q = 'x^(N+1) * substvec(P,['x,'y],[N^s/'x,Y]);
     d = N^(s*(2*N+1)) * (-1)^(N+1);
   , \\ Atkin
     Q = subst(P,'y,Y);
     d = (X-Y)^(N+1));
   polresultant(subst(P,'y,X), Q) / d;
 }
 @eprog

Function: ellmul
Class: basic
Section: elliptic_curves
C-Name: ellmul
Prototype: GGG
Help: ellmul(E,z,n): n times the point z on elliptic curve E (n in Z).
Doc: 
 computes $[n]z$, where $z$ is a point on the elliptic curve $E$. The
 exponent $n$ is in $\Z$, or may be a complex quadratic integer if the curve $E$
 has complex multiplication by $n$ (if not, an error message is issued).
 \bprog
 ? Ei = ellinit([1,0]); z = [0,0];
 ? ellmul(Ei, z, 10)
 %2 = [0]     \\ unsurprising: z has order 2
 ? ellmul(Ei, z, I)
 %3 = [0, 0]  \\ Ei has complex multiplication by Z[i]
 ? ellmul(Ei, z, quadgen(-4))
 %4 = [0, 0]  \\ an alternative syntax for the same query
 ? Ej  = ellinit([0,1]); z = [-1,0];
 ? ellmul(Ej, z, I)
   ***   at top-level: ellmul(Ej,z,I)
   ***                 ^--------------
   *** ellmul: not a complex multiplication in ellmul.
 ? ellmul(Ej, z, 1+quadgen(-3))
 %6 = [1 - w, 0]
 @eprog
 The simple-minded algorithm for the CM case assumes that we are in
 characteristic $0$, and that the quadratic order to which $n$ belongs has
 small discriminant.

Function: ellneg
Class: basic
Section: elliptic_curves
C-Name: ellneg
Prototype: GG
Help: ellneg(E,z): opposite of the point z on elliptic curve E.
Doc: 
 Opposite of the point $z$ on elliptic curve $E$.

Function: ellnonsingularmultiple
Class: basic
Section: elliptic_curves
C-Name: ellnonsingularmultiple
Prototype: GG
Help: ellnonsingularmultiple(E,P): given E/Q and P in E(Q), returns the pair
 [R,n] where n is the least positive integer such that R = [n]P has
 everywhere good reduction. More precisely, its image in a minimal model
 is everywhere non-singular.
Doc: given an elliptic curve $E/\Q$ (more precisely, a model defined over $\Q$
 of a curve) and a rational point $P \in E(\Q)$, returns the pair $[R,n]$,
 where $n$ is the least positive integer such that $R := [n]P$ has good
 reduction at every prime. More precisely, its image in a minimal model is
 everywhere non-singular.
 \bprog
 ? e = ellinit("57a1"); P = [2,-2];
 ? ellnonsingularmultiple(e, P)
 %2 = [[1, -1], 2]
 ? e = ellinit("396b2"); P = [35, -198];
 ? [R,n] = ellnonsingularmultiple(e, P);
 ? n
 %5 = 12
 @eprog

Function: ellorder
Class: basic
Section: elliptic_curves
C-Name: ellorder
Prototype: GGDG
Help: ellorder(E,z,{o}): order of the point z on the elliptic curve E over
 a number field or a finite field, 0 if non-torsion. The parameter o,
 if present, represents a non-zero multiple of the order of z.
Doc: gives the order of the point $z$ on the elliptic
 curve $E$, defined over a finite field or a number field.
 Return (the impossible value) zero if the point has infinite order.
 \bprog
 ? E = ellinit([-157^2,0]);  \\ the "157-is-congruent" curve
 ? P = [2,2]; ellorder(E, P)
 %2 = 2
 ? P = ellheegner(E); ellorder(E, P) \\ infinite order
 %3 = 0
 ? K = nfinit(polcyclo(11,t)); E=ellinit("11a3", K); T = elltors(E);
 ? ellorder(E, T.gen[1])
 %5 = 25
 ? E = ellinit(ellfromj(ffgen(5^10)));
 ? ellcard(E)
 %7 = 9762580
 ? P = random(E); ellorder(E, P)
 %8 = 4881290
 ? p = 2^160+7; E = ellinit([1,2], p);
 ? N = ellcard(E)
 %9 = 1461501637330902918203686560289225285992592471152
 ? o = [N, factor(N)];
 ? for(i=1,100, ellorder(E,random(E)))
 time = 260 ms.
 @eprog
 The parameter $o$, is now mostly useless, and kept for backward
 compatibility. If present, it represents a non-zero multiple of the order
 of $z$, see \secref{se:DLfun}; the preferred format for this parameter is
 \kbd{[ord, factor(ord)]}, where \kbd{ord} is the cardinality of the curve.
 It is no longer needed since PARI is now able to compute it over large
 finite fields (was restricted to small prime fields at the time this feature
 was introduced), \emph{and} caches the result in $E$ so that it is computed
 and factored only once. Modifying the last example, we see that including
 this extra parameter provides no improvement:
 \bprog
 ? o = [N, factor(N)];
 ? for(i=1,100, ellorder(E,random(E),o))
 time = 260 ms.
 @eprog
Variant: The obsolete form \fun{GEN}{orderell}{GEN e, GEN z} should no longer be
 used.

Function: ellordinate
Class: basic
Section: elliptic_curves
C-Name: ellordinate
Prototype: GGp
Help: ellordinate(E,x): y-coordinates corresponding to x-ordinate x on
 elliptic curve E.
Doc: 
 gives a 0, 1 or 2-component vector containing
 the $y$-coordinates of the points of the curve $E$ having $x$ as
 $x$-coordinate.

Function: ellpadicL
Class: basic
Section: elliptic_curves
C-Name: ellpadicL
Prototype: GGLDGD0,L,DG
Help: ellpadicL(E, p, n, {s = 0}, {r = 0}, {D = 1}): returns the value
 on a character of Z_p^* represented by an integer s or a vector [s1,s2]
 of the derivative of order r of the p-adic L-function of
 the elliptic curve E (twisted by D, if present).
Doc: Returns the value (or $r$-th derivative) on a character $\chi^s$ of
 $\Z_p^*$ of the $p$-adic $L$-function of the elliptic curve $E/\Q$, twisted by
 $D$, given modulo $p^n$.
 
 \misctitle{Characters} The set of continuous characters of
 $\text{Gal}(\Q(\mu_{p^{\infty}})/ \Q)$ is identified to $\Z_p^*$ via the
 cyclotomic character $\chi$ with values in $\overline{\Q_p}^*$. Denote by
 $\tau:\Z_p^*\to\Z_p^*$ the Teichm\"uller character, with values
 in the $(p-1)$-th roots of $1$ for $p\neq 2$, and $\{-1,1\}$ for $p = 2$;
 finally, let
 $\langle\chi\rangle =\chi \tau^{-1}$, with values in $1 + 2p\Z_p$.
 In GP, the continuous character of
 $\text{Gal}(\Q(\mu_{p^{\infty}})/ \Q)$ given by $\langle\chi\rangle^{s_1}
 \tau^{s_2}$ is represented by the pair of integers $s=(s_1,s_2)$, with $s_1
 \in \Z_p$ and $s_2 \bmod p-1$ for $p > 2$, (resp. mod $2$ for $p = 2$); $s$
 may be also an integer, representing $(s,s)$ or $\chi^s$.
 
 \misctitle{The $p$-adic $L$ function}
 The $p$-adic $L$ function $L_p$ is defined on the set of continuous
 characters of $\text{Gal}(\Q(\mu_{p^{\infty}})/ \Q)$, as $\int_{\Z_p^*}
 \chi^s d \mu$ for a certain $p$-adic distribution $\mu$ on $\Z_p^*$. The
 derivative is given by
 $$L_p^{(r)}(E, \chi^s) = \int_{\Z_p^*} \log_p^r(a) \chi^s(a) d\mu(a).$$
 More precisely:
 
 \item When $E$ has good supersingular reduction, $L_p$ takes its
 values in $\Q_p \otimes H^1_{dR}(E/\Q)$ and satisfies
 $$(1-p^{-1} F)^{-2} L_p(E, \chi^0)= (L(E,1) / \Omega) \cdot \omega$$
 where $F$ is the Frobenius, $L(E,1)$ is the value of the complex $L$
 function at $1$, $\omega$ is the N\'eron differential
 and $\Omega$ the attached period on $E(\R)$. Here, $\chi^0$ represents
 the trivial character.
 
 The function returns the components of $L_p^{(r)}(E,\chi^s)$ in
 the basis $(\omega, F(\omega))$.
 
 \item When $E$ has ordinary good reduction, this method only defines
 the projection of $L_p(E,\chi^s)$ on the $\alpha$-eigenspace,
 where $\alpha$ is the unit eigenvalue for $F$. This is what the function
 returns. We have
 $$(1- \alpha^{-1})^{-2} L_{p,\alpha}(E,\chi^0)= L(E,1) / \Omega.$$
 
 Two supersingular examples:
 \bprog
 ? cxL(e) = bestappr( ellL1(e) / e.omega[1] );
 
 ? e = ellinit("17a1"); p=3; \\ supersingular, a3 = 0
 ? L = ellpadicL(e,p,4);
 ? F = [0,-p;1,ellap(e,p)]; \\ Frobenius matrix in the basis (omega,F(omega))
 ? (1-p^(-1)*F)^-2 * L / cxL(e)
 %5 = [1 + O(3^5), O(3^5)]~ \\ [1,0]~
 
 ? e = ellinit("116a1"); p=3; \\ supersingular, a3 != 0~
 ? L = ellpadicL(e,p,4);
 ? F = [0,-p; 1,ellap(e,p)];
 ? (1-p^(-1)*F)^-2*L~ / cxL(e)
 %9 = [1 + O(3^4), O(3^5)]~
 @eprog
 
 Good ordinary reduction:
 \bprog
 ? e = ellinit("17a1"); p=5; ap = ellap(e,p)
 %1 = -2 \\ ordinary
 ? L = ellpadicL(e,p,4)
 %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4)
 ? al = padicappr(x^2 - ap*x + p, ap + O(p^7))[1];
 ? (1-al^(-1))^(-2) * L / cxL(e)
 %4 = 1 + O(5^4)
 @eprog
 
 Twist and Teichm\"uller:
 \bprog
 ? e = ellinit("17a1"); p=5; \\ ordinary
 \\ 2nd derivative at tau^1, twist by -7
 ? ellpadicL(e, p, 4, [0,1], 2, -7)
 %2 = 2*5^2 + 5^3 + O(5^4)
 @eprog
 
 This function is a special case of \tet{mspadicL}, and it also appears
 as the first term of \tet{mspadicseries}:
 \bprog
 ? e = ellinit("17a1"); p=5;
 ? L = ellpadicL(e,p,4)
 %2 = 4 + 3*5 + 4*5^2 + 2*5^3 + O(5^4)
 ? [M,phi] = msfromell(e, 1);
 ? Mp = mspadicinit(M, p, 4);
 ? mu = mspadicmoments(Mp, phi);
 ? mspadicL(mu)
 %6 = 4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6)
 ? mspadicseries(mu)
 %7 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + O(5^6))
       + (3 + 3*5 + 5^2 + 5^3 + O(5^4))*x
       + (2 + 3*5 + 5^2 + O(5^3))*x^2
       + (3 + 4*5 + 4*5^2 + O(5^3))*x^3
       + (3 + 2*5 + O(5^2))*x^4 + O(x^5)
 @eprog\noindent These are more cumbersome than \kbd{ellpadicL} but allow to
 compute at different characters, or successive derivatives, or to
 twist by a quadratic character essentially for the cost of a single call to
 \kbd{ellpadicL} due to precomputations.

Function: ellpadicfrobenius
Class: basic
Section: elliptic_curves
C-Name: ellpadicfrobenius
Prototype: GLL
Help: ellpadicfrobenius(E,p,n): matrix of the Frobenius at p>2 in the standard
 basis of H^1_dR(E) to absolute p-adic precision p^n.
Doc: If $p>2$ is a prime and $E$ is a elliptic curve on $\Q$ with good
 reduction at $p$, return the matrix of the Frobenius endomorphism $\varphi$ on
 the crystalline module $D_p(E)= \Q_p \otimes H^1_{dR}(E/\Q)$ with respect to
 the basis of the given model $(\omega, \eta=x\*\omega)$, where
 $\omega = dx/(2\*y+a_1\*x+a_3)$ is the invariant differential.
 The characteristic polynomial of $\varphi$ is $x^2 - a_p\*x + p$.
 The matrix is computed to absolute $p$-adic precision $p^n$.
 
 \bprog
 ? E = ellinit([1,-1,1,0,0]);
 ? F = ellpadicfrobenius(E,5,3);
 ? lift(F)
 %3 =
 [120 29]
 
 [ 55  5]
 ? charpoly(F)
 %4 = x^2 + O(5^3)*x + (5 + O(5^3))
 ? ellap(E, 5)
 %5 = 0
 @eprog

Function: ellpadicheight
Class: basic
Section: elliptic_curves
C-Name: ellpadicheight0
Prototype: GGLGDG
Help: ellpadicheight(E, p,n, P,{Q}): E elliptic curve/Q, P in E(Q),
 p prime, n an integer; returns the cyclotomic p-adic heights of P.
 Resp. the value of the attached bilinear form at (P,Q).
Doc: cyclotomic $p$-adic height of the rational point $P$ on the elliptic curve
 $E$ (defined over $\Q$), given to $n$ $p$-adic digits.
 If the argument $Q$ is present, computes the value of the bilinear
 form $(h(P+Q)-h(P-Q)) / 4$.
 
 Let $D_{dR}(E) := H^1_{dR}(E) \otimes_\Q \Q_p$ be the $\Q_p$ vector space
 spanned by $\omega$
 (invariant differential $dx/(2y+a_1x+a3)$ related to the given model) and
 $\eta = x \omega$. Then the cyclotomic $p$-adic height associates to
 $P\in E(\Q)$ an element $f \omega + g\eta$ in $D_{dR}$.
 This routine returns the vector $[f, g]$ to $n$ $p$-adic digits.
 
 If $P\in E(\Q)$ is in the kernel of reduction mod $p$ and if its reduction
 at all finite places is non singular, then $g = -(\log_E P)^2$, where
 $\log_E$ is the logarithm for the formal group of $E$ at $p$.
 
 If furthermore the model is of the form $Y^2 = X^3 + a X + b$ and $P = (x,y)$,
 then
   $$ f = \log_p(\kbd{denominator}(x)) - 2 \log_p(\sigma(P))$$
 where $\sigma(P)$ is given by \kbd{ellsigma}$(E,P)$.
 
 Recall (\emph{Advanced topics in the arithmetic of elliptic
 curves}, Theorem~3.2) that the local height function over the complex numbers
 is of the form
   $$ \lambda(z) = -\log (|\kbd{E.disc}|) / 6 + \Re(z \eta(z)) - 2 \log(
   \sigma(z). $$
 (N.B. our normalization for local and global heights is twice that of
 Silverman's).
 \bprog
  ? E = ellinit([1,-1,1,0,0]); P = [0,0];
  ? ellpadicheight(E,5,4, P)
  %2 = [3*5 + 5^2 + 2*5^3 + O(5^4), 5^2 + 4*5^4 + O(5^6)]
  ? E = ellinit("11a1"); P = [5,5]; \\ torsion point
  ? ellpadicheight(E,19,6, P)
  %4 = O(19^6)
  ? E = ellinit([0,0,1,-4,2]); P = [-2,1];
  ? ellpadicheight(E,3,5, P)
  %6 = [2*3^2 + 2*3^3 + 3^4 + O(3^5), 2*3^2 + 3^4 + 2*3^5 + 3^6 + O(3^7)]
  ? ellpadicheight(E,3,5, P, elladd(E,P,P))
 @eprog
 
 One can replace the parameter $p$ prime by a vector $[p,[a,b]]$, in which
 case the routine returns the $p$-adic number $af + bg$.
 
 When $E$ has good ordinary reduction at $p$, the ``canonical''
 $p$-adic height is given by
 \bprog
 s2 = ellpadics2(E,p,n);
 ellpadicheight(E, [p,[1,-s2]], n, P)
 @eprog\noindent Since $s_2$ does not depend on $P$, it is preferable to
 compute it only once:
 \bprog
 ? E = ellinit("5077a1"); p = 5; n = 7;
 ? s2 = ellpadics2(E,p,n);
 ? M = ellpadicheightmatrix(E,[p,[1,-s2]], n, E.gen);
 ? matdet(M)   \\ p-adic regulator
 %4 = 5 + 5^2 + 4*5^3 + 2*5^4 + 2*5^5 + 5^6 + O(5^7)
 @eprog

Function: ellpadicheightmatrix
Class: basic
Section: elliptic_curves
C-Name: ellpadicheightmatrix
Prototype: GGLG
Help: ellpadicheightmatrix(E,p,n,v): gives the height-pairing matrix for vector
 of points v on elliptic curve E.
Doc: $v$ being a vector of points, this function outputs the Gram matrix of
 $v$ with respect to the cyclotomic $p$-adic height, given to $n$ $p$-adic
 digits; in other words, the $(i,j)$ component of the matrix is equal to
 \kbd{ellpadicheight}$(E,p,n, v[i],v[j]) = [f,g]$.
 
 See \tet{ellpadicheight}; in particular one can replace the parameter $p$
 prime by a vector $[p,[a,b]]$, in which case the routine returns the matrix
 containing the $p$-adic numbers $af + bg$.

Function: ellpadiclog
Class: basic
Section: elliptic_curves
C-Name: ellpadiclog
Prototype: GGLG
Help: ellpadiclog(E,p,n,P): returns the logarithm of P (in the kernel of
 reduction) to absolute p-adic precision p^n.
Doc: Given $E$ defined over $K = \Q$ or $\Q_p$ and $P = [x,y]$ on $E(K)$ in the
 kernel of reduction mod $p$, let $t(P) = -x/y$ be the formal group
 parameter; this function returns $L(t)$, where $L$ denotes the formal
 logarithm (mapping the formal group of $E$  to the additive formal group)
 attached to the canonical invariant differential:
 $dL = dx/(2y + a_1x + a_3)$.

Function: ellpadics2
Class: basic
Section: elliptic_curves
C-Name: ellpadics2
Prototype: GGL
Help: ellpadics2(E,p,n): returns s2 to absolute p-adic precision p^n.
Doc: If $p>2$ is a prime and $E/\Q$ is a elliptic curve with ordinary good
 reduction at $p$, returns the slope of the unit eigenvector
 of \kbd{ellpadicfrobenius(E,p,n)}, i.e. the action of Frobenius $\varphi$ on
 the crystalline module $D_p(E)= \Q_p \otimes H^1_{dR}(E/\Q)$ in the basis of
 the given model $(\omega, \eta=x\*\omega)$, where $\omega$ is the invariant
 differential $dx/(2\*y+a_1\*x+a_3)$. In other words, $\eta + s_2\omega$
 is an eigenvector for the unit eigenvalue of $\varphi$.
 
 This slope is the unique $c \in 3^{-1}\Z_p$ such that the odd solution
   $\sigma(t) = t + O(t^2)$ of
 $$ - d(\dfrac{1}{\sigma} \dfrac{d \sigma}{\omega})
  = (x(t) + c) \omega$$
 is in $t\Z_p[[t]]$.
 
 It is equal to $b_2/12 - E_2/12$ where $E_2$ is the value of the Katz
 $p$-adic Eisenstein series of weight 2 on $(E,\omega)$. This is
 used to construct a canonical $p$-adic height when $E$ has good ordinary
 reduction at $p$ as follows
 \bprog
 s2 = ellpadics2(E,p,n);
 h(E,p,n, P, s2) = ellpadicheight(E, [p,[1,-s2]],n, P);
 @eprog\noindent Since $s_2$ does not depend on the point $P$, we compute it
 only once.

Function: ellperiods
Class: basic
Section: elliptic_curves
C-Name: ellperiods
Prototype: GD0,L,p
Help: ellperiods(w, {flag = 0}): w describes a complex period lattice ([w1,w2]
 or an ellinit structure). Returns normalized periods [W1,W2] generating the
 same lattice such that tau := W1/W2 satisfies Im(tau) > 0 and lies in the
 standard fundamental domain for SL2. If flag is 1, the return value is
 [[W1,W2], [eta1,eta2]], where eta1, eta2 are the quasi-periods attached to
 [W1,W2], satisfying eta1 W2 - eta2 W1 = 2 I Pi.
Doc: Let $w$ describe a complex period lattice ($w = [w_1,w_2]$
 or an \kbd{ellinit} structure). Returns normalized periods $[W_1,W_2]$ generating
 the same lattice such that $\tau := W_1/W_2$ has positive imaginary part
 and lies in the standard fundamental domain for $\text{SL}_2(\Z)$.
 
 If $\fl = 1$, the function returns $[[W_1,W_2], [\eta_1,\eta_2]]$, where
 $\eta_1$ and $\eta_2$ are the quasi-periods attached to
 $[W_1,W_2]$, satisfying $\eta_1 W_2 - \eta_2 W_1 = 2 i \pi$.
 
 The output of this function is meant to be used as the first argument
 given to ellwp, ellzeta, ellsigma or elleisnum. Quasi-periods are
 needed by ellzeta and ellsigma only.

Function: ellpointtoz
Class: basic
Section: elliptic_curves
C-Name: zell
Prototype: GGp
Help: ellpointtoz(E,P): lattice point z corresponding to the point P on the
 elliptic curve E.
Doc: 
 if $E/\C \simeq \C/\Lambda$ is a complex elliptic curve ($\Lambda =
 \kbd{E.omega}$),
 computes a complex number $z$, well-defined modulo the lattice $\Lambda$,
 corresponding to the point $P$; i.e.~such that
  $P = [\wp_\Lambda(z),\wp'_\Lambda(z)]$
 satisfies the equation
 $$y^2 = 4x^3 - g_2 x - g_3,$$
 where $g_2$, $g_3$ are the elliptic invariants.
 
 If $E$ is defined over $\R$ and $P\in E(\R)$, we have more precisely, $0 \leq
 \Re(t) < w1$ and $0 \leq \Im(t) < \Im(w2)$, where $(w1,w2)$ are the real and
 complex periods of $E$.
 \bprog
 ? E = ellinit([0,1]); P = [2,3];
 ? z = ellpointtoz(E, P)
 %2 = 3.5054552633136356529375476976257353387
 ? ellwp(E, z)
 %3 = 2.0000000000000000000000000000000000000
 ? ellztopoint(E, z) - P
 %4 = [2.548947057811923643 E-57, 7.646841173435770930 E-57]
 ? ellpointtoz(E, [0]) \\ the point at infinity
 %5 = 0
 @eprog
 
 If $E/\Q_p$ has multiplicative reduction, then $E/\bar{\Q_p}$ is analytically
 isomorphic to $\bar{\Q}_p^*/q^\Z$ (Tate curve) for some $p$-adic integer $q$.
 The behaviour is then as follows:
 
 \item If the reduction is split ($E.\kbd{tate[2]}$ is a \typ{PADIC}), we have
 an isomorphism $\phi: E(\Q_p) \simeq \Q_p^*/q^\Z$ and the function returns
 $\phi(P)\in \Q_p$.
 
 \item If the reduction is \emph{not} split ($E.\kbd{tate[2]}$ is a
 \typ{POLMOD}), we only have an isomorphism $\phi: E(K) \simeq K^*/q^\Z$ over
 the unramified quadratic extension $K/\Q_p$. In this case, the output
 $\phi(P)\in K$ is a \typ{POLMOD}.
 \bprog
 ? E = ellinit([0,-1,1,0,0], O(11^5)); P = [0,0];
 ? [u2,u,q] = E.tate; type(u) \\ split multiplicative reduction
 %2 = "t_PADIC"
 ? ellmul(E, P, 5)  \\ P has order 5
 %3 = [0]
 ? z = ellpointtoz(E, [0,0])
 %4 = 3 + 11^2 + 2*11^3 + 3*11^4 + 6*11^5 + 10*11^6 + 8*11^7 + O(11^8)
 ? z^5
 %5 = 1 + O(11^9)
 ? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1];
 ? z = ellpointtoz(E,[x,y]); \\ t_POLMOD of t_POL with t_PADIC coeffs
 ? liftint(z) \\ lift all p-adics
 %8 = Mod(8*u + 7, u^2 + 437)
 @eprog

Function: ellpow
Class: basic
Section: elliptic_curves
C-Name: ellmul
Prototype: GGG
Help: ellpow(E,z,n): deprecated alias for ellmul.
Doc: deprecated alias for \kbd{ellmul}.
Obsolete: 2012-06-06

Function: ellrootno
Class: basic
Section: elliptic_curves
C-Name: ellrootno
Prototype: lGDG
Help: ellrootno(E,{p}): root number for the L-function of the elliptic
 curve E/Q at a prime p (including 0, for the infinite place); global root
 number if p is omitted.
Doc: $E$ being an \kbd{ell} structure over $\Q$ as output by \kbd{ellinit},
 this function computes the local root number of its $L$-series at the place
 $p$ (at the infinite place if $p = 0$). If $p$ is omitted, return the global
 root number. Note that the global root number is the sign of the functional
 equation and conjecturally is the parity of the rank of the
 \idx{Mordell-Weil group}. The equation for $E$ needs not be minimal at $p$,
 but if the model is already minimal the function will run faster.

Function: ellsea
Class: basic
Section: elliptic_curves
C-Name: ellsea
Prototype: GD0,U,
Help: ellsea(E,{tors=0}): computes the order of the group E(Fq)
 for the elliptic curve E, defined over a finite field,
 using SEA algorithm, with early abort for curves with non prime orders.
Doc: Let $E$ be an \var{ell} structure as output by \kbd{ellinit}, defined over
 a finite field $\F_q$. This low-level function computes the order of the
 group $E(\F_q)$ using the SEA algorithm; compared to the high-level
 function \kbd{ellcard}, which includes SEA among its choice of algorithms,
 the \kbd{tors} argument allows to speed up a search for curves having almost
 prime order.
 When \kbd{tors} is set to a non-zero value, the function returns $0$ as soon
 as it detects that the order has a small prime factor not dividing \kbd{tors};
 SEA considers modular polynomials of increasing prime degree $\ell$ and we
 return $0$ as soon as we hit an $\ell$ (coprime to \kbd{tors}) dividing
 $\#E(\F_q)$:
 \bprog
 ? ellsea(ellinit([1,1], 2^56+3477), 1)
 %1 = 72057594135613381
 ? forprime(p=2^128,oo, q = ellcard(ellinit([1,1],p)); if(isprime(q),break))
 time = 6,571 ms.
 ? forprime(p=2^128,oo, q = ellsea(ellinit([1,1],p),1);if(isprime(q),break))
 time = 522 ms.
 @eprog\noindent
 In particular, set \kbd{tors} to $1$ if you want a curve with prime order,
 to $2$ if you want to allow a cofactor which is a power of two (e.g. for
 Edwards's curves), etc. The early exit on bad curves yields a massive
 speedup compared to running the cardinal algorithm to completion.
 
 The following function returns a curve of prime order over $\F_p$.
 \bprog
 cryptocurve(p) =
 {
   while(1,
     my(E, N, j = Mod(random(p), p));
     E = ellinit(ellfromj(j));
     N = ellsea(E, 1); if(!N, continue);
     if (isprime(N), return(E));
     \\ try the quadratic twist for free
     if (isprime(2*p+2 - N), return(ellinit(elltwist(E))));
   );
 }
 ? p = randomprime([2^255, 2^256]);
 ? E = cryptocurve(p); \\ insist on prime order
 %2 = 47,447ms
 @eprog\noindent The same example without early abort (using \kbd{ellsea(E,1)}
 instead of \kbd{ellsea(E)}) runs for about 5 minutes before finding a
 suitable curve.
 
 The availability of the \kbd{seadata} package will speed up the computation,
 and is strongly recommended. The generic function \kbd{ellcard} should be
 preferred when you only want to compute the cardinal of a given curve without
 caring about it having almost prime order:
 
 \item If the characteristic is too small ($p \leq 7$) or the field
 cardinality is tiny ($q \leq 523$) the generic algorithm
 \kbd{ellcard} is used instead and the \kbd{tors} argument is ignored.
 (The reason for this is that SEA is not implemented for $p \leq 7$ and
 that if $q \leq 523$ it is likely to run into an infinite loop.)
 
 \item If the field cardinality is smaller than about $2^{50}$, the
 generic algorithm will be faster.
 
 \item Contrary to \kbd{ellcard}, \kbd{ellsea} does not store the computed
 cardinality in $E$.

Function: ellsearch
Class: basic
Section: elliptic_curves
C-Name: ellsearch
Prototype: G
Help: ellsearch(N): returns all curves in the elldata database matching
 constraint N:  given name (N = "11a1" or [11,0,1]),
 given isogeny class (N = "11a" or [11,0]), or
 given conductor (N = 11, "11", or [11]).
Doc: This function finds all curves in the \tet{elldata} database satisfying
 the constraint defined by the argument $N$:
 
 \item if $N$ is a character string, it selects a given curve, e.g.
 \kbd{"11a1"}, or curves in the given isogeny class, e.g. \kbd{"11a"}, or
 curves with given conductor, e.g. \kbd{"11"};
 
 \item if $N$ is a vector of integers, it encodes the same constraints
 as the character string above, according to the \tet{ellconvertname}
 correspondance, e.g. \kbd{[11,0,1]} for \kbd{"11a1"}, \kbd{[11,0]} for
 \kbd{"11a"} and \kbd{[11]} for \kbd{"11"};
 
 \item if $N$ is an integer, curves with conductor $N$ are selected.
 
 If $N$ codes a full curve name, for instance \kbd{"11a1"} or \kbd{[11,0,1]},
 the output format is $[N, [a_1,a_2,a_3,a_4,a_6], G]$ where
 $[a_1,a_2,a_3,a_4,a_6]$ are the coefficients of the Weierstrass equation of
 the curve and $G$ is a $\Z$-basis of the free part of the
 \idx{Mordell-Weil group} attached to the curve.
 \bprog
 ? ellsearch("11a3")
 %1 = ["11a3", [0, -1, 1, 0, 0], []]
 ? ellsearch([11,0,3])
 %2 = ["11a3", [0, -1, 1, 0, 0], []]
 @eprog\noindent
 
 If $N$ is not a full curve name, then the output is a vector of all matching
 curves in the above format:
 \bprog
 ? ellsearch("11a")
 %1 = [["11a1", [0, -1, 1, -10, -20], []],
       ["11a2", [0, -1, 1, -7820, -263580], []],
       ["11a3", [0, -1, 1, 0, 0], []]]
 ? ellsearch("11b")
 %2 = []
 @eprog
Variant: Also available is \fun{GEN}{ellsearchcurve}{GEN N} that only
 accepts complete curve names (as \typ{STR}).

Function: ellsigma
Class: basic
Section: elliptic_curves
C-Name: ellsigma
Prototype: GDGD0,L,p
Help: ellsigma(L,{z='x},{flag=0}): computes the value at z of the Weierstrass
 sigma function attached to the lattice w, as given by ellperiods(,1).
 If flag = 1, returns an arbitrary determination of the logarithm of sigma.
Doc: Computes the value at $z$ of the Weierstrass $\sigma$ function attached to
 the lattice $L$ as given by \tet{ellperiods}$(,1)$: including quasi-periods
 is useful, otherwise there are recomputed from scratch for each new $z$.
 $$ \sigma(z, L) = z \prod_{\omega\in L^*} \left(1 -
 \dfrac{z}{\omega}\right)e^{\dfrac{z}{\omega} + \dfrac{z^2}{2\omega^2}}.$$
 It is also possible to directly input $L = [\omega_1,\omega_2]$,
 or an elliptic curve $E$ as given by \kbd{ellinit} ($L = \kbd{E.omega}$).
 \bprog
 ? w = ellperiods([1,I], 1);
 ? ellsigma(w, 1/2)
 %2 = 0.47494937998792065033250463632798296855
 ? E = ellinit([1,0]);
 ? ellsigma(E) \\ at 'x, implicitly at default seriesprecision
 %4 = x + 1/60*x^5 - 1/10080*x^9 - 23/259459200*x^13 + O(x^17)
 @eprog
 
 If $\fl=1$, computes an arbitrary determination of $\log(\sigma(z))$.

Function: ellsub
Class: basic
Section: elliptic_curves
C-Name: ellsub
Prototype: GGG
Help: ellsub(E,z1,z2): difference of the points z1 and z2 on elliptic curve E.
Doc: 
 difference of the points $z1$ and $z2$ on the
 elliptic curve corresponding to $E$.

Function: elltaniyama
Class: basic
Section: elliptic_curves
C-Name: elltaniyama
Prototype: GDP
Help: elltaniyama(E, {d = seriesprecision}): modular parametrization of
 elliptic curve E/Q.
Doc: 
 computes the modular parametrization of the elliptic curve $E/\Q$,
 where $E$ is an \kbd{ell} structure as output by \kbd{ellinit}. This returns
 a two-component vector $[u,v]$ of power series, given to $d$ significant
 terms (\tet{seriesprecision} by default), characterized by the following two
 properties. First the point $(u,v)$ satisfies the equation of the elliptic
 curve. Second, let $N$ be the conductor of $E$ and $\Phi: X_0(N)\to E$
 be a modular parametrization; the pullback by $\Phi$ of the
 N\'eron differential $du/(2v+a_1u+a_3)$ is equal to $2i\pi
 f(z)dz$, a holomorphic differential form. The variable used in the power
 series for $u$ and $v$ is $x$, which is implicitly understood to be equal to
 $\exp(2i\pi z)$.
 
 The algorithm assumes that $E$ is a \emph{strong} \idx{Weil curve}
 and that the Manin constant is equal to 1: in fact, $f(x) = \sum_{n > 0}
 \kbd{ellan}(E, n) x^n$.

Function: elltatepairing
Class: basic
Section: elliptic_curves
C-Name: elltatepairing
Prototype: GGGG
Help: elltatepairing(E, P, Q, m): computes the Tate pairing of the two points
 P and Q on the elliptic curve E. The point P must be of m-torsion.
Doc: Computes the Tate pairing of the two points $P$ and $Q$ on the elliptic
 curve $E$. The point $P$ must be of $m$-torsion.

Function: elltors
Class: basic
Section: elliptic_curves
C-Name: elltors
Prototype: G
Help: elltors(E): torsion subgroup of elliptic curve E: order, structure,
 generators.
Doc: 
 if $E$ is an elliptic curve defined over a number field or a finite field,
 outputs the torsion subgroup of $E$ as a 3-component vector \kbd{[t,v1,v2]},
 where \kbd{t} is the order of the torsion group, \kbd{v1} gives the structure
 of the torsion group as a product of cyclic groups (sorted by decreasing
 order), and \kbd{v2} gives generators for these cyclic groups. $E$ must be an
 \kbd{ell} structure as output by \kbd{ellinit}.
 \bprog
 ?  E = ellinit([-1,0]);
 ?  elltors(E)
 %1 = [4, [2, 2], [[0, 0], [1, 0]]]
 @eprog\noindent
 Here, the torsion subgroup is isomorphic to $\Z/2\Z \times \Z/2\Z$, with
 generators $[0,0]$ and $[1,0]$.

Function: elltwist
Class: basic
Section: elliptic_curves
C-Name: elltwist
Prototype: GDG
Help: elltwist(E,{P}): returns the coefficients [a1,a2,a3,a4,a6] of
 the twist of the elliptic curve E by the quadratic extension defined by
 P (when P is a polynomial of degree 2) or quadpoly(P) (when P is an integer).
 If E is defined over a finite field, then P can be omitted.
Doc: returns the coefficients $[a_1,a_2,a_3,a_4,a_6]$ of the twist of the
 elliptic curve $E$ by the quadratic extension of the coefficient ring
 defined by $P$ (when $P$ is a polynomial) or \kbd{quadpoly(P)} when $P$ is an
 integer.  If $E$ is defined over a finite field, then $P$ can be omitted,
 in which case a random model of the unique non-trivial twist is returned.
 If $E$ is defined over a number field, the model should be replaced by a
 minimal model (if one exists).
 
 Example: Twist by discriminant $-3$:
 \bprog
 ? elltwist(ellinit([0,a2,0,a4,a6]),-3)
 %1 = [0,-3*a2,0,9*a4,-27*a6]
 @eprog
 Twist by the Artin-Shreier extension given by $x^2+x+T$ in
 characteristic $2$:
 \bprog
 ? lift(elltwist(ellinit([a1,a2,a3,a4,a6]*Mod(1,2)),x^2+x+T))
 %1 = [a1,a2+a1^2*T,a3,a4,a6+a3^2*T]
 @eprog
 Twist of an elliptic curve defined over a finite field:
 \bprog
 ? E=ellinit([1,7]*Mod(1,19));lift(elltwist(E))
 %1 = [0,0,0,11,12]
 @eprog

Function: ellweilpairing
Class: basic
Section: elliptic_curves
C-Name: ellweilpairing
Prototype: GGGG
Help: ellweilpairing(E, P, Q, m): computes the Weil pairing of the two points
 of m-torsion P and Q on the elliptic curve E.
Doc: Computes the Weil pairing of the two points of $m$-torsion $P$ and $Q$
 on the elliptic curve $E$.

Function: ellwp
Class: basic
Section: elliptic_curves
C-Name: ellwp0
Prototype: GDGD0,L,p
Help: ellwp(w,{z='x},{flag=0}): computes the value at z of the Weierstrass P
 function attached to the lattice w, as given by ellperiods. Optional flag
 means 0 (default), compute only P(z), 1 compute [P(z),P'(z)].
Doc: Computes the value at $z$ of the Weierstrass $\wp$ function attached to
 the lattice $w$ as given by \tet{ellperiods}. It is also possible to
 directly input $w = [\omega_1,\omega_2]$, or an elliptic curve $E$ as given
 by \kbd{ellinit} ($w = \kbd{E.omega}$).
 \bprog
 ? w = ellperiods([1,I]);
 ? ellwp(w, 1/2)
 %2 = 6.8751858180203728274900957798105571978
 ? E = ellinit([1,1]);
 ? ellwp(E, 1/2)
 %4 = 3.9413112427016474646048282462709151389
 @eprog\noindent One can also compute the series expansion around $z = 0$:
 \bprog
 ? E = ellinit([1,0]);
 ? ellwp(E)              \\ 'x implicitly at default seriesprecision
 %5 = x^-2 - 1/5*x^2 + 1/75*x^6 - 2/4875*x^10 + O(x^14)
 ? ellwp(E, x + O(x^12)) \\ explicit precision
 %6 = x^-2 - 1/5*x^2 + 1/75*x^6 + O(x^9)
 @eprog
 
 Optional \fl\ means 0 (default): compute only $\wp(z)$, 1: compute
 $[\wp(z),\wp'(z)]$.
Variant: For $\fl = 0$, we also have
 \fun{GEN}{ellwp}{GEN w, GEN z, long prec}, and
 \fun{GEN}{ellwpseries}{GEN E, long v, long precdl} for the power series in
 variable $v$.

Function: ellxn
Class: basic
Section: elliptic_curves
C-Name: ellxn
Prototype: GLDn
Help: ellxn(E,n,{v='x}): polynomials [phi_n, (psi_n)^2] in variable v,
 where x([n]P) = phi_n/(psi_n)^2.
Doc: In standard notation, for any affine point $P = (v,w)$ on the
 curve $E$, we have
 $$[n]P = (\phi_n(P)\psi_n(P) : \omega_n(P) : \psi_n(P)^3)$$
 for some polynomials $\phi_n,\omega_n,\psi_n$ in
 $\Z[a_1,a_2,a_3,a_4,a_6][v,w]$. This function returns
 $[\phi_n(P),\psi_n(P)^2]$, which give the numerator and denominator of
 the abcissa of $[n]P$ and depend only on $v$.

Function: ellzeta
Class: basic
Section: elliptic_curves
C-Name: ellzeta
Prototype: GDGp
Help: ellzeta(w,{z='x}): computes the value at z of the Weierstrass Zeta
 function attached to the lattice w, as given by ellperiods(,1).
Doc: Computes the value at $z$ of the Weierstrass $\zeta$ function attached to
 the lattice $w$ as given by \tet{ellperiods}$(,1)$: including quasi-periods
 is useful, otherwise there are recomputed from scratch for each new $z$.
 $$ \zeta(z, L) = \dfrac{1}{z} + z^2\sum_{\omega\in L^*}
 \dfrac{1}{\omega^2(z-\omega)}.$$
 It is also possible to directly input $w = [\omega_1,\omega_2]$,
 or an elliptic curve $E$ as given by \kbd{ellinit} ($w = \kbd{E.omega}$).
 The quasi-periods of $\zeta$, such that
 $$\zeta(z + a\omega_1 + b\omega_2) = \zeta(z) + a\eta_1 + b\eta_2 $$
 for integers $a$ and $b$ are obtained as $\eta_i = 2\zeta(\omega_i/2)$.
 Or using directly \tet{elleta}.
 \bprog
 ? w = ellperiods([1,I],1);
 ? ellzeta(w, 1/2)
 %2 = 1.5707963267948966192313216916397514421
 ? E = ellinit([1,0]);
 ? ellzeta(E, E.omega[1]/2)
 %4 = 0.84721308479397908660649912348219163647
 @eprog\noindent One can also compute the series expansion around $z = 0$
 (the quasi-periods are useless in this case):
 \bprog
 ? E = ellinit([0,1]);
 ? ellzeta(E) \\ at 'x, implicitly at default seriesprecision
 %4 = x^-1 + 1/35*x^5 - 1/7007*x^11 + O(x^15)
 ? ellzeta(E, x + O(x^20)) \\ explicit precision
 %5 = x^-1 + 1/35*x^5 - 1/7007*x^11 + 1/1440257*x^17 + O(x^18)
 @eprog\noindent

Function: ellztopoint
Class: basic
Section: elliptic_curves
C-Name: pointell
Prototype: GGp
Help: ellztopoint(E,z): inverse of ellpointtoz. Returns the coordinates of
 point P on the curve E corresponding to a complex or p-adic z.
Doc: 
 $E$ being an \var{ell} as output by
 \kbd{ellinit}, computes the coordinates $[x,y]$ on the curve $E$
 corresponding to the complex or $p$-adic parameter $z$. Hence this is the
 inverse function of \kbd{ellpointtoz}.
 
 \item If $E$ is defined over a $p$-adic field and has multiplicative
 reduction, then $z$ is understood as an element on the
 Tate curve $\bar{Q}_p^* / q^\Z$.
 \bprog
 ? E = ellinit([0,-1,1,0,0], O(11^5));
 ? [u2,u,q] = E.tate; type(u)
 %2 = "t_PADIC" \\ split multiplicative reduction
 ? z = ellpointtoz(E, [0,0])
 %3 = 3 + 11^2 + 2*11^3 + 3*11^4 + 6*11^5 + 10*11^6 + 8*11^7 + O(11^8)
 ? ellztopoint(E,z)
 %4 = [O(11^9), O(11^9)]
 
 ? E = ellinit(ellfromj(1/4), O(2^6)); x=1/2; y=ellordinate(E,x)[1];
 ? z = ellpointtoz(E,[x,y]); \\ non-split: t_POLMOD with t_PADIC coefficients
 ? P = ellztopoint(E, z);
 ? P[1] \\ y coordinate is analogous, more complicated
 %8 = Mod(O(2^4)*x + (2^-1 + O(2^5)), x^2 + (1 + 2^2 + 2^4 + 2^5 + O(2^7)))
 @eprog
 
 \item If $E$ is defined over the complex numbers (for instance over $\Q$),
 $z$ is understood as a complex number in $\C/\Lambda_E$. If the
 short Weierstrass equation is $y^2 = 4x^3 - g_2x - g_3$, then $[x,y]$
 represents the Weierstrass $\wp$-function\sidx{Weierstrass $\wp$-function}
 and its derivative. For a general Weierstrass equation we have
 $$x = \wp(z) - b_2/12,\quad y = \wp'(z) - (a_1 x + a_3)/2.$$
 If $z$ is in the lattice defining $E$ over $\C$, the result is the point at
 infinity $[0]$.
 \bprog
 ? E = ellinit([0,1]); P = [2,3];
 ? z = ellpointtoz(E, P)
 %2 = 3.5054552633136356529375476976257353387
 ? ellwp(E, z)
 %3 = 2.0000000000000000000000000000000000000
 ? ellztopoint(E, z) - P
 %4 = [2.548947057811923643 E-57, 7.646841173435770930 E-57]
 ? ellztopoint(E, 0)
 %5 = [0] \\ point at infinity
 @eprog

Function: erfc
Class: basic
Section: transcendental
C-Name: gerfc
Prototype: Gp
Help: erfc(x): complementary error function.
Doc: complementary error function, analytic continuation of
 $(2/\sqrt\pi)\int_x^\infty e^{-t^2}\,dt = \kbd{incgam}(1/2,x^2)/\sqrt\pi$,
 where the latter expression extends the function definition from real $x$ to
 all complex $x \neq 0$.

Function: errname
Class: basic
Section: programming/specific
C-Name: errname
Prototype: G
Help: errname(E): returns the type of the error message E.
Description: 
 (gen):errtyp err_get_num($1)
Doc: returns the type of the error message \kbd{E} as a string.

Function: error
Class: basic
Section: programming/specific
C-Name: error0
Prototype: vs*
Help: error({str}*): abort script with error message str.
Description: 
 (error):void  pari_err(0, $1)
 (?gen,...):void  pari_err(e_MISC, "${2 format_string}"${2 format_args})
Doc: outputs its argument list (each of
 them interpreted as a string), then interrupts the running \kbd{gp} program,
 returning to the input prompt. For instance
 \bprog
 error("n = ", n, " is not squarefree!")
 @eprog\noindent
  % \syn{NO}

Function: eta
Class: basic
Section: transcendental
C-Name: eta0
Prototype: GD0,L,p
Help: eta(z,{flag=0}): if flag=0, returns prod(n=1,oo, 1-q^n), where
 q = exp(2 i Pi z) if z is a complex scalar (belonging to the upper half plane);
 q = z if z is a p-adic number or can be converted to a power series.
 If flag is non-zero, the function only applies to complex scalars and returns
 the true eta function, with the factor q^(1/24) included.
Doc: Variants of \idx{Dedekind}'s $\eta$ function.
 If $\fl = 0$, return $\prod_{n=1}^\infty(1-q^n)$, where $q$ depends on $x$
 in the following way:
 
 \item $q = e^{2i\pi x}$ if $x$ is a \emph{complex number} (which must then
 have positive imaginary part); notice that the factor $q^{1/24}$ is
 missing!
 
 \item $q = x$ if $x$ is a \typ{PADIC}, or can be converted to a
 \emph{power series} (which must then have positive valuation).
 
 If $\fl$ is non-zero, $x$ is converted to a complex number and we return the
 true $\eta$ function, $q^{1/24}\prod_{n=1}^\infty(1-q^n)$,
 where $q = e^{2i\pi x}$.
Variant: 
 Also available is \fun{GEN}{trueeta}{GEN x, long prec} ($\fl=1$).

Function: eulerphi
Class: basic
Section: number_theoretical
C-Name: eulerphi
Prototype: G
Help: eulerphi(x): Euler's totient function of x.
Description: 
 (gen):int        eulerphi($1)
Doc: Euler's $\phi$ (totient)\sidx{Euler totient function} function of the
 integer $|x|$, in other words $|(\Z/x\Z)^*|$.
 \bprog
 ? eulerphi(40)
 %1 = 16
 @eprog\noindent
 According to this definition we let $\phi(0) := 2$, since $\Z^* = \{-1,1\}$;
 this is consistent with \kbd{znstar(0)}: we have
 \kbd{znstar$(n)$.no = eulerphi(n)} for all $n\in\Z$.

Function: eval
Class: basic
Section: polynomials
C-Name: geval_gp
Prototype: GC
Help: eval(x): evaluation of x, replacing variables by their value.
Description: 
 (gen):gen      geval($1)
Doc: replaces in $x$ the formal variables by the values that
 have been assigned to them after the creation of $x$. This is mainly useful
 in GP, and not in library mode. Do not confuse this with substitution (see
 \kbd{subst}).
 
 If $x$ is a character string, \kbd{eval($x$)} executes $x$ as a GP
 command, as if directly input from the keyboard, and returns its
 output.
 \bprog
 ? x1 = "one"; x2 = "two";
 ? n = 1; eval(Str("x", n))
 %2 = "one"
 ? f = "exp"; v = 1;
 ? eval(Str(f, "(", v, ")"))
 %4 = 2.7182818284590452353602874713526624978
 @eprog\noindent Note that the first construct could be implemented in a
 simpler way by using a vector \kbd{x = ["one","two"]; x[n]}, and the second
 by using a closure \kbd{f = exp; f(v)}. The final example is more interesting:
 \bprog
 ? genmat(u,v) = matrix(u,v,i,j, eval( Str("x",i,j) ));
 ? genmat(2,3)   \\ generic 2 x 3 matrix
 %2 =
 [x11 x12 x13]
 
 [x21 x22 x23]
 @eprog
 
 A syntax error in the evaluation expression raises an \kbd{e\_SYNTAX}
 exception, which can be trapped as usual:
 \bprog
 ? 1a
  ***   syntax error, unexpected variable name, expecting $end or ';': 1a
  ***                                                                   ^-
 ? E(expr) =
   {
     iferr(eval(expr),
           e, print("syntax error"),
           errname(e) == "e_SYNTAX");
   }
 ? E("1+1")
 %1 = 2
 ? E("1a")
 syntax error
 @eprog
 \synt{geval}{GEN x}.

Function: exp
Class: basic
Section: transcendental
C-Name: gexp
Prototype: Gp
Help: exp(x): exponential of x.
Description: 
 (real):real         mpexp($1)
 (mp):mp:prec        gexp($1, $prec)
 (gen):gen:prec      gexp($1, $prec)
Doc: exponential of $x$.
 $p$-adic arguments with positive valuation are accepted.
Variant: For a \typ{PADIC} $x$, the function
 \fun{GEN}{Qp_exp}{GEN x} is also available.

Function: expm1
Class: basic
Section: transcendental
C-Name: gexpm1
Prototype: Gp
Help: expm1(x): exp(x)-1.
Description: 
 (real):real         mpexpm1($1)
Doc: return $\exp(x)-1$, computed in a way that is also accurate
 when the real part of $x$ is near $0$.
 A naive direct computation would suffer from catastrophic cancellation;
 PARI's direct computation of $\exp(x)$ alleviates this well known problem at
 the expense of computing $\exp(x)$ to a higher accuracy when $x$ is small.
 Using \kbd{expm1} is recommended instead:
 \bprog
 ? default(realprecision, 10000); x = 1e-100;
 ? a = expm1(x);
 time = 4 ms.
 ? b = exp(x)-1;
 time = 28 ms.
 ? default(realprecision, 10040); x = 1e-100;
 ? c = expm1(x);  \\ reference point
 ? abs(a-c)/c  \\ relative error in expm1(x)
 %7 = 0.E-10017
 ? abs(b-c)/c  \\ relative error in exp(x)-1
 %8 = 1.7907031188259675794 E-9919
 @eprog\noindent As the example above shows, when $x$ is near $0$,
 \kbd{expm1} is both faster and more accurate than \kbd{exp(x)-1}.

Function: extern
Class: basic
Section: programming/specific
C-Name: gpextern
Prototype: s
Help: extern(str): execute shell command str, and feeds the result to GP (as
 if loading from file).
Doc: the string \var{str} is the name of an external command (i.e.~one you
 would type from your UNIX shell prompt). This command is immediately run and
 its output fed into \kbd{gp}, just as if read from a file.

Function: externstr
Class: basic
Section: programming/specific
C-Name: externstr
Prototype: s
Help: externstr(str): execute shell command str, and returns the result as a
 vector of GP strings, one component per output line.
Doc: the string \var{str} is the name of an external command (i.e.~one you
 would type from your UNIX shell prompt). This command is immediately run and
 its output is returned as a vector of GP strings, one component per output
 line.

Function: factor
Class: basic
Section: number_theoretical
C-Name: gp_factor0
Prototype: GDG
Help: factor(x,{lim}): factorization of x. lim is optional and can be set
 whenever x is of (possibly recursive) rational type. If lim is set return
 partial factorization, using primes < lim.
Description: 
 (int, ?-1):vec        Z_factor($1)
 (gen, ?-1):vec        factor($1)
 (gen, small):vec      factor0($1, $2)
Doc: general factorization function, where $x$ is a
 rational (including integers), a complex number with rational
 real and imaginary parts, or a rational function (including polynomials).
 The result is a two-column matrix: the first contains the irreducibles
 dividing $x$ (rational or Gaussian primes, irreducible polynomials),
 and the second the exponents. By convention, $0$ is factored as $0^1$.
 
 \misctitle{$\Q$ and $\Q(i)$}
 See \tet{factorint} for more information about the algorithms used.
 The rational or Gaussian primes are in fact \var{pseudoprimes}
 (see \kbd{ispseudoprime}), a priori not rigorously proven primes. In fact,
 any factor which is $\leq 2^{64}$ (whose norm is $\leq 2^{64}$ for an
 irrational Gaussian prime) is a genuine prime. Use \kbd{isprime} to prove
 primality of other factors, as in
 \bprog
 ? fa = factor(2^2^7 + 1)
 %1 =
 [59649589127497217 1]
 
 [5704689200685129054721 1]
 
 ? isprime( fa[,1] )
 %2 = [1, 1]~   \\ both entries are proven primes
 @eprog\noindent
 Another possibility is to set the global default \tet{factor_proven}, which
 will perform a rigorous primality proof for each pseudoprime factor.
 
 A \typ{INT} argument \var{lim} can be added, meaning that we look only for
 prime factors $p < \var{lim}$. The limit \var{lim} must be non-negative.
 In this case, all but the last factor are proven primes, but the remaining
 factor may actually be a proven composite! If the remaining factor is less
 than $\var{lim}^2$, then it is prime.
 \bprog
 ? factor(2^2^7 +1, 10^5)
 %3 =
 [340282366920938463463374607431768211457 1]
 @eprog\noindent
 \misctitle{Deprecated feature} Setting $\var{lim}=0$ is the same
 as setting it to $\kbd{primelimit} + 1$. Don't use this: it is unwise to
 rely on global variables when you can specify an explicit argument.
 \smallskip
 
 This routine uses trial division and perfect power tests, and should not be
 used for huge values of \var{lim} (at most $10^9$, say):
 \kbd{factorint(, 1 + 8)} will in general be faster. The latter does not
 guarantee that all small
 prime factors are found, but it also finds larger factors, and in a much more
 efficient way.
 \bprog
 ? F = (2^2^7 + 1) * 1009 * 100003; factor(F, 10^5)  \\ fast, incomplete
 time = 0 ms.
 %4 =
 [1009 1]
 
 [34029257539194609161727850866999116450334371 1]
 
 ? factor(F, 10^9)    \\ very slow
 time = 6,892 ms.
 %6 =
 [1009 1]
 
 [100003 1]
 
 [340282366920938463463374607431768211457 1]
 
 ? factorint(F, 1+8)  \\ much faster, all small primes were found
 time = 12 ms.
 %7 =
 [1009 1]
 
 [100003 1]
 
 [340282366920938463463374607431768211457 1]
 
 ? factor(F)   \\ complete factorisation
 time = 112 ms.
 %8 =
 [1009 1]
 
 [100003 1]
 
 [59649589127497217 1]
 
 [5704689200685129054721 1]
 @eprog\noindent Over $\Q$, the prime factors are sorted in increasing order.
 
 \misctitle{Rational functions}
 The polynomials or rational functions to be factored must have scalar
 coefficients. In particular PARI does not know how to factor
 \emph{multivariate} polynomials. The following domains are currently
 supported: $\Q$, $\R$, $\C$, $\Q_p$, finite fields and number fields. See
 \tet{factormod} and \tet{factorff} for the algorithms used over finite
 fields, \tet{nffactor} for the algorithms over number fields. The irreducible
 factors are sorted by increasing degree.
 
 The routine guesses a sensible ring over which to factor: the
 smallest ring containing all coefficients, taking into account quotient
 structures induced by \typ{INTMOD}s and \typ{POLMOD}s (e.g.~if a coefficient
 in $\Z/n\Z$ is known, all rational numbers encountered are first mapped to
 $\Z/n\Z$; different moduli will produce an error). Factoring modulo a
 non-prime number is not supported; to factor in $\Q_p$, use \typ{PADIC}
 coefficients not \typ{INTMOD} modulo $p^n$.
 \bprog
 ? T = x^2+1;
 ? factor(T);                         \\ over Q
 ? factor(T*Mod(1,3))                 \\ over F_3
 ? factor(T*ffgen(ffinit(3,2,'t))^0)  \\ over F_{3^2}
 ? factor(T*Mod(Mod(1,3), t^2+t+2))   \\ over F_{3^2}, again
 ? factor(T*(1 + O(3^6))              \\ over Q_3, precision 6
 ? factor(T*1.)                       \\ over R, current precision
 ? factor(T*(1.+0.*I))                \\ over C
 ? factor(T*Mod(1, y^3-2))            \\ over Q(2^{1/3})
 @eprog\noindent In most cases, it is clearer and simpler to call an
 explicit variant than to rely on the generic \kbd{factor} function and
 the above detection mechanism:
 \bprog
 ? factormod(T, 3)           \\ over F_3
 ? factorff(T, 3, t^2+t+2))  \\ over F_{3^2}
 ? factorpadic(T, 3,6)       \\ over Q_3, precision 6
 ? nffactor(y^3-2, T)        \\ over Q(2^{1/3})
 ? polroots(T)               \\ over C
 ? polrootsreal(T)           \\ over R (real polynomial)
 @eprog
 
 \misctitle{Note about inseparable polynomials} Polynomials with inexact
 coefficients (e.g. floating point or $p$-adic numbers) are assumed to be
 squarefree: in fact, there exist a squarefree polynomial arbitrarily close
 to the input, and they cannot be distinguished at the input accuracy. This
 means that irreducible factors are repeated according to their apparent
 multiplicity. On the contrary, using a specialized function such as
 \kbd{factorpadic} with an \emph{exact} rational input yields the correct
 multiplicity when the (now exact) input is not separable. Compare:
 \bprog
 ? factor(z^2 * (1 + O(5^2)))
 %1 =
 [(1 + O(5^2))*z + O(5^2) 1]
 
 [(1 + O(5^2))*z + O(5^2) 1]
 ? factorpadic(z^2, 5, 2)
 %2 =
 [1 + O(5^2))*z + O(5^2) 2]
 @eprog
 
 \misctitle{Note about contents}
 Factorization of polynomials is done up to
 multiplication by a constant. In particular, the factors of rational
 polynomials will have integer coefficients, and the content of a polynomial
 or rational function is discarded and not included in the factorization. If
 needed, you can always ask for the content explicitly:
 \bprog
 ? factor(t^2 + 5/2*t + 1)
 %1 =
 [2*t + 1 1]
 
 [t + 2 1]
 
 ? content(t^2 + 5/2*t + 1)
 %2 = 1/2
 @eprog
Variant: This function should only be used by the \kbd{gp} interface. Use
 directly \fun{GEN}{factor}{GEN x} or \fun{GEN}{boundfact}{GEN x, ulong lim}.
 The obsolete function \fun{GEN}{factor0}{GEN x, long lim} is kept for
 backward compatibility.

Function: factorback
Class: basic
Section: number_theoretical
C-Name: factorback2
Prototype: GDG
Help: factorback(f,{e}): given a factorisation f, gives the factored
 object back. If this is a prime ideal factorisation you must supply the
 corresponding number field as last argument. If e is present, f has to be a
 vector of the same length, and we return the product of the f[i]^e[i].
Description: 
 (gen):gen      factorback($1)
 (gen,):gen     factorback($1)
 (gen,gen):gen  factorback2($1, $2)
Doc: gives back the factored object
 corresponding to a factorization. The integer $1$ corresponds to the empty
 factorization.
 
 If $e$ is present, $e$ and $f$ must be vectors of the same length ($e$ being
 integral), and the corresponding factorization is the product of the
 $f[i]^{e[i]}$.
 
 If not, and $f$ is vector, it is understood as in the preceding case with $e$
 a vector of 1s: we return the product of the $f[i]$. Finally, $f$ can be a
 regular factorization, as produced with any \kbd{factor} command. A few
 examples:
 \bprog
 ? factor(12)
 %1 =
 [2 2]
 
 [3 1]
 
 ? factorback(%)
 %2 = 12
 ? factorback([2,3], [2,1])   \\ 2^3 * 3^1
 %3 = 12
 ? factorback([5,2,3])
 %4 = 30
 @eprog
Variant: Also available is \fun{GEN}{factorback}{GEN f} (case $e = \kbd{NULL}$).

Function: factorcantor
Class: basic
Section: number_theoretical
C-Name: factcantor
Prototype: GG
Help: factorcantor(x,p): factorization mod p of the polynomial x using
 Cantor-Zassenhaus.
Doc: factors the polynomial $x$ modulo the
 prime $p$, using distinct degree plus
 \idx{Cantor-Zassenhaus}\sidx{Zassenhaus}. The coefficients of $x$ must be
 operation-compatible with $\Z/p\Z$. The result is a two-column matrix, the
 first column being the irreducible polynomials dividing $x$, and the second
 the exponents. If you want only the \emph{degrees} of the irreducible
 polynomials (for example for computing an $L$-function), use
 $\kbd{factormod}(x,p,1)$. Note that the \kbd{factormod} algorithm is
 usually faster than \kbd{factorcantor}.

Function: factorff
Class: basic
Section: number_theoretical
C-Name: factorff
Prototype: GDGDG
Help: factorff(x,{p},{a}): factorization of the polynomial x in the finite field
 F_p[X]/a(X)F_p[X].
Doc: factors the polynomial $x$ in the field
 $\F_q$ defined by the irreducible polynomial $a$ over $\F_p$. The
 coefficients of $x$ must be operation-compatible with $\Z/p\Z$. The result
 is a two-column matrix: the first column contains the irreducible factors of
 $x$, and the second their exponents. If all the coefficients of $x$ are in
 $\F_p$, a much faster algorithm is applied, using the computation of
 isomorphisms between finite fields.
 
 Either $a$ or $p$ can omitted (in which case both are ignored) if x has
 \typ{FFELT} coefficients; the function then becomes identical to \kbd{factor}:
 \bprog
 ? factorff(x^2 + 1, 5, y^2+3)  \\ over F_5[y]/(y^2+3) ~ F_25
 %1 =
 [Mod(Mod(1, 5), Mod(1, 5)*y^2 + Mod(3, 5))*x
  + Mod(Mod(2, 5), Mod(1, 5)*y^2 + Mod(3, 5)) 1]
 
 [Mod(Mod(1, 5), Mod(1, 5)*y^2 + Mod(3, 5))*x
  + Mod(Mod(3, 5), Mod(1, 5)*y^2 + Mod(3, 5)) 1]
 ? t = ffgen(y^2 + Mod(3,5), 't); \\ a generator for F_25 as a t_FFELT
 ? factorff(x^2 + 1)   \\ not enough information to determine the base field
  ***   at top-level: factorff(x^2+1)
  ***                 ^---------------
  *** factorff: incorrect type in factorff.
 ? factorff(x^2 + t^0) \\ make sure a coeff. is a t_FFELT
 %3 =
 [x + 2 1]
 
 [x + 3 1]
 ? factorff(x^2 + t + 1)
 %11 =
 [x + (2*t + 1) 1]
 
 [x + (3*t + 4) 1]
 @eprog\noindent
 Notice that the second syntax is easier to use and much more readable.

Function: factorial
Class: basic
Section: number_theoretical
C-Name: mpfactr
Prototype: Lp
Help: factorial(x): factorial of x, the result being given as a real number.
Doc: factorial of $x$. The expression $x!$ gives a result which is an integer,
 while $\kbd{factorial}(x)$ gives a real number.
Variant: \fun{GEN}{mpfact}{long x} returns $x!$ as a \typ{INT}.

Function: factorint
Class: basic
Section: number_theoretical
C-Name: factorint
Prototype: GD0,L,
Help: factorint(x,{flag=0}): factor the integer x. flag is optional, whose
 binary digits mean 1: avoid MPQS, 2: avoid first-stage ECM (may fall back on
 it later), 4: avoid Pollard-Brent Rho and Shanks SQUFOF, 8: skip final ECM
 (huge composites will be declared prime).
Doc: factors the integer $n$ into a product of
 pseudoprimes (see \kbd{ispseudoprime}), using a combination of the
 \idx{Shanks SQUFOF} and \idx{Pollard Rho} method (with modifications due to
 Brent), \idx{Lenstra}'s \idx{ECM} (with modifications by Montgomery), and
 \idx{MPQS} (the latter adapted from the \idx{LiDIA} code with the kind
 permission of the LiDIA maintainers), as well as a search for pure powers.
 The output is a two-column matrix as for \kbd{factor}: the first column
 contains the ``prime'' divisors of $n$, the second one contains the
 (positive) exponents.
 
 By convention $0$ is factored as $0^1$, and $1$ as the empty factorization;
 also the divisors are by default not proven primes is they are larger than
 $2^{64}$, they only failed the BPSW compositeness test (see
 \tet{ispseudoprime}). Use \kbd{isprime} on the result if you want to
 guarantee primality or set the \tet{factor_proven} default to $1$.
 Entries of the private prime tables (see \tet{addprimes}) are also included
 as is.
 
 This gives direct access to the integer factoring engine called by most
 arithmetical functions. \fl\ is optional; its binary digits mean 1: avoid
 MPQS, 2: skip first stage ECM (we may still fall back to it later), 4: avoid
 Rho and SQUFOF, 8: don't run final ECM (as a result, a huge composite may be
 declared to be prime). Note that a (strong) probabilistic primality test is
 used; thus composites might not be detected, although no example is known.
 
 You are invited to play with the flag settings and watch the internals at
 work by using \kbd{gp}'s \tet{debug} default parameter (level 3 shows
 just the outline, 4 turns on time keeping, 5 and above show an increasing
 amount of internal details).

Function: factormod
Class: basic
Section: number_theoretical
C-Name: factormod0
Prototype: GGD0,L,
Help: factormod(x,p,{flag=0}): factors the polynomial x modulo the prime p, using Berlekamp. flag is optional, and can be 0: default or 1:
 only the degrees of the irreducible factors are given.
Doc: factors the polynomial $x$ modulo the prime integer $p$, using
 \idx{Berlekamp}. The coefficients of $x$ must be operation-compatible with
 $\Z/p\Z$. The result is a two-column matrix, the first column being the
 irreducible polynomials dividing $x$, and the second the exponents. If $\fl$
 is non-zero, outputs only the \emph{degrees} of the irreducible polynomials
 (for example, for computing an $L$-function). A different algorithm for
 computing the mod $p$ factorization is \kbd{factorcantor} which is sometimes
 faster.

Function: factornf
Class: basic
Section: number_fields
C-Name: polfnf
Prototype: GG
Help: factornf(x,t): this function is obsolete, use nffactor.
Doc: This function is obsolete, use \kbd{nffactor}.
 
 factorization of the univariate polynomial $x$
 over the number field defined by the (univariate) polynomial $t$. $x$ may
 have coefficients in $\Q$ or in the number field. The algorithm reduces to
 factorization over $\Q$ (\idx{Trager}'s trick). The direct approach of
 \tet{nffactor}, which uses \idx{van Hoeij}'s method in a relative setting, is
 in general faster.
 
 The main variable of $t$ must be of \emph{lower} priority than that of $x$
 (see \secref{se:priority}). However if non-rational number field elements
 occur (as polmods or polynomials) as coefficients of $x$, the variable of
 these polmods \emph{must} be the same as the main variable of $t$. For
 example
 
 \bprog
 ? factornf(x^2 + Mod(y, y^2+1), y^2+1);
 ? factornf(x^2 + y, y^2+1); \\@com these two are OK
 ? factornf(x^2 + Mod(z,z^2+1), y^2+1)
   ***   at top-level: factornf(x^2+Mod(z,z
   ***                 ^--------------------
   *** factornf: inconsistent data in rnf function.
 ? factornf(x^2 + z, y^2+1)
   ***   at top-level: factornf(x^2+z,y^2+1
   ***                 ^--------------------
   *** factornf: incorrect variable in rnf function.
 @eprog
Obsolete: 2016-08-08

Function: factorpadic
Class: basic
Section: polynomials
C-Name: factorpadic
Prototype: GGL
Help: factorpadic(pol,p,r): p-adic factorization of the polynomial pol
 to precision r.
Doc: $p$-adic factorization
 of the polynomial \var{pol} to precision $r$, the result being a
 two-column matrix as in \kbd{factor}. Note that this is not the same
 as a factorization over $\Z/p^r\Z$ (polynomials over that ring do not form a
 unique factorization domain, anyway), but approximations in $\Q/p^r\Z$ of
 the true factorization in $\Q_p[X]$.
 \bprog
 ? factorpadic(x^2 + 9, 3,5)
 %1 =
 [(1 + O(3^5))*x^2 + O(3^5)*x + (3^2 + O(3^5)) 1]
 ? factorpadic(x^2 + 1, 5,3)
 %2 =
 [  (1 + O(5^3))*x + (2 + 5 + 2*5^2 + O(5^3)) 1]
 
 [(1 + O(5^3))*x + (3 + 3*5 + 2*5^2 + O(5^3)) 1]
 @eprog\noindent
 The factors are normalized so that their leading coefficient is a power of
 $p$. The method used is a modified version of the \idx{round 4} algorithm of
 \idx{Zassenhaus}.
 
 If \var{pol} has inexact \typ{PADIC} coefficients, this is not always
 well-defined; in this case, the polynomial is first made integral by dividing
 out the $p$-adic content,  then lifted to $\Z$ using \tet{truncate}
 coefficientwise.
 Hence we actually factor exactly a polynomial which is only $p$-adically
 close to the input. To avoid pitfalls, we advise to only factor polynomials
 with exact rational coefficients.
 
 \synt{factorpadic}{GEN f,GEN p, long r} . The function \kbd{factorpadic0} is
 deprecated, provided for backward compatibility.

Function: ffgen
Class: basic
Section: number_theoretical
C-Name: ffgen
Prototype: GDn
Help: ffgen(q,{v}): return a generator X mod P(X) for the finite field with
 q elements. If v is given, the variable name is used to display g, else the
 variable 'x' is used. Alternative syntax, q = P(X) an irreducible
 polynomial with t_INTMOD
 coefficients, return the generator X mod P(X) of the finite field defined
 by P. If v is given, the variable name is used to display g, else the
 variable of the polynomial P is used.
Doc: return a \typ{FFELT} generator for the finite field with $q$ elements;
 $q = p^f$ must be a prime power. This functions computes an irreducible
 monic polynomial $P\in\F_p[X]$ of degree~$f$ (via \tet{ffinit}) and
 returns $g = X \pmod{P(X)}$. If \kbd{v} is given, the variable name is used
 to display $g$, else the variable $x$ is used.
 \bprog
 ? g = ffgen(8, 't);
 ? g.mod
 %2 = t^3 + t^2 + 1
 ? g.p
 %3 = 2
 ? g.f
 %4 = 3
 ? ffgen(6)
  ***   at top-level: ffgen(6)
  ***                 ^--------
  *** ffgen: not a prime number in ffgen: 6.
 @eprog\noindent Alternative syntax: instead of a prime power $q=p^f$, one may
 input the pair $[p,f]$:
 \bprog
 ? g = ffgen([2,4], 't);
 ? g.p
 %2 = 2
 ? g.mod
 %3 = t^4 + t^3 + t^2 + t + 1
 @eprog\noindent Finally, one may input
 directly the polynomial $P$ (monic, irreducible, with \typ{INTMOD}
 coefficients), and the function returns the generator $g = X \pmod{P(X)}$,
 inferring $p$ from the coefficients of $P$. If \kbd{v} is given, the
 variable name is used to display $g$, else the variable of the polynomial
 $P$ is used. If $P$ is not irreducible, we create an invalid object and
 behaviour of functions dealing with the resulting \typ{FFELT}
 is undefined; in fact, it is much more costly to test $P$ for
 irreducibility than it would be to produce it via \kbd{ffinit}.
Variant: 
 To create a generator for a prime finite field, the function
 \fun{GEN}{p_to_GEN}{GEN p, long v} returns \kbd{1+ffgen(x*Mod(1,p),v)}.

Function: ffinit
Class: basic
Section: number_theoretical
C-Name: ffinit
Prototype: GLDn
Help: ffinit(p,n,{v='x}): monic irreducible polynomial of degree n over F_p[v].
Description: 
 (int, small, ?var):pol        ffinit($1, $2, $3)
Doc: computes a monic polynomial of degree $n$ which is irreducible over
  $\F_p$, where $p$ is assumed to be prime. This function uses a fast variant
  of Adleman and Lenstra's algorithm.
 
 It is useful in conjunction with \tet{ffgen}; for instance if
 \kbd{P = ffinit(3,2)}, you can represent elements in $\F_{3^2}$ in term of
 \kbd{g = ffgen(P,'t)}. This can be abbreviated as
 \kbd{g = ffgen(3\pow2, 't)}, where the defining polynomial $P$ can be later
 recovered as \kbd{g.mod}.

Function: fflog
Class: basic
Section: number_theoretical
C-Name: fflog
Prototype: GGDG
Help: fflog(x,g,{o}): return the discrete logarithm of the finite field
 element x in base g. If present, o must represents the multiplicative
 order of g. If no o is given, assume that g is a primitive root.
Doc: discrete logarithm of the finite field element $x$ in base $g$, i.e.~
 an $e$ in $\Z$ such that $g^e = o$. If
 present, $o$ represents the multiplicative order of $g$, see
 \secref{se:DLfun}; the preferred format for
 this parameter is \kbd{[ord, factor(ord)]}, where \kbd{ord} is the
 order of $g$. It may be set as a side effect of calling \tet{ffprimroot}.
 
 If no $o$ is given, assume that $g$ is a primitive root. The result is
 undefined if $e$ does not exist. This function uses
 
 \item a combination of generic discrete log algorithms (see \tet{znlog})
 
 \item a cubic sieve index calculus algorithm for large fields of degree at
 least $5$.
 
 \item Coppersmith's algorithm for fields of characteristic at most $5$.
 
 \bprog
 ? t = ffgen(ffinit(7,5));
 ? o = fforder(t)
 %2 = 5602   \\@com \emph{not} a primitive root.
 ? fflog(t^10,t)
 %3 = 10
 ? fflog(t^10,t, o)
 %4 = 10
 ? g = ffprimroot(t, &o);
 ? o   \\ order is 16806, bundled with its factorization matrix
 %6 = [16806, [2, 1; 3, 1; 2801, 1]]
 ? fforder(g, o)
 %7 = 16806
 ? fflog(g^10000, g, o)
 %8 = 10000
 @eprog

Function: ffnbirred
Class: basic
Section: number_theoretical
C-Name: ffnbirred0
Prototype: GLD0,L,
Help: ffnbirred(q,n,{fl=0}): number of monic irreducible polynomials over F_q, of
 degree n (fl=0, default) or at most n (fl=1).
Description: 
 (int, small, ?0):int      ffnbirred($1, $2)
 (int, small, 1):int       ffsumnbirred($1, $2)
 (int, small, ?small):int  ffnbirred0($1, $2, $3)
Doc: computes the number of monic irreducible polynomials over $\F_q$ of degree exactly $n$,
 ($\fl=0$ or omitted) or at most $n$ ($\fl=1$).
Variant: Also available are
  \fun{GEN}{ffnbirred}{GEN q, long n} (for $\fl=0$)
  and \fun{GEN}{ffsumnbirred}{GEN q, long n} (for $\fl=1$).

Function: fforder
Class: basic
Section: number_theoretical
C-Name: fforder
Prototype: GDG
Help: fforder(x,{o}): multiplicative order of the finite field element x.
 Optional o represents a multiple of the order of the element.
Doc: multiplicative order of the finite field element $x$.  If $o$ is
 present, it represents a multiple of the order of the element,
 see \secref{se:DLfun}; the preferred format for
 this parameter is \kbd{[N, factor(N)]}, where \kbd{N} is the cardinality
 of the multiplicative group of the underlying finite field.
 \bprog
 ? t = ffgen(ffinit(nextprime(10^8), 5));
 ? g = ffprimroot(t, &o);  \\@com o will be useful!
 ? fforder(g^1000000, o)
 time = 0 ms.
 %5 = 5000001750000245000017150000600250008403
 ? fforder(g^1000000)
 time = 16 ms. \\@com noticeably slower, same result of course
 %6 = 5000001750000245000017150000600250008403
 @eprog

Function: ffprimroot
Class: basic
Section: number_theoretical
C-Name: ffprimroot
Prototype: GD&
Help: ffprimroot(x, {&o}): return a primitive root of the multiplicative group
 of the definition field of the finite field element x (not necessarily the
 same as the field generated by x). If present, o is set to [ord, fa], where
 ord is the order of the group, and fa its factorization
 (useful in fflog and fforder).
Doc: return a primitive root of the multiplicative
 group of the definition field of the finite field element $x$ (not necessarily
 the same as the field generated by $x$). If present, $o$ is set to
 a vector \kbd{[ord, fa]}, where \kbd{ord} is the order of the group
 and \kbd{fa} its factorisation \kbd{factor(ord)}. This last parameter is
 useful in \tet{fflog} and \tet{fforder}, see \secref{se:DLfun}.
 \bprog
 ? t = ffgen(ffinit(nextprime(10^7), 5));
 ? g = ffprimroot(t, &o);
 ? o[1]
 %3 = 100000950003610006859006516052476098
 ? o[2]
 %4 =
 [2 1]
 
 [7 2]
 
 [31 1]
 
 [41 1]
 
 [67 1]
 
 [1523 1]
 
 [10498781 1]
 
 [15992881 1]
 
 [46858913131 1]
 
 ? fflog(g^1000000, g, o)
 time = 1,312 ms.
 %5 = 1000000
 @eprog

Function: fibonacci
Class: basic
Section: number_theoretical
C-Name: fibo
Prototype: L
Help: fibonacci(x): fibonacci number of index x (x C-integer).
Doc: $x^{\text{th}}$ Fibonacci number.

Function: floor
Class: basic
Section: conversions
C-Name: gfloor
Prototype: G
Help: floor(x): floor of x = largest integer <= x.
Description: 
 (small):small:parens   $1
 (int):int:copy:parens  $1
 (real):int             floorr($1)
 (mp):int               mpfloor($1)
 (gen):gen              gfloor($1)
Doc: 
 floor of $x$. When $x$ is in $\R$, the result is the
 largest integer smaller than or equal to $x$. Applied to a rational function,
 $\kbd{floor}(x)$ returns the Euclidean quotient of the numerator by the
 denominator.

Function: fold
Class: basic
Section: programming/specific
C-Name: fold0
Prototype: GG
Help: fold(f, A): return f(...f(f(A[1],A[2]),A[3]),...,A[#A]).
Wrapper: (GG)
Description: 
  (closure,gen):gen    genfold(${1 cookie}, ${1 wrapper}, $2)
Doc: Apply the \typ{CLOSURE} \kbd{f} of arity $2$ to the entries of \kbd{A},
 in order to return \kbd{f(\dots f(f(A[1],A[2]),A[3])\dots ,A[\#A])}.
 \bprog
 ? fold((x,y)->x*y, [1,2,3,4])
 %1 = 24
 ? fold((x,y)->[x,y], [1,2,3,4])
 %2 = [[[1, 2], 3], 4]
 ? fold((x,f)->f(x), [2,sqr,sqr,sqr])
 %3 = 256
 ? fold((x,y)->(x+y)/(1-x*y),[1..5])
 %4 = -9/19
 ? bestappr(tan(sum(i=1,5,atan(i))))
 %5 = -9/19
 @eprog
Variant: Also available is
 \fun{GEN}{genfold}{void *E, GEN (*fun)(void*,GEN, GEN), GEN A}.

Function: for
Class: basic
Section: programming/control
C-Name: forpari
Prototype: vV=GGI
Help: for(X=a,b,seq): the sequence is evaluated, X going from a up to b.
 If b is set to +oo, the loop will not stop.
Doc: evaluates \var{seq}, where
 the formal variable $X$ goes from $a$ to $b$. Nothing is done if $a>b$.
 $a$ and $b$ must be in $\R$. If $b$ is set to \kbd{+oo}, the loop will not
 stop; it is expected that the caller will break out of the loop itself at some
 point, using \kbd{break} or \kbd{return}.

Function: forcomposite
Class: basic
Section: programming/control
C-Name: forcomposite
Prototype: vV=GDGI
Help: forcomposite(n=a,{b},seq): the sequence is evaluated, n running over the
 composite numbers between a and b. Omitting b runs through composites >= a.
Iterator: 
 (gen,gen,?gen) (forcomposite, _forcomposite_init, _forcomposite_next)
Doc: evaluates \var{seq},
 where the formal variable $n$ ranges over the composite numbers between the
 non-negative real numbers $a$ to $b$, including $a$ and $b$ if they are
 composite. Nothing is done if $a>b$.
 \bprog
 ? forcomposite(n = 0, 10, print(n))
 4
 6
 8
 9
 10
 @eprog\noindent Omitting $b$ means we will run through all composites $\geq a$,
 starting an infinite loop; it is expected that the user will break out of
 the loop himself at some point, using \kbd{break} or \kbd{return}.
 
 Note that the value of $n$ cannot be modified within \var{seq}:
 \bprog
 ? forcomposite(n = 2, 10, n = [])
  ***   at top-level: forcomposite(n=2,10,n=[])
  ***                                      ^---
  ***   index read-only: was changed to [].
 @eprog

Function: fordiv
Class: basic
Section: programming/control
C-Name: fordiv
Prototype: vGVI
Help: fordiv(n,X,seq): the sequence is evaluated, X running over the
 divisors of n.
Doc: evaluates \var{seq}, where
 the formal variable $X$ ranges through the divisors of $n$
 (see \tet{divisors}, which is used as a subroutine). It is assumed that
 \kbd{factor} can handle $n$, without negative exponents. Instead of $n$,
 it is possible to input a factorization matrix, i.e. the output of
 \kbd{factor(n)}.
 
 This routine uses \kbd{divisors} as a subroutine, then loops over the
 divisors. In particular, if $n$ is an integer, divisors are sorted by
 increasing size.
 
 To avoid storing all divisors, possibly using a lot of memory, the following
 (much slower) routine loops over the divisors using essentially constant
 space:
 \bprog
 FORDIV(N)=
 { my(P, E);
 
   P = factor(N); E = P[,2]; P = P[,1];
   forvec( v = vector(#E, i, [0,E[i]]),
   X = factorback(P, v)
   \\ ...
 );
 }
 ? for(i=1,10^5, FORDIV(i))
 time = 3,445 ms.
 ? for(i=1,10^5, fordiv(i, d, ))
 time = 490 ms.
 @eprog

Function: forell
Class: basic
Section: programming/control
C-Name: forell0
Prototype: vVLLID0,L,
Help: forell(E,a,b,seq,{flag=0}): execute seq for each elliptic curves E of
 conductor between a and b in the elldata database. If flag is non-zero, select
 only the first curve in each isogeny class.
Wrapper: (,,,vG,)
Description: 
 (,small,small,closure,?small):void forell(${4 cookie}, ${4 wrapper}, $2, $3, $5)
Doc: evaluates \var{seq}, where the formal variable $E = [\var{name}, M, G]$
 ranges through all elliptic curves of conductors from $a$ to $b$. In this
 notation \var{name} is the curve name in Cremona's elliptic  curve  database,
 $M$ is the minimal model, $G$ is a $\Z$-basis of the free part of the
 Mordell-Weil group $E(\Q)$. If flag is non-zero, select
 only the first curve in each isogeny class.
 \bprog
 ? forell(E, 1, 500, my([name,M,G] = E); \
     if (#G > 1, print(name)))
 389a1
 433a1
 446d1
 ? c = 0; forell(E, 1, 500, c++); c   \\ number of curves
 %2 = 2214
 ? c = 0; forell(E, 1, 500, c++, 1); c \\ number of isogeny classes
 %3 = 971
 @eprog\noindent
 The \tet{elldata} database must be installed and contain data for the
 specified conductors.
 
 \synt{forell}{void *data, long (*call)(void*,GEN), long a, long b, long flag}.

Function: forpart
Class: basic
Section: programming/control
C-Name: forpart0
Prototype: vV=GIDGDG
Help: forpart(X=k,seq,{a=k},{n=k}): evaluate seq where the Vecsmall X
 goes over the partitions of k. Optional parameter n (n=nmax or n=[nmin,nmax])
 restricts the length of the partition. Optional parameter a (a=amax or
 a=[amin,amax]) restricts the range of the parts. Zeros are removed unless one
 sets amin=0 to get X of fixed length nmax (=k by default).
Iterator: 
 (gen,small,?gen,?gen)         (forpart, _forpart_init, _forpart_next)
Wrapper: (,vG,,)
Description: 
 (small,closure,?gen,?gen):void forpart(${2 cookie}, ${2 wrapper}, $1, $3, $4)
Doc: evaluate \var{seq} over the partitions $X=[x_1,\dots x_n]$ of the
 integer $k$, i.e.~increasing sequences $x_1\leq x_2\dots \leq x_n$ of sum
 $x_1+\dots + x_n=k$. By convention, $0$ admits only the empty partition and
 negative numbers have no partitions. A partition is given by a
 \typ{VECSMALL}, where parts are sorted in nondecreasing order:
 \bprog
 ? forpart(X=3, print(X))
 Vecsmall([3])
 Vecsmall([1, 2])
 Vecsmall([1, 1, 1])
 @eprog\noindent Optional parameters $n$ and $a$ are as follows:
 
 \item $n=\var{nmax}$ (resp. $n=[\var{nmin},\var{nmax}]$) restricts
 partitions to length less than $\var{nmax}$ (resp. length between
 $\var{nmin}$ and $nmax$), where the \emph{length} is the number of nonzero
 entries.
 
 \item $a=\var{amax}$ (resp. $a=[\var{amin},\var{amax}]$) restricts the parts
 to integers less than $\var{amax}$ (resp. between $\var{amin}$ and
 $\var{amax}$).
 
 By default, parts are positive and we remove zero entries unless $amin\leq0$,
 in which case $X$ is of constant length $\var{nmax}$.
 \bprog
 \\ at most 3 non-zero parts, all <= 4
 ? forpart(v=5,print(Vec(v)),4,3)
 [1, 4]
 [2, 3]
 [1, 1, 3]
 [1, 2, 2]
 
 \\ between 2 and 4 parts less than 5, fill with zeros
 ? forpart(v=5,print(Vec(v)),[0,5],[2,4])
 [0, 0, 1, 4]
 [0, 0, 2, 3]
 [0, 1, 1, 3]
 [0, 1, 2, 2]
 [1, 1, 1, 2]
 @eprog\noindent
 The behavior is unspecified if $X$ is modified inside the loop.
 
 \synt{forpart}{void *data, long (*call)(void*,GEN), long k, GEN a, GEN n}.

Function: forprime
Class: basic
Section: programming/control
C-Name: forprime
Prototype: vV=GDGI
Help: forprime(p=a,{b},seq): the sequence is evaluated, p running over the
 primes between a and b. Omitting b runs through primes >= a.
Iterator: 
 (*notype,small,small) (forprime, _u_forprime_init, _u_forprime_next)
 (*small,gen,?gen)    (forprime, _u_forprime_init, _u_forprime_next)
 (*int,gen,?gen)      (forprime, _forprime_init, _forprime_next_)
 (gen,gen,?gen)       (forprime, _forprime_init, _forprime_next_)
Doc: evaluates \var{seq},
 where the formal variable $p$ ranges over the prime numbers between the real
 numbers $a$ to $b$, including $a$ and $b$ if they are prime. More precisely,
 the value of
 $p$ is incremented to \kbd{nextprime($p$ + 1)}, the smallest prime strictly
 larger than $p$, at the end of each iteration. Nothing is done if $a>b$.
 \bprog
 ? forprime(p = 4, 10, print(p))
 5
 7
 @eprog\noindent Setting $b$ to \kbd{+oo} means we will run through all primes
 $\geq a$, starting an infinite loop; it is expected that the caller will break
 out of the loop itself at some point, using \kbd{break} or \kbd{return}.
 
 Note that the value of $p$ cannot be modified within \var{seq}:
 \bprog
 ? forprime(p = 2, 10, p = [])
  ***   at top-level: forprime(p=2,10,p=[])
  ***                                   ^---
  ***   prime index read-only: was changed to [].
 @eprog

Function: forqfvec
Class: basic
Section: linear_algebra
C-Name: forqfvec0
Prototype: vVGDGI
Help: forqfvec(v,q,b,expr): q being a square and symmetric integral matrix
 representing an positive definite quadratic form, evaluate expr
 for all vectors v such that q(v)<=b.
Doc: $q$ being a square and symmetric integral matrix representing a positive
 definite
 quadratic form, evaluate \kbd{expr} for all vector $v$ such that $q(v)\leq b$.
 The formal variable $v$ runs through all such vectors in turn.
 \bprog
 ? forqfvec(v, [3,2;2,3], 3, print(v))
 [0, 1]~
 [1, 0]~
 [-1, 1]~
 @eprog
Variant: The following function is also available:
 \fun{void}{forqfvec}{void *E, long (*fun)(void *, GEN, GEN, double), GEN q, GEN b}:
 Evaluate \kbd{fun(E,w,v,m)} on all $v$ such that $q(v)<b$, where $v$ is a
 \typ{VECSMALL} and $m=q(v)$ is a C double. The function \kbd{fun} must
 return $0$, unless \kbd{forqfvec} should stop, in which case, it should
 return $1$.

Function: forstep
Class: basic
Section: programming/control
C-Name: forstep
Prototype: vV=GGGI
Help: forstep(X=a,b,s,seq): the sequence is evaluated, X going from a to b
 in steps of s (can be a vector of steps). If b is set to +oo the loop will
 not stop.
Doc: evaluates \var{seq},
 where the formal variable $X$ goes from $a$ to $b$, in increments of $s$.
 Nothing is done if $s>0$ and $a>b$ or if $s<0$ and $a<b$. $s$ must be in
 $\R^*$ or a vector of steps $[s_1,\dots,s_n]$. In the latter case, the
 successive steps are used in the order they appear in $s$.
 
 \bprog
 ? forstep(x=5, 20, [2,4], print(x))
 5
 7
 11
 13
 17
 19
 @eprog\noindent Setting $b$ to \kbd{+oo} will start an infinite loop; it is
 expected that the caller will break out of the loop itself at some point,
 using \kbd{break} or \kbd{return}.

Function: forsubgroup
Class: basic
Section: programming/control
C-Name: forsubgroup0
Prototype: vV=GDGI
Help: forsubgroup(H=G,{bound},seq): execute seq for each subgroup H of the
 abelian group G, whose index is bounded by bound if not omitted. H is given
 as a left divisor of G in HNF form.
Wrapper: (,,vG)
Description: 
 (gen,?gen,closure):void  forsubgroup(${3 cookie}, ${3 wrapper}, $1, $2)
Doc: evaluates \var{seq} for
 each subgroup $H$ of the \emph{abelian} group $G$ (given in
 SNF\sidx{Smith normal form} form or as a vector of elementary divisors).
 
 If \var{bound} is present, and is a positive integer, restrict the output to
 subgroups of index less than \var{bound}. If \var{bound} is a vector
 containing a single positive integer $B$, then only subgroups of index
 exactly equal to $B$ are computed
 
 The subgroups are not ordered in any
 obvious way, unless $G$ is a $p$-group in which case Birkhoff's algorithm
 produces them by decreasing index. A \idx{subgroup} is given as a matrix
 whose columns give its generators on the implicit generators of $G$. For
 example, the following prints all subgroups of index less than 2 in $G =
 \Z/2\Z g_1 \times \Z/2\Z g_2$:
 
 \bprog
 ? G = [2,2]; forsubgroup(H=G, 2, print(H))
 [1; 1]
 [1; 2]
 [2; 1]
 [1, 0; 1, 1]
 @eprog\noindent
 The last one, for instance is generated by $(g_1, g_1 + g_2)$. This
 routine is intended to treat huge groups, when \tet{subgrouplist} is not an
 option due to the sheer size of the output.
 
 For maximal speed the subgroups have been left as produced by the algorithm.
 To print them in canonical form (as left divisors of $G$ in HNF form), one
 can for instance use
 \bprog
 ? G = matdiagonal([2,2]); forsubgroup(H=G, 2, print(mathnf(concat(G,H))))
 [2, 1; 0, 1]
 [1, 0; 0, 2]
 [2, 0; 0, 1]
 [1, 0; 0, 1]
 @eprog\noindent
 Note that in this last representation, the index $[G:H]$ is given by the
 determinant. See \tet{galoissubcyclo} and \tet{galoisfixedfield} for
 applications to \idx{Galois} theory.
 
 \synt{forsubgroup}{void *data, long (*call)(void*,GEN), GEN G, GEN bound}.

Function: forvec
Class: basic
Section: programming/control
C-Name: forvec
Prototype: vV=GID0,L,
Help: forvec(X=v,seq,{flag=0}): v being a vector of two-component vectors of
 length n, the sequence is evaluated with X[i] going from v[i][1] to v[i][2]
 for i=n,..,1 if flag is zero or omitted. If flag = 1 (resp. flag = 2),
 restrict to increasing (resp. strictly increasing) sequences.
Iterator: (gen,gen,?small) (forvec, _forvec_init, _forvec_next)
Doc: Let $v$ be an $n$-component
 vector (where $n$ is arbitrary) of two-component vectors $[a_i,b_i]$
 for $1\le i\le n$, where all entries $a_i$, $b_i$ are real numbers.
 This routine lets $X$ vary over the $n$-dimensional hyperrectangle
 given by $v$, that is, $X$ is an $n$-dimensional vector taking
 successively its entries $X[i]$ in the range $[a_i,b_i]$ with lexicographic
 ordering. (The component with the highest index moves the fastest.)
 The type of $X$ is the same as the type of $v$: \typ{VEC} or \typ{COL}.
 
 The expression \var{seq} is evaluated with the successive values of $X$.
 
 If $\fl=1$, generate only nondecreasing vectors $X$, and
 if $\fl=2$, generate only strictly increasing vectors $X$.
 \bprog
 ? forvec (X=[[0,1],[-1,1]], print(X));
 [0, -1]
 [0, 0]
 [0, 1]
 [1, -1]
 [1, 0]
 [1, 1]
 ? forvec (X=[[0,1],[-1,1]], print(X), 1);
 [0, 0]
 [0, 1]
 [1, 1]
 ? forvec (X=[[0,1],[-1,1]], print(X), 2)
 [0, 1]
 @eprog

Function: frac
Class: basic
Section: conversions
C-Name: gfrac
Prototype: G
Help: frac(x): fractional part of x = x-floor(x).
Doc: 
 fractional part of $x$. Identical to
 $x-\text{floor}(x)$. If $x$ is real, the result is in $[0,1[$.

Function: fromdigits
Class: basic
Section: conversions
C-Name: fromdigits
Prototype: GDG
Help: fromdigits(x,{b=10}): gives the integer formed by the elements of x seen
 as the digits of a number in base b.
Doc: gives the integer formed by the elements of $x$ seen as the digits of a
 number in base $b$ ($b = 10$ by default).  This is the reverse of \kbd{digits}:
 \bprog
 ? digits(1234,5)
 %1 = [1,4,4,1,4]
 ? fromdigits([1,4,4,1,4],5)
 %2 = 1234
 @eprog\noindent By convention, $0$ has no digits:
 \bprog
 ? fromdigits([])
 %3 = 0
 @eprog

Function: galoisexport
Class: basic
Section: number_fields
C-Name: galoisexport
Prototype: GD0,L,
Help: galoisexport(gal,{flag}): gal being a Galois group as output by
 galoisinit, output a string representing the underlying permutation group in
 GAP notation (default) or Magma notation (flag = 1).
Doc: \var{gal} being be a Galois group as output by \tet{galoisinit},
 export the underlying permutation group as a string suitable
 for (no flags or $\fl=0$) GAP or ($\fl=1$) Magma. The following example
 compute the index of the underlying abstract group in the GAP library:
 \bprog
 ? G = galoisinit(x^6+108);
 ? s = galoisexport(G)
 %2 = "Group((1, 2, 3)(4, 5, 6), (1, 4)(2, 6)(3, 5))"
 ? extern("echo \"IdGroup("s");\" | gap -q")
 %3 = [6, 1]
 ? galoisidentify(G)
 %4 = [6, 1]
 @eprog\noindent
 This command also accepts subgroups returned by \kbd{galoissubgroups}.
 
 To \emph{import} a GAP permutation into gp (for \tet{galoissubfields} for
 instance), the following GAP function may be useful:
 \bprog
 PermToGP := function(p, n)
   return Permuted([1..n],p);
 end;
 
 gap> p:= (1,26)(2,5)(3,17)(4,32)(6,9)(7,11)(8,24)(10,13)(12,15)(14,27)
   (16,22)(18,28)(19,20)(21,29)(23,31)(25,30)
 gap> PermToGP(p,32);
 [ 26, 5, 17, 32, 2, 9, 11, 24, 6, 13, 7, 15, 10, 27, 12, 22, 3, 28, 20, 19,
   29, 16, 31, 8, 30, 1, 14, 18, 21, 25, 23, 4 ]
 @eprog

Function: galoisfixedfield
Class: basic
Section: number_fields
C-Name: galoisfixedfield
Prototype: GGD0,L,Dn
Help: galoisfixedfield(gal,perm,{flag},{v=y}): gal being a Galois group as
 output by galoisinit and perm a subgroup, an element of gal.group or a vector
 of such elements, return [P,x] such that P is a polynomial defining the fixed
 field of gal[1] by the subgroup generated by perm, and x is a root of P in gal
 expressed as a polmod in gal.pol. If flag is 1 return only P. If flag is 2
 return [P,x,F] where F is the factorization of gal.pol over the field
 defined by P, where the variable v stands for a root of P.
Description: 
 (gen, gen, ?small, ?var):vec        galoisfixedfield($1, $2, $3, $4)
Doc: \var{gal} being be a Galois group as output by \tet{galoisinit} and
 \var{perm} an element of $\var{gal}.group$, a vector of such elements
 or a subgroup of \var{gal} as returned by galoissubgroups,
 computes the fixed field of \var{gal} by the automorphism defined by the
 permutations \var{perm} of the roots $\var{gal}.roots$. $P$ is guaranteed to
 be squarefree modulo $\var{gal}.p$.
 
 If no flags or $\fl=0$, output format is the same as for \tet{nfsubfield},
 returning $[P,x]$ such that $P$ is a polynomial defining the fixed field, and
 $x$ is a root of $P$ expressed as a polmod in $\var{gal}.pol$.
 
 If $\fl=1$ return only the polynomial $P$.
 
 If $\fl=2$ return $[P,x,F]$ where $P$ and $x$ are as above and $F$ is the
 factorization of $\var{gal}.pol$ over the field defined by $P$, where
 variable $v$ ($y$ by default) stands for a root of $P$. The priority of $v$
 must be less than the priority of the variable of $\var{gal}.pol$ (see
 \secref{se:priority}). Example:
 
 \bprog
 ? G = galoisinit(x^4+1);
 ? galoisfixedfield(G,G.group[2],2)
 %2 = [x^2 + 2, Mod(x^3 + x, x^4 + 1), [x^2 - y*x - 1, x^2 + y*x - 1]]
 @eprog\noindent
 computes the factorization  $x^4+1=(x^2-\sqrt{-2}x-1)(x^2+\sqrt{-2}x-1)$

Function: galoisgetpol
Class: basic
Section: number_fields
C-Name: galoisgetpol
Prototype: LD0,L,D1,L,
Help: galoisgetpol(a,{b},{s}): query the galpol package for a polynomial with
 Galois group isomorphic to GAP4(a,b), totally real if s=1 (default) and
 totally complex if s=2.  The output is a vector [pol, den] where pol is the
 polynomial and den is the common denominator of the conjugates expressed
 as a polynomial in a root of pol. If b and s are omitted, return the number of
 isomorphism classes of groups of order a.
Description: 
 (small):int               galoisnbpol($1)
 (small,):int              galoisnbpol($1)
 (small,,):int             galoisnbpol($1)
 (small,small,small):vec   galoisgetpol($1, $2 ,$3)
Doc: Query the galpol package for a polynomial with Galois group isomorphic to
 GAP4(a,b), totally real if $s=1$ (default) and totally complex if $s=2$. The
 output is a vector [\kbd{pol}, \kbd{den}] where
 
 \item  \kbd{pol} is the polynomial of degree $a$
 
 \item \kbd{den} is the denominator of \kbd{nfgaloisconj(pol)}.
 Pass it as an optional argument to \tet{galoisinit} or \tet{nfgaloisconj} to
 speed them up:
 \bprog
 ? [pol,den] = galoisgetpol(64,4,1);
 ? G = galoisinit(pol);
 time = 352ms
 ? galoisinit(pol, den);  \\ passing 'den' speeds up the computation
 time = 264ms
 ? % == %`
 %4 = 1  \\ same answer
 @eprog
 If $b$ and $s$ are omitted, return the number of isomorphism classes of
 groups of order $a$.
Variant: Also available is \fun{GEN}{galoisnbpol}{long a} when $b$ and $s$
 are omitted.

Function: galoisidentify
Class: basic
Section: number_fields
C-Name: galoisidentify
Prototype: G
Help: galoisidentify(gal): gal being a Galois group as output by galoisinit,
 output the isomorphism class of the underlying abstract group as a
 two-components vector [o,i], where o is the group order, and i is the group
 index in the GAP4 small group library.
Doc: \var{gal} being be a Galois group as output by \tet{galoisinit},
 output the isomorphism class of the underlying abstract group as a
 two-components vector $[o,i]$, where $o$ is the group order, and $i$ is the
 group index in the GAP4 Small Group library, by Hans Ulrich Besche, Bettina
 Eick and Eamonn O'Brien.
 
 This command also accepts subgroups returned by \kbd{galoissubgroups}.
 
 The current implementation is limited to degree less or equal to $127$.
 Some larger ``easy'' orders are also supported.
 
 The output is similar to the output of the function \kbd{IdGroup} in GAP4.
 Note that GAP4 \kbd{IdGroup} handles all groups of order less than $2000$
 except $1024$, so you can use \tet{galoisexport} and GAP4 to identify large
 Galois groups.

Function: galoisinit
Class: basic
Section: number_fields
C-Name: galoisinit
Prototype: GDG
Help: galoisinit(pol,{den}): pol being a polynomial or a number field as
 output by nfinit defining a Galois extension of Q, compute the Galois group
 and all necessary information for computing fixed fields. den is optional
 and has the same meaning as in nfgaloisconj(,4)(see manual).
Description: 
 (gen, ?int):gal        galoisinit($1, $2)
Doc: computes the Galois group
 and all necessary information for computing the fixed fields of the
 Galois extension $K/\Q$ where $K$ is the number field defined by
 $\var{pol}$ (monic irreducible polynomial in $\Z[X]$ or
 a number field as output by \tet{nfinit}). The extension $K/\Q$ must be
 Galois with Galois group ``weakly'' super-solvable, see below;
 returns 0 otherwise. Hence this permits to quickly check whether a polynomial
 of order strictly less than $36$ is Galois or not.
 
 The algorithm used is an improved version of the paper
 ``An efficient algorithm for the computation of Galois automorphisms'',
 Bill Allombert, Math.~Comp, vol.~73, 245, 2001, pp.~359--375.
 
 A group $G$ is said to be ``weakly'' super-solvable if there exists a
 normal series
 
 $\{1\} = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_{n-1}
 \triangleleft H_n$
 
 such that each $H_i$ is normal in $G$ and for $i<n$, each quotient group
 $H_{i+1}/H_i$ is cyclic, and either $H_n=G$ (then $G$ is super-solvable) or
 $G/H_n$ is isomorphic to either $A_4$ or $S_4$.
 
 In practice, almost all small groups are WKSS, the exceptions having order
 36(1 exception), 48(2), 56(1), 60(1), 72(5), 75(1), 80(1), 96(10) and $\geq
 108$.
 
 This function is a prerequisite for most of the \kbd{galois}$xxx$ routines.
 For instance:
 
 \bprog
 P = x^6 + 108;
 G = galoisinit(P);
 L = galoissubgroups(G);
 vector(#L, i, galoisisabelian(L[i],1))
 vector(#L, i, galoisidentify(L[i]))
 @eprog
 
 The output is an 8-component vector \var{gal}.
 
 $\var{gal}[1]$ contains the polynomial \var{pol}
 (\kbd{\var{gal}.pol}).
 
 $\var{gal}[2]$ is a three-components vector $[p,e,q]$ where $p$ is a
 prime number (\kbd{\var{gal}.p}) such that \var{pol} totally split
 modulo $p$ , $e$ is an integer and $q=p^e$ (\kbd{\var{gal}.mod}) is the
 modulus of the roots in \kbd{\var{gal}.roots}.
 
 $\var{gal}[3]$ is a vector $L$ containing the $p$-adic roots of
 \var{pol} as integers implicitly modulo \kbd{\var{gal}.mod}.
 (\kbd{\var{gal}.roots}).
 
 $\var{gal}[4]$ is the inverse of the Vandermonde matrix of the
 $p$-adic roots of \var{pol}, multiplied by $\var{gal}[5]$.
 
 $\var{gal}[5]$ is a multiple of the least common denominator of the
 automorphisms expressed as polynomial in a root of \var{pol}.
 
 $\var{gal}[6]$ is the Galois group $G$ expressed as a vector of
 permutations of $L$ (\kbd{\var{gal}.group}).
 
 $\var{gal}[7]$ is a generating subset $S=[s_1,\ldots,s_g]$ of $G$
 expressed as a vector of permutations of $L$ (\kbd{\var{gal}.gen}).
 
 $\var{gal}[8]$ contains the relative orders $[o_1,\ldots,o_g]$ of
 the generators of $S$ (\kbd{\var{gal}.orders}).
 
 Let $H_n$ be as above, we have the following properties:
 
 \quad\item if $G/H_n\simeq A_4$ then $[o_1,\ldots,o_g]$ ends by
 $[2,2,3]$.
 
 \quad\item if $G/H_n\simeq S_4$ then $[o_1,\ldots,o_g]$ ends by
 $[2,2,3,2]$.
 
 \quad\item for $1\leq i \leq g$ the subgroup of $G$ generated by
 $[s_1,\ldots,s_g]$ is normal, with the exception of $i=g-2$ in the
 $A_4$ case and of $i=g-3$ in the $S_A$ case.
 
 \quad\item the relative order $o_i$ of $s_i$ is its order in the
 quotient group $G/\langle s_1,\ldots,s_{i-1}\rangle$, with the same
 exceptions.
 
 \quad\item for any $x\in G$ there exists a unique family
 $[e_1,\ldots,e_g]$ such that (no exceptions):
 
 -- for $1\leq i \leq g$ we have $0\leq e_i<o_i$
 
 -- $x=g_1^{e_1}g_2^{e_2}\ldots g_n^{e_n}$
 
 If present $den$ must be a suitable value for $\var{gal}[5]$.

Function: galoisisabelian
Class: basic
Section: number_fields
C-Name: galoisisabelian
Prototype: GD0,L,
Help: galoisisabelian(gal,{flag=0}): gal being as output by galoisinit,
 return 0 if gal is not abelian, the HNF matrix of gal over gal.gen if
 flag=0, 1 if flag is 1, and the SNF of gal is flag=2.
Doc: \var{gal} being as output by \kbd{galoisinit}, return $0$ if
 \var{gal} is not an abelian group, and the HNF matrix of \var{gal} over
 \kbd{gal.gen} if $fl=0$, $1$ if $fl=1$.
 
 This command also accepts subgroups returned by \kbd{galoissubgroups}.

Function: galoisisnormal
Class: basic
Section: number_fields
C-Name: galoisisnormal
Prototype: lGG
Help: galoisisnormal(gal,subgrp): gal being as output by galoisinit,
 and subgrp a subgroup of gal as output by galoissubgroups,
 return 1 if subgrp is a normal subgroup of gal, else return 0.
Doc: \var{gal} being as output by \kbd{galoisinit}, and \var{subgrp} a subgroup
 of \var{gal} as output by \kbd{galoissubgroups},return $1$ if \var{subgrp} is a
 normal subgroup of \var{gal}, else return 0.
 
 This command also accepts subgroups returned by \kbd{galoissubgroups}.

Function: galoispermtopol
Class: basic
Section: number_fields
C-Name: galoispermtopol
Prototype: GG
Help: galoispermtopol(gal,perm): gal being a Galois group as output by
 galoisinit and perm a element of gal.group, return the polynomial defining
 the corresponding Galois automorphism.
Doc: \var{gal} being a
 Galois group as output by \kbd{galoisinit} and \var{perm} a element of
 $\var{gal}.group$, return the polynomial defining the Galois
 automorphism, as output by \kbd{nfgaloisconj}, attached to the
 permutation \var{perm} of the roots $\var{gal}.roots$. \var{perm} can
 also be a vector or matrix, in this case, \kbd{galoispermtopol} is
 applied to all components recursively.
 
 \noindent Note that
 \bprog
 G = galoisinit(pol);
 galoispermtopol(G, G[6])~
 @eprog\noindent
 is equivalent to \kbd{nfgaloisconj(pol)}, if degree of \var{pol} is greater
 or equal to $2$.

Function: galoissubcyclo
Class: basic
Section: number_fields
C-Name: galoissubcyclo
Prototype: GDGD0,L,Dn
Help: galoissubcyclo(N,H,{fl=0},{v}): compute a polynomial (in variable v)
 defining the subfield of Q(zeta_n) fixed by the subgroup H of (Z/nZ)*. N can
 be an integer n, znstar(n) or bnrinit(bnfinit(y),[n,[1]],1). H can be given
 by a generator, a set of generator given by a vector or a HNF matrix (see
 manual). If flag is 1, output only the conductor of the abelian extension.
 If flag is 2 output [pol,f] where pol is the polynomial and f the conductor.
Doc: computes the subextension
 of $\Q(\zeta_n)$ fixed by the subgroup $H \subset (\Z/n\Z)^*$. By the
 Kronecker-Weber theorem, all abelian number fields can be generated in this
 way (uniquely if $n$ is taken to be minimal).
 
 \noindent The pair $(n, H)$ is deduced from the parameters $(N, H)$ as follows
 
 \item $N$ an integer: then $n = N$; $H$ is a generator, i.e. an
 integer or an integer modulo $n$; or a vector of generators.
 
 \item $N$ the output of \kbd{znstar($n$)}. $H$ as in the first case
 above, or a matrix, taken to be a HNF left divisor of the SNF for $(\Z/n\Z)^*$
 (of type \kbd{$N$.cyc}), giving the generators of $H$ in terms of \kbd{$N$.gen}.
 
 \item $N$ the output of \kbd{bnrinit(bnfinit(y), $m$, 1)} where $m$ is a
 module. $H$ as in the first case, or a matrix taken to be a HNF left
 divisor of the SNF for the ray class group modulo $m$
 (of type \kbd{$N$.cyc}), giving the generators of $H$ in terms of \kbd{$N$.gen}.
 
 In this last case, beware that $H$ is understood relatively to $N$; in
 particular, if the infinite place does not divide the module, e.g if $m$ is
 an integer, then it is not a subgroup of $(\Z/n\Z)^*$, but of its quotient by
 $\{\pm 1\}$.
 
 If $fl=0$, compute a polynomial (in the variable \var{v}) defining
 the subfield of $\Q(\zeta_n)$ fixed by the subgroup \var{H} of $(\Z/n\Z)^*$.
 
 If $fl=1$, compute only the conductor of the abelian extension, as a module.
 
 If $fl=2$, output $[pol, N]$, where $pol$ is the polynomial as output when
 $fl=0$ and $N$ the conductor as output when $fl=1$.
 
 The following function can be used to compute all subfields of
 $\Q(\zeta_n)$ (of exact degree \kbd{d}, if \kbd{d} is set):
 \bprog
 polsubcyclo(n, d = -1)=
 { my(bnr,L,IndexBound);
   IndexBound = if (d < 0, n, [d]);
   bnr = bnrinit(bnfinit(y), [n,[1]], 1);
   L = subgrouplist(bnr, IndexBound, 1);
   vector(#L,i, galoissubcyclo(bnr,L[i]));
 }
 @eprog\noindent
 Setting \kbd{L = subgrouplist(bnr, IndexBound)} would produce subfields of exact
 conductor $n\infty$.

Function: galoissubfields
Class: basic
Section: number_fields
C-Name: galoissubfields
Prototype: GD0,L,Dn
Help: galoissubfields(G,{flag=0},{v}): output all the subfields of G. flag
 has the same meaning as for galoisfixedfield.
Doc: outputs all the subfields of the Galois group \var{G}, as a vector.
 This works by applying \kbd{galoisfixedfield} to all subgroups. The meaning of
 \var{flag} is the same as for \kbd{galoisfixedfield}.

Function: galoissubgroups
Class: basic
Section: number_fields
C-Name: galoissubgroups
Prototype: G
Help: galoissubgroups(G): output all the subgroups of G.
Doc: outputs all the subgroups of the Galois group \kbd{gal}. A subgroup is a
 vector [\var{gen}, \var{orders}], with the same meaning
 as for $\var{gal}.gen$ and $\var{gal}.orders$. Hence \var{gen} is a vector of
 permutations generating the subgroup, and \var{orders} is the relatives
 orders of the generators. The cardinality of a subgroup is the product of the
 relative orders. Such subgroup can be used instead of a Galois group in the
 following command: \kbd{galoisisabelian}, \kbd{galoissubgroups},
 \kbd{galoisexport} and \kbd{galoisidentify}.
 
 To get the subfield fixed by a subgroup \var{sub} of \var{gal}, use
 \bprog
 galoisfixedfield(gal,sub[1])
 @eprog

Function: gamma
Class: basic
Section: transcendental
C-Name: ggamma
Prototype: Gp
Help: gamma(s): gamma function at s, a complex or p-adic number, or a series.
Doc: For $s$ a complex number, evaluates Euler's gamma
 function \sidx{gamma-function}
 $$\Gamma(s)=\int_0^\infty t^{s-1}\exp(-t)\,dt.$$
 Error if $s$ is a non-positive integer, where $\Gamma$ has a pole.
 
 For $s$ a \typ{PADIC}, evaluates the Morita gamma function at $s$, that
 is the unique continuous $p$-adic function on the $p$-adic integers
 extending $\Gamma_p(k)=(-1)^k \prod_{j<k}'j$, where the prime means that $p$
 does not divide $j$.
 \bprog
 ? gamma(1/4 + O(5^10))
 %1= 1 + 4*5 + 3*5^4 + 5^6 + 5^7 + 4*5^9 + O(5^10)
 ? algdep(%,4)
 %2 = x^4 + 4*x^2 + 5
 @eprog
Variant: For a \typ{PADIC} $x$, the function \fun{GEN}{Qp_gamma}{GEN x} is
 also available.

Function: gammah
Class: basic
Section: transcendental
C-Name: ggammah
Prototype: Gp
Help: gammah(x): gamma of x+1/2 (x integer).
Doc: gamma function evaluated at the argument $x+1/2$.

Function: gammamellininv
Class: basic
Section: transcendental
C-Name: gammamellininv
Prototype: GGD0,L,b
Help: gammamellininv(G,t,{m=0}): returns G(t), where G is as output
 by gammamellininvinit. The alternative syntax gammamellininv(A,t,m)
 is also available.
Doc: returns the value at $t$ of the inverse Mellin transform
 $G$ initialized by \tet{gammamellininvinit}.
 \bprog
 ? G = gammamellininvinit([0]);
 ? gammamellininv(G, 2) - 2*exp(-Pi*2^2)
 %2 = -4.484155085839414627 E-44
 @eprog
 
 The alternative shortcut
 \bprog
   gammamellininv(A,t,m)
 @eprog\noindent for
 \bprog
   gammamellininv(gammamellininvinit(A,m), t)
 @eprog\noindent is available.

Function: gammamellininvasymp
Class: basic
Section: transcendental
C-Name: gammamellininvasymp
Prototype: GDPD0,L,
Help: gammamellininvasymp(A,n,{m=0}): return the first n terms of the
 asymptotic expansion at infinity of the m-th derivative K^m(t) of the
 inverse Mellin transform of the function
 f(s)=Gamma_R(s+a_1)*...*Gamma_R(s+a_d), where Vga is the vector [a_1,...,a_d]
 and Gamma_R(s)=Pi^(-s/2)*gamma(s/2). The result is a vector [M[1]...M[n]]
 with M[1]=1, such that
 K^m(t) = \sqrt{2^{d+1}/d}t^{a+m(2/d-1)}e^{-d pi t^{2/d}}\sum_{n\ge0}M[n+1]
 (pi t^{2n/d})^{-n}, with a = (1-d+sum_ja_j)/d.
Doc: Return the first $n$ terms of the asymptotic expansion at infinity
 of the $m$-th derivative $K^{(m)}(t)$ of the inverse Mellin transform of the
 function
 $$f(s) = \Gamma_\R(s+a_1)\*\ldots\*\Gamma_\R(s+a_d)\;,$$
 where \kbd{A} is the vector $[a_1,\ldots,a_d]$ and
 $\Gamma_\R(s)=\pi^{-s/2}\*\Gamma(s/2)$ (Euler's \kbd{gamma}).
 The result is a vector
 $[M[1]...M[n]]$ with M[1]=1, such that
 $$K^{(m)}(t)=\sqrt{2^{d+1}/d}t^{a+m(2/d-1)}e^{-d\pi t^{2/d}}
    \sum_{n\ge0} M[n+1] (\pi t^{2/d})^{-n} $$
 with $a=(1-d+\sum_{1\le j\le d}a_j)/d$.

Function: gammamellininvinit
Class: basic
Section: transcendental
C-Name: gammamellininvinit
Prototype: GD0,L,b
Help: gammamellininvinit(A,{m=0}): initialize data for the computation by
 gammamellininv() of the m-th derivative of the inverse Mellin transform
 of the function f(s) = Gamma_R(s+a1)*...*Gamma_R(s+ad), where
 A is the vector [a1,...,ad] and Gamma_R(s) = Pi^(-s/2)*gamma(s/2).
Doc: initialize data for the computation by \tet{gammamellininv} of
 the $m$-th derivative of the inverse Mellin transform of the function
 $$f(s) = \Gamma_\R(s+a_1)\*\ldots\*\Gamma_\R(s+a_d)$$
 where \kbd{A} is the vector $[a_1,\ldots,a_d]$ and
 $\Gamma_\R(s)=\pi^{-s/2}\*\Gamma(s/2)$ (Euler's \kbd{gamma}). This is the
 special case of Meijer's $G$ functions used to compute $L$-values via the
 approximate functional equation.
 
 \misctitle{Caveat} Contrary to the PARI convention, this function
 guarantees an \emph{absolute} (rather than relative) error bound.
 
 For instance, the inverse Mellin transform of $\Gamma_\R(s)$ is
 $2\exp(-\pi z^2)$:
 \bprog
 ? G = gammamellininvinit([0]);
 ? gammamellininv(G, 2) - 2*exp(-Pi*2^2)
 %2 = -4.484155085839414627 E-44
 @eprog
 The inverse Mellin transform of $\Gamma_\R(s+1)$ is
 $2 z\exp(-\pi z^2)$, and its second derivative is
 $ 4\pi z \exp(-\pi z^2)(2\pi z^2 - 3)$:
 \bprog
 ? G = gammamellininvinit([1], 2);
 ? a(z) = 4*Pi*z*exp(-Pi*z^2)*(2*Pi*z^2-3);
 ? b(z) = gammamellininv(G,z);
 ? t(z) = b(z) - a(z);
 ? t(3/2)
 %3 = -1.4693679385278593850 E-39
 @eprog

Function: gcd
Class: basic
Section: number_theoretical
C-Name: ggcd0
Prototype: GDG
Help: gcd(x,{y}): greatest common divisor of x and y.
Description: 
 (small, small):small   cgcd($1, $2)
 (int, int):int         gcdii($1, $2)
 (gen):gen              content($1)
 (gen, gen):gen         ggcd($1, $2)
Doc: creates the greatest common divisor of $x$ and $y$.
 If you also need the $u$ and $v$ such that $x*u + y*v = \gcd(x,y)$,
 use the \tet{bezout} function. $x$ and $y$ can have rather quite general
 types, for instance both rational numbers. If $y$ is omitted and $x$ is a
 vector, returns the $\text{gcd}$ of all components of $x$, i.e.~this is
 equivalent to \kbd{content(x)}.
 
 When $x$ and $y$ are both given and one of them is a vector/matrix type,
 the GCD is again taken recursively on each component, but in a different way.
 If $y$ is a vector, resp.~matrix, then the result has the same type as $y$,
 and components equal to \kbd{gcd(x, y[i])}, resp.~\kbd{gcd(x, y[,i])}. Else
 if $x$ is a vector/matrix the result has the same type as $x$ and an
 analogous definition. Note that for these types, \kbd{gcd} is not
 commutative.
 
 The algorithm used is a naive \idx{Euclid} except for the following inputs:
 
 \item integers: use modified right-shift binary (``plus-minus''
 variant).
 
 \item univariate polynomials with coefficients in the same number
 field (in particular rational): use modular gcd algorithm.
 
 \item general polynomials: use the \idx{subresultant algorithm} if
 coefficient explosion is likely (non modular coefficients).
 
 If $u$ and $v$ are polynomials in the same variable with \emph{inexact}
 coefficients, their gcd is defined to be scalar, so that
 \bprog
 ? a = x + 0.0; gcd(a,a)
 %1 = 1
 ? b = y*x + O(y); gcd(b,b)
 %2 = y
 ? c = 4*x + O(2^3); gcd(c,c)
 %3 = 4
 @eprog\noindent A good quantitative check to decide whether such a
 gcd ``should be'' non-trivial, is to use \tet{polresultant}: a value
 close to $0$ means that a small deformation of the inputs has non-trivial gcd.
 You may also use \tet{gcdext}, which does try to compute an approximate gcd
 $d$ and provides $u$, $v$ to check whether $u x + v y$ is close to $d$.
Variant: Also available are \fun{GEN}{ggcd}{GEN x, GEN y}, if \kbd{y} is not
 \kbd{NULL}, and \fun{GEN}{content}{GEN x}, if $\kbd{y} = \kbd{NULL}$.

Function: gcdext
Class: basic
Section: number_theoretical
C-Name: gcdext0
Prototype: GG
Help: gcdext(x,y): returns [u,v,d] such that d=gcd(x,y) and u*x+v*y=d.
Doc: Returns $[u,v,d]$ such that $d$ is the gcd of $x,y$,
 $x*u+y*v=\gcd(x,y)$, and $u$ and $v$ minimal in a natural sense.
 The arguments must be integers or polynomials. \sidx{extended gcd}
 \sidx{Bezout relation}
 \bprog
 ? [u, v, d] = gcdext(32,102)
 %1 = [16, -5, 2]
 ? d
 %2 = 2
 ? gcdext(x^2-x, x^2+x-2)
 %3 = [-1/2, 1/2, x - 1]
 @eprog
 
 If $x,y$ are polynomials in the same variable and \emph{inexact}
 coefficients, then compute $u,v,d$ such that $x*u+y*v = d$, where $d$
 approximately divides both and $x$ and $y$; in particular, we do not obtain
 \kbd{gcd(x,y)} which is \emph{defined} to be a scalar in this case:
 \bprog
 ? a = x + 0.0; gcd(a,a)
 %1 = 1
 
 ? gcdext(a,a)
 %2 = [0, 1, x + 0.E-28]
 
 ? gcdext(x-Pi, 6*x^2-zeta(2))
 %3 = [-6*x - 18.8495559, 1, 57.5726923]
 @eprog\noindent For inexact inputs, the output is thus not well defined
 mathematically, but you obtain explicit polynomials to check whether the
 approximation is close enough for your needs.

Function: genus2red
Class: basic
Section: elliptic_curves
C-Name: genus2red
Prototype: GDG
Help: genus2red(PQ,{p}): let PQ be a polynomial P, resp. a vector [P,Q] of
 polynomials, with rational coefficients.  Determines the reduction at p > 2
 of the (proper, smooth) hyperelliptic curve C/Q of genus 2 defined by
 y^2 = P, resp. y^2 + Q*y = P. More precisely, determines the special fiber X_p
 of the minimal regular model X of C over Z.
Doc: Let $PQ$ be a polynomial $P$, resp. a vector $[P,Q]$ of polynomials, with
 rational coefficients.
 Determines the reduction at $p > 2$ of the (proper, smooth) genus~2
 curve $C/\Q$, defined by the hyperelliptic equation $y^2 = P(x)$, resp.
 $y^2 + Q(x)*y = P(x)$.
 (The special fiber $X_p$ of the minimal regular model $X$ of $C$ over $\Z$.)
 
 If $p$ is omitted, determines the reduction type for all (odd) prime
 divisors of the discriminant.
 
 \noindent This function was rewritten from an implementation of Liu's
 algorithm by Cohen and Liu (1994), \kbd{genus2reduction-0.3}, see
 \url{http://www.math.u-bordeaux.fr/~liu/G2R/}.
 
 \misctitle{CAVEAT} The function interface may change: for the
 time being, it returns $[N,\var{FaN}, T, V]$
 where $N$ is either the local conductor at $p$ or the
 global conductor, \var{FaN} is its factorization, $y^2 = T$ defines a
 minimal model over $\Z[1/2]$ and $V$ describes the reduction type at the
 various considered~$p$. Unfortunately, the program is not complete for
 $p = 2$, and we may return the odd part of the conductor only: this is the
 case if the factorization includes the (impossible) term $2^{-1}$; if the
 factorization contains another power of $2$, then this is the exact local
 conductor at $2$ and $N$ is the global conductor.
 
 \bprog
 ? default(debuglevel, 1);
 ? genus2red(x^6 + 3*x^3 + 63, 3)
 (potential) stable reduction: [1, []]
 reduction at p: [III{9}] page 184, [3, 3], f = 10
 %1 = [59049, Mat([3, 10]), x^6 + 3*x^3 + 63, [3, [1, []],
        ["[III{9}] page 184", [3, 3]]]]
 ? [N, FaN, T, V] = genus2red(x^3-x^2-1, x^2-x);  \\ X_1(13), global reduction
 p = 13
 (potential) stable reduction: [5, [Mod(0, 13), Mod(0, 13)]]
 reduction at p: [I{0}-II-0] page 159, [], f = 2
 ? N
 %3 = 169
 ? FaN
 %4 = Mat([13, 2])   \\ in particular, good reduction at 2 !
 ? T
 %5 = x^6 + 58*x^5 + 1401*x^4 + 18038*x^3 + 130546*x^2 + 503516*x + 808561
 ? V
 %6 = [[13, [5, [Mod(0, 13), Mod(0, 13)]], ["[I{0}-II-0] page 159", []]]]
 @eprog\noindent
 We now first describe the format of the vector $V = V_p$ in the case where
 $p$ was specified (local reduction at~$p$): it is a triple $[p, \var{stable},
 \var{red}]$. The component $\var{stable} = [\var{type}, \var{vecj}]$ contains
 information about the stable reduction after a field extension;
 depending on \var{type}s, the stable reduction is
 
 \item 1: smooth (i.e. the curve has potentially good reduction). The
       Jacobian $J(C)$ has potentially good reduction.
 
 \item 2: an elliptic curve $E$ with an ordinary double point; \var{vecj}
 contains $j$ mod $p$, the modular invariant of $E$. The (potential)
 semi-abelian reduction of $J(C)$ is the extension of an elliptic curve (with
 modular invariant $j$ mod $p$) by a torus.
 
 \item 3: a projective line with two ordinary double points. The Jacobian
 $J(C)$ has potentially multiplicative reduction.
 
 \item 4: the union of two projective lines crossing transversally at three
 points. The Jacobian $J(C)$ has potentially multiplicative reduction.
 
 \item 5: the union of two elliptic curves $E_1$ and $E_2$ intersecting
 transversally at one point; \var{vecj} contains their modular invariants
 $j_1$ and $j_2$, which may live in a quadratic extension of $\F_p$ and need
 not be distinct. The Jacobian $J(C)$ has potentially good reduction,
 isomorphic to the product of the reductions of $E_1$ and $E_2$.
 
 \item 6: the union of an elliptic curve $E$ and a projective line which has
 an ordinary double point, and these two components intersect transversally
 at one point; \var{vecj} contains $j$ mod $p$, the modular invariant of $E$.
 The (potential) semi-abelian reduction of $J(C)$ is the extension of an
 elliptic curve (with modular invariant $j$ mod $p$) by a torus.
 
 \item 7: as in type 6, but the two components are both singular. The
 Jacobian $J(C)$ has potentially multiplicative reduction.
 
 The component $\var{red} = [\var{NUtype}, \var{neron}]$ contains two data
 concerning the reduction at $p$ without any ramified field extension.
 
 The \var{NUtype} is a \typ{STR} describing the reduction at $p$ of $C$,
 following Namikawa-Ueno, \emph{The complete classification of fibers in
 pencils of curves of genus two}, Manuscripta Math., vol. 9, (1973), pages
 143-186. The reduction symbol is followed by the corresponding page number
 or page range in this article.
 
 The second datum \var{neron} is the group of connected components (over an
 algebraic closure of $\F_p$) of the N\'eron model of $J(C)$, given as a
 finite abelian group (vector of elementary divisors).
 \smallskip
 If $p = 2$, the \var{red} component may be omitted altogether (and
 replaced by \kbd{[]}, in the case where the program could not compute it.
 When $p$ was not specified, $V$ is the vector of all $V_p$, for all
 considered $p$.
 
 \misctitle{Notes about Namikawa-Ueno types}
 
 \item A lower index is denoted between braces: for instance,
  \kbd{[I\obr2\cbr-II-5]} means \kbd{[I\_2-II-5]}.
 
 \item If $K$ and $K'$ are Kodaira symbols for singular fibers of elliptic
 curves, then \kbd{[$K$-$K'$-m]} and \kbd{[$K'$-$K$-m]} are the same.
 
 We define a total ordering on Kodaira symbol by fixing $\kbd{I} < \kbd{I*} <
 \kbd{II} < \kbd{II*}, \dots$. If the reduction type is the same, we order by
 the number of components, e.g. $\kbd{I}_2 < \kbd{I}_4$, etc.
 Then we normalize our output so that $K \leq K'$.
 
 \item \kbd{[$K$-$K'$-$-1$]}  is \kbd{[$K$-$K'$-$\alpha$]} in the notation of
 Namikawa-Ueno.
 
 \item The figure \kbd{[2I\_0-m]} in Namikawa-Ueno, page 159, must be denoted
 by \kbd{[2I\_0-(m+1)]}.

Function: getabstime
Class: basic
Section: programming/specific
C-Name: getabstime
Prototype: l
Help: getabstime(): time (in milliseconds) since startup.
Doc: returns the CPU time (in milliseconds) elapsed since \kbd{gp} startup.
 This provides a reentrant version of \kbd{gettime}:
 \bprog
 my (t = getabstime());
 ...
 print("Time: ", getabstime() - t);
 @eprog
 For a version giving wall-clock time, see \tet{getwalltime}.

Function: getenv
Class: basic
Section: programming/specific
C-Name: gp_getenv
Prototype: s
Help: getenv(s): value of the environment variable s, 0 if it is not defined.
Doc: return the value of the environment variable \kbd{s} if it is defined, otherwise return 0.

Function: getheap
Class: basic
Section: programming/specific
C-Name: getheap
Prototype: 
Help: getheap(): 2-component vector giving the current number of objects in
 the heap and the space they occupy (in long words).
Doc: returns a two-component row vector giving the
 number of objects on the heap and the amount of memory they occupy in long
 words. Useful mainly for debugging purposes.

Function: getrand
Class: basic
Section: programming/specific
C-Name: getrand
Prototype: 
Help: getrand(): current value of random number seed.
Doc: returns the current value of the seed used by the
 pseudo-random number generator \tet{random}. Useful mainly for debugging
 purposes, to reproduce a specific chain of computations. The returned value
 is technical (reproduces an internal state array), and can only be used as an
 argument to \tet{setrand}.

Function: getstack
Class: basic
Section: programming/specific
C-Name: getstack
Prototype: l
Help: getstack(): current value of stack pointer avma.
Doc: returns the current value of $\kbd{top}-\kbd{avma}$, i.e.~the number of
 bytes used up to now on the stack. Useful mainly for debugging purposes.

Function: gettime
Class: basic
Section: programming/specific
C-Name: gettime
Prototype: l
Help: gettime(): time (in milliseconds) since last call to gettime.
Doc: returns the CPU time (in milliseconds) used since either the last call to
 \kbd{gettime}, or to the beginning of the containing GP instruction (if
 inside \kbd{gp}), whichever came last.
 
 For a reentrant version, see \tet{getabstime}.
 
 For a version giving wall-clock time, see \tet{getwalltime}.

Function: getwalltime
Class: basic
Section: programming/specific
C-Name: getwalltime
Prototype: 
Help: getwalltime(): time (in milliseconds) since the UNIX Epoch.
Doc: returns the time (in milliseconds) elapsed since the UNIX Epoch
 (1970-01-01 00:00:00 (UTC)).
 \bprog
 my (t = getwalltime());
 ...
 print("Time: ", getwalltime() - t);
 @eprog

Function: global
Class: basic
Section: programming/specific
Help: global(list of variables): obsolete. Scheduled for deletion.
Doc: obsolete. Scheduled for deletion.
 % \syn{NO}
Obsolete: 2007-10-03

Function: hammingweight
Class: basic
Section: conversions
C-Name: hammingweight
Prototype: lG
Help: hammingweight(x): returns the Hamming weight of x.
Doc: 
 If $x$ is a \typ{INT}, return the binary Hamming weight of $|x|$. Otherwise
 $x$ must be of type \typ{POL}, \typ{VEC}, \typ{COL}, \typ{VECSMALL}, or
 \typ{MAT} and the function returns the number of non-zero coefficients of
 $x$.
 \bprog
 ? hammingweight(15)
 %1 = 4
 ? hammingweight(x^100 + 2*x + 1)
 %2 = 3
 ? hammingweight([Mod(1,2), 2, Mod(0,3)])
 %3 = 2
 ? hammingweight(matid(100))
 %4 = 100
 @eprog

Function: hilbert
Class: basic
Section: number_theoretical
C-Name: hilbert
Prototype: lGGDG
Help: hilbert(x,y,{p}): Hilbert symbol at p of x,y.
Doc: \idx{Hilbert symbol} of $x$ and $y$ modulo the prime $p$, $p=0$ meaning
 the place at infinity (the result is undefined if $p\neq 0$ is not prime).
 
 It is possible to omit $p$, in which case we take $p = 0$ if both $x$
 and $y$ are rational, or one of them is a real number. And take $p = q$
 if one of $x$, $y$ is a \typ{INTMOD} modulo $q$ or a $q$-adic. (Incompatible
 types will raise an error.)

Function: hyperellcharpoly
Class: basic
Section: elliptic_curves
C-Name: hyperellcharpoly
Prototype: G
Help: hyperellcharpoly(X): X being a non-singular hyperelliptic curve defined
 over a finite field, return the characteristic polynomial of the Frobenius
 automorphism.  X can be given either by a squarefree polynomial P such that
 X:y^2=P(x) or by a vector [P,Q] such that X:y^2+Q(x)*y=P(x) and Q^2+4P is
 squarefree.
Doc: 
 $X$ being a non-singular hyperelliptic curve defined over a finite field,
 return the characteristic polynomial of the Frobenius automorphism.
 $X$ can be given either by a squarefree polynomial $P$ such that
 $X: y^2 = P(x)$ or by a vector $[P,Q]$ such that
 $X: y^2 + Q(x)\*y = P(x)$ and $Q^2+4\*P$ is squarefree.

Function: hyperellpadicfrobenius
Class: basic
Section: elliptic_curves
C-Name: hyperellpadicfrobenius
Prototype: GUL
Help: hyperellpadicfrobenius(Q,p,n): Q being a  rational polynomial of degree
 d and X being the curve defined by y^2=Q(x), return the matrix of the
 Frobenius at p>=d in the standard basis of H^1_dR(X) to absolute p-adic
 precision p^n.
Doc: 
 Let $X$ be the curve defined by $y^2=Q(x)$, where  $Q$ is a polynomial of
 degree $d$ over $\Q$ and $p\ge d$ a prime such that $X$ has good reduction
 at $p$ return the matrix of the Frobenius endomorphism $\varphi$ on the
 crystalline module $D_p(X) = \Q_p \otimes H^1_{dR}(X/\Q)$ with respect to the
 basis of the given model $(\omega, x\*\omega,\ldots,x^{g-1}\*\omega)$, where
 $\omega = dx/(2\*y)$ is the invariant differential, where $g$ is the genus of
 $X$ (either $d=2\*g+1$ or $d=2\*g+2$).  The characteristic polynomial of
 $\varphi$ is the numerator of the zeta-function of the reduction of the curve
 $X$ modulo $p$. The matrix is computed to absolute $p$-adic precision $p^n$.

Function: hyperu
Class: basic
Section: transcendental
C-Name: hyperu
Prototype: GGGp
Help: hyperu(a,b,x): U-confluent hypergeometric function.
Doc: $U$-confluent hypergeometric function with
 parameters $a$ and $b$. The parameters $a$ and $b$ can be complex but
 the present implementation requires $x$ to be positive.

Function: idealadd
Class: basic
Section: number_fields
C-Name: idealadd
Prototype: GGG
Help: idealadd(nf,x,y): sum of two ideals x and y in the number field
 defined by nf.
Doc: sum of the two ideals $x$ and $y$ in the number field $\var{nf}$. The
 result is given in HNF.
 \bprog
  ? K = nfinit(x^2 + 1);
  ? a = idealadd(K, 2, x + 1)  \\ ideal generated by 2 and 1+I
  %2 =
  [2 1]
 
  [0 1]
  ? pr = idealprimedec(K, 5)[1];  \\ a prime ideal above 5
  ? idealadd(K, a, pr)     \\ coprime, as expected
  %4 =
  [1 0]
 
  [0 1]
 @eprog\noindent
 This function cannot be used to add arbitrary $\Z$-modules, since it assumes
 that its arguments are ideals:
 \bprog
   ? b = Mat([1,0]~);
   ? idealadd(K, b, b)     \\ only square t_MATs represent ideals
   *** idealadd: non-square t_MAT in idealtyp.
   ? c = [2, 0; 2, 0]; idealadd(K, c, c)   \\ non-sense
   %6 =
   [2 0]
 
   [0 2]
   ? d = [1, 0; 0, 2]; idealadd(K, d, d)   \\ non-sense
   %7 =
   [1 0]
 
   [0 1]
 
 @eprog\noindent In the last two examples, we get wrong results since the
 matrices $c$ and $d$ do not correspond to an ideal: the $\Z$-span of their
 columns (as usual interpreted as coordinates with respect to the integer basis
 \kbd{K.zk}) is not an $O_K$-module. To add arbitrary $\Z$-modules generated
 by the columns of matrices $A$ and $B$, use \kbd{mathnf(concat(A,B))}.

Function: idealaddtoone
Class: basic
Section: number_fields
C-Name: idealaddtoone0
Prototype: GGDG
Help: idealaddtoone(nf,x,{y}): if y is omitted, when the sum of the ideals
 in the number field K defined by nf and given in the vector x is equal to
 Z_K, gives a vector of elements of the corresponding ideals who sum to 1.
 Otherwise, x and y are ideals, and if they sum up to 1, find one element in
 each of them such that the sum is 1.
Doc: $x$ and $y$ being two co-prime
 integral ideals (given in any form), this gives a two-component row vector
 $[a,b]$ such that $a\in x$, $b\in y$ and $a+b=1$.
 
 The alternative syntax $\kbd{idealaddtoone}(\var{nf},v)$, is supported, where
 $v$ is a $k$-component vector of ideals (given in any form) which sum to
 $\Z_K$. This outputs a $k$-component vector $e$ such that $e[i]\in x[i]$ for
 $1\le i\le k$ and $\sum_{1\le i\le k}e[i]=1$.

Function: idealappr
Class: basic
Section: number_fields
C-Name: idealappr0
Prototype: GGD0,L,
Help: idealappr(nf,x,{flag}): x being a fractional ideal, gives an element
 b such that v_p(b)=v_p(x) for all prime ideals p dividing x, and v_p(b)>=0
 for all other p; x may also be a prime ideal factorization with possibly
 zero exponents. flag is deprecated (ignored), kept for backward compatibility
Doc: if $x$ is a fractional ideal
 (given in any form), gives an element $\alpha$ in $\var{nf}$ such that for
 all prime ideals $\goth{p}$ such that the valuation of $x$ at $\goth{p}$ is
 non-zero, we have $v_{\goth{p}}(\alpha)=v_{\goth{p}}(x)$, and
 $v_{\goth{p}}(\alpha)\ge0$ for all other $\goth{p}$.
 
 The argument $x$ may also be given as a prime ideal factorization, as
 output by \kbd{idealfactor}, but allowing zero exponents.
 This yields an element $\alpha$ such that for all prime ideals $\goth{p}$
 occurring in $x$, $v_{\goth{p}}(\alpha) = v_{\goth{p}}(x)$;
 for all other prime ideals, $v_{\goth{p}}(\alpha)\ge0$.
 
 flag is deprecated (ignored), kept for backward compatibility
Variant: Use directly \fun{GEN}{idealappr}{GEN nf, GEN x} since \fl is ignored.

Function: idealchinese
Class: basic
Section: number_fields
C-Name: idealchinese
Prototype: GGDG
Help: idealchinese(nf,x,{y}): x being a prime ideal factorization and y a
 vector of elements, gives an element b such that v_p(b-y_p)>=v_p(x) for all
 prime ideals p dividing x, and v_p(b)>=0 for all other p. If y is omitted,
 return a data structure which can be used in place of x in later calls.
Doc: $x$ being a prime ideal factorization
 (i.e.~a 2 by 2 matrix whose first column contains prime ideals, and the second
 column integral exponents), $y$ a vector of elements in $\var{nf}$ indexed by
 the ideals in $x$, computes an element $b$ such that
 
 $v_{\goth{p}}(b - y_{\goth{p}}) \geq v_{\goth{p}}(x)$ for all prime ideals
 in $x$ and $v_{\goth{p}}(b)\geq 0$ for all other $\goth{p}$.
 
 \bprog
 ? K = nfinit(t^2-2);
 ? x = idealfactor(K, 2^2*3)
 %2 =
 [[2, [0, 1]~, 2, 1, [0, 2; 1, 0]] 4]
 
 [           [3, [3, 0]~, 1, 2, 1] 1]
 ? y = [t,1];
 ? idealchinese(K, x, y)
 %4 = [4, -3]~
 @eprog
 
 The argument $x$ may also be of the form $[x, s]$ where the first component
 is as above and $s$ is a vector of signs, with $r_1$ components
 $s_i$ in $\{-1,0,1\}$:
 if $\sigma_i$ denotes the $i$-th real embedding of the number field,
 the element $b$ returned satisfies further
 $s_i \kbd{sign}(\sigma_i(b)) \geq 0$ for all $i$. In other words, the sign is
 fixed to $s_i$ at the $i$-th embedding whenever $s_i$ is non-zero.
 \bprog
 ? idealchinese(K, [x, [1,1]], y)
 %5 = [16, -3]~
 ? idealchinese(K, [x, [-1,-1]], y)
 %6 = [-20, -3]~
 ? idealchinese(K, [x, [1,-1]], y)
 %7 = [4, -3]~
 @eprog
 
 If $y$ is omitted, return a data structure which can be used in
 place of $x$ in later calls and allows to solve many chinese remainder
 problems for a given $x$ more efficiently.
 \bprog
 ? C = idealchinese(K, [x, [1,1]]);
 ? idealchinese(K, C, y) \\ as above
 %9 = [16, -3]~
 ? for(i=1,10^4, idealchinese(K,C,y))  \\ ... but faster !
 time = 80 ms.
 ? for(i=1,10^4, idealchinese(K,[x,[1,1]],y))
 time = 224 ms.
 @eprog
 Finally, this structure is itself allowed in place of $x$, the
 new $s$ overriding the one already present in the structure. This allows to
 initialize for different sign conditions more efficiently when the underlying
 ideal factorization remains the same.
 \bprog
 ? D = idealchinese(K, [C, [1,-1]]);   \\ replaces [1,1]
 ? idealchinese(K, D, y)
 %13 = [4, -3]~
 ? for(i=1,10^4,idealchinese(K,[C,[1,-1]]))
 time = 40 ms.   \\ faster than starting from scratch
 ? for(i=1,10^4,idealchinese(K,[x,[1,-1]]))
 time = 128 ms.
 @eprog
Variant: Also available is
 \fun{GEN}{idealchineseinit}{GEN nf, GEN x} when $y = \kbd{NULL}$.

Function: idealcoprime
Class: basic
Section: number_fields
C-Name: idealcoprime
Prototype: GGG
Help: idealcoprime(nf,x,y): gives an element b in nf such that b. x is an
 integral ideal coprime to the integral ideal y.
Doc: given two integral ideals $x$ and $y$
 in the number field $\var{nf}$, returns a $\beta$ in the field,
 such that $\beta\cdot x$ is an integral ideal coprime to $y$.

Function: idealdiv
Class: basic
Section: number_fields
C-Name: idealdiv0
Prototype: GGGD0,L,
Help: idealdiv(nf,x,y,{flag=0}): quotient x/y of two ideals x and y in HNF
 in the number field nf. If (optional) flag is non-null, the quotient is
 supposed to be an integral ideal (slightly faster).
Description: 
 (gen, gen, gen, ?0):gen        idealdiv($1, $2, $3)
 (gen, gen, gen, 1):gen         idealdivexact($1, $2, $3)
 (gen, gen, gen, #small):gen    $"invalid flag in idealdiv"
 (gen, gen, gen, small):gen     idealdiv0($1, $2, $3, $4)
Doc: quotient $x\cdot y^{-1}$ of the two ideals $x$ and $y$ in the number
 field $\var{nf}$. The result is given in HNF.
 
 If $\fl$ is non-zero, the quotient $x \cdot y^{-1}$ is assumed to be an
 integral ideal. This can be much faster when the norm of the quotient is
 small even though the norms of $x$ and $y$ are large.
Variant: Also available are \fun{GEN}{idealdiv}{GEN nf, GEN x, GEN y}
 ($\fl=0$) and \fun{GEN}{idealdivexact}{GEN nf, GEN x, GEN y} ($\fl=1$).

Function: idealfactor
Class: basic
Section: number_fields
C-Name: idealfactor
Prototype: GG
Help: idealfactor(nf,x): factorization of the ideal x into prime ideals in the
 number field nf.
Doc: factors into prime ideal powers the
 ideal $x$ in the number field $\var{nf}$. The output format is similar to the
 \kbd{factor} function, and the prime ideals are represented in the form
 output by the \kbd{idealprimedec} function.

Function: idealfactorback
Class: basic
Section: number_fields
C-Name: idealfactorback
Prototype: GGDGD0,L,
Help: idealfactorback(nf,f,{e},{flag = 0}): given a factorisation f, gives the
 ideal product back. If e is present, f has to be a
 vector of the same length, and we return the product of the f[i]^e[i]. If
 flag is non-zero, perform idealred along the way.
Doc: gives back the ideal corresponding to a factorization. The integer $1$
 corresponds to the empty factorization.
 If $e$ is present, $e$ and $f$ must be vectors of the same length ($e$ being
 integral), and the corresponding factorization is the product of the
 $f[i]^{e[i]}$.
 
 If not, and $f$ is vector, it is understood as in the preceding case with $e$
 a vector of 1s: we return the product of the $f[i]$. Finally, $f$ can be a
 regular factorization, as produced by \kbd{idealfactor}.
 \bprog
 ? nf = nfinit(y^2+1); idealfactor(nf, 4 + 2*y)
 %1 =
 [[2, [1, 1]~, 2, 1, [1, 1]~] 2]
 
 [[5, [2, 1]~, 1, 1, [-2, 1]~] 1]
 
 ? idealfactorback(nf, %)
 %2 =
 [10 4]
 
 [0  2]
 
 ? f = %1[,1]; e = %1[,2]; idealfactorback(nf, f, e)
 %3 =
 [10 4]
 
 [0  2]
 
 ? % == idealhnf(nf, 4 + 2*y)
 %4 = 1
 @eprog
 If \kbd{flag} is non-zero, perform ideal reductions (\tet{idealred}) along the
 way. This is most useful if the ideals involved are all \emph{extended}
 ideals (for instance with trivial principal part), so that the principal parts
 extracted by \kbd{idealred} are not lost. Here is an example:
 \bprog
 ? f = vector(#f, i, [f[i], [;]]);  \\ transform to extended ideals
 ? idealfactorback(nf, f, e, 1)
 %6 = [[1, 0; 0, 1], [2, 1; [2, 1]~, 1]]
 ? nffactorback(nf, %[2])
 %7 = [4, 2]~
 @eprog
 The extended ideal returned in \kbd{\%6} is the trivial ideal $1$, extended
 with a principal generator given in factored form. We use \tet{nffactorback}
 to recover it in standard form.

Function: idealfrobenius
Class: basic
Section: number_fields
C-Name: idealfrobenius
Prototype: GGG
Help: idealfrobenius(nf,gal,pr): returns the Frobenius element (pr|nf/Q)
 attached to the unramified prime ideal pr in prid format, in the Galois
 group gal of the number field nf.
Doc: Let $K$ be the number field defined by $nf$ and assume $K/\Q$ be a
 Galois extension with Galois group given \kbd{gal=galoisinit(nf)},
 and that \var{pr} is an unramified prime ideal $\goth{p}$ in \kbd{prid}
 format.
 This function returns a permutation of \kbd{gal.group} which defines
 the Frobenius element $\Frob_{\goth{p}}$ attached to $\goth{p}$.
 If $p$ is the unique prime number in $\goth{p}$, then
 $\Frob(x)\equiv x^p\mod\goth{p}$ for all $x\in\Z_K$.
 \bprog
 ? nf = nfinit(polcyclo(31));
 ? gal = galoisinit(nf);
 ? pr = idealprimedec(nf,101)[1];
 ? g = idealfrobenius(nf,gal,pr);
 ? galoispermtopol(gal,g)
 %5 = x^8
 @eprog\noindent This is correct since $101\equiv 8\mod{31}$.

Function: idealhnf
Class: basic
Section: number_fields
C-Name: idealhnf0
Prototype: GGDG
Help: idealhnf(nf,u,{v}): hermite normal form of the ideal u in the number
 field nf if v is omitted. If called as idealhnf(nf,u,v), the ideal
 is given as uZ_K + vZ_K in the number field K defined by nf.
Doc: gives the \idx{Hermite normal form} of the ideal $u\Z_K+v\Z_K$, where $u$
 and $v$ are elements of the number field $K$ defined by \var{nf}.
 \bprog
 ? nf = nfinit(y^3 - 2);
 ? idealhnf(nf, 2, y+1)
 %2 =
 [1 0 0]
 
 [0 1 0]
 
 [0 0 1]
 ? idealhnf(nf, y/2, [0,0,1/3]~)
 %3 =
 [1/3 0 0]
 
 [0 1/6 0]
 
 [0 0 1/6]
 @eprog
 
 If $b$ is omitted, returns the HNF of the ideal defined by $u$: $u$ may be an
 algebraic number (defining a principal ideal), a maximal ideal (as given by
 \kbd{idealprimedec} or \kbd{idealfactor}), or a matrix whose columns give
 generators for the ideal. This last format is a little complicated, but
 useful to reduce general modules to the canonical form once in a while:
 
 \item if strictly less than $N = [K:\Q]$ generators are given, $u$
 is the $\Z_K$-module they generate,
 
 \item if $N$ or more are given, it is \emph{assumed} that they form a
 $\Z$-basis of the ideal, in particular that the matrix has maximal rank $N$.
 This acts as \kbd{mathnf} since the $\Z_K$-module structure is (taken for
 granted hence) not taken into account in this case.
 \bprog
 ? idealhnf(nf, idealprimedec(nf,2)[1])
 %4 =
 [2 0 0]
 
 [0 1 0]
 
 [0 0 1]
 ? idealhnf(nf, [1,2;2,3;3,4])
 %5 =
 [1 0 0]
 
 [0 1 0]
 
 [0 0 1]
 @eprog\noindent Finally, when $K$ is quadratic with discriminant $D_K$, we
 allow $u =$ \kbd{Qfb(a,b,c)}, provided $b^2 - 4ac = D_K$. As usual,
 this represents the ideal $a \Z + (1/2)(-b + \sqrt{D_K}) \Z$.
 \bprog
 ? K = nfinit(x^2 - 60); K.disc
 %1 = 60
 ? idealhnf(K, qfbprimeform(60,2))
 %2 =
 [2 1]
 
 [0 1]
 ? idealhnf(K, Qfb(1,2,3))
   ***   at top-level: idealhnf(K,Qfb(1,2,3
   ***                 ^--------------------
   *** idealhnf: Qfb(1, 2, 3) has discriminant != 60 in idealhnf.
 @eprog
Variant: Also available is \fun{GEN}{idealhnf}{GEN nf, GEN a}.

Function: idealintersect
Class: basic
Section: number_fields
C-Name: idealintersect
Prototype: GGG
Help: idealintersect(nf,A,B): intersection of two ideals A and B in the
 number field defined by nf.
Doc: intersection of the two ideals
 $A$ and $B$ in the number field $\var{nf}$. The result is given in HNF.
 \bprog
 ? nf = nfinit(x^2+1);
 ? idealintersect(nf, 2, x+1)
 %2 =
 [2 0]
 
 [0 2]
 @eprog
 
 This function does not apply to general $\Z$-modules, e.g.~orders, since its
 arguments are replaced by the ideals they generate. The following script
 intersects $\Z$-modules $A$ and $B$ given by matrices of compatible
 dimensions with integer coefficients:
 \bprog
 ZM_intersect(A,B) =
 { my(Ker = matkerint(concat(A,B)));
   mathnf( A * Ker[1..#A,] )
 }
 @eprog

Function: idealinv
Class: basic
Section: number_fields
C-Name: idealinv
Prototype: GG
Help: idealinv(nf,x): inverse of the ideal x in the number field nf.
Description: 
 (gen, gen):gen        idealinv($1, $2)
Doc: inverse of the ideal $x$ in the
 number field $\var{nf}$, given in HNF. If $x$ is an extended
 ideal\sidx{ideal (extended)}, its principal part is suitably
 updated: i.e. inverting $[I,t]$, yields $[I^{-1}, 1/t]$.

Function: ideallist
Class: basic
Section: number_fields
C-Name: ideallist0
Prototype: GLD4,L,
Help: ideallist(nf,bound,{flag=4}): vector of vectors L of all idealstar of
 all ideals of norm<=bound. If (optional) flag is present, its binary digits
 are toggles meaning 1: give generators; 2: add units; 4: give only the
 ideals and not the bid.
Doc: computes the list
 of all ideals of norm less or equal to \var{bound} in the number field
 \var{nf}. The result is a row vector with exactly \var{bound} components.
 Each component is itself a row vector containing the information about
 ideals of a given norm, in no specific order, depending on the value of
 $\fl$:
 
 The possible values of $\fl$ are:
 
 \quad 0: give the \var{bid} attached to the ideals, without generators.
 
 \quad 1: as 0, but include the generators in the \var{bid}.
 
 \quad 2: in this case, \var{nf} must be a \var{bnf} with units. Each
 component is of the form $[\var{bid},U]$, where \var{bid} is as case 0
 and $U$ is a vector of discrete logarithms of the units. More precisely, it
 gives the \kbd{ideallog}s with respect to \var{bid} of \kbd{bnf.tufu}.
 This structure is technical, and only meant to be used in conjunction with
 \tet{bnrclassnolist} or \tet{bnrdisclist}.
 
 \quad 3: as 2, but include the generators in the \var{bid}.
 
 \quad 4: give only the HNF of the ideal.
 
 \bprog
 ? nf = nfinit(x^2+1);
 ? L = ideallist(nf, 100);
 ? L[1]
 %3 = [[1, 0; 0, 1]]  \\@com A single ideal of norm 1
 ? #L[65]
 %4 = 4               \\@com There are 4 ideals of norm 4 in $\Z[i]$
 @eprog
 If one wants more information, one could do instead:
 \bprog
 ? nf = nfinit(x^2+1);
 ? L = ideallist(nf, 100, 0);
 ? l = L[25]; vector(#l, i, l[i].clgp)
 %3 = [[20, [20]], [16, [4, 4]], [20, [20]]]
 ? l[1].mod
 %4 = [[25, 18; 0, 1], []]
 ? l[2].mod
 %5 = [[5, 0; 0, 5], []]
 ? l[3].mod
 %6 = [[25, 7; 0, 1], []]
 @eprog\noindent where we ask for the structures of the $(\Z[i]/I)^*$ for all
 three ideals of norm $25$. In fact, for all moduli with finite part of norm
 $25$ and trivial Archimedean part, as the last 3 commands show. See
 \tet{ideallistarch} to treat general moduli.

Function: ideallistarch
Class: basic
Section: number_fields
C-Name: ideallistarch
Prototype: GGG
Help: ideallistarch(nf,list,arch): list is a vector of vectors of bid's as
 output by ideallist. Return a vector of vectors with the same number of
 components as the original list. The leaves give information about
 moduli whose finite part is as in original list, in the same order, and
 Archimedean part is now arch. The information contained is of the same kind
 as was present in the input.
Doc: 
 \var{list} is a vector of vectors of bid's, as output by \tet{ideallist} with
 flag $0$ to $3$. Return a vector of vectors with the same number of
 components as the original \var{list}. The leaves give information about
 moduli whose finite part is as in original list, in the same order, and
 Archimedean part is now \var{arch} (it was originally trivial). The
 information contained is of the same kind as was present in the input; see
 \tet{ideallist}, in particular the meaning of \fl.
 
 \bprog
 ? bnf = bnfinit(x^2-2);
 ? bnf.sign
 %2 = [2, 0]                         \\@com two places at infinity
 ? L = ideallist(bnf, 100, 0);
 ? l = L[98]; vector(#l, i, l[i].clgp)
 %4 = [[42, [42]], [36, [6, 6]], [42, [42]]]
 ? La = ideallistarch(bnf, L, [1,1]); \\@com add them to the modulus
 ? l = La[98]; vector(#l, i, l[i].clgp)
 %6 = [[168, [42, 2, 2]], [144, [6, 6, 2, 2]], [168, [42, 2, 2]]]
 @eprog
 Of course, the results above are obvious: adding $t$ places at infinity will
 add $t$ copies of $\Z/2\Z$ to $(\Z_K/f)^*$. The following application
 is more typical:
 \bprog
 ? L = ideallist(bnf, 100, 2);        \\@com units are required now
 ? La = ideallistarch(bnf, L, [1,1]);
 ? H = bnrclassnolist(bnf, La);
 ? H[98];
 %4 = [2, 12, 2]
 @eprog

Function: ideallog
Class: basic
Section: number_fields
C-Name: ideallog
Prototype: DGGG
Help: ideallog({nf},x,bid): if bid is a big ideal, as given by
 idealstar(nf,D,...), gives the vector of exponents on the generators bid.gen
 (even if these generators have not been explicitly computed).
Doc: $\var{nf}$ is a number field,
 \var{bid} is as output by \kbd{idealstar(nf, D, \dots)} and $x$ a
 non-necessarily integral element of \var{nf} which must have valuation
 equal to 0 at all prime ideals in the support of $\kbd{D}$. This function
 computes the discrete logarithm of $x$ on the generators given in
 \kbd{\var{bid}.gen}. In other words, if $g_i$ are these generators, of orders
 $d_i$ respectively, the result is a column vector of integers $(x_i)$ such
 that $0\le x_i<d_i$ and
 $$x \equiv \prod_i g_i^{x_i} \pmod{\ ^*D}\enspace.$$
 Note that when the support of \kbd{D} contains places at infinity, this
 congruence implies also sign conditions on the attached real embeddings.
 See \tet{znlog} for the limitations of the underlying discrete log algorithms.
 
 When \var{nf} is omitted, take it to be the rational number field. In that
 case, $x$ must be a \typ{INT} and \var{bid} must have been initialized by
 \kbd{idealstar(,N)}.
Variant: Also available is
 \fun{GEN}{Zideallog}{GEN bid, GEN x} when \kbd{nf} is \kbd{NULL}.

Function: idealmin
Class: basic
Section: number_fields
C-Name: idealmin
Prototype: GGDG
Help: idealmin(nf,ix,{vdir}): pseudo-minimum of the ideal ix in the direction
 vdir in the number field nf.
Doc: \emph{This function is useless and kept for backward compatibility only,
 use \kbd{idealred}}. Computes a pseudo-minimum of the ideal $x$ in the
 direction \var{vdir} in the number field \var{nf}.

Function: idealmul
Class: basic
Section: number_fields
C-Name: idealmul0
Prototype: GGGD0,L,
Help: idealmul(nf,x,y,{flag=0}): product of the two ideals x and y in the
 number field nf. If (optional) flag is non-nul, reduce the result.
Description: 
 (gen, gen, gen, ?0):gen        idealmul($1, $2, $3)
 (gen, gen, gen, 1):gen         idealmulred($1, $2, $3)
 (gen, gen, gen, #small):gen    $"invalid flag in idealmul"
 (gen, gen, gen, small):gen     idealmul0($1, $2, $3, $4)
Doc: ideal multiplication of the ideals $x$ and $y$ in the number field
 \var{nf}; the result is the ideal product in HNF. If either $x$ or $y$
 are extended ideals\sidx{ideal (extended)}, their principal part is suitably
 updated: i.e. multiplying $[I,t]$, $[J,u]$ yields $[IJ, tu]$; multiplying
 $I$ and $[J, u]$ yields $[IJ, u]$.
 \bprog
 ? nf = nfinit(x^2 + 1);
 ? idealmul(nf, 2, x+1)
 %2 =
 [4 2]
 
 [0 2]
 ? idealmul(nf, [2, x], x+1)        \\ extended ideal * ideal
 %3 = [[4, 2; 0, 2], x]
 ? idealmul(nf, [2, x], [x+1, x])   \\ two extended ideals
 %4 = [[4, 2; 0, 2], [-1, 0]~]
 @eprog\noindent
 If $\fl$ is non-zero, reduce the result using \kbd{idealred}.
Variant: 
 \noindent See also
 \fun{GEN}{idealmul}{GEN nf, GEN x, GEN y} ($\fl=0$) and
 \fun{GEN}{idealmulred}{GEN nf, GEN x, GEN y} ($\fl\neq0$).

Function: idealnorm
Class: basic
Section: number_fields
C-Name: idealnorm
Prototype: GG
Help: idealnorm(nf,x): norm of the ideal x in the number field nf.
Doc: computes the norm of the ideal~$x$ in the number field~$\var{nf}$.

Function: idealnumden
Class: basic
Section: number_fields
C-Name: idealnumden
Prototype: GG
Help: idealnumden(nf,x): returns [A,B], where A,B are coprime integer ideals
 such that x = A/B.
Doc: returns $[A,B]$, where $A,B$ are coprime integer ideals
 such that $x = A/B$, in the number field $\var{nf}$.
 \bprog
 ? nf = nfinit(x^2+1);
 ? idealnumden(nf, (x+1)/2)
 %2 = [[1, 0; 0, 1], [2, 1; 0, 1]]
 @eprog

Function: idealpow
Class: basic
Section: number_fields
C-Name: idealpow0
Prototype: GGGD0,L,
Help: idealpow(nf,x,k,{flag=0}): k-th power of the ideal x in HNF in the
 number field nf. If (optional) flag is non-null, reduce the result.
Doc: computes the $k$-th power of
 the ideal $x$ in the number field $\var{nf}$; $k\in\Z$.
 If $x$ is an extended
 ideal\sidx{ideal (extended)}, its principal part is suitably
 updated: i.e. raising $[I,t]$ to the $k$-th power, yields $[I^k, t^k]$.
 
 If $\fl$ is non-zero, reduce the result using \kbd{idealred}, \emph{throughout
 the (binary) powering process}; in particular, this is \emph{not} the same
 as $\kbd{idealpow}(\var{nf},x,k)$ followed by reduction.
Variant: 
 \noindent See also
 \fun{GEN}{idealpow}{GEN nf, GEN x, GEN k} and
 \fun{GEN}{idealpows}{GEN nf, GEN x, long k} ($\fl = 0$).
 Corresponding to $\fl=1$ is \fun{GEN}{idealpowred}{GEN nf, GEN vp, GEN k}.

Function: idealprimedec
Class: basic
Section: number_fields
C-Name: idealprimedec_limit_f
Prototype: GGD0,L,
Help: idealprimedec(nf,p,{f=0}): prime ideal decomposition of the prime number
 p in the number field nf as a vector of prime ideals. If f is present
 and non-zero, restrict the result to primes of residue degree <= f.
Description: 
 (gen, gen):vec idealprimedec($1, $2)
 (gen, gen, ?small):vec idealprimedec_limit_f($1, $2, $3)
Doc: computes the prime ideal
 decomposition of the (positive) prime number $p$ in the number field $K$
 represented by \var{nf}. If a non-prime $p$ is given the result is undefined.
 If $f$ is present and non-zero, restrict the result to primes of residue
 degree $\leq f$.
 
 The result is a vector of \tev{prid} structures, each representing one of the
 prime ideals above $p$ in the number field $\var{nf}$. The representation
 $\kbd{pr}=[p,a,e,f,\var{mb}]$ of a prime ideal means the following: $a$
 is an algebraic integer in the maximal order $\Z_K$ and the prime ideal is
 equal to $\goth{p} = p\Z_K + a\Z_K$;
 $e$ is the ramification index; $f$ is the residual index;
 finally, \var{mb} is the multiplication table attached to the algebraic
 integer $b$ is such that $\goth{p}^{-1}=\Z_K+ b/ p\Z_K$, which is used
 internally to compute valuations. In other words if $p$ is inert,
 then \var{mb} is the integer $1$, and otherwise it is a square \typ{MAT}
 whose $j$-th column is $b \cdot \kbd{nf.zk[j]}$.
 
 The algebraic number $a$ is guaranteed to have a
 valuation equal to 1 at the prime ideal (this is automatic if $e>1$).
 
 The components of \kbd{pr} should be accessed by member functions: \kbd{pr.p},
 \kbd{pr.e}, \kbd{pr.f}, and \kbd{pr.gen} (returns the vector $[p,a]$):
 \bprog
 ? K = nfinit(x^3-2);
 ? P = idealprimedec(K, 5);
 ? #P       \\ 2 primes above 5 in Q(2^(1/3))
 %3 = 2
 ? [p1,p2] = P;
 ? [p1.e, p1.f]    \\ the first is unramified of degree 1
 %5 = [1, 1]
 ? [p2.e, p2.f]    \\ the second is unramified of degree 2
 %6 = [1, 2]
 ? p1.gen
 %7 = [5, [2, 1, 0]~]
 ? nfbasistoalg(K, %[2])  \\ a uniformizer for p1
 %8 = Mod(x + 2, x^3 - 2)
 ? #idealprimedec(K, 5, 1) \\ restrict to f = 1
 %9 = 1            \\ now only p1
 @eprog

Function: idealprincipalunits
Class: basic
Section: number_fields
C-Name: idealprincipalunits
Prototype: GGL
Help: idealprincipalunits(nf,pr,k): returns the structure [no, cyc, gen]
 of the multiplicative group (1 + pr) / (1 + pr^k).
Doc: given a prime ideal in \tet{idealprimedec} format,
 returns the multiplicative group $(1 + \var{pr}) / (1 + \var{pr}^k)$ as an
 abelian group. This function is much faster than \tet{idealstar} when the
 norm of \var{pr} is large, since it avoids (useless) work in the
 multiplicative group of the residue field.
 \bprog
 ? K = nfinit(y^2+1);
 ? P = idealprimedec(K,2)[1];
 ? G = idealprincipalunits(K, P, 20);
 ? G.cyc
 %4 = [512, 256, 4]   \\ Z/512 x Z/256 x Z/4
 ? G.gen
 %5 = [[-1, -2]~, 1021, [0, -1]~] \\ minimal generators of given order
 @eprog

Function: idealramgroups
Class: basic
Section: number_fields
C-Name: idealramgroups
Prototype: GGG
Help: idealramgroups(nf,gal,pr): let pr be a prime ideal in prid format, and
 gal the Galois group of the number field nf, return a vector g such that g[1]
 is the decomposition group of pr, g[2] is the inertia group, g[i] is the
 (i-2)th ramification group of pr, all trivial subgroups being omitted.
Doc: Let $K$ be the number field defined by \var{nf} and assume that $K/\Q$ is
 Galois with Galois group $G$ given by \kbd{gal=galoisinit(nf)}.
 Let \var{pr} be the prime ideal $\goth{P}$ in prid format.
 This function returns a vector $g$ of subgroups of \kbd{gal}
 as follow:
 
 \item \kbd{g[1]} is the decomposition group of $\goth{P}$,
 
 \item \kbd{g[2]} is $G_0(\goth{P})$, the inertia group of $\goth{P}$,
 
 and for $i\geq 2$,
 
 \item \kbd{g[i]} is $G_{i-2}(\goth{P})$, the $i-2$-th
 \idx{ramification group} of $\goth{P}$.
 
 \noindent The length of $g$ is the number of non-trivial groups in the
 sequence, thus is $0$ if $e=1$ and $f=1$, and $1$ if $f>1$ and $e=1$.
 The following function computes the cardinality of a subgroup of $G$,
 as given by the components of $g$:
 \bprog
 card(H) =my(o=H[2]); prod(i=1,#o,o[i]);
 @eprog
 \bprog
 ? nf=nfinit(x^6+3); gal=galoisinit(nf); pr=idealprimedec(nf,3)[1];
 ? g = idealramgroups(nf, gal, pr);
 ? apply(card,g)
 %3 = [6, 6, 3, 3, 3] \\ cardinalities of the G_i
 @eprog
 
 \bprog
 ? nf=nfinit(x^6+108); gal=galoisinit(nf); pr=idealprimedec(nf,2)[1];
 ? iso=idealramgroups(nf,gal,pr)[2]
 %5 = [[Vecsmall([2, 3, 1, 5, 6, 4])], Vecsmall([3])]
 ? nfdisc(galoisfixedfield(gal,iso,1))
 %6 = -3
 @eprog\noindent The field fixed by the inertia group of $2$ is not ramified at
 $2$.

Function: idealred
Class: basic
Section: number_fields
C-Name: idealred0
Prototype: GGDG
Help: idealred(nf,I,{v=0}): LLL reduction of the ideal I in the number
 field nf along direction v, in HNF.
Doc: \idx{LLL} reduction of
 the ideal $I$ in the number field $K$ attached to \var{nf}, along the
 direction $v$. The $v$ parameter is best left omitted, but if it is present,
 it must be an $\kbd{nf.r1} + \kbd{nf.r2}$-component vector of
 \emph{non-negative} integers. (What counts is the relative magnitude of the
 entries: if all entries are equal, the effect is the same as if the vector
 had been omitted.)
 
 This function finds an $a\in K^*$ such that $J = (a)I$ is
 ``small'' and integral (see the end for technical details).
 The result is the Hermite normal form of
 the ``reduced'' ideal $J$.
 \bprog
 ? K = nfinit(y^2+1);
 ? P = idealprimedec(K,5)[1];
 ? idealred(K, P)
 %3 =
 [1 0]
 
 [0 1]
 @eprog\noindent More often than not, a \idx{principal ideal} yields the unit
 ideal as above. This is a quick and dirty way to check if ideals are principal,
 but it is not a necessary condition: a non-trivial result does not prove that
 the ideal is non-principal. For guaranteed results, see \kbd{bnfisprincipal},
 which requires the computation of a full \kbd{bnf} structure.
 
 If the input is an extended ideal $[I,s]$, the output is $[J, sa]$; in
 this way, one keeps track of the principal ideal part:
 \bprog
 ? idealred(K, [P, 1])
 %5 = [[1, 0; 0, 1], [2, -1]~]
 @eprog\noindent
 meaning that $P$ is generated by $[2, -1]~$. The number field element in the
 extended part is an algebraic number in any form \emph{or} a factorization
 matrix (in terms of number field elements, not ideals!). In the latter case,
 elements stay in factored form, which is a convenient way to avoid
 coefficient explosion; see also \tet{idealpow}.
 
 \misctitle{Technical note} The routine computes an LLL-reduced
 basis for the lattice $I^(-1)$ equipped with the quadratic
 form
 $$|| x ||_v^2 = \sum_{i=1}^{r_1+r_2} 2^{v_i}\varepsilon_i|\sigma_i(x)|^2,$$
 where as usual the $\sigma_i$ are the (real and) complex embeddings and
 $\varepsilon_i = 1$, resp.~$2$, for a real, resp.~complex place. The element
 $a$ is simply the first vector in the LLL basis. The only reason you may want
 to try to change some directions and set some $v_i\neq 0$ is to randomize
 the elements found for a fixed ideal, which is heuristically useful in index
 calculus algorithms like \tet{bnfinit} and \tet{bnfisprincipal}.
 
 \misctitle{Even more technical note} In fact, the above is a white lie.
 We do not use $||\cdot||_v$ exactly but a rescaled rounded variant which
 gets us faster and simpler LLLs. There's no harm since we are not using any
 theoretical property of $a$ after all, except that it belongs to $I^(-1)$
 and that $a I$ is ``expected to be small''.

Function: idealstar
Class: basic
Section: number_fields
C-Name: idealstar0
Prototype: DGGD1,L,
Help: idealstar({nf},N,{flag=1}): gives the structure of (Z_K/N)^*, where N is
 a modulus (an ideal in any form or a vector [f0, foo], where f0 is an ideal
 and foo is a {0,1}-vector with r1 components. flag is
 optional, and can be 0: simply gives the structure as an abelian group, i.e.
 a 3-component vector [h,d,g] where h is the order, d the orders of the cyclic
 factors and g the generators;
 if flag=1 (default), gives a bid structure used in ideallog
 to compute discrete logarithms; underlying generators are well-defined but not
 explicitly computed, which saves time; if flag=2, same as with flag=1 except
 that the generators are also given.
 If nf is omitted, N must be an integer and
 we return the structure of (Z/NZ)^*.
Doc: outputs a \kbd{bid} structure,
 necessary for computing in the finite abelian group $G = (\Z_K/N)^*$. Here,
 \var{nf} is a number field and $N$ is a \var{modulus}: either an ideal in any
 form, or a row vector whose first component is an ideal and whose second
 component is a row vector of $r_1$ 0 or 1. Ideals can also be given
 by a factorization into prime ideals, as produced by \tet{idealfactor}.
 
 This \var{bid} is used in \tet{ideallog} to compute discrete logarithms. It
 also contains useful information which can be conveniently retrieved as
 \kbd{\var{bid}.mod} (the modulus),
 \kbd{\var{bid}.clgp} ($G$ as a finite abelian group),
 \kbd{\var{bid}.no} (the cardinality of $G$),
 \kbd{\var{bid}.cyc} (elementary divisors) and
 \kbd{\var{bid}.gen} (generators).
 
 If $\fl=1$ (default), the result is a \kbd{bid} structure without
 generators: they are well defined but not explicitly computed, which saves
 time.
 
 If $\fl=2$, as $\fl=1$, but including generators.
 
 If $\fl=0$, only outputs $(\Z_K/N)^*$ as an abelian group,
 i.e as a 3-component vector $[h,d,g]$: $h$ is the order, $d$ is the vector of
 SNF\sidx{Smith normal form} cyclic components and $g$ the corresponding
 generators.
 
 If \var{nf} is omitted, we take it to be the rational number fields, $N$ must
 be an integer and we return the structure of $(\Z/N\Z)^*$. In other words
 \kbd{idealstar(, N, flag)} is short for
 \bprog
   idealstar(nfinit(x), N, flag)
 @eprog\noindent but much faster. The alternative syntax \kbd{znstar(N, flag)}
 is also available for the same effect, but due to an unfortunate historical
 oversight, the default value of \kbd{flag} is different in the two
 functions (\kbd{znstar} does not initialize by default).
Variant: Instead the above hardcoded numerical flags, one should rather use
 \fun{GEN}{Idealstar}{GEN nf, GEN ideal, long flag}, where \kbd{flag} is
 an or-ed combination of \tet{nf_GEN} (include generators) and \tet{nf_INIT}
 (return a full \kbd{bid}, not a group), possibly $0$. This offers
 one more combination: gen, but no init.

Function: idealtwoelt
Class: basic
Section: number_fields
C-Name: idealtwoelt0
Prototype: GGDG
Help: idealtwoelt(nf,x,{a}): two-element representation of an ideal x in the
 number field nf. If (optional) a is non-zero, first element will be equal to a.
Doc: computes a two-element
 representation of the ideal $x$ in the number field $\var{nf}$, combining a
 random search and an approximation theorem; $x$ is an ideal
 in any form (possibly an extended ideal, whose principal part is ignored)
 
 \item When called as \kbd{idealtwoelt(nf,x)}, the result is a row vector
 $[a,\alpha]$ with two components such that $x=a\Z_K+\alpha\Z_K$ and $a$ is
 chosen to be the positive generator of $x\cap\Z$, unless $x$ was given as a
 principal ideal (in which case we may choose $a = 0$). The algorithm
 uses a fast lazy factorization of $x\cap \Z$ and runs in randomized
 polynomial time.
 
 \item When called as \kbd{idealtwoelt(nf,x,a)} with an explicit non-zero $a$
 supplied as third argument, the function assumes that $a \in x$ and returns
 $\alpha\in x$ such that $x = a\Z_K + \alpha\Z_K$. Note that we must factor
 $a$ in this case, and the algorithm is generally much slower than the
 default variant.
Variant: Also available are
 \fun{GEN}{idealtwoelt}{GEN nf, GEN x} and
 \fun{GEN}{idealtwoelt2}{GEN nf, GEN x, GEN a}.

Function: idealval
Class: basic
Section: number_fields
C-Name: gpidealval
Prototype: GGG
Help: idealval(nf,x,pr): valuation at pr given in idealprimedec format of the
 ideal x in the number field nf.
Doc: gives the valuation of the ideal $x$ at the prime ideal \var{pr} in the
 number field $\var{nf}$, where \var{pr} is in \kbd{idealprimedec} format.
 The valuation of the $0$ ideal is \kbd{+oo}.
Variant: Also available is
 \fun{long}{idealval}{GEN nf, GEN x, GEN pr}, which returns
 \tet{LONG_MAX} if $x = 0$ and the valuation as a \kbd{long} integer.

Function: if
Class: basic
Section: programming/control
C-Name: ifpari
Prototype: GDEDE
Help: if(a,{seq1},{seq2}): if a is nonzero, seq1 is evaluated, otherwise seq2.
 seq1 and seq2 are optional, and if seq2 is omitted, the preceding comma can
 be omitted also.
Doc: evaluates the expression sequence \var{seq1} if $a$ is non-zero, otherwise
 the expression \var{seq2}. Of course, \var{seq1} or \var{seq2} may be empty:
 
 \kbd{if ($a$,\var{seq})} evaluates \var{seq} if $a$ is not equal to zero
 (you don't have to write the second comma), and does nothing otherwise,
 
 \kbd{if ($a$,,\var{seq})} evaluates \var{seq} if $a$ is equal to zero, and
 does nothing otherwise. You could get the same result using the \kbd{!}
 (\kbd{not}) operator: \kbd{if (!$a$,\var{seq})}.
 
 The value of an \kbd{if} statement is the value of the branch that gets
 evaluated: for instance
 \bprog
 x = if(n % 4 == 1, y, z);
 @eprog\noindent sets $x$ to $y$ if $n$ is $1$ modulo $4$, and to $z$
 otherwise.
 
 Successive 'else' blocks can be abbreviated in a single compound \kbd{if}
 as follows:
 \bprog
 if (test1, seq1,
     test2, seq2,
     ...
     testn, seqn,
     seqdefault);
 @eprog\noindent is equivalent to
 \bprog
 if (test1, seq1
          , if (test2, seq2
                     , ...
                       if (testn, seqn, seqdefault)...));
 @eprog For instance, this allows to write traditional switch / case
 constructions:
 \bprog
 if (x == 0, do0(),
     x == 1, do1(),
     x == 2, do2(),
     dodefault());
 @eprog
 
 \misctitle{Remark}
 The boolean operators \kbd{\&\&} and \kbd{||} are evaluated
 according to operator precedence as explained in \secref{se:operators}, but,
 contrary to other operators, the evaluation of the arguments is stopped
 as soon as the final truth value has been determined. For instance
 \bprog
 if (x != 0 && f(1/x), ...)
 @eprog
 \noindent is a perfectly safe statement.
 
 \misctitle{Remark} Functions such as \kbd{break} and \kbd{next} operate on
 \emph{loops}, such as \kbd{for$xxx$}, \kbd{while}, \kbd{until}. The \kbd{if}
 statement is \emph{not} a loop. (Obviously!)

Function: iferr
Class: basic
Section: programming/control
C-Name: iferrpari
Prototype: EVEDE
Help: iferr(seq1,E,seq2,{pred}): evaluates the expression sequence seq1. If
 an error occurs, set the formal parameter E set to the error data.
 If pred is not present or evaluates to true, catch the error and evaluate
 seq2. Both pred and seq2 can reference E.
Doc: evaluates the expression sequence \var{seq1}. If an error occurs,
 set the formal parameter \var{E} set to the error data.
 If \var{pred} is not present or evaluates to true, catch the error
 and evaluate \var{seq2}. Both \var{pred} and \var{seq2} can reference \var{E}.
 The error type is given by \kbd{errname(E)}, and other data can be
 accessed using the \tet{component} function. The code \var{seq2} should check
 whether the error is the one expected. In the negative the error can be
 rethrown using \tet{error(E)} (and possibly caught by an higher \kbd{iferr}
 instance). The following uses \kbd{iferr} to implement Lenstra's ECM factoring
  method
 \bprog
 ? ecm(N, B = 1000!, nb = 100)=
   {
     for(a = 1, nb,
       iferr(ellmul(ellinit([a,1]*Mod(1,N)), [0,1]*Mod(1,N), B),
         E, return(gcd(lift(component(E,2)),N)),
         errname(E)=="e_INV" && type(component(E,2)) == "t_INTMOD"))
   }
 ? ecm(2^101-1)
 %2 = 7432339208719
 @eprog
 The return value of \kbd{iferr} itself is the value of \var{seq2} if an
 error occurs, and the value of \var{seq1} otherwise. We now describe the
 list of valid error types, and the attached error data \var{E}; in each
 case, we list in order the components of \var{E}, accessed via
 \kbd{component(E,1)}, \kbd{component(E,2)}, etc.
 
  \misctitle{Internal errors, ``system'' errors}
 
  \item \kbd{"e\_ARCH"}. A requested feature $s$ is not available on this
  architecture or operating system.
  \var{E} has one component (\typ{STR}): the missing feature name $s$.
 
  \item \kbd{"e\_BUG"}. A bug in the PARI library, in function $s$.
  \var{E} has one component (\typ{STR}): the function name $s$.
 
  \item \kbd{"e\_FILE"}. Error while trying to open a file.
  \var{E} has two components, 1 (\typ{STR}): the file type (input, output,
  etc.), 2 (\typ{STR}): the file name.
 
  \item \kbd{"e\_IMPL"}. A requested feature $s$ is not implemented.
  \var{E} has one component, 1 (\typ{STR}): the feature name $s$.
 
  \item \kbd{"e\_PACKAGE"}. Missing optional package $s$.
  \var{E} has one component, 1 (\typ{STR}): the package name $s$.
 
  \misctitle{Syntax errors, type errors}
 
  \item \kbd{"e\_DIM"}. The dimensions of arguments $x$ and $y$ submitted
  to function $s$ does not match up.
  E.g., multiplying matrices of inconsistent dimension, adding vectors of
  different lengths,\dots
  \var{E} has three component, 1 (\typ{STR}): the function name $s$, 2: the
  argument $x$, 3: the argument $y$.
 
  \item \kbd{"e\_FLAG"}. A flag argument is out of bounds in function $s$.
  \var{E} has one component, 1 (\typ{STR}): the function name $s$.
 
  \item \kbd{"e\_NOTFUNC"}. Generated by the PARI evaluator; tried to use a
 \kbd{GEN} $x$ which is not a \typ{CLOSURE} in a function call syntax (as in
 \kbd{f = 1; f(2);}).
  \var{E} has one component, 1: the offending \kbd{GEN} $x$.
 
  \item \kbd{"e\_OP"}. Impossible operation between two objects than cannot
  be typecast to a sensible common domain for deeper reasons than a type
  mismatch, usually for arithmetic reasons. As in \kbd{O(2) + O(3)}: it is
  valid to add two \typ{PADIC}s, provided the underlying prime is the same; so
  the addition is not forbidden a priori for type reasons, it only becomes so
  when inspecting the objects and trying to perform the operation.
  \var{E} has three components, 1 (\typ{STR}): the operator name \var{op},
  2: first argument, 3: second argument.
 
  \item \kbd{"e\_TYPE"}. An argument $x$ of function $s$ had an unexpected type.
  (As in \kbd{factor("blah")}.)
  \var{E} has two components, 1 (\typ{STR}): the function name $s$,
  2: the offending argument $x$.
 
  \item \kbd{"e\_TYPE2"}. Forbidden operation between two objects than cannot be
  typecast to a sensible common domain, because their types do not match up.
  (As in \kbd{Mod(1,2) + Pi}.)
  \var{E} has three components, 1 (\typ{STR}): the operator name \var{op},
  2: first argument, 3: second argument.
 
  \item \kbd{"e\_PRIORITY"}. Object $o$ in function $s$ contains
  variables whose priority is incompatible with the expected operation.
  E.g.~\kbd{Pol([x,1], 'y)}: this raises an error because it's not possible to
  create a polynomial whose coefficients involve variables with higher priority
  than the main variable. $E$ has four components: 1 (\typ{STR}): the function
  name $s$, 2: the offending argument $o$, 3 (\typ{STR}): an operator
  $\var{op}$ describing the priority error, 4 (\typ{POL}):
  the variable $v$ describing the priority error. The argument
  satisfies $\kbd{variable}(x)~\var{op} \kbd{variable}(v)$.
 
  \item \kbd{"e\_VAR"}. The variables of arguments $x$ and $y$ submitted
  to function $s$ does not match up. E.g., considering the algebraic number
  \kbd{Mod(t,t\pow2+1)} in \kbd{nfinit(x\pow2+1)}.
  \var{E} has three component, 1 (\typ{STR}): the function name $s$, 2
  (\typ{POL}): the argument $x$, 3 (\typ{POL}): the argument $y$.
 
  \misctitle{Overflows}
 
  \item \kbd{"e\_COMPONENT"}. Trying to access an inexistent component in a
  vector/matrix/list in a function: the index is less than $1$ or greater
  than the allowed length.
  \var{E} has four components,
  1 (\typ{STR}): the function name
  2 (\typ{STR}): an operator $\var{op}$ ($<$ or $>$),
  2 (\typ{GEN}): a numerical limit $l$ bounding the allowed range,
  3 (\kbd{GEN}): the index $x$. It satisfies $x$ \var{op} $l$.
 
  \item \kbd{"e\_DOMAIN"}. An argument is not in the function's domain.
  \var{E} has five components, 1 (\typ{STR}): the function name,
  2 (\typ{STR}): the mathematical name of the out-of-domain argument
  3 (\typ{STR}): an operator $\var{op}$ describing the domain error,
  4 (\typ{GEN}): the numerical limit $l$ describing the domain error,
  5 (\kbd{GEN}): the out-of-domain argument $x$. The argument satisfies $x$
  \var{op} $l$, which prevents it from belonging to the function's domain.
 
  \item \kbd{"e\_MAXPRIME"}. A function using the precomputed list of prime
  numbers ran out of primes.
  \var{E} has one component, 1 (\typ{INT}): the requested prime bound, which
  overflowed \kbd{primelimit} or $0$ (bound is unknown).
 
  \item \kbd{"e\_MEM"}. A call to \tet{pari_malloc} or \tet{pari_realloc}
  failed. \var{E} has no component.
 
  \item \kbd{"e\_OVERFLOW"}. An object in function $s$ becomes too large to be
  represented within PARI's hardcoded limits. (As in \kbd{2\pow2\pow2\pow10} or
  \kbd{exp(1e100)}, which overflow in \kbd{lg} and \kbd{expo}.)
  \var{E} has one component, 1 (\typ{STR}): the function name $s$.
 
  \item \kbd{"e\_PREC"}. Function $s$ fails because input accuracy is too low.
  (As in \kbd{floor(1e100)} at default accuracy.)
  \var{E} has one component, 1 (\typ{STR}): the function name $s$.
 
  \item \kbd{"e\_STACK"}. The PARI stack overflows.
  \var{E} has no component.
 
  \misctitle{Errors triggered intentionally}
 
  \item \kbd{"e\_ALARM"}. A timeout, generated by the \tet{alarm} function.
  \var{E} has one component (\typ{STR}): the error message to print.
 
  \item \kbd{"e\_USER"}. A user error, as triggered by
  \tet{error}($g_1,\dots,g_n)$.
  \var{E} has one component, 1 (\typ{VEC}): the vector of $n$ arguments given
  to \kbd{error}.
 
  \misctitle{Mathematical errors}
 
  \item \kbd{"e\_CONSTPOL"}. An argument of function $s$ is a constant
  polynomial, which does not make sense. (As in \kbd{galoisinit(Pol(1))}.)
  \var{E} has one component, 1 (\typ{STR}): the function name $s$.
 
  \item \kbd{"e\_COPRIME"}. Function $s$ expected coprime arguments,
  and did receive $x,y$, which were not.
  \var{E} has three component, 1 (\typ{STR}): the function name $s$,
  2: the argument $x$, 3: the argument $y$.
 
  \item \kbd{"e\_INV"}. Tried to invert a non-invertible object $x$ in
  function $s$.
  \var{E} has two components, 1 (\typ{STR}): the function name $s$,
  2: the non-invertible $x$. If $x = \kbd{Mod}(a,b)$
  is a \typ{INTMOD} and $a$ is not $0$ mod $b$, this allows to factor
  the modulus, as \kbd{gcd}$(a,b)$ is a non-trivial divisor of $b$.
 
  \item \kbd{"e\_IRREDPOL"}. Function $s$ expected an irreducible polynomial,
  and did receive $T$, which was not. (As in \kbd{nfinit(x\pow2-1)}.)
  \var{E} has two component, 1 (\typ{STR}): the function name $s$,
  2 (\typ{POL}): the polynomial $x$.
 
  \item \kbd{"e\_MISC"}. Generic uncategorized error.
  \var{E} has one component (\typ{STR}): the error message to print.
 
  \item \kbd{"e\_MODULUS"}. moduli $x$ and $y$ submitted to function $s$ are
  inconsistent. As in
  \bprog
    nfalgtobasis(nfinit(t^3-2), Mod(t,t^2+1)
  @eprog\noindent
  \var{E} has three component, 1 (\typ{STR}): the function $s$,
  2: the argument $x$, 3: the argument $x$.
 
  \item \kbd{"e\_PRIME"}. Function $s$ expected a prime number,
  and did receive $p$, which was not. (As in \kbd{idealprimedec(nf, 4)}.)
  \var{E} has two component, 1 (\typ{STR}): the function name $s$,
  2: the argument $p$.
 
  \item \kbd{"e\_ROOTS0"}. An argument of function $s$ is a zero polynomial,
  and we need to consider its roots. (As in \kbd{polroots(0)}.) \var{E} has
  one component, 1 (\typ{STR}): the function name $s$.
 
  \item \kbd{"e\_SQRTN"}. Trying to compute an $n$-th root of $x$, which does
  not exist, in function $s$. (As in \kbd{sqrt(Mod(-1,3))}.)
  \var{E} has two components, 1 (\typ{STR}): the function name $s$,
  2: the argument $x$.

Function: imag
Class: basic
Section: conversions
C-Name: gimag
Prototype: G
Help: imag(x): imaginary part of x.
Doc: imaginary part of $x$. When $x$ is a quadratic number, this is the
 coefficient of $\omega$ in the ``canonical'' integral basis $(1,\omega)$.

Function: incgam
Class: basic
Section: transcendental
C-Name: incgam0
Prototype: GGDGp
Help: incgam(s,x,{g}): incomplete gamma function. g is optional and is the
 precomputed value of gamma(s).
Doc: incomplete gamma function $\int_x^\infty e^{-t}t^{s-1}\,dt$, extended by
 analytic continuation to all complex $x, s$ not both $0$. The relative error
 is bounded in terms of the precision of $s$ (the accuracy of $x$ is ignored
 when determining the output precision). When $g$ is given, assume that
 $g=\Gamma(s)$. For small $|x|$, this will speed up the computation.
Variant: Also available is \fun{GEN}{incgam}{GEN s, GEN x, long prec}.

Function: incgamc
Class: basic
Section: transcendental
C-Name: incgamc
Prototype: GGp
Help: incgamc(s,x): complementary incomplete gamma function.
Doc: complementary incomplete gamma function.
 The arguments $x$ and $s$ are complex numbers such that $s$ is not a pole of
 $\Gamma$ and $|x|/(|s|+1)$ is not much larger than 1 (otherwise the
 convergence is very slow). The result returned is $\int_0^x
 e^{-t}t^{s-1}\,dt$.

Function: inline
Class: basic
Section: programming/specific
Help: inline(x,...,z): declares x,...,z as inline variables [EXPERIMENTAL].
Doc: (Experimental) declare $x,\ldots, z$ as inline variables. Such variables
 behave like lexically scoped variable (see my()) but with unlimited scope.
 It is however possible to exit the scope by using \kbd{uninline()}.
 When used in a GP script, it is recommended to call \kbd{uninline()} before
 the script's end to avoid inline variables leaking outside the script.

Function: input
Class: basic
Section: programming/specific
C-Name: gp_input
Prototype: 
Help: input(): read an expression from the input file or standard input.
Doc: reads a string, interpreted as a GP expression,
 from the input file, usually standard input (i.e.~the keyboard). If a
 sequence of expressions is given, the result is the result of the last
 expression of the sequence. When using this instruction, it is useful to
 prompt for the string by using the \kbd{print1} function. Note that in the
 present version 2.19 of \kbd{pari.el}, when using \kbd{gp} under GNU Emacs (see
 \secref{se:emacs}) one \emph{must} prompt for the string, with a string
 which ends with the same prompt as any of the previous ones (a \kbd{"? "}
 will do for instance).

Function: install
Class: basic
Section: programming/specific
C-Name: gpinstall
Prototype: vrrD"",r,D"",s,
Help: install(name,code,{gpname},{lib}): load from dynamic library 'lib' the
 function 'name'. Assign to it the name 'gpname' in this GP session, with
 prototype 'code'. If 'lib' is omitted, all symbols known to gp
 (includes the whole 'libpari.so' and possibly others) are available.
 If 'gpname' is omitted, use 'name'.
Doc: loads from dynamic library \var{lib} the function \var{name}. Assigns to it
 the name \var{gpname} in this \kbd{gp} session, with \emph{prototype}
 \var{code} (see below). If \var{gpname} is omitted, uses \var{name}.
 If \var{lib} is omitted, all symbols known to \kbd{gp} are available: this
 includes the whole of \kbd{libpari.so} and possibly others (such as
 \kbd{libc.so}).
 
 Most importantly, \kbd{install} gives you access to all non-static functions
 defined in the PARI library. For instance, the function
 \bprog
   GEN addii(GEN x, GEN y)
 @eprog\noindent adds two PARI integers, and is not directly accessible under
 \kbd{gp} (it is eventually called by the \kbd{+} operator of course):
 \bprog
 ? install("addii", "GG")
 ? addii(1, 2)
 %1 = 3
 @eprog\noindent
 It also allows to add external functions to the \kbd{gp} interpreter.
 For instance, it makes the function \tet{system} obsolete:
 \bprog
 ? install(system, vs, sys,/*omitted*/)
 ? sys("ls gp*")
 gp.c            gp.h            gp_rl.c
 @eprog\noindent This works because \kbd{system} is part of \kbd{libc.so},
 which is linked to \kbd{gp}. It is also possible to compile a shared library
 yourself and provide it to gp in this way: use \kbd{gp2c}, or do it manually
 (see the \kbd{modules\_build} variable in \kbd{pari.cfg} for hints).
 
 Re-installing a function will print a warning and update the prototype code
 if needed. However, it will not reload a symbol from the library, even if the
 latter has been recompiled.
 
 \misctitle{Prototype} We only give a simplified description here, covering
 most functions, but there are many more possibilities. The full documentation
 is available in \kbd{libpari.dvi}, see
 \bprog
   ??prototype
 @eprog
 
 \item First character \kbd{i}, \kbd{l}, \kbd{v} : return type int / long /
 void. (Default: \kbd{GEN})
 
 \item One letter for each mandatory argument, in the same order as they appear
 in the argument list: \kbd{G} (\kbd{GEN}), \kbd{\&}
 (\kbd{GEN*}), \kbd{L} (\kbd{long}), \kbd{s} (\kbd{char *}), \kbd{n}
 (variable).
 
  \item \kbd{p} to supply \kbd{realprecision} (usually \kbd{long prec} in the
  argument list), \kbd{P} to supply \kbd{seriesprecision}
  (usually \kbd{long precdl}).
 
  \noindent We also have special constructs for optional arguments and default
  values:
 
  \item \kbd{DG} (optional \kbd{GEN}, \kbd{NULL} if omitted),
 
  \item \kbd{D\&} (optional \kbd{GEN*}, \kbd{NULL} if omitted),
 
  \item \kbd{Dn} (optional variable, $-1$ if omitted),
 
 For instance the prototype corresponding to
 \bprog
   long issquareall(GEN x, GEN *n = NULL)
 @eprog\noindent is \kbd{lGD\&}.
 
 \misctitle{Caution} This function may not work on all systems, especially
 when \kbd{gp} has been compiled statically. In that case, the first use of an
 installed function will provoke a Segmentation Fault (this should never
 happen with a dynamically linked executable). If you intend to use this
 function, please check first on some harmless example such as the one above
 that it works properly on your machine.

Function: intcirc
Class: basic
Section: sums
C-Name: intcirc0
Prototype: V=GGEDGp
Help: intcirc(X=a,R,expr,{tab}): numerical integration of expr on the circle
 |z-a|=R, divided by 2*I*Pi. tab is as in intnum.
Wrapper: (,,G)
Description: 
  (gen,gen,gen,?gen):gen:prec intcirc(${3 cookie}, ${3 wrapper}, $1, $2, $4, $prec)
Doc: numerical
 integration of $(2i\pi)^{-1}\var{expr}$ with respect to $X$ on the circle
 $|X-a| = R$.
 In other words, when \var{expr} is a meromorphic
 function, sum of the residues in the corresponding disk; \var{tab} is as in
 \kbd{intnum}, except that if computed with \kbd{intnuminit} it should be with
 the endpoints \kbd{[-1, 1]}.
 
 \bprog
 ? \p105
 ? intcirc(s=1, 0.5, zeta(s)) - 1
 time = 496 ms.
 %1 = 1.2883911040127271720 E-101 + 0.E-118*I
 @eprog
 
 \synt{intcirc}{void *E, GEN (*eval)(void*,GEN), GEN a,GEN R,GEN tab, long prec}.

Function: intformal
Class: basic
Section: polynomials
C-Name: integ
Prototype: GDn
Help: intformal(x,{v}): formal integration of x with respect to v, or to the
 main variable of x if v is omitted.
Doc: \idx{formal integration} of $x$ with respect to the variable $v$ (wrt.
 the main variable if $v$ is omitted). Since PARI cannot represent
 logarithmic or arctangent terms, any such term in the result will yield an
 error:
 \bprog
  ? intformal(x^2)
  %1 = 1/3*x^3
  ? intformal(x^2, y)
  %2 = y*x^2
  ? intformal(1/x)
    ***   at top-level: intformal(1/x)
    ***                 ^--------------
    *** intformal: domain error in intformal: residue(series, pole) != 0
 @eprog
 The argument $x$ can be of any type. When $x$ is a rational function, we
 assume that the base ring is an integral domain of characteristic zero.
 
   By  definition,   the main variable of a \typ{POLMOD} is the main variable
 among the  coefficients  from  its  two  polynomial  components
 (representative and modulus); in other words, assuming a polmod represents an
 element of $R[X]/(T(X))$, the variable $X$ is a mute variable and the
 integral is taken with respect to the main variable used in the base ring $R$.
 In particular, it is meaningless to integrate with respect to the main
 variable of \kbd{x.mod}:
 \bprog
 ? intformal(Mod(1,x^2+1), 'x)
 *** intformal: incorrect priority in intformal: variable x = x
 @eprog

Function: intfuncinit
Class: basic
Section: sums
C-Name: intfuncinit0
Prototype: V=GGED0,L,p
Help: intfuncinit(t=a,b,f,{m=0}): initialize tables for integrations
 from a to b using a weight f(t). For integral transforms such
 as Fourier or Mellin transforms.
Wrapper: (,,G)
Description: 
  (gen,gen,gen,?small):gen:prec intfuncinit(${3 cookie}, ${3 wrapper}, $1, $2, $4, $prec)
Doc: initialize tables for use with integral transforms such Fourier,
 Laplace or Mellin transforms, in order to compute
 $$ \int_a^b f(t) k(t,z) \, dt $$
 for some kernel $k(t,z)$.
 The endpoints $a$ and $b$ are coded as in \kbd{intnum}, $f$ is the
 function to which the integral transform is to be applied and the
 non-negative integer $m$ is as in \kbd{intnum}: multiply the number of
 sampling points roughly by $2^m$, hopefully increasing the accuracy. This
 function is particularly useful when the function $f$ is hard to compute,
 such as a gamma product.
 
 \misctitle{Limitation} the endpoints $a$ and $b$ must be at infinity,
 with the same asymptotic behaviour. Oscillating types are not supported.
 This is easily overcome by integrating vectors of functions, see example
 below.
 
 \misctitle{Examples}
 
 \item numerical Fourier transform
 $$F(z) = \int_{-\infty}^{+\infty} f(t)e^{-2i\pi z t}\, dt. $$
 First the easy case, assume that $f$ decrease exponentially:
 \bprog
    f(t) = exp(-t^2);
    A = [-oo,1];
    B = [+oo,1];
    \p200
    T = intfuncinit(t = A,B , f(t));
    F(z) =
    { my(a = -2*I*Pi*z);
      intnum(t = A,B, exp(a*t), T);
    }
    ? F(1) - sqrt(Pi)*exp(-Pi^2)
    %1 = -1.3... E-212
 @eprog\noindent
 Now the harder case, $f$ decrease slowly: we must specify the oscillating
 behaviour. Thus, we cannot precompute usefully since everything depends on
 the point we evaluate at:
 \bprog
    f(t) = 1 / (1+ abs(t));
    \p200
    \\ Fourier cosine transform
    FC(z) =
    { my(a = 2*Pi*z);
      intnum(t = [-oo, a*I], [+oo, a*I], cos(a*t)*f(t));
    }
    FC(1)
 @eprog
 \item Fourier coefficients: we must integrate over a period, but
 \kbd{intfuncinit} does not support finite endpoints.
 The solution is to integrate a vector of functions !
 \bprog
 FourierSin(f, T, k) =  \\ first k sine Fourier coeffs
 {
   my (w = 2*Pi/T);
   my (v = vector(k+1));
   intnum(t = -T/2, T/2,
      my (z = exp(I*w*t));
      v[1] = z;
      for (j = 2, k, v[j] = v[j-1]*z);
      f(t) * imag(v)) * 2/T;
 }
 FourierSin(t->sin(2*t), 2*Pi, 10)
 @eprog\noindent The same technique can be used instead of \kbd{intfuncinit}
 to integrate $f(t) k(t,z)$ whenever the list of $z$-values is known
 beforehand.
 
 Note that the above code includes an unrelated optimization: the
 $\sin(j w t)$ are computed as imaginary parts of $\exp(i j w t)$ and the
 latter by successive multiplications.
 
 \item numerical Mellin inversion
 $$F(z) = (2i\pi)^{-1} \int_{c -i\infty}^{c+i\infty} f(s)z^{-s}\, ds
  = (2\pi)^{-1} \int_{-\infty}^{+\infty}
     f(c + i t)e^{-\log z(c + it)}\, dt. $$
 We take $c = 2$ in the program below:
 \bprog
    f(s) = gamma(s)^3;  \\ f(c+it) decrease as exp(-3Pi|t|/2)
    c = 2; \\ arbitrary
    A = [-oo,3*Pi/2];
    B = [+oo,3*Pi/2];
    T = intfuncinit(t=A,B, f(c + I*t));
    F(z) =
    { my (a = -log(z));
      intnum(t=A,B, exp(a*I*t), T)*exp(a*c) / (2*Pi);
    }
 @eprog
 
 \synt{intfuncinit}{void *E, GEN (*eval)(void*,GEN), GEN a,GEN b,long m, long prec}.

Function: intnum
Class: basic
Section: sums
C-Name: intnum0
Prototype: V=GGEDGp
Help: intnum(X=a,b,expr,{tab}): numerical integration of expr from a to b with
 respect to X. Plus/minus infinity is coded as +oo/-oo. Finally tab is
 either omitted (let the program choose the integration step), a non-negative
 integer m (divide integration step by 2^m), or data precomputed with
 intnuminit.
Wrapper: (,,G)
Description: 
  (gen,gen,gen,?gen):gen:prec intnum(${3 cookie}, ${3 wrapper}, $1, $2, $4, $prec)
Doc: numerical integration
 of \var{expr} on $]a,b[$ with respect to $X$, using the
 double-exponential method, and thus $O(D\log D)$ evaluation of
 the integrand in precision $D$. The integrand may have values
 belonging to a vector space over the real numbers; in particular, it can be
 complex-valued or vector-valued. But it is assumed that the function is
 regular on $]a,b[$. If the endpoints $a$ and $b$ are finite and the
 function is regular there, the situation is simple:
 \bprog
 ? intnum(x = 0,1, x^2)
 %1 = 0.3333333333333333333333333333
 ? intnum(x = 0,Pi/2, [cos(x), sin(x)])
 %2 = [1.000000000000000000000000000, 1.000000000000000000000000000]
 @eprog\noindent
 An endpoint equal to $\pm\infty$ is coded as \kbd{+oo} or \kbd{-oo}, as
 expected:
 \bprog
 ? intnum(x = 1,+oo, 1/x^2)
 %3 = 1.000000000000000000000000000
 @eprog\noindent
 In basic usage, it is assumed that the function does not decrease
 exponentially fast at infinity:
 \bprog
 ? intnum(x=0,+oo, exp(-x))
   ***   at top-level: intnum(x=0,+oo,exp(-
   ***                 ^--------------------
   *** exp: overflow in expo().
 @eprog\noindent
 We shall see in a moment how to avoid that last problem, after describing
 the last \emph{optional} argument \var{tab}.
 
 \misctitle{The \var{tab} argument}
 The routine uses weights $w_i$, which are mostly independent of the function
 being integrated, evaluated at many sampling points $x_i$ and
 approximates the integral by $\sum w_i f(x_i)$. If \var{tab} is
 
 \item a non-negative integer $m$, we multiply the number of sampling points
 by $2^m$, hopefully increasing accuracy. Note that the running time
 increases roughly by a factor $2^m$. One may try consecutive values of $m$
 until they give the same value up to an accepted error.
 
 \item a set of integration tables containing precomputed $x_i$ and $w_i$
 as output by \tet{intnuminit}. This is useful if several integrations of
 the same type are performed (on the same kind of interval and functions,
 for a given accuracy): we skip a precomputation of $O(D\log D)$
 elementary functions in accuracy $D$, whose running time has the same order
 of magnitude as the evaluation of the integrand. This is in particular
 useful for multivariate integrals.
 
 \misctitle{Specifying the behavior at endpoints}
 This is done as follows. An endpoint $a$ is either given as such (a scalar,
 real or complex, \kbd{oo} or \kbd{-oo} for $\pm\infty$), or as a two
 component vector $[a,\alpha]$, to indicate the behavior of the integrand in a
 neighborhood of $a$.
 
 If $a$ is finite, the code $[a,\alpha]$ means the function has a
 singularity of the form $(x-a)^{\alpha}$, up to logarithms. (If $\alpha \ge
 0$, we only assume the function is regular, which is the default assumption.)
 If a wrong singularity exponent is used, the result will lose a catastrophic
 number of decimals:
 \bprog
 ? intnum(x=0, 1, x^(-1/2))         \\@com assume $x^{-1/2}$ is regular at 0
 %1 = 1.9999999999999999999999999999827931660
 ? intnum(x=[0,-1/2], 1, x^(-1/2))  \\@com no, it's not
 %2 = 2.0000000000000000000000000000000000000
 ? intnum(x=[0,-1/10], 1, x^(-1/2)) \\@com using a wrong exponent is bad
 %3 = 1.9999999999999999999999999999999901912
 @eprog
 
 If $a$ is $\pm\infty$, which is coded as \kbd{+oo} or \kbd{-oo},
 the situation is more complicated, and $[\pm\kbd{oo},\alpha]$ means:
 
 \item $\alpha=0$ (or no $\alpha$ at all, i.e. simply $\pm\kbd{oo}$)
 assumes that the integrand tends to zero moderately quickly, at least as
 $O(x^{-2})$ but not exponentially fast.
 
 \item $\alpha>0$ assumes that the function tends to zero exponentially fast
 approximately as $\exp(-\alpha x)$. This includes oscillating but quickly
 decreasing functions such as $\exp(-x)\sin(x)$.
 \bprog
 ? intnum(x=0, +oo, exp(-2*x))
   ***   at top-level: intnum(x=0,+oo,exp(-
   ***                 ^--------------------
   *** exp: exponent (expo) overflow
 ? intnum(x=0, [+oo, 2], exp(-2*x))  \\@com OK!
 %1 = 0.50000000000000000000000000000000000000
 ? intnum(x=0, [+oo, 3], exp(-2*x))  \\@com imprecise exponent, still OK !
 %2 = 0.50000000000000000000000000000000000000
 ? intnum(x=0, [+oo, 10], exp(-2*x)) \\@com wrong exponent $\Rightarrow$ disaster
 %3 = 0.49999999999952372962457451698256707393
 @eprog\noindent As the last exemple shows, the exponential decrease rate
 \emph{must} be indicated to avoid overflow, but the method is robust enough
 for a rough guess to be acceptable.
 
 \item $\alpha<-1$ assumes that the function tends to $0$ slowly, like
 $x^{\alpha}$. Here the algorithm is less robust and it is essential to give a
 sharp $\alpha$, unless $\alpha \le -2$ in which case we use
 the default algorithm as if $\alpha$ were missing (or equal to $0$).
 \bprog
 ? intnum(x=1, +oo, x^(-3/2))         \\ default
 %1 = 1.9999999999999999999999999999646391207
 ? intnum(x=1, [+oo,-3/2], x^(-3/2))  \\ precise decrease rate
 %2 = 2.0000000000000000000000000000000000000
 ? intnum(x=1, [+oo,-11/10], x^(-3/2)) \\ worse than default
 %3 = 2.0000000000000000000000000089298011973
 @eprog
 
 \smallskip The last two codes are reserved for oscillating functions.
 Let $k > 0$ real, and $g(x)$ a non-oscillating function tending slowly to $0$
 (e.g. like a negative power of $x$), then
 
 \item $\alpha=k * I$ assumes that the function behaves like $\cos(kx)g(x)$.
 
 \item $\alpha=-k* I$ assumes that the function behaves like $\sin(kx)g(x)$.
 
 \noindent Here it is critical to give the exact value of $k$. If the
 oscillating part is not a pure sine or cosine, one must expand it into a
 Fourier series, use the above codings, and sum the resulting contributions.
 Otherwise you will get nonsense. Note that $\cos(kx)$, and similarly
 $\sin(kx)$, means that very function, and not a translated version such as
 $\cos(kx+a)$.
 
 \misctitle{Note} If $f(x)=\cos(kx)g(x)$ where $g(x)$ tends to zero
 exponentially fast as $\exp(-\alpha x)$, it is up to the user to choose
 between $[\pm\kbd{oo},\alpha]$ and $[\pm\kbd{oo},k* I]$, but a good rule of
 thumb is that
 if the oscillations are weaker than the exponential decrease, choose
 $[\pm\kbd{oo},\alpha]$, otherwise choose $[\pm\kbd{oo},k*I]$, although the
 latter can
 reasonably be used in all cases, while the former cannot. To take a specific
 example, in the inverse Mellin transform, the integrand is almost always a
 product of an exponentially decreasing and an oscillating factor. If we
 choose the oscillating type of integral we perhaps obtain the best results,
 at the expense of having to recompute our functions for a different value of
 the variable $z$ giving the transform, preventing us to use a function such
 as \kbd{intfuncinit}. On the other hand using the exponential type of
 integral, we obtain less accurate results, but we skip expensive
 recomputations. See \kbd{intfuncinit} for more explanations.
 
 \smallskip
 
 We shall now see many examples to get a feeling for what the various
 parameters achieve. All examples below assume precision is set to $115$
 decimal digits. We first type
 \bprog
 ? \p 115
 @eprog
 
 \misctitle{Apparent singularities} In many cases, apparent singularities
 can be ignored. For instance, if $f(x) = 1
 /(\exp(x)-1) - \exp(-x)/x$, then $\int_0^\infty f(x)\,dx=\gamma$, Euler's
 constant \kbd{Euler}. But
 
 \bprog
 ? f(x) = 1/(exp(x)-1) - exp(-x)/x
 ? intnum(x = 0, [oo,1],  f(x)) - Euler
 %1 = 0.E-115
 @eprog\noindent
 But close to $0$ the function $f$ is computed with an enormous loss of
 accuracy, and we are in fact lucky that it get multiplied by weights which are
 sufficiently close to $0$ to hide this:
 \bprog
 ? f(1e-200)
 %2 = -3.885337784451458142 E84
 @eprog
 
 A more robust solution is to define the function differently near special
 points, e.g. by a Taylor expansion
 \bprog
 ? F = truncate( f(t + O(t^10)) ); \\@com expansion around t = 0
 ? poldegree(F)
 %4 = 7
 ? g(x) = if (x > 1e-18, f(x), subst(F,t,x)); \\@com note that $7 \cdot 18 > 105$
 ? intnum(x = 0, [oo,1],  g(x)) - Euler
 %2 = 0.E-115
 @eprog\noindent It is up to the user to determine constants such as the
 $10^{-18}$ and $10$ used above.
 
 \misctitle{True singularities} With true singularities the result is worse.
 For instance
 
 \bprog
 ? intnum(x = 0, 1,  x^(-1/2)) - 2
 %1 = -3.5... E-68 \\@com only $68$ correct decimals
 
 ? intnum(x = [0,-1/2], 1,  x^(-1/2)) - 2
 %2 = 0.E-114 \\@com better
 @eprog
 
 \misctitle{Oscillating functions}
 
 \bprog
 ? intnum(x = 0, oo, sin(x) / x) - Pi/2
 %1 = 16.19.. \\@com nonsense
 ? intnum(x = 0, [oo,1], sin(x)/x) - Pi/2
 %2 = -0.006.. \\@com bad
 ? intnum(x = 0, [oo,-I], sin(x)/x) - Pi/2
 %3 = 0.E-115 \\@com perfect
 ? intnum(x = 0, [oo,-I], sin(2*x)/x) - Pi/2  \\@com oops, wrong $k$
 %4 = 0.06...
 ? intnum(x = 0, [oo,-2*I], sin(2*x)/x) - Pi/2
 %5 = 0.E-115 \\@com perfect
 
 ? intnum(x = 0, [oo,-I], sin(x)^3/x) - Pi/4
 %6 = -0.0008... \\@com bad
 ? sin(x)^3 - (3*sin(x)-sin(3*x))/4
 %7 = O(x^17)
 @eprog\noindent
 We may use the above linearization and compute two oscillating integrals with
 endpoints \kbd{[oo, -I]} and \kbd{[oo, -3*I]} respectively, or
 notice the obvious change of variable, and reduce to the single integral
 ${1\over 2}\int_0^\infty \sin(x)/x\,dx$. We finish with some more complicated
 examples:
 
 \bprog
 ? intnum(x = 0, [oo,-I], (1-cos(x))/x^2) - Pi/2
 %1 = -0.0003... \\@com bad
 ? intnum(x = 0, 1, (1-cos(x))/x^2) \
 + intnum(x = 1, oo, 1/x^2) - intnum(x = 1, [oo,I], cos(x)/x^2) - Pi/2
 %2 = 0.E-115 \\@com perfect
 
 ? intnum(x = 0, [oo, 1], sin(x)^3*exp(-x)) - 0.3
 %3 = -7.34... E-55 \\@com bad
 ? intnum(x = 0, [oo,-I], sin(x)^3*exp(-x)) - 0.3
 %4 = 8.9... E-103 \\@com better. Try higher $m$
 ? tab = intnuminit(0,[oo,-I], 1); \\@com double number of sampling points
 ? intnum(x = 0, oo, sin(x)^3*exp(-x), tab) - 0.3
 %6 = 0.E-115 \\@com perfect
 @eprog
 
 \misctitle{Warning} Like \tet{sumalt}, \kbd{intnum} often assigns a
 reasonable value to diverging integrals. Use these values at your own risk!
 For example:
 
 \bprog
 ? intnum(x = 0, [oo, -I], x^2*sin(x))
 %1 = -2.0000000000...
 @eprog\noindent
 Note the formula
 $$ \int_0^\infty \sin(x)/x^s\,dx = \cos(\pi s/2) \Gamma(1-s)\;, $$
 a priori valid only for $0 < \Re(s) < 2$, but the right hand side provides an
 analytic continuation which may be evaluated at $s = -2$\dots
 
 \misctitle{Multivariate integration}
 Using successive univariate integration with respect to different formal
 parameters, it is immediate to do naive multivariate integration. But it is
 important to use a suitable \kbd{intnuminit} to precompute data for the
 \emph{internal} integrations at least!
 
 For example, to compute the double integral on the unit disc $x^2+y^2\le1$
 of the function $x^2+y^2$, we can write
 \bprog
 ? tab = intnuminit(-1,1);
 ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab),tab) - Pi/2
 %2 = -7.1... E-115 \\@com OK
 
 @eprog\noindent
 The first \var{tab} is essential, the second optional. Compare:
 
 \bprog
 ? tab = intnuminit(-1,1);
 time = 4 ms.
 ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2));
 time = 3,092 ms. \\@com slow
 ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab), tab);
 time = 252 ms.  \\@com faster
 ? intnum(x=-1,1, intnum(y=-sqrt(1-x^2),sqrt(1-x^2), x^2+y^2, tab));
 time = 261 ms.  \\@com the \emph{internal} integral matters most
 @eprog
 
 \synt{intnum}{void *E, GEN (*eval)(void*,GEN), GEN a,GEN b,GEN tab, long prec},
 where an omitted \var{tab} is coded as \kbd{NULL}.

Function: intnumgauss
Class: basic
Section: sums
C-Name: intnumgauss0
Prototype: V=GGEDGp
Help: intnumgauss(X=a,b,expr,{tab}): numerical integration of expr from
 a to b, a compact interval, with respect to X using Gauss-Legendre
 quadrature. tab is either omitted (and will be recomputed) or
 precomputed with intnumgaussinit.
Wrapper: (,,G)
Description: 
  (gen,gen,gen,?gen):gen:prec intnumgauss(${3 cookie}, ${3 wrapper}, $1, $2, $4, $prec)
Doc: numerical integration of \var{expr} on the compact interval $[a,b]$ with
 respect to $X$ using Gauss-Legendre quadrature; \kbd{tab} is either omitted
 or precomputed with \kbd{intnumgaussinit}. As a convenience, it can be an
 integer $n$ in which case we call
 \kbd{intnumgaussinit}$(n)$ and use $n$-point quadrature.
 \bprog
 ? test(n, b = 1) = T=intnumgaussinit(n);\
     intnumgauss(x=-b,b, 1/(1+x^2),T) - 2*atan(b);
 ? test(0) \\ default
 %1 = -9.490148553624725335 E-22
 ? test(40)
 %2 = -6.186629001816965717 E-31
 ? test(50)
 %3 = -1.1754943508222875080 E-38
 ? test(50, 2) \\ double interval length
 %4 = -4.891779568527713636 E-21
 ? test(90, 2) \\ n must almost be doubled as well!
 %5 = -9.403954806578300064 E-38
 @eprog\noindent On the other hand, we recommend to split the integral
 and change variables rather than increasing $n$ too much:
 \bprog
 ? f(x) = 1/(1+x^2);
 ? b = 100;
 ? intnumgauss(x=0,1, f(x)) + intnumgauss(x=1,1/b, f(1/x)*(-1/x^2)) - atan(b)
 %3 = -1.0579449157400587572 E-37
 @eprog

Function: intnumgaussinit
Class: basic
Section: sums
C-Name: intnumgaussinit
Prototype: D0,L,p
Help: intnumgaussinit({n}): initialize tables for n-point Gauss-Legendre
 integration on a compact interval.
Doc: initialize tables for $n$-point Gauss-Legendre integration of
 a smooth function $f$ lon a compact
 interval $[a,b]$ at current \kbd{realprecision}. If $n$ is omitted, make a
 default choice $n \approx \kbd{realprecision}$, suitable for analytic
 functions on $[-1,1]$. The error is bounded by
 $$
    \dfrac{(b-a)^{2n+1} (n!)^4}{(2n+1)[(2n)!]^3} f^{(2n)} (\xi) ,
    \qquad a < \xi < b
 $$
 so, if the interval length increases, $n$ should be increased as well.
 \bprog
 ? T = intnumgaussinit();
 ? intnumgauss(t=-1,1,exp(t), T) - exp(1)+exp(-1)
 %1 = -5.877471754111437540 E-39
 ? intnumgauss(t=-10,10,exp(t), T) - exp(10)+exp(-10)
 %2 = -8.358367809712546836 E-35
 ? intnumgauss(t=-1,1,1/(1+t^2), T) - Pi/2
 %3 = -9.490148553624725335 E-22
 
 ? T = intnumgaussinit(50);
 ? intnumgauss(t=-1,1,1/(1+t^2), T) - Pi/2
 %5 = -1.1754943508222875080 E-38
 ? intnumgauss(t=-5,5,1/(1+t^2), T) - 2*atan(5)
 %6 = -1.2[...]E-8
 @eprog
 On the other hand, we recommend to split the integral and change variables
 rather than increasing $n$ too much, see \tet{intnumgauss}.

Function: intnuminit
Class: basic
Section: sums
C-Name: intnuminit
Prototype: GGD0,L,p
Help: intnuminit(a,b,{m=0}): initialize tables for integrations from a to b.
 See help for intnum for coding of a and b. Possible types: compact interval,
 semi-compact (one extremity at + or - infinity) or R, and very slowly, slowly
 or exponentially decreasing, or sine or cosine oscillating at infinities.
Doc: initialize tables for integration from
 $a$ to $b$, where $a$ and $b$ are coded as in \kbd{intnum}. Only the
 compactness, the possible existence of singularities, the speed of decrease
 or the oscillations at infinity are taken into account, and not the values.
 For instance {\tt intnuminit(-1,1)} is equivalent to {\tt intnuminit(0,Pi)},
 and {\tt intnuminit([0,-1/2],oo)} is equivalent to
 {\tt intnuminit([-1,-1/2], -oo)}; on the other hand, the order matters
 and
 {\tt intnuminit([0,-1/2], [1,-1/3])} is \emph{not} equivalent to
 {\tt intnuminit([0,-1/3], [1,-1/2])} !
 
 If $m$ is present, it must be non-negative and we multiply the default
 number of sampling points by $2^m$ (increasing the running time by a
 similar factor).
 
 The result is technical and liable to change in the future, but we document
 it here for completeness. Let $x=\phi(t)$, $t\in ]-\infty,\infty[$ be an
 internally chosen change of variable, achieving double exponential decrease of
 the integrand at infinity. The integrator \kbd{intnum} will compute
 $$ h \sum_{|n| < N} \phi'(nh) F(\phi(nh)) $$
 for some integration step $h$ and truncation parameter $N$.
 In basic use, let
 \bprog
 [h, x0, w0, xp, wp, xm, wm] = intnuminit(a,b);
 @eprog
 
 \item $h$ is the integration step
 
 \item $x_0 = \phi(0)$  and $w_0 = \phi'(0)$,
 
 \item \var{xp} contains the $\phi(nh)$, $0 < n < N$,
 
 \item \var{xm} contains the $\phi(nh)$, $0 < -n < N$, or is empty.
 
 \item \var{wp} contains the $\phi'(nh)$, $0 < n < N$,
 
 \item \var{wm} contains the $\phi'(nh)$, $0 < -n < N$, or is empty.
 
 The arrays \var{xm} and \var{wm} are left empty when $\phi$ is an odd
 function. In complicated situations when non-default behaviour is specified at
 end points, \kbd{intnuminit} may return up to $3$ such arrays, corresponding
 to a splitting of up to $3$ integrals of basic type.
 
 If the functions to be integrated later are of the form $F = f(t) k(t,z)$
 for some kernel $k$ (e.g. Fourier, Laplace, Mellin, \dots), it is
 useful to also precompute the values of $f(\phi(nh))$, which is accomplished
 by \tet{intfuncinit}. The hard part is to determine the behaviour
 of $F$ at endpoints, depending on $z$.

Function: intnumromb
Class: basic
Section: sums
C-Name: intnumromb0_bitprec
Prototype: V=GGED0,L,b
Help: intnumromb(X=a,b,expr,{flag=0}): numerical integration of expr (smooth in
 ]a,b[) from a to b with respect to X. flag is optional and mean 0: default.
 expr can be evaluated exactly on [a,b]; 1: general function; 2: a or b can be
 plus or minus infinity (chosen suitably), but of same sign; 3: expr has only
 limits at a or b.
Wrapper: (,,G)
Description: 
  (gen,gen,gen,?small):gen:prec intnumromb(${3 cookie}, ${3 wrapper}, $1, $2, $4, $bitprec)
Doc: numerical integration of \var{expr} (smooth in $]a,b[$), with respect to
 $X$. Suitable for low accuracy; if \var{expr} is very regular (e.g. analytic
 in a large region) and high accuracy is desired, try \tet{intnum} first.
 
 Set $\fl=0$ (or omit it altogether) when $a$ and $b$ are not too large, the
 function is smooth, and can be evaluated exactly everywhere on the interval
 $[a,b]$.
 
 If $\fl=1$, uses a general driver routine for doing numerical integration,
 making no particular assumption (slow).
 
 $\fl=2$ is tailored for being used when $a$ or $b$ are infinite using the
 change of variable $t = 1/X$. One \emph{must} have $ab>0$, and in fact if
 for example $b=+\infty$, then it is preferable to have $a$ as large as
 possible, at least $a\ge1$.
 
 If $\fl=3$, the function is allowed to be undefined
 at $a$ (but right continuous) or $b$ (left continuous),
 for example the function $\sin(x)/x$ between $x=0$ and $1$.
 
 The user should not require too much accuracy: \tet{realprecision} about
 30 decimal digits (\tet{realbitprecision} about 100 bits) is OK,
 but not much more. In addition, analytical cleanup of the integral must have
 been done: there must be no singularities in the interval or at the
 boundaries. In practice this can be accomplished with a change of
 variable. Furthermore, for improper integrals, where one or both of the
 limits of integration are plus or minus infinity, the function must decrease
 sufficiently rapidly at infinity, which can often be accomplished through
 integration by parts. Finally, the function to be integrated should not be
 very small (compared to the current precision) on the entire interval. This
 can of course be accomplished by just multiplying by an appropriate constant.
 
 Note that \idx{infinity} can be represented with essentially no loss of
 accuracy by an appropriate huge number. However beware of real underflow
 when dealing with rapidly decreasing functions. For example, in order to
 compute the $\int_0^\infty e^{-x^2}\,dx$ to 28 decimal digits, then one can
 set infinity equal to 10 for example, and certainly not to \kbd{1e1000}.
 
 \synt{intnumromb_bitprec}{void *E, GEN (*eval)(void*,GEN), GEN a, GEN b, long flag, long bitprec},
 where $\kbd{eval}(x, E)$ returns the value of the function at $x$.
 You may store any additional information required by \kbd{eval} in $E$, or set
 it to \kbd{NULL}. The historical variant
 \synt{intnumromb}{\dots, long prec}, where \kbd{prec} is expressed in words,
 not bits, is obsolete and should no longer be used.

Function: isfundamental
Class: basic
Section: number_theoretical
C-Name: isfundamental
Prototype: lG
Help: isfundamental(x): true(1) if x is a fundamental discriminant
 (including 1), false(0) if not.
Description: 
 (int):bool       Z_isfundamental($1)
 (gen):bool       isfundamental($1)
Doc: true (1) if $x$ is equal to 1 or to the discriminant of a quadratic
 field, false (0) otherwise.

Function: ispolygonal
Class: basic
Section: number_theoretical
C-Name: ispolygonal
Prototype: lGGD&
Help: ispolygonal(x,s,{&N}): true(1) if x is an s-gonal number, false(0) if
 not (s > 2). If N is given set it to n if x is the n-th s-gonal number.
Doc: true (1) if the integer $x$ is an s-gonal number, false (0) if not.
 The parameter $s > 2$ must be a \typ{INT}. If $N$ is given, set it to $n$
 if $x$ is the $n$-th $s$-gonal number.
 \bprog
 ? ispolygonal(36, 3, &N)
 %1 = 1
 ? N
 @eprog

Function: ispower
Class: basic
Section: number_theoretical
C-Name: ispower
Prototype: lGDGD&
Help: ispower(x,{k},{&n}): if k > 0 is given, return true (1) if x is a k-th
 power, false (0) if not. If k is omitted, return the maximal k >= 2 such
 that x = n^k is a perfect power, or 0 if no such k exist.
 If n is present, and the function returns a non-zero result, set n to the
 k-th root of x.
Description: 
 (int):small       Z_isanypower($1, NULL)
 (int, &int):small Z_isanypower($1, &$2)
Doc: if $k$ is given, returns true (1) if $x$ is a $k$-th power, false
 (0) if not. What it means to be a $k$-th power depends on the type of
 $x$; see \tet{issquare} for details.
 
 If $k$ is omitted, only integers and fractions are allowed for $x$ and the
 function returns the maximal $k \geq 2$ such that $x = n^k$ is a perfect
 power, or 0 if no such $k$ exist; in particular \kbd{ispower(-1)},
 \kbd{ispower(0)}, and \kbd{ispower(1)} all return $0$.
 
 If a third argument $\&n$ is given and $x$ is indeed a $k$-th power, sets
 $n$ to a $k$-th root of $x$.
 
 \noindent For a \typ{FFELT} \kbd{x}, instead of omitting \kbd{k} (which is
 not allowed for this type), it may be natural to set
 \bprog
 k = (x.p ^ x.f - 1) / fforder(x)
 @eprog
Variant: Also available is
 \fun{long}{gisanypower}{GEN x, GEN *pty} ($k$ omitted).

Function: ispowerful
Class: basic
Section: number_theoretical
C-Name: ispowerful
Prototype: lG
Help: ispowerful(x): true(1) if x is a powerful integer (valuation at all
 primes dividing x is greater than 1), false(0) if not.
Doc: true (1) if $x$ is a powerful integer, false (0) if not;
 an integer is powerful if and only if its valuation at all primes dividing
 $x$ is greater than 1.
 \bprog
 ? ispowerful(50)
 %1 = 0
 ? ispowerful(100)
 %2 = 1
 ? ispowerful(5^3*(10^1000+1)^2)
 %3 = 1
 @eprog

Function: isprime
Class: basic
Section: number_theoretical
C-Name: gisprime
Prototype: GD0,L,
Help: isprime(x,{flag=0}): true(1) if x is a (proven) prime number, false(0)
 if not. If flag is 0 or omitted, use a combination of algorithms. If flag is
 1, the primality is certified by the Pocklington-Lehmer Test. If flag is 2,
 the primality is certified using the APRCL test.
Description: 
 (int, ?0):bool        isprime($1)
 (gen, ?small):gen     gisprime($1, $2)
Doc: true (1) if $x$ is a prime
 number, false (0) otherwise. A prime number is a positive integer having
 exactly two distinct divisors among the natural numbers, namely 1 and
 itself.
 
 This routine proves or disproves rigorously that a number is prime, which can
 be very slow when $x$ is indeed prime and has more than $1000$ digits, say.
 Use \tet{ispseudoprime} to quickly check for compositeness. See also
 \kbd{factor}. It accepts vector/matrices arguments, and is then applied
 componentwise.
 
 If $\fl=0$, use a combination of Baillie-PSW pseudo primality test (see
 \tet{ispseudoprime}), Selfridge ``$p-1$'' test if $x-1$ is smooth enough, and
 Adleman-Pomerance-Rumely-Cohen-Lenstra (APRCL) for general $x$.
 
 If $\fl=1$, use Selfridge-Pocklington-Lehmer ``$p-1$'' test and output a
 primality certificate as follows: return
 
 \item 0 if $x$ is composite,
 
 \item 1 if $x$ is small enough that passing Baillie-PSW test guarantees
 its primality (currently $x < 2^{64}$, as checked by Jan Feitsma),
 
 \item $2$ if $x$ is a large prime whose primality could only sensibly be
 proven (given the algorithms implemented in PARI) using the APRCL test.
 
 \item Otherwise ($x$ is large and $x-1$ is smooth) output a three column
 matrix as a primality certificate. The first column contains prime
 divisors $p$ of $x-1$ (such that $\prod p^{v_p(x-1)} > x^{1/3}$), the second
 the corresponding elements $a_p$ as in Proposition~8.3.1 in GTM~138 , and the
 third the output of isprime(p,1).
 
 The algorithm fails if one of the pseudo-prime factors is not prime, which is
 exceedingly unlikely and well worth a bug report. Note that if you monitor
 \kbd{isprime} at a high enough debug level, you may see warnings about
 untested integers being declared primes. This is normal: we ask for partial
 factorisations (sufficient to prove primality if the unfactored part is not
 too large), and \kbd{factor} warns us that the cofactor hasn't been tested.
 It may or may not be tested later, and may or may not be prime. This does
 not affect the validity of the whole \kbd{isprime} procedure.
 
 If $\fl=2$, use APRCL.

Function: isprimepower
Class: basic
Section: number_theoretical
C-Name: isprimepower
Prototype: lGD&
Help: isprimepower(x,{&n}): if x = p^k is a prime power (p prime, k > 0),
 return k, else return 0. If n is present, and the function returns a non-zero
 result, set n to p, the k-th root of x.
Doc: if $x = p^k$ is a prime power ($p$ prime, $k > 0$), return $k$, else
 return 0. If a second argument $\&n$ is given and $x$ is indeed
 the $k$-th power of a prime $p$, sets $n$ to $p$.

Function: ispseudoprime
Class: basic
Section: number_theoretical
C-Name: gispseudoprime
Prototype: GD0,L,
Help: ispseudoprime(x,{flag}): true(1) if x is a strong pseudoprime, false(0)
 if not. If flag is 0 or omitted, use BPSW test, otherwise use strong
 Rabin-Miller test for flag randomly chosen bases.
Description: 
 (int,?0):bool      BPSW_psp($1)
 (int,#small):bool  millerrabin($1,$2)
 (int,small):bool   ispseudoprime($1, $2)
 (gen,?small):gen   gispseudoprime($1, $2)
Doc: true (1) if $x$ is a strong pseudo
 prime (see below), false (0) otherwise. If this function returns false, $x$
 is not prime; if, on the other hand it returns true, it is only highly likely
 that $x$ is a prime number. Use \tet{isprime} (which is of course much
 slower) to prove that $x$ is indeed prime.
 The function accepts vector/matrices arguments, and is then applied
 componentwise.
 
 If $\fl = 0$, checks whether $x$ has no small prime divisors (up to $101$
 included) and is a Baillie-Pomerance-Selfridge-Wagstaff pseudo prime.
 Such a pseudo prime passes a Rabin-Miller test for base $2$,
 followed by a Lucas test for the sequence $(P,-1)$, $P$ smallest
 positive integer such that $P^2 - 4$ is not a square mod $x$).
 
 There are no known composite numbers passing the above test, although it is
 expected that infinitely many such numbers exist. In particular, all
 composites $\leq 2^{64}$ are correctly detected (checked using
 \url{http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html}).
 
 If $\fl > 0$, checks whether $x$ is a strong Miller-Rabin pseudo prime  for
 $\fl$ randomly chosen bases (with end-matching to catch square roots of $-1$).

Function: ispseudoprimepower
Class: basic
Section: number_theoretical
C-Name: ispseudoprimepower
Prototype: lGD&
Help: ispseudoprimepower(x,{&n}): if x = p^k is a pseudo-prime power (p
 pseudo-prime, k > 0),
 return k, else return 0. If n is present, and the function returns a non-zero
 result, set n to p, the k-th root of x.
Doc: if $x = p^k$ is a pseudo-prime power ($p$ pseudo-prime as per
 \tet{ispseudoprime}, $k > 0$), return $k$, else
 return 0. If a second argument $\&n$ is given and $x$ is indeed
 the $k$-th power of a prime $p$, sets $n$ to $p$.
 
 More precisely, $k$ is always the largest integer such that $x = n^k$ for
 some integer $n$ and, when $n \leq  2^{64}$ the function returns $k > 0$ if and
 only if $n$ is indeed prime. When $n > 2^{64}$ is larger than the threshold,
 the function may return $1$ even though $n$ is composite: it only passed
 an \kbd{ispseudoprime(n)} test.

Function: issquare
Class: basic
Section: number_theoretical
C-Name: issquareall
Prototype: lGD&
Help: issquare(x,{&n}): true(1) if x is a square, false(0) if not. If n is
 given puts the exact square root there if it was computed.
Description: 
 (int):bool        Z_issquare($1)
 (gen):bool        issquare($1)
 (int, &int):bool  Z_issquareall($1, &$2)
 (gen, &gen):bool  issquareall($1, &$2)
Doc: true (1) if $x$ is a square, false (0)
 if not. What ``being a square'' means depends on the type of $x$: all
 \typ{COMPLEX} are squares, as well as all non-negative \typ{REAL}; for
 exact types such as \typ{INT}, \typ{FRAC} and \typ{INTMOD}, squares are
 numbers of the form $s^2$ with $s$ in $\Z$, $\Q$ and $\Z/N\Z$ respectively.
 \bprog
 ? issquare(3)          \\ as an integer
 %1 = 0
 ? issquare(3.)         \\ as a real number
 %2 = 1
 ? issquare(Mod(7, 8))  \\ in Z/8Z
 %3 = 0
 ? issquare( 5 + O(13^4) )  \\ in Q_13
 %4 = 0
 @eprog
 If $n$ is given, a square root of $x$ is put into $n$.
 \bprog
 ? issquare(4, &n)
 %1 = 1
 ? n
 %2 = 2
 @eprog
 For polynomials, either we detect that the characteristic is 2 (and check
 directly odd and even-power monomials) or we assume that $2$ is invertible
 and check whether squaring the truncated power series for the square root
 yields the original input.
 
 For \typ{POLMOD} $x$, we only support \typ{POLMOD}s of \typ{INTMOD}s
 encoding finite fields, assuming without checking that the intmod modulus
 $p$ is prime and that the polmod modulus is irreducible modulo $p$.
 \bprog
 ? issquare(Mod(Mod(2,3), x^2+1), &n)
 %1 = 1
 ? n
 %2 = Mod(Mod(2, 3)*x, Mod(1, 3)*x^2 + Mod(1, 3))
 @eprog
Variant: Also available is \fun{long}{issquare}{GEN x}. Deprecated
 GP-specific functions \fun{GEN}{gissquare}{GEN x} and
 \fun{GEN}{gissquareall}{GEN x, GEN *pt} return \kbd{gen\_0} and \kbd{gen\_1}
 instead of a boolean value.

Function: issquarefree
Class: basic
Section: number_theoretical
C-Name: issquarefree
Prototype: lG
Help: issquarefree(x): true(1) if x is squarefree, false(0) if not.
Description: 
 (gen):bool       issquarefree($1)
Doc: true (1) if $x$ is squarefree, false (0) if not. Here $x$ can be an
 integer or a polynomial with coefficients in an integral domain.
 \bprog
 ? issquarefree(12)
 %1 = 0
 ? issquarefree(6)
 %2 = 1
 ? issquarefree(x^3+x^2)
 %3 = 0
 ? issquarefree(Mod(1,4)*(x^2+x+1))    \\ Z/4Z is not a domain !
  ***   at top-level: issquarefree(Mod(1,4)*(x^2+x+1))
  ***                 ^--------------------------------
  *** issquarefree: impossible inverse in Fp_inv: Mod(2, 4).
 @eprog\noindent A polynomial is declared squarefree if \kbd{gcd}$(x,x')$ is
 $1$. In particular a non-zero polynomial with inexact coefficients is
 considered to be squarefree. Note that this may be inconsistent with
 \kbd{factor}, which first rounds the input to some exact approximation before
 factoring in the apropriate domain; this is correct when the input is not
 close to an inseparable polynomial (the resultant of $x$ and $x'$ is not
 close to $0$).

Function: istotient
Class: basic
Section: number_theoretical
C-Name: istotient
Prototype: lGD&
Help: istotient(x,{&N}): true(1) if x = eulerphi(n) for some integer n,
 false(0) if not. If N is given, set N = n as well.
Doc: true (1) if $x = \phi(n)$ for some integer $n$, false (0)
 if not.
 \bprog
 ? istotient(14)
 %1 = 0
 ? istotient(100)
 %2 = 0
 @eprog
 If $N$ is given, set $N = n$ as well.
 \bprog
 ? istotient(4, &n)
 %1 = 1
 ? n
 %2 = 10
 @eprog

Function: kill
Class: basic
Section: programming/specific
C-Name: kill0
Prototype: vr
Help: kill(sym): restores the symbol sym to its ``undefined'' status and kill
 attached help messages.
Doc: restores the symbol \kbd{sym} to its ``undefined'' status, and deletes any
 help messages attached to \kbd{sym} using \kbd{addhelp}. Variable names
 remain known to the interpreter and keep their former priority: you cannot
 make a variable ``less important" by killing it!
 \bprog
 ? z = y = 1; y
 %1 = 1
 ? kill(y)
 ? y            \\ restored to ``undefined'' status
 %2 = y
 ? variable()
 %3 = [x, y, z] \\ but the variable name y is still known, with y > z !
 @eprog\noindent
 For the same reason, killing a user function (which is an ordinary
 variable holding a \typ{CLOSURE}) does not remove its name from the list of
 variable names.
 
 If the symbol is attached to a variable --- user functions being an
 important special case ---, one may use the \idx{quote} operator
 \kbd{a = 'a} to reset variables to their starting values. However, this
 will not delete a help message attached to \kbd{a}, and is also slightly
 slower than \kbd{kill(a)}.
 \bprog
 ? x = 1; addhelp(x, "foo"); x
 %1 = 1
 ? x = 'x; x   \\ same as 'kill', except we don't delete help.
 %2 = x
 ? ?x
 foo
 @eprog\noindent
 On the other hand, \kbd{kill} is the only way to remove aliases and installed
 functions.
 \bprog
 ? alias(fun, sin);
 ? kill(fun);
 
 ? install(addii, GG);
 ? kill(addii);
 @eprog

Function: kronecker
Class: basic
Section: number_theoretical
C-Name: kronecker
Prototype: lGG
Help: kronecker(x,y): kronecker symbol (x/y).
Description: 
 (small, small):small  kross($1, $2)
 (int, small):small    krois($1, $2)
 (small, int):small    krosi($1, $2)
 (gen, gen):small      kronecker($1, $2)
Doc: 
 \idx{Kronecker symbol} $(x|y)$, where $x$ and $y$ must be of type integer. By
 definition, this is the extension of \idx{Legendre symbol} to $\Z \times \Z$
 by total multiplicativity in both arguments with the following special rules
 for $y = 0, -1$ or $2$:
 
 \item $(x|0) = 1$ if $|x| = 1$ and $0$ otherwise.
 
 \item $(x|-1) = 1$ if $x \geq 0$ and $-1$ otherwise.
 
 \item $(x|2) = 0$ if $x$ is even and $1$ if $x = 1,-1 \mod 8$ and $-1$
 if $x=3,-3 \mod 8$.

Function: lambertw
Class: basic
Section: transcendental
C-Name: glambertW
Prototype: Gp
Help: lambertw(y): solution of the implicit equation x*exp(x)=y.
Doc: Lambert $W$ function, solution of the implicit equation $xe^x=y$,
 for $y > 0$.

Function: lcm
Class: basic
Section: number_theoretical
C-Name: glcm0
Prototype: GDG
Help: lcm(x,{y}): least common multiple of x and y, i.e. x*y / gcd(x,y)
 up to units.
Description: 
 (int, int):int lcmii($1, $2)
 (gen):gen      glcm0($1, NULL)
 (gen, gen):gen glcm($1, $2)
Doc: least common multiple of $x$ and $y$, i.e.~such
 that $\lcm(x,y)*\gcd(x,y) = x*y$, up to units. If $y$ is omitted and $x$
 is a vector, returns the $\text{lcm}$ of all components of $x$.
 For integer arguments, return the non-negative \text{lcm}.
 
 When $x$ and $y$ are both given and one of them is a vector/matrix type,
 the LCM is again taken recursively on each component, but in a different way.
 If $y$ is a vector, resp.~matrix, then the result has the same type as $y$,
 and components equal to \kbd{lcm(x, y[i])}, resp.~\kbd{lcm(x, y[,i])}. Else
 if $x$ is a vector/matrix the result has the same type as $x$ and an
 analogous definition. Note that for these types, \kbd{lcm} is not
 commutative.
 
 Note that \kbd{lcm(v)} is quite different from
 \bprog
 l = v[1]; for (i = 1, #v, l = lcm(l, v[i]))
 @eprog\noindent
 Indeed, \kbd{lcm(v)} is a scalar, but \kbd{l} may not be (if one of
 the \kbd{v[i]} is a vector/matrix). The computation uses a divide-conquer tree
 and should be much more efficient, especially when using the GMP
 multiprecision kernel (and more subquadratic algorithms become available):
 \bprog
 ? v = vector(10^5, i, random);
 ? lcm(v);
 time = 546 ms.
 ? l = v[1]; for (i = 1, #v, l = lcm(l, v[i]))
 time = 4,561 ms.
 @eprog

Function: length
Class: basic
Section: conversions
C-Name: glength
Prototype: lG
Help: length(x): number of non code words in x, number of characters for a
 string.
Description: 
 (vecsmall):lg      lg($1)
 (vec):lg           lg($1)
 (pol):small        lgpol($1)
 (gen):small        glength($1)
Doc: length of $x$; \kbd{\#}$x$ is a shortcut for \kbd{length}$(x)$.
 This is mostly useful for
 
 \item vectors: dimension (0 for empty vectors),
 
 \item lists: number of entries (0 for empty lists),
 
 \item matrices: number of columns,
 
 \item character strings: number of actual characters (without
 trailing \kbd{\bs 0}, should you expect it from $C$ \kbd{char*}).
 \bprog
  ? #"a string"
  %1 = 8
  ? #[3,2,1]
  %2 = 3
  ? #[]
  %3 = 0
  ? #matrix(2,5)
  %4 = 5
  ? L = List([1,2,3,4]); #L
  %5 = 4
 @eprog
 
 The routine is in fact defined for arbitrary GP types, but is awkward and
 useless in other cases: it returns the number of non-code words in $x$, e.g.
 the effective length minus 2 for integers since the \typ{INT} type has two code
 words.

Function: lex
Class: basic
Section: operators
C-Name: lexcmp
Prototype: iGG
Help: lex(x,y): compare x and y lexicographically (1 if x>y, 0 if x=y, -1 if x<y).
Doc: gives the result of a lexicographic comparison
 between $x$ and $y$ (as $-1$, $0$ or $1$). This is to be interpreted in quite
 a wide sense: It is admissible to compare objects of different types
 (scalars, vectors, matrices), provided the scalars can be compared, as well
 as vectors/matrices of different lengths. The comparison is recursive.
 
 In case all components are equal up to the smallest length of the operands,
 the more complex is considered to be larger. More precisely, the longest is
 the largest; when lengths are equal, we have matrix $>$ vector $>$ scalar.
 For example:
 \bprog
 ? lex([1,3], [1,2,5])
 %1 = 1
 ? lex([1,3], [1,3,-1])
 %2 = -1
 ? lex([1], [[1]])
 %3 = -1
 ? lex([1], [1]~)
 %4 = 0
 @eprog

Function: lfun
Class: basic
Section: l_functions
C-Name: lfun0
Prototype: GGD0,L,b
Help: lfun(L,s,{D=0}): compute the L-function value L(s), or
 if D is set, the derivative of order D at s. L is either an
 Lmath, an Ldata or an Linit.
Description: 
 (gen,gen):gen:prec       lfun($1, $2, $bitprec)
 (gen,gen,?0):gen:prec    lfun($1, $2, $bitprec)
 (gen,gen,small):gen:prec lfun0($1, $2, $3, $bitprec)
Doc: compute the L-function value $L(s)$, or if \kbd{D} is set, the
 derivative of order \kbd{D} at $s$. The parameter
 \kbd{L} is either an Lmath, an Ldata (created by \kbd{lfuncreate}, or an
 Linit (created by \kbd{lfuninit}), preferrably the latter if many values
 are to be computed.
 
 The argument $s$ is also allowed to be a power series; for instance, if $s =
 \alpha + x + O(x^n)$, the function returns the Taylor expansion of order $n$
 around $\alpha$. The result is given with absolute error less than $2^{-B}$,
 where $B = \text{realbitprecision}$.
 
 \misctitle{Caveat} The requested precision has a major impact on runtimes.
 It is advised to manipulate precision via \tet{realbitprecision} as
  explained above instead of \tet{realprecision} as the latter allows less
 granularity: \kbd{realprecision} increases by increments of 64 bits, i.e. 19
 decimal digits at a time.
 
 \bprog
 ? lfun(x^2+1, 2)  \\ Lmath: Dedekind zeta for Q(i) at 2
 %1 = 1.5067030099229850308865650481820713960
 
 ? L = lfuncreate(ellinit("5077a1")); \\ Ldata: Hasse-Weil zeta function
 ? lfun(L, 1+x+O(x^4))  \\ zero of order 3 at the central point
 %3 = 0.E-58 - 5.[...] E-40*x + 9.[...] E-40*x^2 + 1.7318[...]*x^3 + O(x^4)
 
 \\ Linit: zeta(1/2+it), |t| < 100, and derivative
 ? L = lfuninit(1, [100], 1);
 ? T = lfunzeros(L, [1,25]);
 %5 = [14.134725[...], 21.022039[...]]
 ? z = 1/2 + I*T[1];
 ? abs( lfun(L, z) )
 %7 = 8.7066865533412207420780392991125136196 E-39
 ? abs( lfun(L, z, 1) )
 %8 = 0.79316043335650611601389756527435211412  \\ simple zero
 @eprog

Function: lfunabelianrelinit
Class: basic
Section: l_functions
C-Name: lfunabelianrelinit
Prototype: GGGGD0,L,b
Help: lfunabelianrelinit(bnfL,bnfK,polrel,sdom,{der=0}): returns the
  Linit structure attached to the Dedekind zeta function of the number field
  L, given a subfield K such that L/K is abelian, where polrel defines
  L over K. The priority of the variable
  of bnfK must be lower than that of polrel; bnfL is the absolute polynomial
  corresponding to polrel, and sdom and der are as in lfuninit.
Doc: returns the \kbd{Linit} structure attached to the Dedekind zeta function
  of the number field $L$ (see \tet{lfuninit}), given a subfield $K$ such that
  $L/K$ is abelian.
  Here \kbd{polrel} defines $L$ over $K$, as usual with the priority of the
  variable of \kbd{bnfK} lower than that of \kbd{polrel}.
  \kbd{sdom} and \kbd{der} are as in \kbd{lfuninit}.
  \bprog
  ? D = -47; K = bnfinit(y^2-D);
  ? rel = quadhilbert(D); T = rnfequation(K.pol, rel); \\ degree 10
  ? L = lfunabelianrelinit(T,K,rel, [2,0,0]); \\ at 2
  time = 84 ms.
  ? lfun(L, 2)
  %4 = 1.0154213394402443929880666894468182650
  ? lfun(T, 2) \\ using parisize > 300MB
  time = 652 ms.
  %5 = 1.0154213394402443929880666894468182656
  @eprog\noindent As the example shows, using the (abelian) relative structure
  is more efficient than a direct computation. The difference becomes drastic
  as the absolute degree increases while the subfield degree remains constant.

Function: lfunan
Class: basic
Section: l_functions
C-Name: lfunan
Prototype: GLp
Help: lfunan(L,n): compute the first n terms of the Dirichlet series
  attached to the L-function given by L (Lmath, Ldata or Linit).
Doc: Compute the first $n$ terms of the Dirichlet series attached to the
  $L$-function given by \kbd{L} (\kbd{Lmath}, \kbd{Ldata} or \kbd{Linit}).
  \bprog
  ? lfunan(1, 10)  \\ Riemann zeta
  %1 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
  ? lfunan(5, 10)  \\ Dirichlet L-function for kronecker(5,.)
  %2 = [1, -1, -1, 1, 0, 1, -1, -1, 1, 0]
  @eprog

Function: lfunartin
Class: basic
Section: l_functions
C-Name: lfunartin
Prototype: GGGL
Help: lfunartin(nf,gal,M,n): returns the Ldata structure attached to the
 Artin L-function attached to the representation R of the Galois group
 of the extension K/Q, defined over the cyclotomic field Q(zeta_n),
 where nf is the nfinit structure attached to K,
 gal is the galoisinit structure attached to K/Q, and M is the vector
 of the image of the generators G.gen by R. The elements of M are matrices
 with polynomial entries, whose variable is understood as the complex
 number exp(2*I*Pi/n).
Doc: returns the \kbd{Ldata} structure attached to the
 Artin $L$-function attached to the representation $\rho$ of the Galois group
 of the extension $K/\Q$, defined over the cyclotomic field $\Q(\zeta_n)$,
 where \var{nf} is the nfinit structure attached to $K$,
 \var{gal} is the galoisinit structure attached to $K/\Q$, and $M$ is
 the vector of the image of the generators \kbd{\var{gal}.gen} by $\rho$.
 The elements of $M$ are matrices with polynomial entries, whose variable
 is understood as the complex number $\exp(2\*i\*\pi/n)$.
 
 In the following example we build the Artin $L$-functions attached to the two
 irreducible degree $2$ representations of the dihedral group $D_{10}$ defined
 over $\Q(\zeta_5)$, for the extension $H/\Q$ where $H$ is the Hilbert class
 field of $\Q(\sqrt{-47})$.
 We show numerically some identities involving Dedekind $\zeta$ functions and
 Hecke $L$ series.
 \bprog
 ? P = quadhilbert(-47);
 ? N = nfinit(nfsplitting(P));
 ? G = galoisinit(N);
 ? L1 = lfunartin(N,G, [[a,0;0,a^-1],[0,1;1,0]], 5);
 ? L2 = lfunartin(N,G, [[a^2,0;0,a^-2],[0,1;1,0]], 5);
 ? s = 1 + x + O(x^4);
 ? lfun(1,s)*lfun(-47,s)*lfun(L1,s)^2*lfun(L2,s)^2 - lfun(N,s)
 %6 ~ 0
 ? lfun(1,s)*lfun(L1,s)*lfun(L2,s) - lfun(P,s)
 %7 ~ 0
 ? bnr = bnrinit(bnfinit(x^2+47),1,1);
 ? lfun([bnr,[1]], s) - lfun(L1, s)
 %9 ~ 0
 ? lfun([bnr,[1]], s) - lfun(L1, s)
 %10 ~ 0
 @eprog
 The first identity is the factorisation of the regular representation of
 $D_{10}$, the second the factorisation of the natural representation of
 $D_{10}\subset S_5$, the next two are the expressions of the degree $2$
 representations as induced from degree $1$ representations.

Function: lfuncheckfeq
Class: basic
Section: l_functions
C-Name: lfuncheckfeq
Prototype: lGDGb
Help: lfuncheckfeq(L,{t}): given an L-function (Lmath, Ldata or Linit),
 check whether the functional equation is satisfied. If the function has
 poles, the polar part must be specified. The program returns a bit accuracy
 which should be a large negative value close to the current bit accuracy.
 If t is given, it checks the functional equation for the theta function
 at t and 1/t.
Doc: Given the data attached to an $L$-function (\kbd{Lmath}, \kbd{Ldata}
 or \kbd{Linit}), check whether the functional equation is satisfied.
 This is most useful for an \kbd{Ldata} constructed ``by hand'', via
 \kbd{lfuncreate}, to detect mistakes.
 
 If the function has poles, the polar part must be specified. The routine
 returns a bit accuracy $b$ such that $|w - \hat{w}| < 2^{b}$, where $w$ is
 the root number contained in \kbd{data}, and $\hat{w}$ is a computed value
 derived from $\overline{\theta}(t)$ and $\theta(1/t)$ at some $t\neq 0$ and
 the assumed functional equation. Of course, a large negative value of the
 order of \kbd{realbitprecision} is expected.
 
 If $t$ is given, it should be close to the unit disc for efficiency and
 such that $\overline{\theta}(t) \neq 0$. We then check the functional
 equation at that $t$.
 \bprog
 ? \pb 128       \\ 128 bits of accuracy
 ? default(realbitprecision)
 %1 = 128
 ? L = lfuncreate(1);  \\ Riemann zeta
 ? lfuncheckfeq(L)
 %3 = -124
 @eprog\noindent i.e. the given data is consistent to within 4 bits for the
 particular check consisting of estimating the root number from all other
 given quantities. Checking away from the unit disc will either fail with
 a precision error, or give disappointing results (if $\theta(1/t)$ is
 large it will be computed with a large absolute error)
 \bprog
 ? lfuncheckfeq(L, 2+I)
 %4 = -115
 ? lfuncheckfeq(L,10)
  ***   at top-level: lfuncheckfeq(L,10)
  ***                 ^------------------
  *** lfuncheckfeq: precision too low in lfuncheckfeq.
 @eprog

Function: lfunconductor
Class: basic
Section: l_functions
C-Name: lfunconductor
Prototype: GDGD0,L,b
Help: lfunconductor(L,{ab=[1,10000]},{flag=0}): give the conductor
  of the given L-function; ab = [a,b] is the interval where we expect
  to find the conductor.
  If flag=0 (default), give either the conductor found as an integer, or a
  vector (possibly empty) of conductors found. If flag=1, same but give the
  computed floating point approximations to the conductors found, without
  rounding to integers.
  If flag=2, give all the conductors found, even those far from integers.
  Note: this program is heuristic and should only be used if the primes
  dividing the conductor are unknown. If they are known, a direct search
  through possible prime exponents using lfuncheckfeq will be more efficient.
Doc: Compute the conductor of the given $L$-function
  (if the structure contains a conductor, it is ignored);
  $\kbd{ab} = [a,b]$ is the interval where we expect to find the conductor;
  it may be given as a single scalar $b$, in which case we look in $[1,b]$.
  Increasing \kbd{ab} slows down the program but gives better accuracy for the
  result.
 
  If \kbd{flag} is $0$ (default), give either the conductor found as an
  integer, or a vector (possibly empty) of conductors found. If \kbd{flag} is
  $1$, same but give the computed floating point approximations to the
  conductors found, without rounding to integers. It \kbd{flag} is $2$, give
  all the conductors found, even those far from integers.
 
  \misctitle{Caveat} This is a heuristic program and the result is not
  proven in any way:
  \bprog
  ? L = lfuncreate(857); \\ Dirichlet L function for kronecker(857,.)
  ? \p19
    realprecision = 19 significant digits
  ? lfunconductor(L)
  %2 = [17, 857]
  ? lfunconductor(L,,1) \\ don't round
  %3 = [16.99999999999999999, 857.0000000000000000]
 
  ? \p38
    realprecision = 38 significant digits
  ? lfunconductor(L)
  %4 = 857
  @eprog
 
  \misctitle{Note} This program should only be used if the primes dividing the
  conductor are unknown, which is rare. If they are known, a direct
  search through possible prime exponents using \kbd{lfuncheckfeq} will
  be more efficient and rigorous:
  \bprog
  ? E = ellinit([0,0,0,4,0]); /* Elliptic curve y^2 = x^3+4x */
  ? E.disc  \\ |disc E| = 2^12
  %2 = -4096
  \\ create Ldata by hand. Guess that root number is 1 and conductor N
  ? L(N) = lfuncreate([n->ellan(E,n), 0, [0,1], 1, N, 1]);
  ? fordiv(E.disc, d, print(d,": ",lfuncheckfeq(L(d))))
  1: 0
  2: 0
  4: -1
  8: -2
  16: -3
  32: -127
  64: -3
  128: -2
  256: -2
  512: -1
  1024: -1
  2048: 0
  4096: 0
  ? lfunconductor(L(1)) \\ lfunconductor ignores conductor = 1 in Ldata !
  %5 = 32
  @eprog\noindent The above code assumed that root number was $1$;
  had we set it to $-1$, none of the \kbd{lfuncheckfeq} values would have been
  acceptable:
  \bprog
  ? L2(N) = lfuncreate([n->ellan(E,n), 0, [0,1], 1, N, -1]);
  ? [ lfuncheckfeq(L2(d)) | d<-divisors(E.disc) ]
  %7 = [0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, -1, -1]
  @eprog

Function: lfuncost
Class: basic
Section: l_functions
C-Name: lfuncost0
Prototype: GDGD0,L,b
Help: lfuncost(L,{sdom},{der=0}): estimate the cost of running
 lfuninit(L,sdom,der) at current bit precision. Returns [t,b], to indicate
 that t coefficients a_n will be computed at bit accuracy b. Subsequent
 evaluation of lfun at s evaluates a polynomial of degree t at exp(h s).
 If L is already an Linit, then sdom and der are ignored.
Doc: estimate the cost of running
 \kbd{lfuninit(L,sdom,der)} at current bit precision. Returns $[t,b]$, to
 indicate that $t$ coefficients $a_n$ will be computed, as well as $t$ values of
 \tet{gammamellininv}, all at bit accuracy $b$.
 A subsequent call to \kbd{lfun} at $s$ evaluates a polynomial of degree $t$
 at $\exp(h s)$ for some real parameter $h$, at the same bit accuracy $b$.
 If $L$ is already an \kbd{Linit}, then \var{sdom} and \var{der} are ignored
 and are best left omitted; the bit accuracy is also inferred from $L$: in
 short we get an estimate of the cost of using that particular \kbd{Linit}.
 
 \bprog
 ? \pb 128
 ? lfuncost(1, [100]) \\ for zeta(1/2+I*t), |t| < 100
 %1 = [7, 242]  \\ 7 coefficients, 242 bits
 ? lfuncost(1, [1/2, 100]) \\ for zeta(s) in the critical strip, |Im s| < 100
 %2 = [7, 246]  \\ now 246 bits
 ? lfuncost(1, [100], 10) \\ for zeta(1/2+I*t), |t| < 100
 %3 = [8, 263]  \\ 10th derivative increases the cost by a small amount
 ? lfuncost(1, [10^5])
 %3 = [158, 113438]  \\ larger imaginary part: huge accuracy increase
 
 ? L = lfuncreate(polcyclo(5)); \\ Dedekind zeta for Q(zeta_5)
 ? lfuncost(L, [100]) \\ at s = 1/2+I*t), |t| < 100
 %5 = [11457, 582]
 ? lfuncost(L, [200]) \\ twice higher
 %6 = [36294, 1035]
 ? lfuncost(L, [10^4])  \\ much higher: very costly !
 %7 = [70256473, 45452]
 ? \pb 256
 ? lfuncost(L, [100]); \\ doubling bit accuracy
 %8 = [17080, 710]
 @eprog\noindent In fact, some $L$ functions can be factorized algebraically
 by the \kbd{lfuninit} call, e.g. the Dedekind zeta function of abelian
 fields, leading to much faster evaluations than the above upper bounds.
 In that case, the function returns a vector of costs as above for each
 individual function in the product actually evaluated:
 \bprog
 ? L = lfuncreate(polcyclo(5)); \\ Dedekind zeta for Q(zeta_5)
 ? lfuncost(L, [100])  \\ a priori cost
 %2 = [11457, 582]
 ? L = lfuninit(L, [100]); \\ actually perform all initializations
 ? lfuncost(L)
 %4 = [[16, 242], [16, 242], [7, 242]]
 @eprog\noindent The Dedekind function of this abelian quartic field
 is the product of four Dirichlet $L$-functions attached to the trivial
 character, a non-trivial real character and two complex conjugate
 characters. The non-trivial characters happen to have the same conductor
 (hence same evaluation costs), and correspond to two evaluations only
 since the two conjugate characters are evaluated simultaneously.
 For a total of three $L$-functions evaluations, which explains the three
 components above. Note that the actual cost is much lower than the a priori
 cost in this case.
Variant: Also available is
 \fun{GEN}{lfuncost}{GEN L, GEN dom, long der, long bitprec}
 when $L$ is \emph{not} an \kbd{Linit}; the return value is a \typ{VECSMALL}
 in this case.

Function: lfuncreate
Class: basic
Section: l_functions
C-Name: lfuncreate
Prototype: G
Help: lfuncreate(obj): given either an object such as a polynomial, elliptic
 curve, Dirichlet or Hecke character, eta quotient, etc., or an explicit
 6 or 7 component vector [dir,real,Vga,k,N,eps,r],
 create the Ldata structure necessary for lfun computation.
Doc: This low-level routine creates \tet{Ldata} structures, needed by
 \var{lfun} functions, describing an $L$-function and its functional equation.
 You are urged to use a high-level constructor when one is available,
 and this function accepts them, see \kbd{??lfun}:
 \bprog
 ? L = lfuncreate(1); \\ Riemann zeta
 ? L = lfuncreate(5); \\ Dirichlet L-function for quadratic character (5/.)
 ? L = lfuncreate(x^2+1); \\ Dedekind zeta for Q(i)
 ? L = lfuncreate(ellinit([0,1])); \\ L-function of E/Q: y^2=x^3+1
 @eprog\noindent One can then use, e.g., \kbd{Lfun(L,s)} to directly
 evaluate the respective $L$-functions at $s$, or \kbd{lfuninit(L, [c,w,h]}
 to initialize computations in the rectangular box $\Re(s-c) \leq w$,
 $\Im(s) \leq h$.
 
 We now describe the low-level interface, used to input non-builtin
 $L$-functions. The input is now a $6$ or $7$ component vector
 $V=[a,astar,Vga,k,N,eps,poles]$, whose components are as follows:
 
 \item \kbd{V[1]=a} encodes the Dirichlet series coefficients. The
 preferred format is a closure of arity 1: \kbd{n->vector(n,i,a(i))} giving
 the vector of the first $n$ coefficients. The closure is allowed to return
 a vector of more than $n$ coefficients (only the first $n$ will be
 considered) or even less than $n$, in which case loss of accuracy will occur
 and a warning that \kbd{\#an} is less than expected is issued. This
 allows to precompute and store a fixed large number of Dirichlet
 coefficients in a vector $v$ and use the closure \kbd{n->v}, which
 does not depend on $n$. As a shorthand for this latter case, you can input
 the vector $v$ itself instead of the closure.
 
 A second format is limited to multiplicative $L$ functions affording an
 Euler product. It is a closure of arity 2 \kbd{(p,d)->L(p)} giving the local
 factor $L_p$ at $p$ as a rational function, to be evaluated at $p^{-s}$ as in
 \kbd{direuler}; $d$ is set to the floor of $\log_p(n)$, where $n$ is the
 total number of Dirichlet coefficients $(a_1,\dots,a_n)$ that will be
 computed in this way. This parameter $d$ allows to compute only part of $L_p$
 when $p$ is large and $L_p$ expensive to compute, but it can of course be
 ignored by the closure.
 
 Finally one can describe separately the generic Dirichlet coefficients
 and the bad local factors by setting $\kbd{dir} = [an, [p_1,L^{-1}_{p_1}],
 \dots,[p_k,L^{-1}_{p_k}]]$, where \kbd{an} describes the generic coefficients
 in one of the two formats above, except that coefficients $a_n$ with
 $p_i \mid n$ for some $i \leq k$ will be ignored. The subsequent pairs $[p,
 L_p^{-1}]$ give the bad primes and corresponding \emph{inverse} local
 factors.
 
 \item \kbd{V[2]=astar} is the Dirichlet series coefficients of the dual
 function, encoded as \kbd{a} above. The sentinel values $0$ and $1$ may
 be used for the special cases where $a = a^*$ and $a = \overline{a^*}$,
 respectively.
 
 \item \kbd{V[3]=Vga} is the vector of $\alpha_j$ such that the gamma
 factor of the $L$-function is equal to
 $$\gamma_A(s)=\prod_{1\le j\le d}\Gamma_{\R}(s+\alpha_j),$$
 where $\Gamma_{\R}(s)=\pi^{-s/2}\Gamma(s/2)$.
 This same syntax is used in the \kbd{gammamellininv} functions.
 In particular the length $d$ of \kbd{Vga} is the degree of the $L$-function.
 In the present implementation, the $\alpha_j$ are assumed to be exact
 rational numbers. However when calling theta functions with \emph{complex}
 (as opposed to real) arguments, determination problems occur which may
 give wrong results when the $\alpha_j$ are not integral.
 
 \item \kbd{V[4]=k} is a positive integer $k$. The functional equation relates
 values at $s$ and $k-s$. For instance, for an Artin $L$-series such as a
 Dedekind zeta function we have $k = 1$, for an elliptic curve $k = 2$, and
 for a modular form, $k$ is its weight. For motivic $L$-functions, the
 \emph{motivic} weight $w$ is $w = k-1$.
 
 \item \kbd{V[5]=N} is the conductor, an integer $N\ge1$, such that
 $\Lambda(s)=N^{s/2}\gamma_A(s)L(s)$ with $\gamma_A(s)$ as above.
 
 \item \kbd{V[6]=eps} is the root number $\varepsilon$, i.e., the
 complex number (usually of modulus $1$) such that
 $\Lambda(a, k-s) = \varepsilon \Lambda(a^*, s)$.
 
 \item The last optional component \kbd{V[7]=poles} encodes the poles of the
 $L$ or $\Lambda$-functions, and is omitted if they have no poles.
 A polar part is given by a list of $2$-component vectors
 $[\beta,P_{\beta}(x)]$, where
 $\beta$ is a pole and the power series $P_{\beta}(x)$ describes
 the attached polar part, such that $L(s) - P_\beta(s-\beta)$ is holomorphic
 in a neighbourhood of $\beta$. For instance $P_\beta = r/x+O(1)$ for a
 simple pole at $\beta$ or $r_1/x^2+r_2/x+O(1)$ for a double pole.
 The type of the list describing the polar part allows to distinguish between
 $L$ and $\Lambda$: a \typ{VEC} is attached to $L$, and a \typ{COL}
 is attached to $\Lambda$.
 
 The latter is mandatory unless $a = \overline{a^*}$ (coded by \kbd{astar}
 equal to $0$ or $1$): otherwise, the poles of $L^*$ cannot be infered from
 the poles of $L$ ! (Whereas the functional equation allows to deduce
 the polar part of $\Lambda^*$ from the polar part of $\Lambda$.)
 The special coding $\kbd{poles} = r$ a complex scalar is available in this
 case, to describe a $L$ function with at most a single simple pole at $s =
 k$ and residue $r$. (This is the usual situation, for instance for Dedekind
 zeta functions.) This value $r$ can be set to $0$ if unknown, and it will be
 computed.

Function: lfundiv
Class: basic
Section: l_functions
C-Name: lfundiv
Prototype: GGb
Help: lfundiv(L1,L2): creates the Ldata structure (without
  initialization) corresponding to the quotient of the Dirichlet series
  given by L1 and L2.
Doc: creates the \kbd{Ldata} structure (without initialization) corresponding
  to the quotient of the Dirichlet series $L_1$ and $L_2$ given by
 \kbd{L1} and \kbd{L2}. Assume that $v_z(L_1) \geq v_z(L_2)$ at all
 complex numbers $z$: the construction may not create new poles, nor increase
 the order of existing ones.

Function: lfunetaquo
Class: basic
Section: l_functions
C-Name: lfunetaquo
Prototype: G
Help: lfunetaquo(M): returns the Ldata structure attached to the
 modular form z->prod(i=1,#M[,1],eta(M[i,1]*z)^M[i,2]).
Doc: returns the \kbd{Ldata} structure attached to the $L$ function
 attached to the modular form
 $z\mapsto \prod_{i=1}^n \eta(M_{i,1}\*z)^{M_{i,2}}$
 It is currently assumed that $f$ is a self-dual cuspidal form on
 $\Gamma_0(N)$ for some $N$.
 For instance, the $L$-function $\sum \tau(n) n^{-s}$
 attached to Ramanujan's $\Delta$ function is encoded as follows
 \bprog
 ? L = lfunetaquo(Mat([1,24]));
 ? lfunan(L, 100)  \\ first 100 values of tau(n)
 @eprog

Function: lfungenus2
Class: basic
Section: l_functions
C-Name: lfungenus2
Prototype: G
Help: lfungenus2(F): returns the Ldata structure attached to the
 L-function attached to the genus-2 curve defined by y^2=F(x)
 or y^2+Q(x)*y=P(x) if F=[P,Q].
 Currently, only odd conductors are supported, and the model needs to
 be minimal at 2.
Doc: returns the \kbd{Ldata} structure attached to the $L$ function
 attached to the genus-2 curve defined by $y^2=F(x)$ or
 $y^2+Q(x)\*y=P(x)$ if $F=[P,Q]$.
 Currently, the model needs to be minimal at 2, and if the conductor
 is even, its valuation at $2$ might be incorrect (a warning is issued).

Function: lfunhardy
Class: basic
Section: l_functions
C-Name: lfunhardy
Prototype: GGb
Help: lfunhardy(L,t): variant of the Hardy L-function attached to L, used for
 plotting on the critical line.
Doc: Variant of the Hardy $Z$-function given by \kbd{L}, used for
 plotting or locating zeros of $L(k/2+it)$ on the critical line.
 The precise definition is as
 follows: if as usual $k/2$ is the center of the critical strip, $d$ is the
 degree, $\alpha_j$ the entries of \kbd{Vga} giving the gamma factors,
 and $\varepsilon$ the root number, then if we set
 $s = k/2+it = \rho e^{i\theta}$ and
 $E=(d(k/2-1)+\sum_{1\le j\le d}\alpha_j)/2$, the computed function at $t$ is
 equal to
 $$Z(t) = \varepsilon^{-1/2}\Lambda(s) \cdot |s|^{-E}e^{dt\theta/2}\;,$$
 which is a real function of $t$ for self-dual $\Lambda$,
 vanishing exactly when $L(k/2+it)$ does on the critical line. The
 normalizing factor $|s|^{-E}e^{dt\theta/2}$ compensates the
 exponential decrease of $\gamma_A(s)$ as $t\to\infty$ so that
 $Z(t) \approx 1$.
 
 \bprog
 ? T = 100; \\ maximal height
 ? L = lfuninit(1, [T]); \\ initialize for zeta(1/2+it), |t|<T
 ? \p19 \\ no need for large accuracy
 ? ploth(t = 0, T, lfunhardy(L,t))
 @eprog\noindent Using \kbd{lfuninit} is critical for this particular
 applications since thousands of values are computed. Make sure to initialize
 up to the maximal $t$ needed: otherwise expect to see many warnings for
 unsufficient initialization and suffer major slowdowns.

Function: lfuninit
Class: basic
Section: l_functions
C-Name: lfuninit0
Prototype: GGD0,L,b
Help: lfuninit(L,sdom,{der=0}): precompute data
 for evaluating the L-function given by 'L' (and its derivatives
 of order der, if set) in rectangular domain sdom = [center,w,h]
 centered on the real axis, |Re(s)-center| <= w, |Im(s)| <= h,
 where all three components of sdom are real and w,h are non-negative.
 The subdomain [k/2, 0, h] on the critical line can be encoded as [h] for
 brevity.
Doc: initalization function for all functions linked to the
 computation of the $L$-function $L(s)$ encoded by \kbd{L}, where
 $s$ belongs to the rectangular domain $\kbd{sdom} = [\var{center},w,h]$
 centered on the real axis, $|\Re(s)-\var{center}| \leq w$, $|\Im(s)| \leq h$,
 where all three components of \kbd{sdom} are real and $w$, $h$ are
 non-negative. \kbd{der} is the maximum order of derivation that will be used.
 The subdomain $[k/2, 0, h]$ on the critical line (up to height $h$)
 can be encoded as $[h]$ for brevity. The subdomain $[k/2, w, h]$
 centered on the critical line can be encoded as $[w, h]$ for brevity.
 
 The argument \kbd{L} is an \kbd{Lmath}, an \kbd{Ldata} or an \kbd{Linit}. See
 \kbd{??Ldata} and \kbd{??lfuncreate} for how to create it.
 
 The height $h$ of the domain is a \emph{crucial} parameter: if you only
 need $L(s)$ for real $s$, set $h$ to~0.
 The running time is roughly proportional to
 $$(B / d+\pi h/4)^{d/2+3}N^{1/2},$$
 where $B$ is the default bit accuracy, $d$ is the degree of the
 $L$-function, and $N$ is the conductor (the exponent $d/2+3$ is reduced
 to $d/2+2$ when $d=1$ and $d=2$). There is also a dependency on $w$,
 which is less crucial, but make sure to use the smallest rectangular
 domain that you need.
 \bprog
 ? L0 = lfuncreate(1); \\ Riemann zeta
 ? L = lfuninit(L0, [1/2, 0, 100]); \\ for zeta(1/2+it), |t| < 100
 ? lfun(L, 1/2 + I)
 ? L = lfuninit(L0, [100]); \\ same as above !
 @eprog

Function: lfunlambda
Class: basic
Section: l_functions
C-Name: lfunlambda0
Prototype: GGD0,L,b
Help: lfunlambda(L,s,{D=0}): compute the completed L function Lambda(s),
 or if D is set, the derivative of order D at s. L is either
 an Lmath, an Ldata or an Linit.
Doc: compute the completed $L$-function $\Lambda(s) = N^{s/2}\gamma(s)L(s)$,
 or if \kbd{D} is set, the derivative of order \kbd{D} at $s$.
 The parameter \kbd{L} is either an \kbd{Lmath}, an \kbd{Ldata} (created by
 \kbd{lfuncreate}, or an \kbd{Linit} (created by \kbd{lfuninit}), preferrably the
 latter if many values are to be computed.
 
 The result is given with absolute error less than $2^{-B}|\gamma(s)N^{s/2}|$,
 where $B = \text{realbitprecision}$.

Function: lfunmfspec
Class: basic
Section: l_functions
C-Name: lfunmfspec
Prototype: Gb
Help: lfunmfspec(L): L corresponding to a modular form, returns
  [valeven,valodd,omminus,omplus], where valeven (resp., valodd) is the vector
  of even (resp., odd) periods, and omminus and omplus the corresponding
  real numbers omega^- and omega^+. For the moment, only for modular forms of even weight.
Doc: returns \kbd{[valeven,valodd,omminus,omplus]},
  where \kbd{valeven} (resp., \kbd{valodd}) is the vector of even (resp., odd)
  periods of the modular form given by \kbd{L}, and \kbd{omminus} and
  \kbd{omplus} the corresponding real numbers $\omega^-$ and $\omega^+$
  normalized in a noncanonical way. For the moment, only for modular forms of even weight.

Function: lfunmul
Class: basic
Section: l_functions
C-Name: lfunmul
Prototype: GGb
Help: lfunmul(L1,L2): creates the Ldata structure (without
  initialization) corresponding to the product of the Dirichlet series
  given by L1 and L2.
Doc: creates the \kbd{Ldata} structure (without initialization) corresponding
  to the product of the Dirichlet series given by \kbd{L1} and
  \kbd{L2}.

Function: lfunorderzero
Class: basic
Section: l_functions
C-Name: lfunorderzero
Prototype: lGD-1,L,b
Help: lfunorderzero(L, {m = -1}): computes the order of the possible zero
 of the L-function at the center k/2 of the critical strip. If $m$ is
 given and has a non-negative value, assumes the order is at most $m$.
Doc: Computes the order of the possible zero of the $L$-function at the
 center $k/2$ of the critical strip; return $0$ if $L(k/2)$ does not vanish.
 
 If $m$ is given and has a non-negative value, assumes the order is at most $m$.
 Otherwise, the algorithm chooses a sensible default:
 
 \item if the $L$ argument is an \kbd{Linit}, assume that a multiple zero at
 $s = k / 2$ has order less than or equal to the maximal allowed derivation
 order.
 
 \item else assume the order is less than $4$.
 
 You may explicitly increase this value using optional argument~$m$; this
 overrides the default value above. (Possibly forcing a recomputation
 of the \kbd{Linit}.)

Function: lfunqf
Class: basic
Section: l_functions
C-Name: lfunqf
Prototype: Gp
Help: lfunqf(Q): returns the Ldata structure attached to the
 theta function of the lattice attached to the definite positive quadratic
 form Q.
Doc: returns the \kbd{Ldata} structure attached to the $\Theta$ function
 of the lattice attached to the definite positive quadratic form $Q$.
 \bprog
 ? L = lfunqf(matid(2));
 ? lfunqf(L,2)
 %2 = 6.0268120396919401235462601927282855839
 ? lfun(x^2+1,2)*4
 %3 = 6.0268120396919401235462601927282855839
 @eprog

Function: lfunrootres
Class: basic
Section: l_functions
C-Name: lfunrootres
Prototype: Gb
Help: lfunrootres(data): given the Ldata attached to an L-function (or the
 output of lfunthetainit), compute the root number and the
 residues. In the present implementation, if the polar part is not already
 known completely, at most a single pole is allowed.
 The output is a 3-component vector [r,R,w], where r is the residue of L(s)
 at the unique pole (0 if no pole), R is the residue of Lambda(s), and w is
 the root number.
Doc: Given the \kbd{Ldata} attached to an $L$-function (or the output of
 \kbd{lfunthetainit}), compute the root number and the residues.
 The output is a 3-component vector $[r,R,w]$, where $r$ is the
 residue of $L(s)$ at the unique pole, $R$ is the residue of $\Lambda(s)$,
 and $w$ is the root number. In the present implementation,
 
 \item either the polar part must be completely known (and is then arbitrary):
 the function determines the root number,
 
 \bprog
 ? L = lfunmul(1,1); \\ zeta^2
 ? [r,R,w] = lfunrootres(L);
 ? r  \\ single pole at 1, double
 %3 = [[1, 1.[...]*x^-2 + 1.1544[...]*x^-1 + O(x^0)]]
 ? w
 %4 = 1
 ? R \\ double pole at 0 and 1
 %5 = [[1,[...]], [0,[...]]
 @eprog
 
 \item or at most a single pole is allowed: the function computes both
 the root number and the residue ($0$ if no pole).

Function: lfuntheta
Class: basic
Section: l_functions
C-Name: lfuntheta
Prototype: GGD0,L,b
Help: lfuntheta(data,t,{m=0}): compute the value of the m-th derivative
 at t of the theta function attached to the L-function given by data.
 data can be either the standard L-function data, or the output of
 lfunthetainit.
Doc: compute the value of the $m$-th derivative
 at $t$ of the theta function attached to the $L$-function given by \kbd{data}.
  \kbd{data} can be either the standard $L$-function data, or the output of
 \kbd{lfunthetainit}.
 The theta function is defined by the formula
 $\Theta(t)=\sum_{n\ge1}a(n)K(nt/\sqrt(N))$, where $a(n)$ are the coefficients
 of the Dirichlet series, $N$ is the conductor, and $K$ is the inverse Mellin
 transform of the gamma product defined by the \kbd{Vga} component.
 Its Mellin transform is equal to $\Lambda(s)-P(s)$, where $\Lambda(s)$
 is the completed $L$-function and the rational function $P(s)$ its polar part.
 In particular, if the $L$-function is the $L$-function of a modular form
 $f(\tau)=\sum_{n\ge0}a(n)q^n$ with $q=\exp(2\pi i\tau)$, we have
 $\Theta(t)=2(f(it/\sqrt{N})-a(0))$. Note that an easy theorem on modular
 forms implies that $a(0)$ can be recovered by the formula $a(0)=-L(f,0)$.

Function: lfunthetacost
Class: basic
Section: l_functions
C-Name: lfunthetacost0
Prototype: lGDGD0,L,b
Help: lfunthetacost(L,{tdom},{m=0}): estimates the cost of running
 lfunthetainit(L,tdom,m) at current bit precision. Returns the number of
 coefficients an that would be computed. Subsequent evaluation of lfuntheta
 computes that many values of gammamellininv.
 If L is already an Linit, then tdom and m are ignored.
Doc: This function estimates the cost of running
 \kbd{lfunthetainit(L,tdom,m)} at current bit precision. Returns the number of
 coefficients $a_n$ that would be computed. This also estimates the
 cost of a subsequent evaluation \kbd{lfuntheta}, which must compute
 that many values of \kbd{gammamellininv} at the current bit precision.
 If $L$ is already an \kbd{Linit}, then \var{tdom} and $m$ are ignored
 and are best left omitted: we get an estimate of the cost of using that
 particular \kbd{Linit}.
 
 \bprog
 ? \pb 1000
 ? L = lfuncreate(1); \\ Riemann zeta
 ? lfunthetacost(L); \\ cost for theta(t), t real >= 1
 %1 = 15
 ? lfunthetacost(L, 1 + I); \\ cost for theta(1+I). Domain error !
  ***   at top-level: lfunthetacost(1,1+I)
  ***                 ^--------------------
  *** lfunthetacost: domain error in lfunthetaneed: arg t > 0.785
 ? lfunthetacost(L, 1 + I/2) \\ for theta(1+I/2).
 %2 = 23
 ? lfunthetacost(L, 1 + I/2, 10) \\ for theta^((10))(1+I/2).
 %3 = 24
 ? lfunthetacost(L, [2, 1/10]) \\ cost for theta(t), |t| >= 2, |arg(t)| < 1/10
 %4 = 8
 
 ? L = lfuncreate( ellinit([1,1]) );
 ? lfunthetacost(L)  \\ for t >= 1
 %6 = 2471
 @eprog

Function: lfunthetainit
Class: basic
Section: l_functions
C-Name: lfunthetainit
Prototype: GDGD0,L,b
Help: lfunthetainit(L,{tdom},{m=0}): precompute data for evaluating
  the m-th derivative of theta functions with argument in domain tdom
  (by default t is real >= 1).
Doc: Initalization function for evaluating the $m$-th derivative of theta
 functions with argument $t$ in domain \var{tdom}. By default (\var{tdom}
 omitted), $t$ is real, $t \geq 1$. Otherwise, \var{tdom} may be
 
 \item a positive real scalar $\rho$: $t$ is real, $t \geq \rho$.
 
 \item a non-real complex number: compute at this particular $t$; this
 allows to compute $\theta(z)$ for any complex $z$ satisfying $|z|\geq |t|$
 and $|\arg z| \leq |\arg t|$; we must have $|2 \arg z / d| < \pi/2$, where
 $d$ is the degree of the $\Gamma$ factor.
 
 \item a pair $[\rho,\alpha]$: assume that $|t| \geq \rho$ and $|\arg t| \leq
 \alpha$; we must have $|2\alpha / d| < \pi/2$, where $d$ is the degree of
 the $\Gamma$ factor.
 
 \bprog
 ? \p500
 ? L = lfuncreate(1); \\ Riemann zeta
 ? t = 1+I/2;
 ? lfuntheta(L, t); \\ direct computation
 time = 30 ms.
 ? T = lfunthetainit(L, 1+I/2);
 time = 30 ms.
 ? lfuntheta(T, t); \\ instantaneous
 @eprog\noindent The $T$ structure would allow to quickly compute $\theta(z)$
 for any $z$ in the cone delimited by $t$ as explained above. On the other hand
 \bprog
 ? lfuntheta(T,I)
  ***   at top-level: lfuntheta(T,I)
  ***                 ^--------------
  *** lfuntheta: domain error in lfunthetaneed: arg t > 0.785398163397448
 @eprog
 The initialization is equivalent to
 \bprog
 ? lfunthetainit(L, [abs(t), arg(t)])
 @eprog

Function: lfunzeros
Class: basic
Section: l_functions
C-Name: lfunzeros
Prototype: GGD8,L,b
Help: lfunzeros(L,lim,{divz=8}): lim being
 either an upper limit or a real interval, computes an ordered list of
 zeros of L(s) on the critical line up to the given upper limit or in the
 given interval. Use a naive algorithm which may miss some zeros.
 To use a finer search mesh, set divz to some integral value
 larger than the default (= 8).
Doc: \kbd{lim} being either a positive upper limit or a non-empty real
 interval inside $[0,+\infty[$, computes an
 ordered list of zeros of $L(s)$ on the critical line up to the given
 upper limit or in the given interval. Use a naive algorithm which may miss
 some zeros: it assumes that two consecutive zeros at height $T \geq 1$
 differ at least by $2\pi/\omega$, where
 $$\omega := \kbd{divz} \cdot \big(d\log(T/2\pi) +d+ 2\log(N/(\pi/2)^d)\big).$$
 To use a finer search mesh, set divz to some integral value
 larger than the default (= 8).
 \bprog
 ? lfunzeros(1, 30) \\ zeros of Rieman zeta up to height 30
 %1 = [14.134[...], 21.022[...], 25.010[...]]
 ? #lfunzeros(1, [100,110])  \\ count zeros with 100 <= Im(s) <= 110
 %2 = 4
 @eprog\noindent The algorithm also assumes that all zeros are simple except
 possibly on the real axis at $s = k/2$ and that there are no poles in the
 search interval. (The possible zero at $s = k/2$ is repeated according to
 its multiplicity.)
 
 Should you pass an \kbd{Linit} argument to the function, beware that the
 algorithm needs at least
 \bprog
    L = lfuninit(Ldata, T+1)
 @eprog\noindent where $T$ is the upper bound of the interval defined by
 \kbd{lim}: this allows to detect zeros near $T$. Make sure that your
 \kbd{Linit} domain contains this one. The algorithm assumes
 that a multiple zero at $s = k / 2$ has order less than or equal to
 the maximal derivation order allowed by the \kbd{Linit}. You may increase
 that value in the \kbd{Linit} but this is costly: only do it for zeros
 of low height or in \kbd{lfunorderzero} instead.

Function: lift
Class: basic
Section: conversions
C-Name: lift0
Prototype: GDn
Help: lift(x,{v}):
 if v is omitted, lifts elements of Z/nZ to Z, of Qp to Q, and of K[x]/(P) to
 K[x]. Otherwise lift only polmods with main variable v.
Description: 
 (pol):pol        lift($1)
 (vec):vec        lift($1)
 (gen):gen        lift($1)
 (pol, var):pol        lift0($1, $2)
 (vec, var):vec        lift0($1, $2)
 (gen, var):gen        lift0($1, $2)
Doc: 
 if $v$ is omitted, lifts intmods from $\Z/n\Z$ in $\Z$,
 $p$-adics from $\Q_p$ to $\Q$ (as \tet{truncate}), and polmods to
 polynomials. Otherwise, lifts only polmods whose modulus has main
 variable~$v$. \typ{FFELT} are not lifted, nor are List elements: you may
 convert the latter to vectors first, or use \kbd{apply(lift,L)}. More
 generally, components for which such lifts are meaningless (e.g. character
 strings) are copied verbatim.
 \bprog
 ? lift(Mod(5,3))
 %1 = 2
 ? lift(3 + O(3^9))
 %2 = 3
 ? lift(Mod(x,x^2+1))
 %3 = x
 ? lift(Mod(x,x^2+1))
 %4 = x
 @eprog
 Lifts are performed recursively on an object components, but only
 by \emph{one level}: once a \typ{POLMOD} is lifted, the components of
 the result are \emph{not} lifted further.
 \bprog
 ? lift(x * Mod(1,3) + Mod(2,3))
 %4 = x + 2
 ? lift(x * Mod(y,y^2+1) + Mod(2,3))
 %5 = y*x + Mod(2, 3)   \\@com do you understand this one?
 ? lift(x * Mod(y,y^2+1) + Mod(2,3), 'x)
 %6 = Mod(y, y^2 + 1)*x + Mod(Mod(2, 3), y^2 + 1)
 ? lift(%, y)
 %7 = y*x + Mod(2, 3)
 @eprog\noindent To recursively lift all components not only by one level,
 but as long as possible, use \kbd{liftall}. To lift only \typ{INTMOD}s and
 \typ{PADIC}s components, use \tet{liftint}. To lift only \typ{POLMOD}s
 components, use \tet{liftpol}. Finally, \tet{centerlift} allows to lift
 \typ{INTMOD}s and \typ{PADIC}s using centered residues (lift of smallest
 absolute value).
Variant: Also available is \fun{GEN}{lift}{GEN x} corresponding to
 \kbd{lift0(x,-1)}.

Function: liftall
Class: basic
Section: conversions
C-Name: liftall
Prototype: G
Help: liftall(x): lifts every element of Z/nZ to Z, of Qp to Q, and of
 K[x]/(P) to K[x].
Description: 
 (pol):pol        liftall($1)
 (vec):vec        liftall($1)
 (gen):gen        liftall($1)
Doc: 
 recursively lift all components of $x$ from $\Z/n\Z$ to $\Z$,
 from $\Q_p$ to $\Q$ (as \tet{truncate}), and polmods to
 polynomials. \typ{FFELT} are not lifted, nor are List elements: you may
 convert the latter to vectors first, or use \kbd{apply(liftall,L)}. More
 generally, components for which such lifts are meaningless (e.g. character
 strings) are copied verbatim.
 \bprog
 ? liftall(x * (1 + O(3)) + Mod(2,3))
 %1 = x + 2
 ? liftall(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2))
 %2 = y*x + 2*z
 @eprog

Function: liftint
Class: basic
Section: conversions
C-Name: liftint
Prototype: G
Help: liftint(x): lifts every element of Z/nZ to Z, of Qp to Q, and of
 K[x]/(P) to K[x].
Description: 
 (pol):pol        liftint($1)
 (vec):vec        liftint($1)
 (gen):gen        liftint($1)
Doc: recursively lift all components of $x$ from $\Z/n\Z$ to $\Z$ and
 from $\Q_p$ to $\Q$ (as \tet{truncate}).
 \typ{FFELT} are not lifted, nor are List elements: you may
 convert the latter to vectors first, or use \kbd{apply(liftint,L)}. More
 generally, components for which such lifts are meaningless (e.g. character
 strings) are copied verbatim.
 \bprog
 ? liftint(x * (1 + O(3)) + Mod(2,3))
 %1 = x + 2
 ? liftint(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2))
 %2 = Mod(y, y^2 + 1)*x + Mod(Mod(2*z, z^2), y^2 + 1)
 @eprog

Function: liftpol
Class: basic
Section: conversions
C-Name: liftpol
Prototype: G
Help: liftpol(x): lifts every polmod component of x to polynomials.
Description: 
 (pol):pol        liftpol($1)
 (vec):vec        liftpol($1)
 (gen):gen        liftpol($1)
Doc: recursively lift all components of $x$ which are polmods to
 polynomials. \typ{FFELT} are not lifted, nor are List elements: you may
 convert the latter to vectors first, or use \kbd{apply(liftpol,L)}. More
 generally, components for which such lifts are meaningless (e.g. character
 strings) are copied verbatim.
 \bprog
 ? liftpol(x * (1 + O(3)) + Mod(2,3))
 %1 = (1 + O(3))*x + Mod(2, 3)
 ? liftpol(x * Mod(y,y^2+1) + Mod(2,3)*Mod(z,z^2))
 %2 = y*x + Mod(2, 3)*z
 @eprog

Function: limitnum
Class: basic
Section: sums
C-Name: limitnum0
Prototype: GD0,L,DGp
Help: limitnum(expr,{k = 20},{alpha=1}): numerical limit of sequence expr
 using Lagrange-Zagier extrapolation; k is a multiplier so that we extrapolate
 from expr(k*n). Assume u(n) ~ sum a_i n^(-alpha*i). flag=2, assuming that the
 asymptotic expansion is in powers of 1/n^2.
Doc: Lagrange-Zagier numerical extrapolation of \var{expr}, corresponding to a
 sequence
 $u_n$, either given by a closure \kbd{n->u(n)} or by a vector of values
 I.e., assuming that $u_n$ tends to a finite limit $\ell$, try to determine
 $\ell$. This routine is purely numerical and heuristic, thus may or may not
 work on your examples; $k$ is ignored if $u$ is given by a vector,
 and otherwise is a multiplier such that we extrapolate from $u(kn)$.
 
 Assume that $u_n$ has an asymptotic expansion in $n^{-\alpha}$ :
 $$u_n = \ell + \sum_{i\geq 1} a_i n^{-i\alpha}$$
 for some $a_i$.
 \bprog
 ? limitnum(n -> n*sin(1/n))
 %1 = 1.0000000000000000000000000000000000000
 
 ? limitnum(n -> (1+1/n)^n) - exp(1)
 %2 = 0.E-37
 
 ? limitnum(n -> 2^(4*n+1)*(n!)^4 / (2*n)! /(2*n+1)! )
 %3 = 3.1415926535897932384626433832795028842
 ? Pi
 %4 = 3.1415926535897932384626433832795028842
 @eprog\noindent
 If $u_n$ is given by a vector, it must be long enough for the extrapolation
 to make sense: at least $k$ times the current \kbd{realprecision}. The
 preferred format is thus a closure, although it becomes inconvenient
 when $u_n$ cannot be directly computed in time polynomial in $\log n$,
 for instance if it is defined as a sum or by induction. In that case,
 passing a vector of values is the best option. It usually pays off to
 interpolate $u(kn)$ for some $k > 1$:
 \bprog
 ? limitnum(vector(10,n,(1+1/n)^n))
  ***                 ^--------------------
  *** limitnum: non-existent component in limitnum: index < 20
 \\ at this accuracy, we must have at least 20 values
 ? limitnum(vector(20,n,(1+1/n)^n)) - exp(1)
 %5 = -2.05... E-20
 ? limitnum(vector(20,n, m=10*n;(1+1/m)^m)) - exp(1) \\ better accuracy
 %6 = 0.E-37
 
 ? v = vector(20); s = 0;
 ? for(i=1,#v, s += 1/i; v[i]= s - log(i));
 ? limitnum(v) - Euler
 %9 = -1.6... E-19
 
 ? V = vector(200); s = 0;
 ? for(i=1,#V, s += 1/i; V[i]= s);
 ? v = vector(#V \ 10, i, V[10*i] - log(10*i));
 ? limitnum(v) - Euler
 %13 = 6.43... E-29
 @eprog
 
 \synt{limitnum}{void *E, GEN (*u)(void *,GEN,long), long muli, GEN alpha, long prec}, where \kbd{u(E, n, prec)} must return $u(n)$ in precision \kbd{prec}.
 Also available is
 \fun{GEN}{limitnum0}{GEN u, long muli, GEN alpha, long prec}, where $u$
 must be a vector of sufficient length as above.

Function: lindep
Class: basic
Section: linear_algebra
C-Name: lindep0
Prototype: GD0,L,
Help: lindep(v,{flag=0}): integral linear dependencies between components of v.
 flag is optional, and can be 0: default, guess a suitable
 accuracy, or positive: accuracy to use for the computation, in decimal
 digits.
Doc: \sidx{linear dependence} finds a small non-trivial integral linear
 combination between components of $v$. If none can be found return an empty
 vector.
 
 If $v$ is a vector with real/complex entries we use a floating point
 (variable precision) LLL algorithm. If $\fl = 0$ the accuracy is chosen
 internally using a crude heuristic. If $\fl > 0$ the computation is done with
 an accuracy of $\fl$ decimal digits. To get meaningful results in the latter
 case, the parameter $\fl$ should be smaller than the number of correct
 decimal digits in the input.
 
 \bprog
 ? lindep([sqrt(2), sqrt(3), sqrt(2)+sqrt(3)])
 %1 = [-1, -1, 1]~
 @eprog
 
 If $v$ is $p$-adic, $\fl$ is ignored and the algorithm LLL-reduces a
 suitable (dual) lattice.
 \bprog
 ? lindep([1, 2 + 3 + 3^2 + 3^3 + 3^4 + O(3^5)])
 %2 = [1, -2]~
 @eprog
 
 If $v$ is a matrix (or a vector of column vectors, or a vector of row
 vectors), $\fl$ is ignored and the function returns a non trivial kernel
 vector if one exists, else an empty vector.
 \bprog
 ? lindep([1,2,3;4,5,6;7,8,9])
 %3 = [1, -2, 1]~
 ? lindep([[1,0], [2,0]])
 %4 = [2, -1]~
 ? lindep([[1,0], [0,1]])
 %5 = []~
 @eprog
 
 If $v$ contains polynomials or power series over some base field, finds a
 linear relation with coefficients in the field.
 \bprog
 ? lindep([x*y, x^2 + y, x^2*y + x*y^2, 1])
 %4 = [y, y, -1, -y^2]~
 @eprog\noindent For better control, it is preferable to use \typ{POL} rather
 than \typ{SER} in the input, otherwise one gets a linear combination which is
 $t$-adically small, but not necessarily $0$. Indeed, power series are first
 converted to the minimal absolute accuracy occurring among the entries of $v$
 (which can cause some coefficients to be ignored), then truncated to
 polynomials:
 \bprog
 ? v = [t^2+O(t^4), 1+O(t^2)]; L=lindep(v)
 %1 = [1, 0]~
 ? v*L
 %2 = t^2+O(t^4)  \\ small but not 0
 @eprog
Variant: Also available are \fun{GEN}{lindep}{GEN v} (real/complex entries,
 $\fl=0$), \fun{GEN}{lindep2}{GEN v, long flag} (real/complex entries)
 \fun{GEN}{padic_lindep}{GEN v} ($p$-adic entries) and
 \fun{GEN}{Xadic_lindep}{GEN v} (polynomial entries).
 Finally \fun{GEN}{deplin}{GEN v} returns a non-zero kernel vector for a
 \typ{MAT} input.

Function: listcreate
Class: basic
Section: programming/specific
C-Name: listcreate_gp
Prototype: D0,L,
Help: listcreate({n}): this function is obsolete, use List().
Description: 
 (?gen):list        mklist()
Doc: This function is obsolete, use \kbd{List}.
 
 Creates an empty list. This routine used to have a mandatory argument,
 which is now ignored (for backward compatibility).
 % \syn{NO}
Obsolete: 2007-08-10

Function: listinsert
Class: basic
Section: programming/specific
C-Name: listinsert
Prototype: WGL
Help: listinsert(L,x,n): insert x at index n in list L, shifting the
 remaining elements to the right.
Description: 
 (list, gen, small):gen        listinsert($1, $2, $3)
Doc: inserts the object $x$ at
 position $n$ in $L$ (which must be of type \typ{LIST}). This has
 complexity $O(\#L - n + 1)$: all the
 remaining elements of \var{list} (from position $n+1$ onwards) are shifted
 to the right.

Function: listkill
Class: basic
Section: programming/specific
C-Name: listkill
Prototype: vG
Help: listkill(L): obsolete, retained for backward compatibility.
Doc: obsolete, retained for backward compatibility. Just use \kbd{L = List()}
 instead of \kbd{listkill(L)}. In most cases, you won't even need that, e.g.
 local variables are automatically cleared when a user function returns.
Obsolete: 2007-08-10

Function: listpop
Class: basic
Section: programming/specific
C-Name: listpop0
Prototype: vWD0,L,
Help: listpop(list,{n}): removes n-th element from list. If n is
 omitted or greater than the current list length, removes last element.
Description: 
 (list, small):void     listpop($1, $2)
Doc: 
 removes the $n$-th element of the list
 \var{list} (which must be of type \typ{LIST}). If $n$ is omitted,
 or greater than the list current length, removes the last element.
 If the list is already empty, do nothing. This runs in time $O(\#L - n + 1)$.

Function: listput
Class: basic
Section: programming/specific
C-Name: listput0
Prototype: WGD0,L,
Help: listput(list,x,{n}): sets n-th element of list equal to x. If n is
 omitted or greater than the current list length, appends x.
Description: 
 (list, gen, small):gen        listput($1, $2, $3)
Doc: 
 sets the $n$-th element of the list
 \var{list} (which must be of type \typ{LIST}) equal to $x$. If $n$ is omitted,
 or greater than the list length, appends $x$. The function returns the
 inserted element.
 \bprog
 ? L = List();
 ? listput(L, 1)
 %2 = 1
 ? listput(L, 2)
 %3 = 2
 ? L
 %4 = List([1, 2])
 @eprog
 
 You may put an element into an occupied cell (not changing the
 list length), but it is easier to use the standard \kbd{list[n] = x}
 construct.
 \bprog
 ? listput(L, 3, 1) \\ insert at position 1
 %5 = 3
 ? L
 %6 = List([3, 2])
 ? L[2] = 4 \\ simpler
 %7 = List([3, 4])
 ? L[10] = 1  \\ can't insert beyond the end of the list
  ***   at top-level: L[10]=1
  ***                  ^------
  ***   non-existent component: index > 2
 ? listput(L, 1, 10) \\ but listput can
 %8 = 1
 ? L
 %9 = List([3, 2, 1])
 @eprog
 
 This function runs in time $O(\#L)$ in the worst case (when the list must
 be reallocated), but in time $O(1)$ on average: any number of successive
 \kbd{listput}s run in time $O(\#L)$, where $\#L$ denotes the list
 \emph{final} length.

Function: listsort
Class: basic
Section: programming/specific
C-Name: listsort
Prototype: vWD0,L,
Help: listsort(L,{flag=0}): sort the list L in place. If flag is non-zero,
 suppress all but one occurence of each element in list.
Doc: sorts the \typ{LIST} \var{list} in place, with respect to the (somewhat
 arbitrary) universal comparison function \tet{cmp}. In particular, the
 ordering is the same as for sets and \tet{setsearch} can be used on a sorted
 list.
 \bprog
 ? L = List([1,2,4,1,3,-1]); listsort(L); L
 %1 = List([-1, 1, 1, 2, 3, 4])
 ? setsearch(L, 4)
 %2 = 6
 ? setsearch(L, -2)
 %3 = 0
 @eprog\noindent This is faster than the \kbd{vecsort} command since the list
 is sorted in place: no copy is made. No value returned.
 
 If $\fl$ is non-zero, suppresses all repeated coefficients.

Function: lngamma
Class: basic
Section: transcendental
C-Name: glngamma
Prototype: Gp
Help: lngamma(x): logarithm of the gamma function of x.
Doc: principal branch of the logarithm of the gamma function of $x$. This
 function is analytic on the complex plane with non-positive integers
 removed, and can have much larger arguments than \kbd{gamma} itself.
 
 For $x$ a power series such that $x(0)$ is not a pole of \kbd{gamma},
 compute the Taylor expansion. (PARI only knows about regular power series
 and can't include logarithmic terms.)
 \bprog
 ? lngamma(1+x+O(x^2))
 %1 = -0.57721566490153286060651209008240243104*x + O(x^2)
 ? lngamma(x+O(x^2))
  ***   at top-level: lngamma(x+O(x^2))
  ***                 ^-----------------
  *** lngamma: domain error in lngamma: valuation != 0
 ? lngamma(-1+x+O(x^2))
  *** lngamma: Warning: normalizing a series with 0 leading term.
  ***   at top-level: lngamma(-1+x+O(x^2))
  ***                 ^--------------------
  *** lngamma: domain error in intformal: residue(series, pole) != 0
 @eprog

Function: local
Class: basic
Section: programming/specific
Help: local(x,...,z): declare x,...,z as (dynamically scoped) local variables.

Function: localbitprec
Class: basic
Section: programming/specific
C-Name: localbitprec
Prototype: vL
Help: localbitprec(p): set the real precision to p bits in the dynamic scope.
Doc: set the real precision to $p$ bits in the dynamic scope. All computations
 are performed as if \tet{realbitprecision} was $p$:
 transcendental constants (e.g.~\kbd{Pi}) and
 conversions from exact to floating point inexact data use $p$ bits, as well as
 iterative routines implicitly using a floating point
 accuracy as a termination criterion (e.g.~\tet{solve} or \tet{intnum}).
 But \kbd{realbitprecision} itself is unaffected
 and is ``unmasked'' when we exit the dynamic (\emph{not} lexical) scope.
 In effect, this is similar to
 \bprog
 my(bit = default(realbitprecision));
 default(realbitprecision,p);
 ...
 default(realbitprecision, bit);
 @eprog\noindent but is both less cumbersome, cleaner (no need to manipulate
 a global variable, which in fact never changes and is only temporarily masked)
 and more robust: if the above computation is interrupted or an exception
 occurs, \kbd{realbitprecision} will not be restored as intended.
 
 Such \kbd{localbitprec} statements can be nested, the innermost one taking
 precedence as expected. Beware that \kbd{localbitprec} follows the semantic of
 \tet{local}, not \tet{my}: a subroutine called from \kbd{localbitprec} scope
 uses the local accuracy:
 \bprog
 ? f()=bitprecision(1.0);
 ? f()
 %2 = 128
 ? localbitprec(1000); f()
 %3 = 1024
 @eprog\noindent Note that the bit precision of \emph{data} (\kbd{1.0} in the
 above example) increases by steps of 64 (32 on a 32-bit machine) so we get
 $1024$ instead of the expected $1000$; \kbd{localbitprec} bounds the
 relative error exactly as specified in functions that support that
 granularity (e.g.~\kbd{lfun}), and rounded to the next multiple of 64
 (resp.~32) everywhere else.
 
 \misctitle{Warning} Changing \kbd{realbitprecision} or \kbd{realprecision}
 in programs is deprecated in favor of \kbd{localbitprec} and
 \kbd{localprec}. Think about the \kbd{realprecision} and
 \kbd{realbitprecision} defaults as interactive commands for the \kbd{gp}
 interpreter, best left out of GP programs. Indeed, the above rules imply that
 mixing both constructs yields surprising results:
 
 \bprog
 ? \p38
 ? localprec(19); default(realprecision,1000);  Pi
 %1 = 3.141592653589793239
 ? \p
   realprecision = 1001 significant digits (1000 digits displayed)
 @eprog\noindent Indeed, \kbd{realprecision} itself is ignored within
 \kbd{localprec} scope, so \kbd{Pi} is computed to a low accuracy. And when
 we leave the \kbd{localprec} scope, \kbd{realprecision} only regains precedence,
 it is not ``restored'' to the original value.
 %\syn{NO}

Function: localprec
Class: basic
Section: programming/specific
C-Name: localprec
Prototype: vL
Help: localprec(p): set the real precision to p in the dynamic scope.
Doc: set the real precision to $p$ in the dynamic scope. All computations
 are performed as if \tet{realprecision} was $p$:
 transcendental constants (e.g.~\kbd{Pi}) and
 conversions from exact to floating point inexact data use $p$ decimal
 digits, as well as iterative routines implicitly using a floating point
 accuracy as a termination criterion (e.g.~\tet{solve} or \tet{intnum}).
 But \kbd{realprecision} itself is unaffected
 and is ``unmasked'' when we exit the dynamic (\emph{not} lexical) scope.
 In effect, this is similar to
 \bprog
 my(prec = default(realprecision));
 default(realprecision,p);
 ...
 default(realprecision, prec);
 @eprog\noindent but is both less cumbersome, cleaner (no need to manipulate
 a global variable, which in fact never changes and is only temporarily masked)
 and more robust: if the above computation is interrupted or an exception
 occurs, \kbd{realprecision} will not be restored as intended.
 
 Such \kbd{localprec} statements can be nested, the innermost one taking
 precedence as expected. Beware that \kbd{localprec} follows the semantic of
 \tet{local}, not \tet{my}: a subroutine called from \kbd{localprec} scope
 uses the local accuracy:
 \bprog
 ? f()=precision(1.);
 ? f()
 %2 = 38
 ? localprec(19); f()
 %3 = 19
 @eprog\noindent
 \misctitle{Warning} Changing \kbd{realprecision} itself in programs is
 now deprecated in favor of \kbd{localprec}. Think about the
 \kbd{realprecision} default as an interactive command for the \kbd{gp}
 interpreter, best left out of GP programs. Indeed, the above rules
 imply that mixing both constructs yields surprising results:
 \bprog
 ? \p38
 ? localprec(19); default(realprecision,100);  Pi
 %1 = 3.141592653589793239
 ? \p
     realprecision = 115 significant digits (100 digits displayed)
 @eprog\noindent Indeed, \kbd{realprecision} itself is ignored within
 \kbd{localprec} scope, so \kbd{Pi} is computed to a low accuracy. And when
 we leave \kbd{localprec} scope, \kbd{realprecision} only regains precedence,
 it is not ``restored'' to the original value.
 %\syn{NO}

Function: log
Class: basic
Section: transcendental
C-Name: glog
Prototype: Gp
Help: log(x): natural logarithm of x.
Description: 
 (gen):gen:prec        glog($1, $prec)
Doc: principal branch of the natural logarithm of
 $x \in \C^*$, i.e.~such that $\Im(\log(x))\in{} ]-\pi,\pi]$.
 The branch cut lies
 along the negative real axis, continuous with quadrant 2, i.e.~such that
 $\lim_{b\to 0^+} \log (a+bi) = \log a$ for $a \in\R^*$. The result is complex
 (with imaginary part equal to $\pi$) if $x\in \R$ and $x < 0$. In general,
 the algorithm uses the formula
 $$\log(x) \approx {\pi\over 2\text{agm}(1, 4/s)} - m \log 2, $$
 if $s = x 2^m$ is large enough. (The result is exact to $B$ bits provided
 $s > 2^{B/2}$.) At low accuracies, the series expansion near $1$ is used.
 
 $p$-adic arguments are also accepted for $x$, with the convention that
 $\log(p)=0$. Hence in particular $\exp(\log(x))/x$ is not in general equal to
 1 but to a $(p-1)$-th root of unity (or $\pm1$ if $p=2$) times a power of $p$.
Variant: For a \typ{PADIC} $x$, the function
 \fun{GEN}{Qp_log}{GEN x} is also available.

Function: logint
Class: basic
Section: number_theoretical
C-Name: logint0
Prototype: lGGD&
Help: logint(x,b,{&z}): return the largest integer e so that b^e <= x, where the
 parameters b > 1 and x > 0 are both integers. If the parameter z is present,
 set it to b^e.
Description: 
 (gen,2):small        expi($1)
 (gen,gen,&int):small logint0($1, $2, &$3)
Doc: Return the largest integer $e$ so that $b^e \leq x$, where the
 parameters $b > 1$ and $x > 0$ are both integers. If the parameter $z$ is
 present, set it to $b^e$.
 \bprog
 ? logint(1000, 2)
 %1 = 9
 ? 2^9
 %2 = 512
 ? logint(1000, 2, &z)
 %3 = 9
 ? z
 %4 = 512
 @eprog\noindent The number of digits used to write $b$ in base $x$ is
 \kbd{1 + logint(x,b)}:
 \bprog
 ? #digits(1000!, 10)
 %5 = 2568
 ? logint(1000!, 10)
 %6 = 2567
 @eprog\noindent This function may conveniently replace
 \bprog
   floor( log(x) / log(b) )
 @eprog\noindent which may not give the correct answer since PARI
 does not guarantee exact rounding.

Function: mapdelete
Class: basic
Section: programming/specific
C-Name: mapdelete
Prototype: vGG
Help: mapdelete(M,x): removes x from the domain of the map M.
Doc: removes $x$ from the domain of the map $M$.
 \bprog
 ? M = Map(["a",1; "b",3; "c",7]);
 ? mapdelete(M,"b");
 ? Mat(M)
 ["a" 1]
 
 ["c" 7]
 @eprog

Function: mapget
Class: basic
Section: programming/specific
C-Name: mapget
Prototype: GG
Help: mapget(M,x): returns the image of x by the map M.
Doc: Returns the image of $x$ by the map $M$.
 \bprog
 ? M=Map(["a",23;"b",43]);
 ? mapget(M,"a")
 %2 = 23
 ? mapget(M,"b")
 %3 = 43
 @eprog\noindent Raises an exception when the key $x$ is not present in $M$.
 \bprog
 ? mapget(M,"c")
   ***   at top-level: mapget(M,"c")
   ***                 ^-------------
   *** mapget: non-existent component in mapget: index not in map
 @eprog

Function: mapisdefined
Class: basic
Section: programming/specific
C-Name: mapisdefined
Prototype: iGGD&
Help: mapisdefined(M,x,{&z}): true (1) if x has an image by the map M,
 false (0) otherwise.
 If z is present, set it to the image of x, if it exists.
Doc: Returns true ($1$) if \kbd{x} has an image by the map $M$, false ($0$)
 otherwise. If \kbd{z} is present, set \kbd{z} to the image of $x$, if it exists.
 \bprog
 ? M1 = Map([1, 10; 2, 20]);
 ? mapisdefined(M1,3)
 %1 = 0
 ? mapisdefined(M1, 1, &z)
 %2 = 1
 ? z
 %3 = 10
 @eprog
 
 \bprog
 ? M2 = Map(); N = 19;
 ? for (a=0, N-1, mapput(M2, a^3%N, a));
 ? {for (a=0, N-1,
      if (mapisdefined(M2, a, &b),
        printf("%d is the cube of %d mod %d\n",a,b,N)));}
 0 is the cube of 0 mod 19
 1 is the cube of 11 mod 19
 7 is the cube of 9 mod 19
 8 is the cube of 14 mod 19
 11 is the cube of 17 mod 19
 12 is the cube of 15 mod 19
 18 is the cube of 18 mod 19
 @eprog

Function: mapput
Class: basic
Section: programming/specific
C-Name: mapput
Prototype: vWGG
Help: mapput(M,x,y): associates x to y in the map M.
Doc: Associates $x$ to $y$ in the map $M$. The value $y$ can be retrieved
 with \tet{mapget}.
 \bprog
 ? M = Map();
 ? mapput(M, "foo", 23);
 ? mapput(M, 7718, "bill");
 ? mapget(M, "foo")
 %4 = 23
 ? mapget(M, 7718)
 %5 = "bill"
 ? Vec(M)  \\ keys
 %6 = [7718, "foo"]
 ? Mat(M)
 %7 =
 [ 7718 "bill"]
 
 ["foo"     23]
 @eprog

Function: matadjoint
Class: basic
Section: linear_algebra
C-Name: matadjoint0
Prototype: GD0,L,
Help: matadjoint(M,{flag=0}): adjoint matrix of M using Leverrier-Faddeev's
 algorithm. If flag is 1, compute the characteristic polynomial independently
 first.
Doc: 
 \idx{adjoint matrix} of $M$, i.e.~a matrix $N$
 of cofactors of $M$, satisfying $M*N=\det(M)*\Id$. $M$ must be a
 (non-necessarily invertible) square matrix of dimension $n$.
 If $\fl$ is 0 or omitted, we try to use Leverrier-Faddeev's algorithm,
 which assumes that $n!$ invertible. If it fails or $\fl = 1$,
 compute $T = \kbd{charpoly}(M)$ independently first and return
 $(-1)^{n-1} (T(x)-T(0))/x$ evaluated at $M$.
 \bprog
 ? a = [1,2,3;3,4,5;6,7,8] * Mod(1,4);
 %2 =
 [Mod(1, 4) Mod(2, 4) Mod(3, 4)]
 
 [Mod(3, 4) Mod(0, 4) Mod(1, 4)]
 
 [Mod(2, 4) Mod(3, 4) Mod(0, 4)]
 @eprog\noindent
 Both algorithms use $O(n^4)$ operations in the base ring, and are usually
 slower than computing the characteristic polynomial or the inverse of $M$
 directly.
Variant: Also available are
 \fun{GEN}{adj}{GEN x} (\fl=0) and
 \fun{GEN}{adjsafe}{GEN x} (\fl=1).

Function: matalgtobasis
Class: basic
Section: number_fields
C-Name: matalgtobasis
Prototype: GG
Help: matalgtobasis(nf,x): nfalgtobasis applied to every element of the
 vector or matrix x.
Doc: This function is deprecated, use \kbd{apply}.
 
 $\var{nf}$ being a number field in \kbd{nfinit} format, and $x$ a
 (row or column) vector or matrix, apply \tet{nfalgtobasis} to each entry
 of $x$.
Obsolete: 2016-08-08

Function: matbasistoalg
Class: basic
Section: number_fields
C-Name: matbasistoalg
Prototype: GG
Help: matbasistoalg(nf,x): nfbasistoalg applied to every element of the
 matrix or vector x.
Doc: This function is deprecated, use \kbd{apply}.
 
 $\var{nf}$ being a number field in \kbd{nfinit} format, and $x$ a
 (row or column) vector or matrix, apply \tet{nfbasistoalg} to each entry
 of $x$.
Obsolete: 2016-08-08

Function: matcompanion
Class: basic
Section: linear_algebra
C-Name: matcompanion
Prototype: G
Help: matcompanion(x): companion matrix to polynomial x.
Doc: 
 the left companion matrix to the non-zero polynomial $x$.

Function: matconcat
Class: basic
Section: linear_algebra
C-Name: matconcat
Prototype: G
Help: matconcat(v): concatenate the entries of v and return the resulting
 matrix.
Doc: returns a \typ{MAT} built from the entries of $v$, which may
 be a \typ{VEC} (concatenate horizontally), a \typ{COL} (concatenate
 vertically), or a \typ{MAT} (concatenate vertically each column, and
 concatenate vertically the resulting matrices). The entries of $v$ are always
 considered as matrices: they can themselves be \typ{VEC} (seen as a row
 matrix), a \typ{COL} seen as a column matrix), a \typ{MAT}, or a scalar (seen
 as an $1 \times 1$ matrix).
 \bprog
 ? A=[1,2;3,4]; B=[5,6]~; C=[7,8]; D=9;
 ? matconcat([A, B]) \\ horizontal
 %1 =
 [1 2 5]
 
 [3 4 6]
 ? matconcat([A, C]~) \\ vertical
 %2 =
 [1 2]
 
 [3 4]
 
 [7 8]
 ? matconcat([A, B; C, D]) \\ block matrix
 %3 =
 [1 2 5]
 
 [3 4 6]
 
 [7 8 9]
 @eprog\noindent
 If the dimensions of the entries to concatenate do not match up, the above
 rules are extended as follows:
 
 \item each entry $v_{i,j}$ of $v$ has a natural length and height: $1 \times
 1$ for a scalar, $1 \times n$ for a \typ{VEC} of length $n$, $n \times 1$
 for a \typ{COL}, $m \times n$ for an $m\times n$ \typ{MAT}
 
 \item let $H_i$ be the maximum over $j$ of the lengths of the $v_{i,j}$,
 let $L_j$ be the maximum over $i$ of the heights of the $v_{i,j}$.
 The dimensions of the $(i,j)$-th block in the concatenated matrix are
 $H_i \times L_j$.
 
 \item a scalar $s = v_{i,j}$ is considered as $s$ times an identity matrix
 of the block dimension $\min (H_i,L_j)$
 
 \item blocks are extended by 0 columns on the right and 0 rows at the
 bottom, as needed.
 
 \bprog
 ? matconcat([1, [2,3]~, [4,5,6]~]) \\ horizontal
 %4 =
 [1 2 4]
 
 [0 3 5]
 
 [0 0 6]
 ? matconcat([1, [2,3], [4,5,6]]~) \\ vertical
 %5 =
 [1 0 0]
 
 [2 3 0]
 
 [4 5 6]
 ? matconcat([B, C; A, D]) \\ block matrix
 %6 =
 [5 0 7 8]
 
 [6 0 0 0]
 
 [1 2 9 0]
 
 [3 4 0 9]
 ? U=[1,2;3,4]; V=[1,2,3;4,5,6;7,8,9];
 ? matconcat(matdiagonal([U, V])) \\ block diagonal
 %7 =
 [1 2 0 0 0]
 
 [3 4 0 0 0]
 
 [0 0 1 2 3]
 
 [0 0 4 5 6]
 
 [0 0 7 8 9]
 @eprog

Function: matdet
Class: basic
Section: linear_algebra
C-Name: det0
Prototype: GD0,L,
Help: matdet(x,{flag=0}): determinant of the matrix x using an appropriate
 algorithm depending on the coefficients. If (optional) flag is set to 1, use
 classical Gaussian elimination (usually worse than the default).
Description: 
 (gen, ?0):gen           det($1)
 (gen, 1):gen            det2($1)
 (gen, #small):gen       $"incorrect flag in matdet"
 (gen, small):gen        det0($1, $2)
Doc: determinant of the square matrix $x$.
 
 If $\fl=0$, uses an appropriate algorithm depending on the coefficients:
 
 \item integer entries: modular method due to Dixon, Pernet and Stein.
 
 \item real or $p$-adic entries: classical Gaussian elimination using maximal
 pivot.
 
 \item intmod entries: classical Gaussian elimination using first non-zero
 pivot.
 
 \item other cases: Gauss-Bareiss.
 
 If $\fl=1$, uses classical Gaussian elimination with appropriate pivoting
 strategy (maximal pivot for real or $p$-adic coefficients). This is usually
 worse than the default.
Variant: Also available are \fun{GEN}{det}{GEN x} ($\fl=0$),
 \fun{GEN}{det2}{GEN x} ($\fl=1$) and \fun{GEN}{ZM_det}{GEN x} for integer
 entries.

Function: matdetint
Class: basic
Section: linear_algebra
C-Name: detint
Prototype: G
Help: matdetint(B): some multiple of the determinant of the lattice
 generated by the columns of B (0 if not of maximal rank). Useful with
 mathnfmod.
Doc: 
 Let $B$ be an $m\times n$ matrix with integer coefficients. The
 \emph{determinant} $D$ of the lattice generated by the columns of $B$ is
 the square root of $\det(B^T B)$ if $B$ has maximal rank $m$, and $0$
 otherwise.
 
 This function uses the Gauss-Bareiss algorithm to compute a positive
 \emph{multiple} of $D$. When $B$ is square, the function actually returns
 $D = |\det B|$.
 
 This function is useful in conjunction with \kbd{mathnfmod}, which needs to
 know such a multiple. If the rank is maximal and the matrix non-square,
 you can obtain $D$ exactly using
 \bprog
   matdet( mathnfmod(B, matdetint(B)) )
 @eprog\noindent
 Note that as soon as one of the dimensions gets large ($m$ or $n$ is larger
 than 20, say), it will often be much faster to use \kbd{mathnf(B, 1)} or
 \kbd{mathnf(B, 4)} directly.

Function: matdiagonal
Class: basic
Section: linear_algebra
C-Name: diagonal
Prototype: G
Help: matdiagonal(x): creates the diagonal matrix whose diagonal entries are
 the entries of the vector x.
Doc: $x$ being a vector, creates the diagonal matrix
 whose diagonal entries are those of $x$.
 \bprog
 ? matdiagonal([1,2,3]);
 %1 =
 [1 0 0]
 
 [0 2 0]
 
 [0 0 3]
 @eprog\noindent Block diagonal matrices are easily created using
 \tet{matconcat}:
 \bprog
 ? U=[1,2;3,4]; V=[1,2,3;4,5,6;7,8,9];
 ? matconcat(matdiagonal([U, V]))
 %1 =
 [1 2 0 0 0]
 
 [3 4 0 0 0]
 
 [0 0 1 2 3]
 
 [0 0 4 5 6]
 
 [0 0 7 8 9]
 @eprog

Function: mateigen
Class: basic
Section: linear_algebra
C-Name: mateigen
Prototype: GD0,L,p
Help: mateigen(x,{flag=0}): complex eigenvectors of the matrix x given as
 columns of a matrix H. If flag=1, return [L,H], where L contains the
 eigenvalues and H the corresponding eigenvectors.
Doc: returns the (complex) eigenvectors of $x$ as columns of a matrix.
 If $\fl=1$, return $[L,H]$, where $L$ contains the
 eigenvalues and $H$ the corresponding eigenvectors; multiple eigenvalues are
 repeated according to the eigenspace dimension (which may be less
 than the eigenvalue multiplicity in the characteristic polynomial).
 
 This function first computes the characteristic polynomial of $x$ and
 approximates its complex roots $(\lambda_i)$, then tries to compute the
 eigenspaces as kernels of the $x - \lambda_i$. This algorithm is
 ill-conditioned and is likely to miss kernel vectors if some roots of the
 characteristic polynomial are close, in particular if it has multiple roots.
 \bprog
 ? A = [13,2; 10,14]; mateigen(A)
 %1 =
 [-1/2 2/5]
 
 [   1   1]
 ? [L,H] = mateigen(A, 1);
 ? L
 %3 = [9, 18]
 ? H
 %4 =
 [-1/2 2/5]
 
 [   1   1]
 @eprog\noindent
 For symmetric matrices, use \tet{qfjacobi} instead; for Hermitian matrices,
 compute
 \bprog
  A = real(x);
  B = imag(x);
  y = matconcat([A, -B; B, A]);
 @eprog\noindent and apply \kbd{qfjacobi} to $y$.
Variant: Also available is \fun{GEN}{eigen}{GEN x, long prec} ($\fl = 0$)

Function: matfrobenius
Class: basic
Section: linear_algebra
C-Name: matfrobenius
Prototype: GD0,L,Dn
Help: matfrobenius(M,{flag},{v='x}): return the Frobenius form of the square
 matrix M. If flag is 1, return only the elementary divisors as a vector of
 polynomials in the variable v. If flag is 2, return a two-components vector
 [F,B] where F is the Frobenius form and B is the basis change so that
 M=B^-1*F*B.
Doc: returns the Frobenius form of
 the square matrix \kbd{M}. If $\fl=1$, returns only the elementary divisors as
 a vector of polynomials in the variable \kbd{v}.  If $\fl=2$, returns a
 two-components vector [F,B] where \kbd{F} is the Frobenius form and \kbd{B} is
 the basis change so that $M=B^{-1}FB$.

Function: mathess
Class: basic
Section: linear_algebra
C-Name: hess
Prototype: G
Help: mathess(x): Hessenberg form of x.
Doc: returns a matrix similar to the square matrix $x$, which is in upper Hessenberg
 form (zero entries below the first subdiagonal).

Function: mathilbert
Class: basic
Section: linear_algebra
C-Name: mathilbert
Prototype: L
Help: mathilbert(n): Hilbert matrix of order n.
Doc: $x$ being a \kbd{long}, creates the
 \idx{Hilbert matrix}of order $x$, i.e.~the matrix whose coefficient
 ($i$,$j$) is $1/ (i+j-1)$.

Function: mathnf
Class: basic
Section: linear_algebra
C-Name: mathnf0
Prototype: GD0,L,
Help: mathnf(M,{flag=0}): (upper triangular) Hermite normal form of M, basis
 for the lattice formed by the columns of M. flag is optional whose value
 range from 0 to 3 have a binary meaning. Bit 1: complete output, returns
 a 2-component vector [H,U] such that H is the HNF of M, and U is an
 invertible matrix such that MU=H. Bit 2: allow polynomial entries, otherwise
 assume that M is integral. These use a naive algorithm; larger values
 correspond to more involved algorithms and are restricted to integer
 matrices; flag = 4: returns [H,U] using LLL reduction along the way;
 flag = 5: return [H,U,P] where P is a permutation of row indices such that
 P applied to M U is H.
Doc: let $R$ be a Euclidean ring, equal to $\Z$ or to $K[X]$ for some field
 $K$. If $M$ is a (not necessarily square) matrix with entries in $R$, this
 routine finds the \emph{upper triangular} \idx{Hermite normal form} of $M$.
 If the rank of $M$ is equal to its number of rows, this is a square
 matrix. In general, the columns of the result form a basis of the $R$-module
 spanned by the columns of $M$.
 
 The values $0,1,2,3$ of $\fl$ have a binary meaning, analogous to the one
 in \tet{matsnf}; in this case, binary digits of $\fl$ mean:
 
 \item 1 (complete output): if set, outputs $[H,U]$, where $H$ is the Hermite
 normal form of $M$, and $U$ is a transformation matrix such that $MU=[0|H]$.
 The matrix $U$ belongs to $\text{GL}(R)$. When $M$ has a large kernel, the
 entries of $U$ are in general huge.
 
 \item 2 (generic input): \emph{Deprecated}. If set, assume that $R = K[X]$ is
 a polynomial ring; otherwise, assume that $R = \Z$. This flag is now useless
 since the routine always checks whether the matrix has integral entries.
 
 \noindent For these 4 values, we use a naive algorithm, which behaves well
 in small dimension only. Larger values correspond to different algorithms,
 are restricted to \emph{integer} matrices, and all output the unimodular
 matrix $U$. From now on all matrices have integral entries.
 
 \item $\fl=4$, returns $[H,U]$ as in ``complete output'' above, using a
 variant of \idx{LLL} reduction along the way. The matrix $U$ is provably
 small in the $L_2$ sense, and in general close to optimal; but the
 reduction is in general slow, although provably polynomial-time.
 
 If $\fl=5$, uses Batut's algorithm and output $[H,U,P]$, such that $H$ and
 $U$ are as before and $P$ is a permutation of the rows such that $P$ applied
 to $MU$ gives $H$. This is in general faster than $\fl=4$ but the matrix $U$
 is usually worse; it is heuristically smaller than with the default algorithm.
 
 When the matrix is dense and the dimension is large (bigger than 100, say),
 $\fl = 4$ will be fastest. When $M$ has maximal rank, then
 \bprog
   H = mathnfmod(M, matdetint(M))
 @eprog\noindent will be even faster. You can then recover $U$ as $M^{-1}H$.
 
 \bprog
 ? M = matrix(3,4,i,j,random([-5,5]))
 %1 =
 [ 0 2  3  0]
 
 [-5 3 -5 -5]
 
 [ 4 3 -5  4]
 
 ? [H,U] = mathnf(M, 1);
 ? U
 %3 =
 [-1 0 -1 0]
 
 [ 0 5  3 2]
 
 [ 0 3  1 1]
 
 [ 1 0  0 0]
 
 ? H
 %5 =
 [19 9 7]
 
 [ 0 9 1]
 
 [ 0 0 1]
 
 ? M*U
 %6 =
 [0 19 9 7]
 
 [0  0 9 1]
 
 [0  0 0 1]
 @eprog
 
 For convenience, $M$ is allowed to be a \typ{VEC}, which is then
 automatically converted to a \typ{MAT}, as per the \tet{Mat} function.
 For instance to solve the generalized extended gcd problem, one may use
 \bprog
 ? v = [116085838, 181081878, 314252913,10346840];
 ? [H,U] = mathnf(v, 1);
 ? U
 %2 =
 [ 103 -603    15  -88]
 
 [-146   13 -1208  352]
 
 [  58  220   678 -167]
 
 [-362 -144   381 -101]
 ? v*U
 %3 = [0, 0, 0, 1]
 @eprog\noindent This also allows to input a matrix as a \typ{VEC} of
 \typ{COL}s of the same length (which \kbd{Mat} would concatenate to
 the \typ{MAT} having those columns):
 \bprog
 ? v = [[1,0,4]~, [3,3,4]~, [0,-4,-5]~]; mathnf(v)
 %1 =
 [47 32 12]
 
 [ 0  1  0]
 
 [ 0  0  1]
 @eprog
Variant: Also available are \fun{GEN}{hnf}{GEN M} ($\fl=0$) and
 \fun{GEN}{hnfall}{GEN M} ($\fl=1$). To reduce \emph{huge} relation matrices
 (sparse with small entries, say dimension $400$ or more), you can use the
 pair \kbd{hnfspec} / \kbd{hnfadd}. Since this is quite technical and the
 calling interface may change, they are not documented yet. Look at the code
 in \kbd{basemath/hnf\_snf.c}.

Function: mathnfmod
Class: basic
Section: linear_algebra
C-Name: hnfmod
Prototype: GG
Help: mathnfmod(x,d): (upper triangular) Hermite normal form of x, basis for
 the lattice formed by the columns of x, where d is a multiple of the
 non-zero determinant of this lattice.
Doc: if $x$ is a (not necessarily square) matrix of
 maximal rank with integer entries, and $d$ is a multiple of the (non-zero)
 determinant of the lattice spanned by the columns of $x$, finds the
 \emph{upper triangular} \idx{Hermite normal form} of $x$.
 
 If the rank of $x$ is equal to its number of rows, the result is a square
 matrix. In general, the columns of the result form a basis of the lattice
 spanned by the columns of $x$. Even when $d$ is known, this is in general
 slower than \kbd{mathnf} but uses much less memory.

Function: mathnfmodid
Class: basic
Section: linear_algebra
C-Name: hnfmodid
Prototype: GG
Help: mathnfmodid(x,d): (upper triangular) Hermite normal form of x
 concatenated with matdiagonal(d).
Doc: outputs the (upper triangular)
 \idx{Hermite normal form} of $x$ concatenated with the diagonal
 matrix with diagonal $d$. Assumes that $x$ has integer entries.
 Variant: if $d$ is an integer instead of a vector, concatenate $d$ times the
 identity matrix.
 \bprog
 ? m=[0,7;-1,0;-1,-1]
 %1 =
 [ 0  7]
 
 [-1  0]
 
 [-1 -1]
 ? mathnfmodid(m, [6,2,2])
 %2 =
 [2 1 1]
 
 [0 1 0]
 
 [0 0 1]
 ? mathnfmodid(m, 10)
 %3 =
 [10 7 3]
 
 [ 0 1 0]
 
 [ 0 0 1]
 @eprog

Function: mathouseholder
Class: basic
Section: linear_algebra
C-Name: mathouseholder
Prototype: GG
Help: mathouseholder(Q,v): applies a sequence Q of Householder transforms
 to the vector or matrix v.
Doc: \sidx{Householder transform}applies a sequence $Q$ of Householder
 transforms, as returned by \kbd{matqr}$(M,1)$ to the vector or matrix $v$.

Function: matid
Class: basic
Section: linear_algebra
C-Name: matid
Prototype: L
Help: matid(n): identity matrix of order n.
Description: 
 (small):vec    matid($1)
Doc: creates the $n\times n$ identity matrix.

Function: matimage
Class: basic
Section: linear_algebra
C-Name: matimage0
Prototype: GD0,L,
Help: matimage(x,{flag=0}): basis of the image of the matrix x. flag is
 optional and can be set to 0 or 1, corresponding to two different algorithms.
Description: 
 (gen, ?0):vec           image($1)
 (gen, 1):vec            image2($1)
 (gen, #small)           $"incorrect flag in matimage"
 (gen, small):vec        matimage0($1, $2)
Doc: gives a basis for the image of the
 matrix $x$ as columns of a matrix. A priori the matrix can have entries of
 any type. If $\fl=0$, use standard Gauss pivot. If $\fl=1$, use
 \kbd{matsupplement} (much slower: keep the default flag!).
Variant: Also available is \fun{GEN}{image}{GEN x} ($\fl=0$).

Function: matimagecompl
Class: basic
Section: linear_algebra
C-Name: imagecompl
Prototype: G
Help: matimagecompl(x): vector of column indices not corresponding to the
 indices given by the function matimage.
Description: 
 (gen):vecsmall                imagecompl($1)
Doc: gives the vector of the column indices which
 are not extracted by the function \kbd{matimage}, as a permutation
 (\typ{VECSMALL}). Hence the number of
 components of \kbd{matimagecompl(x)} plus the number of columns of
 \kbd{matimage(x)} is equal to the number of columns of the matrix $x$.

Function: matindexrank
Class: basic
Section: linear_algebra
C-Name: indexrank
Prototype: G
Help: matindexrank(x): gives two extraction vectors (rows and columns) for
 the matrix x such that the extracted matrix is square of maximal rank.
Doc: $x$ being a matrix of rank $r$, returns a vector with two
 \typ{VECSMALL} components $y$ and $z$ of length $r$ giving a list of rows
 and columns respectively (starting from 1) such that the extracted matrix
 obtained from these two vectors using $\tet{vecextract}(x,y,z)$ is
 invertible.

Function: matintersect
Class: basic
Section: linear_algebra
C-Name: intersect
Prototype: GG
Help: matintersect(x,y): intersection of the vector spaces whose bases are
 the columns of x and y.
Doc: $x$ and $y$ being two matrices with the same
 number of rows each of whose columns are independent, finds a basis of the
 $\Q$-vector space equal to the intersection of the spaces spanned by the
 columns of $x$ and $y$ respectively. The faster function
 \tet{idealintersect} can be used to intersect fractional ideals (projective
 $\Z_K$ modules of rank $1$); the slower but much more general function
 \tet{nfhnf} can be used to intersect general $\Z_K$-modules.

Function: matinverseimage
Class: basic
Section: linear_algebra
C-Name: inverseimage
Prototype: GG
Help: matinverseimage(x,y): an element of the inverse image of the vector y
 by the matrix x if one exists, the empty vector otherwise.
Doc: given a matrix $x$ and
 a column vector or matrix $y$, returns a preimage $z$ of $y$ by $x$ if one
 exists (i.e such that $x z = y$), an empty vector or matrix otherwise. The
 complete inverse image is $z + \text{Ker} x$, where a basis of the kernel of
 $x$ may be obtained by \kbd{matker}.
 \bprog
 ? M = [1,2;2,4];
 ? matinverseimage(M, [1,2]~)
 %2 = [1, 0]~
 ? matinverseimage(M, [3,4]~)
 %3 = []~    \\@com no solution
 ? matinverseimage(M, [1,3,6;2,6,12])
 %4 =
 [1 3 6]
 
 [0 0 0]
 ? matinverseimage(M, [1,2;3,4])
 %5 = [;]    \\@com no solution
 ? K = matker(M)
 %6 =
 [-2]
 
 [1]
 @eprog

Function: matisdiagonal
Class: basic
Section: linear_algebra
C-Name: isdiagonal
Prototype: iG
Help: matisdiagonal(x): true(1) if x is a diagonal matrix, false(0)
 otherwise.
Doc: returns true (1) if $x$ is a diagonal matrix, false (0) if not.

Function: matker
Class: basic
Section: linear_algebra
C-Name: matker0
Prototype: GD0,L,
Help: matker(x,{flag=0}): basis of the kernel of the matrix x. flag is
 optional, and may be set to 0: default; non-zero: x is known to have
 integral entries.
Description: 
 (gen, ?0):vec           ker($1)
 (gen, 1):vec            keri($1)
 (gen, #small)           $"incorrect flag in matker"
 (gen, small):vec        matker0($1, $2)
Doc: gives a basis for the kernel of the matrix $x$ as columns of a matrix.
 The matrix can have entries of any type, provided they are compatible with
 the generic arithmetic operations ($+$, $\times$ and $/$).
 
 If $x$ is known to have integral entries, set $\fl=1$.
Variant: Also available are \fun{GEN}{ker}{GEN x} ($\fl=0$),
 \fun{GEN}{keri}{GEN x} ($\fl=1$).

Function: matkerint
Class: basic
Section: linear_algebra
C-Name: matkerint0
Prototype: GD0,L,
Help: matkerint(x,{flag=0}): LLL-reduced Z-basis of the kernel of the matrix
 x with integral entries. flag is deprecated, and may be set to 0 or 1
 for backward compatibility.
Doc: gives an \idx{LLL}-reduced $\Z$-basis
 for the lattice equal to the kernel of the matrix $x$ with rational entries.
 
 \fl is deprecated, kept for backward compatibility.
Variant: Use directly \fun{GEN}{kerint}{GEN x} if $x$ is known to have
 integer entries, and \tet{Q_primpart} first otherwise.

Function: matmuldiagonal
Class: basic
Section: linear_algebra
C-Name: matmuldiagonal
Prototype: GG
Help: matmuldiagonal(x,d): product of matrix x by diagonal matrix whose
 diagonal coefficients are those of the vector d, equivalent but faster than
 x*matdiagonal(d).
Doc: product of the matrix $x$ by the diagonal
 matrix whose diagonal entries are those of the vector $d$. Equivalent to,
 but much faster than $x*\kbd{matdiagonal}(d)$.

Function: matmultodiagonal
Class: basic
Section: linear_algebra
C-Name: matmultodiagonal
Prototype: GG
Help: matmultodiagonal(x,y): product of matrices x and y, knowing that the
 result will be a diagonal matrix. Much faster than general multiplication in
 that case.
Doc: product of the matrices $x$ and $y$ assuming that the result is a
 diagonal matrix. Much faster than $x*y$ in that case. The result is
 undefined if $x*y$ is not diagonal.

Function: matpascal
Class: basic
Section: linear_algebra
C-Name: matqpascal
Prototype: LDG
Help: matpascal(n,{q}): Pascal triangle of order n if q is omitted. q-Pascal
 triangle otherwise.
Doc: creates as a matrix the lower triangular
 \idx{Pascal triangle} of order $x+1$ (i.e.~with binomial coefficients
 up to $x$). If $q$ is given, compute the $q$-Pascal triangle (i.e.~using
 $q$-binomial coefficients).
Variant: Also available is \fun{GEN}{matpascal}{GEN x}.

Function: matqr
Class: basic
Section: linear_algebra
C-Name: matqr
Prototype: GD0,L,p
Help: matqr(M,{flag=0}): returns [Q,R], the QR-decomposition of the square
 invertible matrix M. If flag=1, Q is given as a sequence of Householder
 transforms (faster and stabler).
Doc: returns $[Q,R]$, the \idx{QR-decomposition} of the square invertible
 matrix $M$ with real entries: $Q$ is orthogonal and $R$ upper triangular. If
 $\fl=1$, the orthogonal matrix is returned as a sequence of Householder
 transforms: applying such a sequence is stabler and faster than
 multiplication by the corresponding $Q$ matrix.\sidx{Householder transform}
 More precisely, if
 \bprog
   [Q,R] = matqr(M);
   [q,r] = matqr(M, 1);
 @eprog\noindent then $r = R$ and \kbd{mathouseholder}$(q, M)$ is
 (close to) $R$; furthermore
 \bprog
   mathouseholder(q, matid(#M)) == Q~
 @eprog\noindent the inverse of $Q$. This function raises an error if the
 precision is too low or $x$ is singular.

Function: matrank
Class: basic
Section: linear_algebra
C-Name: rank
Prototype: lG
Help: matrank(x): rank of the matrix x.
Doc: rank of the matrix $x$.

Function: matrix
Class: basic
Section: linear_algebra
C-Name: matrice
Prototype: GGDVDVDE
Help: matrix(m,n,{X},{Y},{expr=0}): mXn matrix of expression expr, the row
 variable X going from 1 to m and the column variable Y going from 1 to n. By
 default, fill with 0s.
Doc: creation of the
 $m\times n$ matrix whose coefficients are given by the expression
 \var{expr}. There are two formal parameters in \var{expr}, the first one
 ($X$) corresponding to the rows, the second ($Y$) to the columns, and $X$
 goes from 1 to $m$, $Y$ goes from 1 to $n$. If one of the last 3 parameters
 is omitted, fill the matrix with zeroes.
 %\syn{NO}

Function: matrixqz
Class: basic
Section: linear_algebra
C-Name: matrixqz0
Prototype: GDG
Help: matrixqz(A,{p=0}): if p>=0, transforms the rational or integral mxn (m>=n)
 matrix A into an integral matrix with gcd of maximal determinants coprime to
 p. If p=-1, finds a basis of the intersection with Z^n of the lattice spanned
 by the columns of A. If p=-2, finds a basis of the intersection with Z^n of
 the Q-vector space spanned by the columns of A.
Doc: $A$ being an $m\times n$ matrix in $M_{m,n}(\Q)$, let
 $\text{Im}_\Q A$ (resp.~$\text{Im}_\Z A$) the $\Q$-vector space
 (resp.~the $\Z$-module) spanned by the columns of $A$. This function has
 varying behavior depending on the sign of $p$:
 
 If $p \geq 0$, $A$ is assumed to have maximal rank $n\leq m$. The function
 returns a matrix $B\in M_{m,n}(\Z)$, with $\text{Im}_\Q B = \text{Im}_\Q A$,
 such that the GCD of all its $n\times n$ minors is coprime to
 $p$; in particular, if $p = 0$ (default), this GCD is $1$.
 \bprog
 ? minors(x) = vector(#x[,1], i, matdet(x[^i,]));
 ? A = [3,1/7; 5,3/7; 7,5/7]; minors(A)
 %1 = [4/7, 8/7, 4/7]   \\ determinants of all 2x2 minors
 ? B = matrixqz(A)
 %2 =
 [3 1]
 
 [5 2]
 
 [7 3]
 ? minors(%)
 %3 = [1, 2, 1]   \\ B integral with coprime minors
 @eprog
 
 If $p=-1$, returns the HNF basis of the lattice $\Z^n \cap \text{Im}_\Z A$.
 
 If $p=-2$, returns the HNF basis of the lattice $\Z^n \cap \text{Im}_\Q A$.
 \bprog
 ? matrixqz(A,-1)
 %4 =
 [8 5]
 
 [4 3]
 
 [0 1]
 
 ? matrixqz(A,-2)
 %5 =
 [2 -1]
 
 [1 0]
 
 [0 1]
 @eprog

Function: matsize
Class: basic
Section: linear_algebra
C-Name: matsize
Prototype: G
Help: matsize(x): number of rows and columns of the vector/matrix x as a
 2-vector.
Doc: $x$ being a vector or matrix, returns a row vector
 with two components, the first being the number of rows (1 for a row vector),
 the second the number of columns (1 for a column vector).

Function: matsnf
Class: basic
Section: linear_algebra
C-Name: matsnf0
Prototype: GD0,L,
Help: matsnf(X,{flag=0}): Smith normal form (i.e. elementary divisors) of
 the matrix X, expressed as a vector d. Binary digits of flag mean 1: returns
 [u,v,d] where d=u*X*v, otherwise only the diagonal d is returned, 2: allow
 polynomial entries, otherwise assume X is integral, 4: removes all
 information corresponding to entries equal to 1 in d.
Doc: if $X$ is a (singular or non-singular) matrix outputs the vector of
 \idx{elementary divisors} of $X$, i.e.~the diagonal of the
 \idx{Smith normal form} of $X$, normalized so that $d_n \mid d_{n-1} \mid
 \ldots \mid d_1$.
 
 The binary digits of \fl\ mean:
 
 1 (complete output): if set, outputs $[U,V,D]$, where $U$ and $V$ are two
 unimodular matrices such that $UXV$ is the diagonal matrix $D$. Otherwise
 output only the diagonal of $D$. If $X$ is not a square matrix, then $D$
 will be a square diagonal matrix padded with zeros on the left or the top.
 
 2 (generic input): if set, allows polynomial entries, in which case the
 input matrix must be square. Otherwise, assume that $X$ has integer
 coefficients with arbitrary shape.
 
 4 (cleanup): if set, cleans up the output. This means that elementary
 divisors equal to $1$ will be deleted, i.e.~outputs a shortened vector $D'$
 instead of $D$. If complete output was required, returns $[U',V',D']$ so
 that $U'XV' = D'$ holds. If this flag is set, $X$ is allowed to be of the
 form `vector of elementary divisors' or $[U,V,D]$ as would normally be output with the cleanup flag
 unset.

Function: matsolve
Class: basic
Section: linear_algebra
C-Name: gauss
Prototype: GG
Help: matsolve(M,B): solution of MX=B (M matrix, B column vector).
Doc: $M$ being an invertible matrix and $B$ a column
 vector, finds the solution $X$ of $MX=B$, using Dixon $p$-adic lifting method
 if $M$ and $B$ are integral and Gaussian elimination otherwise. This
 has the same effect as, but is faster, than $M^{-1}*B$.
Variant: For integral input, the function
 \fun{GEN}{ZM_gauss}{GEN M,GEN B} is also available.

Function: matsolvemod
Class: basic
Section: linear_algebra
C-Name: matsolvemod0
Prototype: GGGD0,L,
Help: matsolvemod(M,D,B,{flag=0}): one solution of system of congruences
 MX=B mod D (M matrix, B and D column vectors). If (optional) flag is
 non-null return all solutions.
Doc: $M$ being any integral matrix,
 $D$ a column vector of non-negative integer moduli, and $B$ an integral
 column vector, gives a small integer solution to the system of congruences
 $\sum_i m_{i,j}x_j\equiv b_i\pmod{d_i}$ if one exists, otherwise returns
 zero. Shorthand notation: $B$ (resp.~$D$) can be given as a single integer,
 in which case all the $b_i$ (resp.~$d_i$) above are taken to be equal to $B$
 (resp.~$D$).
 \bprog
 ? M = [1,2;3,4];
 ? matsolvemod(M, [3,4]~, [1,2]~)
 %2 = [-2, 0]~
 ? matsolvemod(M, 3, 1) \\ M X = [1,1]~ over F_3
 %3 = [-1, 1]~
 ? matsolvemod(M, [3,0]~, [1,2]~) \\ x + 2y = 1 (mod 3), 3x + 4y = 2 (in Z)
 %4 = [6, -4]~
 @eprog
 If $\fl=1$, all solutions are returned in the form of a two-component row
 vector $[x,u]$, where $x$ is a small integer solution to the system of
 congruences and $u$ is a matrix whose columns give a basis of the homogeneous
 system (so that all solutions can be obtained by adding $x$ to any linear
 combination of columns of $u$). If no solution exists, returns zero.
Variant: Also available are \fun{GEN}{gaussmodulo}{GEN M, GEN D, GEN B}
 ($\fl=0$) and \fun{GEN}{gaussmodulo2}{GEN M, GEN D, GEN B} ($\fl=1$).

Function: matsupplement
Class: basic
Section: linear_algebra
C-Name: suppl
Prototype: G
Help: matsupplement(x): supplement the columns of the matrix x to an
 invertible matrix.
Doc: assuming that the columns of the matrix $x$
 are linearly independent (if they are not, an error message is issued), finds
 a square invertible matrix whose first columns are the columns of $x$,
 i.e.~supplement the columns of $x$ to a basis of the whole space.
 \bprog
 ? matsupplement([1;2])
 %1 =
 [1 0]
 
 [2 1]
 @eprog
 Raises an error if $x$ has 0 columns, since (due to a long standing design
 bug), the dimension of the ambient space (the number of rows) is unknown in
 this case:
 \bprog
 ? matsupplement(matrix(2,0))
   ***   at top-level: matsupplement(matrix
   ***                 ^--------------------
   *** matsupplement: sorry, suppl [empty matrix] is not yet implemented.
 @eprog

Function: mattranspose
Class: basic
Section: linear_algebra
C-Name: gtrans
Prototype: G
Help: mattranspose(x): x~ = transpose of x.
Doc: transpose of $x$ (also $x\til$).
 This has an effect only on vectors and matrices.

Function: max
Class: basic
Section: operators
C-Name: gmax
Prototype: GG
Help: max(x,y): maximum of x and y.
Description: 
 (small, small):small  maxss($1, $2)
 (small, int):int      gmaxsg($1, $2)
 (int, small):int      gmaxgs($1, $2)
 (int, int):int        gmax($1, $2)
 (small, mp):mp        gmaxsg($1, $2)
 (mp, small):mp        gmaxgs($1, $2)
 (mp, mp):mp           gmax($1, $2)
 (small, gen):gen      gmaxsg($1, $2)
 (gen, small):gen      gmaxgs($1, $2)
 (gen, gen):gen        gmax($1, $2)
Doc: creates the maximum of $x$ and $y$ when they can be compared.

Function: min
Class: basic
Section: operators
C-Name: gmin
Prototype: GG
Help: min(x,y): minimum of x and y.
Description: 
 (small, small):small  minss($1, $2)
 (small, int):int      gminsg($1, $2)
 (int, small):int      gmings($1, $2)
 (int, int):int        gmin($1, $2)
 (small, mp):mp        gminsg($1, $2)
 (mp, small):mp        gmings($1, $2)
 (mp, mp):mp           gmin($1, $2)
 (small, gen):gen      gminsg($1, $2)
 (gen, small):gen      gmings($1, $2)
 (gen, gen):gen        gmin($1, $2)
Doc: creates the minimum of $x$ and $y$ when they can be compared.

Function: minpoly
Class: basic
Section: linear_algebra
C-Name: minpoly
Prototype: GDn
Help: minpoly(A,{v='x}): minimal polynomial of the matrix or polmod A.
Doc: \idx{minimal polynomial}
 of $A$ with respect to the variable $v$., i.e. the monic polynomial $P$
 of minimal degree (in the variable $v$) such that $P(A) = 0$.

Function: modreverse
Class: basic
Section: number_fields
C-Name: modreverse
Prototype: G
Help: modreverse(z): reverse polmod of the polmod z, if it exists.
Doc: let $z = \kbd{Mod(A, T)}$ be a polmod, and $Q$ be its minimal
 polynomial, which must satisfy $\text{deg}(Q) = \text{deg}(T)$.
 Returns a ``reverse polmod'' \kbd{Mod(B, Q)}, which is a root of $T$.
 
 This is quite useful when one changes the generating element in algebraic
 extensions:
 \bprog
 ? u = Mod(x, x^3 - x -1); v = u^5;
 ? w = modreverse(v)
 %2 = Mod(x^2 - 4*x + 1, x^3 - 5*x^2 + 4*x - 1)
 @eprog\noindent
 which means that $x^3 - 5x^2 + 4x -1$ is another defining polynomial for the
 cubic field
 $$\Q(u) = \Q[x]/(x^3 - x - 1) = \Q[x]/(x^3 - 5x^2 + 4x - 1) = \Q(v),$$
 and that $u \to v^2 - 4v + 1$ gives an explicit isomorphism. From this, it is
 easy to convert elements between the $A(u)\in \Q(u)$ and $B(v)\in \Q(v)$
 representations:
 \bprog
 ? A = u^2 + 2*u + 3; subst(lift(A), 'x, w)
 %3 = Mod(x^2 - 3*x + 3, x^3 - 5*x^2 + 4*x - 1)
 ? B = v^2 + v + 1;   subst(lift(B), 'x, v)
 %4 = Mod(26*x^2 + 31*x + 26, x^3 - x - 1)
 @eprog
 If the minimal polynomial of $z$ has lower degree than expected, the routine
 fails
 \bprog
 ? u = Mod(-x^3 + 9*x, x^4 - 10*x^2 + 1)
 ? modreverse(u)
  *** modreverse: domain error in modreverse: deg(minpoly(z)) < 4
  ***   Break loop: type 'break' to go back to GP prompt
 break> Vec( dbg_err() ) \\ ask for more info
 ["e_DOMAIN", "modreverse", "deg(minpoly(z))", "<", 4,
   Mod(-x^3 + 9*x, x^4 - 10*x^2 + 1)]
 break> minpoly(u)
 x^2 - 8
 @eprog

Function: moebius
Class: basic
Section: number_theoretical
C-Name: moebius
Prototype: lG
Help: moebius(x): Moebius function of x.
Doc: \idx{Moebius} $\mu$-function of $|x|$. $x$ must be of type integer.

Function: msatkinlehner
Class: basic
Section: modular_symbols
C-Name: msatkinlehner
Prototype: GLDG
Help: msatkinlehner(M,Q,{H}): M being a full modular symbol space of level N,
 as given by msinit, let Q | N, (Q,N/Q) = 1, and let H be a subspace stable
 under the Atkin-Lehner involution w_Q. Return the matrix of w_Q
 acting on H (M if omitted).
Doc: Let $M$ be a full modular symbol space of level $N$,
 as given by \kbd{msinit}, let $Q \mid N$, $(Q,N/Q) = 1$,
 and let $H$ be a subspace stable under the Atkin-Lehner involution $w_Q$.
 Return the matrix of $w_Q$ acting on $H$ ($M$ if omitted).
 \bprog
 ? M = msinit(36,2); \\ M_2(Gamma_0(36))
 ? w = msatkinlehner(M,4); w^2 == 1
 %2 = 1
 ? #w   \\ involution acts on a 13-dimensional space
 %3 = 13
 ? M = msinit(36,2, -1); \\ M_2(Gamma_0(36))^-
 ? w = msatkinlehner(M,4); w^2 == 1
 %5 = 1
 ? #w
 %6 = 4
 @eprog

Function: mscuspidal
Class: basic
Section: modular_symbols
C-Name: mscuspidal
Prototype: GD0,L,
Help: mscuspidal(M, {flag=0}): M being a full modular symbol space, as given
 by msinit, return its cuspidal part S. If flag = 1, return [S,E] its
 decomposition into Eisenstein and cuspidal parts.
Doc: 
 $M$ being a full modular symbol space, as given by \kbd{msinit},
 return its cuspidal part $S$. If $\fl = 1$, return
 $[S,E]$ its decomposition into cuspidal and Eisenstein parts.
 
 A subspace is given by a structure allowing quick projection and
 restriction of linear operators; its first component is
 a matrix with integer coefficients whose columns form a $\Q$-basis of
 the subspace.
 \bprog
 ? M = msinit(2,8, 1); \\ M_8(Gamma_0(2))^+
 ? [S,E] = mscuspidal(M, 1);
 ? E[1]  \\ 2-dimensional
 %3 =
 [0 -10]
 
 [0 -15]
 
 [0  -3]
 
 [1   0]
 
 ? S[1]  \\ 1-dimensional
 %4 =
 [ 3]
 
 [30]
 
 [ 6]
 
 [-8]
 @eprog

Function: mseisenstein
Class: basic
Section: modular_symbols
C-Name: mseisenstein
Prototype: G
Help: mseisenstein(M): M being a full modular symbol space, as given by msinit,
 return its Eisenstein subspace.
Doc: 
 $M$ being a full modular symbol space, as given by \kbd{msinit},
 return its Eisenstein subspace.
 A subspace is given by a structure allowing quick projection and
 restriction of linear operators; its first component is
 a matrix with integer coefficients whose columns form a $\Q$-basis of
 the subspace.
 This is the same basis as given by the second component of
 \kbd{mscuspidal}$(M, 1)$.
 \bprog
 ? M = msinit(2,8, 1); \\ M_8(Gamma_0(2))^+
 ? E = mseisenstein(M);
 ? E[1]  \\ 2-dimensional
 %3 =
 [0 -10]
 
 [0 -15]
 
 [0  -3]
 
 [1   0]
 
 ? E == mscuspidal(M,1)[2]
 %4 = 1
 @eprog

Function: mseval
Class: basic
Section: modular_symbols
C-Name: mseval
Prototype: GGDG
Help: mseval(M,s,{p}): M being a full modular symbol space, as given by
 msinit, s being a modular symbol from M and p being a path between two
 elements in P^1(Q), return s(p).
Doc: Let $\Delta:=\text{Div}^0(\P^1 (\Q))$.
 Let $M$ be a full modular symbol space, as given by \kbd{msinit},
 let $s$ be a modular symbol from $M$, i.e. an element
 of $\Hom_G(\Delta, V)$, and let $p=[a,b] \in \Delta$ be a path between
 two elements in $\P^1(\Q)$, return $s(p)\in V$. The path extremities $a$ and
 $b$ may be given as \typ{INT}, \typ{FRAC} or $\kbd{oo} = (1:0)$.
 The symbol $s$ is either
 
 \item a \typ{COL} coding an element of a modular symbol subspace in terms of
 the fixed basis of $\Hom_G(\Delta,V)$ chosen in $M$; if $M$ was
 initialized with a non-zero \emph{sign} ($+$ or $-$), then either the
 basis for the full symbol space or the $\pm$-part can be used (the dimension
 being used to distinguish the two).
 
 \item a \typ{VEC} $(v_i)$ of elements of $V$, where the $v_i = s(g_i)$ give
 the image of the generators $g_i$ of $\Delta$, see \tet{mspathgens}.
 We assume that $s$ is a proper symbol, i.e.~that the $v_i$ satisfy
 the \kbd{mspathgens} relations.
 
 If $p$ is omitted, convert the symbol $s$ to the second form: a vector of
 the $s(g_i)$.
 \bprog
 ? M = msinit(2,8,1); \\ M_8(Gamma_0(2))^+
 ? g = mspathgens(M)[1]
 %2 = [[+oo, 0], [0, 1]]
 ? N = msnew(M)[1]; #N \\ Q-basis of new subspace, dimension 1
 %3 = 1
 ? s = N[,1]         \\ t_COL representation
 %4 = [-3, 6, -8]~
 ? S = mseval(M, s)   \\ t_VEC representation
 %5 = [64*x^6-272*x^4+136*x^2-8, 384*x^5+960*x^4+192*x^3-672*x^2-432*x-72]
 ? mseval(M,s, g[1])
 %6 = 64*x^6 - 272*x^4 + 136*x^2 - 8
 ? mseval(M,S, g[1])
 %7 = 64*x^6 - 272*x^4 + 136*x^2 - 8
 @eprog\noindent Note that the symbol should have values in
 $V = \Q[x,y]_{k-2}$, we return the de-homogenized values corresponding to $y
 = 1$ instead.

Function: msfromcusp
Class: basic
Section: modular_symbols
C-Name: msfromcusp
Prototype: GG
Help: msfromcusp(M, c): returns the modular symbol attached to the cusp
 c, where M is a modular symbol space of level N.
Doc: returns the modular symbol attached to the cusp
 $c$, where $M$ is a modular symbol space of level $N$, attached to
 $G = \Gamma_0(N)$. The cusp $c$ in  $\P^1(\Q)/G$
 can be given either as \kbd{oo} ($=(1:0)$), as a rational number $a/b$
 ($=(a:b)$). The attached symbol maps the path $[b] - [a] \in
 \text{Div}^0 (\P^1(\Q))$ to $E_c(b) - E_c(a)$, where $E_c(r)$ is
 $0$ when $r \neq c$ and $X^{k-2} \mid \gamma_r$ otherwise, where
 $\gamma_r \cdot r = (1:0)$. These symbol span the  Eisenstein subspace
 of $M$.
 \bprog
 ? M = msinit(2,8);  \\  M_8(Gamma_0(2))
 ? E =  mseisenstein(M);
 ? E[1] \\ two-dimensional
 %3 =
 [0 -10]
 
 [0 -15]
 
 [0  -3]
 
 [1   0]
 
 ? s = msfromcusp(M,oo)
 %4 = [0, 0, 0, 1]~
 ? mseval(M, s)
 %5 = [1, 0]
 ? s = msfromcusp(M,1)
 %6 = [-5/16, -15/32, -3/32, 0]~
 ? mseval(M,s)
 %7 = [-x^6, -6*x^5 - 15*x^4 - 20*x^3 - 15*x^2 - 6*x - 1]
 @eprog
 In case $M$ was initialized with a non-zero \emph{sign}, the symbol is given
 in terms of the fixed basis of the whole symbol space, not the $+$ or $-$
 part (to which it need not belong).
 \bprog
 ? M = msinit(2,8, 1);  \\  M_8(Gamma_0(2))^+
 ? E =  mseisenstein(M);
 ? E[1] \\ still two-dimensional, in a smaller space
 %3 =
 [ 0 -10]
 
 [ 0   3]
 
 [-1   0]
 
 ? s = msfromcusp(M,oo) \\ in terms of the basis for M_8(Gamma_0(2)) !
 %4 = [0, 0, 0, 1]~
 ? mseval(M, s) \\ same symbol as before
 %5 = [1, 0]
 @eprog

Function: msfromell
Class: basic
Section: modular_symbols
C-Name: msfromell
Prototype: GD0,L,
Help: msfromell(E, {sign=0}): return the [M, x], where M is msinit(N,2)
 and x is the modular symbol in M attached to the elliptic curve E/Q.
Doc: Let $E/\Q$ be an elliptic curve of conductor $N$. For $\varepsilon =
 \pm1$, we define the (cuspidal, new) modular symbol $x^\varepsilon$ in
 $H^1_c(X_0(N),\Q)^\varepsilon$  attached to
 $E$. For all primes $p$ not dividing $N$ we have
 $T_p(x^\varepsilon) =  a_p x^\varepsilon$, where $a_p = p+1-\#E(\F_p)$.
 
 Let $\Omega^+ = \kbd{E.omega[1]}$ be the real period of $E$
 (integration of the N\'eron differential $dx/(2y+a_1x+a3)$ on the connected
 component of $E(\R)$, i.e.~the generator of $H_1(E,\Z)^+$) normalized by
 $\Omega^+>0$. Let $i\Omega^-$ the integral on a generator of $H_1(E,\Z)^-$ with
 $\Omega^- \in \R_{>0}$. If $c_\infty$ is the number of connected
 components of $E(\R)$, $\Omega^-$ is equal to
 $(-2/c_\infty) \times \kbd{imag(E.omega[2])}$.
 The complex modular symbol is defined by
 $$F: \delta \to  2i\pi \int_{\delta} f(z) dz$$
 The modular symbols $x^\varepsilon$ are normalized so that
 $ F = x^+ \Omega^+ + x^- i\Omega^-$.
 In particular, we have
 $$ x^+([0]-[\infty]) = L(E,1) / \Omega^+,$$
 which defines $x^{\pm}$ unless $L(E,1)=0$.
 Furthermore, for all fundamental discriminants $D$ such
 that $\varepsilon \cdot D > 0$, we also have
 $$\sum_{0\leq a<|D|} (D|a) x^\varepsilon([a/|D|]-[\infty])
    = L(E,(D|.),1) / \Omega^{\varepsilon},$$
 where $(D|.)$ is the Kronecker symbol.
 The period $\Omega^-$ is also $2/c_\infty \times$ the real period
 of the twist $E^{(-4)} = \kbd{elltwist(E,-4)}$.
 
 This function returns the pair $[M, x]$, where $M$ is
 \kbd{msinit}$(N,2)$ and $x$ is $x^{\var{sign}}$ as above when $\var{sign}=
 \pm1$, and $x = [x^+,x^-]$ when \var{sign} is $0$.
 The modular symbols $x^\pm$ are given as a \typ{COL} (in terms
 of the fixed basis of $\Hom_G(\Delta,\Q)$ chosen in $M$).
 \bprog
 ? E=ellinit([0,-1,1,-10,-20]);  \\ X_0(11)
 ? [M,xp]= msfromell(E,1);
 ? xp
 %3 = [1/5, -1/2, -1/2]~
 ? [M,x]= msfromell(E);
 ? x    \\ both x^+ and x^-
 %5 = [[1/5, -1/2, -1/2]~, [0, 1/2, -1/2]~]
 ? p = 23; (mshecke(M,p) - ellap(E,p))*x[1]
 %6 = [0, 0, 0]~ \\ true at all primes, including p = 11; same for x[2]
 @eprog

Function: msfromhecke
Class: basic
Section: modular_symbols
C-Name: msfromhecke
Prototype: GGDG
Help: msfromhecke(M, v, {H}): given a msinit M and a vector v
 of pairs [p, P] (where p is prime and P is a polynomial with integer
 coefficients), return a basis of all modular symbols such that
 P(Tp) * s = 0. If H is present, it must be a Hecke-stable subspace
 and we restrict to s in H.
Doc: given a msinit $M$ and a vector $v$ of pairs $[p, P]$ (where $p$ is prime
 and $P$ is a polynomial with integer coefficients), return a basis of all
 modular symbols such that $P(T_p)(s) = 0$. If $H$ is present, it must
 be a Hecke-stable subspace and we restrict to $s \in H$. When $T_p$ has
 a rational eigenvalue and $P(x) = x-a_p$ has degree $1$, we also accept the
 integer $a_p$ instead of $P$.
 \bprog
 ? E = ellinit([0,-1,1,-10,-20]) \\11a1
 ? ellap(E,2)
 %2 = -2
 ? ellap(E,3)
 %3 = -1
 ? M = msinit(11,2);
 ? S = msfromhecke(M, [[2,-2],[3,-1]])
 %5 =
 [ 1  1]
 
 [-5  0]
 
 [ 0 -5]
 ? mshecke(M, 2, S)
 %6 =
 [-2  0]
 
 [ 0 -2]
 
 ? M = msinit(23,4);
 ? S = msfromhecke(M, [[5, x^4-14*x^3-244*x^2+4832*x-19904]]);
 ? factor( charpoly(mshecke(M,5,S)) )
 %9 =
 [x^4 - 14*x^3 - 244*x^2 + 4832*x - 19904 2]
 @eprog

Function: msgetlevel
Class: basic
Section: modular_symbols
C-Name: msgetlevel
Prototype: lG
Help: msgetlevel(M): M being a full modular symbol space, as given by msinit, return its level N.
Doc: $M$ being a full modular symbol space, as given by \kbd{msinit}, return
 its level $N$.

Function: msgetsign
Class: basic
Section: modular_symbols
C-Name: msgetsign
Prototype: lG
Help: msgetsign(M): M being a full modular symbol space, as given by msinit, return its sign.
Doc: $M$ being a full modular symbol space, as given by \kbd{msinit}, return
 its sign: $\pm1$ or 0 (unset).
 \bprog
 ? M = msinit(11,4, 1);
 ? msgetsign(M)
 %2 = 1
 ? M = msinit(11,4);
 ? msgetsign(M)
 %4 = 0
 @eprog

Function: msgetweight
Class: basic
Section: modular_symbols
C-Name: msgetweight
Prototype: lG
Help: msgetweight(M): M being a full modular symbol space, as given by msinit, return its weight k.
Doc: $M$ being a full modular symbol space, as given by \kbd{msinit}, return
 its weight $k$.
 \bprog
 ? M = msinit(11,4);
 ? msgetweight(M)
 %2 = 4
 @eprog

Function: mshecke
Class: basic
Section: modular_symbols
C-Name: mshecke
Prototype: GLDG
Help: mshecke(M,p,{H}): M being a full modular symbol space, as given by msinit,
 p being a prime number, and H being a Hecke-stable subspace (M if omitted),
 return the matrix of T_p acting on H (U_p if p divides the level).
Doc: $M$ being a full modular symbol space, as given by \kbd{msinit},
 $p$ being a prime number, and $H$ being a Hecke-stable subspace ($M$ if
 omitted) return the matrix of $T_p$ acting on $H$
 ($U_p$ if $p$ divides $N$). Result is undefined if $H$ is not stable
 by $T_p$ (resp.~$U_p$).
 \bprog
 ? M = msinit(11,2); \\ M_2(Gamma_0(11))
 ? T2 = mshecke(M,2)
 %2 =
 [3  0  0]
 
 [1 -2  0]
 
 [1  0 -2]
 ? M = msinit(11,2, 1); \\ M_2(Gamma_0(11))^+
 ? T2 = mshecke(M,2)
 %4 =
 [ 3  0]
 
 [-1 -2]
 
 ? N = msnew(M)[1] \\ Q-basis of new cuspidal subspace
 %5 =
 [-2]
 
 [-5]
 
 ? p = 1009; mshecke(M, p, N) \\ action of T_1009 on N
 %6 =
 [-10]
 ? ellap(ellinit("11a1"), p)
 %7 = -10
 @eprog

Function: msinit
Class: basic
Section: modular_symbols
C-Name: msinit
Prototype: GGD0,L,
Help: msinit(G, V, {sign=0}): given G a finite index subgroup of SL(2,Z)
 and a finite dimensional representation V of GL(2,Q), creates a space of
 modular symbols, the G-module Hom_G(Div^0(P^1 Q), V). This is canonically
 isomorphic to H^1_c(X(G), V), and allows to compute modular forms for G.
 If sign is present and non-zero, it must be +1 or -1 and we consider
 the subspace defined by Ker (Sigma - sign), where Sigma is induced by
 [-1,0;0,1]. Currently the only supported groups are the Gamma_0(N), coded by
 the integer N. The only supported representation is V_k = Q[X,Y]_{k-2}, coded
 by the integer k >= 2.
Doc: given $G$ a finite index subgroup of $\text{SL}(2,\Z)$
 and a finite dimensional representation $V$ of $\text{GL}(2,\Q)$, creates a
 space of modular symbols, the $G$-module $\Hom_G(\text{Div}^0(\P^1
 (\Q)), V)$. This is canonically isomorphic to $H^1_c(X(G), V)$, and allows to
 compute modular forms for $G$. If \emph{sign} is present and non-zero, it
 must be $\pm1$ and we consider the subspace defined by $\text{Ker} (\sigma -
 \var{sign})$, where $\sigma$ is induced by \kbd{[-1,0;0,1]}. Currently the
 only supported groups are the $\Gamma_0(N)$, coded by the integer $N > 1$.
 The only supported representation is $V_k = \Q[X,Y]_{k-2}$, coded by the
 integer $k \geq 2$.

Function: msissymbol
Class: basic
Section: modular_symbols
C-Name: msissymbol
Prototype: lGG
Help: msissymbol(M,s): M being a full modular symbol space, as given by msinit,
 check whether s is a modular symbol attached to M.
Doc: 
 $M$ being a full modular symbol space, as given by \kbd{msinit},
 check whether $s$ is a modular symbol attached to $M$.
 \bprog
 ? M = msinit(7,8, 1); \\ M_8(Gamma_0(7))^+
 ? N = msnew(M)[1];
 ? s = N[,1];
 ? msissymbol(M, s)
 %4 = 1
 ? S = mseval(M,s);
 ? msissymbol(M, S)
 %6 = 1
 ? [g,R] = mspathgens(M); g
 %7 = [[+oo, 0], [0, 1/2], [1/2, 1]]
 ? #R  \\ 3 relations among the generators g_i
 %8 = 3
 ? T = S; T[3]++; \\ randomly perturb S(g_3)
 ? msissymbol(M, T)
 %10 = 0  \\ no longer satisfies the relations
 @eprog

Function: msnew
Class: basic
Section: modular_symbols
C-Name: msnew
Prototype: G
Help: msnew(M): M being a full modular symbol space, as given by msinit,
 return its new cuspidal subspace.
Doc: 
 $M$ being a full modular symbol space, as given by \kbd{msinit},
 return the \emph{new} part of its cuspidal subspace. A subspace is given by
 a structure allowing quick projection and restriction of linear operators;
 its first component is a matrix with integer coefficients whose columns form
 a $\Q$-basis of the subspace.
 \bprog
 ? M = msinit(11,8, 1); \\ M_8(Gamma_0(11))^+
 ? N = msnew(M);
 ? #N[1]  \\ 6-dimensional
 %3 = 6
 @eprog

Function: msomseval
Class: basic
Section: modular_symbols
C-Name: msomseval
Prototype: GGG
Help: msomseval(Mp, PHI, path):
 return the vectors of moments of the p-adic distribution attached
 to the path 'path' via the overconvergent modular symbol 'PHI'.
Doc: return the vectors of moments of the $p$-adic distribution attached
 to the path \kbd{path} by the overconvergent modular symbol \kbd{PHI}.
 \bprog
 ? M = msinit(3,6,1);
 ? Mp= mspadicinit(M,5,10);
 ? phi = [5,-3,-1]~;
 ? msissymbol(M,phi)
 %4 = 1
 ? PHI = mstooms(Mp,phi);
 ? ME = msomseval(Mp,PHI,[oo, 0]);
 @eprog

Function: mspadicL
Class: basic
Section: modular_symbols
C-Name: mspadicL
Prototype: GDGD0,L,
Help: mspadicL(mu, {s = 0}, {r = 0}): given
 mu from mspadicmoments (p-adic distributions attached to an
 overconvergent symbol PHI) returns the value on a
 character of Z_p^* represented by s of the derivative of order r of the
 p-adic L-function attached to PHI.
Doc: Returns the value (or $r$-th derivative)
 on a character $\chi^s$ of $\Z_p^*$ of the $p$-adic $L$-function
 attached to \kbd{mu}.
 
 Let $\Phi$ be the $p$-adic distribution-valued overconvergent symbol
 attached to a modular symbol $\phi$ for $\Gamma_0(N)$ (eigenvector for
 $T_N(p)$ for the eigenvalue $a_p$). Then $L_p(\Phi,\chi^s)=L_p(\mu,s)$ is the
 $p$-adic $L$ function defined by
 $$L_p(\Phi,\chi^s)= \int_{\Z_p^*} \chi^s(z) d\mu(z)$$
 where $\mu$ is the distribution on $\Z_p^*$ defined by the restriction of
 $\Phi([\infty]-[0])$ to $\Z_p^*$. The $r$-th derivative is taken in
 direction $\langle \chi\rangle$:
 $$L_p^{(r)}(\Phi,\chi^s)= \int_{\Z_p^*} \chi^s(z) (\log z)^r d\mu(z).$$
 In the argument list,
 
 \item \kbd{mu} is as returned by \tet{mspadicmoments} (distributions
 attached to $\Phi$ by restriction to discs $a + p^\nu\Z_p$, $(a,p)=1$).
 
 \item $s=[s_1,s_2]$ with $s_1 \in \Z \subset \Z_p$ and $s_2 \bmod p-1$ or
 $s_2 \bmod 2$ for $p=2$, encoding the $p$-adic character $\chi^s := \langle
 \chi \rangle^{s_1} \tau^{s_2}$; here $\chi$ is the cyclotomic character from
 $\text{Gal}(\Q_p(\mu_{p^\infty})/\Q_p)$ to $\Z_p^*$, and $\tau$ is the
 Teichm\"uller character (for $p>2$ and the character of order 2 on
 $(\Z/4\Z)^*$ if $p=2$); for convenience, the character $[s,s]$ can also be
 represented by the integer $s$.
 
 When $a_p$ is a $p$-adic unit, $L_p$ takes its values in $\Q_p$.
 When $a_p$ is not a unit, it takes its values in the
 two-dimensional $\Q_p$-vector space $D_{cris}(M(\phi))$ where $M(\phi)$ is
 the ``motive'' attached to $\phi$, and we return the two $p$-adic components
 with respect to some fixed $\Q_p$-basis.
 \bprog
 ? M = msinit(3,6,1); phi=[5, -3, -1]~;
 ? msissymbol(M,phi)
 %2 = 1
 ? Mp = mspadicinit(M, 5, 4);
 ? mu = mspadicmoments(Mp, phi); \\ no twist
 \\ End of initializations
 
 ? mspadicL(mu,0) \\ L_p(chi^0)
 %5 = 5 + 2*5^2 + 2*5^3 + 2*5^4 + ...
 ? mspadicL(mu,1) \\ L_p(chi), zero for parity reasons
 %6 = [O(5^13)]~
 ? mspadicL(mu,2) \\ L_p(chi^2)
 %7 = 3 + 4*5 + 4*5^2 + 3*5^5 + ...
 ? mspadicL(mu,[0,2]) \\ L_p(tau^2)
 %8 = 3 + 5 + 2*5^2 + 2*5^3 + ...
 ? mspadicL(mu, [1,0]) \\ L_p(<chi>)
 %9 = 3*5 + 2*5^2 + 5^3 + 2*5^7 + 5^8 + 5^10 + 2*5^11 + O(5^13)
 ? mspadicL(mu,0,1) \\ L_p'(chi^0)
 %10 = 2*5 + 4*5^2 + 3*5^3 + ...
 ? mspadicL(mu, 2, 1) \\ L_p'(chi^2)
 %11 = 4*5 + 3*5^2 + 5^3 + 5^4 + ...
 @eprog
 
 Now several quadratic twists: \tet{mstooms} is indicated.
 \bprog
 ? PHI = mstooms(Mp,phi);
 ? mu = mspadicmoments(Mp, PHI, 12); \\ twist by 12
 ? mspadicL(mu)
 %14 = 5 + 5^2 + 5^3 + 2*5^4 + ...
 ? mu = mspadicmoments(Mp, PHI, 8); \\ twist by 8
 ? mspadicL(mu)
 %16 = 2 + 3*5 + 3*5^2 + 2*5^4 + ...
 ? mu = mspadicmoments(Mp, PHI, -3); \\ twist by -3 < 0
 ? mspadicL(mu)
 %18 = O(5^13) \\ always 0, phi is in the + part and D < 0
 @eprog
 
 One can locate interesting symbols of level $N$ and weight $k$ with
 \kbd{msnew} and \kbd{mssplit}. Note that instead of a symbol, one can
 input a 1-dimensional Hecke-subspace from \kbd{mssplit}: the function will
 automatically use the underlying basis vector.
 \bprog
 ? M=msinit(5,4,1); \\ M_4(Gamma_0(5))^+
 ? L = mssplit(M, msnew(M)); \\ list of irreducible Hecke-subspaces
 ? phi = L[1]; \\ one Galois orbit of newforms
 ? #phi[1] \\... this one is rational
 %4 = 1
 ? Mp = mspadicinit(M, 3, 4);
 ? mu = mspadicmoments(Mp, phi);
 ? mspadicL(mu)
 %7 = 1 + 3 + 3^3 + 3^4 + 2*3^5 + 3^6 + O(3^9)
 
 ? M = msinit(11,8, 1); \\ M_8(Gamma_0(11))^+
 ? Mp = mspadicinit(M, 3, 4);
 ? L = mssplit(M, msnew(M));
 ? phi = L[1]; #phi[1] \\ ... this one is two-dimensional
 %11 = 2
 ? mu = mspadicmoments(Mp, phi);
  ***   at top-level: mu=mspadicmoments(Mp,ph
  ***                    ^--------------------
  *** mspadicmoments: incorrect type in mstooms [dim_Q (eigenspace) > 1]
 @eprog

Function: mspadicinit
Class: basic
Section: modular_symbols
C-Name: mspadicinit
Prototype: GLLD-1,L,
Help: mspadicinit(M, p, n, {flag}): M being a full modular symbol space,
 as given by msinit and a prime p, initialize
 technical data needed to compute with overconvergent modular symbols
 (modulo p^n). If flag is unset, allow all symbols; if flag = 0, restrict
 to ordinary symbols; else initialize for symbols phi such that
 Tp(phi) = a_p * phi, with v_p(a_p) >= flag.
Doc: $M$ being a full modular symbol space, as given by \kbd{msinit}, and $p$
 a prime, initialize technical data needed to compute with overconvergent
 modular symbols, modulo $p^n$. If $\fl$ is unset, allow
 all symbols; else initialize only for a restricted range of symbols
 depending on $\fl$: if $\fl = 0$ restrict to ordinary symbols, else
 restrict to symbols $\phi$ such that $T_p(\phi) = a_p \phi$,
 with $v_p(a_p) \geq \fl$, which is faster as $\fl$ increases.
 (The fastest initialization is obtained for $\fl = 0$ where we only allow
 ordinary symbols.) For supersingular eigensymbols, such that $p\mid a_p$, we
 must further assume that $p$ does not divide the level.
 \bprog
 ? E = ellinit("11a1");
 ? [M,phi] = msfromell(E,1);
 ? ellap(E,3)
 %3 = -1
 ? Mp = mspadicinit(M, 3, 10, 0); \\ commit to ordinary symbols
 ? PHI = mstooms(Mp,phi);
 @eprog
 
 If we restrict the range of allowed symbols with \fl (for faster
 initialization), exceptions will occur if $v_p(a_p)$ violates this bound:
 \bprog
 ? E = ellinit("15a1");
 ? [M,phi] = msfromell(E,1);
 ? ellap(E,7)
 %3 = 0
 ? Mp = mspadicinit(M,7,5,0); \\ restrict to ordinary symbols
 ? PHI = mstooms(Mp,phi)
 ***   at top-level: PHI=mstooms(Mp,phi)
 ***                     ^---------------
 *** mstooms: incorrect type in mstooms [v_p(ap) > mspadicinit flag] (t_VEC).
 ? Mp = mspadicinit(M,7,5); \\ no restriction
 ? PHI = mstooms(Mp,phi);
 @eprog\noindent This function uses $O(N^2(n+k)^2p)$ memory, where $N$ is the
 level of $M$.

Function: mspadicmoments
Class: basic
Section: modular_symbols
C-Name: mspadicmoments
Prototype: GGD1,L,
Help: mspadicmoments(Mp, PHI, {D = 1}): given Mp from mspadicinit, an
 overconvergent eigensymbol PHI, and optionally a fundamental discriminant
 D coprime to p, return the moments of the p-1 distributions
 PHI^D([0]-[oo]) | (a + pZp), 0 < a < p. To be used by mspadicL and
 mspadicseries.
Doc: given \kbd{Mp} from \kbd{mspadicinit}, an overconvergent
 eigensymbol \kbd{PHI} from \kbd{mstooms} and a fundamental discriminant
 $D$ coprime to $p$,
 let $\kbd{PHI}^D$ denote the twisted symbol. This function computes
 the distribution $\mu = \kbd{PHI}^D([0] - \infty]) \mid \Z_p^*$ restricted
 to $\Z_p^*$. More precisely, it returns
 the moments of the $p-1$ distributions $\kbd{PHI}^D([0]-[\infty])
 \mid (a + p\Z_p)$, $0 < a < p$.
 We also allow \kbd{PHI} to be given as a classical
 symbol, which is then lifted to an overconvergent symbol by \kbd{mstooms};
 but this is wasteful if more than one twist is later needed.
 
 The returned data $\mu$ ($p$-adic distributions attached to \kbd{PHI})
 can then be used in \tet{mspadicL} or \tet{mspadicseries}.
 This precomputation allows to quickly compute derivatives of different
 orders or values at different characters.
 \bprog
 ? M = msinit(3,6, 1);
 ? phi = [5,-3,-1]~;
 ? msissymbol(M, phi)
 %3 = 1
 ? p = 5; mshecke(M,p) * phi  \\ eigenvector of T_5, a_5 = 6
 %4 = [30, -18, -6]~
 ? Mp = mspadicinit(M, p, 10, 0); \\ restrict to ordinary symbols, mod p^10
 ? PHI = mstooms(Mp, phi);
 ? mu = mspadicmoments(Mp, PHI);
 ? mspadicL(mu)
 %8 = 5 + 2*5^2 + 2*5^3 + ...
 ? mu = mspadicmoments(Mp, PHI, 12); \\ twist by 12
 ? mspadicL(mu)
 %10 = 5 + 5^2 + 5^3 + 2*5^4 + ...
 @eprog

Function: mspadicseries
Class: basic
Section: modular_symbols
C-Name: mspadicseries
Prototype: GD0,L,
Help: mspadicseries(mu, {i=0}): given mu from mspadicmoments,
 returns the attached p-adic series with maximal p-adic precision, depending
 on the precision of M (i-th Teichmueller component, if present).
Doc: Let $\Phi$ be the $p$-adic distribution-valued overconvergent symbol
 attached to a modular symbol $\phi$ for $\Gamma_0(N)$ (eigenvector for
 $T_N(p)$ for the eigenvalue $a_p$).
 If $\mu$ is the distribution on $\Z_p^*$ defined by the restriction of
 $\Phi([\infty]-[0])$ to $\Z_p^*$, let
 $$\hat{L}_p(\mu,\tau^{i})(x)
   = \int_{\Z_p^*} \tau^i(t) (1+x)^{\log_p(t)/\log_p(u)}d\mu(t)$$
 Here, $\tau$ is the Teichm\"uller character and $u$ is a specific
 multiplicative generator of $1+2p\Z_p$. (Namely $1+p$ if $p>2$ or $5$
 if $p=2$.) To explain
 the formula, let $G_\infty := \text{Gal}(\Q(\mu_{p^{\infty}})/ \Q)$,
 let $\chi:G_\infty\to \Z_p^*$ be the cyclotomic character (isomorphism)
 and $\gamma$ the element of $G_\infty$ such that $\chi(\gamma)=u$;
 then
 $\chi(\gamma)^{\log_p(t)/\log_p(u)}= \langle t \rangle$.
 
 The $p$-padic precision of individual terms is maximal given the precision of
 the overconvergent symbol $\mu$.
 \bprog
 ? [M,phi] = msfromell(ellinit("17a1"),1);
 ? Mp = mspadicinit(M, 5,7);
 ? mu = mspadicmoments(Mp, phi,1); \\ overconvergent symbol
 ? mspadicseries(mu)
 %4 = (4 + 3*5 + 4*5^2 + 2*5^3 + 2*5^4 + 5^5 + 4*5^6 + 3*5^7 + O(5^9)) \
  + (3 + 3*5 + 5^2 + 5^3 + 2*5^4 + 5^6 + O(5^7))*x \
  + (2 + 3*5 + 5^2 + 4*5^3 + 2*5^4 + O(5^5))*x^2 \
  + (3 + 4*5 + 4*5^2 + O(5^3))*x^3 \
  + (3 + O(5))*x^4 + O(x^5)
 @eprog\noindent
 An example with non-zero Teichm\"uller:
 \bprog
 ? [M,phi] = msfromell(ellinit("11a1"),1);
 ? Mp = mspadicinit(M, 3,10);
 ? mu = mspadicmoments(Mp, phi,1);
 ? mspadicseries(mu, 2)
 %4 = (2 + 3 + 3^2 + 2*3^3 + 2*3^5 + 3^6 + 3^7 + 3^10 + 3^11 + O(3^12)) \
  + (1 + 3 + 2*3^2 + 3^3 + 3^5 + 2*3^6 + 2*3^8 + O(3^9))*x \
  + (1 + 2*3 + 3^4 + 2*3^5 + O(3^6))*x^2 \
  + (3 + O(3^2))*x^3 + O(x^4)
 @eprog\noindent
 Supersingular example (not checked)
 \bprog
 ? E = ellinit("17a1"); ellap(E,3)
 %1 = 0
 ? [M,phi] = msfromell(E,1);
 ? Mp = mspadicinit(M, 3,7);
 ? mu = mspadicmoments(Mp, phi,1);
 ? mspadicseries(mu)
 %5 = [(2*3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + O(3^7)) \
  + (2 + 3^3 + O(3^5))*x \
  + (1 + 2*3 + O(3^2))*x^2 + O(x^3),\
  (3^-1 + 1 + 3 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + O(3^7)) \
  + (1 + 2*3 + 2*3^2 + 3^3 + 2*3^4 + O(3^5))*x \
  + (3^-2 + 3^-1 + O(3^2))*x^2 + O(3^-2)*x^3 + O(x^4)]
 @eprog\noindent
 Example with a twist:
 \bprog
 ? E = ellinit("11a1");
 ? [M,phi] = msfromell(E,1);
 ? Mp = mspadicinit(M, 3,10);
 ? mu = mspadicmoments(Mp, phi,5); \\ twist by 5
 ? L = mspadicseries(mu)
 %5 = (2*3^2 + 2*3^4 + 3^5 + 3^6 + 2*3^7 + 2*3^10 + O(3^12)) \
  + (2*3^2 + 2*3^6 + 3^7 + 3^8 + O(3^9))*x \
  + (3^3 + O(3^6))*x^2 + O(3^2)*x^3 + O(x^4)
 ? mspadicL(mu)
 %6 = [2*3^2 + 2*3^4 + 3^5 + 3^6 + 2*3^7 + 2*3^10 + O(3^12)]~
 ? ellpadicL(E,3,10,,5)
 %7 = 2 + 2*3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^6 + 2*3^7 + O(3^10)
 ? mspadicseries(mu,1) \\ must be 0
 %8 = O(3^12) + O(3^9)*x + O(3^6)*x^2 + O(3^2)*x^3 + O(x^4)
 @eprog

Function: mspathgens
Class: basic
Section: modular_symbols
C-Name: mspathgens
Prototype: G
Help: mspathgens(M): M being a full modular symbol space, as given by
 msinit, return a set of Z[G]-generators for Div^0(P^1 Q). The output
 is [g,R], where g is a minimal system of generators and R the vector of
 Z[G]-relations between the given generators.
Doc: Let $\Delta:=\text{Div}^0(\P^1(\Q))$.
 Let $M$ being a full modular symbol space, as given by \kbd{msinit},
 return a set of $\Z[G]$-generators for $\Delta$. The output
 is $[g,R]$, where $g$ is a minimal system of generators and $R$
 the vector of $\Z[G]$-relations between the given generators. A
 relation is coded by a vector of pairs $[a_i,i]$ with $a_i\in \Z[G]$
 and $i$ the index of a generator, so that $\sum_i a_i g[i] = 0$.
 
 An element $[v]-[u]$ in $\Delta$ is coded by the ``path'' $[u,v]$,
 where \kbd{oo} denotes the point at infinity $(1:0)$ on the projective
 line.
 An element of $\Z[G]$ is coded by a ``factorization matrix'': the first
 column contains distinct elements of $G$, and the second integers:
 \bprog
 ? M = msinit(11,8); \\ M_8(Gamma_0(11))
 ? [g,R] = mspathgens(M);
 ? g
 %3 = [[+oo, 0], [0, 1/3], [1/3, 1/2]] \\ 3 paths
 ? #R  \\ a single relation
 %4 = 1
 ? r = R[1]; #r \\ ...involving all 3 generators
 %5 = 3
 ? r[1]
 %6 = [[1, 1; [1, 1; 0, 1], -1], 1]
 ? r[2]
 %7 = [[1, 1; [7, -2; 11, -3], -1], 2]
 ? r[3]
 %8 = [[1, 1; [8, -3; 11, -4], -1], 3]
 @eprog\noindent
 The given relation is of the form $\sum_i (1-\gamma_i) g_i = 0$, with
 $\gamma_i\in \Gamma_0(11)$. There will always be a single relation involving
 all generators (corresponding to a round trip along all cusps), then
 relations involving a single generator (corresponding to $2$ and $3$-torsion
 elements in the group:
 \bprog
 ? M = msinit(2,8); \\ M_8(Gamma_0(2))
 ? [g,R] = mspathgens(M);
 ? g
 %3 = [[+oo, 0], [0, 1]]
 @eprog\noindent
 Note that the output depends only on the group $G$, not on the
 representation $V$.

Function: mspathlog
Class: basic
Section: modular_symbols
C-Name: mspathlog
Prototype: GG
Help: mspathlog(M,p): M being a full modular symbol space, as given by
 msinit and p being a path between two elements in P^1(Q), return (p_i)
 in Z[G] such that p = \sum p_i g_i, and the g_i are fixed Z[G]-generators
 for Div^0(P^1 Q), see mspathgens.
Doc: Let $\Delta:=\text{Div}^0(\P^1(\Q))$.
 Let $M$ being a full modular symbol space, as given by \kbd{msinit},
 encoding fixed $\Z[G]$-generators $(g_i)$ of $\Delta$ (see \tet{mspathgens}).
 A path $p=[a,b]$ between two elements in $\P^1(\Q)$ corresponds to
 $[b]-[a]\in \Delta$. The path extremities $a$ and $b$ may be given as
 \typ{INT}, \typ{FRAC} or $\kbd{oo} = (1:0)$.
 
 Returns $(p_i)$ in $\Z[G]$ such that $p = \sum_i p_i g_i$.
 \bprog
 ? M = msinit(2,8); \\ M_8(Gamma_0(2))
 ? [g,R] = mspathgens(M);
 ? g
 %3 = [[+oo, 0], [0, 1]]
 ? p = mspathlog(M, [1/2,2/3]);
 ? p[1]
 %5 =
 [[1, 0; 2, 1] 1]
 
 ? p[2]
 %6 =
 [[1, 0; 0, 1] 1]
 
 [[3, -1; 4, -1] 1]
 
 @eprog\noindent
 Note that the output depends only on the group $G$, not on the
 representation $V$.

Function: msqexpansion
Class: basic
Section: modular_symbols
C-Name: msqexpansion
Prototype: GGDP
Help: msqexpansion(M,projH,{B = seriesprecision}): M being a full modular
 symbol space, as given by msinit, and projH being a projector on a
 Hecke-simple subspace, return the Fourier coefficients [a_n, n <= B]
 of the corresponding normalized newform. If B omitted, use seriesprecision.
Doc: 
 $M$ being a full modular symbol space, as given by \kbd{msinit},
 and \var{projH} being a projector on a Hecke-simple subspace (as given
 by \tet{mssplit}), return the Fourier coefficients $a_n$, $n\leq B$ of the
 corresponding normalized newform. If $B$ is omitted, use
 \kbd{seriesprecision}.
 
 This function uses a naive $O(B^2 d^3)$
 algorithm, where $d = O(kN)$ is the dimension of $M_k(\Gamma_0(N))$.
 \bprog
 ? M = msinit(11,2, 1); \\ M_2(Gamma_0(11))^+
 ? L = mssplit(M, msnew(M));
 ? msqexpansion(M,L[1], 20)
 %3 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2]
 ? ellan(ellinit("11a1"), 20)
 %4 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2]
 @eprog\noindent The shortcut \kbd{msqexpansion(M, s, B)} is available for
 a symbol $s$, provided it is a Hecke eigenvector:
 \bprog
 ? E = ellinit("11a1");
 ? [M,s]=msfromell(E);
 ? msqexpansion(M,s,10)
 %3 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
 ? ellan(E, 10)
 %4 = [1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
 @eprog

Function: mssplit
Class: basic
Section: modular_symbols
C-Name: mssplit
Prototype: GGD0,L,
Help: mssplit(M,H,{dimlim}): M being a full modular symbol space, as given by
 msinit, and H being a subspace, split H into Hecke-simple subspaces.
 If dimlim is present and positive, restrict to dim <= dimlim.
Doc: 
 Let $M$ denote a full modular symbol space, as given by \kbd{msinit}$(N,k,1)$
 or $\kbd{msinit}(N,k,-1)$ and let $H$ be a Hecke-stable subspace of
 \kbd{msnew}$(M)$. This function split $H$ into Hecke-simple subspaces. If
 \kbd{dimlim} is present and positive, restrict to subspaces of dimension
 $\leq \kbd{dimlim}$. A subspace is given by a structure allowing quick
 projection and restriction of linear operators; its first component is a
 matrix with integer coefficients whose columns form a $\Q$-basis of the
 subspace.
 
 \bprog
 ? M = msinit(11,8, 1); \\ M_8(Gamma_0(11))^+
 ? L = mssplit(M, msnew(M));
 ? #L
 %3 = 2
 ? f = msqexpansion(M,L[1],5); f[1].mod
 %4 = x^2 + 8*x - 44
 ? lift(f)
 %5 = [1, x, -6*x - 27, -8*x - 84, 20*x - 155]
 ? g = msqexpansion(M,L[2],5); g[1].mod
 %6 = x^4 - 558*x^2 + 140*x + 51744
 @eprog\noindent To a Hecke-simple subspace corresponds an orbit of
 (normalized) newforms, defined over a number field. In the above example,
 we printed the polynomials defining the said fields, as well as the first
 5 Fourier coefficients (at the infinite cusp) of one such form.

Function: msstar
Class: basic
Section: modular_symbols
C-Name: msstar
Prototype: GDG
Help: msstar(M,{H}): M being a full modular symbol space,
 as given by msinit, return the matrix of the * involution, induced by
 complex conjugation, acting on the (stable) subspace H (M if omitted).
Doc: $M$ being a full modular symbol space, as given by \kbd{msinit},
 return the matrix of the \kbd{*} involution, induced by complex conjugation,
 acting on the (stable) subspace $H$ ($M$ if omitted).
 \bprog
 ? M = msinit(11,2); \\ M_2(Gamma_0(11))
 ? w = msstar(M);
 ? w^2 == 1
 %3 = 1
 @eprog

Function: mstooms
Class: basic
Section: modular_symbols
C-Name: mstooms
Prototype: GG
Help: mstooms(Mp, phi): given Mp from mspadicinit, lift the
 (classical) eigen symbol phi to a distribution-valued overconvergent symbol
 in the sense of Pollack and Stevens.
 The resulting overconvergent eigensymbol can then be used in
 mspadicmoments, then mspadicL or mspadicseries.
Doc: given \kbd{Mp} from \kbd{mspadicinit}, lift the (classical) eigen symbol
 \kbd{phi} to a $p$-adic distribution-valued overconvergent symbol in the
 sense of Pollack and Stevens. More precisely, let $\phi$ belong to the space
 $W$ of modular symbols of level $N$, $v_p(N) \leq 1$, and weight $k$ which is
 an eigenvector for the Hecke operator $T_N(p)$ for a non-zero eigenvalue
 $a_p$ and let $N_0 = \text{lcm}(N,p)$.
 
 Under the action of $T_{N_0}(p)$, $\phi$ generates a subspace $W_\phi$ of
 dimension $1$ (if $p\mid N$) or $2$ (if $p$ does not divide $N$) in the
 space of modular symbols of level $N_0$.
 
 Let $V_p=[p,0;0,1]$ and $C_p=[a_p,p^{k-1};-1,0]$.
 When $p$ does not divide $N$ and $a_p$ is divisible by $p$, \kbd{mstooms}
 returns the lift $\Phi$ of $(\phi,\phi|_k V_p)$ such that
  $$T_{N_0}(p) \Phi = C_p \Phi$$
 
 When $p$ does not divide $N$ and $a_p$ is not divisible by $p$, \kbd{mstooms}
 returns the lift $\Phi$ of $\phi - \alpha^{-1} \phi|_k V_p$
 which is an eigenvector of $T_{N_0}(p)$ for the unit eigenvalue
 where $\alpha^2 - a_p \alpha + p^{k-1}=0$.
 
 The resulting overconvergent eigensymbol can then be used in
 \tet{mspadicmoments}, then \tet{mspadicL} or \tet{mspadicseries}.
 \bprog
 ? M = msinit(3,6, 1); p = 5;
 ? Tp = mshecke(M, p); factor(charpoly(Tp))
 %2 =
 [x - 3126 2]
 
 [   x - 6 1]
 ? phi = matker(Tp - 6)[,1] \\ generator of p-Eigenspace, a_p = 6
 %3 = [5, -3, -1]~
 ? Mp = mspadicinit(M, p, 10, 0); \\ restrict to ordinary symbols, mod p^10
 ? PHI = mstooms(Mp, phi);
 ? mu = mspadicmoments(Mp, PHI);
 ? mspadicL(mu)
 %7 = 5 + 2*5^2 + 2*5^3 + ...
 @eprog
 A non ordinary symbol.
 \bprog
 ? M = msinit(4,6,1); p = 3;
 ? Tp = mshecke(M, p); factor(charpoly(Tp))
 %2 =
 [x - 244 3]
 
 [ x + 12 1]
 ? phi = matker(Tp + 12)[,1] \\ a_p = -12 is divisible by p = 3
 %3 = [-1/32, -1/4, -1/32, 1]~
 ? msissymbol(M,phi)
 %4 = 1
 ? Mp = mspadicinit(M,3,5,0);
 ? PHI = mstooms(Mp,phi);
  ***   at top-level: PHI=mstooms(Mp,phi)
  ***                     ^---------------
  *** mstooms: incorrect type in mstooms [v_p(ap) > mspadicinit flag] (t_VEC).
 ? Mp = mspadicinit(M,3,5,1);
 ? PHI = mstooms(Mp,phi);
 @eprog

Function: my
Class: basic
Section: programming/specific
Help: my(x,...,z): declare x,...,z as lexically-scoped local variables.

Function: newtonpoly
Class: basic
Section: number_fields
C-Name: newtonpoly
Prototype: GG
Help: newtonpoly(x,p): Newton polygon of polynomial x with respect to the
 prime p.
Doc: gives the vector of the slopes of the Newton
 polygon of the polynomial $x$ with respect to the prime number $p$. The $n$
 components of the vector are in decreasing order, where $n$ is equal to the
 degree of $x$. Vertical slopes occur iff the constant coefficient of $x$ is
 zero and are denoted by \kbd{+oo}.

Function: next
Class: basic
Section: programming/control
C-Name: next0
Prototype: D1,L,
Help: next({n=1}): interrupt execution of current instruction sequence, and
 start another iteration from the n-th innermost enclosing loops.
Doc: interrupts execution of current $seq$,
 resume the next iteration of the innermost enclosing loop, within the
 current function call (or top level loop). If $n$ is specified, resume at
 the $n$-th enclosing loop. If $n$ is bigger than the number of enclosing
 loops, all enclosing loops are exited.

Function: nextprime
Class: basic
Section: number_theoretical
C-Name: nextprime
Prototype: G
Help: nextprime(x): smallest pseudoprime >= x.
Description: 
 (gen):int        nextprime($1)
Doc: finds the smallest pseudoprime (see
 \tet{ispseudoprime}) greater than or equal to $x$. $x$ can be of any real
 type. Note that if $x$ is a pseudoprime, this function returns $x$ and not
 the smallest pseudoprime strictly larger than $x$. To rigorously prove that
 the result is prime, use \kbd{isprime}.

Function: nfalgtobasis
Class: basic
Section: number_fields
C-Name: algtobasis
Prototype: GG
Help: nfalgtobasis(nf,x): transforms the algebraic number x into a column
 vector on the integral basis nf.zk.
Doc: Given an algebraic number $x$ in the number field $\var{nf}$,
 transforms it to a column vector on the integral basis \kbd{\var{nf}.zk}.
 \bprog
 ? nf = nfinit(y^2 + 4);
 ? nf.zk
 %2 = [1, 1/2*y]
 ? nfalgtobasis(nf, [1,1]~)
 %3 = [1, 1]~
 ? nfalgtobasis(nf, y)
 %4 = [0, 2]~
 ? nfalgtobasis(nf, Mod(y, y^2+4))
 %5 = [0, 2]~
 @eprog
 This is the inverse function of \kbd{nfbasistoalg}.

Function: nfbasis
Class: basic
Section: number_fields
C-Name: nfbasis_gp
Prototype: G
Help: nfbasis(T): integral basis of the field Q[a], where a is
 a root of the polynomial T, using the round 4 algorithm. An argument
 [T,listP] is possible, where listP is a list of primes (to get an
 order which is maximal at certain primes only) or a prime bound.
Doc: 
 Let $T(X)$ be an irreducible polynomial with integral coefficients. This
 function returns an \idx{integral basis} of the number field defined by $T$,
 that is a $\Z$-basis of its maximal order. The basis elements are given as
 elements in $\Q[X]/(T)$:
 \bprog
 ? nfbasis(x^2 + 1)
 %1 = [1, x]
 @eprog
 This function uses a modified version of the \idx{round 4} algorithm,
 due to David \idx{Ford}, Sebastian \idx{Pauli} and Xavier \idx{Roblot}.
 
 \misctitle{Local basis, orders maximal at certain primes}
 
 Obtaining the maximal order is hard: it requires factoring the discriminant
 $D$ of $T$. Obtaining an order which is maximal at a finite explicit set of
 primes is easy, but it may then be a strict suborder of the maximal order. To
 specify that we are interested in a given set of places only, we can replace
 the argument $T$ by an argument $[T,\var{listP}]$, where \var{listP} encodes
 the primes we are interested in: it must be a factorization matrix, a vector
 of integers or a single integer.
 
 \item Vector: we assume that it contains distinct \emph{prime} numbers.
 
 \item Matrix: we assume that it is a two-column matrix of a
 (partial) factorization of $D$; namely the first column contains
 distinct \emph{primes} and the second one the valuation of $D$ at each of
 these primes.
 
 \item Integer $B$: this is replaced by the vector of primes up to $B$. Note
 that the function will use at least $O(B)$ time: a small value, about
 $10^5$, should be enough for most applications. Values larger than $2^{32}$
 are not supported.
 
 In all these cases, the primes may or may not divide the discriminant $D$
 of $T$. The function then returns a $\Z$-basis of an order whose index is
 not divisible by any of these prime numbers. The result is actually a global
 integral basis if all prime divisors of the \emph{field} discriminant are
 included! Note that \kbd{nfinit} has built-in support for such
 a check:
 \bprog
 ? K = nfinit([T, listP]);
 ? nfcertify(K)   \\ we computed an actual maximal order
 %2 = [];
 @eprog\noindent The first line initializes a number field structure
 incorporating \kbd{nfbasis([T, listP]} in place of a proven integral basis.
 The second line certifies that the resulting structure is correct. This
 allows to create an \kbd{nf} structure attached to the number field $K =
 \Q[X]/(T)$, when the discriminant of $T$ cannot be factored completely,
 whereas the prime divisors of $\disc K$ are known.
 
 Of course, if \var{listP} contains a single prime number $p$,
 the function returns a local integral basis for $\Z_p[X]/(T)$:
 \bprog
 ? nfbasis(x^2+x-1001)
 %1 = [1, 1/3*x - 1/3]
 ? nfbasis( [x^2+x-1001, [2]] )
 %2 = [1, x]
 @eprog
 
 \misctitle{The Buchmann-Lenstra algorithm}
 
 We now complicate the picture: it is in fact allowed to include
 \emph{composite} numbers instead of primes
 in \kbd{listP} (Vector or Matrix case), provided they are pairwise coprime.
 The result will still be a correct integral basis \emph{if}
 the field discriminant factors completely over the actual primes in the list.
 Adding a composite $C$ such that $C^2$ \emph{divides} $D$ may help because
 when we consider $C$ as a prime and run the algorithm, two good things can
 happen: either we
 succeed in proving that no prime dividing $C$ can divide the index
 (without actually needing to find those primes), or the computation
 exhibits a non-trivial zero divisor, thereby factoring $C$ and
 we go on with the refined factorization. (Note that including a $C$
 such that $C^2$ does not divide $D$ is useless.) If neither happen, then the
 computed basis need not generate the maximal order. Here is an example:
 \bprog
 ? B = 10^5;
 ? P = factor(poldisc(T), B)[,1]; \\ primes <= B dividing D + cofactor
 ? basis = nfbasis([T, listP])
 ? disc = nfdisc([T, listP])
 @eprog\noindent We obtain the maximal order and its discriminant if the
 field discriminant factors
 completely over the primes less than $B$ (together with the primes
 contained in the \tet{addprimes} table). This can be tested as follows:
 \bprog
   check = factor(disc, B);
   lastp = check[-1..-1,1];
   if (lastp > B && !setsearch(addprimes(), lastp),
     warning("nf may be incorrect!"))
 @eprog\noindent
 This is a sufficient but not a necessary condition, hence the warning,
 instead of an error. N.B. \kbd{lastp} is the last entry
 in the first column of the \kbd{check} matrix, i.e. the largest prime
 dividing \kbd{nf.disc} if $\leq B$ or if it belongs to the prime table.
 
 The function \tet{nfcertify} speeds up and automates the above process:
 \bprog
 ? B = 10^5;
 ? nf = nfinit([T, B]);
 ? nfcertify(nf)
 %3 = []      \\ nf is unconditionally correct
 ? basis = nf.zk;
 ? disc = nf.disc;
 @eprog
 
 \synt{nfbasis}{GEN T, GEN *d, GEN listP = NULL}, which returns the order
 basis, and where \kbd{*d} receives the order discriminant.

Function: nfbasistoalg
Class: basic
Section: number_fields
C-Name: basistoalg
Prototype: GG
Help: nfbasistoalg(nf,x): transforms the column vector x on the integral
 basis into an algebraic number.
Doc: Given an algebraic number $x$ in the number field \var{nf}, transforms it
 into \typ{POLMOD} form.
 \bprog
 ? nf = nfinit(y^2 + 4);
 ? nf.zk
 %2 = [1, 1/2*y]
 ? nfbasistoalg(nf, [1,1]~)
 %3 = Mod(1/2*y + 1, y^2 + 4)
 ? nfbasistoalg(nf, y)
 %4 = Mod(y, y^2 + 4)
 ? nfbasistoalg(nf, Mod(y, y^2+4))
 %5 = Mod(y, y^2 + 4)
 @eprog
 This is the inverse function of \kbd{nfalgtobasis}.

Function: nfcertify
Class: basic
Section: number_fields
C-Name: nfcertify
Prototype: G
Help: nfcertify(nf): returns a vector of composite integers used to certify
 nf.zk and nf.disc unconditionally (both are correct when the output
 is the empty vector).
Doc: $\var{nf}$ being as output by
 \kbd{nfinit}, checks whether the integer basis is known unconditionally.
 This is in particular useful when the argument to \kbd{nfinit} was of the
 form $[T, \kbd{listP}]$, specifying a finite list of primes when
 $p$-maximality had to be proven, or a list of coprime integers to which
 Buchmann-Lenstra algorithm was to be applied.
 
 The function returns a vector of coprime composite integers. If this vector
 is empty, then \kbd{nf.zk} and \kbd{nf.disc} are correct. Otherwise, the
 result is dubious. In order to obtain a certified result, one must completely
 factor each of the given integers, then \kbd{addprime} each of their prime
 factors, then check whether \kbd{nfdisc(nf.pol)} is equal to \kbd{nf.disc}.

Function: nfcompositum
Class: basic
Section: number_fields
C-Name: nfcompositum
Prototype: GGGD0,L,
Help: nfcompositum(nf,P,Q,{flag=0}): vector of all possible compositums
 of the number fields defined by the polynomials P and Q; flag is
 optional, whose binary digits mean 1: output for each compositum, not only
 the compositum polynomial pol, but a vector [R,a,b,k] where a (resp. b) is a
 root of P (resp. Q) expressed as a polynomial modulo R, and a small integer k
 such that al2+k*al1 is the chosen root of R; 2: assume that the number
 fields defined by P and Q are linearly disjoint.
Doc: Let \var{nf} be a number field structure attached to the field $K$
 and let \sidx{compositum} $P$ and $Q$
 be squarefree polynomials in $K[X]$ in the same variable. Outputs
 the simple factors of the \'etale $K$-algebra $A = K[X, Y] / (P(X), Q(Y))$.
 The factors are given by a list of polynomials $R$ in $K[X]$, attached to
 the number field $K[X]/ (R)$, and sorted by increasing degree (with respect
 to lexicographic ordering for factors of equal degrees). Returns an error if
 one of the polynomials is not squarefree.
 
 Note that it is more efficient to reduce to the case where $P$ and $Q$ are
 irreducible first. The routine will not perform this for you, since it may be
 expensive, and the inputs are irreducible in most applications anyway. In
 this case, there will be a single factor $R$ if and only if the number
 fields defined by $P$ and $Q$ are linearly disjoint (their intersection is
 $K$).
 
 The binary digits of $\fl$ mean
 
 1: outputs a vector of 4-component vectors $[R,a,b,k]$, where $R$
 ranges through the list of all possible compositums as above, and $a$
 (resp. $b$) expresses the root of $P$ (resp. $Q$) as an element of
 $K[X]/(R)$. Finally, $k$ is a small integer such that $b + ka = X$ modulo
 $R$.
 
 2: assume that $P$ and $Q$ define number fields that are linearly disjoint:
 both polynomials are irreducible and the corresponding number fields
 have no common subfield besides $K$. This allows to save a costly
 factorization over $K$. In this case return the single simple factor
 instead of a vector with one element.
 
 A compositum is often defined by a complicated polynomial, which it is
 advisable to reduce before further work. Here is an example involving
 the field $K(\zeta_5, 5^{1/10})$, $K=\Q(\sqrt{5})$:
 \bprog
 ? K = nfinit(y^2-5);
 ? L = nfcompositum(K, x^5 - y, polcyclo(5), 1); \\@com list of $[R,a,b,k]$
 ? [R, a] = L[1];  \\@com pick the single factor, extract $R,a$ (ignore $b,k$)
 ? lift(R)         \\@com defines the compositum
 %4 = x^10 + (-5/2*y + 5/2)*x^9 + (-5*y + 20)*x^8 + (-20*y + 30)*x^7 + \
 (-45/2*y + 145/2)*x^6 + (-71/2*y + 121/2)*x^5 + (-20*y + 60)*x^4 +    \
 (-25*y + 5)*x^3 + 45*x^2 + (-5*y + 15)*x + (-2*y + 6)
 ? a^5 - y         \\@com a fifth root of $y$
 %5 = 0
 ? [T, X] = rnfpolredbest(K, R, 1);
 ? lift(T)     \\@com simpler defining polynomial for $K[x]/(R)$
 %7 = x^10 + (-11/2*y + 25/2)
 ? liftall(X)  \\ @com root of $R$ in $K[x]/(T(x))$
 %8 = (3/4*y + 7/4)*x^7 + (-1/2*y - 1)*x^5 + 1/2*x^2 + (1/4*y - 1/4)
 ? a = subst(a.pol, 'x, X);  \\@com \kbd{a} in the new coordinates
 ? liftall(a)
 %10 = (-3/4*y - 7/4)*x^7 - 1/2*x^2
 ? a^5 - y
 %11 = 0
 @eprog
 
 The main variables of $P$ and $Q$ must be the same and have higher priority
 than that of \var{nf} (see~\kbd{varhigher} and~\kbd{varlower}).

Function: nfdetint
Class: basic
Section: number_fields
C-Name: nfdetint
Prototype: GG
Help: nfdetint(nf,x): multiple of the ideal determinant of the pseudo
 generating set x.
Doc: given a pseudo-matrix $x$, computes a
 non-zero ideal contained in (i.e.~multiple of) the determinant of $x$. This
 is particularly useful in conjunction with \kbd{nfhnfmod}.

Function: nfdisc
Class: basic
Section: number_fields
C-Name: nfdisc
Prototype: G
Help: nfdisc(T): discriminant of the number field defined by
 the polynomial T. An argument [T,listP] is possible, where listP is a list
 of primes or a prime bound.
Doc: \idx{field discriminant} of the number field defined by the integral,
 preferably monic, irreducible polynomial $T(X)$. Returns the discriminant of
 the number field $\Q[X]/(T)$, using the Round $4$ algorithm.
 
 \misctitle{Local discriminants, valuations at certain primes}
 
 As in \kbd{nfbasis}, the argument $T$ can be replaced by $[T,\var{listP}]$,
 where \kbd{listP} is as in \kbd{nfbasis}: a vector of
 pairwise coprime integers (usually distinct primes), a factorization matrix,
 or a single integer. In that case, the function returns the discriminant of
 an order whose basis is given by \kbd{nfbasis(T,listP)}, which need not be
 the maximal order, and whose valuation at a prime entry in \kbd{listP} is the
 same as the valuation of the field discriminant.
 
 In particular, if \kbd{listP} is $[p]$ for a prime $p$, we can
 return the $p$-adic discriminant of the maximal order of $\Z_p[X]/(T)$,
 as a power of $p$, as follows:
 \bprog
 ? padicdisc(T,p) = p^valuation(nfdisc(T,[p]), p);
 ? nfdisc(x^2 + 6)
 %2 = -24
 ? padicdisc(x^2 + 6, 2)
 %3 = 8
 ? padicdisc(x^2 + 6, 3)
 %4 = 3
 @eprog
 
 \synt{nfdisc}{GEN T} (\kbd{listP = NULL}). Also available is
 \fun{GEN}{nfbasis}{GEN T, GEN *d, GEN listP = NULL}, which returns the order
 basis, and where \kbd{*d} receives the order discriminant.

Function: nfeltadd
Class: basic
Section: number_fields
C-Name: nfadd
Prototype: GGG
Help: nfeltadd(nf,x,y): element x+y in nf.
Doc: 
 given two elements $x$ and $y$ in
 \var{nf}, computes their sum $x+y$ in the number field $\var{nf}$.

Function: nfeltdiv
Class: basic
Section: number_fields
C-Name: nfdiv
Prototype: GGG
Help: nfeltdiv(nf,x,y): element x/y in nf.
Doc: given two elements $x$ and $y$ in
 \var{nf}, computes their quotient $x/y$ in the number field $\var{nf}$.

Function: nfeltdiveuc
Class: basic
Section: number_fields
C-Name: nfdiveuc
Prototype: GGG
Help: nfeltdiveuc(nf,x,y): gives algebraic integer q such that x-qy is small.
Doc: given two elements $x$ and $y$ in
 \var{nf}, computes an algebraic integer $q$ in the number field $\var{nf}$
 such that the components of $x-qy$ are reasonably small. In fact, this is
 functionally identical to \kbd{round(nfdiv(\var{nf},x,y))}.

Function: nfeltdivmodpr
Class: basic
Section: number_fields
C-Name: nfdivmodpr
Prototype: GGGG
Help: nfeltdivmodpr(nf,x,y,pr): this function is obsolete, use nfmodpr.
Doc: this function is obsolete, use \kbd{nfmodpr}.
 
 Given two elements $x$
 and $y$ in \var{nf} and \var{pr} a prime ideal in \kbd{modpr} format (see
 \tet{nfmodprinit}), computes their quotient $x / y$ modulo the prime ideal
 \var{pr}.
Obsolete: 2016-08-09
Variant: This function is normally useless in library mode. Project your
 inputs to the residue field using \kbd{nf\_to\_Fq}, then work there.

Function: nfeltdivrem
Class: basic
Section: number_fields
C-Name: nfdivrem
Prototype: GGG
Help: nfeltdivrem(nf,x,y): gives [q,r] such that r=x-qy is small.
Doc: given two elements $x$ and $y$ in
 \var{nf}, gives a two-element row vector $[q,r]$ such that $x=qy+r$, $q$ is
 an algebraic integer in $\var{nf}$, and the components of $r$ are
 reasonably small.

Function: nfeltmod
Class: basic
Section: number_fields
C-Name: nfmod
Prototype: GGG
Help: nfeltmod(nf,x,y): gives r such that r=x-qy is small with q algebraic
 integer.
Doc: 
 given two elements $x$ and $y$ in
 \var{nf}, computes an element $r$ of $\var{nf}$ of the form $r=x-qy$ with
 $q$ and algebraic integer, and such that $r$ is small. This is functionally
 identical to
 $$\kbd{x - nfmul(\var{nf},round(nfdiv(\var{nf},x,y)),y)}.$$

Function: nfeltmul
Class: basic
Section: number_fields
C-Name: nfmul
Prototype: GGG
Help: nfeltmul(nf,x,y): element x.y in nf.
Doc: 
 given two elements $x$ and $y$ in
 \var{nf}, computes their product $x*y$ in the number field $\var{nf}$.

Function: nfeltmulmodpr
Class: basic
Section: number_fields
C-Name: nfmulmodpr
Prototype: GGGG
Help: nfeltmulmodpr(nf,x,y,pr): this function is obsolete, use nfmodpr.
Doc: this function is obsolete, use \kbd{nfmodpr}.
 
 Given two elements $x$ and
 $y$ in \var{nf} and \var{pr} a prime ideal in \kbd{modpr} format (see
 \tet{nfmodprinit}), computes their product $x*y$ modulo the prime ideal
 \var{pr}.
Obsolete: 2016-08-09
Variant: This function is normally useless in library mode. Project your
 inputs to the residue field using \kbd{nf\_to\_Fq}, then work there.

Function: nfeltnorm
Class: basic
Section: number_fields
C-Name: nfnorm
Prototype: GG
Help: nfeltnorm(nf,x): norm of x.
Doc: returns the absolute norm of $x$.

Function: nfeltpow
Class: basic
Section: number_fields
C-Name: nfpow
Prototype: GGG
Help: nfeltpow(nf,x,k): element x^k in nf.
Doc: given an element $x$ in \var{nf}, and a positive or negative integer $k$,
 computes $x^k$ in the number field $\var{nf}$.
Variant: \fun{GEN}{nfinv}{GEN nf, GEN x} correspond to $k = -1$, and
 \fun{GEN}{nfsqr}{GEN nf,GEN x} to $k = 2$.

Function: nfeltpowmodpr
Class: basic
Section: number_fields
C-Name: nfpowmodpr
Prototype: GGGG
Help: nfeltpowmodpr(nf,x,k,pr): this function is obsolete, use nfmodpr.
Doc: this function is obsolete, use \kbd{nfmodpr}.
 
 Given an element $x$ in \var{nf}, an integer $k$ and a prime ideal
 \var{pr} in \kbd{modpr} format
 (see \tet{nfmodprinit}), computes $x^k$ modulo the prime ideal \var{pr}.
Obsolete: 2016-08-09
Variant: This function is normally useless in library mode. Project your
 inputs to the residue field using \kbd{nf\_to\_Fq}, then work there.

Function: nfeltreduce
Class: basic
Section: number_fields
C-Name: nfreduce
Prototype: GGG
Help: nfeltreduce(nf,a,id): gives r such that a-r is in the ideal id and r
 is small.
Doc: given an ideal \var{id} in
 Hermite normal form and an element $a$ of the number field $\var{nf}$,
 finds an element $r$ in $\var{nf}$ such that $a-r$ belongs to the ideal
 and $r$ is small.

Function: nfeltreducemodpr
Class: basic
Section: number_fields
C-Name: nfreducemodpr
Prototype: GGG
Help: nfeltreducemodpr(nf,x,pr): this function is obsolete, use nfmodpr.
Doc: this function is obsolete, use \kbd{nfmodpr}.
 
 Given an element $x$ of the number field $\var{nf}$ and a prime ideal
 \var{pr} in \kbd{modpr} format compute a canonical representative for the
 class of $x$ modulo \var{pr}.
Obsolete: 2016-08-09
Variant: This function is normally useless in library mode. Project your
 inputs to the residue field using \kbd{nf\_to\_Fq}, then work there.

Function: nfelttrace
Class: basic
Section: number_fields
C-Name: nftrace
Prototype: GG
Help: nfelttrace(nf,x): trace of x.
Doc: returns the absolute trace of $x$.

Function: nfeltval
Class: basic
Section: number_fields
C-Name: gpnfvalrem
Prototype: GGGD&
Help: nfeltval(nf,x,pr,{&y}): valuation of element x at the prime pr as output
 by idealprimedec.
Doc: given an element $x$ in
 \var{nf} and a prime ideal \var{pr} in the format output by
 \kbd{idealprimedec}, computes the valuation $v$ at \var{pr} of the
 element $x$. The valuation of $0$ is \kbd{+oo}.
 \bprog
 ? nf = nfinit(x^2 + 1);
 ? P = idealprimedec(nf, 2)[1];
 ? nfeltval(nf, x+1, P)
 %3 = 1
 @eprog\noindent
 This particular valuation can also be obtained using
 \kbd{idealval(\var{nf},x,\var{pr})}, since $x$ is then converted to a
 principal ideal.
 
 If the $y$ argument is present, sets $y = x \tau^v$, where $\tau$ is a
 fixed ``anti-uniformizer'' for \var{pr}: its valuation at \var{pr} is $-1$;
 its valuation is $0$ at other prime ideals dividing \kbd{\var{pr}.p} and
 nonnegative at all other primes. In other words $y$ is the part of $x$
 coprime to \var{pr}. If $x$ is an algebraic integer, so is $y$.
 \bprog
 ? nfeltval(nf, x+1, P, &y); y
 %4 = [0, 1]~
 @eprog
 For instance if $x = \prod_i x_i^{e_i}$ is known to be coprime to \var{pr},
 where the $x_i$ are algebraic integers and $e_i\in\Z$ then,
 if $v_i = \kbd{nfeltval}(\var{nf}, x_i, \var{pr}, \&y_i)$, we still
 have $x = \prod_i y_i^{e_i}$, where the $y_i$ are still algebraic integers
 but now all of them are coprime to \var{pr}. They can then be mapped to
 the residue field of \var{pr} more efficiently than if the product had
 been expanded beforehand: we can reduce mod \var{pr} after each ring
 operation.
Variant: Also available is
 \fun{long}{nfvalrem}{GEN nf, GEN x, GEN pr, GEN *y = NULL}, which returns
 \tet{LONG_MAX} if $x = 0$ and the valuation as a \kbd{long} integer.

Function: nffactor
Class: basic
Section: number_fields
C-Name: nffactor
Prototype: GG
Help: nffactor(nf,T): factor polynomial T in number field nf.
Doc: factorization of the univariate
 polynomial $T$ over the number field $\var{nf}$ given by \kbd{nfinit}; $T$
 has coefficients in $\var{nf}$ (i.e.~either scalar, polmod, polynomial or
 column vector). The factors are sorted by increasing degree.
 
 The main variable of $\var{nf}$ must be of \emph{lower}
 priority than that of $T$, see \secref{se:priority}. However if
 the polynomial defining the number field occurs explicitly  in the
 coefficients of $T$ as modulus of a \typ{POLMOD} or as a \typ{POL}
 coefficient, its main variable must be \emph{the same} as the main variable
 of $T$. For example,
 \bprog
 ? nf = nfinit(y^2 + 1);
 ? nffactor(nf, x^2 + y); \\@com OK
 ? nffactor(nf, x^2 + Mod(y, y^2+1)); \\ @com OK
 ? nffactor(nf, x^2 + Mod(z, z^2+1)); \\ @com WRONG
 @eprog
 
 It is possible to input a defining polynomial for \var{nf}
 instead, but this is in general less efficient since parts of an \kbd{nf}
 structure will then be computed internally. This is useful in two
 situations: when you do not need the \kbd{nf} elsewhere, or when you cannot
 initialize an \kbd{nf} due to integer factorization difficulties when
 attempting to compute the field discriminant and maximal order. In all
 cases, the function runs in polynomial time using Belabas's variant
 of \idx{van Hoeij}'s algorithm, which copes with hundreds of modular factors.
 
 \misctitle{Caveat} \kbd{nfinit([T, listP])} allows to compute in polynomial
 time a conditional \var{nf} structure, which sets \kbd{nf.zk} to an order
 which is not guaranteed to be maximal at all primes. Always either use
 \kbd{nfcertify} first (which may not run in polynomial time) or make sure
 to input \kbd{nf.pol} instead of the conditional \var{nf}: \kbd{nffactor} is
 able to recover in polynomial time in this case, instead of potentially
 missing a factor.

Function: nffactorback
Class: basic
Section: number_fields
C-Name: nffactorback
Prototype: GGDG
Help: nffactorback(nf,f,{e}): given a factorisation f, returns
 the factored object back as an nf element.
Doc: gives back the \var{nf} element corresponding to a factorization.
 The integer $1$ corresponds to the empty factorization.
 
 If $e$ is present, $e$ and $f$ must be vectors of the same length ($e$ being
 integral), and the corresponding factorization is the product of the
 $f[i]^{e[i]}$.
 
 If not, and $f$ is vector, it is understood as in the preceding case with $e$
 a vector of 1s: we return the product of the $f[i]$. Finally, $f$ can be a
 regular factorization matrix.
 \bprog
 ? nf = nfinit(y^2+1);
 ? nffactorback(nf, [3, y+1, [1,2]~], [1, 2, 3])
 %2 = [12, -66]~
 ? 3 * (I+1)^2 * (1+2*I)^3
 %3 = 12 - 66*I
 @eprog

Function: nffactormod
Class: basic
Section: number_fields
C-Name: nffactormod
Prototype: GGG
Help: nffactormod(nf,Q,pr): this routine is obsolete, use nfmodpr and
 factorff. Factor polynomial Q modulo prime ideal pr
 in number field nf.
Doc: this routine is obsolete, use \kbd{nfmodpr} and \kbd{factorff}.
 
 Factors the univariate polynomial $Q$ modulo the prime ideal \var{pr} in
 the number field $\var{nf}$. The coefficients of $Q$ belong to the number
 field (scalar, polmod, polynomial, even column vector) and the main variable
 of $\var{nf}$ must be of lower priority than that of $Q$ (see
 \secref{se:priority}). The prime ideal \var{pr} is either in
 \tet{idealprimedec} or (preferred) \tet{modprinit} format. The coefficients
 of the polynomial factors are lifted to elements of \var{nf}:
 \bprog
 ? K = nfinit(y^2+1);
 ? P = idealprimedec(K, 3)[1];
 ? nffactormod(K, x^2 + y*x + 18*y+1, P)
 %3 =
 [x + (2*y + 1) 1]
 
 [x + (2*y + 2) 1]
 ? P = nfmodprinit(K, P);  \\ convert to nfmodprinit format
 ? nffactormod(K, x^2 + y*x + 18*y+1)
 %5 =
 [x + (2*y + 1) 1]
 
 [x + (2*y + 2) 1]
 @eprog\noindent Same result, of course, here about 10\% faster due to the
 precomputation.
Obsolete: 2016-09-18

Function: nfgaloisapply
Class: basic
Section: number_fields
C-Name: galoisapply
Prototype: GGG
Help: nfgaloisapply(nf,aut,x): apply the Galois automorphism aut to the object
 x (element or ideal) in the number field nf.
Doc: let $\var{nf}$ be a
 number field as output by \kbd{nfinit}, and let \var{aut} be a \idx{Galois}
 automorphism of $\var{nf}$ expressed by its image on the field generator
 (such automorphisms can be found using \kbd{nfgaloisconj}). The function
 computes the action of the automorphism \var{aut} on the object $x$ in the
 number field; $x$ can be a number field element, or an ideal (possibly
 extended). Because of possible confusion with elements and ideals, other
 vector or matrix arguments are forbidden.
  \bprog
  ? nf = nfinit(x^2+1);
  ? L = nfgaloisconj(nf)
  %2 = [-x, x]~
  ? aut = L[1]; /* the non-trivial automorphism */
  ? nfgaloisapply(nf, aut, x)
  %4 = Mod(-x, x^2 + 1)
  ? P = idealprimedec(nf,5); /* prime ideals above 5 */
  ? nfgaloisapply(nf, aut, P[2]) == P[1]
  %6 = 0 \\ !!!!
  ? idealval(nf, nfgaloisapply(nf, aut, P[2]), P[1])
  %7 = 1
 @eprog\noindent The surprising failure of the equality test (\kbd{\%7}) is
 due to the fact that although the corresponding prime ideals are equal, their
 representations are not. (A prime ideal is specified by a uniformizer, and
 there is no guarantee that applying automorphisms yields the same elements
 as a direct \kbd{idealprimedec} call.)
 
 The automorphism can also be given as a column vector, representing the
 image of \kbd{Mod(x, nf.pol)} as an algebraic number. This last
 representation is more efficient and should be preferred if a given
 automorphism must be used in many such calls.
 \bprog
  ? nf = nfinit(x^3 - 37*x^2 + 74*x - 37);
  ? aut = nfgaloisconj(nf)[2]; \\ @com an automorphism in basistoalg form
  %2 = -31/11*x^2 + 1109/11*x - 925/11
  ? AUT = nfalgtobasis(nf, aut); \\ @com same in algtobasis form
  %3 = [16, -6, 5]~
  ? v = [1, 2, 3]~; nfgaloisapply(nf, aut, v) == nfgaloisapply(nf, AUT, v)
  %4 = 1 \\ @com same result...
  ? for (i=1,10^5, nfgaloisapply(nf, aut, v))
  time = 463 ms.
  ? for (i=1,10^5, nfgaloisapply(nf, AUT, v))
  time = 343 ms.  \\ @com but the latter is faster
 @eprog

Function: nfgaloisconj
Class: basic
Section: number_fields
C-Name: galoisconj0
Prototype: GD0,L,DGp
Help: nfgaloisconj(nf,{flag=0},{d}): list of conjugates of a root of the
 polynomial x=nf.pol in the same number field. flag is optional (set to 0 by
 default), meaning 0: use combination of flag 4 and 1, always complete; 1:
 use nfroots; 4: use Allombert's algorithm, complete if the field is Galois of
 degree <= 35 (see manual for details). nf can be simply a polynomial.
Doc: $\var{nf}$ being a number field as output by \kbd{nfinit}, computes the
 conjugates of a root $r$ of the non-constant polynomial $x=\var{nf}[1]$
 expressed as polynomials in $r$. This also makes sense when the number field
 is not \idx{Galois} since some conjugates may lie in the field.
 $\var{nf}$ can simply be a polynomial.
 
 If no flags or $\fl=0$, use a combination of flag $4$ and $1$ and the result
 is always complete. There is no point whatsoever in using the other flags.
 
 If $\fl=1$, use \kbd{nfroots}: a little slow, but guaranteed to work in
 polynomial time.
 
 If $\fl=4$, use \kbd{galoisinit}: very fast, but only applies to (most)
 Galois fields. If the field is Galois with weakly super-solvable Galois
 group (see \tet{galoisinit}), return the complete list of automorphisms, else
 only the identity element. If present, $d$ is assumed to be a multiple of the
 least common denominator of the conjugates expressed as polynomial in a root
 of \var{pol}.
 
 This routine can only compute $\Q$-automorphisms, but it may be used to get
 $K$-automorphism for any base field $K$ as follows:
 \bprog
 rnfgaloisconj(nfK, R) = \\ K-automorphisms of L = K[X] / (R)
 {
   my(polabs, N,al,S, ala,k, vR);
   R *= Mod(1, nfK.pol); \\ convert coeffs to polmod elts of K
   vR = variable(R);
   al = Mod(variable(nfK.pol),nfK.pol);
   [polabs,ala,k] = rnfequation(nfK, R, 1);
   Rt = if(k==0,R,subst(R,vR,vR-al*k));
   N = nfgaloisconj(polabs) % Rt; \\ Q-automorphisms of L
   S = select(s->subst(Rt, vR, Mod(s,Rt)) == 0, N);
   if (k==0, S, apply(s->subst(s,vR,vR+k*al)-k*al,S));
 }
 K  = nfinit(y^2 + 7);
 rnfgaloisconj(K, x^4 - y*x^3 - 3*x^2 + y*x + 1)  \\ K-automorphisms of L
 @eprog
Variant: Use directly
 \fun{GEN}{galoisconj}{GEN nf, GEN d}, corresponding to $\fl = 0$, the others
 only have historical interest.

Function: nfgrunwaldwang
Class: basic
Section: number_fields
C-Name: nfgrunwaldwang
Prototype: GGGGDn
Help: nfgrunwaldwang(nf,Lpr,Ld,pl,{v='x}): a polynomial in the variable v
 defining a cyclic extension of nf (given in nf or bnf form) with local
 behaviour prescribed by Lpr, Ld and pl: the extension has local degree a
 multiple of Ld[i] at the prime Lpr[i], and the extension is complex at the
 i-th real place of nf if pl[i]=-1 (no condition if pl[i]=0). The extension
 has degree the LCM of the local degrees.
Doc: Given \var{nf} a number field in \var{nf} or \var{bnf} format,
 a \typ{VEC} \var{Lpr} of primes of \var{nf} and a \typ{VEC} \var{Ld} of
 positive integers of the same length, a \typ{VECSMALL} \var{pl} of length
 $r_1$ the number of real places of \var{nf}, computes a polynomial with
 coefficients in \var{nf} defining a cyclic extension of \var{nf} of
 minimal degree satisfying certain local conditions:
 
 \item at the prime \kbd{Lpr[i]}, the extension has local degree a multiple of
 \kbd{Ld[i]};
 
 \item at the $i$-th real place of \var{nf}, it is complex if $pl[i]=-1$
 (no condition if $pl[i]=0$).
 
 The extension has degree the LCM of the local degrees. Currently, the degree
 is restricted to be a prime power for the search, and to be prime for the
 construction because of the \kbd{rnfkummer} restrictions.
 
 When \var{nf} is $\Q$, prime integers are accepted instead of \kbd{prid}
 structures. However, their primality is not checked and the behaviour is
 undefined if you provide a composite number.
 
 \misctitle{Warning} If the number field \var{nf} does not contain the $n$-th
 roots of unity where $n$ is the degree of the extension to be computed,
 triggers the computation of the \var{bnf} of $nf(\zeta_n)$, which may be
 costly.
 
 \bprog
 ? nf = nfinit(y^2-5);
 ? pr = idealprimedec(nf,13)[1];
 ? pol = nfgrunwaldwang(nf, [pr], [2], [0,-1], 'x)
 %3 = x^2 + Mod(3/2*y + 13/2, y^2 - 5)
 @eprog

Function: nfhilbert
Class: basic
Section: number_fields
C-Name: nfhilbert0
Prototype: lGGGDG
Help: nfhilbert(nf,a,b,{pr}): if pr is omitted, global Hilbert symbol (a,b) in
 nf, that is 1 if X^2-aY^2-bZ^2 has a non-trivial solution (X,Y,Z) in nf, -1
 otherwise. Otherwise compute the local symbol modulo the prime ideal pr.
Doc: if \var{pr} is omitted,
 compute the global quadratic \idx{Hilbert symbol} $(a,b)$ in $\var{nf}$, that
 is $1$ if $x^2 - a y^2 - b z^2$ has a non trivial solution $(x,y,z)$ in
 $\var{nf}$, and $-1$ otherwise. Otherwise compute the local symbol modulo
 the prime ideal \var{pr}, as output by \kbd{idealprimedec}.
Variant: 
 Also available is \fun{long}{nfhilbert}{GEN bnf,GEN a,GEN b} (global
 quadratic Hilbert symbol).

Function: nfhnf
Class: basic
Section: number_fields
C-Name: nfhnf0
Prototype: GGD0,L,
Help: nfhnf(nf,x,{flag=0}): if x=[A,I], gives a pseudo-basis [B,J] of the module
 sum A_jI_j. If flag is non-zero, return [[B,J], U], where U is the
 transformation matrix such that AU = [0|B].
Doc: given a pseudo-matrix $(A,I)$, finds a
 pseudo-basis $(B,J)$ in \idx{Hermite normal form} of the module it generates.
 If $\fl$ is non-zero, also return the transformation matrix $U$ such that
 $AU = [0|B]$.
Variant: Also available:
 
 \fun{GEN}{nfhnf}{GEN nf, GEN x} ($\fl = 0$).
 
 \fun{GEN}{rnfsimplifybasis}{GEN bnf, GEN x} simplifies the pseudo-basis
 given by $x = (A,I)$. The ideals in the list $I$ are integral, primitive and
 either trivial (equal to the full ring of integer) or non-principal.

Function: nfhnfmod
Class: basic
Section: number_fields
C-Name: nfhnfmod
Prototype: GGG
Help: nfhnfmod(nf,x,detx): if x=[A,I], and detx is a multiple of the ideal
 determinant of x, gives a pseudo-basis of the module sum A_jI_j.
Doc: given a pseudo-matrix $(A,I)$
 and an ideal \var{detx} which is contained in (read integral multiple of) the
 determinant of $(A,I)$, finds a pseudo-basis in \idx{Hermite normal form}
 of the module generated by $(A,I)$. This avoids coefficient explosion.
 \var{detx} can be computed using the function \kbd{nfdetint}.

Function: nfinit
Class: basic
Section: number_fields
C-Name: nfinit0
Prototype: GD0,L,p
Help: nfinit(pol,{flag=0}): pol being a nonconstant irreducible polynomial,
 gives the vector: [pol,[r1,r2],discf,index,[M,MC,T2,T,different] (see
 manual),r1+r2 first roots, integral basis, matrix of power basis in terms of
 integral basis, multiplication table of basis]. flag is optional and can be
 set to 0: default; 1: do not compute different; 2: first use polred to find
 a simpler polynomial; 3: outputs a two-element vector [nf,Mod(a,P)], where
 nf is as in 2 and Mod(a,P) is a polmod equal to Mod(x,pol) and P=nf.pol.
Description: 
 (gen, ?0):nf:prec       nfinit0($1, 0, $prec)
 (gen, 1):nf:prec        nfinit0($1, 1, $prec)
 (gen, 2):nf:prec        nfinit0($1, 2, $prec)
 (gen, 3):gen:prec       nfinit0($1, 3, $prec)
 (gen, 4):nf:prec        nfinit0($1, 4, $prec)
 (gen, 5):gen:prec       nfinit0($1, 5, $prec)
 (gen, #small):void      $"incorrect flag in nfinit"
 (gen, small):gen:prec   nfinit0($1, $2, $prec)
Doc: \var{pol} being a non-constant,
 preferably monic, irreducible polynomial in $\Z[X]$, initializes a
 \emph{number field} structure (\kbd{nf}) attached to the field $K$ defined
 by \var{pol}. As such, it's a technical object passed as the first argument
 to most \kbd{nf}\var{xxx} functions, but it contains some information which
 may be directly useful. Access to this information via \emph{member
 functions} is preferred since the specific data organization given below
 may change in the future. Currently, \kbd{nf} is a row vector with 9
 components:
 
 $\var{nf}[1]$ contains the polynomial \var{pol} (\kbd{\var{nf}.pol}).
 
 $\var{nf}[2]$ contains $[r1,r2]$ (\kbd{\var{nf}.sign}, \kbd{\var{nf}.r1},
 \kbd{\var{nf}.r2}), the number of real and complex places of $K$.
 
 $\var{nf}[3]$ contains the discriminant $d(K)$ (\kbd{\var{nf}.disc}) of $K$.
 
 $\var{nf}[4]$ contains the index of $\var{nf}[1]$ (\kbd{\var{nf}.index}),
 i.e.~$[\Z_K : \Z[\theta]]$, where $\theta$ is any root of $\var{nf}[1]$.
 
 $\var{nf}[5]$ is a vector containing 7 matrices $M$, $G$, \var{roundG}, $T$,
 $MD$, $TI$, $MDI$ useful for certain computations in the number field $K$.
 
 \quad\item $M$ is the $(r1+r2)\times n$ matrix whose columns represent
 the numerical values of the conjugates of the elements of the integral
 basis.
 
 \quad\item $G$ is an $n\times n$ matrix such that $T2 = {}^t G G$,
 where $T2$ is the quadratic form $T_2(x) = \sum |\sigma(x)|^2$, $\sigma$
 running over the embeddings of $K$ into $\C$.
 
 \quad\item \var{roundG} is a rescaled copy of $G$, rounded to nearest
 integers.
 
 \quad\item $T$ is the $n\times n$ matrix whose coefficients are
 $\text{Tr}(\omega_i\omega_j)$ where the $\omega_i$ are the elements of the
 integral basis. Note also that $\det(T)$ is equal to the discriminant of the
 field $K$. Also, when understood as an ideal, the matrix $T^{-1}$
 generates the codifferent ideal.
 
 \quad\item The columns of $MD$ (\kbd{\var{nf}.diff}) express a $\Z$-basis
 of the different of $K$ on the integral basis.
 
 \quad\item $TI$ is equal to the primitive part of $T^{-1}$, which has integral
 coefficients.
 
 \quad\item Finally, $MDI$ is a two-element representation (for faster
 ideal product) of $d(K)$ times the codifferent ideal
 (\kbd{\var{nf}.disc$*$\var{nf}.codiff}, which is an integral ideal). $MDI$
 is only used in \tet{idealinv}.
 
 $\var{nf}[6]$ is the vector containing the $r1+r2$ roots
 (\kbd{\var{nf}.roots}) of $\var{nf}[1]$ corresponding to the $r1+r2$
 embeddings of the number field into $\C$ (the first $r1$ components are real,
 the next $r2$ have positive imaginary part).
 
 $\var{nf}[7]$ is an integral basis for $\Z_K$ (\kbd{\var{nf}.zk}) expressed
 on the powers of~$\theta$. Its first element is guaranteed to be $1$. This
 basis is LLL-reduced with respect to $T_2$ (strictly speaking, it is a
 permutation of such a basis, due to the condition that the first element be
 $1$).
 
 $\var{nf}[8]$ is the $n\times n$ integral matrix expressing the power
 basis in terms of the integral basis, and finally
 
 $\var{nf}[9]$ is the $n\times n^2$ matrix giving the multiplication table
 of the integral basis.
 
 If a non monic polynomial is input, \kbd{nfinit} will transform it into a
 monic one, then reduce it (see $\fl=3$). It is allowed, though not very
 useful given the existence of \tet{nfnewprec}, to input a \var{nf} or a
 \var{bnf} instead of a polynomial. It is also allowed to
 input a \var{rnf}, in which case an \kbd{nf} structure attached to the
 absolute defining polynomial \kbd{polabs} is returned (\fl is then ignored).
 
 \bprog
 ? nf = nfinit(x^3 - 12); \\ initialize number field Q[X] / (X^3 - 12)
 ? nf.pol   \\ defining polynomial
 %2 = x^3 - 12
 ? nf.disc  \\ field discriminant
 %3 = -972
 ? nf.index \\ index of power basis order in maximal order
 %4 = 2
 ? nf.zk    \\ integer basis, lifted to Q[X]
 %5 = [1, x, 1/2*x^2]
 ? nf.sign  \\ signature
 %6 = [1, 1]
 ? factor(abs(nf.disc ))  \\ determines ramified primes
 %7 =
 [2 2]
 
 [3 5]
 ? idealfactor(nf, 2)
 %8 =
 [[2, [0, 0, -1]~, 3, 1, [0, 1, 0]~] 3]  \\ @com $\goth{p}_2^3$
 @eprog
 
 \misctitle{Huge discriminants, helping nfdisc}
 
 In case \var{pol} has a huge discriminant which is difficult to factor,
 it is hard to compute from scratch the maximal order. The special input
 format $[\var{pol}, B]$ is also accepted where \var{pol} is a polynomial as
 above and $B$ has one of the following forms
 
 \item an integer basis, as would be computed by \tet{nfbasis}: a vector of
 polynomials with first element $1$. This is useful if the maximal order is
 known in advance.
 
 \item an argument \kbd{listP} which specifies a list of primes (see
 \tet{nfbasis}). Instead of the maximal order, \kbd{nfinit} then computes an
 order which is maximal at these particular primes as well as the primes
 contained in the private prime table (see \tet{addprimes}). The result is
 unconditionaly correct when the discriminant \kbd{nf.disc} factors
 completely over this set of primes. The function \tet{nfcertify} automates
 this:
 \bprog
 ? pol = polcompositum(x^5 - 101, polcyclo(7))[1];
 ? nf = nfinit( [pol, 10^3] );
 ? nfcertify(nf)
 %3 = []
 @eprog\noindent A priori, \kbd{nf.zk} defines an order which is only known
 to be maximal at all primes $\leq 10^3$ (no prime $\leq 10^3$ divides
 \kbd{nf.index}). The certification step proves the correctness of the
 computation. Had it failed, that particular \kbd{nf} structure could
 not have been trusted and may have caused routines using it to fail randomly.
 One particular functions that remains trustworthy in all cases is
 \kbd{idealprimedec} when applied to a prime included in the above list
 of primes.
 \medskip
 
 If $\fl=2$: \var{pol} is changed into another polynomial $P$ defining the same
 number field, which is as simple as can easily be found using the
 \tet{polredbest} algorithm, and all the subsequent computations are done
 using this new polynomial. In particular, the first component of the result
 is the modified polynomial.
 
 If $\fl=3$, apply \kbd{polredbest} as in case 2, but outputs
 $[\var{nf},\kbd{Mod}(a,P)]$, where $\var{nf}$ is as before and
 $\kbd{Mod}(a,P)=\kbd{Mod}(x,\var{pol})$ gives the change of
 variables. This is implicit when \var{pol} is not monic: first a linear change
 of variables is performed, to get a monic polynomial, then \kbd{polredbest}.
Variant: Also available are
 \fun{GEN}{nfinit}{GEN x, long prec} ($\fl = 0$),
 \fun{GEN}{nfinitred}{GEN x, long prec} ($\fl = 2$),
 \fun{GEN}{nfinitred2}{GEN x, long prec} ($\fl = 3$).
 Instead of the above hardcoded numerical flags in \kbd{nfinit0}, one should
 rather use
 
 \fun{GEN}{nfinitall}{GEN x, long flag, long prec}, where \fl\ is an
 or-ed combination of
 
 \item \tet{nf_RED}: find a simpler defining polynomial,
 
 \item \tet{nf_ORIG}: if \tet{nf_RED} set, also return the change of variable,
 
 \item \tet{nf_ROUND2}: \emph{Deprecated}. Slow down the routine by using an
 obsolete normalization algorithm (do not use this one!),
 
 \item \tet{nf_PARTIALFACT}: \emph{Deprecated}. Lazy factorization of the
 polynomial discriminant. Result is conditional unless \kbd{nfcertify}
 can certify it.

Function: nfisideal
Class: basic
Section: number_fields
C-Name: isideal
Prototype: lGG
Help: nfisideal(nf,x): true(1) if x is an ideal in the number field nf,
 false(0) if not.
Doc: returns 1 if $x$ is an ideal in the number field $\var{nf}$, 0 otherwise.

Function: nfisincl
Class: basic
Section: number_fields
C-Name: nfisincl
Prototype: GG
Help: nfisincl(x,y): tests whether the number field x is isomorphic to a
 subfield of y (where x and y are either polynomials or number fields as
 output by nfinit). Return 0 if not, and otherwise all the isomorphisms. If y
 is a number field, a faster algorithm is used.
Doc: tests whether the number field $K$ defined
 by the polynomial $x$ is conjugate to a subfield of the field $L$ defined
 by $y$ (where $x$ and $y$ must be in $\Q[X]$). If they are not, the output
 is the number 0. If they are, the output is a vector of polynomials, each
 polynomial $a$ representing an embedding of $K$ into $L$, i.e.~being such
 that $y\mid x\circ a$.
 
 If $y$ is a number field (\var{nf}), a much faster algorithm is used
 (factoring $x$ over $y$ using \tet{nffactor}). Before version 2.0.14, this
 wasn't guaranteed to return all the embeddings, hence was triggered by a
 special flag. This is no longer the case.

Function: nfisisom
Class: basic
Section: number_fields
C-Name: nfisisom
Prototype: GG
Help: nfisisom(x,y): as nfisincl but tests whether x is isomorphic to y.
Doc: as \tet{nfisincl}, but tests for isomorphism. If either $x$ or $y$ is a
 number field, a much faster algorithm will be used.

Function: nfislocalpower
Class: basic
Section: number_fields
C-Name: nfislocalpower
Prototype: lGGGG
Help: nfislocalpower(nf,pr,a,n): true(1) if a is an n-th power in
 the local field K_v, false(0) if not.
Doc: Let \var{nf} be a number field structure attached to $K$,
 let $a \in K$ and let \var{pr} be a \var{prid} attched to the
 maximal ideal $v$. Return $1$ if $a$ is an $n$-th power in the completed
 local field $K_v$, and $0$ otherwise.
 \bprog
 ? K = nfinit(y^2+1);
 ? P = idealprimedec(K,2)[1]; \\ the ramified prime above 2
 ? nfislocalpower(K,P,-1, 2) \\ -1 is a square
 %3 = 1
 ? nfislocalpower(K,P,-1, 4) \\ ... but not a 4-th power
 %4 = 0
 ? nfislocalpower(K,P,2, 2)  \\ 2 is not a square
 %5 = 0
 
 ? Q = idealprimedec(K,5)[1]; \\ a prime above 5
 ? nfislocalpower(K,Q, [0, 32]~, 30)  \\ 32*I is locally a 30-th power
 %7 = 1
 @eprog

Function: nfkermodpr
Class: basic
Section: number_fields
C-Name: nfkermodpr
Prototype: GGG
Help: nfkermodpr(nf,x,pr): this function is obsolete, use nfmodpr.
Doc: this function is obsolete, use \kbd{nfmodpr}.
 
 Kernel of the matrix $a$ in $\Z_K/\var{pr}$, where \var{pr} is in
 \key{modpr} format (see \kbd{nfmodprinit}).
Obsolete: 2016-08-09
Variant: This function is normally useless in library mode. Project your
 inputs to the residue field using \kbd{nfM\_to\_FqM}, then work there.

Function: nfmodpr
Class: basic
Section: number_fields
C-Name: nfmodpr
Prototype: GGG
Help: nfmodpr(nf,x,pr): map x to the residue field mod pr.
Doc: map $x$ to the residue field modulo \var{pr}, to a \typ{FFELT}.
 The argument \var{pr} is either a maximal ideal in \kbd{idealprimedec}
 format or, preferably, a \kbd{modpr} structure from \tet{nfmodprinit}. The
 function \tet{nfmodprlift} allows to lift back to $\Z_K$.
 
 Note that the function applies to number field elements and not to
 vector / matrices / polynomials of such. Use \kbd{apply} to convert
 recursive structures.
 
 \bprog
 ? K = nfinit(y^3-250);
 ? P = idealprimedec(K, 5)[2]
 ? modP = nfmodprinit(K,P);
 ? K.zk
 %4 = [1, 1/5*y, 1/25*y^2]
 ? apply(t->nfmodpr(K,t,modP), K.zk)
 %5 = [1, y, 2*y + 1]
 @eprog

Function: nfmodprinit
Class: basic
Section: number_fields
C-Name: nfmodprinit
Prototype: GG
Help: nfmodprinit(nf,pr): transform the prime ideal pr into modpr format
 necessary for all operations mod pr in the number field nf.
Doc: transforms the prime ideal \var{pr} into \tet{modpr} format necessary
 for all operations modulo \var{pr} in the number field \var{nf}.
 The functions \tet{nfmodpr} and \tet{nfmodprlift} allow to project
 to and lift from the residue field.

Function: nfmodprlift
Class: basic
Section: number_fields
C-Name: nfmodprlift
Prototype: GGG
Help: nfmodprlift(nf,x,pr): lift x from residue field mod pr to nf.
Doc: lift the \typ{FFELT} $x$ (from \tet{nfmodpr}) to the residue field
 modulo \var{pr}. Vectors and matrices are also supported. For polynomials,
 use \kbd{apply} and the present function.
 
 The argument \kbd{pr} is either a maximal ideal in \kbd{idealprimedec}
 format or, preferably, a \kbd{modpr} structure from \tet{nfmodprinit}.
 There are no compatibility checks to try and decide whether $x$ is attached
 the same residue field as defined by \kbd{pr}: the result is undefined
 if not.
 
 The function \tet{nfmodpr} allows to reduce to the residue field.
 \bprog
 ? K = nfinit(y^3-250);
 ? P = idealprimedec(K, 5)[2]
 ? modP = nfmodprinit(K,P);
 ? K.zk
 %4 = [1, 1/5*y, 1/25*y^2]
 ? apply(t->nfmodpr(K,t,modP), K.zk)
 %5 = [1, y, 2*y + 1]
 ? nfmodprlift(K, %, modP)
 %6 = [1, 1/5*y, 2/5*y + 1]
 ? nfeltval(K, %[3] - K.zk[3], P)
 %7 = 1
 @eprog

Function: nfnewprec
Class: basic
Section: number_fields
C-Name: nfnewprec
Prototype: Gp
Help: nfnewprec(nf): transform the number field data nf into new data using
 the current (usually larger) precision.
Doc: transforms the number field $\var{nf}$
 into the corresponding data using current (usually larger) precision. This
 function works as expected if \var{nf} is in fact a \var{bnf} or a \var{bnr}
 (update structure to current precision) but may be quite slow: many
 generators of principal ideals have to be computed; note that in this latter
 case, the \var{bnf} must contain fundamental units.
Variant: See also \fun{GEN}{bnfnewprec}{GEN bnf, long prec} and
 \fun{GEN}{bnrnewprec}{GEN bnr, long prec}.

Function: nfroots
Class: basic
Section: number_fields
C-Name: nfroots
Prototype: DGG
Help: nfroots({nf},x): roots of polynomial x belonging to nf (Q if
 omitted) without multiplicity.
Doc: roots of the polynomial $x$ in the
 number field $\var{nf}$ given by \kbd{nfinit} without multiplicity (in $\Q$
 if $\var{nf}$ is omitted). $x$ has coefficients in the number field (scalar,
 polmod, polynomial, column vector). The main variable of $\var{nf}$ must be
 of lower priority than that of $x$ (see \secref{se:priority}). However if the
 coefficients of the number field occur explicitly (as polmods) as
 coefficients of $x$, the variable of these polmods \emph{must} be the same as
 the main variable of $t$ (see \kbd{nffactor}).
 
 It is possible to input a defining polynomial for \var{nf}
 instead, but this is in general less efficient since parts of an \kbd{nf}
 structure will then be computed internally. This is useful in two
 situations: when you do not need the \kbd{nf} elsewhere, or when you cannot
 initialize an \kbd{nf} due to integer factorization difficulties when
 attempting to compute the field discriminant and maximal order.
 
 \misctitle{Caveat} \kbd{nfinit([T, listP])} allows to compute in polynomial
 time a conditional \var{nf} structure, which sets \kbd{nf.zk} to an order
 which is not guaranteed to be maximal at all primes. Always either use
 \kbd{nfcertify} first (which may not run in polynomial time) or make sure
 to input \kbd{nf.pol} instead of the conditional \var{nf}: \kbd{nfroots} is
 able to recover in polynomial time in this case, instead of potentially
 missing a factor.
Variant: See also \fun{GEN}{nfrootsQ}{GEN x},
 corresponding to $\kbd{nf} = \kbd{NULL}$.

Function: nfrootsof1
Class: basic
Section: number_fields
C-Name: rootsof1
Prototype: G
Help: nfrootsof1(nf): number of roots of unity and primitive root of unity
 in the number field nf.
Doc: Returns a two-component vector $[w,z]$ where $w$ is the number of roots of
 unity in the number field \var{nf}, and $z$ is a primitive $w$-th root
 of unity.
 \bprog
 ? K = nfinit(polcyclo(11));
 ? nfrootsof1(K)
 %2 = [22, [0, 0, 0, 0, 0, -1, 0, 0, 0, 0]~]
 ? z = nfbasistoalg(K, %[2])   \\ in algebraic form
 %3 = Mod(-x^5, x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)
 ? [lift(z^11), lift(z^2)]     \\ proves that the order of z is 22
 %4 = [-1, -x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1]
 @eprog
 This function guesses the number $w$ as the gcd of the $\#k(v)^*$ for
 unramified $v$ above odd primes, then computes the roots in \var{nf}
 of the $w$-th cyclotomic polynomial: the algorithm is polynomial time with
 respect to the field degree and the bitsize of the multiplication table in
 \var{nf} (both of them polynomially bounded in terms of the size of the
 discriminant). Fields of degree up to $100$ or so should require less than
 one minute.
Variant: Also available is \fun{GEN}{rootsof1_kannan}{GEN nf}, that computes
 all algebraic integers of $T_2$ norm equal to the field degree
 (all roots of $1$, by Kronecker's theorem). This is in general a little
 faster than the default when there \emph{are} roots of $1$ in the field
 (say twice faster), but can be much slower (say, \emph{days} slower), since
 the algorithm is a priori exponential in the field degree.

Function: nfsnf
Class: basic
Section: number_fields
C-Name: nfsnf0
Prototype: GGD0,L,
Help: nfsnf(nf,x,{flag=0}): if x=[A,I,J], outputs D=[d_1,...d_n] Smith normal
 form of x. If flag is non-zero return [D,U,V], where UAV = Id.
Doc: given a torsion $\Z_K$-module $x$ attached to the square integral
 invertible pseudo-matrix $(A,I,J)$, returns an ideal list
 $D=[d_1,\dots,d_n]$ which is the \idx{Smith normal form} of $x$. In other
 words, $x$ is isomorphic to $\Z_K/d_1\oplus\cdots\oplus\Z_K/d_n$ and $d_i$
 divides $d_{i-1}$ for $i\ge2$. If $\fl$ is non-zero return $[D,U,V]$, where
 $UAV$ is the identity.
 
 See \secref{se:ZKmodules} for the definition of integral pseudo-matrix;
 briefly, it is input as a 3-component row vector $[A,I,J]$ where
 $I = [b_1,\dots,b_n]$ and $J = [a_1,\dots,a_n]$ are two ideal lists,
 and $A$ is a square $n\times n$ matrix with columns $(A_1,\dots,A_n)$,
 seen as elements in $K^n$ (with canonical basis $(e_1,\dots,e_n)$).
 This data defines the $\Z_K$ module $x$ given by
 $$ (b_1e_1\oplus\cdots\oplus b_ne_n) / (a_1A_1\oplus\cdots\oplus a_nA_n)
 \enspace, $$
 The integrality condition is $a_{i,j} \in b_i a_j^{-1}$ for all $i,j$. If it
 is not satisfied, then the $d_i$ will not be integral. Note that every
 finitely generated torsion module is isomorphic to a module of this form and
 even with $b_i=Z_K$ for all $i$.
Variant: Also available:
 
 \fun{GEN}{nfsnf}{GEN nf, GEN x} ($\fl = 0$).

Function: nfsolvemodpr
Class: basic
Section: number_fields
C-Name: nfsolvemodpr
Prototype: GGGG
Help: nfsolvemodpr(nf,a,b,P): this function is obsolete, use nfmodpr.
Doc: this function is obsolete, use \kbd{nfmodpr}.
 
 Let $P$ be a prime ideal in \key{modpr} format (see \kbd{nfmodprinit}),
 let $a$ be a matrix, invertible over the residue field, and let $b$ be
 a column vector or matrix. This function returns a solution of $a\cdot x =
 b$; the coefficients of $x$ are lifted to \var{nf} elements.
 \bprog
 ? K = nfinit(y^2+1);
 ? P = idealprimedec(K, 3)[1];
 ? P = nfmodprinit(K, P);
 ? a = [y+1, y; y, 0]; b = [1, y]~
 ? nfsolvemodpr(K, a,b, P)
 %5 = [1, 2]~
 @eprog
Obsolete: 2016-08-09
Variant: This function is normally useless in library mode. Project your
 inputs to the residue field using \kbd{nfM\_to\_FqM}, then work there.

Function: nfsplitting
Class: basic
Section: number_fields
C-Name: nfsplitting
Prototype: GDG
Help: nfsplitting(nf,{d}): defining polynomial over Q for the splitting field of
 the number field nf; if d is given, it must be a multiple of the splitting
 field degree.
Doc: defining polynomial over~$\Q$ for the splitting field of \var{nf};
 if $d$ is given, it must be a multiple of the splitting field degree.
 Instead of~\kbd{nf}, it is possible to input a defining (irreducible)
 polynomial $T$ for~\kbd{nf}, but in general this is less efficient.
 
 \bprog
 ? K = nfinit(x^3-2);
 ? nfsplitting(K)
 %2 = x^6 + 108
 ?  nfsplitting(x^8-2)
 %3 = x^16 + 272*x^8 + 64
 @eprog
 \noindent
 Specifying the degree of the splitting field can make the computation faster.
 \bprog
 ? nfsplitting(x^17-123);
 time = 3,607 ms.
 ? poldegree(%)
 %2 = 272
 ? nfsplitting(x^17-123,272);
 time = 150 ms.
 ? nfsplitting(x^17-123,273);
  *** nfsplitting: Warning: ignoring incorrect degree bound 273
 time = 3,611 ms.
 @eprog
 \noindent
 The complexity of the algorithm is polynomial in the degree $d$ of the
 splitting field and the bitsize of $T$; if $d$ is large the result will
 likely be unusable, e.g. \kbd{nfinit} will not be an option:
 \bprog
 ? nfsplitting(x^6-x-1)
 [... degree 720 polynomial deleted ...]
 time = 11,020 ms.
 @eprog

Function: nfsubfields
Class: basic
Section: number_fields
C-Name: nfsubfields
Prototype: GD0,L,
Help: nfsubfields(pol,{d=0}): find all subfields of degree d of number field
 defined by pol (all subfields if d is null or omitted). Result is a vector of
 subfields, each being given by [g,h], where g is an absolute equation and h
 expresses one of the roots of g in terms of the root x of the polynomial
 defining nf.
Doc: finds all subfields of degree
 $d$ of the number field defined by the (monic, integral) polynomial
 \var{pol} (all subfields if $d$ is null or omitted). The result is a vector
 of subfields, each being given by $[g,h]$, where $g$ is an absolute equation
 and $h$ expresses one of the roots of $g$ in terms of the root $x$ of the
 polynomial defining $\var{nf}$. This routine uses J.~Kl\"uners's algorithm
 in the general case, and B.~Allombert's \tet{galoissubfields} when \var{nf}
 is Galois (with weakly supersolvable Galois group).\sidx{Galois}\sidx{subfield}

Function: norm
Class: basic
Section: conversions
C-Name: gnorm
Prototype: G
Help: norm(x): norm of x.
Doc: 
 algebraic norm of $x$, i.e.~the product of $x$ with
 its conjugate (no square roots are taken), or conjugates for polmods. For
 vectors and matrices, the norm is taken componentwise and hence is not the
 $L^2$-norm (see \kbd{norml2}). Note that the norm of an element of
 $\R$ is its square, so as to be compatible with the complex norm.

Function: norml2
Class: basic
Section: linear_algebra
C-Name: gnorml2
Prototype: G
Help: norml2(x): square of the L2-norm of x.
Doc: square of the $L^2$-norm of $x$. More precisely,
 if $x$ is a scalar, $\kbd{norml2}(x)$ is defined to be the square
 of the complex modulus of $x$ (real \typ{QUAD}s are not supported).
 If $x$ is a polynomial, a (row or column) vector or a matrix, \kbd{norml2($x$)} is
 defined recursively as $\sum_i \kbd{norml2}(x_i)$, where $(x_i)$ run through
 the components of $x$. In particular, this yields the usual $\sum |x_i|^2$
 (resp.~$\sum |x_{i,j}|^2$) if $x$ is a polynomial or vector (resp.~matrix) with
 complex components.
 
 \bprog
 ? norml2( [ 1, 2, 3 ] )      \\ vector
 %1 = 14
 ? norml2( [ 1, 2; 3, 4] )   \\ matrix
 %2 = 30
 ? norml2( 2*I + x )
 %3 = 5
 ? norml2( [ [1,2], [3,4], 5, 6 ] )   \\ recursively defined
 %4 = 91
 @eprog

Function: normlp
Class: basic
Section: linear_algebra
C-Name: gnormlp
Prototype: GDGp
Help: normlp(x,{p=oo}): Lp-norm of x; sup norm if p is omitted.
Doc: 
 $L^p$-norm of $x$; sup norm if $p$ is omitted or \kbd{+oo}. More precisely,
 if $x$ is a scalar, \kbd{normlp}$(x, p)$ is defined to be \kbd{abs}$(x)$.
 If $x$ is a polynomial, a (row or column) vector or a matrix:
 
 \item  if $p$ is omitted or \kbd{+oo}, then \kbd{normlp($x$)} is defined
 recursively as $\max_i \kbd{normlp}(x_i))$, where $(x_i)$ run through the
 components of~$x$. In particular, this yields the usual sup norm if $x$ is a
 polynomial or vector with complex components.
 
 \item otherwise, \kbd{normlp($x$, $p$)} is defined recursively as $(\sum_i
 \kbd{normlp}^p(x_i,p))^{1/p}$. In particular, this yields the usual $(\sum
 |x_i|^p)^{1/p}$ if $x$ is a polynomial or vector with complex components.
 
 \bprog
 ? v = [1,-2,3]; normlp(v)      \\ vector
 %1 = 3
 ? normlp(v, +oo)               \\ same, more explicit
 %2 = 3
 ? M = [1,-2;-3,4]; normlp(M)   \\ matrix
 %3 = 4
 ? T = (1+I) + I*x^2; normlp(T)
 %4 = 1.4142135623730950488016887242096980786
 ? normlp([[1,2], [3,4], 5, 6])   \\ recursively defined
 %5 = 6
 
 ? normlp(v, 1)
 %6 = 6
 ? normlp(M, 1)
 %7 = 10
 ? normlp(T, 1)
 %8 = 2.4142135623730950488016887242096980786
 @eprog

Function: numbpart
Class: basic
Section: number_theoretical
C-Name: numbpart
Prototype: G
Help: numbpart(n): number of partitions of n.
Doc: gives the number of unrestricted partitions of
 $n$, usually called $p(n)$ in the literature; in other words the number of
 nonnegative integer solutions to $a+2b+3c+\cdots=n$. $n$ must be of type
 integer and $n<10^{15}$ (with trivial values $p(n) = 0$ for $n < 0$ and
 $p(0) = 1$). The algorithm uses the Hardy-Ramanujan-Rademacher formula.
 To explicitly enumerate them, see \tet{partitions}.

Function: numdiv
Class: basic
Section: number_theoretical
C-Name: numdiv
Prototype: G
Help: numdiv(x): number of divisors of x.
Description: 
 (gen):int        numdiv($1)
Doc: number of divisors of $|x|$. $x$ must be of type integer.

Function: numerator
Class: basic
Section: conversions
C-Name: numer
Prototype: G
Help: numerator(x): numerator of x.
Doc: 
 numerator of $x$. The meaning of this
 is clear when $x$ is a rational number or function. If $x$ is an integer
 or a polynomial, it is treated as a rational number or function,
 respectively, and the result is $x$ itself. For polynomials, you
 probably want to use
 \bprog
 numerator( content(x) )
 @eprog\noindent
 instead.
 
 In other cases, \kbd{numerator(x)} is defined to be
 \kbd{denominator(x)*x}. This is the case when $x$ is a vector or a
 matrix, but also for \typ{COMPLEX} or \typ{QUAD}. In particular since a
 \typ{PADIC} or \typ{INTMOD} has  denominator $1$, its numerator is
 itself.
 
 \misctitle{Warning} Multivariate objects are created according to variable
 priorities, with possibly surprising side effects ($x/y$ is a polynomial, but
 $y/x$ is a rational function). See \secref{se:priority}.

Function: numtoperm
Class: basic
Section: conversions
C-Name: numtoperm
Prototype: LG
Help: numtoperm(n,k): permutation number k (mod n!) of n letters (n
 C-integer).
Doc: generates the $k$-th permutation (as a row vector of length $n$) of the
 numbers $1$ to $n$. The number $k$ is taken modulo $n!\,$, i.e.~inverse
 function of \tet{permtonum}. The numbering used is the standard lexicographic
 ordering, starting at $0$.

Function: omega
Class: basic
Section: number_theoretical
C-Name: omega
Prototype: lG
Help: omega(x): number of distinct prime divisors of x.
Doc: number of distinct prime divisors of $|x|$. $x$ must be of type integer.
 \bprog
 ? factor(392)
 %1 =
 [2 3]
 
 [7 2]
 
 ? omega(392)
 %2 = 2;  \\ without multiplicity
 ? bigomega(392)
 %3 = 5;  \\ = 3+2, with multiplicity
 @eprog

Function: oo
Class: basic
Section: conversions
C-Name: mkoo
Prototype: 
Help: oo=oo(): infinity.
Description: 
Doc: returns an object meaning $+\infty$, for use in functions such as
 \kbd{intnum}. It can be negated (\kbd{-oo} represents $-\infty$), and
 compared to real numbers (\typ{INT}, \typ{FRAC}, \typ{REAL}), with the
 expected meaning: $+\infty$ is greater than any real number and $-\infty$ is
 smaller.

Function: padicappr
Class: basic
Section: polynomials
C-Name: padicappr
Prototype: GG
Help: padicappr(pol,a): p-adic roots of the polynomial pol congruent to a mod p.
Doc: vector of $p$-adic roots of the
 polynomial $pol$ congruent to the $p$-adic number $a$ modulo $p$, and with
 the same $p$-adic precision as $a$. The number $a$ can be an ordinary
 $p$-adic number (type \typ{PADIC}, i.e.~an element of $\Z_p$) or can be an
 integral element of a finite extension of $\Q_p$, given as a \typ{POLMOD}
 at least one of whose coefficients is a \typ{PADIC}. In this case, the result
 is the vector of roots belonging to the same extension of $\Q_p$ as $a$.
Variant: Also available is \fun{GEN}{Zp_appr}{GEN f, GEN a} when $a$ is a
 \typ{PADIC}.

Function: padicfields
Class: basic
Section: polynomials
C-Name: padicfields0
Prototype: GGD0,L,
Help: padicfields(p, N, {flag=0}): returns polynomials generating all
 the extensions of degree N of the field of p-adic rational numbers; N is
 allowed to be a 2-component vector [n,d], in which case, returns the
 extensions of degree n and discriminant p^d. flag is optional,
 and can be 0: default, 1: return also the ramification index, the residual
 degree, the valuation of the discriminant and the number of conjugate fields,
 or 2: return only the number of extensions in a fixed algebraic closure.
Doc: returns a vector of polynomials generating all the extensions of degree
 $N$ of the field $\Q_p$ of $p$-adic rational numbers; $N$ is
 allowed to be a 2-component vector $[n,d]$, in which case we return the
 extensions of degree $n$ and discriminant $p^d$.
 
 The list is minimal in the sense that two different polynomials generate
 non-isomorphic extensions; in particular, the number of polynomials is the
 number of classes of non-isomorphic extensions. If $P$ is a polynomial in this
 list, $\alpha$ is any root of $P$ and $K = \Q_p(\alpha)$, then $\alpha$
 is the sum of a uniformizer and a (lift of a) generator of the residue field
 of $K$; in particular, the powers of $\alpha$ generate the ring of $p$-adic
 integers of $K$.
 
 If $\fl = 1$, replace each polynomial $P$ by a vector $[P, e, f, d, c]$
 where $e$ is the ramification index, $f$ the residual degree, $d$ the
 valuation of the discriminant, and $c$ the number of conjugate fields.
 If $\fl = 2$, only return the \emph{number} of extensions in a fixed
 algebraic closure (Krasner's formula), which is much faster.
Variant: Also available is
 \fun{GEN}{padicfields}{GEN p, long n, long d, long flag}, which computes
 extensions of $\Q_p$ of degree $n$ and discriminant $p^d$.

Function: padicprec
Class: basic
Section: conversions
C-Name: gppadicprec
Prototype: GG
Help: padicprec(x,p):
 return the absolute p-adic precision of object x.
Doc: returns the absolute $p$-adic precision of the object $x$; this is the
 minimum precision of the components of $x$. The result is \tet{+oo} if $x$
 is an exact object (as a $p$-adic):
 \bprog
 ? padicprec((1 + O(2^5)) * x + (2 + O(2^4)), 2)
 %1 = 4
 ? padicprec(x + 2, 2)
 %2 = +oo
 ? padicprec(2 + x + O(x^2), 2)
 %3 = +oo
 @eprog\noindent The function raises an exception if it encounters
 an object incompatible with $p$-adic computations:
 \bprog
 ? padicprec(O(3), 2)
  ***   at top-level: padicprec(O(3),2)
  ***                 ^-----------------
  *** padicprec: inconsistent moduli in padicprec: 3 != 2
 
 ? padicprec(1.0, 2)
  ***   at top-level: padicprec(1.0,2)
  ***                 ^----------------
  *** padicprec: incorrect type in padicprec (t_REAL).
 @eprog
Variant: Also available is the function \fun{long}{padicprec}{GEN x, GEN p},
 which returns \tet{LONG_MAX} if $x = 0$ and the $p$-adic precision as a
 \kbd{long} integer.

Function: parapply
Class: basic
Section: programming/parallel
C-Name: parapply
Prototype: GG
Help: parapply(f, x): parallel evaluation of f on the elements of x.
Doc: parallel evaluation of \kbd{f} on the elements of \kbd{x}.
 The function \kbd{f} must not access global variables or variables
 declared with local(), and must be free of side effects.
 \bprog
 parapply(factor,[2^256 + 1, 2^193 - 1])
 @eprog
 factors $2^{256} + 1$ and $2^{193} - 1$ in parallel.
 \bprog
 {
   my(E = ellinit([1,3]), V = vector(12,i,randomprime(2^200)));
   parapply(p->ellcard(E,p), V)
 }
 @eprog
 computes the order of $E(\F_p)$ for $12$ random primes of $200$ bits.

Function: pareval
Class: basic
Section: programming/parallel
C-Name: pareval
Prototype: G
Help: pareval(x): parallel evaluation of the elements of the vector of
 closures x.
Doc: parallel evaluation of the elements of \kbd{x}, where \kbd{x} is a
 vector of closures. The closures must be of arity $0$, must not access
 global variables or variables declared with \kbd{local} and must be
 free of side effects.

Function: parfor
Class: basic
Section: programming/parallel
C-Name: parfor0
Prototype: vV=GDGJDVDI
Help: parfor(i=a,{b},expr1,{r},{expr2}):
 evaluates the expression expr1 in parallel for all i between a and b
 (if b is set to +oo, the loop will not stop), resulting in as many
 values; if the formal variables r and expr2 are present, evaluate
 sequentially expr2, in which r has been replaced by the different results
 obtained for expr1 and i with the corresponding arguments.
Description: 
 (gen,gen,closure):void parfor($1, $2, $3, NULL, NULL)
Doc: evaluates in parallel the expression \kbd{expr1} in the formal
 argument $i$ running from $a$ to $b$.
 If $b$ is set to \kbd{+oo}, the loop runs indefinitely.
 If $r$ and \kbd{expr2} are present, the expression \kbd{expr2} in the
 formal variables $r$ and $i$ is evaluated with $r$ running through all
 the different results obtained for \kbd{expr1} and $i$ takes the
 corresponding argument.
 
 The computations of \kbd{expr1} are \emph{started} in increasing order
 of $i$; otherwise said, the computation for $i=c$ is started after those
 for $i=1, \ldots, c-1$ have been started, but before the computation for
 $i=c+1$ is started. Notice that the order of \emph{completion}, that is,
 the order in which the different $r$ become available, may be different;
 \kbd{expr2} is evaluated sequentially on each $r$ as it appears.
 
 The following example computes the sum of the squares of the integers
 from $1$ to $10$ by computing the squares in parallel and is equivalent
 to \kbd{parsum (i=1, 10, i\^{}2)}:
 \bprog
 ? s=0;
 ? parfor (i=1, 10, i^2, r, s=s+r)
 ? s
 %3 = 385
 @eprog
 More precisely, apart from a potentially different order of evaluation
 due to the parallelism, the line containing \kbd{parfor} is equivalent to
 \bprog
 ? my (r); for (i=1, 10, r=i^2; s=s+r)
 @eprog
 The sequentiality of the evaluation of \kbd{expr2} ensures that the
 variable \kbd{s} is not modified concurrently by two different additions,
 although the order in which the terms are added is non-deterministic.
 
 It is allowed for \kbd{expr2} to exit the loop using
 \kbd{break}/\kbd{next}/\kbd{return}. If that happens for $i=c$,
 then the evaluation of \kbd{expr1} and \kbd{expr2} is continued
 for all values $i<c$, and the return value is the one obtained for
 the smallest $i$ causing an interruption in \kbd{expr2} (it may be
 undefined if this is a \kbd{break}/\kbd{next}).
 In that case, using side-effects
 in \kbd{expr2} may lead to undefined behavior, as the exact
 number of values of $i$ for which it is executed is non-deterministic.
 The following example computes \kbd{nextprime(1000)} in parallel:
 \bprog
 ? parfor (i=1000, , isprime (i), r, if (r, return (i)))
 %1 = 1009
 @eprog
 
 %\syn{NO}

Function: parforprime
Class: basic
Section: programming/parallel
C-Name: parforprime0
Prototype: vV=GDGJDVDI
Help: parforprime(p=a,{b},expr1,{r},{expr2}):
 evaluates the expression expr1 in parallel for all primes p between a and b
 (if b is set to +oo, the loop will not stop), resulting in as many
 values; if the formal variables r and expr2 are present, evaluate
 sequentially expr2, in which r has been replaced by the different results
 obtained for expr1 and p with the corresponding arguments.
Description: 
 (gen,gen,closure):void parforprime($1, $2, $3, NULL, NULL)
Doc: 
 behaves exactly as \kbd{parfor}, but loops only over prime values $p$.
 Precisely, the functions evaluates in parallel the expression \kbd{expr1}
 in the formal
 argument $p$ running through the primes from $a$ to $b$.
 If $b$ is set to \kbd{+oo}, the loop runs indefinitely.
 If $r$ and \kbd{expr2} are present, the expression \kbd{expr2} in the
 formal variables $r$ and $p$ is evaluated with $r$ running through all
 the different results obtained for \kbd{expr1} and $p$ takes the
 corresponding argument.
 
 It is allowed fo \kbd{expr2} to exit the loop using
 \kbd{break}/\kbd{next}/\kbd{return}; see the remarks in the documentation
 of \kbd{parfor} for details.
 
 %\syn{NO}

Function: parforvec
Class: basic
Section: programming/parallel
C-Name: parforvec0
Prototype: vV=GJDVDID0,L,
Help: parforvec(X=v,expr1,{j},{expr2},{flag}): evaluates the sequence expr2
 (dependent on X and j) for X as generated by forvec, in random order,
 computed in parallel. Substitute for j the value of expr1 (dependent on X).
Description: 
 (gen,closure,,,?small):void parforvec($1, $2, $5, NULL, NULL)
Doc: evaluates the sequence \kbd{expr2} (dependent on $X$ and $j$) for $X$
 as generated by \kbd{forvec}, in random order, computed in parallel. Substitute
 for $j$ the value of \kbd{expr1} (dependent on $X$).
 
 It is allowed fo \kbd{expr2} to exit the loop using
 \kbd{break}/\kbd{next}/\kbd{return}, however in that case, \kbd{expr2} will
 still be evaluated for all remaining value of $p$ less than the current one,
 unless a subsequent \kbd{break}/\kbd{next}/\kbd{return} happens.
 %\syn{NO}

Function: parselect
Class: basic
Section: programming/parallel
C-Name: parselect
Prototype: GGD0,L,
Help: parselect(f, A, {flag = 0}): (parallel select) selects elements of A
 according to the selection function f which is tested in parallel. If flag
 is 1, return the indices of those elements (indirect selection).
Doc: selects elements of $A$ according to the selection function $f$, done in
 parallel.  If \fl is $1$, return the indices of those elements (indirect
 selection) The function \kbd{f} must not access global variables or
 variables declared with local(), and must be free of side effects.

Function: parsum
Class: basic
Section: programming/parallel
C-Name: parsum
Prototype: V=GGJDG
Help: parsum(i=a,b,expr,{x}): x plus the sum (X goes from a to b) of
 expression expr, evaluated in parallel (in random order).
Description: 
 (gen,gen,closure,?gen):gen parsum($1, $2, $3, $4)
Doc: sum of expression \var{expr}, initialized at $x$, the formal parameter
 going from $a$ to $b$, evaluated in parallel in random order.
 The expression \kbd{expr} must not access global variables or
 variables declared with \kbd{local()}, and must be free of side effects.
 \bprog
 parsum(i=1,1000,ispseudoprime(2^prime(i)-1))
 @eprog
 returns the numbers of prime numbers among the first $1000$ Mersenne numbers.
 %\syn{NO}

Function: partitions
Class: basic
Section: number_theoretical
C-Name: partitions
Prototype: LDGDG
Help: partitions(k,{a=k},{n=k})): vector of partitions of the integer k.
 You can restrict the length of the partitions with parameter n (n=nmax or
 n=[nmin,nmax]), or the range of the parts with parameter a (a=amax
 or a=[amin,amax]). By default remove zeros, but one can set amin=0 to get X of
 fixed length nmax (=k by default).
Doc: returns the vector of partitions of the integer $k$ as a sum of positive
 integers (parts); for $k < 0$, it returns the empty set \kbd{[]}, and for $k
 = 0$ the trivial partition (no parts). A partition is given by a
 \typ{VECSMALL}, where parts are sorted in nondecreasing order:
 \bprog
 ? partitions(3)
 %1 = [Vecsmall([3]), Vecsmall([1, 2]), Vecsmall([1, 1, 1])]
 @eprog\noindent correspond to $3$, $1+2$ and $1+1+1$. The number
 of (unrestricted) partitions of $k$ is given
 by \tet{numbpart}:
 \bprog
 ? #partitions(50)
 %1 = 204226
 ? numbpart(50)
 %2 = 204226
 @eprog
 
 \noindent Optional parameters $n$ and $a$ are as follows:
 
 \item $n=\var{nmax}$ (resp. $n=[\var{nmin},\var{nmax}]$) restricts
 partitions to length less than $\var{nmax}$ (resp. length between
 $\var{nmin}$ and $nmax$), where the \emph{length} is the number of nonzero
 entries.
 
 \item $a=\var{amax}$ (resp. $a=[\var{amin},\var{amax}]$) restricts the parts
 to integers less than $\var{amax}$ (resp. between $\var{amin}$ and
 $\var{amax}$).
 \bprog
 ? partitions(4, 2)  \\ parts bounded by 2
 %1 = [Vecsmall([2, 2]), Vecsmall([1, 1, 2]), Vecsmall([1, 1, 1, 1])]
 ? partitions(4,, 2) \\ at most 2 parts
 %2 = [Vecsmall([4]), Vecsmall([1, 3]), Vecsmall([2, 2])]
 ? partitions(4,[0,3], 2) \\ at most 2 parts
 %3 = [Vecsmall([4]), Vecsmall([1, 3]), Vecsmall([2, 2])]
 @eprog\noindent
 By default, parts are positive and we remove zero entries unless
 $amin\leq0$, in which case $nmin$ is ignored and $X$ is of constant length
 $\var{nmax}$:
 \bprog
 ? partitions(4, [0,3])  \\ parts between 0 and 3
 %1 = [Vecsmall([0, 0, 1, 3]), Vecsmall([0, 0, 2, 2]),\
       Vecsmall([0, 1, 1, 2]), Vecsmall([1, 1, 1, 1])]
 @eprog

Function: parvector
Class: basic
Section: programming/parallel
C-Name: parvector
Prototype: LVJ
Help: parvector(N,i,expr): as vector(N,i,expr) but the evaluations of expr are
 done in parallel.
Description: 
  (small,,closure):vec    parvector($1, $3)
Doc: As \kbd{vector(N,i,expr)} but the evaluations of \kbd{expr} are done in
 parallel. The expression \kbd{expr} must not access global variables or
 variables declared with \kbd{local()}, and must be free of side effects.
 \bprog
 parvector(10,i,quadclassunit(2^(100+i)+1).no)
 @eprog\noindent
 computes the class numbers in parallel.
 %\syn{NO}

Function: permtonum
Class: basic
Section: conversions
C-Name: permtonum
Prototype: G
Help: permtonum(x): ordinal (between 1 and n!) of permutation x.
Doc: given a permutation $x$ on $n$ elements, gives the number $k$ such that
 $x=\kbd{numtoperm(n,k)}$, i.e.~inverse function of \tet{numtoperm}.
 The numbering used is the standard lexicographic ordering, starting at $0$.

Function: plot
Class: basic
Section: graphic
C-Name: pariplot
Prototype: vV=GGEDGDGp
Help: plot(X=a,b,expr,{Ymin},{Ymax}): crude plot of expression expr, X goes
 from a to b, with Y ranging from Ymin to Ymax. If Ymin (resp. Ymax) is not
 given, the minimum (resp. the maximum) of the expression is used instead.
Doc: crude ASCII plot of the function represented by expression \var{expr}
 from $a$ to $b$, with \var{Y} ranging from \var{Ymin} to \var{Ymax}. If
 \var{Ymin} (resp. \var{Ymax}) is not given, the minimum (resp. the maximum)
 of the computed values of the expression is used instead.

Function: plotbox
Class: highlevel
Section: graphic
C-Name: rectbox
Prototype: vLGG
Help: plotbox(w,x2,y2): if the cursor is at position (x1,y1), draw a box
 with diagonal (x1,y1) and (x2,y2) in rectwindow w (cursor does not move).
Doc: let $(x1,y1)$ be the current position of the virtual cursor. Draw in the
 rectwindow $w$ the outline of the rectangle which is such that the points
 $(x1,y1)$ and $(x2,y2)$ are opposite corners. Only the part of the rectangle
 which is in $w$ is drawn. The virtual cursor does \emph{not} move.

Function: plotclip
Class: highlevel
Section: graphic
C-Name: rectclip
Prototype: vL
Help: plotclip(w): clip the contents of the rectwindow to the bounding box
 (except strings).
Doc: `clips' the content of rectwindow $w$, i.e remove all parts of the
 drawing that would not be visible on the screen. Together with
 \tet{plotcopy} this function enables you to draw on a scratchpad before
 committing the part you're interested in to the final picture.

Function: plotcolor
Class: highlevel
Section: graphic
C-Name: rectcolor
Prototype: vLL
Help: plotcolor(w,c): in rectwindow w, set default color to c. Possible
 values for c are given by the graphcolormap default: factory settings
 are 1=black, 2=blue, 3=sienna, 4=red, 5=green, 6=grey, 7=gainsborough.
Doc: set default color to $c$ in rectwindow $w$.
 This is only implemented for the X-windows, fltk and Qt graphing engines.
 Possible values for $c$ are given by the \tet{graphcolormap} default,
 factory setting are
 
 1=black, 2=blue, 3=violetred, 4=red, 5=green, 6=grey, 7=gainsborough.
 
 but this can be considerably extended.

Function: plotcopy
Class: highlevel
Section: graphic
C-Name: rectcopy_gen
Prototype: vLLGGD0,L,
Help: plotcopy(sourcew,destw,dx,dy,{flag=0}): copy the contents of
 rectwindow sourcew to rectwindow destw with offset (dx,dy). If flag's bit 1
 is set, dx and dy express fractions of the size of the current output
 device, otherwise dx and dy are in pixels. dx and dy are relative positions
 of northwest corners if other bits of flag vanish, otherwise of: 2:
 southwest, 4: southeast, 6: northeast corners.
Doc: copy the contents of rectwindow \var{sourcew} to rectwindow \var{destw}
 with offset (dx,dy). If flag's bit 1 is set, dx and dy express fractions of
 the size of the current output device, otherwise dx and dy are in pixels. dx
 and dy are relative positions of northwest corners if other bits of flag
 vanish, otherwise of: 2: southwest, 4: southeast, 6: northeast corners

Function: plotcursor
Class: highlevel
Section: graphic
C-Name: rectcursor
Prototype: L
Help: plotcursor(w): current position of cursor in rectwindow w.
Doc: give as a 2-component vector the current
 (scaled) position of the virtual cursor corresponding to the rectwindow $w$.

Function: plotdraw
Class: highlevel
Section: graphic
C-Name: rectdraw_flag
Prototype: vGD0,L,
Help: plotdraw(list, {flag=0}): draw vector of rectwindows list at indicated
 x,y positions; list is a vector w1,x1,y1,w2,x2,y2,etc. If flag!=0, x1, y1
 etc. express fractions of the size of the current output device.
Doc: physically draw the rectwindows given in $list$
 which must be a vector whose number of components is divisible by 3. If
 $list=[w1,x1,y1,w2,x2,y2,\dots]$, the windows $w1$, $w2$, etc.~are
 physically placed with their upper left corner at physical position
 $(x1,y1)$, $(x2,y2)$,\dots\ respectively, and are then drawn together.
 Overlapping regions will thus be drawn twice, and the windows are considered
 transparent. Then display the whole drawing in a special window on your
 screen. If $\fl \neq 0$, x1, y1 etc. express fractions of the size of the
 current output device

Function: ploth
Class: highlevel
Section: graphic
C-Name: ploth
Prototype: V=GGEpD0,M,D0,L,\nParametric|1; Recursive|2; no_Rescale|4; no_X_axis|8; no_Y_axis|16; no_Frame|32; no_Lines|64; Points_too|128; Splines|256; no_X_ticks|512; no_Y_ticks|1024; Same_ticks|2048; Complex|4096
Help: ploth(X=a,b,expr,{flags=0},{n=0}): plot of expression expr, X goes
 from a to b in high resolution. Both flags and n are optional. Binary digits
 of flags mean: 1=Parametric, 2=Recursive, 4=no_Rescale, 8=no_X_axis,
 16=no_Y_axis, 32=no_Frame, 64=no_Lines (do not join points), 128=Points_too
 (plot both lines and points), 256=Splines (use cubic splines),
 512=no_X_ticks, 1024= no_Y_ticks, 2048=Same_ticks (plot all ticks with the
 same length), 4096=Complex (the two coordinates of each point are encoded
 as a complex number). n specifies number of reference points on the graph
 (0=use default value). Returns a vector for the bounding box.
Doc: high precision plot of the function $y=f(x)$ represented by the expression
 \var{expr}, $x$ going from $a$ to $b$. This opens a specific window (which is
 killed whenever you click on it), and returns a four-component vector giving
 the coordinates of the bounding box in the form
 $[\var{xmin},\var{xmax},\var{ymin},\var{ymax}]$.
 
 \misctitle{Important note} \kbd{ploth} may evaluate \kbd{expr} thousands of
 times; given the relatively low resolution of plotting devices, few
 significant digits of the result will be meaningful. Hence you should keep
 the current precision to a minimum (e.g.~9) before calling this function.
 
 $n$ specifies the number of reference point on the graph, where a value of 0
 means we use the hardwired default values (1000 for general plot, 1500 for
 parametric plot, and 8 for recursive plot).
 
 If no $\fl$ is given, \var{expr} is either a scalar expression $f(X)$, in which
 case the plane curve $y=f(X)$ will be drawn, or a vector
 $[f_1(X),\dots,f_k(X)]$, and then all the curves $y=f_i(X)$ will be drawn in
 the same window.
 
 \noindent The binary digits of $\fl$ mean:
 
 \item $1 = \kbd{Parametric}$: \tev{parametric plot}. Here \var{expr} must
 be a vector with an even number of components. Successive pairs are then
 understood as the parametric coordinates of a plane curve. Each of these are
 then drawn.
 
 For instance:
 \bprog
 ploth(X=0,2*Pi,[sin(X),cos(X)], "Parametric")
 ploth(X=0,2*Pi,[sin(X),cos(X)])
 ploth(X=0,2*Pi,[X,X,sin(X),cos(X)], "Parametric")
 @eprog\noindent draw successively a circle, two entwined sinusoidal curves
 and a circle cut by the line $y=x$.
 
 \item $2 = \kbd{Recursive}$: \tev{recursive plot}. If this flag is set,
 only \emph{one} curve can be drawn at a time, i.e.~\var{expr} must be either a
 two-component vector (for a single parametric curve, and the parametric flag
 \emph{has} to be set), or a scalar function. The idea is to choose pairs of
 successive reference points, and if their middle point is not too far away
 from the segment joining them, draw this as a local approximation to the
 curve. Otherwise, add the middle point to the reference points. This is
 fast, and usually more precise than usual plot. Compare the results of
 \bprog
 ploth(X=-1,1, sin(1/X), "Recursive")
 ploth(X=-1,1, sin(1/X))
 @eprog\noindent
 for instance. But beware that if you are extremely unlucky, or choose too few
 reference points, you may draw some nice polygon bearing little resemblance
 to the original curve. For instance you should \emph{never} plot recursively
 an odd function in a symmetric interval around 0. Try
 \bprog
 ploth(x = -20, 20, sin(x), "Recursive")
 @eprog\noindent
 to see why. Hence, it's usually a good idea to try and plot the same curve
 with slightly different parameters.
 
 The other values toggle various display options:
 
 \item $4 = \kbd{no\_Rescale}$: do not rescale plot according to the
 computed extrema. This is used in conjunction with \tet{plotscale} when
 graphing multiple functions on a rectwindow (as a \tet{plotrecth} call):
 \bprog
   s = plothsizes();
   plotinit(0, s[2]-1, s[2]-1);
   plotscale(0, -1,1, -1,1);
   plotrecth(0, t=0,2*Pi, [cos(t),sin(t)], "Parametric|no_Rescale")
   plotdraw([0, -1,1]);
 @eprog\noindent
 This way we get a proper circle instead of the distorted ellipse produced by
 \bprog
   ploth(t=0,2*Pi, [cos(t),sin(t)], "Parametric")
 @eprog
 
 \item $8 = \kbd{no\_X\_axis}$: do not print the $x$-axis.
 
 \item $16 = \kbd{no\_Y\_axis}$: do not print the $y$-axis.
 
 \item $32 = \kbd{no\_Frame}$: do not print frame.
 
 \item $64 = \kbd{no\_Lines}$: only plot reference points, do not join them.
 
 \item $128 = \kbd{Points\_too}$: plot both lines and points.
 
 \item $256 = \kbd{Splines}$: use splines to interpolate the points.
 
 \item $512 = \kbd{no\_X\_ticks}$: plot no $x$-ticks.
 
 \item $1024 = \kbd{no\_Y\_ticks}$: plot no $y$-ticks.
 
 \item $2048 = \kbd{Same\_ticks}$: plot all ticks with the same length.
 
 \item $4096 = \kbd{Complex}$: is a parametric plot but where each member of
 \kbd{expr} is considered a complex number encoding the two coordinates of a
 point. For instance:
 \bprog
 ploth(X=0,2*Pi,exp(I*X), "Complex")
 ploth(X=0,2*Pi,[(1+I)*X,exp(I*X)], "Complex")
 @eprog\noindent will draw respectively a circle and a circle cut by the line
 $y=x$.

Function: plothraw
Class: highlevel
Section: graphic
C-Name: plothraw
Prototype: GGD0,L,
Help: plothraw(listx,listy,{flag=0}): plot in high resolution points whose x
 (resp. y) coordinates are in listx (resp. listy). If flag is 1, join points,
 other non-0 flags should be combinations of bits 8,16,32,64,128,256 meaning
 the same as for ploth().
Doc: given \var{listx} and \var{listy} two vectors of equal length, plots (in
 high precision) the points whose $(x,y)$-coordinates are given in
 \var{listx} and \var{listy}. Automatic positioning and scaling is done, but
 with the same scaling factor on $x$ and $y$. If $\fl$ is 1, join points,
 other non-0 flags toggle display options and should be combinations of bits
 $2^k$, $k \geq 3$ as in \kbd{ploth}.

Function: plothsizes
Class: highlevel
Section: graphic
C-Name: plothsizes_flag
Prototype: D0,L,
Help: plothsizes({flag=0}): returns array of 6 elements: terminal width and
 height, sizes for ticks in horizontal and vertical directions, width and
 height of characters. If flag=0, sizes of ticks and characters are in
 pixels, otherwise are fractions of the screen size.
Doc: return data corresponding to the output window
 in the form of a 6-component vector: window width and height, sizes for ticks
 in horizontal and vertical directions (this is intended for the \kbd{gnuplot}
 interface and is currently not significant), width and height of characters.
 
 If $\fl = 0$, sizes of ticks and characters are in
 pixels, otherwise are fractions of the screen size

Function: plotinit
Class: highlevel
Section: graphic
C-Name: initrect_gen
Prototype: vLDGDGD0,L,
Help: plotinit(w,{x},{y},{flag=0}): initialize rectwindow w to size x,y.
 If flag!=0, x and y express fractions of the size of the current output
 device. Omitting x or y means use the full size of the device.
Doc: initialize the rectwindow $w$,
 destroying any rect objects you may have already drawn in $w$. The virtual
 cursor is set to $(0,0)$. The rectwindow size is set to width $x$ and height
 $y$; omitting either $x$ or $y$ means we use the full size of the device
 in that direction.
 If $\fl=0$, $x$ and $y$ represent pixel units. Otherwise, $x$ and $y$
 are understood as fractions of the size of the current output device (hence
 must be between $0$ and $1$) and internally converted to pixels.
 
 The plotting device imposes an upper bound for $x$ and $y$, for instance the
 number of pixels for screen output. These bounds are available through the
 \tet{plothsizes} function. The following sequence initializes in a portable
 way (i.e independent of the output device) a window of maximal size, accessed
 through coordinates in the $[0,1000] \times [0,1000]$ range:
 
 \bprog
 s = plothsizes();
 plotinit(0, s[1]-1, s[2]-1);
 plotscale(0, 0,1000, 0,1000);
 @eprog

Function: plotkill
Class: highlevel
Section: graphic
C-Name: killrect
Prototype: vL
Help: plotkill(w): erase the rectwindow w.
Doc: erase rectwindow $w$ and free the corresponding memory. Note that if you
 want to use the rectwindow $w$ again, you have to use \kbd{plotinit} first
 to specify the new size. So it's better in this case to use \kbd{plotinit}
 directly as this throws away any previous work in the given rectwindow.

Function: plotlines
Class: highlevel
Section: graphic
C-Name: rectlines
Prototype: vLGGD0,L,
Help: plotlines(w,X,Y,{flag=0}): draws an open polygon in rectwindow
 w where X and Y contain the x (resp. y) coordinates of the vertices.
 If X and Y are both single values (i.e not vectors), draw the
 corresponding line (and move cursor). If (optional) flag is non-zero, close
 the polygon.
Doc: draw on the rectwindow $w$
 the polygon such that the (x,y)-coordinates of the vertices are in the
 vectors of equal length $X$ and $Y$. For simplicity, the whole
 polygon is drawn, not only the part of the polygon which is inside the
 rectwindow. If $\fl$ is non-zero, close the polygon. In any case, the
 virtual cursor does not move.
 
 $X$ and $Y$ are allowed to be scalars (in this case, both have to).
 There, a single segment will be drawn, between the virtual cursor current
 position and the point $(X,Y)$. And only the part thereof which
 actually lies within the boundary of $w$. Then \emph{move} the virtual cursor
 to $(X,Y)$, even if it is outside the window. If you want to draw a
 line from $(x1,y1)$ to $(x2,y2)$ where $(x1,y1)$ is not necessarily the
 position of the virtual cursor, use \kbd{plotmove(w,x1,y1)} before using this
 function.

Function: plotlinetype
Class: highlevel
Section: graphic
C-Name: rectlinetype
Prototype: vLL
Help: plotlinetype(w,type): this function is obsolete; no graphing engine
 implement this functionality.
Doc: This function is obsolete and currently a no-op.
 
 Change the type of lines subsequently plotted in rectwindow $w$.
 \var{type} $-2$ corresponds to frames, $-1$ to axes, larger values may
 correspond to something else. $w = -1$ changes highlevel plotting.
Obsolete: 2007-05-11

Function: plotmove
Class: highlevel
Section: graphic
C-Name: rectmove
Prototype: vLGG
Help: plotmove(w,x,y): move cursor to position x,y in rectwindow w.
Doc: move the virtual cursor of the rectwindow $w$ to position $(x,y)$.

Function: plotpoints
Class: highlevel
Section: graphic
C-Name: rectpoints
Prototype: vLGG
Help: plotpoints(w,X,Y): draws in rectwindow w the points whose x
 (resp y) coordinates are in X (resp Y). If X and Y are both
 single values (i.e not vectors), draw the corresponding point (and move
 cursor).
Doc: draw on the rectwindow $w$ the
 points whose $(x,y)$-coordinates are in the vectors of equal length $X$ and
 $Y$ and which are inside $w$. The virtual cursor does \emph{not} move. This
 is basically the same function as \kbd{plothraw}, but either with no scaling
 factor or with a scale chosen using the function \kbd{plotscale}.
 
 As was the case with the \kbd{plotlines} function, $X$ and $Y$ are allowed to
 be (simultaneously) scalar. In this case, draw the single point $(X,Y)$ on
 the rectwindow $w$ (if it is actually inside $w$), and in any case
 \emph{move} the virtual cursor to position $(x,y)$.

Function: plotpointsize
Class: highlevel
Section: graphic
C-Name: rectpointsize
Prototype: vLG
Help: plotpointsize(w,size): change the "size" of following points in
 rectwindow w. w=-1 changes global value.
Doc: This function is obsolete. It is currently a no-op.
 
 Changes the ``size'' of following points in rectwindow $w$. If $w = -1$,
 change it in all rectwindows.
Obsolete: 2007-05-11

Function: plotpointtype
Class: highlevel
Section: graphic
C-Name: rectpointtype
Prototype: vLL
Help: plotpointtype(w,type): this function is obsolete; no graphing engine
 implement this functionality.
Doc: This function is obsolete and currently a no-op.
 
 change the type of points subsequently plotted in rectwindow $w$.
 $\var{type} = -1$ corresponds to a dot, larger values may correspond to
 something else. $w = -1$ changes highlevel plotting.
Obsolete: 2007-05-11

Function: plotrbox
Class: highlevel
Section: graphic
C-Name: rectrbox
Prototype: vLGG
Help: plotrbox(w,dx,dy): if the cursor is at (x1,y1), draw a box with
 diagonal (x1,y1)-(x1+dx,y1+dy) in rectwindow w (cursor does not move).
Doc: draw in the rectwindow $w$ the outline of the rectangle which is such
 that the points $(x1,y1)$ and $(x1+dx,y1+dy)$ are opposite corners, where
 $(x1,y1)$ is the current position of the cursor. Only the part of the
 rectangle which is in $w$ is drawn. The virtual cursor does \emph{not} move.

Function: plotrecth
Class: highlevel
Section: graphic
C-Name: rectploth
Prototype: LV=GGEpD0,M,D0,L,\nParametric|1; Recursive|2; no_Rescale|4; no_X_axis|8; no_Y_axis|16; no_Frame|32; no_Lines|64; Points_too|128; Splines|256; no_X_ticks|512; no_Y_ticks|1024; Same_ticks|2048; Complex|4096
Help: plotrecth(w,X=a,b,expr,{flag=0},{n=0}):
 writes to rectwindow w the curve output of
 ploth(w,X=a,b,expr,flag,n). Returns a vector for the bounding box.
Doc: writes to rectwindow $w$ the curve output of
 \kbd{ploth}$(w,X=a,b,\var{expr},\fl,n)$. Returns a vector for the bounding box.

Function: plotrecthraw
Class: highlevel
Section: graphic
C-Name: rectplothraw
Prototype: LGD0,L,
Help: plotrecthraw(w,data,{flags=0}): plot graph(s) for data in rectwindow
 w, where data is a vector of vectors. If plot is parametric, length of data
 should be even, and pairs of entries give curves to plot. If not, first
 entry gives x-coordinate, and the other ones y-coordinates. Admits the same
 optional flags as plotrecth, save that recursive plot is meaningless.
Doc: plot graph(s) for
 \var{data} in rectwindow $w$. $\fl$ has the same significance here as in
 \kbd{ploth}, though recursive plot is no more significant.
 
 \var{data} is a vector of vectors, each corresponding to a list a coordinates.
 If parametric plot is set, there must be an even number of vectors, each
 successive pair corresponding to a curve. Otherwise, the first one contains
 the $x$ coordinates, and the other ones contain the $y$-coordinates
 of curves to plot.

Function: plotrline
Class: highlevel
Section: graphic
C-Name: rectrline
Prototype: vLGG
Help: plotrline(w,dx,dy): if the cursor is at (x1,y1), draw a line from
 (x1,y1) to (x1+dx,y1+dy) (and move the cursor) in the rectwindow w.
Doc: draw in the rectwindow $w$ the part of the segment
 $(x1,y1)-(x1+dx,y1+dy)$ which is inside $w$, where $(x1,y1)$ is the current
 position of the virtual cursor, and move the virtual cursor to
 $(x1+dx,y1+dy)$ (even if it is outside the window).

Function: plotrmove
Class: highlevel
Section: graphic
C-Name: rectrmove
Prototype: vLGG
Help: plotrmove(w,dx,dy): move cursor to position (dx,dy) relative to the
 present position in the rectwindow w.
Doc: move the virtual cursor of the rectwindow $w$ to position
 $(x1+dx,y1+dy)$, where $(x1,y1)$ is the initial position of the cursor
 (i.e.~to position $(dx,dy)$ relative to the initial cursor).

Function: plotrpoint
Class: highlevel
Section: graphic
C-Name: rectrpoint
Prototype: vLGG
Help: plotrpoint(w,dx,dy): draw a point (and move cursor) at position dx,dy
 relative to present position of the cursor in rectwindow w.
Doc: draw the point $(x1+dx,y1+dy)$ on the rectwindow $w$ (if it is inside
 $w$), where $(x1,y1)$ is the current position of the cursor, and in any case
 move the virtual cursor to position $(x1+dx,y1+dy)$.

Function: plotscale
Class: highlevel
Section: graphic
C-Name: rectscale
Prototype: vLGGGG
Help: plotscale(w,x1,x2,y1,y2): scale the coordinates in rectwindow w so
 that x goes from x1 to x2 and y from y1 to y2 (y2<y1 is allowed).
Doc: scale the local coordinates of the rectwindow $w$ so that $x$ goes from
 $x1$ to $x2$ and $y$ goes from $y1$ to $y2$ ($x2<x1$ and $y2<y1$ being
 allowed). Initially, after the initialization of the rectwindow $w$ using
 the function \kbd{plotinit}, the default scaling is the graphic pixel count,
 and in particular the $y$ axis is oriented downwards since the origin is at
 the upper left. The function \kbd{plotscale} allows to change all these
 defaults and should be used whenever functions are graphed.

Function: plotstring
Class: highlevel
Section: graphic
C-Name: rectstring3
Prototype: vLsD0,L,
Help: plotstring(w,x,{flags=0}): draw in rectwindow w the string
 corresponding to x. Bits 1 and 2 of flag regulate horizontal alignment: left
 if 0, right if 2, center if 1. Bits 4 and 8 regulate vertical alignment:
 bottom if 0, top if 8, v-center if 4. Can insert additional gap between
 point and string: horizontal if bit 16 is set, vertical if bit 32 is set.
Doc: draw on the rectwindow $w$ the String $x$ (see \secref{se:strings}), at
 the current position of the cursor.
 
 \fl\ is used for justification: bits 1 and 2 regulate horizontal alignment:
 left if 0, right if 2, center if 1. Bits 4 and 8 regulate vertical
 alignment: bottom if 0, top if 8, v-center if 4. Can insert additional small
 gap between point and string: horizontal if bit 16 is set, vertical if bit
 32 is set (see the tutorial for an example).

Function: polchebyshev
Class: basic
Section: polynomials
C-Name: polchebyshev_eval
Prototype: LD1,L,DG
Help: polchebyshev(n,{flag=1},{a='x}): Chebychev polynomial of the first (flag
 = 1) or second (flag = 2) kind, of degree n, evaluated at a.
Description: 
 (small,?1,?var):gen polchebyshev1($1,$3)
 (small,2,?var):gen  polchebyshev2($1,$3)
 (small,small,?var):gen polchebyshev($1,$2,$3)
Doc: returns the $n^{\text{th}}$
 \idx{Chebyshev} polynomial of the first kind $T_n$ ($\fl=1$) or the second
 kind $U_n$ ($\fl=2$), evaluated at $a$ (\kbd{'x} by default). Both series of
 polynomials satisfy the 3-term relation
 $$ P_{n+1} = 2xP_n - P_{n-1}, $$
 and are determined by the initial conditions $U_0 = T_0 = 1$, $T_1 = x$,
 $U_1 = 2x$. In fact $T_n' = n U_{n-1}$ and, for all complex numbers $z$, we
 have $T_n(\cos z) = \cos (nz)$ and $U_{n-1}(\cos z) = \sin(nz)/\sin z$.
 If $n \geq 0$, then these polynomials have degree $n$.  For $n < 0$,
 $T_n$ is equal to $T_{-n}$ and $U_n$ is equal to $-U_{-2-n}$.
 In particular, $U_{-1} = 0$.
Variant: Also available are
 \fun{GEN}{polchebyshev}{long n, long flag, long v},
 \fun{GEN}{polchebyshev1}{long n, long v} and
 \fun{GEN}{polchebyshev2}{long n, long v} for $T_n$ and $U_n$ respectively.

Function: polclass
Class: basic
Section: polynomials
C-Name: polclass
Prototype: GD0,L,Dn
Help: polclass(D, {inv = 0}, {x = 'x}): return a polynomial generating the
 Hilbert class field of Q(sqrt(D)) for the discriminant D<0.
Doc: 
 Return a polynomial in $\Z[x]$ generating the Hilbert class field for the
 imaginary quadratic discriminant $D$.  If $inv$ is 0 (the default),
 use the modular $j$-function and return the classical Hilbert polynomial,
 otherwise use a class invariant. The following invariants correspond to
 the different values of $inv$, where $f$ denotes Weber's function
 \kbd{weber}, and $w_{p,q}$ the double eta quotient given by
 $w_{p,q} = \dfrac{ \eta(x/p)\*\eta(x/q) }{ \eta(x)\*\eta(x/{pq}) }$
 
 The invariants $w_{p,q}$ are not allowed unless they satisfy the following
 technical conditions ensuring they do generate the Hilbert class
 field and not a strict subfield:
 
 \item if $p\neq q$, we need them both non-inert, prime to the conductor of
 $\Z[\sqrt{D}]$. Let $P, Q$ be prime ideals  above $p$ and $q$; if both are
 unramified, we further require that $P^{\pm 1} Q^{\pm 1}$ be all distinct in
 the class group of $\Z[\sqrt{D}]$; if both are ramified, we require that $PQ
 \neq 1$ in the class group.
 
 \item if $p = q$, we want it split and prime to the conductor and
 the prime ideal above it must have order $\neq 1, 2, 4$ in the class group.
 
 \noindent Invariants are allowed under the additional conditions on $D$
 listed below.
 
 \item 0 : $j$
 
 \item 1 : $f$, $D = 1 \mod 8$ and $D = 1,2 \mod 3$;
 
 \item 2 : $f^2$, $D = 1 \mod 8$ and $D = 1,2 \mod 3$;
 
 \item 3 : $f^3$, $D = 1 \mod 8$;
 
 \item 4 : $f^4$, $D = 1 \mod 8$ and $D = 1,2 \mod 3$;
 
 \item 5 : $\gamma_2= j^{1/3}$, $D = 1,2 \mod 3$;
 
 \item 6 : $w_{2,3}$, $D = 1 \mod 8$ and $D = 1,2 \mod 3$;
 
 \item 8 : $f^8$, $D = 1 \mod 8$ and $D = 1,2 \mod 3$;
 
 \item 9 : $w_{3,3}$, $D = 1 \mod 2$ and $D = 1,2 \mod 3$;
 
 \item 10: $w_{2,5}$, $D \neq 60 \mod 80$ and $D = 1,2 \mod 3$;
 
 \item 14: $w_{2,7}$, $D = 1 \mod 8$;
 
 \item 15: $w_{3,5}$, $D = 1,2 \mod 3$;
 
 \item 21: $w_{3,7}$, $D = 1 \mod 2$ and $21$ does not divide $D$
 
 \item 23: $w_{2,3}^2$, $D = 1,2 \mod 3$;
 
 \item 24: $w_{2,5}^2$, $D = 1,2 \mod 3$;
 
 \item 26: $w_{2,13}$, $D \neq 156 \mod 208$;
 
 \item 27: $w_{2,7}^2$, $D\neq 28 \mod 112$;
 
 \item 28: $w_{3,3}^2$, $D = 1,2 \mod 3$;
 
 \item 35: $w_{5,7}$, $D = 1,2 \mod 3$;
 
 \item 39: $w_{3,13}$, $D = 1 \mod 2$ and $D = 1,2 \mod 3$;
 
 The algorithm for computing the polynomial does not use the floating point
 approach, which would evaluate a precise modular function in a precise
 complex argument. Instead, it relies on a faster Chinese remainder based
 approach modulo small primes, in which the class invariant is only defined
 algebraically by the modular polynomial relating the modular function to $j$.
 So in fact, any of the several roots of the modular polynomial may actually
 be the class invariant, and more precise assertions cannot be made.
 
 For instance, while \kbd{polclass(D)} returns the minimal polynomial of
 $j(\tau)$ with $\tau$ (any) quadratic integer for the discriminant $D$,
 the polynomial returned by \kbd{polclass(D, 5)} can be the minimal polynomial
 of any of $\gamma_2 (\tau)$, $\zeta_3 \gamma_2 (\tau)$ or
 $\zeta_3^2 \gamma_2 (\tau)$, the three roots of the modular polynomial
 $j = \gamma_2^3$, in which $j$ has been specialised to $j (\tau)$.
 
 The modular polynomial is given by
 $j = {(f^{24}-16)^3 \over f^{24}}$ for Weber's function $f$.
 
 For the double eta quotients of level $N = p q$, all functions are covered
 such that the modular curve $X_0^+ (N)$, the function field of which is
 generated by the functions invariant under $\Gamma^0 (N)$ and the
 Fricke--Atkin--Lehner involution, is of genus $0$ with function field
 generated by (a power of) the double eta quotient $w$.
 This ensures that the full Hilbert class field (and not a proper subfield)
 is generated by class invariants from these double eta quotients.
 Then the modular polynomial is of degree $2$ in $j$, and
 of degree $\psi (N) = (p+1)(q+1)$ in $w$.
 
 \bprog
 ? polclass(-163)
 %1 = x + 262537412640768000
 ? polclass(-51, , 'z)
 %2 = z^2 + 5541101568*z + 6262062317568
 ? polclass(-151,1)
 x^7 - x^6 + x^5 + 3*x^3 - x^2 + 3*x + 1
 @eprog

Function: polcoeff
Class: basic
Section: polynomials
C-Name: polcoeff0
Prototype: GLDn
Help: polcoeff(x,n,{v}): coefficient of degree n of x, or the n-th component
 for vectors or matrices (for which it is simpler to use x[]). With respect
 to the main variable if v is omitted, with respect to the variable v
 otherwise.
Description: 
 (pol, 0):gen:copy       constant_coeff($1)
 (pol, 0,):gen:copy      constant_coeff($1)
 (pol, small):gen:copy   RgX_coeff($1, $2)
 (pol, small,):gen:copy  RgX_coeff($1, $2)
 (gen, small, ?var):gen polcoeff0($1, $2, $3)
Doc: coefficient of degree $n$ of the polynomial $x$, with respect to the
 main variable if $v$ is omitted, with respect to $v$ otherwise.  If $n$
 is greater than the degree, the result is zero.
 
 Naturally applies to scalars (polynomial of degree $0$), as well as to
 rational functions whose denominator is a monomial.
 It also applies to power series: if $n$ is less than the valuation, the result
 is zero. If it is greater than the largest significant degree, then an error
 message is issued.
 
  For greater flexibility, $x$ can be a vector or matrix type and the
  function then returns \kbd{component(x,n)}.

Function: polcompositum
Class: basic
Section: number_fields
C-Name: polcompositum0
Prototype: GGD0,L,
Help: polcompositum(P,Q,{flag=0}): vector of all possible compositums
 of the number fields defined by the polynomials P and Q; flag is
 optional, whose binary digits mean 1: output for each compositum, not only
 the compositum polynomial pol, but a vector [R,a,b,k] where a (resp. b) is a
 root of P (resp. Q) expressed as a polynomial modulo R, and a small integer k
 such that al2+k*al1 is the chosen root of R; 2: assume that the number
 fields defined by P and Q are linearly disjoint.
Doc: \sidx{compositum} $P$ and $Q$
 being squarefree polynomials in $\Z[X]$ in the same variable, outputs
 the simple factors of the \'etale $\Q$-algebra $A = \Q(X, Y) / (P(X), Q(Y))$.
 The factors are given by a list of polynomials $R$ in $\Z[X]$, attached to
 the number field $\Q(X)/ (R)$, and sorted by increasing degree (with respect
 to lexicographic ordering for factors of equal degrees). Returns an error if
 one of the polynomials is not squarefree.
 
 Note that it is more efficient to reduce to the case where $P$ and $Q$ are
 irreducible first. The routine will not perform this for you, since it may be
 expensive, and the inputs are irreducible in most applications anyway. In
 this case, there will be a single factor $R$ if and only if the number
 fields defined by $P$ and $Q$ are linearly disjoint (their intersection is
 $\Q$).
 
 Assuming $P$ is irreducible (of smaller degree than $Q$ for efficiency), it
 is in general much faster to proceed as follows
 \bprog
 nf = nfinit(P); L = nffactor(nf, Q)[,1];
 vector(#L, i, rnfequation(nf, L[i]))
 @eprog\noindent
 to obtain the same result. If you are only interested in the degrees of the
 simple factors, the \kbd{rnfequation} instruction can be replaced by a
 trivial \kbd{poldegree(P) * poldegree(L[i])}.
 
 The binary digits of $\fl$ mean
 
 1: outputs a vector of 4-component vectors $[R,a,b,k]$, where $R$
 ranges through the list of all possible compositums as above, and $a$
 (resp. $b$) expresses the root of $P$ (resp. $Q$) as an element of
 $\Q(X)/(R)$. Finally, $k$ is a small integer such that $b + ka = X$ modulo
 $R$.
 
 2: assume that $P$ and $Q$ define number fields which are linearly disjoint:
 both polynomials are irreducible and the corresponding number fields
 have no common subfield besides $\Q$. This allows to save a costly
 factorization over $\Q$. In this case return the single simple factor
 instead of a vector with one element.
 
 A compositum is often defined by a complicated polynomial, which it is
 advisable to reduce before further work. Here is an example involving
 the field $\Q(\zeta_5, 5^{1/5})$:
 \bprog
 ? L = polcompositum(x^5 - 5, polcyclo(5), 1); \\@com list of $[R,a,b,k]$
 ? [R, a] = L[1];  \\@com pick the single factor, extract $R,a$ (ignore $b,k$)
 ? R               \\@com defines the compositum
 %3 = x^20 + 5*x^19 + 15*x^18 + 35*x^17 + 70*x^16 + 141*x^15 + 260*x^14\
 + 355*x^13 + 95*x^12 - 1460*x^11 - 3279*x^10 - 3660*x^9 - 2005*x^8    \
 + 705*x^7 + 9210*x^6 + 13506*x^5 + 7145*x^4 - 2740*x^3 + 1040*x^2     \
 - 320*x + 256
 ? a^5 - 5         \\@com a fifth root of $5$
 %4 = 0
 ? [T, X] = polredbest(R, 1);
 ? T     \\@com simpler defining polynomial for $\Q[x]/(R)$
 %6 = x^20 + 25*x^10 + 5
 ? X     \\ @com root of $R$ in $\Q[y]/(T(y))$
 %7 = Mod(-1/11*x^15 - 1/11*x^14 + 1/22*x^10 - 47/22*x^5 - 29/11*x^4 + 7/22,\
 x^20 + 25*x^10 + 5)
 ? a = subst(a.pol, 'x, X)  \\@com \kbd{a} in the new coordinates
 %8 = Mod(1/11*x^14 + 29/11*x^4, x^20 + 25*x^10 + 5)
 ? a^5 - 5
 %9 = 0
 @eprog\noindent In the above example, $x^5-5$ and the $5$-th cyclotomic
 polynomial are irreducible over $\Q$; they have coprime degrees so
 define linearly disjoint extensions and we could have started by
 \bprog
 ? [R,a] = polcompositum(x^5 - 5, polcyclo(5), 3); \\@com $[R,a,b,k]$
 @eprog
Variant: Also available are
 \fun{GEN}{compositum}{GEN P, GEN Q} ($\fl = 0$) and
 \fun{GEN}{compositum2}{GEN P, GEN Q} ($\fl = 1$).

Function: polcyclo
Class: basic
Section: polynomials
C-Name: polcyclo_eval
Prototype: LDG
Help: polcyclo(n,{a = 'x}): n-th cyclotomic polynomial evaluated at a.
Description: 
  (small,?var):gen     polcyclo($1,$2)
  (small,gen):gen      polcyclo_eval($1,$2)
Doc: $n$-th cyclotomic polynomial, evaluated at $a$ (\kbd{'x} by default). The
 integer $n$ must be positive.
 
 Algorithm used: reduce to the case where $n$ is squarefree; to compute the
 cyclotomic polynomial, use $\Phi_{np}(x)=\Phi_n(x^p)/\Phi(x)$; to compute
 it evaluated, use $\Phi_n(x) = \prod_{d\mid n} (x^d-1)^{\mu(n/d)}$. In the
 evaluated case, the algorithm assumes that $a^d - 1$ is either $0$ or
 invertible, for all $d\mid n$. If this is not the case (the base ring has
 zero divisors), use \kbd{subst(polcyclo(n),x,a)}.
Variant: The variant \fun{GEN}{polcyclo}{long n, long v} returns the $n$-th
 cyclotomic polynomial in variable $v$.

Function: polcyclofactors
Class: basic
Section: polynomials
C-Name: polcyclofactors
Prototype: G
Help: polcyclofactors(f): returns a vector of polynomials, whose product is
 the product of distinct cyclotomic polynomials dividing f.
Doc: returns a vector of polynomials, whose product is the product of
 distinct cyclotomic polynomials dividing $f$.
 \bprog
 ? f = x^10+5*x^8-x^7+8*x^6-4*x^5+8*x^4-3*x^3+7*x^2+3;
 ? v = polcyclofactors(f)
 %2 = [x^2 + 1, x^2 + x + 1, x^4 - x^3 + x^2 - x + 1]
 ? apply(poliscycloprod, v)
 %3 = [1, 1, 1]
 ? apply(poliscyclo, v)
 %4 = [4, 3, 10]
 @eprog\noindent In general, the polynomials are products of cyclotomic
 polynomials and not themselves irreducible:
 \bprog
 ? g = x^8+2*x^7+6*x^6+9*x^5+12*x^4+11*x^3+10*x^2+6*x+3;
 ? polcyclofactors(g)
 %2 = [x^6 + 2*x^5 + 3*x^4 + 3*x^3 + 3*x^2 + 2*x + 1]
 ? factor(%[1])
 %3 =
 [            x^2 + x + 1 1]
 
 [x^4 + x^3 + x^2 + x + 1 1]
 @eprog

Function: poldegree
Class: basic
Section: polynomials
C-Name: gppoldegree
Prototype: GDn
Help: poldegree(x,{v}): degree of the polynomial or rational function x with
 respect to main variable if v is omitted, with respect to v otherwise.
 For scalar x, return 0 if x is non-zero and -oo otherwise.
Doc: degree of the polynomial $x$ in the main variable if $v$ is omitted, in
 the variable $v$ otherwise.
 
 The degree of $0$ is \kbd{-oo}. The degree of a non-zero scalar is $0$.
 Finally, when $x$ is a non-zero polynomial or rational function, returns the
 ordinary degree of $x$. Raise an error otherwise.
Variant: Also available is
 \fun{long}{poldegree}{GEN x, long v}, which returns \tet{-LONG_MAX} if $x = 0$
 and the degree as a \kbd{long} integer.

Function: poldisc
Class: basic
Section: polynomials
C-Name: poldisc0
Prototype: GDn
Help: poldisc(pol,{v}): discriminant of the polynomial pol, with respect to main
 variable if v is omitted, with respect to v otherwise.
Description: 
 (gen):gen        poldisc0($1, -1)
 (gen, var):gen   poldisc0($1, $2)
Doc: discriminant of the polynomial
 \var{pol} in the main variable if $v$ is omitted, in $v$ otherwise. Uses a
 modular algorithm over $\Z$ or $\Q$, and the \idx{subresultant algorithm}
 otherwise.
 \bprog
 ? T = x^4 + 2*x+1;
 ? poldisc(T)
 %2 = -176
 ? poldisc(T^2)
 %3 = 0
 @eprog
 
 For convenience, the function also applies to types \typ{QUAD} and
 \typ{QFI}/\typ{QFR}:
 \bprog
 ? z = 3*quadgen(8) + 4;
 ? poldisc(z)
 %2 = 8
 ? q = Qfb(1,2,3);
 ? poldisc(q)
 %4 = -8
 @eprog

Function: poldiscreduced
Class: basic
Section: polynomials
C-Name: reduceddiscsmith
Prototype: G
Help: poldiscreduced(f): vector of elementary divisors of Z[a]/f'(a)Z[a],
 where a is a root of the polynomial f.
Doc: reduced discriminant vector of the
 (integral, monic) polynomial $f$. This is the vector of elementary divisors
 of $\Z[\alpha]/f'(\alpha)\Z[\alpha]$, where $\alpha$ is a root of the
 polynomial $f$. The components of the result are all positive, and their
 product is equal to the absolute value of the discriminant of~$f$.

Function: polgalois
Class: basic
Section: number_fields
C-Name: polgalois
Prototype: Gp
Help: polgalois(T): Galois group of the polynomial T (see manual for group
 coding). Return [n, s, k, name] where n is the group order, s the signature,
 k the index and name is the GAP4 name of the transitive group.
Doc: \idx{Galois} group of the non-constant
 polynomial $T\in\Q[X]$. In the present version \vers, $T$ must be irreducible
 and the degree $d$ of $T$ must be less than or equal to 7. If the
 \tet{galdata} package has been installed, degrees 8, 9, 10 and 11 are also
 implemented. By definition, if $K = \Q[x]/(T)$, this computes the action of
 the Galois group of the Galois closure of $K$ on the $d$ distinct roots of
 $T$, up to conjugacy (corresponding to different root orderings).
 
 The output is a 4-component vector $[n,s,k,name]$ with the
 following meaning: $n$ is the cardinality of the group, $s$ is its signature
 ($s=1$ if the group is a subgroup of the alternating group $A_d$, $s=-1$
 otherwise) and name is a character string containing name of the transitive
 group according to the GAP 4 transitive groups library by Alexander Hulpke.
 
 $k$ is more arbitrary and the choice made up to version~2.2.3 of PARI is rather
 unfortunate: for $d > 7$, $k$ is the numbering of the group among all
 transitive subgroups of $S_d$, as given in ``The transitive groups of degree up
 to eleven'', G.~Butler and J.~McKay, \emph{Communications in Algebra}, vol.~11,
 1983,
 pp.~863--911 (group $k$ is denoted $T_k$ there). And for $d \leq 7$, it was ad
 hoc, so as to ensure that a given triple would denote a unique group.
 Specifically, for polynomials of degree $d\leq 7$, the groups are coded as
 follows, using standard notations
 \smallskip
 In degree 1: $S_1=[1,1,1]$.
 \smallskip
 In degree 2: $S_2=[2,-1,1]$.
 \smallskip
 In degree 3: $A_3=C_3=[3,1,1]$, $S_3=[6,-1,1]$.
 \smallskip
 In degree 4: $C_4=[4,-1,1]$, $V_4=[4,1,1]$, $D_4=[8,-1,1]$, $A_4=[12,1,1]$,
 $S_4=[24,-1,1]$.
 \smallskip
 In degree 5: $C_5=[5,1,1]$, $D_5=[10,1,1]$, $M_{20}=[20,-1,1]$,
 $A_5=[60,1,1]$, $S_5=[120,-1,1]$.
 \smallskip
 In degree 6: $C_6=[6,-1,1]$, $S_3=[6,-1,2]$, $D_6=[12,-1,1]$, $A_4=[12,1,1]$,
 $G_{18}=[18,-1,1]$, $S_4^-=[24,-1,1]$, $A_4\times C_2=[24,-1,2]$,
 $S_4^+=[24,1,1]$, $G_{36}^-=[36,-1,1]$, $G_{36}^+=[36,1,1]$,
 $S_4\times C_2=[48,-1,1]$, $A_5=PSL_2(5)=[60,1,1]$, $G_{72}=[72,-1,1]$,
 $S_5=PGL_2(5)=[120,-1,1]$, $A_6=[360,1,1]$, $S_6=[720,-1,1]$.
 \smallskip
 In degree 7: $C_7=[7,1,1]$, $D_7=[14,-1,1]$, $M_{21}=[21,1,1]$,
 $M_{42}=[42,-1,1]$, $PSL_2(7)=PSL_3(2)=[168,1,1]$, $A_7=[2520,1,1]$,
 $S_7=[5040,-1,1]$.
 \smallskip
 This is deprecated and obsolete, but for reasons of backward compatibility,
 we cannot change this behavior yet. So you can use the default
 \tet{new_galois_format} to switch to a consistent naming scheme, namely $k$ is
 always the standard numbering of the group among all transitive subgroups of
 $S_n$. If this default is in effect, the above groups will be coded as:
 \smallskip
 In degree 1: $S_1=[1,1,1]$.
 \smallskip
 In degree 2: $S_2=[2,-1,1]$.
 \smallskip
 In degree 3: $A_3=C_3=[3,1,1]$, $S_3=[6,-1,2]$.
 \smallskip
 In degree 4: $C_4=[4,-1,1]$, $V_4=[4,1,2]$, $D_4=[8,-1,3]$, $A_4=[12,1,4]$,
 $S_4=[24,-1,5]$.
 \smallskip
 In degree 5: $C_5=[5,1,1]$, $D_5=[10,1,2]$, $M_{20}=[20,-1,3]$,
 $A_5=[60,1,4]$, $S_5=[120,-1,5]$.
 \smallskip
 In degree 6: $C_6=[6,-1,1]$, $S_3=[6,-1,2]$, $D_6=[12,-1,3]$, $A_4=[12,1,4]$,
 $G_{18}=[18,-1,5]$, $A_4\times C_2=[24,-1,6]$, $S_4^+=[24,1,7]$,
 $S_4^-=[24,-1,8]$, $G_{36}^-=[36,-1,9]$, $G_{36}^+=[36,1,10]$,
 $S_4\times C_2=[48,-1,11]$, $A_5=PSL_2(5)=[60,1,12]$, $G_{72}=[72,-1,13]$,
 $S_5=PGL_2(5)=[120,-1,14]$, $A_6=[360,1,15]$, $S_6=[720,-1,16]$.
 \smallskip
 In degree 7: $C_7=[7,1,1]$, $D_7=[14,-1,2]$, $M_{21}=[21,1,3]$,
 $M_{42}=[42,-1,4]$, $PSL_2(7)=PSL_3(2)=[168,1,5]$, $A_7=[2520,1,6]$,
 $S_7=[5040,-1,7]$.
 \smallskip
 
 \misctitle{Warning} The method used is that of resolvent polynomials and is
 sensitive to the current precision. The precision is updated internally but,
 in very rare cases, a wrong result may be returned if the initial precision
 was not sufficient.
Variant: To enable the new format in library mode,
 set the global variable \tet{new_galois_format} to $1$.

Function: polgraeffe
Class: basic
Section: polynomials
C-Name: polgraeffe
Prototype: G
Help: polgraeffe(f): returns the Graeffe transform g of f, such that
 g(x^2) = f(x)f(-x).
Doc: returns the \idx{Graeffe} transform $g$ of $f$, such that $g(x^2) = f(x)
 f(-x)$.

Function: polhensellift
Class: basic
Section: polynomials
C-Name: polhensellift
Prototype: GGGL
Help: polhensellift(A, B, p, e): lift the factorization B of A modulo p to a
 factorization modulo p^e using Hensel lift. The factors in B must be
 pairwise relatively prime modulo p.
Doc: given a prime $p$, an integral polynomial $A$ whose leading coefficient
 is a $p$-unit, a vector $B$ of integral polynomials that are monic and
 pairwise relatively prime modulo $p$, and whose product is congruent to
 $A/\text{lc}(A)$ modulo $p$, lift the elements of $B$ to polynomials whose
 product is congruent to $A$ modulo $p^e$.
 
 More generally, if $T$ is an integral polynomial irreducible mod $p$, and
 $B$ is a factorization of $A$ over the finite field $\F_p[t]/(T)$, you can
 lift it to $\Z_p[t]/(T, p^e)$ by replacing the $p$ argument with $[p,T]$:
 \bprog
 ? { T = t^3 - 2; p = 7; A = x^2 + t + 1;
     B = [x + (3*t^2 + t + 1), x + (4*t^2 + 6*t + 6)];
     r = polhensellift(A, B, [p, T], 6) }
 %1 = [x + (20191*t^2 + 50604*t + 75783), x + (97458*t^2 + 67045*t + 41866)]
 ? liftall( r[1] * r[2] * Mod(Mod(1,p^6),T) )
 %2 = x^2 + (t + 1)
 @eprog

Function: polhermite
Class: basic
Section: polynomials
C-Name: polhermite_eval
Prototype: LDG
Help: polhermite(n,{a='x}): Hermite polynomial H(n,v) of degree n, evaluated
 at a.
Description: 
  (small,?var):gen    polhermite($1,$2)
  (small,gen):gen     polhermite_eval($1,$2)
Doc: $n^{\text{th}}$ \idx{Hermite} polynomial $H_n$ evaluated at $a$
 (\kbd{'x} by default), i.e.
 $$ H_n(x) = (-1)^n\*e^{x^2} \dfrac{d^n}{dx^n}e^{-x^2}.$$
Variant: The variant \fun{GEN}{polhermite}{long n, long v} returns the $n$-th
 Hermite polynomial in variable $v$.

Function: polinterpolate
Class: basic
Section: polynomials
C-Name: polint
Prototype: GDGDGD&
Help: polinterpolate(X,{Y},{t = 'x},{&e}): polynomial interpolation at t
 according to data vectors X, Y (i.e. given P of minimal degree
 such that P(X[i]) = Y[i] for all i, return P(t)). If Y is omitted,
 take P such that P(i) = X[i]. If present, e will contain an error estimate on
 the returned value.
Doc: given the data vectors
 $X$ and $Y$ of the same length $n$ ($X$ containing the $x$-coordinates,
 and $Y$ the corresponding $y$-coordinates), this function finds the
 \idx{interpolating polynomial} $P$ of minimal degree passing through these
 points and evaluates it at~$t$. If $Y$ is omitted, the polynomial $P$
 interpolates the $(i,X[i])$. If present, $e$ will contain an error estimate
 on the returned value.

Function: poliscyclo
Class: basic
Section: polynomials
C-Name: poliscyclo
Prototype: lG
Help: poliscyclo(f): returns 0 if f is not a cyclotomic polynomial, and n
 > 0 if f = Phi_n, the n-th cyclotomic polynomial.
Doc: returns 0 if $f$ is not a cyclotomic polynomial, and $n > 0$ if $f =
 \Phi_n$, the $n$-th cyclotomic polynomial.
 \bprog
 ? poliscyclo(x^4-x^2+1)
 %1 = 12
 ? polcyclo(12)
 %2 = x^4 - x^2 + 1
 ? poliscyclo(x^4-x^2-1)
 %3 = 0
 @eprog

Function: poliscycloprod
Class: basic
Section: polynomials
C-Name: poliscycloprod
Prototype: lG
Help: poliscycloprod(f): returns 1 if f is a product of cyclotomic
 polynonials, and 0 otherwise.
Doc: returns 1 if $f$ is a product of cyclotomic polynomial, and $0$
 otherwise.
 \bprog
 ? f = x^6+x^5-x^3+x+1;
 ? poliscycloprod(f)
 %2 = 1
 ? factor(f)
 %3 =
 [  x^2 + x + 1 1]
 
 [x^4 - x^2 + 1 1]
 ? [ poliscyclo(T) | T <- %[,1] ]
 %4 = [3, 12]
 ? polcyclo(3) * polcyclo(12)
 %5 = x^6 + x^5 - x^3 + x + 1
 @eprog

Function: polisirreducible
Class: basic
Section: polynomials
C-Name: isirreducible
Prototype: lG
Help: polisirreducible(pol): true(1) if pol is an irreducible non-constant
 polynomial, false(0) if pol is reducible or constant.
Doc: \var{pol} being a polynomial (univariate in the present version \vers),
 returns 1 if \var{pol} is non-constant and irreducible, 0 otherwise.
 Irreducibility is checked over the smallest base field over which \var{pol}
 seems to be defined.

Function: pollead
Class: basic
Section: polynomials
C-Name: pollead
Prototype: GDn
Help: pollead(x,{v}): leading coefficient of polynomial or series x, or x
 itself if x is a scalar. Error otherwise. With respect to the main variable
 of x if v is omitted, with respect to the variable v otherwise.
Description: 
 (pol):gen:copy         leading_coeff($1)
 (gen):gen              pollead($1, -1)
 (gen, var):gen         pollead($1, $2)
Doc: leading coefficient of the polynomial or power series $x$. This is
  computed with respect to the main variable of $x$ if $v$ is omitted, with
  respect to the variable $v$ otherwise.

Function: pollegendre
Class: basic
Section: polynomials
C-Name: pollegendre_eval
Prototype: LDG
Help: pollegendre(n,{a='x}): legendre polynomial of degree n evaluated at a.
Description: 
  (small,?var):gen    pollegendre($1,$2)
  (small,gen):gen     pollegendre_eval($1,$2)
Doc: $n^{\text{th}}$ \idx{Legendre polynomial} evaluated at $a$ (\kbd{'x} by
 default).
Variant: To obtain the $n$-th Legendre polynomial in variable $v$,
 use \fun{GEN}{pollegendre}{long n, long v}.

Function: polmodular
Class: basic
Section: polynomials
C-Name: polmodular
Prototype: LD0,L,DGDnD0,L,
Help: polmodular(L, {inv = 0}, {x = 'x}, {y = 'y}, {derivs = 0}):
 return the modular polynomial of level L and invariant inv.
Doc: Return the modular polynomial of prime level $L$ in variables $x$ and $y$
 for the modular function specified by \kbd{inv}.  If \kbd{inv} is 0 (the
 default), use the modular $j$ function, if \kbd{inv} is 1 use the
 Weber-$f$ function, and if \kbd{inv} is 5 use $\gamma_2 =
 \sqrt[3]{j}$.
 See \kbd{polclass} for the full list of invariants.
 If $x$ is given as \kbd{Mod(j, p)} or an element $j$ of
 a finite field (as a \typ{FFELT}), then return the modular polynomial of
 level $L$ evaluated at $j$.  If $j$ is from a finite field and
 \kbd{derivs} is non-zero, then return a triple where the
 last two elements are the first and second derivatives of the modular
 polynomial evaluated at $j$.
 \bprog
 ? polmodular(3)
 %1 = x^4 + (-y^3 + 2232*y^2 - 1069956*y + 36864000)*x^3 + ...
 ? polmodular(7, 1, , 'J)
 %2 = x^8 - J^7*x^7 + 7*J^4*x^4 - 8*J*x + J^8
 ? polmodular(7, 5, 7*ffgen(19)^0, 'j)
 %3 = j^8 + 4*j^7 + 4*j^6 + 8*j^5 + j^4 + 12*j^2 + 18*j + 18
 ? polmodular(7, 5, Mod(7,19), 'j)
 %4 = Mod(1, 19)*j^8 + Mod(4, 19)*j^7 + Mod(4, 19)*j^6 + ...
 
 ? u = ffgen(5)^0; T = polmodular(3,0,,'j)*u;
 ? polmodular(3, 0, u,'j,1)
 %6 = [j^4 + 3*j^2 + 4*j + 1, 3*j^2 + 2*j + 4, 3*j^3 + 4*j^2 + 4*j + 2]
 ? subst(T,x,u)
 %7 = j^4 + 3*j^2 + 4*j + 1
 ? subst(T',x,u)
 %8 = 3*j^2 + 2*j + 4
 ? subst(T'',x,u)
 %9 = 3*j^3 + 4*j^2 + 4*j + 2
 @eprog

Function: polrecip
Class: basic
Section: polynomials
C-Name: polrecip
Prototype: G
Help: polrecip(pol): reciprocal polynomial of pol.
Doc: reciprocal polynomial of \var{pol}, i.e.~the coefficients are in
 reverse order. \var{pol} must be a polynomial.

Function: polred
Class: basic
Section: number_fields
C-Name: polred0
Prototype: GD0,L,DG
Help: polred(T,{flag=0}): deprecated, use polredbest. Reduction of the
 polynomial T (gives minimal polynomials only). The following binary digits of
 (optional) flag are significant 1: partial reduction, 2: gives also elements.
Doc: This function is \emph{deprecated}, use \tet{polredbest} instead.
 Finds polynomials with reasonably small coefficients defining subfields of
 the number field defined by $T$. One of the polynomials always defines $\Q$
 (hence is equal to $x-1$), and another always defines the same number field
 as $T$ if $T$ is irreducible.
 
 All $T$ accepted by \tet{nfinit} are also allowed here;
 in particular, the format \kbd{[T, listP]} is recommended, e.g. with
 $\kbd{listP} = 10^5$ or a vector containing all ramified primes. Otherwise,
 the maximal order of $\Q[x]/(T)$ must be computed.
 
 The following binary digits of $\fl$ are significant:
 
 1: Possibly use a suborder of the maximal order. The
 primes dividing the index of the order chosen are larger than
 \tet{primelimit} or divide integers stored in the \tet{addprimes} table.
 This flag is \emph{deprecated}, the \kbd{[T, listP]} format is more
 flexible.
 
 2: gives also elements. The result is a two-column matrix, the first column
 giving primitive elements defining these subfields, the second giving the
 corresponding minimal polynomials.
 \bprog
 ? M = polred(x^4 + 8, 2)
 %1 =
 [1 x - 1]
 
 [1/2*x^2 x^2 + 2]
 
 [1/4*x^3 x^4 + 2]
 
 [x x^4 + 8]
 ? minpoly(Mod(M[2,1], x^4+8))
 %2 = x^2 + 2
 @eprog
 
 \synt{polred}{GEN T} ($\fl = 0$). Also available is
 \fun{GEN}{polred2}{GEN T} ($\fl = 2$). The function \kbd{polred0} is
 deprecated, provided for backward compatibility.
Obsolete: 2013-03-27

Function: polredabs
Class: basic
Section: number_fields
C-Name: polredabs0
Prototype: GD0,L,
Help: polredabs(T,{flag=0}): a smallest generating polynomial of the number
 field for the T2 norm on the roots, with smallest index for the minimal T2
 norm. flag is optional, whose binary digit mean 1: give the element whose
 characteristic polynomial is the given polynomial. 4: give all polynomials
 of minimal T2 norm (give only one of P(x) and P(-x)).
Doc: returns a canonical defining polynomial $P$ for the number field
 $\Q[X]/(T)$ defined by $T$, such that the sum of the squares of the modulus
 of the roots (i.e.~the $T_2$-norm) is minimal. Different $T$ defining
 isomorphic number fields will yield the same $P$. All $T$ accepted by
 \tet{nfinit} are also allowed here, e.g. non-monic polynomials, or pairs
 \kbd{[T, listP]} specifying that a non-maximal order may be used. For
 convenience, any number field structure (\var{nf}, \var{bnf},\dots) can also
 be used instead of $T$.
 \bprog
 ? polredabs(x^2 + 16)
 %1 = x^2 + 1
 ? K = bnfinit(x^2 + 16); polredabs(K)
 %2 = x^2 + 1
 @eprog
 
 \misctitle{Warning 1} Using a \typ{POL} $T$ requires computing
 and fully factoring the discriminant $d_K$ of the maximal order which may be
 very hard. You can use the format \kbd{[T, listP]}, where \kbd{listP}
 encodes a list of known coprime divisors of $\disc(T)$ (see \kbd{??nfbasis}),
 to help the routine, thereby replacing this part of the algorithm by a
 polynomial time computation But this may only compute a suborder of the
 maximal order, when the divisors are not squarefree or do not include all
 primes dividing $d_K$. The routine attempts to certify the result
 independently of this order computation as per \tet{nfcertify}: we try to
 prove that the computed order is maximal. If the certification fails,
 the routine then fully factors the integers returned by \kbd{nfcertify}.
 You can use \tet{polredbest} or \kbd{polredabs(,16)} to avoid this
 factorization step; in both cases, the result is no longer canonical.
 
 \misctitle{Warning 2} Apart from the factorization of the discriminant of
 $T$, this routine runs in polynomial time for a \emph{fixed} degree.
 But the complexity is exponential in the degree: this routine
 may be exceedingly slow when the number field has many subfields, hence a
 lot of elements of small $T_2$-norm. If you do not need a canonical
 polynomial, the function \tet{polredbest} is in general much faster (it runs
 in polynomial time), and tends to return polynomials with smaller
 discriminants.
 
 The binary digits of $\fl$ mean
 
 1: outputs a two-component row vector $[P,a]$, where $P$ is the default
 output and \kbd{Mod(a, P)} is a root of the original $T$.
 
 4: gives \emph{all} polynomials of minimal $T_2$ norm; of the two polynomials
 $P(x)$ and $\pm P(-x)$, only one is given.
 
 16: Possibly use a suborder of the maximal order, \emph{without} attempting to
 certify the result as in Warning 1: we always return a polynomial and never
 $0$. The result is a priori not canonical.
 
 \bprog
 ? T = x^16 - 136*x^14 + 6476*x^12 - 141912*x^10 + 1513334*x^8 \
       - 7453176*x^6 + 13950764*x^4 - 5596840*x^2 + 46225
 ? T1 = polredabs(T); T2 = polredbest(T);
 ? [ norml2(polroots(T1)), norml2(polroots(T2)) ]
 %3 = [88.0000000, 120.000000]
 ? [ sizedigit(poldisc(T1)), sizedigit(poldisc(T2)) ]
 %4 = [75, 67]
 @eprog
Variant: Instead of the above hardcoded numerical flags, one should use an
 or-ed combination of
 
 \item \tet{nf_PARTIALFACT}: possibly use a suborder of the maximal order,
 \emph{without} attempting to certify the result.
 
 \item \tet{nf_ORIG}: return $[P, a]$, where \kbd{Mod(a, P)} is a root of $T$.
 
 \item \tet{nf_RAW}: return $[P, b]$, where \kbd{Mod(b, T)} is a root of $P$.
 The algebraic integer $b$ is the raw result produced by the small vectors
 enumeration in the maximal order; $P$ was computed as the characteristic
 polynomial of \kbd{Mod(b, T)}. \kbd{Mod(a, P)} as in \tet{nf_ORIG}
 is obtained with \tet{modreverse}.
 
 \item \tet{nf_ADDZK}: if $r$ is the result produced with some of the above
 flags (of the form $P$ or $[P,c]$), return \kbd{[r,zk]}, where \kbd{zk} is a
 $\Z$-basis for the maximal order of $\Q[X]/(P)$.
 
 \item \tet{nf_ALL}: return a vector of results of the above form, for all
 polynomials of minimal $T_2$-norm.

Function: polredbest
Class: basic
Section: number_fields
C-Name: polredbest
Prototype: GD0,L,
Help: polredbest(T,{flag=0}): reduction of the polynomial T (gives minimal
 polynomials only). If flag=1, gives also elements.
Doc: finds a polynomial with reasonably
 small coefficients defining the same number field as $T$.
 All $T$ accepted by \tet{nfinit} are also allowed here (e.g. non-monic
 polynomials, \kbd{nf}, \kbd{bnf}, \kbd{[T,Z\_K\_basis]}). Contrary to
 \tet{polredabs}, this routine runs in polynomial time, but it offers no
 guarantee as to the minimality of its result.
 
 This routine computes an LLL-reduced basis for the ring of integers of
 $\Q[X]/(T)$, then examines small linear combinations of the basis vectors,
 computing their characteristic polynomials. It returns the \emph{separable}
 $P$ polynomial of smallest discriminant (the one with lexicographically
 smallest \kbd{abs(Vec(P))} in case of ties). This is a good candidate
 for subsequent number field computations, since it guarantees that
 the denominators of algebraic integers, when expressed in the power basis,
 are reasonably small. With no claim of minimality, though.
 
 It can happen that iterating this functions yields better and better
 polynomials, until it stabilizes:
 \bprog
 ? \p5
 ? P = X^12+8*X^8-50*X^6+16*X^4-3069*X^2+625;
 ? poldisc(P)*1.
 %2 = 1.2622 E55
 ? P = polredbest(P);
 ? poldisc(P)*1.
 %4 = 2.9012 E51
 ? P = polredbest(P);
 ? poldisc(P)*1.
 %6 = 8.8704 E44
 @eprog\noindent In this example, the initial polynomial $P$ is the one
 returned by \tet{polredabs}, and the last one is stable.
 
 If $\fl = 1$: outputs a two-component row vector $[P,a]$,  where $P$ is the
 default output and \kbd{Mod(a, P)} is a root of the original $T$.
 \bprog
 ? [P,a] = polredbest(x^4 + 8, 1)
 %1 = [x^4 + 2, Mod(x^3, x^4 + 2)]
 ? charpoly(a)
 %2 = x^4 + 8
 @eprog\noindent In particular, the map $\Q[x]/(T) \to \Q[x]/(P)$,
 $x\mapsto \kbd{Mod(a,P)}$ defines an isomorphism of number fields, which can
 be computed as
 \bprog
   subst(lift(Q), 'x, a)
 @eprog\noindent if $Q$ is a \typ{POLMOD} modulo $T$; \kbd{b = modreverse(a)}
 returns a \typ{POLMOD} giving the inverse of the above map (which should be
 useless since $\Q[x]/(P)$ is a priori a better representation for the number
 field and its elements).

Function: polredord
Class: basic
Section: number_fields
C-Name: polredord
Prototype: G
Help: polredord(x): this function is obsolete, use polredbest.
Doc: This function is obsolete, use polredbest.
Obsolete: 2008-07-20

Function: polresultant
Class: basic
Section: polynomials
C-Name: polresultant0
Prototype: GGDnD0,L,
Help: polresultant(x,y,{v},{flag=0}): resultant of the polynomials x and y,
 with respect to the main variables of x and y if v is omitted, with respect
 to the variable v otherwise. flag is optional, and can be 0: default,
 uses either the subresultant algorithm, a modular algorithm or Sylvester's
 matrix, depending on the inputs; 1 uses Sylvester's matrix (should always be
 slower than the default).
Doc: resultant of the two
 polynomials $x$ and $y$ with exact entries, with respect to the main
 variables of $x$ and $y$ if $v$ is omitted, with respect to the variable $v$
 otherwise. The algorithm assumes the base ring is a domain. If you also need
 the $u$ and $v$ such that $x*u + y*v = \text{Res}(x,y)$, use the
 \tet{polresultantext} function.
 
 If $\fl=0$ (default), uses the algorithm best suited to the inputs,
 either the \idx{subresultant algorithm} (Lazard/Ducos variant, generic case),
 a modular algorithm (inputs in $\Q[X]$) or Sylvester's matrix (inexact
 inputs).
 
 If $\fl=1$, uses the determinant of Sylvester's matrix instead; this should
 always be slower than the default.

Function: polresultantext
Class: basic
Section: polynomials
C-Name: polresultantext0
Prototype: GGDn
Help: polresultantext(A,B,{v}): return [U,V,R] such that
 R=polresultant(A,B,v) and U*A+V*B = R, where A and B are polynomials.
Doc: finds polynomials $U$ and $V$ such that $A*U + B*V = R$, where $R$ is
 the resultant of $U$ and $V$ with respect to the main variables of $A$ and
 $B$ if $v$ is omitted, and with respect to $v$ otherwise. Returns the row
 vector $[U,V,R]$. The algorithm used (subresultant) assumes that the base
 ring is a domain.
 \bprog
 ? A = x*y; B = (x+y)^2;
 ? [U,V,R] = polresultantext(A, B)
 %2 = [-y*x - 2*y^2, y^2, y^4]
 ? A*U + B*V
 %3 = y^4
 ? [U,V,R] = polresultantext(A, B, y)
 %4 = [-2*x^2 - y*x, x^2, x^4]
 ? A*U+B*V
 %5 = x^4
 @eprog
Variant: Also available is
 \fun{GEN}{polresultantext}{GEN x, GEN y}.

Function: polroots
Class: basic
Section: polynomials
C-Name: roots
Prototype: Gp
Help: polroots(T): complex roots of the polynomial T using
 Schonhage's method, as modified by Gourdon.
Doc: complex roots of the polynomial
 $T$, given as a column vector where each root is repeated according to
 its multiplicity. The precision is given as for transcendental functions: in
 GP it is kept in the variable \kbd{realprecision} and is transparent to the
 user, but it must be explicitly given as a second argument in library mode.
 
 The algorithm used is a modification of Sch\"onhage\sidx{Sch\"onage}'s
 root-finding algorithm, due to and originally implemented by Gourdon.
 It is guaranteed to converge; if furthermore $T$ has rational coefficients,
 roots are guaranteed to the required relative accuracy.

Function: polrootsff
Class: basic
Section: number_theoretical
C-Name: polrootsff
Prototype: GDGDG
Help: polrootsff(x,{p},{a}): returns the roots of the polynomial x in the finite
 field F_p[X]/a(X)F_p[X]. a or p can be omitted if x has t_FFELT coefficients.
Doc: returns the vector of distinct roots of the polynomial $x$ in the field
 $\F_q$ defined by the irreducible polynomial $a$ over $\F_p$. The
 coefficients of $x$ must be operation-compatible with $\Z/p\Z$.
 Either $a$ or $p$ can omitted (in which case both are ignored) if x has
 \typ{FFELT} coefficients:
 \bprog
 ? polrootsff(x^2 + 1, 5, y^2+3)  \\ over F_5[y]/(y^2+3) ~ F_25
 %1 = [Mod(Mod(3, 5), Mod(1, 5)*y^2 + Mod(3, 5)),
       Mod(Mod(2, 5), Mod(1, 5)*y^2 + Mod(3, 5))]
 ? t = ffgen(y^2 + Mod(3,5), 't); \\ a generator for F_25 as a t_FFELT
 ? polrootsff(x^2 + 1)   \\ not enough information to determine the base field
  ***   at top-level: polrootsff(x^2+1)
  ***                 ^-----------------
  *** polrootsff: incorrect type in factorff.
 ? polrootsff(x^2 + t^0) \\ make sure one coeff. is a t_FFELT
 %3 = [3, 2]
 ? polrootsff(x^2 + t + 1)
 %4 = [2*t + 1, 3*t + 4]
 @eprog\noindent
 Notice that the second syntax is easier to use and much more readable.

Function: polrootsmod
Class: basic
Section: polynomials
C-Name: rootmod0
Prototype: GGD0,L,
Help: polrootsmod(pol,p,{flag=0}): roots mod the prime p of the polynomial pol. flag is
 optional, and can be 0: default, or 1: use a naive search, useful for small p.
Description: 
 (pol, int, ?0):vec           rootmod($1, $2)
 (pol, int, 1):vec            rootmod2($1, $2)
 (pol, int, #small):vec       $"Bad flag in polrootsmod"
 (pol, int, small):vec        rootmod0($1, $2, $3)
Doc: row vector of roots modulo $p$ of the polynomial \var{pol}.
 Multiple roots are \emph{not} repeated.
 \bprog
 ? polrootsmod(x^2-1,2)
 %1 = [Mod(1, 2)]~
 @eprog\noindent
 If $p$ is very small, you may set $\fl=1$, which uses a naive search.

Function: polrootspadic
Class: basic
Section: polynomials
C-Name: rootpadic
Prototype: GGL
Help: polrootspadic(x,p,r): p-adic roots of the polynomial x to precision r.
Doc: vector of $p$-adic roots of the polynomial \var{pol}, given to
 $p$-adic precision $r$ $p$ is assumed to be a prime. Multiple roots are
 \emph{not} repeated. Note that this is not the same as the roots in
 $\Z/p^r\Z$, rather it gives approximations in $\Z/p^r\Z$ of the true roots
 living in $\Q_p$.
 \bprog
 ? polrootspadic(x^3 - x^2 + 64, 2, 5)
 %1 = [2^3 + O(2^5), 2^3 + 2^4 + O(2^5), 1 + O(2^5)]~
 @eprog
 If \var{pol} has inexact \typ{PADIC} coefficients, this is not always
 well-defined; in this case, the polynomial is first made integral by dividing
 out the $p$-adic content, then lifted
 to $\Z$ using \tet{truncate} coefficientwise. Hence the roots given are
 approximations of the roots of an exact polynomial which is $p$-adically
 close to the input. To avoid pitfalls, we advise to only factor polynomials
 with exact rational coefficients.

Function: polrootsreal
Class: basic
Section: polynomials
C-Name: realroots
Prototype: GDGp
Help: polrootsreal(T, {ab}): real roots of the polynomial T with rational
 coefficients, using Uspensky's method. In interval ab = [a,b] if present.
Doc: real roots of the polynomial $T$ with rational coefficients, multiple
 roots being included according to their multiplicity. The roots are given
 to a relative accuracy of \kbd{realprecision}. If argument \var{ab} is
 present, it must be a vector $[a,b]$ with two components (of type
 \typ{INT}, \typ{FRAC} or \typ{INFINITY}) and we restrict to roots belonging
 to that closed interval.
 \bprog
 ? \p9
 ? polrootsreal(x^2-2)
 %1 = [-1.41421356, 1.41421356]~
 ? polrootsreal(x^2-2, [1,+oo])
 %2 = [1.41421356]~
 ? polrootsreal(x^2-2, [2,3])
 %3 = []~
 ? polrootsreal((x-1)*(x-2), [2,3])
 %4 = [2.00000000]~
 @eprog\noindent
 The algorithm used is a modification of Uspensky's method (relying on
 Descartes's rule of sign), following Rouillier and Zimmerman's article
 ``Efficient isolation of a polynomial real roots''
 (\url{http://hal.inria.fr/inria-00072518/}). Barring bugs, it is guaranteed
 to converge and to give the roots to the required accuracy.
 
 \misctitle{Remark} If the polynomial $T$ is of the
 form $Q(x^h)$ for some $h\geq 2$ and \var{ab} is omitted, the routine will
 apply the algorithm to $Q$ (restricting to non-negative roots when $h$ is
 even), then take $h$-th roots. On the other hand, if you want to specify
 \var{ab}, you should apply the routine to $Q$ yourself and a suitable
 interval $[a',b']$ using approximate $h$-th roots adapted to your problem:
 the function will not perform this change of variables if \var{ab} is present.

Function: polsturm
Class: basic
Section: polynomials
C-Name: sturmpart
Prototype: lGDGDG
Help: polsturm(T,{ab}): number of real roots of the squarefree polynomial
 T (in the interval ab = [a,b] if present).
Doc: number of real roots of the real squarefree polynomial \var{T}. If
 the argument \var{ab} is present, it must be a vector $[a,b]$ with
 two real components (of type \typ{INT}, \typ{REAL}, \typ{FRAC}
 or  \typ{INFINITY}) and we count roots belonging to that closed interval.
 
 If possible, you should stick to exact inputs, that is avoid \typ{REAL}s in
 $T$ and the bounds $a,b$: the result is then guaranteed and we use a fast
 algorithm (Uspensky's method, relying on Descartes's rule of sign, see
 \tet{polrootsreal}); otherwise, we use Sturm's algorithm and the result
 may be wrong due to round-off errors.
 \bprog
 ? T = (x-1)*(x-2)*(x-3);
 ? polsturm(T)
 %2 = 3
 ? polsturm(T, [-oo,2])
 %3 = 2
 ? polsturm(T, [1/2,+oo])
 %4 = 3
 ? polsturm(T, [1, Pi])  \\ Pi inexact: not recommended !
 %5 = 3
 ? polsturm(T*1., [0, 4])  \\ T*1. inexact: not recommended !
 %6 = 3
 ? polsturm(T^2, [0, 4])  \\ not squarefree
  ***   at top-level: polsturm(T^2,[0,4])
  ***                 ^-------------------
  *** polsturm: domain error in polsturm: issquarefree(pol) = 0
 ? polsturm((T*1.)^2, [0, 4])  \\ not squarefree AND inexact
  ***   at top-level: polsturm((T*1.)^2,[0
  ***                 ^--------------------
  *** polsturm: precision too low in polsturm.
 @eprog\noindent In the last example, the input polynomial is not
 squarefree but there is no way to ascertain it from the given
 floating point approximation: we get a precision error in this case.
 %\syn{NO}
 
 The library syntax is \fun{long}{RgX_sturmpart}{GEN T, GEN ab} or
 \fun{long}{sturm}{GEN T} (for the case \kbd{ab = NULL}). The function
 \fun{long}{sturmpart}{GEN T, GEN a, GEN b} is obsolete and deprecated.

Function: polsubcyclo
Class: basic
Section: polynomials
C-Name: polsubcyclo
Prototype: LLDn
Help: polsubcyclo(n,d,{v='x}): finds an equation (in variable v) for the d-th
 degree subfields of Q(zeta_n). Output is a polynomial, or a vector of
 polynomials if there are several such fields or none.
Doc: gives polynomials (in variable $v$) defining the sub-Abelian extensions
 of degree $d$ of the cyclotomic field $\Q(\zeta_n)$, where $d\mid \phi(n)$.
 
 If there is exactly one such extension the output is a polynomial, else it is
 a vector of polynomials, possibly empty. To get a vector in all cases,
 use \kbd{concat([], polsubcyclo(n,d))}.
 
 The function \tet{galoissubcyclo} allows to specify exactly which
 sub-Abelian extension should be computed.

Function: polsylvestermatrix
Class: basic
Section: polynomials
C-Name: sylvestermatrix
Prototype: GG
Help: polsylvestermatrix(x,y): forms the sylvester matrix attached to the
 two polynomials x and y. Warning: the polynomial coefficients are in
 columns, not in rows.
Doc: forms the Sylvester matrix
 corresponding to the two polynomials $x$ and $y$, where the coefficients of
 the polynomials are put in the columns of the matrix (which is the natural
 direction for solving equations afterwards). The use of this matrix can be
 essential when dealing with polynomials with inexact entries, since
 polynomial Euclidean division doesn't make much sense in this case.

Function: polsym
Class: basic
Section: polynomials
C-Name: polsym
Prototype: GL
Help: polsym(x,n): column vector of symmetric powers of the roots of x up to n.
Doc: creates the column vector of the \idx{symmetric powers} of the roots of the
 polynomial $x$ up to power $n$, using Newton's formula.

Function: poltchebi
Class: basic
Section: polynomials
C-Name: polchebyshev1
Prototype: LDn
Help: poltchebi(n,{v='x}): deprecated alias for polchebyshev.
Doc: deprecated alias for \kbd{polchebyshev}
Obsolete: 2013-04-03

Function: poltschirnhaus
Class: basic
Section: number_fields
C-Name: tschirnhaus
Prototype: G
Help: poltschirnhaus(x): random Tschirnhausen transformation of the
 polynomial x.
Doc: applies a random Tschirnhausen
 transformation to the polynomial $x$, which is assumed to be non-constant
 and separable, so as to obtain a new equation for the \'etale algebra
 defined by $x$. This is for instance useful when computing resolvents,
 hence is used by the \kbd{polgalois} function.

Function: polylog
Class: basic
Section: transcendental
C-Name: polylog0
Prototype: LGD0,L,p
Help: polylog(m,x,{flag=0}): m-th polylogarithm of x. flag is optional, and
 can be 0: default, 1: D_m~-modified m-th polylog of x, 2: D_m-modified m-th
 polylog of x, 3: P_m-modified m-th polylog of x.
Doc: one of the different polylogarithms, depending on \fl:
 
 If $\fl=0$ or is omitted: $m^\text{th}$ polylogarithm of $x$, i.e.~analytic
 continuation of the power series $\text{Li}_m(x)=\sum_{n\ge1}x^n/n^m$
 ($x < 1$). Uses the functional equation linking the values at $x$ and $1/x$
 to restrict to the case $|x|\leq 1$, then the power series when
 $|x|^2\le1/2$, and the power series expansion in $\log(x)$ otherwise.
 
 Using $\fl$, computes a modified $m^\text{th}$ polylogarithm of $x$.
 We use Zagier's notations; let $\Re_m$ denote $\Re$ or $\Im$ depending
 on whether $m$ is odd or even:
 
 If $\fl=1$: compute $\tilde D_m(x)$, defined for $|x|\le1$ by
 $$\Re_m\left(\sum_{k=0}^{m-1} \dfrac{(-\log|x|)^k}{k!}\text{Li}_{m-k}(x)
 +\dfrac{(-\log|x|)^{m-1}}{m!}\log|1-x|\right).$$
 
 If $\fl=2$: compute $D_m(x)$, defined for $|x|\le1$ by
 $$\Re_m\left(\sum_{k=0}^{m-1}\dfrac{(-\log|x|)^k}{k!}\text{Li}_{m-k}(x)
 -\dfrac{1}{2}\dfrac{(-\log|x|)^m}{m!}\right).$$
 
 If $\fl=3$: compute $P_m(x)$, defined for $|x|\le1$ by
 $$\Re_m\left(\sum_{k=0}^{m-1}\dfrac{2^kB_k}{k!}(\log|x|)^k\text{Li}_{m-k}(x)
 -\dfrac{2^{m-1}B_m}{m!}(\log|x|)^m\right).$$
 
 These three functions satisfy the functional equation
 $f_m(1/x) = (-1)^{m-1}f_m(x)$.
Variant: Also available is
 \fun{GEN}{gpolylog}{long m, GEN x, long prec} (\fl = 0).

Function: polzagier
Class: basic
Section: polynomials
C-Name: polzag
Prototype: LL
Help: polzagier(n,m): Zagier's polynomials of index n,m.
Doc: creates Zagier's polynomial $P_n^{(m)}$ used in
 the functions \kbd{sumalt} and \kbd{sumpos} (with $\fl=1$), see
 ``Convergence acceleration of alternating series'', Cohen et al.,
 \emph{Experiment.~Math.}, vol.~9, 2000, pp.~3--12.
 
 If $m < 0$ or $m \ge n$, $P_n^{(m)} = 0$.
 We have
 $P_n := P_n^{(0)}$ is $T_n(2x-1)$, where $T_n$ is the Legendre polynomial of
 the second kind. For $n > m > 0$, $P_n^{(m)}$ is the $m$-th difference with
 step $2$ of the sequence $n^{m+1}P_n$; in this case, it satisfies
 $$2 P_n^{(m)}(sin^2 t) = \dfrac{d^{m+1}}{dt^{m+1}}(\sin(2t)^m \sin(2(n-m)t)).$$
 
 %@article {MR2001m:11222,
 %    AUTHOR = {Cohen, Henri and Rodriguez Villegas, Fernando and Zagier, Don},
 %     TITLE = {Convergence acceleration of alternating series},
 %   JOURNAL = {Experiment. Math.},
 %    VOLUME = {9},
 %      YEAR = {2000},
 %    NUMBER = {1},
 %     PAGES = {3--12},
 %}

Function: powers
Class: basic
Section: operators
C-Name: gpowers0
Prototype: GLDG
Help: powers(x,n,{x0}): return the vector [1,x,...,x^n] if x0 is omitted,
 and [x0, x0*x, ..., x0*x^n] otherwise.
Description: 
 (gen, small):vec  gpowers($1, $2)
Doc: for non-negative $n$, return the vector with $n+1$ components
 $[1,x,\dots,x^n]$ if \kbd{x0} is omitted, and $[x_0, x_0*x, ..., x_0*x^n]$
 otherwise.
 \bprog
 ? powers(Mod(3,17), 4)
 %1 = [Mod(1, 17), Mod(3, 17), Mod(9, 17), Mod(10, 17), Mod(13, 17)]
 ? powers(Mat([1,2;3,4]), 3)
 %2 = [[1, 0; 0, 1], [1, 2; 3, 4], [7, 10; 15, 22], [37, 54; 81, 118]]
 ? powers(3, 5, 2)
 %3 = [2, 6, 18, 54, 162, 486]
 @eprog\noindent When $n < 0$, the function returns the empty vector \kbd{[]}.
Variant: Also available is
 \fun{GEN}{gpowers}{GEN x, long n} when \kbd{x0} is \kbd{NULL}.

Function: precision
Class: basic
Section: conversions
C-Name: precision0
Prototype: GD0,L,
Help: precision(x,{n}): if n is present, return x at precision n. If n is
 omitted, return real precision of object x.
Description: 
 (real):small          prec2ndec(gprecision($1))
 (gen):int             precision0($1, 0)
 (real,0):small        prec2ndec(gprecision($1))
 (gen,0):int           precision0($1, 0)
 (real,#small):real    rtor($1, ndec2prec($2))
 (gen,#small):gen      gprec($1, $2)
 (real,small):real     precision0($1, $2)
 (mp,small):mp         precision0($1, $2)
 (gen,small):gen       precision0($1, $2)
Doc: the function behaves differently according to whether $n$ is
 present and positive or not. If $n$ is missing, the function returns the
 precision in decimal digits of the PARI object $x$. If $x$ is an exact
 object, the function returns \kbd{+oo}.
 
 \bprog
 ? precision(exp(1e-100))
 %1 = 154                \\ 154 significant decimal digits
 ? precision(2 + x)
 %2 = +oo                \\ exact object
 ? precision(0.5 + O(x))
 %3 = 38                 \\ floating point accuracy, NOT series precision
 ? precision( [ exp(1e-100), 0.5 ] )
 %4 = 38                 \\ minimal accuracy among components
 @eprog
 
 If $n$ is present, the function creates a new object equal to $x$ with a new
 floating point precision $n$: $n$ is the number of desired significant
 \emph{decimal} digits. If $n$ is smaller than the precision of a \typ{REAL}
 component of $x$, it is truncated, otherwise it is extended with zeros.
 For exact or non-floating point types, no change.
Variant: Also available are \fun{GEN}{gprec}{GEN x, long n} and
 \fun{long}{precision}{GEN x}. In both, the accuracy is expressed in
 \emph{words} (32-bit or 64-bit depending on the architecture).

Function: precprime
Class: basic
Section: number_theoretical
C-Name: precprime
Prototype: G
Help: precprime(x): largest pseudoprime <= x, 0 if x<=1.
Description: 
 (gen):int        precprime($1)
Doc: finds the largest pseudoprime (see
 \tet{ispseudoprime}) less than or equal to $x$. $x$ can be of any real type.
 Returns 0 if $x\le1$. Note that if $x$ is a prime, this function returns $x$
 and not the largest prime strictly smaller than $x$. To rigorously prove that
 the result is prime, use \kbd{isprime}.

Function: prime
Class: basic
Section: number_theoretical
C-Name: prime
Prototype: L
Help: prime(n): returns the n-th prime (n C-integer).
Doc: the $n^{\text{th}}$ prime number
 \bprog
 ? prime(10^9)
 %1 = 22801763489
 @eprog\noindent Uses checkpointing and a naive $O(n)$ algorithm.

Function: primepi
Class: basic
Section: number_theoretical
C-Name: primepi
Prototype: G
Help: primepi(x): the prime counting function pi(x) = #{p <= x, p prime}.
Description: 
 (gen):int        primepi($1)
Doc: the prime counting function. Returns the number of
 primes $p$, $p \leq x$.
 \bprog
 ? primepi(10)
 %1 = 4;
 ? primes(5)
 %2 = [2, 3, 5, 7, 11]
 ? primepi(10^11)
 %3 = 4118054813
 @eprog\noindent Uses checkpointing and a naive $O(x)$ algorithm.

Function: primes
Class: basic
Section: number_theoretical
C-Name: primes0
Prototype: G
Help: primes(n): returns the vector of the first n primes (integer), or the
 primes in interval n = [a,b].
Doc: creates a row vector whose components are the first $n$ prime numbers.
 (Returns the empty vector for $n \leq 0$.) A \typ{VEC} $n = [a,b]$ is also
 allowed, in which case the primes in $[a,b]$ are returned
 \bprog
 ? primes(10)     \\ the first 10 primes
 %1 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
 ? primes([0,29])  \\ the primes up to 29
 %2 = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
 ? primes([15,30])
 %3 = [17, 19, 23, 29]
 @eprog

Function: print
Class: basic
Section: programming/specific
C-Name: print
Prototype: vs*
Help: print({str}*): outputs its string arguments (in raw format) ending with
 a newline.
Description: 
 (?gen,...):void  pari_printf("${2 format_string}\n"${2 format_args})
Doc: outputs its (string) arguments in raw format, ending with a newline.
 %\syn{NO}

Function: print1
Class: basic
Section: programming/specific
C-Name: print1
Prototype: vs*
Help: print1({str}*): outputs its string arguments (in raw format) without
 ending with newline.
Description: 
 (?gen,...):void  pari_printf("${2 format_string}"${2 format_args})
Doc: outputs its (string) arguments in raw
 format, without ending with a newline. Note that you can still embed newlines
 within your strings, using the \b{n} notation~!
 %\syn{NO}

Function: printf
Class: basic
Section: programming/specific
C-Name: printf0
Prototype: vss*
Help: printf(fmt,{x}*): prints its arguments according to the format fmt.
Doc: This function is based on the C library command of the same name.
 It prints its arguments according to the format \var{fmt}, which specifies how
 subsequent arguments are converted for output. The format is a
 character string composed of zero or more directives:
 
 \item ordinary characters (not \kbd{\%}), printed unchanged,
 
 \item conversions specifications (\kbd{\%} followed by some characters)
 which fetch one argument from the list and prints it according to the
 specification.
 
 More precisely, a conversion specification consists in a \kbd{\%}, one or more
 optional flags (among \kbd{\#}, \kbd{0}, \kbd{-}, \kbd{+}, ` '), an optional
 decimal digit string specifying a minimal field width, an optional precision
 in the form of a period (`\kbd{.}') followed by a decimal digit string, and
 the conversion specifier (among \kbd{d},\kbd{i}, \kbd{o}, \kbd{u},
 \kbd{x},\kbd{X}, \kbd{p}, \kbd{e},\kbd{E}, \kbd{f}, \kbd{g},\kbd{G}, \kbd{s}).
 
 \misctitle{The flag characters} The character \kbd{\%} is followed by zero or
 more of the following flags:
 
 \item \kbd{\#}: the value is converted to an ``alternate form''. For
 \kbd{o} conversion (octal), a \kbd{0} is prefixed to the string. For \kbd{x}
 and \kbd{X} conversions (hexa), respectively \kbd{0x} and \kbd{0X} are
 prepended. For other conversions, the flag is ignored.
 
 \item \kbd{0}: the value should be zero padded. For
 \kbd{d},
 \kbd{i},
 \kbd{o},
 \kbd{u},
 \kbd{x},
 \kbd{X}
 \kbd{e},
 \kbd{E},
 \kbd{f},
 \kbd{F},
 \kbd{g}, and
 \kbd{G} conversions, the value is padded on the left with zeros rather than
 blanks. (If the \kbd{0} and \kbd{-} flags both appear, the \kbd{0} flag is
 ignored.)
 
 \item \kbd{-}: the value is left adjusted on the field boundary. (The
 default is right justification.) The value is padded on the right with
 blanks, rather than on the left with blanks or zeros. A \kbd{-} overrides a
 \kbd{0} if both are given.
 
 \item \kbd{` '} (a space): a blank is left before a positive number
 produced by a signed conversion.
 
 \item \kbd{+}: a sign (+ or -) is placed before a number produced by a
 signed conversion. A \kbd{+} overrides a space if both are used.
 
 \misctitle{The field width} An optional decimal digit string (whose first
 digit is non-zero) specifying a \emph{minimum} field width. If the value has
 fewer characters than the field width, it is padded with spaces on the left
 (or right, if the left-adjustment flag has been given). In no case does a
 small field width cause truncation of a field; if the value is wider than
 the field width, the field is expanded to contain the conversion result.
 Instead of a decimal digit string, one may write \kbd{*} to specify that the
 field width is given in the next argument.
 
 \misctitle{The precision} An optional precision in the form of a period
 (`\kbd{.}') followed by a decimal digit string. This gives
 the number of digits to appear after the radix character for \kbd{e},
 \kbd{E}, \kbd{f}, and \kbd{F} conversions, the maximum number of significant
 digits for \kbd{g} and \kbd{G} conversions, and the maximum number of
 characters to be printed from an \kbd{s} conversion.
 Instead of a decimal digit string, one may write \kbd{*} to specify that the
 field width is given in the next argument.
 
 \misctitle{The length modifier} This is ignored under \kbd{gp}, but
 necessary for \kbd{libpari} programming. Description given here for
 completeness:
 
 \item \kbd{l}: argument is a \kbd{long} integer.
 
 \item \kbd{P}: argument is a \kbd{GEN}.
 
 \misctitle{The conversion specifier} A character that specifies the type of
 conversion to be applied.
 
 \item \kbd{d}, \kbd{i}: a signed integer.
 
 \item \kbd{o}, \kbd{u}, \kbd{x}, \kbd{X}: an unsigned integer, converted
 to unsigned octal (\kbd{o}), decimal (\kbd{u}) or hexadecimal (\kbd{x} or
 \kbd{X}) notation. The letters \kbd{abcdef} are used for \kbd{x}
 conversions;  the letters \kbd{ABCDEF} are used for \kbd{X} conversions.
 
 \item \kbd{e}, \kbd{E}: the (real) argument is converted in the style
 \kbd{[ -]d.ddd e[ -]dd}, where there is one digit before the decimal point,
 and the number of digits after it is equal to the precision; if the
 precision is missing, use the current \kbd{realprecision} for the total
 number of printed digits. If the precision is explicitly 0, no decimal-point
 character appears. An \kbd{E} conversion uses the letter \kbd{E} rather
 than \kbd{e} to introduce the exponent.
 
 \item \kbd{f}, \kbd{F}: the (real) argument is converted in the style
 \kbd{[ -]ddd.ddd}, where the number of digits after the decimal point
 is equal to the precision; if the precision is missing, use the current
 \kbd{realprecision} for the total number of printed digits. If the precision
 is explicitly 0, no decimal-point character appears. If a decimal point
 appears, at least one digit appears before it.
 
 \item \kbd{g}, \kbd{G}: the (real) argument is converted in style
 \kbd{e} or \kbd{f} (or \kbd{E} or \kbd{F} for \kbd{G} conversions)
 \kbd{[ -]ddd.ddd}, where the total number of digits printed
 is equal to the precision; if the precision is missing, use the current
 \kbd{realprecision}. If the precision is explicitly 0, it is treated as 1.
 Style \kbd{e} is used when
 the decimal exponent is $< -4$, to print \kbd{0.}, or when the integer
 part cannot be decided given the known significant digits, and the \kbd{f}
 format otherwise.
 
 \item \kbd{c}: the integer argument is converted to an unsigned char, and the
 resulting character is written.
 
 \item \kbd{s}: convert to a character string. If a precision is given, no
 more than the specified number of characters are written.
 
 \item \kbd{p}: print the address of the argument in hexadecimal (as if by
 \kbd{\%\#x}).
 
 \item \kbd{\%}: a \kbd{\%} is written. No argument is converted. The complete
 conversion specification is \kbd{\%\%}.
 
 \noindent Examples:
 
 \bprog
 ? printf("floor: %d, field width 3: %3d, with sign: %+3d\n", Pi, 1, 2);
 floor: 3, field width 3:   1, with sign:  +2
 
 ? printf("%.5g %.5g %.5g\n",123,123/456,123456789);
 123.00 0.26974 1.2346 e8
 
 ? printf("%-2.5s:%2.5s:%2.5s\n", "P", "PARI", "PARIGP");
 P :PARI:PARIG
 
 \\ min field width and precision given by arguments
 ? x = 23; y=-1/x; printf("x=%+06.2f y=%+0*.*f\n", x, 6, 2, y);
 x=+23.00 y=-00.04
 
 \\ minimum fields width 5, pad left with zeroes
 ? for (i = 2, 5, printf("%05d\n", 10^i))
 00100
 01000
 10000
 100000  \\@com don't truncate fields whose length is larger than the minimum width
 ? printf("%.2f  |%06.2f|", Pi,Pi)
 3.14  |  3.14|
 @eprog\noindent All numerical conversions apply recursively to the entries
 of vectors and matrices:
 \bprog
 ? printf("%4d", [1,2,3]);
 [   1,   2,   3]
 ? printf("%5.2f", mathilbert(3));
 [ 1.00  0.50  0.33]
 
 [ 0.50  0.33  0.25]
 
 [ 0.33  0.25  0.20]
 @eprog
 \misctitle{Technical note} Our implementation of \tet{printf}
 deviates from the C89 and C99 standards in a few places:
 
 \item whenever a precision is missing, the current \kbd{realprecision} is
 used to determine the number of printed digits (C89: use 6 decimals after
 the radix character).
 
 \item in conversion style \kbd{e}, we do not impose that the
 exponent has at least two digits; we never write a \kbd{+} sign in the
 exponent; 0 is printed in a special way, always as \kbd{0.E\var{exp}}.
 
 \item in conversion style \kbd{f}, we switch to style \kbd{e} if the
 exponent is greater or equal to the precision.
 
 \item in conversion \kbd{g} and \kbd{G}, we do not remove trailing zeros
  from the fractional part of the result; nor a trailing decimal point;
  0 is printed in a special way, always as \kbd{0.E\var{exp}}.
 %\syn{NO}

Function: printsep
Class: basic
Section: programming/specific
C-Name: printsep
Prototype: vss*
Help: printsep(sep,{str}*): outputs its string arguments (in raw format),
 separated by 'sep', ending with a newline.
Doc: outputs its (string) arguments in raw format, ending with a newline.
 Successive entries are separated by \var{sep}:
 \bprog
 ? printsep(":", 1,2,3,4)
 1:2:3:4
 @eprog
 %\syn{NO}

Function: printsep1
Class: basic
Section: programming/specific
C-Name: printsep1
Prototype: vss*
Help: printsep1(sep,{str}*): outputs its string arguments (in raw format),
 separated by 'sep', without ending with a newline.
Doc: outputs its (string) arguments in raw format, without ending with a
 newline.  Successive entries are separated by \var{sep}:
 \bprog
 ? printsep1(":", 1,2,3,4);print("|")
 1:2:3:4
 @eprog
 %\syn{NO}

Function: printtex
Class: basic
Section: programming/specific
C-Name: printtex
Prototype: vs*
Help: printtex({str}*): outputs its string arguments in TeX format.
Doc: outputs its (string) arguments in \TeX\ format. This output can then be
 used in a \TeX\ manuscript.
 The printing is done on the standard output. If you want to print it to a
 file you should use \kbd{writetex} (see there).
 
 Another possibility is to enable the \tet{log} default
 (see~\secref{se:defaults}).
 You could for instance do:\sidx{logfile}
 %
 \bprog
 default(logfile, "new.tex");
 default(log, 1);
 printtex(result);
 @eprog
 %\syn{NO}

Function: prod
Class: basic
Section: sums
C-Name: produit
Prototype: V=GGEDG
Help: prod(X=a,b,expr,{x=1}): x times the product (X runs from a to b) of
 expression.
Doc: product of expression
 \var{expr}, initialized at $x$, the formal parameter $X$ going from $a$ to
 $b$. As for \kbd{sum}, the main purpose of the initialization parameter $x$
 is to force the type of the operations being performed. For example if it is
 set equal to the integer 1, operations will start being done exactly. If it
 is set equal to the real $1.$, they will be done using real numbers having
 the default precision. If it is set equal to the power series $1+O(X^k)$ for
 a certain $k$, they will be done using power series of precision at most $k$.
 These are the three most common initializations.
 
 \noindent As an extreme example, compare
 
 \bprog
 ? prod(i=1, 100, 1 - X^i);  \\@com this has degree $5050$ !!
 time = 128 ms.
 ? prod(i=1, 100, 1 - X^i, 1 + O(X^101))
 time = 8 ms.
 %2 = 1 - X - X^2 + X^5 + X^7 - X^12 - X^15 + X^22 + X^26 - X^35 - X^40 + \
 X^51 + X^57 - X^70 - X^77 + X^92 + X^100 + O(X^101)
 @eprog\noindent
 Of course, in  this specific case, it is faster to use \tet{eta},
 which is computed using Euler's formula.
 \bprog
 ? prod(i=1, 1000, 1 - X^i, 1 + O(X^1001));
 time = 589 ms.
 ? \ps1000
 seriesprecision = 1000 significant terms
 ? eta(X) - %
 time = 8ms.
 %4 = O(X^1001)
 @eprog
 
 \synt{produit}{GEN a, GEN b, char *expr, GEN x}.

Function: prodeuler
Class: basic
Section: sums
C-Name: prodeuler0
Prototype: V=GGEp
Help: prodeuler(X=a,b,expr): Euler product (X runs over the primes between a
 and b) of real or complex expression.
Doc: product of expression \var{expr},
 initialized at 1. (i.e.~to a \emph{real} number equal to 1 to the current
 \kbd{realprecision}), the formal parameter $X$ ranging over the prime numbers
 between $a$ and $b$.\sidx{Euler product}
 
 \synt{prodeuler}{void *E, GEN (*eval)(void*,GEN), GEN a,GEN b, long prec}.

Function: prodinf
Class: basic
Section: sums
C-Name: prodinf0
Prototype: V=GED0,L,p
Help: prodinf(X=a,expr,{flag=0}): infinite product (X goes from a to
 infinity) of real or complex expression. flag can be 0 (default) or 1, in
 which case compute the product of the 1+expr instead.
Wrapper: (,G)
Description: 
  (gen,gen,?small):gen:prec prodinf(${2 cookie}, ${2 wrapper}, $1, $3, $prec)
Doc: \idx{infinite product} of
 expression \var{expr}, the formal parameter $X$ starting at $a$. The evaluation
 stops when the relative error of the expression minus 1 is less than the
 default precision. In particular, non-convergent products result in infinite
 loops. The expressions must always evaluate to an element of $\C$.
 
 If $\fl=1$, do the product of the ($1+\var{expr}$) instead.
 
 \synt{prodinf}{void *E, GEN (*eval)(void*,GEN), GEN a, long prec}
 ($\fl=0$), or \tet{prodinf1} with the same arguments ($\fl=1$).

Function: psdraw
Class: highlevel
Section: graphic
C-Name: postdraw_flag
Prototype: vGD0,L,
Help: psdraw(list, {flag=0}): same as plotdraw, except that the output is a
 PostScript program in psfile (pari.ps by default), and flag!=0 scales the
 plot from size of the current output device to the standard PostScript
 plotting size.
Doc: same as \kbd{plotdraw}, except that the output is a PostScript program
 appended to the \kbd{psfile}, and flag!=0 scales the plot from size of the
 current output device to the standard PostScript plotting size

Function: psi
Class: basic
Section: transcendental
C-Name: gpsi
Prototype: Gp
Help: psi(x): psi-function at x.
Doc: the $\psi$-function of $x$, i.e.~the logarithmic derivative
 $\Gamma'(x)/\Gamma(x)$.

Function: psploth
Class: highlevel
Section: graphic
C-Name: postploth
Prototype: V=GGEpD0,L,D0,L,
Help: psploth(X=a,b,expr,{flags=0},{n=0}): same as ploth, except that the
 output is a PostScript program in psfile (pari.ps by default).
Doc: same as \kbd{ploth}, except that the output is a PostScript program
 appended to the \kbd{psfile}.

Function: psplothraw
Class: highlevel
Section: graphic
C-Name: postplothraw
Prototype: GGD0,L,
Help: psplothraw(listx,listy,{flag=0}): same as plothraw, except that the
 output is a postscript program in psfile (pari.ps by default).
Doc: same as \kbd{plothraw}, except that the output is a PostScript program
 appended to the \kbd{psfile}.

Function: qfauto
Class: basic
Section: linear_algebra
C-Name: qfauto0
Prototype: GDG
Help: qfauto(G,{fl}): automorphism group of the positive definite quadratic
 form G.
Doc: 
 $G$ being a square and symmetric matrix with integer entries representing a
 positive definite quadratic form, outputs the automorphism group of the
 associate lattice.
 Since this requires computing the minimal vectors, the computations can
 become very lengthy as the dimension grows. $G$ can also be given by an
 \kbd{qfisominit} structure.
 See \kbd{qfisominit} for the meaning of \var{fl}.
 
 The output is a two-components vector $[o,g]$ where $o$ is the group order
 and $g$ is the list of generators (as a vector). For each generator $H$,
 the equality $G={^t}H\*G\*H$ holds.
 
 The interface of this function is experimental and will likely change in the
 future.
 
 This function implements an algorithm of Plesken and Souvignier, following
 Souvignier's implementation.
Variant: The function \fun{GEN}{qfauto}{GEN G, GEN fl} is also available
 where $G$ is a vector of \kbd{zm} matrices.

Function: qfautoexport
Class: basic
Section: linear_algebra
C-Name: qfautoexport
Prototype: GD0,L,
Help: qfautoexport(qfa,{flag}): qfa being an automorphism group as output by
 qfauto, output a string representing the underlying matrix group in
 GAP notation (default) or Magma notation (flag = 1).
Doc: \var{qfa} being an automorphism group as output by
 \tet{qfauto}, export the underlying matrix group as a string suitable
 for (no flags or $\fl=0$) GAP or ($\fl=1$) Magma. The following example
 computes the size of the matrix group using GAP:
 \bprog
 ? G = qfauto([2,1;1,2])
 %1 = [12, [[-1, 0; 0, -1], [0, -1; 1, 1], [1, 1; 0, -1]]]
 ? s = qfautoexport(G)
 %2 = "Group([[-1, 0], [0, -1]], [[0, -1], [1, 1]], [[1, 1], [0, -1]])"
 ? extern("echo \"Order("s");\" | gap -q")
 %3 = 12
 @eprog

Function: qfbclassno
Class: basic
Section: number_theoretical
C-Name: qfbclassno0
Prototype: GD0,L,
Help: qfbclassno(D,{flag=0}): class number of discriminant D using Shanks's
 method by default. If (optional) flag is set to 1, use Euler products.
Doc: ordinary class number of the quadratic order of discriminant $D$, for
 ``small'' values of $D$.
 
 \item if  $D > 0$ or $\fl = 1$, use a $O(|D|^{1/2})$
 algorithm (compute $L(1,\chi_D)$ with the approximate functional equation).
 This is slower than \tet{quadclassunit} as soon as $|D| \approx 10^2$ or
 so and is not meant to be used for large $D$.
 
 \item if $D < 0$ and $\fl = 0$ (or omitted), use a $O(|D|^{1/4})$
 algorithm (Shanks's baby-step/giant-step method). It should
 be faster than \tet{quadclassunit} for small values of $D$, say
 $|D| < 10^{18}$.
 
 \misctitle{Important warning} In the latter case, this function only
 implements part of \idx{Shanks}'s method (which allows to speed it up
 considerably). It gives unconditionnally correct results for $|D| < 2\cdot
 10^{10}$, but may give incorrect results for larger values if the class
 group has many cyclic factors. We thus recommend to double-check results
 using the function \kbd{quadclassunit}, which is about 2 to 3 times slower in
 the above range, assuming GRH. We currently have no counter-examples but
 they should exist: we'd appreciate a bug report if you find one.
 
 \misctitle{Warning} Contrary to what its name implies, this routine does not
 compute the number of classes of binary primitive forms of discriminant $D$,
 which is equal to the \emph{narrow} class number. The two notions are the same
 when $D < 0$ or the fundamental unit $\varepsilon$ has negative norm; when $D
 > 0$ and $N\varepsilon > 0$, the number of classes of forms is twice the
 ordinary class number. This is a problem which we cannot fix for backward
 compatibility reasons. Use the following routine if you are only interested
 in the number of classes of forms:
 \bprog
 QFBclassno(D) =
 qfbclassno(D) * if (D < 0 || norm(quadunit(D)) < 0, 1, 2)
 @eprog\noindent
 Here are a few examples:
 \bprog
 ? qfbclassno(400000028)
 time = 3,140 ms.
 %1 = 1
 ? quadclassunit(400000028).no
 time = 20 ms. \\@com{ much faster}
 %2 = 1
 ? qfbclassno(-400000028)
 time = 0 ms.
 %3 = 7253 \\@com{ correct, and fast enough}
 ? quadclassunit(-400000028).no
 time = 0 ms.
 %4 = 7253
 @eprog\noindent
 See also \kbd{qfbhclassno}.
Variant: The following functions are also available:
 
 \fun{GEN}{classno}{GEN D} ($\fl = 0$)
 
 \fun{GEN}{classno2}{GEN D} ($\fl = 1$).
 
 \noindent Finally
 
 \fun{GEN}{hclassno}{GEN D} computes the class number of an imaginary
 quadratic field by counting reduced forms, an $O(|D|)$ algorithm.

Function: qfbcompraw
Class: basic
Section: number_theoretical
C-Name: qfbcompraw
Prototype: GG
Help: qfbcompraw(x,y): Gaussian composition without reduction of the binary
 quadratic forms x and y.
Doc: \idx{composition} of the binary quadratic forms $x$ and $y$, without
 \idx{reduction} of the result. This is useful e.g.~to compute a generating
 element of an ideal. The result is undefined if $x$ and $y$ do not have the
 same discriminant.

Function: qfbhclassno
Class: basic
Section: number_theoretical
C-Name: hclassno
Prototype: G
Help: qfbhclassno(x): Hurwitz-Kronecker class number of x>0.
Doc: \idx{Hurwitz class number} of $x$, where
 $x$ is non-negative and congruent to 0 or 3 modulo 4. For $x > 5\cdot
 10^5$, we assume the GRH, and use \kbd{quadclassunit} with default
 parameters.

Function: qfbil
Class: basic
Section: linear_algebra
C-Name: qfbil
Prototype: GGDG
Help: qfbil(x,y,{q}): this function is obsolete, use qfeval.
Doc: this function is obsolete, use \kbd{qfeval}.
Obsolete: 2016-08-08

Function: qfbnucomp
Class: basic
Section: number_theoretical
C-Name: nucomp
Prototype: GGG
Help: qfbnucomp(x,y,L): composite of primitive positive definite quadratic
 forms x and y using nucomp and nudupl, where L=[|D/4|^(1/4)] is precomputed.
Doc: \idx{composition} of the primitive positive
 definite binary quadratic forms $x$ and $y$ (type \typ{QFI}) using the NUCOMP
 and NUDUPL algorithms of \idx{Shanks}, \`a la Atkin. $L$ is any positive
 constant, but for optimal speed, one should take $L=|D/4|^{1/4}$, i.e.
 \kbd{sqrtnint(abs(D)>>2,4)}, where $D$ is the common discriminant of $x$ and
 $y$. When $x$ and $y$ do not have the same discriminant, the result is
 undefined.
 
 The current implementation is slower than the generic routine for small $D$,
 and becomes faster when $D$ has about $45$ bits.
Variant: Also available is \fun{GEN}{nudupl}{GEN x, GEN L} when $x=y$.

Function: qfbnupow
Class: basic
Section: number_theoretical
C-Name: nupow
Prototype: GGDG
Help: qfbnupow(x,n,{L}): n-th power of primitive positive definite quadratic
 form x using nucomp and nudupl.
Doc: $n$-th power of the primitive positive definite
 binary quadratic form $x$ using \idx{Shanks}'s NUCOMP and NUDUPL algorithms;
 if set, $L$ should be equal to \kbd{sqrtnint(abs(D)>>2,4)}, where $D < 0$ is
 the discriminant of $x$.
 
 The current implementation is slower than the generic routine for small
 discriminant $D$, and becomes faster for $D \approx 2^{45}$.

Function: qfbpowraw
Class: basic
Section: number_theoretical
C-Name: qfbpowraw
Prototype: GL
Help: qfbpowraw(x,n): n-th power without reduction of the binary quadratic
 form x.
Doc: $n$-th power of the binary quadratic form
 $x$, computed without doing any \idx{reduction} (i.e.~using \kbd{qfbcompraw}).
 Here $n$ must be non-negative and $n<2^{31}$.

Function: qfbprimeform
Class: basic
Section: number_theoretical
C-Name: primeform
Prototype: GGp
Help: qfbprimeform(x,p): returns the prime form of discriminant x, whose
 first coefficient is p.
Doc: prime binary quadratic form of discriminant
 $x$ whose first coefficient is $p$, where $|p|$ is a prime number.
 By abuse of notation,
 $p = \pm 1$ is also valid and returns the unit form. Returns an
 error if $x$ is not a quadratic residue mod $p$, or if $x < 0$ and $p < 0$.
 (Negative definite \typ{QFI} are not implemented.) In the case where $x>0$,
 the ``distance'' component of the form is set equal to zero according to the
 current precision.

Function: qfbred
Class: basic
Section: number_theoretical
C-Name: qfbred0
Prototype: GD0,L,DGDGDG
Help: qfbred(x,{flag=0},{d},{isd},{sd}): reduction of the binary
 quadratic form x. All other args. are optional. The arguments d, isd and
 sd, if
 present, supply the values of the discriminant, floor(sqrt(d)) and sqrt(d)
 respectively. If d<0, its value is not used and all references to Shanks's
 distance hereafter are meaningless. flag can be any of 0: default, uses
 Shanks's distance function d; 1: use d, do a single reduction step; 2: do
 not use d; 3: do not use d, single reduction step.
Doc: reduces the binary quadratic form $x$ (updating Shanks's distance function
 if $x$ is indefinite). The binary digits of $\fl$ are toggles meaning
 
 \quad 1: perform a single \idx{reduction} step
 
 \quad 2: don't update \idx{Shanks}'s distance
 
 The arguments $d$, \var{isd}, \var{sd}, if present, supply the values of the
 discriminant, $\floor{\sqrt{d}}$, and $\sqrt{d}$ respectively
 (no checking is done of these facts). If $d<0$ these values are useless,
 and all references to Shanks's distance are irrelevant.
Variant: Also available are
 
 \fun{GEN}{redimag}{GEN x} (for definite $x$),
 
 \noindent and for indefinite forms:
 
 \fun{GEN}{redreal}{GEN x}
 
 \fun{GEN}{rhoreal}{GEN x} (= \kbd{qfbred(x,1)}),
 
 \fun{GEN}{redrealnod}{GEN x, GEN isd} (= \kbd{qfbred(x,2,,isd)}),
 
 \fun{GEN}{rhorealnod}{GEN x, GEN isd} (= \kbd{qfbred(x,3,,isd)}).

Function: qfbredsl2
Class: basic
Section: number_theoretical
C-Name: qfbredsl2
Prototype: GDG
Help: qfbredsl2(x,{data}): reduction of the binary quadratic form x, return
 [y,g] where y is reduced and g in Sl(2,Z) is such that g.x = y; data, if
 present, must be equal to [D, sqrtint(D)], where D > 0 is the discriminant
 of x.
Doc: 
 reduction of the (real or imaginary) binary quadratic form $x$, return
 $[y,g]$ where $y$ is reduced and $g$ in $\text{SL}(2,\Z)$ is such that
  $g \cdot x = y$; \var{data}, if
 present, must be equal to $[D, \kbd{sqrtint}(D)]$, where $D > 0$ is the
 discriminant of $x$. In case $x$ is \typ{QFR}, the distance component is
 unaffected.

Function: qfbsolve
Class: basic
Section: number_theoretical
C-Name: qfbsolve
Prototype: GG
Help: qfbsolve(Q,p): return [x,y] so that Q(x,y)=p where Q is a binary
 quadratic form and p a prime number, or 0 if there is no solution.
Doc: Solve the equation $Q(x,y)=p$ over the integers,
 where $Q$ is a binary quadratic form and $p$ a prime number.
 
 Return $[x,y]$ as a two-components vector, or zero if there is no solution.
 Note that this function returns only one solution and not all the solutions.
 
 Let $D = \disc Q$. The algorithm used runs in probabilistic polynomial time
 in $p$ (through the computation of a square root of $D$ modulo $p$); it is
 polynomial time in $D$ if $Q$ is imaginary, but exponential time if $Q$ is
 real (through the computation of a full cycle of reduced forms). In the
 latter case, note that \tet{bnfisprincipal} provides a solution in heuristic
 subexponential time in $D$ assuming the GRH.

Function: qfeval
Class: basic
Section: linear_algebra
C-Name: qfeval0
Prototype: DGGDG
Help: qfeval({q},x,{y}): evaluate the binary quadratic form q (symmetric matrix)
 at x; if y is present, evaluate the polar form at (x,y);
 if q omitted, use the standard Euclidean form.
Doc: evaluate the binary quadratic form $q$ (given by a symmetric matrix)
 at the vector $x$; if $y$ is present, evaluate the polar form at $(x,y)$;
 if $q$ omitted, use the standard Euclidean scalar product, corresponding to
 the identity matrix.
 
 Roughly equivalent to \kbd{x\til * q * y}, but a little faster and
 more convenient (does not distinguish between column and row vectors):
 \bprog
 ? x = [1,2,3]~; y = [-1,3,1]~; q = [1,2,3;2,2,-1;3,-1,9];
 ? qfeval(q,x,y)
 %2 = 23
 ? for(i=1,10^6, qfeval(q,x,y))
 time = 661ms
 ? for(i=1,10^6, x~*q*y)
 time = 697ms
 @eprog\noindent The speedup is noticeable for the quadratic form,
 compared to \kbd{x\til * q * x}, since we save almost half the
 operations:
 \bprog
 ? for(i=1,10^6, qfeval(q,x))
 time = 487ms
 @eprog\noindent The special case $q = \text{Id}$ is handled faster if we
 omit $q$ altogether:
 \bprog
 ? qfeval(,x,y)
 %1 = 2
 ? q = matid(#x);
 ? for(i=1,10^6, qfeval(q,x,y))
 time = 529 ms.
 ? for(i=1,10^6, qfeval(,x,y))
 time = 228 ms.
 ? for(i=1,10^6, x~*y)
 time = 274 ms.
 @eprog
 
 We also allow \typ{MAT}s of compatible dimensions for $x$,
 and return \kbd{x\til * q * x} in this case as well:
 \bprog
 ? M = [1,2,3;4,5,6;7,8,9]; qfeval(,M) \\ Gram matrix
 %5 =
 [66  78  90]
 
 [78  93 108]
 
 [90 108 126]
 
 ? q = [1,2,3;2,2,-1;3,-1,9];
 ? for(i=1,10^6, qfeval(q,M))
 time = 2,008 ms.
 ? for(i=1,10^6, M~*q*M)
 time = 2,368 ms.
 
 ? for(i=1,10^6, qfeval(,M))
 time = 1,053 ms.
 ? for(i=1,10^6, M~*M)
 time = 1,171 ms.
 @eprog
 
 If $q$ is a \typ{QFI} or \typ{QFR}, it is implicitly converted to the
 attached symmetric \typ{MAT}. This is done more
 efficiently than by direct conversion, since we avoid introducing a
 denominator $2$ and rational arithmetic:
 \bprog
 ? q = Qfb(2,3,4); x = [2,3];
 ? qfeval(q, x)
 %2 = 62
 ? Q = Mat(q)
 %3 =
  [  2 3/2]
 
  [3/2   4]
 ? qfeval(Q, x)
 %4 = 62
 ? for (i=1, 10^6, qfeval(q,x))
 time = 758 ms.
 ? for (i=1, 10^6, qfeval(Q,x))
 time = 1,110 ms.
 @eprog
 Finally, when $x$ is a \typ{MAT} with \emph{integral} coefficients, we allow
 a \typ{QFI} or \typ{QFR} for $q$ and return the binary
 quadratic form $q \circ M$. Again, the conversion to \typ{MAT} is less
 efficient in this case:
 \bprog
 ? q = Qfb(2,3,4); Q = Mat(q); x = [1,2;3,4];
 ? qfeval(q, x)
 %2 = Qfb(47, 134, 96)
 ? qfeval(Q,x)
 %3 =
 [47 67]
 
 [67 96]
 ? for (i=1, 10^6, qfeval(q,x))
 time = 701 ms.
 ? for (i=1, 10^6, qfeval(Q,x))
 time = 1,639 ms.
 @eprog

Function: qfgaussred
Class: basic
Section: linear_algebra
C-Name: qfgaussred
Prototype: G
Help: qfgaussred(q): square reduction of the (symmetric) matrix q (returns a
 square matrix whose i-th diagonal term is the coefficient of the i-th square
 in which the coefficient of the i-th variable is 1).
Doc: 
 \idx{decomposition into squares} of the
 quadratic form represented by the symmetric matrix $q$. The result is a
 matrix whose diagonal entries are the coefficients of the squares, and the
 off-diagonal entries on each line represent the bilinear forms. More
 precisely, if $(a_{ij})$ denotes the output, one has
 $$ q(x) = \sum_i a_{ii} (x_i + \sum_{j \neq i} a_{ij} x_j)^2 $$
 \bprog
 ? qfgaussred([0,1;1,0])
 %1 =
 [1/2 1]
 
 [-1 -1/2]
 @eprog\noindent This means that $2xy = (1/2)(x+y)^2 - (1/2)(x-y)^2$.
 Singular matrices are supported, in which case some diagonal coefficients
 will vanish:
 \bprog
 ? qfgaussred([1,1;1,1])
 %1 =
 [1 1]
 
 [1 0]
 @eprog\noindent This means that $x^2 + 2xy + y^2 = (x+y)^2$.
Variant: \fun{GEN}{qfgaussred_positive}{GEN q} assumes that $q$ is
  positive definite and is a little faster; returns \kbd{NULL} if a vector
  with negative norm occurs (non positive matrix or too many rounding errors).

Function: qfisom
Class: basic
Section: linear_algebra
C-Name: qfisom0
Prototype: GGDG
Help: qfisom(G,H,{fl}): find an isomorphism between the integral positive
 definite quadratic forms G and H if it exists. G can also be given by a
 qfisominit structure which is preferable if several forms need to be compared
 to G.
Doc: 
 $G$, $H$ being square and symmetric matrices with integer entries representing
 positive definite quadratic forms, return an invertible matrix $S$ such that
 $G={^t}S\*H\*S$. This defines a isomorphism between the corresponding lattices.
 Since this requires computing the minimal vectors, the computations can
 become very lengthy as the dimension grows.
 See \kbd{qfisominit} for the meaning of \var{fl}.
 
 $G$ can also be given by an \kbd{qfisominit} structure which is preferable if
 several forms $H$ need to be compared to $G$.
 
 This function implements an algorithm of Plesken and Souvignier, following
 Souvignier's implementation.
Variant: Also available is \fun{GEN}{qfisom}{GEN G, GEN H, GEN fl}
 where $G$ is a vector of \kbd{zm}, and $H$ is a \kbd{zm}.

Function: qfisominit
Class: basic
Section: linear_algebra
C-Name: qfisominit0
Prototype: GDGDG
Help: qfisominit(G,{fl},{m}): G being a square and symmetric matrix representing an
 integral positive definite quadratic form, this function returns a structure
 allowing to compute isomorphisms between G and other quadratic form faster.
Doc: 
 $G$ being a square and symmetric matrix with integer entries representing a
 positive definite quadratic form, return an \kbd{isom} structure allowing to
 compute isomorphisms between $G$ and other quadratic forms faster.
 
 The interface of this function is experimental and will likely change in future
 release.
 
 If present, the optional parameter \var{fl} must be a \typ{VEC} with two
 components. It allows to specify the invariants used, which can make the
 computation faster or slower. The components are
 
 \item \kbd{fl[1]} Depth of scalar product combination to use.
 
 \item \kbd{fl[2]} Maximum level of Bacher polynomials to use.
 
 If present, $m$ must be the set of vectors of norm up to the maximal of the
 diagonal entry of $G$, either as a matrix or as given by \kbd{qfminim}.
 Otherwise this function computes the minimal vectors so it become very
 lengthy as the dimension of $G$ grows.
Variant: Also available is
 \fun{GEN}{qfisominit}{GEN F, GEN fl}
 where $F$ is a vector of \kbd{zm}.

Function: qfjacobi
Class: basic
Section: linear_algebra
C-Name: jacobi
Prototype: Gp
Help: qfjacobi(A): eigenvalues and orthogonal matrix of eigenvectors of the
 real symmetric matrix A.
Doc: apply Jacobi's eigenvalue algorithm to the real symmetric matrix $A$.
 This returns $[L, V]$, where
 
 \item $L$ is the vector of (real) eigenvalues of $A$, sorted in increasing
 order,
 
 \item $V$ is the corresponding orthogonal matrix of eigenvectors of $A$.
 
 \bprog
 ? \p19
 ? A = [1,2;2,1]; mateigen(A)
 %1 =
 [-1 1]
 
 [ 1 1]
 ? [L, H] = qfjacobi(A);
 ? L
 %3 = [-1.000000000000000000, 3.000000000000000000]~
 ? H
 %4 =
 [ 0.7071067811865475245 0.7071067811865475244]
 
 [-0.7071067811865475244 0.7071067811865475245]
 ? norml2( (A-L[1])*H[,1] )       \\ approximate eigenvector
 %5 = 9.403954806578300064 E-38
 ? norml2(H*H~ - 1)
 %6 = 2.350988701644575016 E-38   \\ close to orthogonal
 @eprog

Function: qflll
Class: basic
Section: linear_algebra
C-Name: qflll0
Prototype: GD0,L,
Help: qflll(x,{flag=0}): LLL reduction of the vectors forming the matrix x
 (gives the unimodular transformation matrix T such that x*T is LLL-reduced). flag is
 optional, and can be 0: default, 1: assumes x is integral, 2: assumes x is
 integral, returns a partially reduced basis,
 4: assumes x is integral, returns [K,T] where K is the integer kernel of x
 and T the LLL reduced image, 5: same as 4 but x may have polynomial
 coefficients, 8: same as 0 but x may have polynomial coefficients.
Description: 
 (vec, ?0):vec       lll($1)
 (vec, 1):vec        lllint($1)
 (vec, 2):vec        lllintpartial($1)
 (vec, 4):vec        lllkerim($1)
 (vec, 5):vec        lllkerimgen($1)
 (vec, 8):vec        lllgen($1)
 (vec, #small):vec   $"Bad flag in qflll"
 (vec, small):vec    qflll0($1, $2)
Doc: \idx{LLL} algorithm applied to the
 \emph{columns} of the matrix $x$. The columns of $x$ may be linearly
 dependent. The result is a unimodular transformation matrix $T$ such that $x
 \cdot T$ is an LLL-reduced basis of the lattice generated by the column
 vectors of $x$. Note that if $x$ is not of maximal rank $T$ will not be
 square. The LLL parameters are $(0.51,0.99)$, meaning that the Gram-Schmidt
 coefficients for the final basis satisfy $\mu_{i,j} \leq |0.51|$, and the
 Lov\'{a}sz's constant is $0.99$.
 
 If $\fl=0$ (default), assume that $x$ has either exact (integral or
 rational) or real floating point entries. The matrix is rescaled, converted
 to integers and the behavior is then as in $\fl = 1$.
 
 If $\fl=1$, assume that $x$ is integral. Computations involving Gram-Schmidt
 vectors are approximate, with precision varying as needed (Lehmer's trick,
 as generalized by Schnorr). Adapted from Nguyen and Stehl\'e's algorithm
 and Stehl\'e's code (\kbd{fplll-1.3}).
 
 If $\fl=2$, $x$ should be an integer matrix whose columns are linearly
 independent. Returns a partially reduced basis for $x$, using an unpublished
 algorithm by Peter Montgomery: a basis is said to be \emph{partially reduced}
 if $|v_i \pm v_j| \geq |v_i|$ for any two distinct basis vectors $v_i, \,
 v_j$.
 
 This is faster than $\fl=1$, esp. when one row is huge compared
 to the other rows (knapsack-style), and should quickly produce relatively
 short vectors. The resulting basis is \emph{not} LLL-reduced in general.
 If LLL reduction is eventually desired, avoid this partial reduction:
 applying LLL to the partially reduced matrix is significantly \emph{slower}
 than starting from a knapsack-type lattice.
 
 If $\fl=4$, as $\fl=1$, returning a vector $[K, T]$ of matrices: the
 columns of $K$ represent a basis of the integer kernel of $x$
 (not LLL-reduced in general) and $T$ is the transformation
 matrix such that $x\cdot T$ is an LLL-reduced $\Z$-basis of the image
 of the matrix $x$.
 
 If $\fl=5$, case as case $4$, but $x$ may have polynomial coefficients.
 
 If $\fl=8$, same as case $0$, but $x$ may have polynomial coefficients.
Variant: Also available are \fun{GEN}{lll}{GEN x} ($\fl=0$),
 \fun{GEN}{lllint}{GEN x} ($\fl=1$), and \fun{GEN}{lllkerim}{GEN x} ($\fl=4$).

Function: qflllgram
Class: basic
Section: linear_algebra
C-Name: qflllgram0
Prototype: GD0,L,
Help: qflllgram(G,{flag=0}): LLL reduction of the lattice whose gram matrix
 is G (gives the unimodular transformation matrix). flag is optional and can
 be 0: default,1: assumes x is integral, 4: assumes x is integral,
 returns [K,T],  where K is the integer kernel of x
 and T the LLL reduced image, 5: same as 4 but x may have polynomial
 coefficients, 8: same as 0 but x may have polynomial coefficients.
Doc: same as \kbd{qflll}, except that the
 matrix $G = \kbd{x\til * x}$ is the Gram matrix of some lattice vectors $x$,
 and not the coordinates of the vectors themselves. In particular, $G$ must
 now be a square symmetric real matrix, corresponding to a positive
 quadratic form (not necessarily definite: $x$ needs not have maximal rank).
 The result is a unimodular
 transformation matrix $T$ such that $x \cdot T$ is an LLL-reduced basis of
 the lattice generated by the column vectors of $x$. See \tet{qflll} for
 further details about the LLL implementation.
 
 If $\fl=0$ (default), assume that $G$ has either exact (integral or
 rational) or real floating point entries. The matrix is rescaled, converted
 to integers and the behavior is then as in $\fl = 1$.
 
 If $\fl=1$, assume that $G$ is integral. Computations involving Gram-Schmidt
 vectors are approximate, with precision varying as needed (Lehmer's trick,
 as generalized by Schnorr). Adapted from Nguyen and Stehl\'e's algorithm
 and Stehl\'e's code (\kbd{fplll-1.3}).
 
 $\fl=4$: $G$ has integer entries, gives the kernel and reduced image of $x$.
 
 $\fl=5$: same as $4$, but $G$ may have polynomial coefficients.
Variant: Also available are \fun{GEN}{lllgram}{GEN G} ($\fl=0$),
 \fun{GEN}{lllgramint}{GEN G} ($\fl=1$), and \fun{GEN}{lllgramkerim}{GEN G}
 ($\fl=4$).

Function: qfminim
Class: basic
Section: linear_algebra
C-Name: qfminim0
Prototype: GDGDGD0,L,p
Help: qfminim(x,{b},{m},{flag=0}): x being a square and symmetric
 matrix representing a positive definite quadratic form, this function
 deals with the vectors of x whose norm is less than or equal to b,
 enumerated using the Fincke-Pohst algorithm, storing at most m vectors (no
 limit if m is omitted). The function searches for
 the minimal non-zero vectors if b is omitted. The precise behavior
 depends on flag. 0: returns at most 2m vectors (unless m omitted), returns
 [N,M,mat] where N is the number of vectors enumerated, M the maximum norm among
 these, and mat lists half the vectors (the other half is given by -mat). 1:
 ignores m and returns the first vector whose norm is less than b. 2: as 0
 but uses a more robust, slower implementation, valid for non integral
 quadratic forms.
Doc: $x$ being a square and symmetric matrix representing a positive definite
 quadratic form, this function deals with the vectors of $x$ whose norm is
 less than or equal to $b$, enumerated using the Fincke-Pohst algorithm,
 storing at most $m$ vectors (no limit if $m$ is omitted). The function
 searches for the minimal non-zero vectors if $b$ is omitted. The behavior is
 undefined if $x$ is not positive definite (a ``precision too low'' error is
 most likely, although more precise error messages are possible). The precise
 behavior depends on $\fl$.
 
 If $\fl=0$ (default), returns at most $2m$ vectors. The result is a
 three-component vector, the first component being the number of vectors
 enumerated (which may be larger than $2m$), the second being the maximum
 norm found, and the last vector
 is a matrix whose columns are found vectors, only one being given for each
 pair $\pm v$ (at most $m$ such pairs, unless $m$ was omitted). The vectors
 are returned in no particular order.
 
 If $\fl=1$, ignores $m$ and returns $[N,v]$, where $v$ is a non-zero vector
 of length $N \leq b$, or $[]$ if no non-zero vector has length $\leq b$.
 If no explicit $b$ is provided, return a vector of smallish norm
 (smallest vector in an LLL-reduced basis).
 
 In these two cases, $x$ must have \emph{integral} entries. The
 implementation uses low precision floating point computations for maximal
 speed, which gives incorrect result when $x$ has large entries. (The
 condition is checked in the code and the routine raises an error if
 large rounding errors occur.) A more robust, but much slower,
 implementation is chosen if the following flag is used:
 
 If $\fl=2$, $x$ can have non integral real entries. In this case, if $b$
 is omitted, the ``minimal'' vectors only have approximately the same norm.
 If $b$ is omitted, $m$ is an upper bound for the number of vectors that
 will be stored and returned, but all minimal vectors are nevertheless
 enumerated. If $m$ is omitted, all vectors found are stored and returned;
 note that this may be a huge vector!
 
 \bprog
 ? x = matid(2);
 ? qfminim(x)  \\@com 4 minimal vectors of norm 1: $\pm[0,1]$, $\pm[1,0]$
 %2 = [4, 1, [0, 1; 1, 0]]
 ? { x =
 [4, 2, 0, 0, 0,-2, 0, 0, 0, 0, 0, 0, 1,-1, 0, 0, 0, 1, 0,-1, 0, 0, 0,-2;
  2, 4,-2,-2, 0,-2, 0, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1, 0, 1,-1,-1;
  0,-2, 4, 0,-2, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 0, 0, 1,-1,-1, 0, 0;
  0,-2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1,-1, 0, 0, 0, 1,-1, 0, 1,-1, 1, 0;
  0, 0,-2, 0, 4, 0, 0, 0, 1,-1, 0, 0, 1, 0, 0, 0,-2, 0, 0,-1, 1, 1, 0, 0;
 -2, -2,0, 0, 0, 4,-2, 0,-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,-1, 1, 1;
  0, 0, 0, 0, 0,-2, 4,-2, 0, 0, 0, 0, 0, 1, 0, 0, 0,-1, 0, 0, 0, 1,-1, 0;
  0, 0, 0, 0, 0, 0,-2, 4, 0, 0, 0, 0,-1, 0, 0, 0, 0, 0,-1,-1,-1, 0, 1, 0;
  0, 0, 0, 0, 1,-1, 0, 0, 4, 0,-2, 0, 1, 1, 0,-1, 0, 1, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0,-1, 0, 0, 0, 0, 4, 0, 0, 1, 1,-1, 1, 0, 0, 0, 1, 0, 0, 1, 0;
  0, 0, 0, 0, 0, 0, 0, 0,-2, 0, 4,-2, 0,-1, 0, 0, 0,-1, 0,-1, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0,-2, 4,-1, 1, 0, 0,-1, 1, 0, 1, 1, 1,-1, 0;
  1, 0,-1, 1, 1, 0, 0,-1, 1, 1, 0,-1, 4, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1,-1;
 -1,-1, 1,-1, 0, 0, 1, 0, 1, 1,-1, 1, 0, 4, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 1, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0;
  0, 0, 1, 0,-2, 0, 0, 0, 0, 0, 0,-1, 0, 0, 0, 0, 4, 1, 1, 1, 0, 0, 1, 1;
  1, 0, 0, 1, 0, 0,-1, 0, 1, 0,-1, 1, 1, 0, 0, 0, 1, 4, 0, 1, 1, 0, 1, 0;
  0, 0, 0,-1, 0, 1, 0,-1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 4, 0, 1, 1, 0, 1;
 -1, -1,1, 0,-1, 1, 0,-1, 0, 1,-1, 1, 0, 1, 0, 0, 1, 1, 0, 4, 0, 0, 1, 1;
  0, 0,-1, 1, 1, 0, 0,-1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 4, 1, 0, 1;
  0, 1,-1,-1, 1,-1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 4, 0, 1;
  0,-1, 0, 1, 0, 1,-1, 1, 0, 1, 0,-1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 4, 1;
 -2,-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,-1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 4]; }
 ? qfminim(x,,0)  \\ the Leech lattice has 196560 minimal vectors of norm 4
 time = 648 ms.
 %4 = [196560, 4, [;]]
 ? qfminim(x,,0,2); \\ safe algorithm. Slower and unnecessary here.
 time = 18,161 ms.
 %5 = [196560, 4.000061035156250000, [;]]
 @eprog\noindent\sidx{Leech lattice}\sidx{minimal vector}
 In the last example, we store 0 vectors to limit memory use. All minimal
 vectors are nevertheless enumerated. Provided \kbd{parisize} is about 50MB,
 \kbd{qfminim(x)} succeeds in 2.5 seconds.
Variant: Also available are
 \fun{GEN}{minim}{GEN x, GEN b = NULL, GEN m = NULL} ($\fl=0$),
 \fun{GEN}{minim2}{GEN x, GEN b = NULL, GEN m = NULL} ($\fl=1$).
 \fun{GEN}{minim_raw}{GEN x, GEN b = NULL, GEN m = NULL} (do not perform LLL
 reduction on x and return \kbd{NULL} on accuracy error).

Function: qfnorm
Class: basic
Section: linear_algebra
C-Name: qfnorm
Prototype: GDG
Help: qfnorm(x,{q}): this function is obsolete, use qfeval.
Doc: this function is obsolete, use \kbd{qfeval}.
Obsolete: 2016-08-08

Function: qforbits
Class: basic
Section: linear_algebra
C-Name: qforbits
Prototype: GG
Help: qforbits(G,V): return the orbits of V under the action of the group
 of linear transformation generated by the set G, which must stabilize V.
Doc: return the orbits of $V$ under the action of the group
 of linear transformation generated by the set $G$.
 It is assumed that $G$ contains minus identity, and only one vector
 in $\{v, -v\}$ should be given.
 If $G$ does not stabilize $V$, the function return $0$.
 
 In the example below, we compute representatives and lengths of the orbits of
 the vectors of norm $\leq 3$ under the automorphisms of the lattice $A_1^6$.
 \bprog
 ?  Q=matid(6); G=qfauto(Q); V=qfminim(Q,3);
 ?  apply(x->[x[1],#x],qforbits(G,V))
 %2 = [[[0,0,0,0,0,1]~,6],[[0,0,0,0,1,-1]~,30],[[0,0,0,1,-1,-1]~,80]]
 @eprog

Function: qfparam
Class: basic
Section: linear_algebra
C-Name: qfparam
Prototype: GGD0,L,
Help: qfparam(G, sol, {flag = 0}):
 coefficients of binary quadratic forms that parametrize the
 solutions of the ternary quadratic form G, using the particular
 solution sol.
Doc: coefficients of binary quadratic forms that parametrize the
 solutions of the ternary quadratic form $G$, using the particular
 solution~\var{sol}.
 \fl is optional and can be 1, 2, or 3, in which case the \fl-th form is
 reduced. The default is \fl=0 (no reduction).
 \bprog
 ? G = [1,0,0;0,1,0;0,0,-34];
 ? M = qfparam(G, qfsolve(G))
 %2 =
 [ 3 -10 -3]
 
 [-5  -6  5]
 
 [ 1   0  1]
 @eprog
 Indeed, the solutions can be parametrized as
 $$(3x^2 - 10xy - 3y^2)^2  + (-5x^2 - 6xy + 5y^2)^2 -34(x^2 + y^2)^2 = 0.$$
 \bprog
 ? v = y^2 * M*[1,x/y,(x/y)^2]~
 %3 = [3*x^2 - 10*y*x - 3*y^2, -5*x^2 - 6*y*x + 5*y^2, -x^2 - y^2]~
 ? v~*G*v
 %4 = 0
 @eprog

Function: qfperfection
Class: basic
Section: linear_algebra
C-Name: perf
Prototype: G
Help: qfperfection(G): rank of matrix of xx~ for x minimal vectors of a gram
 matrix G.
Doc: 
 $G$ being a square and symmetric matrix with
 integer entries representing a positive definite quadratic form, outputs the
 perfection rank of the form. That is, gives the rank of the family of the $s$
 symmetric matrices $v_iv_i^t$, where $s$ is half the number of minimal
 vectors and the $v_i$ ($1\le i\le s$) are the minimal vectors.
 
 Since this requires computing the minimal vectors, the computations can
 become very lengthy as the dimension of $x$ grows.

Function: qfrep
Class: basic
Section: linear_algebra
C-Name: qfrep0
Prototype: GGD0,L,
Help: qfrep(q,B,{flag=0}): vector of (half) the number of vectors of norms
 from 1 to B for the integral and definite quadratic form q. If flag is 1,
 count vectors of even norm from 1 to 2B.
Doc: 
 $q$ being a square and symmetric matrix with integer entries representing a
 positive definite quadratic form, count the vectors representing successive
 integers.
 
 \item If $\fl = 0$, count all vectors. Outputs the vector whose $i$-th
 entry, $1 \leq i \leq B$ is half the number of vectors $v$ such that $q(v)=i$.
 
 \item If $\fl = 1$, count vectors of even norm. Outputs the vector
 whose $i$-th entry, $1 \leq i \leq B$ is half the number of vectors such
 that $q(v) = 2i$.
 
 \bprog
 ? q = [2, 1; 1, 3];
 ? qfrep(q, 5)
 %2 = Vecsmall([0, 1, 2, 0, 0]) \\ 1 vector of norm 2, 2 of norm 3, etc.
 ? qfrep(q, 5, 1)
 %3 = Vecsmall([1, 0, 0, 1, 0]) \\ 1 vector of norm 2, 0 of norm 4, etc.
 @eprog\noindent
 This routine uses a naive algorithm based on \tet{qfminim}, and
 will fail if any entry becomes larger than $2^{31}$ (or $2^{63}$).

Function: qfsign
Class: basic
Section: linear_algebra
C-Name: qfsign
Prototype: G
Help: qfsign(x): signature of the symmetric matrix x.
Doc: 
 returns $[p,m]$ the signature of the quadratic form represented by the
 symmetric matrix $x$. Namely, $p$ (resp.~$m$) is the number of positive
 (resp.~negative) eigenvalues of $x$. The result is computed using Gaussian
 reduction.

Function: qfsolve
Class: basic
Section: linear_algebra
C-Name: qfsolve
Prototype: G
Help: qfsolve(G): solve over Q the quadratic equation X^t G X = 0, where
 G is a symmetric matrix.
Doc: Given a square symmetric matrix $G$ of dimension $n \geq 1$, solve over
 $\Q$ the quadratic equation $X^tGX = 0$. The matrix $G$ must have rational
 coefficients. The solution might be a single non-zero vector (vectorv) or a
 matrix (whose columns generate a totally isotropic subspace).
 
 If no solution exists, returns an integer, that can be a prime $p$ such that
 there is no local solution at $p$, or $-1$ if there is no real solution,
 or $-2$ if $n = 2$ and $-\det G$ is positive but not a square (which implies
 there is a real solution, but no local solution at some $p$ dividing $\det G$).
 \bprog
 ? G = [1,0,0;0,1,0;0,0,-34];
 ? qfsolve(G)
 %1 = [-3, -5, 1]~
 ? qfsolve([1,0; 0,2])
 %2 = -1   \\ no real solution
 ? qfsolve([1,0,0;0,3,0; 0,0,-2])
 %3 = 3    \\ no solution in Q_3
 ? qfsolve([1,0; 0,-2])
 %4 = -2   \\ no solution, n = 2
 @eprog

Function: quadclassunit
Class: basic
Section: number_theoretical
C-Name: quadclassunit0
Prototype: GD0,L,DGp
Help: quadclassunit(D,{flag=0},{tech=[]}): compute the structure of the
 class group and the regulator of the quadratic field of discriminant D.
 See manual for the optional technical parameters.
Doc: \idx{Buchmann-McCurley}'s sub-exponential algorithm for computing the
 class group of a quadratic order of discriminant $D$.
 
 This function should be used instead of \tet{qfbclassno} or \tet{quadregula}
 when $D<-10^{25}$, $D>10^{10}$, or when the \emph{structure} is wanted. It
 is a special case of \tet{bnfinit}, which is slower, but more robust.
 
 The result is a vector $v$ whose components should be accessed using member
 functions:
 
 \item \kbd{$v$.no}: the class number
 
 \item \kbd{$v$.cyc}: a vector giving the structure of the class group as a
 product of cyclic groups;
 
 \item \kbd{$v$.gen}: a vector giving generators of those cyclic groups (as
 binary quadratic forms).
 
 \item \kbd{$v$.reg}: the regulator, computed to an accuracy which is the
 maximum of an internal accuracy determined by the program and the current
 default (note that once the regulator is known to a small accuracy it is
 trivial to compute it to very high accuracy, see the tutorial).
 
 The $\fl$ is obsolete and should be left alone. In older versions,
 it supposedly computed the narrow class group when $D>0$, but this did not
 work at all; use the general function \tet{bnfnarrow}.
 
 Optional parameter \var{tech} is a row vector of the form $[c_1, c_2]$,
 where $c_1 \leq c_2$ are non-negative real numbers which control the execution
 time and the stack size, see \ref{se:GRHbnf}. The parameter is used as a
 threshold to balance the relation finding phase against the final linear
 algebra. Increasing the default $c_1$ means that relations are easier
 to find, but more relations are needed and the linear algebra will be
 harder. The default value for $c_1$ is $0$ and means that it is taken equal
 to $c_2$. The parameter $c_2$ is mostly obsolete and should not be changed,
 but we still document it for completeness: we compute a tentative class
 group by generators and relations using a factorbase of prime ideals
 $\leq c_1 (\log |D|)^2$, then prove that ideals of norm
 $\leq c_2 (\log |D|)^2$ do
 not generate a larger group. By default an optimal $c_2$ is chosen, so that
 the result is provably correct under the GRH --- a famous result of Bach
 states that $c_2 = 6$ is fine, but it is possible to improve on this
 algorithmically. You may provide a smaller $c_2$, it will be ignored
 (we use the provably correct
 one); you may provide a larger $c_2$ than the default value, which results
 in longer computing times for equally correct outputs (under GRH).
Variant: If you really need to experiment with the \var{tech} parameter, it is
 usually more convenient to use
 \fun{GEN}{Buchquad}{GEN D, double c1, double c2, long prec}

Function: quaddisc
Class: basic
Section: number_theoretical
C-Name: quaddisc
Prototype: G
Help: quaddisc(x): discriminant of the quadratic field Q(sqrt(x)).
Doc: discriminant of the \'etale algebra $\Q(\sqrt{x})$, where $x\in\Q^*$.
 This is the same as \kbd{coredisc}$(d)$ where $d$ is the integer square-free
 part of $x$, so x=$d f^2$ with $f\in \Q^*$ and $d\in\Z$.
 This returns $0$ for $x = 0$, $1$ for $x$ square and the discriminant of the
 quadratic field $\Q(\sqrt{x})$ otherwise.
 \bprog
 ? quaddisc(7)
 %1 = 28
 ? quaddisc(-7)
 %2 = -7
 @eprog

Function: quadgen
Class: basic
Section: number_theoretical
C-Name: quadgen
Prototype: G
Help: quadgen(D): standard generator of quadratic order of discriminant D.
Doc: creates the quadratic
 number\sidx{omega} $\omega=(a+\sqrt{D})/2$ where $a=0$ if $D\equiv0\mod4$,
 $a=1$ if $D\equiv1\mod4$, so that $(1,\omega)$ is an integral basis for the
 quadratic order of discriminant $D$. $D$ must be an integer congruent to 0 or
 1 modulo 4, which is not a square.

Function: quadhilbert
Class: basic
Section: number_theoretical
C-Name: quadhilbert
Prototype: Gp
Help: quadhilbert(D): relative equation for the Hilbert class field
 of the quadratic field of discriminant D (which can also be a bnf).
Doc: relative equation defining the
 \idx{Hilbert class field} of the quadratic field of discriminant $D$.
 
 If $D < 0$, uses complex multiplication (\idx{Schertz}'s variant).
 
 If $D > 0$ \idx{Stark units} are used and (in rare cases) a
 vector of extensions may be returned whose compositum is the requested class
 field. See \kbd{bnrstark} for details.

Function: quadpoly
Class: basic
Section: number_theoretical
C-Name: quadpoly0
Prototype: GDn
Help: quadpoly(D,{v='x}): quadratic polynomial corresponding to the
 discriminant D, in variable v.
Doc: creates the ``canonical'' quadratic
 polynomial (in the variable $v$) corresponding to the discriminant $D$,
 i.e.~the minimal polynomial of $\kbd{quadgen}(D)$. $D$ must be an integer
 congruent to 0 or 1 modulo 4, which is not a square.

Function: quadray
Class: basic
Section: number_theoretical
C-Name: quadray
Prototype: GGp
Help: quadray(D,f): relative equation for the ray class field of
 conductor f for the quadratic field of discriminant D (which can also be a
 bnf).
Doc: relative equation for the ray
 class field of conductor $f$ for the quadratic field of discriminant $D$
 using analytic methods. A \kbd{bnf} for $x^2 - D$ is also accepted in place
 of $D$.
 
 For $D < 0$, uses the $\sigma$ function and Schertz's method.
 
 For $D>0$, uses Stark's conjecture, and a vector of relative equations may be
 returned. See \tet{bnrstark} for more details.

Function: quadregulator
Class: basic
Section: number_theoretical
C-Name: quadregulator
Prototype: Gp
Help: quadregulator(x): regulator of the real quadratic field of
 discriminant x.
Doc: regulator of the quadratic field of positive discriminant $x$. Returns
 an error if $x$ is not a discriminant (fundamental or not) or if $x$ is a
 square. See also \kbd{quadclassunit} if $x$ is large.

Function: quadunit
Class: basic
Section: number_theoretical
C-Name: quadunit
Prototype: G
Help: quadunit(D): fundamental unit of the quadratic field of discriminant D
 where D must be positive.
Doc: fundamental unit\sidx{fundamental units} of the
 real quadratic field $\Q(\sqrt D)$ where  $D$ is the positive discriminant
 of the field. If $D$ is not a fundamental discriminant, this probably gives
 the fundamental unit of the corresponding order. $D$ must be an integer
 congruent to 0 or 1 modulo 4, which is not a square; the result is a
 quadratic number (see \secref{se:quadgen}).

Function: quit
Class: gp
Section: programming/specific
C-Name: gp_quit
Prototype: vD0,L,
Help: quit({status = 0}): quit, return to the system with exit status
 'status'.
Doc: exits \kbd{gp} and return to the system with exit status
 \kbd{status}, a small integer. A non-zero exit status normally indicates
 abnormal termination. (Note: the system actually sees only
 \kbd{status} mod $256$, see your man pages for \kbd{exit(3)} or \kbd{wait(2)}).

Function: ramanujantau
Class: basic
Section: number_theoretical
C-Name: ramanujantau
Prototype: G
Help: ramanujantau(n): compute the value of Ramanujan's tau function at n,
 assuming the GRH. Algorithm in O(n^{1/2+eps}).
Doc: compute the value of Ramanujan's tau function at an individual $n$,
 assuming the truth of the GRH (to compute quickly class numbers of imaginary
 quadratic fields using \tet{quadclassunit}).
 Algorithm in $\tilde{O}(n^{1/2})$ using $O(\log n)$ space. If all values up
 to $N$ are required, then
 $$\sum \tau(n)q^n = q \prod_{n\geq 1} (1-q^n)^{24}$$
 will produce them in time $\tilde{O}(N)$, against $\tilde{O}(N^{3/2})$ for
 individual calls to \kbd{ramanujantau}; of course the space complexity then
 becomes $\tilde{O}(N)$.
 \bprog
 ? tauvec(N) = Vec(q*eta(q + O(q^N))^24);
 ? N = 10^4; v = tauvec(N);
 time = 26 ms.
 ? ramanujantau(N)
 %3 = -482606811957501440000
 ? w = vector(N, n, ramanujantau(n)); \\ much slower !
 time = 13,190 ms.
 ? v == w
 %4 = 1
 @eprog

Function: random
Class: basic
Section: conversions
C-Name: genrand
Prototype: DG
Help: random({N=2^31}): random object, depending on the type of N.
 Integer between 0 and N-1 (t_INT), int mod N (t_INTMOD), element in a finite
 field (t_FFELT), point on an elliptic curve (ellinit mod p or over a finite
 field).
Description: 
 (?int):int    genrand($1)
 (gen):gen     genrand($1)
Doc: 
 returns a random element in various natural sets depending on the
 argument $N$.
 
 \item \typ{INT}: returns an integer
 uniformly distributed between $0$ and $N-1$. Omitting the argument
 is equivalent to \kbd{random(2\pow31)}.
 
 \item \typ{REAL}: returns a real number in $[0,1[$ with the same accuracy as
 $N$ (whose mantissa has the same number of significant words).
 
 \item \typ{INTMOD}: returns a random intmod for the same modulus.
 
 \item \typ{FFELT}: returns a random element in the same finite field.
 
 \item \typ{VEC} of length $2$, $N = [a,b]$: returns an integer uniformly
 distributed between $a$ and $b$.
 
 \item \typ{VEC} generated by \kbd{ellinit} over a finite field $k$
 (coefficients are \typ{INTMOD}s modulo a prime or \typ{FFELT}s): returns a
 ``random'' $k$-rational \emph{affine} point on the curve. More precisely
 if the curve has a single point (at infinity!) we return it; otherwise
 we return an affine point by drawing an abscissa uniformly at
 random until \tet{ellordinate} succeeds. Note that this is definitely not a
 uniform distribution over $E(k)$, but it should be good enough for
 applications.
 
 \item \typ{POL} return a random polynomial of degree at most the degree of $N$.
 The coefficients are drawn by applying \kbd{random} to the leading
 coefficient of $N$.
 
 \bprog
 ? random(10)
 %1 = 9
 ? random(Mod(0,7))
 %2 = Mod(1, 7)
 ? a = ffgen(ffinit(3,7), 'a); random(a)
 %3 = a^6 + 2*a^5 + a^4 + a^3 + a^2 + 2*a
 ? E = ellinit([3,7]*Mod(1,109)); random(E)
 %4 = [Mod(103, 109), Mod(10, 109)]
 ? E = ellinit([1,7]*a^0); random(E)
 %5 = [a^6 + a^5 + 2*a^4 + 2*a^2, 2*a^6 + 2*a^4 + 2*a^3 + a^2 + 2*a]
 ? random(Mod(1,7)*x^4)
 %6 = Mod(5, 7)*x^4 + Mod(6, 7)*x^3 + Mod(2, 7)*x^2 + Mod(2, 7)*x + Mod(5, 7)
 
 @eprog
 These variants all depend on a single internal generator, and are
 independent from your operating system's random number generators.
 A random seed may be obtained via \tet{getrand}, and reset
 using \tet{setrand}: from a given seed, and given sequence of \kbd{random}s,
 the exact same values will be generated. The same seed is used at each
 startup, reseed the generator yourself if this is a problem. Note that
 internal functions also call the random number generator; adding such a
 function call in the middle of your code will change the numbers produced.
 
 \misctitle{Technical note}
 Up to
 version 2.4 included, the internal generator produced pseudo-random numbers
 by means of linear congruences, which were not well distributed in arithmetic
 progressions. We now
 use Brent's XORGEN algorithm, based on Feedback Shift Registers, see
 \url{http://wwwmaths.anu.edu.au/~brent/random.html}. The generator has period
 $2^{4096}-1$, passes the Crush battery of statistical tests of L'Ecuyer and
 Simard, but is not suitable for cryptographic purposes: one can reconstruct
 the state vector from a small sample of consecutive values, thus predicting
 the entire sequence.
Variant: 
  Also available: \fun{GEN}{ellrandom}{GEN E} and \fun{GEN}{ffrandom}{GEN a}.

Function: randomprime
Class: basic
Section: number_theoretical
C-Name: randomprime
Prototype: DG
Help: randomprime({N = 2^31}): returns a strong pseudo prime in [2, N-1].
Doc: returns a strong pseudo prime (see \tet{ispseudoprime}) in $[2,N-1]$.
 A \typ{VEC} $N = [a,b]$ is also allowed, with $a \leq b$ in which case a
 pseudo prime $a \leq p \leq b$ is returned; if no prime exists in the
 interval, the function will run into an infinite loop. If the upper bound
 is less than $2^{64}$ the pseudo prime returned is a proven prime.

Function: read
Class: basic
Section: programming/specific
C-Name: gp_read_file
Prototype: D"",s,
Help: read({filename}): read from the input file filename. If filename is
 omitted, reread last input file, be it from read() or \r.
Description: 
 (str):gen      gp_read_file($1)
Doc: reads in the file
 \var{filename} (subject to string expansion). If \var{filename} is
 omitted, re-reads the last file that was fed into \kbd{gp}. The return
 value is the result of the last expression evaluated.
 
 If a GP \tet{binary file} is read using this command (see
 \secref{se:writebin}), the file is loaded and the last object in the file
 is returned.
 
 In case the file you read in contains an \tet{allocatemem} statement (to be
 generally avoided), you should leave \kbd{read} instructions by themselves,
 and not part of larger instruction sequences.

Function: readstr
Class: basic
Section: programming/specific
C-Name: readstr
Prototype: D"",s,
Help: readstr({filename}): returns the vector of GP strings containing
 the lines in filename.
Doc: Reads in the file \var{filename} and return a vector of GP strings,
 each component containing one line from the file. If \var{filename} is
 omitted, re-reads the last file that was fed into \kbd{gp}.

Function: readvec
Class: basic
Section: programming/specific
C-Name: gp_readvec_file
Prototype: D"",s,
Help: readvec({filename}): create a vector whose components are the evaluation
 of all the expressions found in the input file filename.
Description: 
 (str):gen      gp_readvec_file($1)
Doc: reads in the file
 \var{filename} (subject to string expansion). If \var{filename} is
 omitted, re-reads the last file that was fed into \kbd{gp}. The return
 value is a vector whose components are the evaluation of all sequences
 of instructions contained in the file. For instance, if \var{file} contains
 \bprog
 1
 2
 3
 @eprog\noindent
 then we will get:
 \bprog
 ? \r a
 %1 = 1
 %2 = 2
 %3 = 3
 ? read(a)
 %4 = 3
 ? readvec(a)
 %5 = [1, 2, 3]
 @eprog
 In general a sequence is just a single line, but as usual braces and
 \kbd{\bs} may be used to enter multiline sequences.
Variant: The underlying library function
 \fun{GEN}{gp_readvec_stream}{FILE *f} is usually more flexible.

Function: real
Class: basic
Section: conversions
C-Name: greal
Prototype: G
Help: real(x): real part of x.
Doc: real part of $x$. In the case where $x$ is a quadratic number, this is the
 coefficient of $1$ in the ``canonical'' integral basis $(1,\omega)$.

Function: removeprimes
Class: basic
Section: number_theoretical
C-Name: removeprimes
Prototype: DG
Help: removeprimes({x=[]}): remove primes in the vector x from the prime table.
 x can also be a single integer. List the current extra primes if x is omitted.
Doc: removes the primes listed in $x$ from
 the prime number table. In particular \kbd{removeprimes(addprimes())} empties
 the extra prime table. $x$ can also be a single integer. List the current
 extra primes if $x$ is omitted.

Function: return
Class: basic
Section: programming/control
C-Name: return0
Prototype: DG
Help: return({x=0}): return from current subroutine with result x.
Doc: returns from current subroutine, with
 result $x$. If $x$ is omitted, return the \kbd{(void)} value (return no
 result, like \kbd{print}).

Function: rnfalgtobasis
Class: basic
Section: number_fields
C-Name: rnfalgtobasis
Prototype: GG
Help: rnfalgtobasis(rnf,x): relative version of nfalgtobasis, where rnf is a
 relative numberfield.
Doc: expresses $x$ on the relative
 integral basis. Here, $\var{rnf}$ is a relative number field extension $L/K$
 as output by \kbd{rnfinit}, and $x$ an element of $L$ in absolute form, i.e.
 expressed as a polynomial or polmod with polmod coefficients, \emph{not} on
 the relative integral basis.

Function: rnfbasis
Class: basic
Section: number_fields
C-Name: rnfbasis
Prototype: GG
Help: rnfbasis(bnf,M): given a projective Z_K-module M as output by
 rnfpseudobasis or rnfsteinitz, gives either a basis of M if it is free, or an
 n+1-element generating set.
Doc: let $K$ the field represented by
 \var{bnf}, as output by \kbd{bnfinit}. $M$ is a projective $\Z_K$-module
 of rank $n$ ($M\otimes K$ is an $n$-dimensional $K$-vector space), given by a
 pseudo-basis of size $n$. The routine returns either a true $\Z_K$-basis of
 $M$ (of size $n$) if it exists, or an $n+1$-element generating set of $M$ if
 not.
 
 It is allowed to use an irreducible polynomial $P$ in $K[X]$ instead of $M$,
 in which case, $M$ is defined as the ring of integers of $K[X]/(P)$, viewed
 as a $\Z_K$-module.

Function: rnfbasistoalg
Class: basic
Section: number_fields
C-Name: rnfbasistoalg
Prototype: GG
Help: rnfbasistoalg(rnf,x): relative version of nfbasistoalg, where rnf is a
 relative numberfield.
Doc: computes the representation of $x$
 as a polmod with polmods coefficients. Here, $\var{rnf}$ is a relative number
 field extension $L/K$ as output by \kbd{rnfinit}, and $x$ an element of
 $L$ expressed on the relative integral basis.

Function: rnfcharpoly
Class: basic
Section: number_fields
C-Name: rnfcharpoly
Prototype: GGGDn
Help: rnfcharpoly(nf,T,a,{var='x}): characteristic polynomial of a
 over nf, where a belongs to the algebra defined by T over nf. Returns a
 polynomial in variable var (x by default).
Doc: characteristic polynomial of
 $a$ over $\var{nf}$, where $a$ belongs to the algebra defined by $T$ over
 $\var{nf}$, i.e.~$\var{nf}[X]/(T)$. Returns a polynomial in variable $v$
 ($x$ by default).
 \bprog
 ? nf = nfinit(y^2+1);
 ? rnfcharpoly(nf, x^2+y*x+1, x+y)
 %2 = x^2 + Mod(-y, y^2 + 1)*x + 1
 @eprog

Function: rnfconductor
Class: basic
Section: number_fields
C-Name: rnfconductor
Prototype: GG
Help: rnfconductor(bnf,pol): conductor of the Abelian extension
 of bnf defined by pol. The result is [conductor,bnr,subgroup],
 where conductor is the conductor itself, bnr the attached bnr
 structure, and subgroup the HNF defining the norm
 group (Artin or Takagi group) on the given generators bnr.gen.
Doc: given $\var{bnf}$
 as output by \kbd{bnfinit}, and \var{pol} a relative polynomial defining an
 \idx{Abelian extension}, computes the class field theory conductor of this
 Abelian extension. The result is a 3-component vector
 $[\var{conductor},\var{bnr},\var{subgroup}]$, where \var{conductor} is
 the conductor of the extension given as a 2-component row vector
 $[f_0,f_\infty]$, \var{bnr} is the attached \kbd{bnr} structure
 and \var{subgroup} is a matrix in HNF defining the subgroup of the ray class
 group on \kbd{bnr.gen}.

Function: rnfdedekind
Class: basic
Section: number_fields
C-Name: rnfdedekind
Prototype: GGDGD0,L,
Help: rnfdedekind(nf,pol,{pr},{flag=0}): relative Dedekind criterion over the
 number field K, represented by nf, applied to the order O_K[X]/(P),
 modulo the prime ideal pr (at all primes if pr omitted, in which case
 flag is automatically set to 1).
 P is assumed to be monic, irreducible, in O_K[X].
 Returns [max,basis,v], where basis is a pseudo-basis of the
 enlarged order, max is 1 iff this order is pr-maximal, and v is the
 valuation at pr of the order discriminant. If flag is set, just return 1 if
 the order is maximal, and 0 if not.
Doc: given a number field $K$ coded by $\var{nf}$ and a monic
 polynomial $P\in \Z_K[X]$, irreducible over $K$ and thus defining a relative
 extension $L$ of $K$, applies \idx{Dedekind}'s criterion to the order
 $\Z_K[X]/(P)$, at the prime ideal \var{pr}. It is possible to set \var{pr}
 to a vector of prime ideals (test maximality at all primes in the vector),
 or to omit altogether, in which case maximality at \emph{all} primes is tested;
 in this situation \fl\ is automatically set to $1$.
 
 The default historic behavior (\fl\ is 0 or omitted and \var{pr} is a
 single prime ideal) is not so useful since
 \kbd{rnfpseudobasis} gives more information and is generally not that
 much slower. It returns a 3-component vector $[\var{max}, \var{basis}, v]$:
 
 \item \var{basis} is a pseudo-basis of an enlarged order $O$ produced by
 Dedekind's criterion, containing the original order $\Z_K[X]/(P)$
 with index a power of \var{pr}. Possibly equal to the original order.
 
 \item \var{max} is a flag equal to 1 if the enlarged order $O$
 could be proven to be \var{pr}-maximal and to 0 otherwise; it may still be
 maximal in the latter case if \var{pr} is ramified in $L$,
 
 \item $v$ is the valuation at \var{pr} of the order discriminant.
 
 If \fl\ is non-zero, on the other hand, we just return $1$ if the order
 $\Z_K[X]/(P)$ is \var{pr}-maximal (resp.~maximal at all relevant primes, as
 described above), and $0$ if not. This is much faster than the default,
 since the enlarged order is not computed.
 \bprog
 ? nf = nfinit(y^2-3); P = x^3 - 2*y;
 ? pr3 = idealprimedec(nf,3)[1];
 ? rnfdedekind(nf, P, pr3)
 %3 = [1, [[1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 1, 1]], 8]
 ? rnfdedekind(nf, P, pr3, 1)
 %4 = 1
 @eprog\noindent In this example, \kbd{pr3} is the ramified ideal above $3$,
 and the order generated by the cube roots of $y$ is already
 \kbd{pr3}-maximal. The order-discriminant has valuation $8$. On the other
 hand, the order is not maximal at the prime above 2:
 \bprog
 ? pr2 = idealprimedec(nf,2)[1];
 ? rnfdedekind(nf, P, pr2, 1)
 %6 = 0
 ? rnfdedekind(nf, P, pr2)
 %7 = [0, [[2, 0, 0; 0, 1, 0; 0, 0, 1], [[1, 0; 0, 1], [1, 0; 0, 1],
      [1, 1/2; 0, 1/2]]], 2]
 @eprog
 The enlarged order is not proven to be \kbd{pr2}-maximal yet. In fact, it
 is; it is in fact the maximal order:
 \bprog
 ? B = rnfpseudobasis(nf, P)
 %8 = [[1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 1, [1, 1/2; 0, 1/2]],
      [162, 0; 0, 162], -1]
 ? idealval(nf,B[3], pr2)
 %9 = 2
 @eprog\noindent
 It is possible to use this routine with non-monic
 $P = \sum_{i\leq n} a_i X^i \in \Z_K[X]$ if $\fl = 1$;
 in this case, we test maximality of Dedekind's order generated by
 $$1, a_n \alpha, a_n\alpha^2 + a_{n-1}\alpha, \dots,
 a_n\alpha^{n-1} + a_{n-1}\alpha^{n-2} + \cdots + a_1\alpha.$$
 The routine will fail if $P$ is $0$ on the projective line over the residue
 field $\Z_K/\kbd{pr}$ (FIXME).

Function: rnfdet
Class: basic
Section: number_fields
C-Name: rnfdet
Prototype: GG
Help: rnfdet(nf,M): given a pseudo-matrix M, compute its determinant.
Doc: given a pseudo-matrix $M$ over the maximal
 order of $\var{nf}$, computes its determinant.

Function: rnfdisc
Class: basic
Section: number_fields
C-Name: rnfdiscf
Prototype: GG
Help: rnfdisc(nf,pol): given a pol with coefficients in nf, gives a
 2-component vector [D,d], where D is the relative ideal discriminant, and d
 is the relative discriminant in nf^*/nf*^2.
Doc: given a number field $\var{nf}$ as
 output by \kbd{nfinit} and a polynomial \var{pol} with coefficients in
 $\var{nf}$ defining a relative extension $L$ of $\var{nf}$, computes the
 relative discriminant of $L$. This is a two-element row vector $[D,d]$, where
 $D$ is the relative ideal discriminant and $d$ is the relative discriminant
 considered as an element of $\var{nf}^*/{\var{nf}^*}^2$. The main variable of
 $\var{nf}$ \emph{must} be of lower priority than that of \var{pol}, see
 \secref{se:priority}.

Function: rnfeltabstorel
Class: basic
Section: number_fields
C-Name: rnfeltabstorel
Prototype: GG
Help: rnfeltabstorel(rnf,x): transforms the element x from absolute to
 relative representation.
Doc: Let $\var{rnf}$ be a relative
 number field extension $L/K$ as output by \kbd{rnfinit} and let $x$ be an
 element of $L$ expressed as a polynomial modulo the absolute equation
 \kbd{\var{rnf}.pol}, or in terms of the absolute $\Z$-basis for $\Z_L$
 if \var{rnf} contains one (as in \kbd{rnfinit(nf,pol,1)}, or after
 a call to \kbd{nfinit(rnf)}).
 Computes $x$ as an element of the relative extension
 $L/K$ as a polmod with polmod coefficients.
 \bprog
 ? K = nfinit(y^2+1); L = rnfinit(K, x^2-y);
 ? L.polabs
 %2 = x^4 + 1
 ? rnfeltabstorel(L, Mod(x, L.polabs))
 %3 = Mod(x, x^2 + Mod(-y, y^2 + 1))
 ? rnfeltabstorel(L, 1/3)
 %4 = 1/3
 ? rnfeltabstorel(L, Mod(x, x^2-y))
 %5 = Mod(x, x^2 + Mod(-y, y^2 + 1))
 
 ? rnfeltabstorel(L, [0,0,0,1]~) \\ Z_L not initialized yet
  ***   at top-level: rnfeltabstorel(L,[0,
  ***                 ^--------------------
  *** rnfeltabstorel: incorrect type in rnfeltabstorel, apply nfinit(rnf).
 ? nfinit(L); \\ initialize now
 ? rnfeltabstorel(L, [0,0,0,1]~)
 %6 = Mod(Mod(y, y^2 + 1)*x, x^2 + Mod(-y, y^2 + 1))
 @eprog

Function: rnfeltdown
Class: basic
Section: number_fields
C-Name: rnfeltdown0
Prototype: GGD0,L,
Help: rnfeltdown(rnf,x,{flag=0}): expresses x on the base field if possible;
 returns an error otherwise.
Doc: $\var{rnf}$ being a relative number
 field extension $L/K$ as output by \kbd{rnfinit} and $x$ being an element of
 $L$ expressed as a polynomial or polmod with polmod coefficients (or as a
 \typ{COL} on \kbd{nfinit(rnf).zk}), computes
 $x$ as an element of $K$ as a \typ{POLMOD} if $\fl = 0$ and as a \typ{COL}
 otherwise. If $x$ is not in $K$, a domain error occurs.
 \bprog
 ? K = nfinit(y^2+1); L = rnfinit(K, x^2-y);
 ? L.pol
 %2 = x^4 + 1
 ? rnfeltdown(L, Mod(x^2, L.pol))
 %3 = Mod(y, y^2 + 1)
 ? rnfeltdown(L, Mod(x^2, L.pol), 1)
 %4 = [0, 1]~
 ? rnfeltdown(L, Mod(y, x^2-y))
 %5 = Mod(y, y^2 + 1)
 ? rnfeltdown(L, Mod(y,K.pol))
 %6 = Mod(y, y^2 + 1)
 ? rnfeltdown(L, Mod(x, L.pol))
  ***   at top-level: rnfeltdown(L,Mod(x,x
  ***                 ^--------------------
  *** rnfeltdown: domain error in rnfeltdown: element not in the base field
 ? rnfeltdown(L, Mod(y, x^2-y), 1) \\ as a t_COL
 %7 = [0, 1]~
 ? rnfeltdown(L, [0,1,0,0]~) \\ not allowed without absolute nf struct
   *** rnfeltdown: incorrect type in rnfeltdown (t_COL).
 ? nfinit(L); \\ add absolute nf structure to L
 ? rnfeltdown(L, [0,1,0,0]~) \\ now OK
 %8 = Mod(y, y^2 + 1)
 @eprog\noindent If we had started with
 \kbd{L = rnfinit(K, x\pow2-y, 1)}, then the final would have worked directly.
Variant: Also available is
 \fun{GEN}{rnfeltdown}{GEN rnf, GEN x} ($\fl = 0$).

Function: rnfeltnorm
Class: basic
Section: number_fields
C-Name: rnfeltnorm
Prototype: GG
Help: rnfeltnorm(rnf,x): returns the relative norm N_{L/K}(x), as an element
 of K.
Doc: $\var{rnf}$ being a relative number field extension $L/K$ as output by
 \kbd{rnfinit} and $x$ being an element of $L$, returns the relative norm
 $N_{L/K}(x)$ as an element of $K$.
 \bprog
 ? K = nfinit(y^2+1); L = rnfinit(K, x^2-y);
 ? rnfeltnorm(L, Mod(x, L.pol))
 %2 = Mod(x, x^2 + Mod(-y, y^2 + 1))
 ? rnfeltnorm(L, 2)
 %3 = 4
 ? rnfeltnorm(L, Mod(x, x^2-y))
 @eprog

Function: rnfeltreltoabs
Class: basic
Section: number_fields
C-Name: rnfeltreltoabs
Prototype: GG
Help: rnfeltreltoabs(rnf,x): transforms the element x from relative to
 absolute representation.
Doc: $\var{rnf}$ being a relative
 number field extension $L/K$ as output by \kbd{rnfinit} and $x$ being an
 element of $L$ expressed as a polynomial or polmod with polmod
 coefficients, computes $x$ as an element of the absolute extension $L/\Q$ as
 a polynomial modulo the absolute equation \kbd{\var{rnf}.pol}.
 \bprog
 ? K = nfinit(y^2+1); L = rnfinit(K, x^2-y);
 ? L.pol
 %2 = x^4 + 1
 ? rnfeltreltoabs(L, Mod(x, L.pol))
 %3 = Mod(x, x^4 + 1)
 ? rnfeltreltoabs(L, Mod(y, x^2-y))
 %4 = Mod(x^2, x^4 + 1)
 ? rnfeltreltoabs(L, Mod(y,K.pol))
 %5 = Mod(x^2, x^4 + 1)
 @eprog

Function: rnfelttrace
Class: basic
Section: number_fields
C-Name: rnfelttrace
Prototype: GG
Help: rnfelttrace(rnf,x): returns the relative trace Tr_{L/K}(x), as an element
 of K.
Doc: $\var{rnf}$ being a relative number field extension $L/K$ as output by
 \kbd{rnfinit} and $x$ being an element of $L$, returns the relative trace
 $Tr_{L/K}(x)$ as an element of $K$.
 \bprog
 ? K = nfinit(y^2+1); L = rnfinit(K, x^2-y);
 ? rnfelttrace(L, Mod(x, L.pol))
 %2 = 0
 ? rnfelttrace(L, 2)
 %3 = 4
 ? rnfelttrace(L, Mod(x, x^2-y))
 @eprog

Function: rnfeltup
Class: basic
Section: number_fields
C-Name: rnfeltup0
Prototype: GGD0,L,
Help: rnfeltup(rnf,x,{flag=0}): expresses x (belonging to the base field) on
 the relative field. As a t_POLMOD if flag = 0 and as a t_COL on the absolute
 field integer basis if flag = 1.
Doc: $\var{rnf}$ being a relative number field extension $L/K$ as output by
 \kbd{rnfinit} and $x$ being an element of $K$, computes $x$ as an element of
 the absolute extension $L/\Q$. As a \typ{POLMOD} modulo \kbd{\var{rnf}.pol}
 if $\fl = 0$ and as a \typ{COL} on the absolute field integer basis if
 $\fl = 1$.
 \bprog
 ? K = nfinit(y^2+1); L = rnfinit(K, x^2-y);
 ? L.pol
 %2 = x^4 + 1
 ? rnfeltup(L, Mod(y, K.pol))
 %3 = Mod(x^2, x^4 + 1)
 ? rnfeltup(L, y)
 %4 = Mod(x^2, x^4 + 1)
 ? rnfeltup(L, [1,2]~) \\ in terms of K.zk
 %5 = Mod(2*x^2 + 1, x^4 + 1)
 ? rnfeltup(L, y, 1) \\ in terms of nfinit(L).zk
 %6 = [0, 1, 0, 0]~
 ? rnfeltup(L, [1,2]~, 1)
 %7 = [1, 2, 0, 0]~
 @eprog

Function: rnfequation
Class: basic
Section: number_fields
C-Name: rnfequation0
Prototype: GGD0,L,
Help: rnfequation(nf,pol,{flag=0}): given a pol with coefficients in nf,
 gives an absolute equation z of the number field defined by pol. flag is
 optional, and can be 0: default, or non-zero, gives [z,al,k], where
 z defines the absolute equation L/Q as in the default behavior,
 al expresses as an element of L a root of the polynomial
 defining the base field nf, and k is a small integer such that
 t = b + k al is a root of z, for b a root of pol.
Doc: given a number field
 $\var{nf}$ as output by \kbd{nfinit} (or simply a polynomial) and a
 polynomial \var{pol} with coefficients in $\var{nf}$ defining a relative
 extension $L$ of $\var{nf}$, computes an absolute equation of $L$ over
 $\Q$.
 
 The main variable of $\var{nf}$ \emph{must} be of lower priority than that
 of \var{pol} (see \secref{se:priority}). Note that for efficiency, this does
 not check whether the relative equation is irreducible over $\var{nf}$, but
 only if it is squarefree. If it is reducible but squarefree, the result will
 be the absolute equation of the \'etale algebra defined by \var{pol}. If
 \var{pol} is not squarefree, raise an \kbd{e\_DOMAIN} exception.
 \bprog
 ? rnfequation(y^2+1, x^2 - y)
 %1 = x^4 + 1
 ? T = y^3-2; rnfequation(nfinit(T), (x^3-2)/(x-Mod(y,T)))
 %2 = x^6 + 108  \\ Galois closure of Q(2^(1/3))
 @eprog
 
 If $\fl$ is non-zero, outputs a 3-component row vector $[z,a,k]$, where
 
 \item $z$ is the absolute equation of $L$ over $\Q$, as in the default
 behavior,
 
 \item $a$ expresses as a \typ{POLMOD} modulo $z$ a root $\alpha$ of the
 polynomial defining the base field $\var{nf}$,
 
 \item $k$ is a small integer such that $\theta = \beta+k\alpha$
 is a root of $z$, where $\beta$ is a root of $\var{pol}$.
 \bprog
 ? T = y^3-2; pol = x^2 +x*y + y^2;
 ? [z,a,k] = rnfequation(T, pol, 1);
 ? z
 %3 = x^6 + 108
 ? subst(T, y, a)
 %4 = 0
 ? alpha= Mod(y, T);
 ? beta = Mod(x*Mod(1,T), pol);
 ? subst(z, x, beta + k*alpha)
 %7 = 0
 @eprog
Variant: Also available are
 \fun{GEN}{rnfequation}{GEN nf, GEN pol} ($\fl = 0$) and
 \fun{GEN}{rnfequation2}{GEN nf, GEN pol} ($\fl = 1$).

Function: rnfhnfbasis
Class: basic
Section: number_fields
C-Name: rnfhnfbasis
Prototype: GG
Help: rnfhnfbasis(bnf,x): given an order x as output by rnfpseudobasis,
 gives either a true HNF basis of the order if it exists, zero otherwise.
Doc: given $\var{bnf}$ as output by
 \kbd{bnfinit}, and either a polynomial $x$ with coefficients in $\var{bnf}$
 defining a relative extension $L$ of $\var{bnf}$, or a pseudo-basis $x$ of
 such an extension, gives either a true $\var{bnf}$-basis of $L$ in upper
 triangular Hermite normal form, if it exists, and returns $0$ otherwise.

Function: rnfidealabstorel
Class: basic
Section: number_fields
C-Name: rnfidealabstorel
Prototype: GG
Help: rnfidealabstorel(rnf,x): transforms the ideal x from absolute to
 relative representation.
Doc: let $\var{rnf}$ be a relative
 number field extension $L/K$ as output by \kbd{rnfinit} and $x$ be an ideal of
 the absolute extension $L/\Q$ given by a $\Z$-basis of elements of $L$.
 Returns the relative pseudo-matrix in HNF giving the ideal $x$ considered as
 an ideal of the relative extension $L/K$, i.e.~as a $\Z_K$-module.
 
 The reason why the input does not use the customary HNF in terms of a fixed
 $\Z$-basis for $\Z_L$ is precisely that no such basis has been explicitly
 specified. On the other hand, if you already computed an (absolute) \kbd{nf}
 structure \kbd{Labs} attached to $L$, and $m$ is in HNF, defining
 an (absolute) ideal with respect to the $\Z$-basis \kbd{Labs.zk}, then
 \kbd{Labs.zk * m} is a suitable $\Z$-basis for the ideal, and
 \bprog
   rnfidealabstorel(rnf, Labs.zk * m)
 @eprog\noindent converts $m$ to a relative ideal.
 \bprog
 ? K = nfinit(y^2+1); L = rnfinit(K, x^2-y); Labs = nfinit(L);
 ? m = idealhnf(Labs, 17, x^3+2);
 ? B = rnfidealabstorel(L, Labs.zk * m)
 %3 = [[1, 8; 0, 1], [[17, 4; 0, 1], 1]]  \\ pseudo-basis for m as Z_K-module
 ? A = rnfidealreltoabs(L, B)
 %4 = [17, x^2 + 4, x + 8, x^3 + 8*x^2]   \\ Z-basis for m in Q[x]/(L.pol)
 ? mathnf(matalgtobasis(Labs, A))
 %5 =
 [17 8 4 2]
 
 [ 0 1 0 0]
 
 [ 0 0 1 0]
 
 [ 0 0 0 1]
 ? % == m
 %6 = 1
 @eprog

Function: rnfidealdown
Class: basic
Section: number_fields
C-Name: rnfidealdown
Prototype: GG
Help: rnfidealdown(rnf,x): finds the intersection of the ideal x with the
 base field.
Doc: let $\var{rnf}$ be a relative number
 field extension $L/K$ as output by \kbd{rnfinit}, and $x$ an ideal of
 $L$, given either in relative form or by a $\Z$-basis of elements of $L$
 (see \secref{se:rnfidealabstorel}). This function returns the ideal of $K$
 below $x$, i.e.~the intersection of $x$ with $K$.

Function: rnfidealfactor
Class: basic
Section: number_fields
C-Name: rnfidealfactor
Prototype: GG
Help: rnfidealfactor(rnf,x): factorization of the ideal x into
 prime ideals in the number field nfinit(rnf).
Doc: factors into prime ideal powers the
 ideal $x$ in the attached absolute number field $L = \kbd{nfinit}(\var{rnf})$.
 The output format is similar to the \kbd{factor} function, and the prime
 ideals are represented in the form output by the \kbd{idealprimedec}
 function for $L$.
 \bprog
 ? rnf = rnfinit(nfinit(y^2+1), x^2-y+1);
 ? rnfidealfactor(rnf, y+1)  \\ P_2^2
 %2 =
 [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 2]
 
 ? rnfidealfactor(rnf, x) \\ P_2
 %3 =
 [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 1]
 
 ? L = nfinit(rnf);
 ? id = idealhnf(L, idealhnf(L, 25, (x+1)^2));
 ? idealfactor(L, id) == rnfidealfactor(rnf, id)
 %6 = 1
 @eprog\noindent Note that ideals of the base field $K$ must be explicitly
 lifted to $L$ via \kbd{rnfidealup} before they can be factored.

Function: rnfidealhnf
Class: basic
Section: number_fields
C-Name: rnfidealhnf
Prototype: GG
Help: rnfidealhnf(rnf,x): relative version of idealhnf, where rnf is a
 relative numberfield.
Doc: $\var{rnf}$ being a relative number
 field extension $L/K$ as output by \kbd{rnfinit} and $x$ being a relative
 ideal (which can be, as in the absolute case, of many different types,
 including of course elements), computes the HNF pseudo-matrix attached to
 $x$, viewed as a $\Z_K$-module.

Function: rnfidealmul
Class: basic
Section: number_fields
C-Name: rnfidealmul
Prototype: GGG
Help: rnfidealmul(rnf,x,y): relative version of idealmul, where rnf is a
 relative numberfield.
Doc: $\var{rnf}$ being a relative number
 field extension $L/K$ as output by \kbd{rnfinit} and $x$ and $y$ being ideals
 of the relative extension $L/K$ given by pseudo-matrices, outputs the ideal
 product, again as a relative ideal.

Function: rnfidealnormabs
Class: basic
Section: number_fields
C-Name: rnfidealnormabs
Prototype: GG
Help: rnfidealnormabs(rnf,x): absolute norm of the ideal x.
Doc: let $\var{rnf}$ be a relative
 number field extension $L/K$ as output by \kbd{rnfinit} and let $x$ be a
 relative ideal (which can be, as in the absolute case, of many different
 types, including of course elements). This function computes the norm of the
 $x$ considered as an ideal of the absolute extension $L/\Q$. This is
 identical to
 \bprog
    idealnorm(rnf, rnfidealnormrel(rnf,x))
 @eprog\noindent but faster.

Function: rnfidealnormrel
Class: basic
Section: number_fields
C-Name: rnfidealnormrel
Prototype: GG
Help: rnfidealnormrel(rnf,x): relative norm of the ideal x.
Doc: let $\var{rnf}$ be a relative
 number field extension $L/K$ as output by \kbd{rnfinit} and let $x$ be a
 relative ideal (which can be, as in the absolute case, of many different
 types, including of course elements). This function computes the relative
 norm of $x$ as an ideal of $K$ in HNF.

Function: rnfidealprimedec
Class: basic
Section: number_fields
C-Name: rnfidealprimedec
Prototype: GG
Help: rnfidealprimedec(rnf,pr): prime ideal decomposition of the maximal
 ideal pr of K in L/K; pr is also allowed to be a prime number p, in which
 case we return a pair of vectors [SK,SL], where SK contains the primes of K
 above p and SL[i] is the vector of primes of L above SK[i].
Doc: let \var{rnf} be a relative number
 field extension $L/K$ as output by \kbd{rnfinit}, and \kbd{pr} a maximal
 ideal of $K$ (\kbd{prid}), this function completes the \var{rnf}
 with a \var{nf} structure attached to $L$ (see \secref{se:rnfinit})
 and returns the prime ideal decomposition of \kbd{pr} in $L/K$.
 \bprog
 ? K = nfinit(y^2+1); rnf = rnfinit(K, x^3+y+1);
 ? P = idealprimedec(K, 2)[1];
 ? S = rnfidealprimedec(rnf, P);
 ? #S
 %4 = 1
 @eprog
 The argument \kbd{pr} is also allowed to be a prime number $p$, in which
 case we return a pair of vectors \kbd{[SK,SL]}, where \kbd{SK} contains
 the primes of $K$ above $p$ and \kbd{SL}$[i]$ is the vector of primes of $L$
 above \kbd{SK}$[i]$.
 \bprog
 ? [SK,SL] = rnfidealprimedec(rnf, 5);
 ? [#SK, vector(#SL,i,#SL[i])]
 %6 = [2, [2, 2]]
 @eprog

Function: rnfidealreltoabs
Class: basic
Section: number_fields
C-Name: rnfidealreltoabs0
Prototype: GGD0,L,
Help: rnfidealreltoabs(rnf,x,{flag=0}): transforms the ideal x from relative to
 absolute representation. As a vector of t_POLMODs if flag = 0 and as an ideal
 in HNF in the absolute field if flag = 1.
Doc: Let $\var{rnf}$ be a relative
 number field extension $L/K$ as output by \kbd{rnfinit} and let $x$ be a
 relative ideal, given as a $\Z_K$-module by a pseudo matrix $[A,I]$.
 This function returns the ideal $x$ as an absolute ideal of $L/\Q$.
 If $\fl = 0$, the result is given by a vector of \typ{POLMOD}s modulo
 \kbd{rnf.pol} forming a $\Z$-basis; if $\fl = 1$, it is given in HNF in terms
 of the fixed $\Z$-basis for $\Z_L$, see \secref{se:rnfinit}.
 \bprog
 ? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y);
 ? P = idealprimedec(K,2)[1];
 ? P = rnfidealup(rnf, P)
 %3 = [2, x^2 + 1, 2*x, x^3 + x]
 ? Prel = rnfidealhnf(rnf, P)
 %4 = [[1, 0; 0, 1], [[2, 1; 0, 1], [2, 1; 0, 1]]]
 ? rnfidealreltoabs(rnf,Prel)
 %5 = [2, x^2 + 1, 2*x, x^3 + x]
 ? rnfidealreltoabs(rnf,Prel,1)
 %6 =
 [2 1 0 0]
 
 [0 1 0 0]
 
 [0 0 2 1]
 
 [0 0 0 1]
 @eprog
 The reason why we do not return by default ($\fl = 0$) the customary HNF in
 terms of a fixed $\Z$-basis for $\Z_L$ is precisely because
 a \var{rnf} does not contain such a basis by default. Completing the
 structure so that it contains a \var{nf} structure for $L$ is polynomial
 time but costly when the absolute degree is large, thus it is not done by
 default. Note that setting $\fl = 1$ will complete the \var{rnf}.
Variant: Also available is
 \fun{GEN}{rnfidealreltoabs}{GEN rnf, GEN x} ($\fl = 0$).

Function: rnfidealtwoelt
Class: basic
Section: number_fields
C-Name: rnfidealtwoelement
Prototype: GG
Help: rnfidealtwoelt(rnf,x): relative version of idealtwoelt, where rnf
 is a relative numberfield.
Doc: $\var{rnf}$ being a relative
 number field extension $L/K$ as output by \kbd{rnfinit} and $x$ being an
 ideal of the relative extension $L/K$ given by a pseudo-matrix, gives a
 vector of two generators of $x$ over $\Z_L$ expressed as polmods with polmod
 coefficients.

Function: rnfidealup
Class: basic
Section: number_fields
C-Name: rnfidealup0
Prototype: GGD0,L,
Help: rnfidealup(rnf,x,{flag=0}): lifts the ideal x (of the base field) to the
 relative field. As a vector of t_POLMODs if flag = 0 and as an ideal in HNF
 in the absolute field if flag = 1.
Doc: let $\var{rnf}$ be a relative number
 field extension $L/K$ as output by \kbd{rnfinit} and let $x$ be an ideal of
 $K$. This function returns the ideal $x\Z_L$ as an absolute ideal of $L/\Q$,
 in the form of a $\Z$-basis. If $\fl = 0$, the result is given by a vector of
 polynomials (modulo \kbd{rnf.pol}); if $\fl = 1$, it is given in HNF in terms
 of the fixed $\Z$-basis for $\Z_L$, see \secref{se:rnfinit}.
 \bprog
 ? K = nfinit(y^2+1); rnf = rnfinit(K, x^2-y);
 ? P = idealprimedec(K,2)[1];
 ? rnfidealup(rnf, P)
 %3 = [2, x^2 + 1, 2*x, x^3 + x]
 ? rnfidealup(rnf, P,1)
 %4 =
 [2 1 0 0]
 
 [0 1 0 0]
 
 [0 0 2 1]
 
 [0 0 0 1]
 @eprog
 The reason why we do not return by default ($\fl = 0$) the customary HNF in
 terms of a fixed $\Z$-basis for $\Z_L$ is precisely because
 a \var{rnf} does not contain such a basis by default. Completing the
 structure so that it contains a \var{nf} structure for $L$ is polynomial
 time but costly when the absolute degree is large, thus it is not done by
 default. Note that setting $\fl = 1$ will complete the \var{rnf}.
Variant: Also available is
  \fun{GEN}{rnfidealup}{GEN rnf, GEN x} ($\fl = 0$).

Function: rnfinit
Class: basic
Section: number_fields
C-Name: rnfinit0
Prototype: GGD0,L,
Help: rnfinit(nf,pol,{flag=0}): pol being an irreducible polynomial
 defined over the number field nf, initializes a vector of data necessary for
 working in relative number fields (rnf functions). See manual for technical
 details.
Doc: $\var{nf}$ being a number field in \kbd{nfinit}
 format considered as base field, and \var{pol} a polynomial defining a relative
 extension over $\var{nf}$, this computes data to work in the
 relative extension. The main variable of \var{pol} must be of higher priority
 (see \secref{se:priority}) than that of $\var{nf}$, and the coefficients of
 \var{pol} must be in $\var{nf}$.
 
 The result is a row vector, whose components are technical. In the following
 description, we let $K$ be the base field defined by $\var{nf}$ and $L/K$
 the extension attached to the \var{rnf}. Furthermore, we let
 $m = [K:\Q]$ the degree of the base field, $n = [L:K]$ the relative degree,
 $r_1$ and $r_2$ the number of real and complex places of $K$. Access to this
 information via \emph{member functions} is preferred since the specific
 data organization specified below will change in the future.
 
 If $\fl = 1$, add an \var{nf} structure attached to $L$ to \var{rnf}.
 This is likely to be very expensive if the absolute degree $mn$ is large,
 but fixes an integer basis for $\Z_L$ as a $\Z$-module and allows to input
 and output elements of $L$ in absolute form: as \typ{COL} for elements,
 as \typ{MAT} in HNF for ideals, as \kbd{prid} for prime ideals. Without such
 a call, elements of $L$ are represented as \typ{POLMOD}, etc.
 Note that a subsequent \kbd{nfinit}$(\var{rnf})$ will also explicitly
 add such a component, and so will the following functions \kbd{rnfidealmul},
 \kbd{rnfidealtwoelt}, \kbd{rnfidealprimedec}, \kbd{rnfidealup} (with flag 1)
 and \kbd{rnfidealreltoabs} (with flag 1). The absolute \var{nf} structure
 attached to $L$ can be recovered using \kbd{nfinit(rnf)}.
 
 $\var{rnf}[1]$(\kbd{rnf.pol}) contains the relative polynomial \var{pol}.
 
 $\var{rnf}[2]$ contains the integer basis $[A,d]$ of $K$, as
 (integral) elements of $L/\Q$. More precisely, $A$ is a vector of
 polynomial with integer coefficients, $d$ is a denominator, and the integer
 basis is given by $A/d$.
 
 $\var{rnf}[3]$ (\kbd{rnf.disc}) is a two-component row vector
 $[\goth{d}(L/K),s]$ where $\goth{d}(L/K)$ is the relative ideal discriminant
 of $L/K$ and $s$ is the discriminant of $L/K$ viewed as an element of
 $K^*/(K^*)^2$, in other words it is the output of \kbd{rnfdisc}.
 
 $\var{rnf}[4]$(\kbd{rnf.index}) is the ideal index $\goth{f}$, i.e.~such
 that $d(pol)\Z_K=\goth{f}^2\goth{d}(L/K)$.
 
 $\var{rnf}[5]$ is currently unused.
 
 $\var{rnf}[6]$ is currently unused.
 
 $\var{rnf}[7]$ (\kbd{rnf.zk}) is the pseudo-basis $(A,I)$ for the maximal
 order $\Z_L$ as a $\Z_K$-module: $A$ is the relative integral pseudo basis
 expressed as polynomials (in the variable of $pol$) with polmod coefficients
 in $\var{nf}$, and the second component $I$ is the ideal list of the
 pseudobasis in HNF.
 
 $\var{rnf}[8]$ is the inverse matrix of the integral basis matrix, with
 coefficients polmods in $\var{nf}$.
 
 $\var{rnf}[9]$ is currently unused.
 
 $\var{rnf}[10]$ (\kbd{rnf.nf}) is $\var{nf}$.
 
 $\var{rnf}[11]$ is an extension of \kbd{rnfequation(K, pol, 1)}. Namely, a
 vector $[P, a, k, \kbd{K.pol}, \kbd{pol}]$ describing the \emph{absolute}
 extension
 $L/\Q$: $P$ is an absolute equation, more conveniently obtained
 as \kbd{rnf.polabs}; $a$ expresses the generator $\alpha = y \mod \kbd{K.pol}$
 of the number field $K$ as an element of $L$, i.e.~a polynomial modulo the
 absolute equation $P$;
 
 $k$ is a small integer such that, if $\beta$ is an abstract root of \var{pol}
 and $\alpha$ the generator of $K$ given above, then $P(\beta + k\alpha) = 0$.
 
 \misctitle{Caveat} Be careful if $k\neq0$ when dealing simultaneously with
 absolute and relative quantities since $L = \Q(\beta + k\alpha) =
 K(\alpha)$, and the generator chosen for the absolute extension is not the
 same as for the relative one. If this happens, one can of course go on
 working, but we advise to change the relative polynomial so that its root
 becomes $\beta + k \alpha$. Typical GP instructions would be
 \bprog
   [P,a,k] = rnfequation(K, pol, 1);
   if (k, pol = subst(pol, x, x - k*Mod(y, K.pol)));
   L = rnfinit(K, pol);
 @eprog
 
 $\var{rnf}[12]$ is by default unused and set equal to 0. This field is used
 to store further information about the field as it becomes available (which
 is rarely needed, hence would be too expensive to compute during the initial
 \kbd{rnfinit} call).
Variant: Also available is
 \fun{GEN}{rnfinit}{GEN nf,GEN pol} ($\fl = 0$).

Function: rnfisabelian
Class: basic
Section: number_fields
C-Name: rnfisabelian
Prototype: lGG
Help: rnfisabelian(nf,T): T being a relative polynomial with coefficients
 in nf, return 1 if it defines an abelian extension, and 0 otherwise.
Doc: $T$ being a relative polynomial with coefficients
 in \var{nf}, return 1 if it defines an abelian extension, and 0 otherwise.
 \bprog
 ? K = nfinit(y^2 + 23);
 ? rnfisabelian(K, x^3 - 3*x - y)
 %2 = 1
 @eprog

Function: rnfisfree
Class: basic
Section: number_fields
C-Name: rnfisfree
Prototype: lGG
Help: rnfisfree(bnf,x): given an order x as output by rnfpseudobasis or
 rnfsteinitz, outputs true (1) or false (0) according to whether the order is
 free or not.
Doc: given $\var{bnf}$ as output by
 \kbd{bnfinit}, and either a polynomial $x$ with coefficients in $\var{bnf}$
 defining a relative extension $L$ of $\var{bnf}$, or a pseudo-basis $x$ of
 such an extension, returns true (1) if $L/\var{bnf}$ is free, false (0) if
 not.

Function: rnfislocalcyclo
Class: basic
Section: number_fields
C-Name: rnfislocalcyclo
Prototype: lG
Help: rnfislocalcyclo(rnf): true(1) if the l-extension attached to rnf
 is locally cyclotomic (locally contained in the Z_l extension of K_v at
 all places v | l), false(0) if not.
Doc: Let \var{rnf} a a relative number field extension $L/K$ as output
 by \kbd{rnfinit} whole degree $[L:K]$ is a power of a prime $\ell$.
 Return $1$ if the $\ell$-extension is locally cyclotomic (locally contained in
 the cyclotomic $\Z_\ell$-extension of $K_v$ at all places $v | \ell$), and
 $0$ if not.
 \bprog
 ? K = nfinit(y^2 + y + 1);
 ? L = rnfinit(K, x^3 - y); /* = K(zeta_9), globally cyclotomic */
 ? rnfislocalcyclo(L)
 %3 = 1
 \\ we expect 3-adic continuity by Krasner's lemma
 ? vector(5, i, rnfislocalcyclo(rnfinit(K, x^3 - y + 3^i)))
 %5 = [0, 1, 1, 1, 1]
 @eprog

Function: rnfisnorm
Class: basic
Section: number_fields
C-Name: rnfisnorm
Prototype: GGD0,L,
Help: rnfisnorm(T,a,{flag=0}): T is as output by rnfisnorminit applied to
 L/K. Tries to tell whether a is a norm from L/K. Returns a vector [x,q]
 where a=Norm(x)*q. Looks for a solution which is a S-integer, with S a list
 of places in K containing the ramified primes, generators of the class group
 of ext, as well as those primes dividing a. If L/K is Galois, omit flag,
 otherwise it is used to add more places to S: all the places above the
 primes p <= flag (resp. p | flag) if flag > 0 (resp. flag < 0). The answer
 is guaranteed (i.e a is a norm iff q=1) if L/K is Galois or, under GRH, if S
 contains all primes less than 12.log(disc(M))^2, where M is the normal
 closure of L/K.
Doc: similar to
 \kbd{bnfisnorm} but in the relative case. $T$ is as output by
 \tet{rnfisnorminit} applied to the extension $L/K$. This tries to decide
 whether the element $a$ in $K$ is the norm of some $x$ in the extension
 $L/K$.
 
 The output is a vector $[x,q]$, where $a = \Norm(x)*q$. The
 algorithm looks for a solution $x$ which is an $S$-integer, with $S$ a list
 of places of $K$ containing at least the ramified primes, the generators of
 the class group of $L$, as well as those primes dividing $a$. If $L/K$ is
 Galois, then this is enough; otherwise, $\fl$ is used to add more primes to
 $S$: all the places above the primes $p \leq \fl$ (resp.~$p|\fl$) if $\fl>0$
 (resp.~$\fl<0$).
 
 The answer is guaranteed (i.e.~$a$ is a norm iff $q = 1$) if the field is
 Galois, or, under \idx{GRH}, if $S$ contains all primes less than
 $12\log^2\left|\disc(M)\right|$, where $M$ is the normal
 closure of $L/K$.
 
 If \tet{rnfisnorminit} has determined (or was told) that $L/K$ is
 \idx{Galois}, and $\fl \neq 0$, a Warning is issued (so that you can set
 $\fl = 1$ to check whether $L/K$ is known to be Galois, according to $T$).
 Example:
 
 \bprog
 bnf = bnfinit(y^3 + y^2 - 2*y - 1);
 p = x^2 + Mod(y^2 + 2*y + 1, bnf.pol);
 T = rnfisnorminit(bnf, p);
 rnfisnorm(T, 17)
 @eprog\noindent
 checks whether $17$ is a norm in the Galois extension $\Q(\beta) /
 \Q(\alpha)$, where $\alpha^3 + \alpha^2 - 2\alpha - 1 = 0$ and $\beta^2 +
 \alpha^2 + 2\alpha + 1 = 0$ (it is).

Function: rnfisnorminit
Class: basic
Section: number_fields
C-Name: rnfisnorminit
Prototype: GGD2,L,
Help: rnfisnorminit(pol,polrel,{flag=2}): let K be defined by a root of pol,
 L/K the extension defined by polrel. Compute technical data needed by
 rnfisnorm to solve norm equations Nx = a, for x in L, and a in K. If flag=0,
 do not care whether L/K is Galois or not; if flag = 1, assume L/K is Galois;
 if flag = 2, determine whether L/K is Galois.
Doc: let $K$ be defined by a root of \var{pol}, and $L/K$ the extension defined
 by the polynomial \var{polrel}. As usual, \var{pol} can in fact be an \var{nf},
 or \var{bnf}, etc; if \var{pol} has degree $1$ (the base field is $\Q$),
 polrel is also allowed to be an \var{nf}, etc. Computes technical data needed
 by \tet{rnfisnorm} to solve norm equations $Nx = a$, for $x$ in $L$, and $a$
 in $K$.
 
 If $\fl = 0$, do not care whether $L/K$ is Galois or not.
 
 If $\fl = 1$, $L/K$ is assumed to be Galois (unchecked), which speeds up
 \tet{rnfisnorm}.
 
 If $\fl = 2$, let the routine determine whether $L/K$ is Galois.

Function: rnfkummer
Class: basic
Section: number_fields
C-Name: rnfkummer
Prototype: GDGD0,L,p
Help: rnfkummer(bnr,{subgp},{d=0}): bnr being as output by bnrinit,
 finds a relative equation for the class field corresponding to the module in
 bnr and the given congruence subgroup (the ray class field if subgp is
 omitted). d can be zero (default), or positive, and in this case the
 output is the list of all relative equations of degree d for the given bnr,
 with the same conductor as (bnr, subgp).
Doc: \var{bnr}
 being as output by \kbd{bnrinit}, finds a relative equation for the
 class field corresponding to the module in \var{bnr} and the given
 congruence subgroup (the full ray class field if \var{subgp} is omitted).
 If $d$ is positive, outputs the list of all relative equations of
 degree $d$ contained in the ray class field defined by \var{bnr}, with
 the \emph{same} conductor as $(\var{bnr}, \var{subgp})$.
 
 \misctitle{Warning} This routine only works for subgroups of prime index. It
 uses Kummer theory, adjoining necessary roots of unity (it needs to compute a
 tough \kbd{bnfinit} here), and finds a generator via Hecke's characterization
 of ramification in Kummer extensions of prime degree. If your extension does
 not have prime degree, for the time being, you have to split it by hand as a
 tower / compositum of such extensions.

Function: rnflllgram
Class: basic
Section: number_fields
C-Name: rnflllgram
Prototype: GGGp
Help: rnflllgram(nf,pol,order): given a pol with coefficients in nf and an
 order as output by rnfpseudobasis or similar, gives [[neworder],U], where
 neworder is a reduced order and U is the unimodular transformation matrix.
Doc: given a polynomial
 \var{pol} with coefficients in \var{nf} defining a relative extension $L$ and
 a suborder \var{order} of $L$ (of maximal rank), as output by
 \kbd{rnfpseudobasis}$(\var{nf},\var{pol})$ or similar, gives
 $[[\var{neworder}],U]$, where \var{neworder} is a reduced order and $U$ is
 the unimodular transformation matrix.

Function: rnfnormgroup
Class: basic
Section: number_fields
C-Name: rnfnormgroup
Prototype: GG
Help: rnfnormgroup(bnr,pol): norm group (or Artin or Takagi group)
 corresponding to the Abelian extension of bnr.bnf defined by pol, where
 the module corresponding to bnr is assumed to be a multiple of the
 conductor. The result is the HNF defining the norm group on the
 generators in bnr.gen.
Doc: 
 \var{bnr} being a big ray
 class field as output by \kbd{bnrinit} and \var{pol} a relative polynomial
 defining an \idx{Abelian extension}, computes the norm group (alias Artin
 or Takagi group) corresponding to the Abelian extension of
 $\var{bnf}=$\kbd{bnr.bnf}
 defined by \var{pol}, where the module corresponding to \var{bnr} is assumed
 to be a multiple of the conductor (i.e.~\var{pol} defines a subextension of
 bnr). The result is the HNF defining the norm group on the given generators
 of \kbd{bnr.gen}. Note that neither the fact that \var{pol} defines an
 Abelian extension nor the fact that the module is a multiple of the conductor
 is checked. The result is undefined if the assumption is not correct,
 but the function will return the empty matrix \kbd{[;]} if it detects a
 problem; it may also not detect the problem and return a wrong result.

Function: rnfpolred
Class: basic
Section: number_fields
C-Name: rnfpolred
Prototype: GGp
Help: rnfpolred(nf,pol): given a pol with coefficients in nf, finds a list
 of relative polynomials defining some subfields, hopefully simpler.
Doc: This function is obsolete: use \tet{rnfpolredbest} instead.
 Relative version of \kbd{polred}. Given a monic polynomial \var{pol} with
 coefficients in $\var{nf}$, finds a list of relative polynomials defining some
 subfields, hopefully simpler and containing the original field. In the present
 version \vers, this is slower and less efficient than \kbd{rnfpolredbest}.
 
 \misctitle{Remark} this function is based on an incomplete reduction
 theory of lattices over number fields, implemented by \kbd{rnflllgram}, which
 deserves to be improved.
Obsolete: 2013-12-28

Function: rnfpolredabs
Class: basic
Section: number_fields
C-Name: rnfpolredabs
Prototype: GGD0,L,
Help: rnfpolredabs(nf,pol,{flag=0}): given a pol with coefficients in nf,
 finds a relative simpler polynomial defining the same field. Binary digits
 of flag mean: 1: return also the element whose characteristic polynomial is
 the given polynomial, 2: return an absolute polynomial, 16: partial
 reduction.
Doc: This function is obsolete: use \tet{rnfpolredbest} instead.
 Relative version of \kbd{polredabs}. Given a monic polynomial \var{pol}
 with coefficients in $\var{nf}$, finds a simpler relative polynomial defining
 the same field. The binary digits of $\fl$ mean
 
 The binary digits of $\fl$ correspond to $1$: add information to convert
 elements to the new representation, $2$: absolute polynomial, instead of
 relative, $16$: possibly use a suborder of the maximal order. More precisely:
 
 0: default, return $P$
 
 1: returns $[P,a]$ where $P$ is the default output and $a$,
 a \typ{POLMOD} modulo $P$, is a root of \var{pol}.
 
 2: returns \var{Pabs}, an absolute, instead of a relative, polynomial.
 Same as but faster than
 \bprog
   rnfequation(nf, rnfpolredabs(nf,pol))
 @eprog
 
 3: returns $[\var{Pabs},a,b]$, where \var{Pabs} is an absolute polynomial
 as above, $a$, $b$ are \typ{POLMOD} modulo \var{Pabs}, roots of \kbd{nf.pol}
 and \var{pol} respectively.
 
 16: possibly use a suborder of the maximal order. This is slower than the
 default when the relative discriminant is smooth, and much faster otherwise.
 See \secref{se:polredabs}.
 
 \misctitle{Warning} In the present implementation, \kbd{rnfpolredabs}
 produces smaller polynomials than \kbd{rnfpolred} and is usually
 faster, but its complexity is still exponential in the absolute degree.
 The function \tet{rnfpolredbest} runs in polynomial time, and  tends  to
 return polynomials with smaller discriminants.
Obsolete: 2013-12-28

Function: rnfpolredbest
Class: basic
Section: number_fields
C-Name: rnfpolredbest
Prototype: GGD0,L,
Help: rnfpolredbest(nf,pol,{flag=0}): given a pol with coefficients in nf,
 finds a relative polynomial P defining the same field, hopefully simpler
 than pol; flag
 can be 0: default, 1: return [P,a], where a is a root of pol
 2: return an absolute polynomial Pabs, 3:
 return [Pabs, a,b], where a is a root of nf.pol and b is a root of pol.
Doc: relative version of \kbd{polredbest}. Given a monic polynomial \var{pol}
 with coefficients in $\var{nf}$, finds a simpler relative polynomial $P$
 defining the same field. As opposed to \tet{rnfpolredabs} this function does
 not return a \emph{smallest} (canonical) polynomial with respect to some
 measure, but it does run in polynomial time.
 
 The binary digits of $\fl$ correspond to $1$: add information to convert
 elements to the new representation, $2$: absolute polynomial, instead of
 relative. More precisely:
 
 0: default, return $P$
 
 1: returns $[P,a]$ where $P$ is the default output and $a$,
 a \typ{POLMOD} modulo $P$, is a root of \var{pol}.
 
 2: returns \var{Pabs}, an absolute, instead of a relative, polynomial.
 Same as but faster than
 \bprog
   rnfequation(nf, rnfpolredbest(nf,pol))
 @eprog
 
 3: returns $[\var{Pabs},a,b]$, where \var{Pabs} is an absolute polynomial
 as above, $a$, $b$ are \typ{POLMOD} modulo \var{Pabs}, roots of \kbd{nf.pol}
 and \var{pol} respectively.
 
 \bprog
 ? K = nfinit(y^3-2); pol = x^2 +x*y + y^2;
 ? [P, a] = rnfpolredbest(K,pol,1);
 ? P
 %3 = x^2 - x + Mod(y - 1, y^3 - 2)
 ? a
 %4 = Mod(Mod(2*y^2+3*y+4,y^3-2)*x + Mod(-y^2-2*y-2,y^3-2),
          x^2 - x + Mod(y-1,y^3-2))
 ? subst(K.pol,y,a)
 %5 = 0
 ? [Pabs, a, b] = rnfpolredbest(K,pol,3);
 ? Pabs
 %7 = x^6 - 3*x^5 + 5*x^3 - 3*x + 1
 ? a
 %8 = Mod(-x^2+x+1, x^6-3*x^5+5*x^3-3*x+1)
 ? b
 %9 = Mod(2*x^5-5*x^4-3*x^3+10*x^2+5*x-5, x^6-3*x^5+5*x^3-3*x+1)
 ? subst(K.pol,y,a)
 %10 = 0
 ? substvec(pol,[x,y],[a,b])
 %11 = 0
 @eprog

Function: rnfpseudobasis
Class: basic
Section: number_fields
C-Name: rnfpseudobasis
Prototype: GG
Help: rnfpseudobasis(nf,pol): given a pol with coefficients in nf, gives a
 4-component vector [A,I,D,d] where [A,I] is a pseudo basis of the maximal
 order in HNF on the power basis, D is the relative ideal discriminant, and d
 is the relative discriminant in nf^*/nf*^2.
Doc: given a number field
 $\var{nf}$ as output by \kbd{nfinit} and a polynomial \var{pol} with
 coefficients in $\var{nf}$ defining a relative extension $L$ of $\var{nf}$,
 computes a pseudo-basis $(A,I)$ for the maximal order $\Z_L$ viewed as a
 $\Z_K$-module, and the relative discriminant of $L$. This is output as a
 four-element row vector $[A,I,D,d]$, where $D$ is the relative ideal
 discriminant and $d$ is the relative discriminant considered as an element of
 $\var{nf}^*/{\var{nf}^*}^2$.

Function: rnfsteinitz
Class: basic
Section: number_fields
C-Name: rnfsteinitz
Prototype: GG
Help: rnfsteinitz(nf,x): given an order x as output by rnfpseudobasis,
 gives [A,I,D,d] where (A,I) is a pseudo basis where all the ideals except
 perhaps the last are trivial.
Doc: given a number field $\var{nf}$ as
 output by \kbd{nfinit} and either a polynomial $x$ with coefficients in
 $\var{nf}$ defining a relative extension $L$ of $\var{nf}$, or a pseudo-basis
 $x$ of such an extension as output for example by \kbd{rnfpseudobasis},
 computes another pseudo-basis $(A,I)$ (not in HNF in general) such that all
 the ideals of $I$ except perhaps the last one are equal to the ring of
 integers of $\var{nf}$, and outputs the four-component row vector $[A,I,D,d]$
 as in \kbd{rnfpseudobasis}. The name of this function comes from the fact
 that the ideal class of the last ideal of $I$, which is well defined, is the
 \idx{Steinitz class} of the $\Z_K$-module $\Z_L$ (its image in $SK_0(\Z_K)$).

Function: round
Class: basic
Section: conversions
C-Name: round0
Prototype: GD&
Help: round(x,{&e}): take the nearest integer to all the coefficients of x.
 If e is present, do not take into account loss of integer part precision,
 and set e = error estimate in bits.
Description: 
 (small):small:parens   $1
 (int):int:copy:parens  $1
 (real):int             roundr($1)
 (mp):int               mpround($1)
 (mp, &small):int       grndtoi($1, &$2)
 (mp, &int):int         round0($1, &$2)
 (gen):gen              ground($1)
 (gen, &small):gen      grndtoi($1, &$2)
 (gen, &int):gen        round0($1, &$2)
Doc: If $x$ is in $\R$, rounds $x$ to the nearest integer (rounding to
 $+\infty$ in case of ties), then and sets $e$ to the number of error bits,
 that is the binary exponent of the difference between the original and the
 rounded value (the ``fractional part''). If the exponent of $x$ is too large
 compared to its precision (i.e.~$e>0$), the result is undefined and an error
 occurs if $e$ was not given.
 
 \misctitle{Important remark} Contrary to the other truncation functions,
 this function operates on every coefficient at every level of a PARI object.
 For example
 $$\text{truncate}\left(\dfrac{2.4*X^2-1.7}{X}\right)=2.4*X,$$
 whereas
 $$\text{round}\left(\dfrac{2.4*X^2-1.7}{X}\right)=\dfrac{2*X^2-2}{X}.$$
 An important use of \kbd{round} is to get exact results after an approximate
 computation, when theory tells you that the coefficients must be integers.
Variant: Also available are \fun{GEN}{grndtoi}{GEN x, long *e} and
 \fun{GEN}{ground}{GEN x}.

Function: select
Class: basic
Section: programming/specific
C-Name: select0
Prototype: GGD0,L,
Help: select(f, A, {flag = 0}): selects elements of A according to the selection
 function f. If flag is 1, return the indices of those elements (indirect
 selection).
Wrapper: (bG)
Description: 
  (gen,gen):gen    genselect(${1 cookie}, ${1 wrapper}, $2)
  (gen,gen,0):gen  genselect(${1 cookie}, ${1 wrapper}, $2)
  (gen,gen,1):vecsmall  genindexselect(${1 cookie}, ${1 wrapper}, $2)
Doc: We first describe the default behavior, when $\fl$ is 0 or omitted.
 Given a vector or list \kbd{A} and a \typ{CLOSURE} \kbd{f}, \kbd{select}
 returns the elements $x$ of \kbd{A} such that $f(x)$ is non-zero. In other
 words, \kbd{f} is seen as a selection function returning a boolean value.
 \bprog
 ? select(x->isprime(x), vector(50,i,i^2+1))
 %1 = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601]
 ? select(x->(x<100), %)
 %2 = [2, 5, 17, 37]
 @eprog\noindent returns the primes of the form $i^2+1$ for some $i\leq 50$,
 then the elements less than 100 in the preceding result. The \kbd{select}
 function also applies to a matrix \kbd{A}, seen as a vector of columns, i.e. it
 selects columns instead of entries, and returns the matrix whose columns are
 the selected ones.
 
 \misctitle{Remark} For $v$ a \typ{VEC}, \typ{COL}, \typ{LIST} or \typ{MAT},
 the alternative set-notations
 \bprog
 [g(x) | x <- v, f(x)]
 [x | x <- v, f(x)]
 [g(x) | x <- v]
 @eprog\noindent
 are available as shortcuts for
 \bprog
 apply(g, select(f, Vec(v)))
 select(f, Vec(v))
 apply(g, Vec(v))
 @eprog\noindent respectively:
 \bprog
 ? [ x | x <- vector(50,i,i^2+1), isprime(x) ]
 %1 = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601]
 @eprog
 
 \noindent If $\fl = 1$, this function returns instead the \emph{indices} of
 the selected elements, and not the elements themselves (indirect selection):
 \bprog
 ? V = vector(50,i,i^2+1);
 ? select(x->isprime(x), V, 1)
 %2 = Vecsmall([1, 2, 4, 6, 10, 14, 16, 20, 24, 26, 36, 40])
 ? vecextract(V, %)
 %3 = [2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601]
 @eprog\noindent
 The following function lists the elements in $(\Z/N\Z)^*$:
 \bprog
 ? invertibles(N) = select(x->gcd(x,N) == 1, [1..N])
 @eprog
 
 \noindent Finally
 \bprog
 ? select(x->x, M)
 @eprog\noindent selects the non-0 entries in \kbd{M}. If the latter is a
 \typ{MAT}, we extract the matrix of non-0 columns. Note that \emph{removing}
 entries instead of selecting them just involves replacing the selection
 function \kbd{f} with its negation:
 \bprog
 ? select(x->!isprime(x), vector(50,i,i^2+1))
 @eprog
 
 \synt{genselect}{void *E, long (*fun)(void*,GEN), GEN a}. Also available
 is \fun{GEN}{genindexselect}{void *E, long (*fun)(void*, GEN), GEN a},
 corresponding to $\fl = 1$.

Function: self
Class: basic
Section: programming/specific
C-Name: pari_self
Prototype: m
Help: self(): return the calling function or closure. Useful for defining
 anonymous recursive functions.
Doc: return the calling function or closure as a \typ{CLOSURE} object.
 This is useful for defining anonymous recursive functions.
 \bprog
 ? (n->if(n==0,1,n*self()(n-1)))(5)
 %1 = 120
 @eprog

Function: seralgdep
Class: basic
Section: linear_algebra
C-Name: seralgdep
Prototype: GLL
Help: seralgdep(s,p,r): find a linear relation between powers (1,s, ..., s^p)
 of the series s, with polynomial coefficients of degree <= r.
Doc: \sidx{algebraic dependence} finds a linear relation between powers $(1,s,
 \dots, s^p)$ of the series $s$, with polynomial coefficients of degree
 $\leq r$. In case no relation is found, return $0$.
 \bprog
 ? s = 1 + 10*y - 46*y^2 + 460*y^3 - 5658*y^4 + 77740*y^5 + O(y^6);
 ? seralgdep(s, 2, 2)
 %2 = -x^2 + (8*y^2 + 20*y + 1)
 ? subst(%, x, s)
 %3 = O(y^6)
 ? seralgdep(s, 1, 3)
 %4 = (-77*y^2 - 20*y - 1)*x + (310*y^3 + 231*y^2 + 30*y + 1)
 ? seralgdep(s, 1, 2)
 %5 = 0
 @eprog\noindent The series main variable must not be $x$, so as to be able
 to express the result as a polynomial in $x$.

Function: serconvol
Class: basic
Section: polynomials
C-Name: convol
Prototype: GG
Help: serconvol(x,y): convolution (or Hadamard product) of two power series.
Doc: convolution (or \idx{Hadamard product}) of the
 two power series $x$ and $y$; in other words if $x=\sum a_k*X^k$ and $y=\sum
 b_k*X^k$ then $\kbd{serconvol}(x,y)=\sum a_k*b_k*X^k$.

Function: serlaplace
Class: basic
Section: polynomials
C-Name: laplace
Prototype: G
Help: serlaplace(x): replaces the power series sum of a_n*x^n/n! by sum of
 a_n*x^n. For the reverse operation, use serconvol(x,exp(X)).
Doc: $x$ must be a power series with non-negative
 exponents or a polynomial. If $x=\sum (a_k/k!)*X^k$ then the result is $\sum
 a_k*X^k$.

Function: serprec
Class: basic
Section: conversions
C-Name: gpserprec
Prototype: Gn
Help: serprec(x,v):
 return the absolute precision x with respect to power series in the variable v.
Doc: returns the absolute precision of $x$ with respect to power series
 in the variable $v$; this is the
 minimum precision of the components of $x$. The result is \tet{+oo} if $x$
 is an exact object (as a series in $v$):
 \bprog
 ? serprec(x + O(y^2), y)
 %1 = 2
 ? serprec(x + 2, x)
 %2 = +oo
 ? serprec(2 + x + O(x^2), y)
 %3 = +oo
 @eprog
Variant: Also available is \fun{long}{serprec}{GEN x, GEN p}, which returns
 \tet{LONG_MAX} if $x = 0$ and the series precision as a \kbd{long} integer.

Function: serreverse
Class: basic
Section: polynomials
C-Name: serreverse
Prototype: G
Help: serreverse(s): reversion of the power series s.
Doc: reverse power series of $s$, i.e. the series $t$ such that $t(s) = x$;
 $s$ must be a power series whose valuation is exactly equal to one.
 \bprog
 ? \ps 8
 ? t = serreverse(tan(x))
 %2 = x - 1/3*x^3 + 1/5*x^5 - 1/7*x^7 + O(x^8)
 ? tan(t)
 %3 = x + O(x^8)
 @eprog

Function: setbinop
Class: basic
Section: linear_algebra
C-Name: setbinop
Prototype: GGDG
Help: setbinop(f,X,{Y}): the set {f(x,y), x in X, y in Y}. If Y is omitted,
 assume that X = Y and that f is symmetric.
Doc: the set whose elements are the f(x,y), where x,y run through X,Y.
 respectively. If $Y$ is omitted, assume that $X = Y$ and that $f$ is symmetric:
 $f(x,y) = f(y,x)$ for all $x,y$ in $X$.
 \bprog
 ? X = [1,2,3]; Y = [2,3,4];
 ? setbinop((x,y)->x+y, X,Y) \\ set X + Y
 %2 = [3, 4, 5, 6, 7]
 ? setbinop((x,y)->x-y, X,Y) \\ set X - Y
 %3 = [-3, -2, -1, 0, 1]
 ? setbinop((x,y)->x+y, X)   \\ set 2X = X + X
 %2 = [2, 3, 4, 5, 6]
 @eprog

Function: setintersect
Class: basic
Section: linear_algebra
C-Name: setintersect
Prototype: GG
Help: setintersect(x,y): intersection of the sets x and y.
Description: 
 (vec, vec):vec        setintersect($1, $2)
Doc: intersection of the two sets $x$ and $y$ (see \kbd{setisset}).
 If $x$ or $y$ is not a set, the result is undefined.

Function: setisset
Class: basic
Section: linear_algebra
C-Name: setisset
Prototype: lG
Help: setisset(x): true(1) if x is a set (row vector with strictly
 increasing entries), false(0) if not.
Doc: 
 returns true (1) if $x$ is a set, false (0) if
 not. In PARI, a set is a row vector whose entries are strictly
 increasing with respect to a (somewhat arbitrary) universal comparison
 function. To convert any object into a set (this is most useful for
 vectors, of course), use the function \kbd{Set}.
 \bprog
 ? a = [3, 1, 1, 2];
 ? setisset(a)
 %2 = 0
 ? Set(a)
 %3 = [1, 2, 3]
 @eprog

Function: setminus
Class: basic
Section: linear_algebra
C-Name: setminus
Prototype: GG
Help: setminus(x,y): set of elements of x not belonging to y.
Description: 
 (vec, vec):vec        setminus($1, $2)
Doc: difference of the two sets $x$ and $y$ (see \kbd{setisset}),
 i.e.~set of elements of $x$ which do not belong to $y$.
 If $x$ or $y$ is not a set, the result is undefined.

Function: setrand
Class: basic
Section: programming/specific
C-Name: setrand
Prototype: vG
Help: setrand(n): reset the seed of the random number generator to n.
Doc: reseeds the random number generator using the seed $n$. No value is
 returned. The seed is either a technical array output by \kbd{getrand}, or a
 small positive integer, used to generate deterministically a suitable state
 array. For instance, running a randomized computation starting by
 \kbd{setrand(1)} twice will generate the exact same output.

Function: setsearch
Class: basic
Section: linear_algebra
C-Name: setsearch
Prototype: lGGD0,L,
Help: setsearch(S,x,{flag=0}): determines whether x belongs to the set (or
 sorted list) S.
 If flag is 0 or omitted, returns 0 if it does not, otherwise returns the index
 j such that x==S[j]. If flag is non-zero, return 0 if x belongs to S,
 otherwise the index j where it should be inserted.
Doc: determines whether $x$ belongs to the set $S$ (see \kbd{setisset}).
 
 We first describe the default behaviour, when $\fl$ is zero or omitted. If $x$
 belongs to the set $S$, returns the index $j$ such that $S[j]=x$, otherwise
 returns 0.
 \bprog
 ? T = [7,2,3,5]; S = Set(T);
 ? setsearch(S, 2)
 %2 = 1
 ? setsearch(S, 4)      \\ not found
 %3 = 0
 ? setsearch(T, 7)      \\ search in a randomly sorted vector
 %4 = 0 \\ WRONG !
 @eprog\noindent
 If $S$ is not a set, we also allow sorted lists with
 respect to the \tet{cmp} sorting function, without repeated entries,
 as per \tet{listsort}$(L,1)$; otherwise the result is undefined.
 \bprog
 ? L = List([1,4,2,3,2]); setsearch(L, 4)
 %1 = 0 \\ WRONG !
 ? listsort(L, 1); L    \\ sort L first
 %2 = List([1, 2, 3, 4])
 ? setsearch(L, 4)
 %3 = 4                 \\ now correct
 @eprog\noindent
 If $\fl$ is non-zero, this function returns the index $j$ where $x$ should be
 inserted, and $0$ if it already belongs to $S$. This is meant to be used for
 dynamically growing (sorted) lists, in conjunction with \kbd{listinsert}.
 \bprog
 ? L = List([1,5,2,3,2]); listsort(L,1); L
 %1 = List([1,2,3,5])
 ? j = setsearch(L, 4, 1)  \\ 4 should have been inserted at index j
 %2 = 4
 ? listinsert(L, 4, j); L
 %3 = List([1, 2, 3, 4, 5])
 @eprog

Function: setunion
Class: basic
Section: linear_algebra
C-Name: setunion
Prototype: GG
Help: setunion(x,y): union of the sets x and y.
Description: 
 (vec, vec):vec        setunion($1, $2)
Doc: union of the two sets $x$ and $y$ (see \kbd{setisset}).
 If $x$ or $y$ is not a set, the result is undefined.

Function: shift
Class: basic
Section: operators
C-Name: gshift
Prototype: GL
Help: shift(x,n): shift x left n bits if n>=0, right -n bits if
 n<0.
Doc: shifts $x$ componentwise left by $n$ bits if $n\ge0$ and right by $|n|$
 bits if $n<0$. May be abbreviated as $x$ \kbd{<<} $n$ or $x$ \kbd{>>} $(-n)$.
 A left shift by $n$ corresponds to multiplication by $2^n$. A right shift of an
 integer $x$ by $|n|$ corresponds to a Euclidean division of $x$ by $2^{|n|}$
 with a remainder of the same sign as $x$, hence is not the same (in general) as
 $x \kbd{\bs} 2^n$.

Function: shiftmul
Class: basic
Section: operators
C-Name: gmul2n
Prototype: GL
Help: shiftmul(x,n): multiply x by 2^n (n>=0 or n<0).
Doc: multiplies $x$ by $2^n$. The difference with
 \kbd{shift} is that when $n<0$, ordinary division takes place, hence for
 example if $x$ is an integer the result may be a fraction, while for shifts
 Euclidean division takes place when $n<0$ hence if $x$ is an integer the result
 is still an integer.

Function: sigma
Class: basic
Section: number_theoretical
C-Name: sumdivk
Prototype: GD1,L,
Help: sigma(x,{k=1}): sum of the k-th powers of the divisors of x. k is
 optional and if omitted is assumed to be equal to 1.
Description: 
 (gen, ?1):int           sumdiv($1)
 (gen, 0):int            numdiv($1)
Doc: sum of the $k^{\text{th}}$ powers of the positive divisors of $|x|$. $x$
 and $k$ must be of type integer.
Variant: Also available is \fun{GEN}{sumdiv}{GEN n}, for $k = 1$.

Function: sign
Class: basic
Section: operators
C-Name: gsigne
Prototype: iG
Help: sign(x): sign of x, of type integer, real or fraction.
Description: 
 (mp):small          signe($1)
 (gen):small        gsigne($1)
Doc: \idx{sign} ($0$, $1$ or $-1$) of $x$, which must be of
 type integer, real or fraction; \typ{QUAD} with positive discriminants and
 \typ{INFINITY} are also supported.

Function: simplify
Class: basic
Section: conversions
C-Name: simplify
Prototype: G
Help: simplify(x): simplify the object x as much as possible.
Doc: 
 this function simplifies $x$ as much as it can. Specifically, a complex or
 quadratic number whose imaginary part is the integer 0 (i.e.~not \kbd{Mod(0,2)}
 or \kbd{0.E-28}) is converted to its real part, and a polynomial of degree $0$
 is converted to its constant term. Simplifications occur recursively.
 
 This function is especially useful before using arithmetic functions,
 which expect integer arguments:
 \bprog
 ? x = 2 + y - y
 %1 = 2
 ? isprime(x)
   ***   at top-level: isprime(x)
   ***                 ^----------
   *** isprime: not an integer argument in an arithmetic function
 ? type(x)
 %2 = "t_POL"
 ? type(simplify(x))
 %3 = "t_INT"
 @eprog
 Note that GP results are simplified as above before they are stored in the
 history. (Unless you disable automatic simplification with \b{y}, that is.)
 In particular
 \bprog
 ? type(%1)
 %4 = "t_INT"
 @eprog

Function: sin
Class: basic
Section: transcendental
C-Name: gsin
Prototype: Gp
Help: sin(x): sine of x.
Doc: sine of $x$.

Function: sinc
Class: basic
Section: transcendental
C-Name: gsinc
Prototype: Gp
Help: sinc(x): sinc function of x.
Doc: cardinal sine of $x$, i.e. $\sin(x)/x$ if $x\neq 0$, $1$ otherwise.
 Note that this function also allows to compute
 $$(1-\cos(x)) / x^2 = \kbd{sinc}(x/2)^2 / 2$$
 accurately near $x = 0$.

Function: sinh
Class: basic
Section: transcendental
C-Name: gsinh
Prototype: Gp
Help: sinh(x): hyperbolic sine of x.
Doc: hyperbolic sine of $x$.

Function: sizebyte
Class: basic
Section: conversions
C-Name: gsizebyte
Prototype: lG
Help: sizebyte(x): number of bytes occupied by the complete tree of the
 object x.
Doc: outputs the total number of bytes occupied by the tree representing the
 PARI object $x$.
Variant: Also available is \fun{long}{gsizeword}{GEN x} returning a
 number of \emph{words}.

Function: sizedigit
Class: basic
Section: conversions
C-Name: sizedigit
Prototype: lG
Help: sizedigit(x): rough upper bound for the number of decimal digits
 of (the components of) $x$. DEPRECATED.
Doc: 
 This function is DEPRECATED, essentially meaningless, and provided for
 backwards compatibility only. Don't use it!
 
 outputs a quick upper bound for the number of decimal digits of (the
 components of) $x$, off by at most $1$. More precisely, for a positive
 integer $x$, it computes (approximately) the ceiling of
 $$\kbd{floor}(1 + \log_2 x) \log_{10}2,$$
 
 To count the number of decimal digits of a positive integer $x$, use
 \kbd{\#digits(x)}. To estimate (recursively) the size of $x$, use
 \kbd{normlp(x)}.
Obsolete: 2015-01-13

Function: solve
Class: basic
Section: sums
C-Name: zbrent0
Prototype: V=GGEp
Help: solve(X=a,b,expr): real root of expression expr (X between a and b),
 where expr(a)*expr(b)<=0.
Wrapper: (,,G)
Description: 
  (gen,gen,gen):gen:prec zbrent(${3 cookie}, ${3 wrapper}, $1, $2, $prec)
Doc: find a real root of expression
 \var{expr} between $a$ and $b$, under the condition
 $\var{expr}(X=a) * \var{expr}(X=b) \le 0$. (You will get an error message
 \kbd{roots must be bracketed in solve} if this does not hold.)
 This routine uses Brent's method and can fail miserably if \var{expr} is
 not defined in the whole of $[a,b]$ (try \kbd{solve(x=1, 2, tan(x))}).
 
 \synt{zbrent}{void *E,GEN (*eval)(void*,GEN),GEN a,GEN b,long prec}.

Function: solvestep
Class: basic
Section: sums
C-Name: solvestep0
Prototype: V=GGGED0,L,p
Help: solvestep(X=a,b,step,expr,{flag=0}): find zeros of a function in the real
 interval [a,b] by naive interval splitting.
Wrapper: (,,,G)
Description: 
  (gen,gen,gen,gen, ?0$):gen:prec solvestep(${4 cookie}, ${4 wrapper}, $1, $2, $3, $5, $prec)
Doc: find zeros of a continuous function in the real interval $[a,b]$ by naive
 interval splitting. This function is heuristic and may or may not find the
 intended zeros. Binary digits of \fl\ mean
 
 \item 1: return as soon as one zero is found, otherwise return all
 zeros found;
 
 \item 2: refine the splitting until at least one zero is found
 (may loop indefinitely if there are no zeros);
 
 \item 4: do a multiplicative search (we must have $a > 0$ and $\var{step} >
 1$), otherwise an additive search; \var{step} is the multiplicative or
 additive step.
 
 \item 8: refine the splitting until at least one zero is very close to an
 integer.
 
 \bprog
 ? solvestep(X=0,10,1,sin(X^2),1)
 %1 = 1.7724538509055160272981674833411451828
 ? solvestep(X=1,12,2,besselj(4,X),4)
 %2 = [7.588342434..., 11.064709488...]
 @eprog\noindent
 
 \synt{solvestep}{void *E, GEN (*eval)(void*,GEN), GEN a,GEN b, GEN step,long flag,long prec}.

Function: sqr
Class: basic
Section: transcendental
C-Name: gsqr
Prototype: G
Help: sqr(x): square of x. NOT identical to x*x.
Description: 
 (int):int        sqri($1)
 (mp):mp          gsqr($1)
 (gen):gen        gsqr($1)
Doc: square of $x$. This operation is not completely
 straightforward, i.e.~identical to $x * x$, since it can usually be
 computed more efficiently (roughly one-half of the elementary
 multiplications can be saved). Also, squaring a $2$-adic number increases
 its precision. For example,
 \bprog
 ? (1 + O(2^4))^2
 %1 = 1 + O(2^5)
 ? (1 + O(2^4)) * (1 + O(2^4))
 %2 = 1 + O(2^4)
 @eprog\noindent
 Note that this function is also called whenever one multiplies two objects
 which are known to be \emph{identical}, e.g.~they are the value of the same
 variable, or we are computing a power.
 \bprog
 ? x = (1 + O(2^4)); x * x
 %3 = 1 + O(2^5)
 ? (1 + O(2^4))^4
 %4 = 1 + O(2^6)
 @eprog\noindent
 (note the difference between \kbd{\%2} and \kbd{\%3} above).

Function: sqrt
Class: basic
Section: transcendental
C-Name: gsqrt
Prototype: Gp
Help: sqrt(x): square root of x.
Description: 
 (real):gen           sqrtr($1)
 (gen):gen:prec       gsqrt($1, $prec)
Doc: principal branch of the square root of $x$, defined as $\sqrt{x} =
 \exp(\log x / 2)$. In particular, we have
 $\text{Arg}(\text{sqrt}(x))\in{} ]-\pi/2, \pi/2]$, and if $x\in \R$ and $x<0$,
 then the result is complex with positive imaginary part.
 
 Intmod a prime $p$, \typ{PADIC} and \typ{FFELT} are allowed as arguments. In
 the first 2 cases (\typ{INTMOD}, \typ{PADIC}), the square root (if it
 exists) which is returned is the one whose first $p$-adic digit is in the
 interval $[0,p/2]$. For other arguments, the result is undefined.
Variant: For a \typ{PADIC} $x$, the function
 \fun{GEN}{Qp_sqrt}{GEN x} is also available.

Function: sqrtint
Class: basic
Section: number_theoretical
C-Name: sqrtint
Prototype: G
Help: sqrtint(x): integer square root of x, where x is a non-negative integer.
Description: 
 (gen):int sqrtint($1)
Doc: returns the integer square root of $x$, i.e. the largest integer $y$
 such that $y^2 \leq x$, where $x$ a non-negative integer.
 \bprog
 ? N = 120938191237; sqrtint(N)
 %1 = 347761
 ? sqrt(N)
 %2 = 347761.68741970412747602130964414095216
 @eprog

Function: sqrtn
Class: basic
Section: transcendental
C-Name: gsqrtn
Prototype: GGD&p
Help: sqrtn(x,n,{&z}): nth-root of x, n must be integer. If present, z is
 set to a suitable root of unity to recover all solutions. If it was not
 possible, z is set to zero.
Doc: principal branch of the $n$th root of $x$,
 i.e.~such that $\text{Arg}(\text{sqrtn}(x))\in{} ]-\pi/n, \pi/n]$. Intmod
 a prime and $p$-adics are allowed as arguments.
 
 If $z$ is present, it is set to a suitable root of unity allowing to
 recover all the other roots. If it was not possible, z is
 set to zero. In the case this argument is present and no $n$th root exist,
 $0$ is returned instead of raising an error.
 \bprog
 ? sqrtn(Mod(2,7), 2)
 %1 = Mod(3, 7)
 ? sqrtn(Mod(2,7), 2, &z); z
 %2 = Mod(6, 7)
 ? sqrtn(Mod(2,7), 3)
   ***   at top-level: sqrtn(Mod(2,7),3)
   ***                 ^-----------------
   *** sqrtn: nth-root does not exist in gsqrtn.
 ? sqrtn(Mod(2,7), 3,  &z)
 %2 = 0
 ? z
 %3 = 0
 @eprog
 
 The following script computes all roots in all possible cases:
 \bprog
 sqrtnall(x,n)=
 { my(V,r,z,r2);
   r = sqrtn(x,n, &z);
   if (!z, error("Impossible case in sqrtn"));
   if (type(x) == "t_INTMOD" || type(x)=="t_PADIC",
     r2 = r*z; n = 1;
     while (r2!=r, r2*=z;n++));
   V = vector(n); V[1] = r;
   for(i=2, n, V[i] = V[i-1]*z);
   V
 }
 addhelp(sqrtnall,"sqrtnall(x,n):compute the vector of nth-roots of x");
 @eprog\noindent
Variant: If $x$ is a \typ{PADIC}, the function
 \fun{GEN}{Qp_sqrtn}{GEN x, GEN n, GEN *z} is also available.

Function: sqrtnint
Class: basic
Section: number_theoretical
C-Name: sqrtnint
Prototype: GL
Help: sqrtnint(x,n): integer n-th root of x, where x is non-negative integer.
Description: 
 (gen,small):int sqrtnint($1, $2)
Doc: returns the integer $n$-th root of $x$, i.e. the largest integer $y$ such
 that $y^n \leq x$, where $x$ is a non-negative integer.
 \bprog
 ? N = 120938191237; sqrtnint(N, 5)
 %1 = 164
 ? N^(1/5)
 %2 = 164.63140849829660842958614676939677391
 @eprog\noindent The special case $n = 2$ is \tet{sqrtint}

Function: stirling
Class: basic
Section: number_theoretical
C-Name: stirling
Prototype: LLD1,L,
Help: stirling(n,k,{flag=1}): if flag=1 (default) return the Stirling number
 of the first kind s(n,k), if flag=2, return the Stirling number of the second
 kind S(n,k).
Doc: \idx{Stirling number} of the first kind $s(n,k)$ ($\fl=1$, default) or
 of the second kind $S(n,k)$ (\fl=2), where $n$, $k$ are non-negative
 integers. The former is $(-1)^{n-k}$ times the
 number of permutations of $n$ symbols with exactly $k$ cycles; the latter is
 the number of ways of partitioning a set of $n$ elements into $k$ non-empty
 subsets. Note that if all $s(n,k)$ are needed, it is much faster to compute
 $$\sum_k s(n,k) x^k = x(x-1)\dots(x-n+1).$$
 Similarly, if a large number of $S(n,k)$ are needed for the same $k$,
 one should use
 $$\sum_n S(n,k) x^n = \dfrac{x^k}{(1-x)\dots(1-kx)}.$$
 (Should be implemented using a divide and conquer product.) Here are
 simple variants for $n$ fixed:
 \bprog
 /* list of s(n,k), k = 1..n */
 vecstirling(n) = Vec( factorback(vector(n-1,i,1-i*'x)) )
 
 /* list of S(n,k), k = 1..n */
 vecstirling2(n) =
 { my(Q = x^(n-1), t);
   vector(n, i, t = divrem(Q, x-i); Q=t[1]; simplify(t[2]));
 }
 @eprog
Variant: Also available are \fun{GEN}{stirling1}{ulong n, ulong k}
 ($\fl=1$) and \fun{GEN}{stirling2}{ulong n, ulong k} ($\fl=2$).

Function: subgrouplist
Class: basic
Section: number_fields
C-Name: subgrouplist0
Prototype: GDGD0,L,
Help: subgrouplist(bnr,{bound},{flag=0}): bnr being as output by bnrinit or
 a list of cyclic components of a finite Abelian group G, outputs the list of
 subgroups of G (of index bounded by bound, if not omitted), given as HNF
 left divisors of the SNF matrix corresponding to G. If flag=0 (default) and
 bnr is as output by bnrinit, gives only the subgroups for which the modulus
 is the conductor.
Doc: \var{bnr} being as output by \kbd{bnrinit} or a list of cyclic components
 of a finite Abelian group $G$, outputs the list of subgroups of $G$. Subgroups
 are given as HNF left divisors of the SNF matrix corresponding to $G$.
 
 If $\fl=0$ (default) and \var{bnr} is as output by \kbd{bnrinit}, gives
 only the subgroups whose modulus is the conductor. Otherwise, the modulus is
 not taken into account.
 
 If \var{bound} is present, and is a positive integer, restrict the output to
 subgroups of index less than \var{bound}. If \var{bound} is a vector
 containing a single positive integer $B$, then only subgroups of index
 exactly equal to $B$ are computed. For instance
 \bprog
 ? subgrouplist([6,2])
 %1 = [[6, 0; 0, 2], [2, 0; 0, 2], [6, 3; 0, 1], [2, 1; 0, 1], [3, 0; 0, 2],
 [1, 0; 0, 2], [6, 0; 0, 1], [2, 0; 0, 1], [3, 0; 0, 1], [1, 0; 0, 1]]
 ? subgrouplist([6,2],3)    \\@com index less than 3
 %2 = [[2, 1; 0, 1], [1, 0; 0, 2], [2, 0; 0, 1], [3, 0; 0, 1], [1, 0; 0, 1]]
 ? subgrouplist([6,2],[3])  \\@com index 3
 %3 = [[3, 0; 0, 1]]
 ? bnr = bnrinit(bnfinit(x), [120,[1]], 1);
 ? L = subgrouplist(bnr, [8]);
 @eprog\noindent
 In the last example, $L$ corresponds to the 24 subfields of
 $\Q(\zeta_{120})$, of degree $8$ and conductor $120\infty$ (by setting \fl,
 we see there are a total of $43$ subgroups of degree $8$).
 \bprog
 ? vector(#L, i, galoissubcyclo(bnr, L[i]))
 @eprog\noindent
 will produce their equations. (For a general base field, you would
 have to rely on \tet{bnrstark}, or \tet{rnfkummer}.)

Function: subst
Class: basic
Section: polynomials
C-Name: gsubst
Prototype: GnG
Help: subst(x,y,z): in expression x, replace the variable y by the
 expression z.
Doc: replace the simple variable $y$ by the argument $z$ in the ``polynomial''
 expression $x$. Every type is allowed for $x$, but if it is not a genuine
 polynomial (or power series, or rational function), the substitution will be
 done as if the scalar components were polynomials of degree zero. In
 particular, beware that:
 
 \bprog
 ? subst(1, x, [1,2; 3,4])
 %1 =
 [1 0]
 
 [0 1]
 
 ? subst(1, x, Mat([0,1]))
   ***   at top-level: subst(1,x,Mat([0,1])
   ***                 ^--------------------
   *** subst: forbidden substitution by a non square matrix.
 @eprog\noindent
 If $x$ is a power series, $z$ must be either a polynomial, a power
 series, or a rational function. Finally, if $x$ is a vector,
 matrix or list, the substitution is applied to each individual entry.
 
 Use the function \kbd{substvec} to replace several variables at once,
 or the function \kbd{substpol} to replace a polynomial expression.

Function: substpol
Class: basic
Section: polynomials
C-Name: gsubstpol
Prototype: GGG
Help: substpol(x,y,z): in expression x, replace the polynomial y by the
 expression z, using remainder decomposition of x.
Doc: replace the ``variable'' $y$ by the argument $z$ in the ``polynomial''
 expression $x$. Every type is allowed for $x$, but the same behavior
 as \kbd{subst} above apply.
 
 The difference with \kbd{subst} is that $y$ is allowed to be any polynomial
 here. The substitution is done moding out all components of $x$
 (recursively) by $y - t$, where $t$ is a new free variable of lowest
 priority. Then substituting $t$ by $z$ in the resulting expression. For
 instance
 \bprog
 ? substpol(x^4 + x^2 + 1, x^2, y)
 %1 = y^2 + y + 1
 ? substpol(x^4 + x^2 + 1, x^3, y)
 %2 = x^2 + y*x + 1
 ? substpol(x^4 + x^2 + 1, (x+1)^2, y)
 %3 = (-4*y - 6)*x + (y^2 + 3*y - 3)
 @eprog
Variant: Further, \fun{GEN}{gdeflate}{GEN T, long v, long d} attempts to
 write $T(x)$ in the form $t(x^d)$, where $x=$\kbd{pol\_x}$(v)$, and returns
 \kbd{NULL} if the substitution fails (for instance in the example \kbd{\%2}
 above).

Function: substvec
Class: basic
Section: polynomials
C-Name: gsubstvec
Prototype: GGG
Help: substvec(x,v,w): in expression x, make a best effort to replace the
 variables v1,...,vn by the expression w1,...,wn.
Doc: $v$ being a vector of monomials of degree 1 (variables),
 $w$ a vector of expressions of the same length, replace in the expression
 $x$ all occurrences of $v_i$ by $w_i$. The substitutions are done
 simultaneously; more precisely, the $v_i$ are first replaced by new
 variables in $x$, then these are replaced by the $w_i$:
 
 \bprog
 ? substvec([x,y], [x,y], [y,x])
 %1 = [y, x]
 ? substvec([x,y], [x,y], [y,x+y])
 %2 = [y, x + y]     \\ not [y, 2*y]
 @eprog

Function: sum
Class: basic
Section: sums
C-Name: somme
Prototype: V=GGEDG
Help: sum(X=a,b,expr,{x=0}): x plus the sum (X goes from a to b) of
 expression expr.
Doc: sum of expression \var{expr},
 initialized at $x$, the formal parameter going from $a$ to $b$. As for
 \kbd{prod}, the initialization parameter $x$ may be given to force the type
 of the operations being performed.
 
 \noindent As an extreme example, compare
 
 \bprog
 ? sum(i=1, 10^4, 1/i); \\@com rational number: denominator has $4345$ digits.
 time = 236 ms.
 ? sum(i=1, 5000, 1/i, 0.)
 time = 8 ms.
 %2 = 9.787606036044382264178477904
 @eprog
 
 \synt{somme}{GEN a, GEN b, char *expr, GEN x}.

Function: sumalt
Class: basic
Section: sums
C-Name: sumalt0
Prototype: V=GED0,L,p
Help: sumalt(X=a,expr,{flag=0}): Cohen-Villegas-Zagier's acceleration of
 alternating series expr, X starting at a. flag is optional, and can be 0:
 default, or 1: uses a slightly different method using Zagier's polynomials.
Wrapper: (,G)
Description: 
  (gen,gen,?0):gen:prec sumalt(${2 cookie}, ${2 wrapper}, $1, $prec)
  (gen,gen,1):gen:prec sumalt2(${2 cookie}, ${2 wrapper}, $1, $prec)
Doc: numerical summation of the series \var{expr}, which should be an
 \idx{alternating series} $(-1)^k a_k$, the formal variable $X$ starting at
 $a$. Use an algorithm of Cohen, Villegas and Zagier (\emph{Experiment. Math.}
 {\bf 9} (2000), no.~1, 3--12).
 
 If $\fl=0$, assuming that the $a_k$ are the moments of a positive
 measure on $[0,1]$, the relative error is $O(3+\sqrt8)^{-n}$ after using
 $a_k$ for $k\leq n$. If \kbd{realprecision} is $p$, we thus set
 $n = \log(10)p/\log(3+\sqrt8)\approx 1.3 p$; besides the time needed to
 compute the $a_k$, $k\leq n$, the algorithm overhead is negligible: time
 $O(p^2)$ and space $O(p)$.
 
 If $\fl=1$, use a variant with more complicated polynomials, see
 \tet{polzagier}. If the $a_k$ are the moments of $w(x)dx$ where $w$
 (or only $xw(x^2)$) is a smooth function extending analytically to the whole
 complex plane, convergence is in $O(14.4^{-n})$. If $xw(x^2)$ extends
 analytically to a smaller region, we still have exponential convergence,
 with worse constants. Usually faster when the computation of $a_k$ is
 expensive. If \kbd{realprecision} is $p$, we thus set
 $n = \log(10)p/\log(14.4)\approx 0.86 p$; besides the time needed to
 compute the $a_k$, $k\leq n$, the algorithm overhead is \emph{not}
 negligible: time $O(p^3)$ and space $O(p^2)$. Thus, even if the analytic
 conditions for rigorous use are met, this variant is only worthwile if the
 $a_k$ are hard to compute, at least $O(p^2)$ individually on average:
 otherwise we gain a small constant factor (1.5, say) in the number of
 needed $a_k$ at the expense of a large overhead.
 
 The conditions for rigorous use are hard to check but the routine is best used
 heuristically: even divergent alternating series can sometimes be summed by
 this method, as well as series which are not exactly alternating (see for
 example \secref{se:user_defined}). It should be used to try and guess the
 value of an infinite sum. (However, see the example at the end of
 \secref{se:userfundef}.)
 
 If the series already converges geometrically,
 \tet{suminf} is often a better choice:
 \bprog
 ? \p28
 ? sumalt(i = 1, -(-1)^i / i)  - log(2)
 time = 0 ms.
 %1 = -2.524354897 E-29
 ? suminf(i = 1, -(-1)^i / i)   \\@com Had to hit \kbd{C-C}
   ***   at top-level: suminf(i=1,-(-1)^i/i)
   ***                                ^------
   *** suminf: user interrupt after 10min, 20,100 ms.
 ? \p1000
 ? sumalt(i = 1, -(-1)^i / i)  - log(2)
 time = 90 ms.
 %2 = 4.459597722 E-1002
 
 ? sumalt(i = 0, (-1)^i / i!) - exp(-1)
 time = 670 ms.
 %3 = -4.03698781490633483156497361352190615794353338591897830587 E-944
 ? suminf(i = 0, (-1)^i / i!) - exp(-1)
 time = 110 ms.
 %4 = -8.39147638 E-1000   \\ @com faster and more accurate
 @eprog
 
 \synt{sumalt}{void *E, GEN (*eval)(void*,GEN),GEN a,long prec}. Also
 available is \tet{sumalt2} with the same arguments ($\fl = 1$).

Function: sumdedekind
Class: basic
Section: number_theoretical
C-Name: sumdedekind
Prototype: GG
Help: sumdedekind(h,k): Dedekind sum attached to h,k.
Doc: returns the \idx{Dedekind sum} attached to the integers $h$ and $k$,
  corresponding to a fast implementation of
  \bprog
   s(h,k) = sum(n = 1, k-1, (n/k)*(frac(h*n/k) - 1/2))
  @eprog

Function: sumdigits
Class: basic
Section: number_theoretical
C-Name: sumdigits0
Prototype: GDG
Help: sumdigits(n,{B=10}): sum of digits in the integer n, when written in
 base B.
Doc: sum of digits in the integer $n$, when written in base $B > 1$.
 \bprog
 ? sumdigits(123456789)
 %1 = 45
 ? sumdigits(123456789, 2)
 %1 = 16
 @eprog\noindent Note that the sum of bits in $n$ is also returned by
 \tet{hammingweight}. This function is much faster than
 \kbd{vecsum(digits(n,B))} when $B$ is $10$ or a power of $2$, and only
 slightly faster in other cases.
Variant: Also available is \fun{GEN}{sumdigits}{GEN n}, for $B = 10$.

Function: sumdiv
Class: basic
Section: sums
C-Name: sumdivexpr
Prototype: GVE
Help: sumdiv(n,X,expr): sum of expression expr, X running over the divisors
 of n.
Doc: sum of expression \var{expr} over the positive divisors of $n$.
 This function is a trivial wrapper essentially equivalent to
 \bprog
   D = divisors(n);
   for (i = 1, #D, X = D[i]; eval(expr))
 @eprog\noindent (except that \kbd{X} is lexically scoped to the \kbd{sumdiv}
 loop). If \var{expr} is a multiplicative function, use \tet{sumdivmult}.
 %\syn{NO}

Function: sumdivmult
Class: basic
Section: sums
C-Name: sumdivmultexpr
Prototype: GVE
Help: sumdivmult(n,d,expr): sum of multiplicative function expr,
 d running over the divisors of n.
Doc: sum of \emph{multiplicative} expression \var{expr} over the positive
 divisors $d$ of $n$. Assume that \var{expr} evaluates to $f(d)$
 where $f$ is multiplicative: $f(1) = 1$ and $f(ab) = f(a)f(b)$ for coprime
 $a$ and $b$.
 %\syn{NO}

Function: sumformal
Class: basic
Section: polynomials
C-Name: sumformal
Prototype: GDn
Help: sumformal(f,{v}): formal sum of f with respect to v, or to the
 main variable of f if v is omitted.
Doc: \idx{formal sum} of the polynomial expression $f$ with respect to the
 main variable if $v$ is omitted, with respect to the variable $v$ otherwise;
 it is assumed that the base ring has characteristic zero. In other words,
 considering $f$ as a polynomial function in the variable $v$,
 returns $F$, a polynomial in $v$ vanishing at $0$, such that $F(b) - F(a)
 = sum_{v = a+1}^b f(v)$:
 \bprog
 ? sumformal(n)  \\ 1 + ... + n
 %1 = 1/2*n^2 + 1/2*n
 ? f(n) = n^3+n^2+1;
 ? F = sumformal(f(n))  \\ f(1) + ... + f(n)
 %3 = 1/4*n^4 + 5/6*n^3 + 3/4*n^2 + 7/6*n
 ? sum(n = 1, 2000, f(n)) == subst(F, n, 2000)
 %4 = 1
 ? sum(n = 1001, 2000, f(n)) == subst(F, n, 2000) - subst(F, n, 1000)
 %5 = 1
 ? sumformal(x^2 + x*y + y^2, y)
 %6 = y*x^2 + (1/2*y^2 + 1/2*y)*x + (1/3*y^3 + 1/2*y^2 + 1/6*y)
 ? x^2 * y + x * sumformal(y) + sumformal(y^2) == %
 %7 = 1
 @eprog

Function: suminf
Class: basic
Section: sums
C-Name: suminf0
Prototype: V=GEp
Help: suminf(X=a,expr): infinite sum (X goes from a to infinity) of real or
 complex expression expr.
Wrapper: (,G)
Description: 
  (gen,gen):gen:prec suminf(${2 cookie}, ${2 wrapper}, $1, $prec)
Doc: \idx{infinite sum} of expression
 \var{expr}, the formal parameter $X$ starting at $a$. The evaluation stops
 when the relative error of the expression is less than the default precision
 for 3 consecutive evaluations. The expressions must always evaluate to a
 complex number.
 
 If the series converges slowly, make sure \kbd{realprecision} is low (even 28
 digits may be too much). In this case, if the series is alternating or the
 terms have a constant sign, \tet{sumalt} and \tet{sumpos} should be used
 instead.
 
 \bprog
 ? \p28
 ? suminf(i = 1, -(-1)^i / i)   \\@com Had to hit \kbd{C-C}
   ***   at top-level: suminf(i=1,-(-1)^i/i)
   ***                                ^------
   *** suminf: user interrupt after 10min, 20,100 ms.
 ? sumalt(i = 1, -(-1)^i / i) - log(2)
 time = 0 ms.
 %1 = -2.524354897 E-29
 @eprog
 
 \synt{suminf}{void *E, GEN (*eval)(void*,GEN), GEN a, long prec}.

Function: sumnum
Class: basic
Section: sums
C-Name: sumnum0
Prototype: V=GEDGp
Help: sumnum(n=a,f,{tab}): numerical summation of f(n) from
 n = a to +infinity using Euler-MacLaurin summation. Assume that f
 corresponds to a series with positive terms and is a C^oo function; a
 must be an integer, and tab, if given, is the output of sumnuminit.
Wrapper: (,G)
Description: 
  (gen,gen,?gen):gen:prec sumnum(${2 cookie}, ${2 wrapper}, $1, $3, $prec)
Doc: Numerical summation of $f(n)$ at high accuracy using Euler-MacLaurin,
 the variable $n$ taking values from $a$ to $+\infty$, where $f$ is assumed to
 have positive values and is a $C^\infty$ function; \kbd{a} must be an integer
 and \kbd{tab}, if given, is the output of \kbd{sumnuminit}. The latter
 precomputes abcissas and weights, speeding up the computation; it also allows
 to specify the behaviour at infinity via \kbd{sumnuminit([+oo, asymp])}.
 \bprog
 ? \p500
 ? z3 = zeta(3);
 ? sumpos(n = 1, n^-3) - z3
 time = 2,332 ms.
 %2 = 2.438468843 E-501
 ? sumnum(n = 1, n^-3) - z3 \\ here slower than sumpos
 time = 2,752 ms.
 %3 = 0.E-500
 @eprog
 
 \misctitle{Complexity}
 The function $f$ will be evaluated at $O(D \log D)$ real arguments,
 where $D \approx \kbd{realprecision} \cdot \log(10)$. The routine is geared
 towards slowly decreasing functions: if $f$ decreases exponentially fast,
 then one of \kbd{suminf} or \kbd{sumpos} should be preferred.
 If $f$ satisfies the stronger hypotheses required for Monien summation,
 i.e. if $f(1/z)$ is holomorphic in a complex neighbourhood of $[0,1]$,
 then \tet{sumnummonien} will be faster since it only requires $O(D/\log D)$
 evaluations:
 \bprog
 ? sumnummonien(n = 1, 1/n^3) - z3
 time = 1,985 ms.
 %3 = 0.E-500
 @eprog\noindent The \kbd{tab} argument precomputes technical data
 not depending on the expression being summed and valid for a given accuracy,
 speeding up immensely later calls:
 \bprog
 ? tab = sumnuminit();
 time = 2,709 ms.
 ? sumnum(n = 1, 1/n^3, tab) - z3 \\ now much faster than sumpos
 time = 40 ms.
 %5 = 0.E-500
 
 ? tabmon = sumnummonieninit(); \\ Monien summation allows precomputations too
 time = 1,781 ms.
 ? sumnummonien(n = 1, 1/n^3, tabmon) - z3
 time = 2 ms.
 %7 = 0.E-500
 @eprog\noindent The speedup due to precomputations becomes less impressive
 when the function $f$ is expensive to evaluate, though:
 \bprog
 ? sumnum(n = 1, lngamma(1+1/n)/n, tab);
 time = 14,180 ms.
 
 ? sumnummonien(n = 1, lngamma(1+1/n)/n, tabmon); \\ fewer evaluations
 time = 717 ms.
 @eprog
 
 \misctitle{Behaviour at infinity}
 By default, \kbd{sumnum} assumes that \var{expr} decreases slowly at infinity,
 but at least like $O(n^{-2})$. If the function decreases like $n^{\alpha}$
 for some $-2 < \alpha < -1$, then it must be indicated via
 \bprog
   tab = sumnuminit([+oo, alpha]); /* alpha < 0 slow decrease */
 @eprog\noindent otherwise loss of accuracy is expected.
 If the functions decreases quickly, like $\exp(-\alpha n)$ for some
 $\alpha > 0$, then it must be indicated via
 \bprog
   tab = sumnuminit([+oo, alpha]); /* alpha  > 0 exponential decrease */
 @eprog\noindent otherwise exponent overflow will occur.
 \bprog
 ? sumnum(n=1,2^-n)
  ***   at top-level: sumnum(n=1,2^-n)
  ***                             ^----
  *** _^_: overflow in expo().
 ? tab = sumnuminit([+oo,log(2)]); sumnum(n=1,2^-n, tab)
 %1 = 1.000[...]
 @eprog
 
 As a shortcut, one can also input
 \bprog
   sumnum(n = [a, asymp], f)
 @eprog\noindent instead of
 \bprog
   tab = sumnuminit(asymp);
   sumnum(n = a, f, tab)
 @eprog
 
 \misctitle{Further examples}
 \bprog
 ? \p200
 ? sumnum(n = 1, n^(-2)) - zeta(2) \\ accurate, fast
 time = 200 ms.
 %1 = -2.376364457868949779 E-212
 ? sumpos(n = 1, n^(-2)) - zeta(2)  \\ even faster
 time = 96 ms.
 %2 = 0.E-211
 ? sumpos(n=1,n^(-4/3)) - zeta(4/3)   \\ now much slower
 time = 13,045 ms.
 %3 = -9.980730723049589073 E-210
 ? sumnum(n=1,n^(-4/3)) - zeta(4/3)  \\ fast but inaccurate
 time = 365 ms.
 %4 = -9.85[...]E-85
 ? sumnum(n=[1,-4/3],n^(-4/3)) - zeta(4/3) \\ with decrease rate, now accurate
 time = 416 ms.
 %5 = -4.134874156691972616 E-210
 
 ? tab = sumnuminit([+oo,-4/3]);
 time = 196 ms.
 ? sumnum(n=1, n^(-4/3), tab) - zeta(4/3) \\ faster with precomputations
 time = 216 ms.
 %5 = -4.134874156691972616 E-210
 ? sumnum(n=1,-log(n)*n^(-4/3), tab) - zeta'(4/3)
 time = 321 ms.
 %7 = 7.224147951921607329 E-210
 @eprog
 
 Note that in the case of slow decrease ($\alpha < 0$), the exact
 decrease rate must be indicated, while in the case of exponential decrease,
 a rough value will do. In fact, for exponentially decreasing functions,
 \kbd{sumnum} is given for completeness and comparison purposes only: one
 of \kbd{suminf} or \kbd{sumpos} should always be preferred.
 \bprog
 ? sumnum(n=[1, 1], 2^-n) \\ pretend we decrease as exp(-n)
 time = 240 ms.
 %8 = 1.000[...] \\ perfect
 ? sumpos(n=1, 2^-n)
 %9 = 1.000[...] \\ perfect and instantaneous
 @eprog
 
 \synt{sumnum}{(void *E, GEN (*eval)(void*, GEN), GEN a, GEN tab, long prec)}
 where an omitted \var{tab} is coded as \kbd{NULL}.

Function: sumnuminit
Class: basic
Section: sums
C-Name: sumnuminit
Prototype: DGp
Help: sumnuminit({asymp}): initialize tables for Euler-MacLaurin delta
 summation of a series with positive terms.
Doc: initialize tables for Euler--MacLaurin delta summation of a series with
 positive terms. If given, \kbd{asymp} is of the form $[\kbd{+oo}, \alpha]$,
 as in \tet{intnum} and indicates the decrease rate at infinity of functions
 to be summed. A positive
 $\alpha > 0$ encodes an exponential decrease of type $\exp(-\alpha n)$ and
 a negative $-2 < \alpha < -1$ encodes a slow polynomial decrease of type
 $n^{\alpha}$.
 \bprog
 ? \p200
 ? sumnum(n=1, n^-2);
 time = 200 ms.
 ? tab = sumnuminit();
 time = 188 ms.
 ? sumnum(n=1, n^-2, tab); \\ faster
 time = 8 ms.
 
 ? tab = sumnuminit([+oo, log(2)]); \\ decrease like 2^-n
 time = 200 ms.
 ? sumnum(n=1, 2^-n, tab)
 time = 44 ms.
 
 ? tab = sumnuminit([+oo, -4/3]); \\ decrease like n^(-4/3)
 time = 200 ms.
 ? sumnum(n=1, n^(-4/3), tab);
 time = 221 ms.
 @eprog

Function: sumnummonien
Class: basic
Section: sums
C-Name: sumnummonien0
Prototype: V=GEDGp
Help: sumnummonien(n=a,f,{tab}): numerical summation from
 n = a to +infinity using Monien summation.
Wrapper: (,G)
Description: 
  (gen,gen,?gen):gen:prec sumnummonien(${2 cookie}, ${2 wrapper}, $1, $3, $prec)
Doc: numerical summation $\sum_{n\geq a} f(n)$ at high accuracy, the variable
 $n$ taking values from the integer $a$ to $+\infty$ using Monien summation,
 which assumes that $f(1/z)$ has a complex analytic continuation in a (complex)
 neighbourhood of the segment $[0,1]$.
 
 The function $f$ is evaluated at $O(D / \log D)$ real arguments,
 where $D \approx \kbd{realprecision} \cdot \log(10)$.
 By default, assume that $f(n) = O(n^{-2})$ and has a non-zero asymptotic
 expansion
 $$f(n) = \sum_{i\geq 2} a_i n^{-i}$$
 at infinity. To handle more complicated behaviours and allow time-saving
 precomputations (for a given \kbd{realprecision}), see \kbd{sumnummonieninit}.

Function: sumnummonieninit
Class: basic
Section: sums
C-Name: sumnummonieninit
Prototype: DGDGDGp
Help: sumnummonieninit({asymp},{w},{n0 = 1}): initialize tables for Monien summation of a series with positive terms.
Doc: initialize tables for Monien summation of a series $\sum_{n\geq n_0}
 f(n)$ where $f(1/z)$ has a complex analytic continuation in a (complex)
 neighbourhood of the segment $[0,1]$.
 
 By default, assume that $f(n) = O(n^{-2})$ and has a non-zero asymptotic
 expansion
 $$f(n) = \sum_{i\geq 2} a_i / n^i$$
 at infinity. Note that the sum starts at $i = 2$! The argument \kbd{asymp}
 allows to specify different expansions:
 
 \item a real number $\alpha > 1$ means
  $$f(n) = \sum_{i\geq 1} a_i / n^{\alpha i}$$
 (Now the summation starts at $1$.)
 
 \item a vector $[\alpha,\beta]$ of reals, where we must have $\alpha > 0$
 and $\alpha + \beta > 1$ to ensure convergence, means that
  $$f(n) = \sum_{i\geq 1} a_i / n^{\alpha i + \beta}$$
 Note that $\kbd{asymp} = [\alpha, \alpha]$ is equivalent to
 $\kbd{asymp}=\alpha$.
 
 \bprog
 ? \p57
 ? s = sumnum(n = 1, sin(1/sqrt(n)) / n); \\ reference point
 
 ? \p38
 ? sumnummonien(n = 1, sin(1/sqrt(n)) / n) - s
 %2 = -0.001[...] \\ completely wrong
 
 ? t = sumnummonieninit([1,1/2]);  \\ f(n) = sum_i 1 / n^(i/2+1)
 ? sumnummonien(n = 1, sin(1/sqrt(n)) / n, t) - s
 %3 = 0.E-37 \\ now correct
 @eprog\noindent (As a matter of fact, in the above summation, the
 result given by \kbd{sumnum} at \kbd{\bs p38} is slighly incorrect,
 so we had to increase the accuracy to \kbd{\bs p57}.)
 
 The argument $w$ is used to sum expressions of the form
 $$ \sum_{n\geq n_0} f(n) w(n),$$
 for varying $f$ \emph{as above}, and fixed weight function $w$, where we
 further assume that the auxiliary sums
 $$g_w(m) = \sum_{n\geq n_0} w(n) / n^{\alpha m + \beta} $$
 converge for all $m\geq 1$. Note that for non-negative integers $k$,
 and weight $w(n) = (\log n)^k$, the function $g_w(m) = \zeta^{(k)}(\alpha m +
 \beta)$ has a simple expression; for general weights, $g_w$ is
 computed using \kbd{sumnum}. The following variants are available
 
 \item an integer $k \geq 0$, to code $w(n) = (\log n)^k$;
 only the cases $k = 0,1$ are presently implemented; due to a poor
 implementation of $\zeta$ derivatives, it is not currently worth it
 to exploit the special shape of $g_w$ when $k > 0$;
 
 \item a \typ{CLOSURE} computing the values $w(n)$, where we
 assume that $w(n) = O(n^\epsilon)$ for all $\epsilon > 0$;
 
 \item a vector $[w, \kbd{fast}]$, where $w$ is a closure as above
 and \kbd{fast} is a scalar;
 we assume that $w(n) = O(n^{\kbd{fast}+\epsilon})$; note that
 $\kbd{w} = [w, 0]$ is equivalent to $\kbd{w} = w$.
 
 \item a vector $[w, \kbd{oo}]$, where $w$ is a closure as above;
 we assume that $w(n)$ decreases exponentially. Note that in this case,
 \kbd{sumnummonien} is provided for completeness and comparison purposes only:
 one of \kbd{suminf} or \kbd{sumpos} should be preferred in practice.
 
 The cases where $w$ is a closure or $w(n) = \log n$ are the only ones where
 $n_0$ is taken into account and stored in the result. The subsequent call to
 \kbd{sumnummonien} \emph{must} use the same value.
 
 \bprog
 ? \p300
 ? sumnummonien(n = 1, n^-2*log(n)) + zeta'(2)
 time = 536 ms.
 %1 = -1.323[...]E-6 \\ completely wrong, f does not satisfy hypotheses !
 ? tab = sumnummonieninit(, 1); \\ codes w(n) = log(n)
 time = 18,316 ms.
 ? sumnummonien(n = 1, n^-2, tab) + zeta'(2)
 time = 44 ms.
 %3 = -5.562684646268003458 E-309  \\ now perfect
 
 ? tab = sumnummonieninit(, n->log(n)); \\ generic, about as fast
 time = 18,693 ms.
 ? sumnummonien(n = 1, n^-2, tab) + zeta'(2)
 time = 40 ms.
 %5 = -5.562684646268003458 E-309  \\ identical result
 @eprog

Function: sumpos
Class: basic
Section: sums
C-Name: sumpos0
Prototype: V=GED0,L,p
Help: sumpos(X=a,expr,{flag=0}): sum of positive (or negative) series expr,
 the formal
 variable X starting at a. flag is optional, and can be 0: default, or 1:
 uses a slightly different method using Zagier's polynomials.
Wrapper: (,G)
Description: 
  (gen,gen,?0):gen:prec sumpos(${2 cookie}, ${2 wrapper}, $1, $prec)
  (gen,gen,1):gen:prec sumpos2(${2 cookie}, ${2 wrapper}, $1, $prec)
Doc: numerical summation of the series \var{expr}, which must be a series of
 terms having the same sign, the formal variable $X$ starting at $a$. The
 algorithm used is Van Wijngaarden's trick for converting such a series into
 an alternating one, then we use \tet{sumalt}. For regular functions, the
 function \kbd{sumnum} is in general much faster once the initializations
 have been made using \kbd{sumnuminit}.
 
 The routine is heuristic and assumes that \var{expr} is more or less a
 decreasing function of $X$. In particular, the result will be completely
 wrong if \var{expr} is 0 too often. We do not check either that all terms
 have the same sign. As \tet{sumalt}, this function should be used to
 try and guess the value of an infinite sum.
 
 If $\fl=1$, use \kbd{sumalt}$(,1)$ instead of \kbd{sumalt}$(,0)$, see
 \secref{se:sumalt}. Requiring more stringent analytic properties for
 rigorous use, but allowing to compute fewer series terms.
 
 To reach accuracy $10^{-p}$, both algorithms require $O(p^2)$ space;
 furthermore, assuming the terms decrease polynomially (in $O(n^{-C})$), both
 need to compute $O(p^2)$ terms. The \kbd{sumpos}$(,1)$ variant has a smaller
 implied constant (roughly 1.5 times smaller). Since the \kbd{sumalt}$(,1)$
 overhead is now small compared to the time needed to compute series terms,
 this last variant should be about 1.5 faster. On the other hand, the
 achieved accuracy may be much worse: as for \tet{sumalt}, since
 conditions for rigorous use are hard to check, the routine is best used
 heuristically.
 
 \synt{sumpos}{void *E, GEN (*eval)(void*,GEN),GEN a,long prec}. Also
 available is \tet{sumpos2} with the same arguments ($\fl = 1$).

Function: system
Class: basic
Section: programming/specific
C-Name: gpsystem
Prototype: vs
Help: system(str): str being a string, execute the system command str.
Doc: \var{str} is a string representing a system command. This command is
 executed, its output written to the standard output (this won't get into your
 logfile), and control returns to the PARI system. This simply calls the C
 \kbd{system} command.

Function: tan
Class: basic
Section: transcendental
C-Name: gtan
Prototype: Gp
Help: tan(x): tangent of x.
Doc: tangent of $x$.

Function: tanh
Class: basic
Section: transcendental
C-Name: gtanh
Prototype: Gp
Help: tanh(x): hyperbolic tangent of x.
Doc: hyperbolic tangent of $x$.

Function: taylor
Class: basic
Section: polynomials
C-Name: tayl
Prototype: GnDP
Help: taylor(x,t,{d=seriesprecision}): taylor expansion of x with respect to
 t, adding O(t^d) to all components of x.
Doc: Taylor expansion around $0$ of $x$ with respect to
 the simple variable $t$. $x$ can be of any reasonable type, for example a
 rational function. Contrary to \tet{Ser}, which takes the valuation into
 account, this function adds $O(t^d)$ to all components of $x$.
 \bprog
 ? taylor(x/(1+y), y, 5)
 %1 = (y^4 - y^3 + y^2 - y + 1)*x + O(y^5)
 ? Ser(x/(1+y), y, 5)
  ***   at top-level: Ser(x/(1+y),y,5)
  ***                 ^----------------
  *** Ser: main variable must have higher priority in gtoser.
 @eprog

Function: teichmuller
Class: basic
Section: transcendental
C-Name: teichmuller
Prototype: GDG
Help: teichmuller(x,{tab}): teichmuller character of p-adic number x. If
 x = [p,n], return the lifts of all teichmuller(i + O(p^n)) for
 i = 1, ..., p-1. Such a vector can be fed back to teichmuller, as the
 optional argument tab, to speed up later computations.
Doc: Teichm\"uller character of the $p$-adic number $x$, i.e. the unique
 $(p-1)$-th root of unity congruent to $x / p^{v_p(x)}$ modulo $p$.
 If $x$ is of the form $[p,n]$, for a prime $p$ and integer $n$,
 return the lifts to $\Z$ of the images of $i + O(p^n)$ for
 $i = 1, \dots, p-1$, i.e. all roots of $1$ ordered  by residue class modulo
 $p$. Such a vector can be fed back to \kbd{teichmuller}, as the
 optional argument \kbd{tab}, to speed up later computations.
 
 \bprog
 ? z = teichmuller(2 + O(101^5))
 %1 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5)
 ? z^100
 %2 = 1 + O(101^5)
 ? T = teichmuller([101, 5]);
 ? teichmuller(2 + O(101^5), T)
 %4 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5)
 @eprog\noindent As a rule of thumb, if more than
 $$p \,/\, 2(\log_2(p) + \kbd{hammingweight}(p))$$
 values of \kbd{teichmuller} are to be computed, then it is worthwile to
 initialize:
 \bprog
 ? p = 101; n = 100; T = teichmuller([p,n]); \\ instantaneous
 ? for(i=1,10^3, vector(p-1, i, teichmuller(i+O(p^n), T)))
 time = 60 ms.
 ? for(i=1,10^3, vector(p-1, i, teichmuller(i+O(p^n))))
 time = 1,293 ms.
 ? 1 + 2*(log(p)/log(2) + hammingweight(p))
 %8 = 22.316[...]
 @eprog\noindent Here the precompuation induces a speedup by a factor
 $1293/ 60 \approx 21.5$.
 
 \misctitle{Caveat}
 If the accuracy of \kbd{tab} (the argument $n$ above) is lower than the
 precision of $x$, the \emph{former} is used, i.e. the cached value is not
 refined to higher accuracy. It the accuracy of \kbd{tab} is larger, then
 the precision of $x$ is used:
 \bprog
 ? Tlow = teichmuller([101, 2]); \\ lower accuracy !
 ? teichmuller(2 + O(101^5), Tlow)
 %10 = 2 + 83*101 + O(101^5)  \\ no longer a root of 1
 
 ? Thigh = teichmuller([101, 10]); \\ higher accuracy
 ? teichmuller(2 + O(101^5), Thigh)
 %12 = 2 + 83*101 + 18*101^2 + 69*101^3 + 62*101^4 + O(101^5)
 @eprog
Variant: 
 Also available are the functions \fun{GEN}{teich}{GEN x} (\kbd{tab} is
 \kbd{NULL}) as well as
 \fun{GEN}{teichmullerinit}{long p, long n}.

Function: theta
Class: basic
Section: transcendental
C-Name: theta
Prototype: GGp
Help: theta(q,z): Jacobi sine theta-function.
Doc: Jacobi sine theta-function
 $$ \theta_1(z, q) = 2q^{1/4} \sum_{n\geq 0} (-1)^n q^{n(n+1)} \sin((2n+1)z).$$

Function: thetanullk
Class: basic
Section: transcendental
C-Name: thetanullk
Prototype: GLp
Help: thetanullk(q,k): k-th derivative at z=0 of theta(q,z).
Doc: $k$-th derivative at $z=0$ of $\kbd{theta}(q,z)$.
Variant: 
 \fun{GEN}{vecthetanullk}{GEN q, long k, long prec} returns the vector
 of all $\dfrac{d^i\theta}{dz^i}(q,0)$ for all odd $i = 1, 3, \dots, 2k-1$.
 \fun{GEN}{vecthetanullk_tau}{GEN tau, long k, long prec} returns
 \kbd{vecthetanullk\_tau} at $q = \exp(2i\pi \kbd{tau})$.

Function: thue
Class: basic
Section: polynomials
C-Name: thue
Prototype: GGDG
Help: thue(tnf,a,{sol}): solve the equation P(x,y)=a, where tnf was created
 with thueinit(P), and sol, if present, contains the solutions of Norm(x)=a
 modulo units in the number field defined by P. If tnf was computed without
 assuming GRH (flag 1 in thueinit), the result is unconditional. If tnf is a
 polynomial, compute thue(thueinit(P,0), a).
Doc: returns all solutions of the equation
 $P(x,y)=a$ in integers $x$ and $y$, where \var{tnf} was created with
 $\kbd{thueinit}(P)$. If present, \var{sol} must contain the solutions of
 $\Norm(x)=a$ modulo units of positive norm in the number field
 defined by $P$ (as computed by \kbd{bnfisintnorm}). If there are infinitely
 many solutions, an error is issued.
 
 It is allowed to input directly the polynomial $P$ instead of a \var{tnf},
 in which case, the function first performs \kbd{thueinit(P,0)}. This is
 very wasteful if more than one value of $a$ is required.
 
 If \var{tnf} was computed without assuming GRH (flag $1$ in \tet{thueinit}),
 then the result is unconditional. Otherwise, it depends in principle of the
 truth of the GRH, but may still be unconditionally correct in some
 favorable cases. The result is conditional on the GRH if
 $a\neq \pm 1$ and, $P$ has a single irreducible rational factor, whose
 attached tentative class number $h$ and regulator $R$ (as computed
 assuming the GRH) satisfy
 
 \item $h > 1$,
 
 \item $R/0.2 > 1.5$.
 
 Here's how to solve the Thue equation $x^{13} - 5y^{13} = - 4$:
 \bprog
 ? tnf = thueinit(x^13 - 5);
 ? thue(tnf, -4)
 %1 = [[1, 1]]
 @eprog\noindent In this case, one checks that \kbd{bnfinit(x\pow13 -5).no}
 is $1$. Hence, the only solution is $(x,y) = (1,1)$, and the result is
 unconditional. On the other hand:
 \bprog
 ? P = x^3-2*x^2+3*x-17; tnf = thueinit(P);
 ? thue(tnf, -15)
 %2 = [[1, 1]]  \\ a priori conditional on the GRH.
 ? K = bnfinit(P); K.no
 %3 = 3
 ? K.reg
 %4 = 2.8682185139262873674706034475498755834
 @eprog
 This time the result is conditional. All results computed using this
 particular \var{tnf} are likewise conditional, \emph{except} for a right-hand
 side of $\pm 1$.
 The above result is in fact correct, so we did not just disprove the GRH:
 \bprog
 ? tnf = thueinit(x^3-2*x^2+3*x-17, 1 /*unconditional*/);
 ? thue(tnf, -15)
 %4 = [[1, 1]]
 @eprog
 Note that reducible or non-monic polynomials are allowed:
 \bprog
 ? tnf = thueinit((2*x+1)^5 * (4*x^3-2*x^2+3*x-17), 1);
 ? thue(tnf, 128)
 %2 = [[-1, 0], [1, 0]]
 @eprog\noindent Reducible polynomials are in fact much easier to handle.

Function: thueinit
Class: basic
Section: polynomials
C-Name: thueinit
Prototype: GD0,L,p
Help: thueinit(P,{flag=0}): initialize the tnf corresponding to P, that will
 be used to solve Thue equations P(x,y) = some-integer. If flag is non-zero,
 certify the result unconditionaly. Otherwise, assume GRH (much faster of
 course).
Doc: initializes the \var{tnf} corresponding to $P$, a non-constant
 univariate polynomial with integer coefficients.
 The result is meant to be used in conjunction with \tet{thue} to solve Thue
 equations $P(X / Y)Y^{\deg P} = a$, where $a$ is an integer. Accordingly,
 $P$ must either have at least two distinct irreducible factors over $\Q$,
 or have one irreducible factor $T$ with degree $>2$ or two conjugate
 complex roots: under these (necessary and sufficient) conditions, the
 equation has finitely many integer solutions.
 \bprog
 ? S = thueinit(t^2+1);
 ? thue(S, 5)
 %2 = [[-2, -1], [-2, 1], [-1, -2], [-1, 2], [1, -2], [1, 2], [2, -1], [2, 1]]
 ? S = thueinit(t+1);
  ***   at top-level: thueinit(t+1)
  ***                 ^-------------
  *** thueinit: domain error in thueinit: P = t + 1
 @eprog\noindent The hardest case is when $\deg P > 2$ and $P$ is irreducible
 with at least one real root. The routine then uses Bilu-Hanrot's algorithm.
 
 If $\fl$ is non-zero, certify results unconditionally. Otherwise, assume
 \idx{GRH}, this being much faster of course. In the latter case, the result
 may still be unconditionally correct, see \tet{thue}. For instance in most
 cases where $P$ is reducible (not a pure power of an irreducible), \emph{or}
 conditional computed class groups are trivial \emph{or} the right hand side
 is $\pm1$, then results are unconditional.
 
 \misctitle{Note} The general philosophy is to disprove the existence of large
 solutions then to enumerate bounded solutions naively. The implementation
 will overflow when there exist huge solutions and the equation has degree
 $> 2$ (the quadratic imaginary case is special, since we can use
 \kbd{bnfisintnorm}):
 \bprog
 ? thue(t^3+2, 10^30)
  ***   at top-level: L=thue(t^3+2,10^30)
  ***                   ^-----------------
  *** thue: overflow in thue (SmallSols): y <= 80665203789619036028928.
 ? thue(x^2+2, 10^30)  \\ quadratic case much easier
 %1 = [[-1000000000000000, 0], [1000000000000000, 0]]
 @eprog
 
 \misctitle{Note} It is sometimes possible to circumvent the above, and in any
 case obtain an important speed-up, if you can write $P = Q(x^d)$ for some $d >
 1$ and $Q$ still satisfying the \kbd{thueinit} hypotheses. You can then solve
 the equation attached to $Q$ then eliminate all solutions $(x,y)$ such that
 either $x$ or $y$ is not a $d$-th power.
 \bprog
 ? thue(x^4+1, 10^40); \\ stopped after 10 hours
 ? filter(L,d) =
     my(x,y); [[x,y] | v<-L, ispower(v[1],d,&x)&&ispower(v[2],d,&y)];
 ? L = thue(x^2+1, 10^40);
 ? filter(L, 2)
 %4 = [[0, 10000000000], [10000000000, 0]]
 @eprog\noindent The last 2 commands use less than 20ms.

Function: trace
Class: basic
Section: linear_algebra
C-Name: gtrace
Prototype: G
Help: trace(x): trace of x.
Doc: this applies to quite general $x$. If $x$ is not a
 matrix, it is equal to the sum of $x$ and its conjugate, except for polmods
 where it is the trace as an algebraic number.
 
 For $x$ a square matrix, it is the ordinary trace. If $x$ is a
 non-square matrix (but not a vector), an error occurs.

Function: trap
Class: basic
Section: programming/specific
C-Name: trap0
Prototype: DrDEDE
Help: trap({e}, {rec}, seq): this function is obsolete, use "iferr".
 Try to execute seq, trapping runtime error e (all of them if e omitted);
 sequence rec is executed if the error occurs and is the result of the command.
Wrapper: (,_,_)
Description: 
 (?str,?closure,?closure):gen trap0($1, $2, $3)
Doc: This function is obsolete, use \tet{iferr}, which has a nicer and much
 more powerful interface. For compatibility's sake we now describe the
 \emph{obsolete} function \tet{trap}.
 
 This function tries to
 evaluate \var{seq}, trapping runtime error $e$, that is effectively preventing
 it from aborting computations in the usual way; the recovery sequence
 \var{rec} is executed if the error occurs and the evaluation of \var{rec}
 becomes the result of the command. If $e$ is omitted, all exceptions are
 trapped. See \secref{se:errorrec} for an introduction to error recovery
 under \kbd{gp}.
 
 \bprog
 ? \\@com trap division by 0
 ? inv(x) = trap (e_INV, INFINITY, 1/x)
 ? inv(2)
 %1 = 1/2
 ? inv(0)
 %2 = INFINITY
 @eprog\noindent
 Note that \var{seq} is effectively evaluated up to the point that produced
 the error, and the recovery sequence is evaluated starting from that same
 context, it does not "undo" whatever happened in the other branch (restore
 the evaluation context):
 \bprog
 ? x = 1; trap (, /* recover: */ x, /* try: */ x = 0; 1/x)
 %1 = 0
 @eprog
 
 \misctitle{Note} The interface is currently not adequate for trapping
 individual exceptions. In the current version \vers, the following keywords
 are recognized, but the name list will be expanded and changed in the
 future (all library mode errors can be trapped: it's a matter of defining
 the keywords to \kbd{gp}):
 
 \kbd{e\_ALARM}: alarm time-out
 
 \kbd{e\_ARCH}: not available on this architecture or operating system
 
 \kbd{e\_STACK}: the PARI stack overflows
 
 \kbd{e\_INV}: impossible inverse
 
 \kbd{e\_IMPL}: not yet implemented
 
 \kbd{e\_OVERFLOW}: all forms of arithmetic overflow, including length
 or exponent overflow (when a larger value is supplied than the
 implementation can handle).
 
 \kbd{e\_SYNTAX}: syntax error
 
 \kbd{e\_MISC}: miscellaneous error
 
 \kbd{e\_TYPE}: wrong type
 
 \kbd{e\_USER}: user error (from the \kbd{error} function)
Obsolete: 2012-01-17

Function: truncate
Class: basic
Section: conversions
C-Name: trunc0
Prototype: GD&
Help: truncate(x,{&e}): truncation of x; when x is a power series,take away
 the O(X^). If e is present, do not take into account loss of integer part
 precision, and set e = error estimate in bits.
Description: 
 (small):small:parens   $1
 (int):int:copy:parens  $1
 (real):int             truncr($1)
 (mp):int               mptrunc($1)
 (mp, &small):int       gcvtoi($1, &$2)
 (mp, &int):int         trunc0($1, &$2)
 (gen):gen              gtrunc($1)
 (gen, &small):gen      gcvtoi($1, &$2)
 (gen, &int):gen        trunc0($1, &$2)
Doc: truncates $x$ and sets $e$ to the number of
 error bits. When $x$ is in $\R$, this means that the part after the decimal
 point is chopped away, $e$ is the binary exponent of the difference between
 the original and the truncated value (the ``fractional part''). If the
 exponent of $x$ is too large compared to its precision (i.e.~$e>0$), the
 result is undefined and an error occurs if $e$ was not given. The function
 applies componentwise on vector / matrices; $e$ is then the maximal number of
 error bits. If $x$ is a rational function, the result is the ``integer part''
 (Euclidean quotient of numerator by denominator) and $e$ is not set.
 
 Note a very special use of \kbd{truncate}: when applied to a power series, it
 transforms it into a polynomial or a rational function with denominator
 a power of $X$, by chopping away the $O(X^k)$. Similarly, when applied to
 a $p$-adic number, it transforms it into an integer or a rational number
 by chopping away the $O(p^k)$.
Variant: The following functions are also available: \fun{GEN}{gtrunc}{GEN x}
 and \fun{GEN}{gcvtoi}{GEN x, long *e}.

Function: type
Class: basic
Section: programming/specific
C-Name: type0
Prototype: G
Help: type(x): return the type of the GEN x.
Description: 
 (gen):typ              typ($1)
Doc: this is useful only under \kbd{gp}. Returns the internal type name of
 the PARI object $x$ as a  string. Check out existing type names with the
 metacommand \b{t}. For example \kbd{type(1)} will return "\typ{INT}".
Variant: The macro \kbd{typ} is usually simpler to use since it returns a
 \kbd{long} that can easily be matched with the symbols \typ{*}. The name
 \kbd{type} was avoided since it is a reserved identifier for some compilers.

Function: unclone
Class: gp2c
Description: 
 (small):void   (void)0 /*unclone*/
 (gen):void     gunclone($1)

Function: uninline
Class: basic
Section: programming/specific
Help: uninline(): forget all inline variables [EXPERIMENTAL].
Doc: (Experimental) Exit the scope of all current \kbd{inline} variables.

Function: until
Class: basic
Section: programming/control
C-Name: untilpari
Prototype: vEI
Help: until(a,seq): evaluate the expression sequence seq until a is nonzero.
Doc: evaluates \var{seq} until $a$ is not
 equal to 0 (i.e.~until $a$ is true). If $a$ is initially not equal to 0,
 \var{seq} is evaluated once (more generally, the condition on $a$ is tested
 \emph{after} execution of the \var{seq}, not before as in \kbd{while}).

Function: valuation
Class: basic
Section: conversions
C-Name: gpvaluation
Prototype: GG
Help: valuation(x,p): valuation of x with respect to p.
Doc: 
 computes the highest
 exponent of $p$ dividing $x$. If $p$ is of type integer, $x$ must be an
 integer, an intmod whose modulus is divisible by $p$, a fraction, a
 $q$-adic number with $q=p$, or a polynomial or power series in which case the
 valuation is the minimum of the valuation of the coefficients.
 
 If $p$ is of type polynomial, $x$ must be of type polynomial or rational
 function, and also a power series if $x$ is a monomial. Finally, the
 valuation of a vector, complex or quadratic number is the minimum of the
 component valuations.
 
 If $x=0$, the result is \kbd{+oo} if $x$ is an exact object. If $x$ is a
 $p$-adic numbers or power series, the result is the exponent of the zero.
 Any other type combinations gives an error.
Variant: Also available is
 \fun{long}{gvaluation}{GEN x, GEN p}, which returns \tet{LONG_MAX} if $x = 0$
 and the valuation as a \kbd{long} integer.

Function: varhigher
Class: basic
Section: conversions
C-Name: varhigher
Prototype: sDn
Help: varhigher(name,{v}): return a variable 'name' whose priority is
 higher than the priority of v (of all existing variables if v is omitted).
Doc: return a variable \emph{name} whose priority is higher
 than the priority of $v$ (of all existing variables if $v$ is omitted).
 This is a counterpart to \tet{varlower}.
 \bprog
 ? Pol([x,x], t)
  ***   at top-level: Pol([x,x],t)
  ***                 ^------------
  *** Pol: incorrect priority in gtopoly: variable x <= t
 ? t = varhigher("t", x);
 ? Pol([x,x], t)
 %3 = x*t + x
 @eprog\noindent This routine is useful since new GP variables directly
 created by the interpreter always have lower priority than existing
 GP variables. When some basic objects already exist in a variable
 that is incompatible with some function requirement, you can now
 create a new variable with a suitable priority instead of changing variables
 in existing objects:
 \bprog
 ? K = nfinit(x^2+1);
 ? rnfequation(K,y^2-2)
  ***   at top-level: rnfequation(K,y^2-2)
  ***                 ^--------------------
  *** rnfequation: incorrect priority in rnfequation: variable y >= x
 ? y = varhigher("y", x);
 ? rnfequation(K, y^2-2)
 %3 = y^4 - 2*y^2 + 9
 @eprog\noindent
 \misctitle{Caution 1}
 The \emph{name} is an arbitrary character string, only used for display
 purposes and need not be related to the GP variable holding the result, nor
 to be a valid variable name. In particular the \emph{name} can
 not be used to retrieve the variable, it is not even present in the parser's
 hash tables.
 \bprog
 ? x = varhigher("#");
 ? x^2
 %2 = #^2
 @eprog
 \misctitle{Caution 2} There are a limited number of variables and if no
 existing variable with the given display name has the requested
 priority, the call to \kbd{varhigher} uses up one such slot. Do not create
 new variables in this way unless it's absolutely necessary,
 reuse existing names instead and choose sensible priority requirements:
 if you only need a variable with higher priority than $x$, state so
 rather than creating a new variable with highest priority.
 \bprog
 \\ quickly use up all variables
 ? n = 0; while(1,varhigher("tmp"); n++)
  ***   at top-level: n=0;while(1,varhigher("tmp");n++)
  ***                             ^-------------------
  *** varhigher: no more variables available.
  ***   Break loop: type 'break' to go back to GP prompt
 break> n
 65510
 \\ infinite loop: here we reuse the same 'tmp'
 ? n = 0; while(1,varhigher("tmp", x); n++)
 @eprog

Function: variable
Class: basic
Section: conversions
C-Name: gpolvar
Prototype: DG
Help: variable({x}): main variable of object x. Gives p for p-adic x, 0
 if no variable can be attached to x. Returns the list of user variables if
 x is omitted.
Description: 
 (pol):var:parens:copy        $var:1
 (gen):gen        gpolvar($1)
Doc: 
 gives the main variable of the object $x$ (the variable with the highest
 priority used in $x$), and $p$ if $x$ is a $p$-adic number. Return $0$ if
 $x$ has no variable attached to it.
 \bprog
 ? variable(x^2 + y)
 %1 = x
 ? variable(1 + O(5^2))
 %2 = 5
 ? variable([x,y,z,t])
 %3 = x
 ? variable(1)
 %4 = 0
 @eprog\noindent The construction
 \bprog
    if (!variable(x),...)
 @eprog\noindent can be used to test whether a variable is attached to $x$.
 
 If $x$ is omitted, returns the list of user variables known to the
 interpreter, by order of decreasing priority. (Highest priority is initially
 $x$, which come first until \tet{varhigher} is used.) If \kbd{varhigher}
 or \kbd{varlower} are used, it is quite possible to end up with different
 variables (with different priorities) printed in the same way: they
 will then appear multiple times in the output:
 \bprog
 ? varhigher("y");
 ? varlower("y");
 ? variable()
 %4 = [y, x, y]
 @eprog\noindent Using \kbd{v = variable()} then \kbd{v[1]}, \kbd{v[2]},
 etc.~allows to recover and use existing variables.
Variant: However, in library mode, this function should not be used for $x$
 non-\kbd{NULL}, since \tet{gvar} is more appropriate. Instead, for
 $x$ a $p$-adic (type \typ{PADIC}), $p$ is $gel(x,2)$; otherwise, use
 \fun{long}{gvar}{GEN x} which returns the variable number of $x$ if
 it exists, \kbd{NO\_VARIABLE} otherwise, which satisfies the property
 $\kbd{varncmp}(\kbd{NO\_VARIABLE}, v) > 0$ for all valid variable number
 $v$, i.e. it has lower priority than any variable.

Function: variables
Class: basic
Section: conversions
C-Name: variables_vec
Prototype: DG
Help: variables({x}): all variables occuring in object x, sorted by
 decreasing priority. Returns the list of user variables if x is omitted.
Doc: 
 returns the list of all variables occuring in object $x$ (all user
 variables known to the interpreter if $x$ is omitted), sorted by
 decreasing priority.
 \bprog
 ? variables([x^2 + y*z + O(t), a+x])
 %1 = [x, y, z, t, a]
 @eprog\noindent The construction
 \bprog
    if (!variables(x),...)
 @eprog\noindent can be used to test whether a variable is attached to $x$.
 
 If \kbd{varhigher} or \kbd{varlower} are used, it is quite possible to end up
 with different variables (with different priorities) printed in the same
 way: they will then appear multiple times in the output:
 \bprog
 ? y1 = varhigher("y");
 ? y2 = varlower("y");
 ? variables(y*y1*y2)
 %4 = [y, y, y]
 @eprog
Variant: 
 Also available is \fun{GEN}{variables_vecsmall}{GEN x} which returns
 the (sorted) variable numbers instead of the attached monomials of degree 1.

Function: varlower
Class: basic
Section: conversions
C-Name: varlower
Prototype: sDn
Help: varlower(name,{v}): return a variable 'name' whose priority is lower
 than the priority of v (of all existing variables if v is omitted.
Doc: return a variable \emph{name} whose priority is lower
 than the priority of $v$ (of all existing variables if $v$ is omitted).
 This is a counterpart to \tet{varhigher}.
 
 New GP variables directly created by the interpreter always
 have lower priority than existing GP variables, but it is not easy
 to check whether an identifier is currently unused, so that the
 corresponding variable has the expected priority when it's created!
 Thus, depending on the session history, the same command may fail or succeed:
 \bprog
 ? t; z;  \\ now t > z
 ? rnfequation(t^2+1,z^2-t)
  ***   at top-level: rnfequation(t^2+1,z^
  ***                 ^--------------------
  *** rnfequation: incorrect priority in rnfequation: variable t >= t
 @eprog\noindent Restart and retry:
 \bprog
 ? z; t;  \\ now z > t
 ? rnfequation(t^2+1,z^2-t)
 %2 = z^4 + 1
 @eprog\noindent It is quite annoying for package authors, when trying to
 define a base ring, to notice that the package may fail for some users
 depending on their session history. The safe way to do this is as follows:
 \bprog
 ? z; t;  \\ In new session: now z > t
 ...
 ? t = varlower("t", 'z);
 ? rnfequation(t^2+1,z^2-2)
 %2 = z^4 - 2*z^2 + 9
 ? variable()
 %3 = [x, y, z, t]
 @eprog
 \bprog
 ? t; z;  \\ In new session: now t > z
 ...
 ? t = varlower("t", 'z); \\ create a new variable, still printed "t"
 ? rnfequation(t^2+1,z^2-2)
 %2 = z^4 - 2*z^2 + 9
 ? variable()
 %3 = [x, y, t, z, t]
 @eprog\noindent Now both constructions succeed. Note that in the
 first case, \kbd{varlower} is essentially a no-op, the existing variable $t$
 has correct priority. While in the second case, two different variables are
 displayed as \kbd{t}, one with higher priority than $z$ (created in the first
  line) and another one with lower priority (created by \kbd{varlower}).
 
 \misctitle{Caution 1}
 The \emph{name} is an arbitrary character string, only used for display
 purposes and need not be related to the GP variable holding the result, nor
 to be a valid variable name. In particular the \emph{name} can
 not be used to retrieve the variable, it is not even present in the parser's
 hash tables.
 \bprog
 ? x = varlower("#");
 ? x^2
 %2 = #^2
 @eprog
 \misctitle{Caution 2} There are a limited number of variables and if no
 existing variable with the given display name has the requested
 priority, the call to \kbd{varlower} uses up one such slot. Do not create
 new variables in this way unless it's absolutely necessary,
 reuse existing names instead and choose sensible priority requirements:
 if you only need a variable with higher priority than $x$, state so
 rather than creating a new variable with highest priority.
 \bprog
 \\ quickly use up all variables
 ? n = 0; while(1,varlower("x"); n++)
  ***   at top-level: n=0;while(1,varlower("x");n++)
  ***                             ^-------------------
  *** varlower: no more variables available.
  ***   Break loop: type 'break' to go back to GP prompt
 break> n
 65510
 \\ infinite loop: here we reuse the same 'tmp'
 ? n = 0; while(1,varlower("tmp", x); n++)
 @eprog

Function: vecextract
Class: basic
Section: linear_algebra
C-Name: extract0
Prototype: GGDG
Help: vecextract(x,y,{z}): extraction of the components of the matrix or
 vector x according to y and z. If z is omitted, y represents columns, otherwise
 y corresponds to rows and z to columns. y and z can be vectors (of indices),
 strings (indicating ranges as in "1..10") or masks (integers whose binary
 representation indicates the indices to extract, from left to right 1, 2, 4,
 8, etc.).
Description: 
 (vec,gen,?gen):vec  extract0($1, $2, $3)
Doc: extraction of components of the vector or matrix $x$ according to $y$.
 In case $x$ is a matrix, its components are the \emph{columns} of $x$. The
 parameter $y$ is a component specifier, which is either an integer, a string
 describing a range, or a vector.
 
 If $y$ is an integer, it is considered as a mask: the binary bits of $y$ are
 read from right to left, but correspond to taking the components from left to
 right. For example, if $y=13=(1101)_2$ then the components 1,3 and 4 are
 extracted.
 
 If $y$ is a vector (\typ{VEC}, \typ{COL} or \typ{VECSMALL}), which must have
 integer entries, these entries correspond to the component numbers to be
 extracted, in the order specified.
 
 If $y$ is a string, it can be
 
 \item a single (non-zero) index giving a component number (a negative
 index means we start counting from the end).
 
 \item a range of the form \kbd{"$a$..$b$"}, where $a$ and $b$ are
 indexes as above. Any of $a$ and $b$ can be omitted; in this case, we take
 as default values $a = 1$ and $b = -1$, i.e.~ the first and last components
 respectively. We then extract all components in the interval $[a,b]$, in
 reverse order if $b < a$.
 
 In addition, if the first character in the string is \kbd{\pow}, the
 complement of the given set of indices is taken.
 
 If $z$ is not omitted, $x$ must be a matrix. $y$ is then the \emph{row}
 specifier, and $z$ the \emph{column} specifier, where the component specifier
 is as explained above.
 
 \bprog
 ? v = [a, b, c, d, e];
 ? vecextract(v, 5)         \\@com mask
 %1 = [a, c]
 ? vecextract(v, [4, 2, 1]) \\@com component list
 %2 = [d, b, a]
 ? vecextract(v, "2..4")    \\@com interval
 %3 = [b, c, d]
 ? vecextract(v, "-1..-3")  \\@com interval + reverse order
 %4 = [e, d, c]
 ? vecextract(v, "^2")      \\@com complement
 %5 = [a, c, d, e]
 ? vecextract(matid(3), "2..", "..")
 %6 =
 [0 1 0]
 
 [0 0 1]
 @eprog
 The range notations \kbd{v[i..j]} and \kbd{v[\pow i]} (for \typ{VEC} or
 \typ{COL}) and \kbd{M[i..j, k..l]} and friends (for \typ{MAT}) implement a
 subset of the above, in a simpler and \emph{faster} way, hence should be
 preferred in most common situations. The following features are not
 implemented in the range notation:
 
 \item reverse order,
 
 \item omitting either $a$ or $b$ in \kbd{$a$..$b$}.

Function: vecmax
Class: basic
Section: operators
C-Name: vecmax0
Prototype: GD&
Help: vecmax(x,{&v}): largest entry in the vector/matrix x. If v
 is present, set it to the index of a largest entry (indirect max).
Description: 
  (gen):gen            vecmax($1)
  (gen, &gen):gen      vecmax0($1, &$2)
Doc: if $x$ is a vector or a matrix, returns the largest entry of $x$,
 otherwise returns a copy of $x$. Error if $x$ is empty.
 
 If $v$ is given, set it to the index of a largest entry (indirect maximum),
 when $x$ is a vector. If $x$ is a matrix, set $v$ to coordinates $[i,j]$
 such that $x[i,j]$ is a largest entry. This flag is ignored if $x$ is not a
 vector or matrix.
 
 \bprog
 ? vecmax([10, 20, -30, 40])
 %1 = 40
 ? vecmax([10, 20, -30, 40], &v); v
 %2 = 4
 ? vecmax([10, 20; -30, 40], &v); v
 %3 = [2, 2]
 @eprog
Variant: When $v$ is not needed, the function \fun{GEN}{vecmax}{GEN x} is
 also available.

Function: vecmin
Class: basic
Section: operators
C-Name: vecmin0
Prototype: GD&
Help: vecmin(x,{&v}): smallest entry in the vector/matrix x. If v is
 present, set it to the index of a smallest
 entry (indirect min).
Description: 
  (gen):gen            vecmin($1)
  (gen, &gen):gen      vecmin0($1, &$2)
Doc: if $x$ is a vector or a matrix, returns the smallest entry of $x$,
 otherwise returns a copy of $x$. Error if $x$ is empty.
 
 If $v$ is given, set it to the index of a smallest entry (indirect minimum),
 when $x$ is a vector. If $x$ is a matrix, set $v$ to coordinates $[i,j]$ such
 that $x[i,j]$ is a smallest entry. This is ignored if $x$ is not a vector or
 matrix.
 
 \bprog
 ? vecmin([10, 20, -30, 40])
 %1 = -30
 ? vecmin([10, 20, -30, 40], &v); v
 %2 = 3
 ? vecmin([10, 20; -30, 40], &v); v
 %3 = [2, 1]
 @eprog
Variant: When $v$ is not needed, the function \fun{GEN}{vecmin}{GEN x} is also
 available.

Function: vecsearch
Class: basic
Section: linear_algebra
C-Name: vecsearch
Prototype: lGGDG
Help: vecsearch(v,x,{cmpf}): determines whether x belongs to the sorted
 vector v. If the comparison function cmpf is explicitly given, assume
 that v was sorted according to vecsort(, cmpf).
Doc: determines whether $x$ belongs to the sorted vector or list $v$: return
 the (positive) index where $x$ was found, or $0$ if it does not belong to
 $v$.
 
 If the comparison function cmpf is omitted, we assume that $v$ is sorted in
 increasing order, according to the standard comparison function \kbd{lex},
 thereby restricting the possible types for $x$ and the elements of $v$
 (integers, fractions, reals, and vectors of such).
 
 If \kbd{cmpf} is present, it is understood as a comparison function and we
 assume that $v$ is sorted according to it, see \tet{vecsort} for how to
 encode comparison functions.
 \bprog
 ? v = [1,3,4,5,7];
 ? vecsearch(v, 3)
 %2 = 2
 ? vecsearch(v, 6)
 %3 = 0 \\ not in the list
 ? vecsearch([7,6,5], 5) \\ unsorted vector: result undefined
 %4 = 0
 @eprog
 
 By abuse of notation, $x$ is also allowed to be a matrix, seen as a vector
 of its columns; again by abuse of notation, a \typ{VEC} is considered
 as part of the matrix, if its transpose is one of the matrix columns.
 \bprog
 ? v = vecsort([3,0,2; 1,0,2]) \\ sort matrix columns according to lex order
 %1 =
 [0 2 3]
 
 [0 2 1]
 ? vecsearch(v, [3,1]~)
 %2 = 3
 ? vecsearch(v, [3,1])  \\ can search for x or x~
 %3 = 3
 ? vecsearch(v, [1,2])
 %4 = 0 \\ not in the list
 @eprog\noindent

Function: vecsort
Class: basic
Section: linear_algebra
C-Name: vecsort0
Prototype: GDGD0,L,
Help: vecsort(x,{cmpf},{flag=0}): sorts the vector of vectors (or matrix) x in
 ascending order, according to the comparison function cmpf, if not omitted.
 (If cmpf is an integer, sort according to the value of the k-th component
 of each entry.) Binary digits of flag (if present) mean: 1: indirect sorting,
 return the permutation instead of the permuted vector, 4: use descending
 instead of ascending order, 8: remove duplicate entries.
Description: 
 (vecsmall,?gen):vecsmall       vecsort0($1, $2, 0)
 (vecsmall,?gen,small):vecsmall vecsort0($1, $2, $3)
 (vec, , ?0):vec                sort($1)
 (vec, , 1):vecsmall            indexsort($1)
 (vec, , 2):vec                 lexsort($1)
 (vec, gen):vec                 vecsort0($1, $2, 0)
 (vec, ?gen, 1):vecsmall        vecsort0($1, $2, 1)
 (vec, ?gen, 3):vecsmall        vecsort0($1, $2, 3)
 (vec, ?gen, 5):vecsmall        vecsort0($1, $2, 5)
 (vec, ?gen, 7):vecsmall        vecsort0($1, $2, 7)
 (vec, ?gen, 9):vecsmall        vecsort0($1, $2, 9)
 (vec, ?gen, 11):vecsmall       vecsort0($1, $2, 11)
 (vec, ?gen, 13):vecsmall       vecsort0($1, $2, 13)
 (vec, ?gen, 15):vecsmall       vecsort0($1, $2, 15)
 (vec, ?gen, #small):vec        vecsort0($1, $2, $3)
 (vec, ?gen, small):gen         vecsort0($1, $2, $3)
Doc: sorts the vector $x$ in ascending order, using a mergesort method.
 $x$ must be a list, vector or matrix (seen as a vector of its columns).
 Note that mergesort is stable, hence the initial ordering of ``equal''
 entries (with respect to the sorting criterion) is not changed.
 
 If \kbd{cmpf} is omitted, we use the standard comparison function
 \kbd{lex}, thereby restricting the possible types for the elements of $x$
 (integers, fractions or reals and vectors of those). If \kbd{cmpf} is
 present, it is understood as a comparison function and we sort according to
 it. The following possibilities exist:
 
 \item an integer $k$: sort according to the value of the $k$-th
 subcomponents of the components of~$x$.
 
 \item a vector: sort lexicographically according to the components listed in
 the vector. For example, if $\kbd{cmpf}=\kbd{[2,1,3]}$, sort with respect to
 the second component, and when these are equal, with respect to the first,
 and when these are equal, with respect to the third.
 
 \item a comparison function (\typ{CLOSURE}), with two arguments $x$ and $y$,
 and returning an integer which is $<0$, $>0$ or $=0$ if $x<y$, $x>y$ or
 $x=y$ respectively. The \tet{sign} function is very useful in this context:
 \bprog
 ? vecsort([3,0,2; 1,0,2]) \\ sort columns according to lex order
 %1 =
 [0 2 3]
 
 [0 2 1]
 ? vecsort(v, (x,y)->sign(y-x))            \\@com reverse sort
 ? vecsort(v, (x,y)->sign(abs(x)-abs(y)))  \\@com sort by increasing absolute value
 ? cmpf(x,y) = my(dx = poldisc(x), dy = poldisc(y)); sign(abs(dx) - abs(dy))
 ? vecsort([x^2+1, x^3-2, x^4+5*x+1], cmpf)
 @eprog\noindent
 The last example used the named \kbd{cmpf} instead of an anonymous function,
 and sorts polynomials with respect to the absolute value of their
 discriminant. A more efficient approach would use precomputations to ensure
 a given discriminant is computed only once:
 \bprog
 ? DISC = vector(#v, i, abs(poldisc(v[i])));
 ? perm = vecsort(vector(#v,i,i), (x,y)->sign(DISC[x]-DISC[y]))
 ? vecextract(v, perm)
 @eprog\noindent Similar ideas apply whenever we sort according to the values
 of a function which is expensive to compute.
 
 \noindent The binary digits of \fl\ mean:
 
 \item 1: indirect sorting of the vector $x$, i.e.~if $x$ is an
 $n$-component vector, returns a permutation of $[1,2,\dots,n]$ which
 applied to the components of $x$ sorts $x$ in increasing order.
 For example, \kbd{vecextract(x, vecsort(x,,1))} is equivalent to
 \kbd{vecsort(x)}.
 
 \item 4: use descending instead of ascending order.
 
 \item 8: remove ``duplicate'' entries with respect to the sorting function
 (keep the first occurring entry).  For example:
 \bprog
   ? vecsort([Pi,Mod(1,2),z], (x,y)->0, 8)   \\@com make everything compare equal
   %1 = [3.141592653589793238462643383]
   ? vecsort([[2,3],[0,1],[0,3]], 2, 8)
   %2 = [[0, 1], [2, 3]]
 @eprog

Function: vecsum
Class: basic
Section: linear_algebra
C-Name: vecsum
Prototype: G
Help: vecsum(v): return the sum of the components of the vector v.
Doc: return the sum of the components of the vector $v$. Return $0$ on an
 empty vector.
 \bprog
 ? vecsum([1,2,3])
 %1 = 6
 ? vecsum([])
 %2 = 0
 @eprog

Function: vector
Class: basic
Section: linear_algebra
C-Name: vecteur
Prototype: GDVDE
Help: vector(n,{X},{expr=0}): row vector with n components of expression
 expr (X ranges from 1 to n). By default, fill with 0s.
Doc: creates a row vector (type
 \typ{VEC}) with $n$ components whose components are the expression
 \var{expr} evaluated at the integer points between 1 and $n$. If one of the
 last two arguments is omitted, fill the vector with zeroes.
 \bprog
 ? vector(3,i, 5*i)
 %1 = [5, 10, 15]
 ? vector(3)
 %2 = [0, 0, 0]
 @eprog
 
 The variable $X$ is lexically scoped to each evaluation of \var{expr}.  Any
 change to $X$ within \var{expr} does not affect subsequent evaluations, it
 still runs 1 to $n$.  A local change allows for example different indexing:
 \bprog
 vector(10, i, i=i-1; f(i)) \\ i = 0, ..., 9
 vector(10, i, i=2*i; f(i)) \\ i = 2, 4, ..., 20
 @eprog\noindent
 This per-element scope for $X$ differs from \kbd{for} loop evaluations,
 as the following example shows:
 \bprog
 n = 3
 v = vector(n); vector(n, i, i++)            ----> [2, 3, 4]
 v = vector(n); for (i = 1, n, v[i] = i++)   ----> [2, 0, 4]
 @eprog\noindent
 %\syn{NO}

Function: vectorsmall
Class: basic
Section: linear_algebra
C-Name: vecteursmall
Prototype: GDVDE
Help: vectorsmall(n,{X},{expr=0}): VECSMALL with n components of expression
 expr (X ranges from 1 to n) which must be small integers. By default, fill
 with 0s.
Doc: creates a row vector of small integers (type
 \typ{VECSMALL}) with $n$ components whose components are the expression
 \var{expr} evaluated at the integer points between 1 and $n$. If one of the
 last two arguments is omitted, fill the vector with zeroes.
 %\syn{NO}

Function: vectorv
Class: basic
Section: linear_algebra
C-Name: vvecteur
Prototype: GDVDE
Help: vectorv(n,{X},{expr=0}): column vector with n components of expression
 expr (X ranges from 1 to n). By default, fill with 0s.
Doc: as \tet{vector}, but returns a column vector (type \typ{COL}).
 %\syn{NO}

Function: version
Class: basic
Section: programming/specific
C-Name: pari_version
Prototype: 
Help: version(): returns the PARI version as [major,minor,patch] or [major,minor,patch,VCSversion].
Doc: returns the current version number as a \typ{VEC} with three integer
 components (major version number, minor version number and patchlevel);
 if your sources were obtained through our version control system, this will
 be followed by further more precise arguments, including
 e.g.~a~\kbd{git} \emph{commit hash}.
 
 This function is present in all versions of PARI following releases 2.3.4
 (stable) and 2.4.3 (testing).
 
 Unless you are working with multiple development versions, you probably only
 care about the 3 first numeric components. In any case, the \kbd{lex} function
 offers a clever way to check against a particular version number, since it will
 compare each successive vector entry, numerically or as strings, and will not
 mind if the vectors it compares have different lengths:
 \bprog
    if (lex(version(), [2,3,5]) >= 0,
      \\ code to be executed if we are running 2.3.5 or more recent.
    ,
      \\ compatibility code
    );
 @eprog\noindent On a number of different machines, \kbd{version()} could return either of
 \bprog
  %1 = [2, 3, 4]    \\ released version, stable branch
  %1 = [2, 4, 3]    \\ released version, testing branch
  %1 = [2, 6, 1, 15174, ""505ab9b"] \\ development
 @eprog
 
 In particular, if you are only working with released versions, the first
 line of the gp introductory message can be emulated by
 \bprog
    [M,m,p] = version();
    printf("GP/PARI CALCULATOR Version %s.%s.%s", M,m,p);
  @eprog\noindent If you \emph{are} working with many development versions of
  PARI/GP, the 4th and/or 5th components can be profitably included in the
  name of your logfiles, for instance.
 
  \misctitle{Technical note} For development versions obtained via \kbd{git},
  the 4th and 5th components are liable to change eventually, but we document
  their current meaning for completeness. The 4th component counts the number
  of reachable commits in the branch (analogous to \kbd{svn}'s revision
  number), and the 5th is the \kbd{git} commit hash. In particular, \kbd{lex}
  comparison still orders correctly development versions with respect to each
  others or to released versions (provided we stay within a given branch,
  e.g. \kbd{master})!

Function: warning
Class: basic
Section: programming/specific
C-Name: warning0
Prototype: vs*
Help: warning({str}*): display warning message str.
Description: 
 (?gen,...):void  pari_warn(warnuser, "${2 format_string}"${2 format_args})
Doc: outputs the message ``user warning''
 and the argument list (each of them interpreted as a string).
 If colors are enabled, this warning will be in a different color,
 making it easy to distinguish.
 \bprog
 warning(n, " is very large, this might take a while.")
 @eprog
 % \syn{NO}

Function: weber
Class: basic
Section: transcendental
C-Name: weber0
Prototype: GD0,L,p
Help: weber(x,{flag=0}): one of Weber's f function of x. flag is optional,
 and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x),
 1: function f1(x)=eta(x/2)/eta(x)
 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x). Note that
 j = (f^24-16)^3/f^24 = (f1^24+16)^3/f1^24 = (f2^24+16)^3/f2^24.
Doc: one of Weber's three $f$ functions.
 If $\fl=0$, returns
 $$f(x)=\exp(-i\pi/24)\cdot\eta((x+1)/2)\,/\,\eta(x) \quad\hbox{such that}\quad
 j=(f^{24}-16)^3/f^{24}\,,$$
 where $j$ is the elliptic $j$-invariant  (see the function \kbd{ellj}).
 If $\fl=1$, returns
 $$f_1(x)=\eta(x/2)\,/\,\eta(x)\quad\hbox{such that}\quad
 j=(f_1^{24}+16)^3/f_1^{24}\,.$$
 Finally, if $\fl=2$, returns
 $$f_2(x)=\sqrt{2}\eta(2x)\,/\,\eta(x)\quad\hbox{such that}\quad
 j=(f_2^{24}+16)^3/f_2^{24}.$$
 Note the identities $f^8=f_1^8+f_2^8$ and $ff_1f_2=\sqrt2$.
Variant: Also available are \fun{GEN}{weberf}{GEN x, long prec},
 \fun{GEN}{weberf1}{GEN x, long prec} and \fun{GEN}{weberf2}{GEN x, long prec}.

Function: whatnow
Class: gp
Section: programming/specific
C-Name: whatnow0
Prototype: vr
Help: whatnow(key): if key was present in GP version 1.39.15, gives
 the new function name.
Description: 
 (str):void             whatnow($1, 0)
Doc: if keyword \var{key} is the name of a function that was present in GP
 version 1.39.15, outputs the new function name and syntax, if it
 changed at all. Functions that where introduced since then, then modified
 are also recognized.
 \bprog
 ? whatnow("mu")
 New syntax: mu(n) ===> moebius(n)
 
 moebius(x): Moebius function of x.
 
 ? whatnow("sin")
 This function did not change
 @eprog When a function was removed and the underlying functionality
 is not available under a compatible interface, no equivalent is mentioned:
 \bprog
 ? whatnow("buchfu")
 This function no longer exists
 @eprog\noindent (The closest equivalent would be to set \kbd{K = bnfinit(T)}
 then access \kbd{K.fu}.)

Function: while
Class: basic
Section: programming/control
C-Name: whilepari
Prototype: vEI
Help: while(a,seq): while a is nonzero evaluate the expression sequence seq.
 Otherwise 0.
Doc: while $a$ is non-zero, evaluates the expression sequence \var{seq}. The
 test is made \emph{before} evaluating the $seq$, hence in particular if $a$
 is initially equal to zero the \var{seq} will not be evaluated at all.

Function: write
Class: basic
Section: programming/specific
C-Name: write0
Prototype: vss*
Help: write(filename,{str}*): appends the remaining arguments (same output as
 print) to filename.
Doc: writes (appends) to \var{filename} the remaining arguments, and appends a
 newline (same output as \kbd{print}).
 %\syn{NO}

Function: write1
Class: basic
Section: programming/specific
C-Name: write1
Prototype: vss*
Help: write1(filename,{str}*): appends the remaining arguments (same output as
 print1) to filename.
Doc: writes (appends) to \var{filename} the remaining arguments without a
 trailing newline (same output as \kbd{print1}).
 %\syn{NO}

Function: writebin
Class: basic
Section: programming/specific
C-Name: gpwritebin
Prototype: vsDG
Help: writebin(filename,{x}): write x as a binary object to file filename.
 If x is omitted, write all session variables.
Doc: writes (appends) to
 \var{filename} the object $x$ in binary format. This format is not human
 readable, but contains the exact internal structure of $x$, and is much
 faster to save/load than a string expression, as would be produced by
 \tet{write}. The binary file format includes a magic number, so that such a
 file can be recognized and correctly input by the regular \tet{read} or \b{r}
 function. If saved objects refer to polynomial variables that are not
 defined in the new session, they will be displayed as \kbd{t$n$} for some
 integer $n$ (the attached variable number).
 Installed functions and history objects can not be saved via this function.
 
 If $x$ is omitted, saves all user variables from the session, together with
 their names. Reading such a ``named object'' back in a \kbd{gp} session will set
 the corresponding user variable to the saved value. E.g after
 \bprog
 x = 1; writebin("log")
 @eprog\noindent
 reading \kbd{log} into a clean session will set \kbd{x} to $1$.
 The relative variables priorities (see \secref{se:priority}) of new variables
 set in this way remain the same (preset variables retain their former
 priority, but are set to the new value). In particular, reading such a
 session log into a clean session will restore all variables exactly as they
 were in the original one.
 
 Just as a regular input file, a binary file can be compressed
 using \tet{gzip}, provided the file name has the standard \kbd{.gz}
 extension.\sidx{binary file}
 
 In the present implementation, the binary files are architecture dependent
 and compatibility with future versions of \kbd{gp} is not guaranteed. Hence
 binary files should not be used for long term storage (also, they are
 larger and harder to compress than text files).

Function: writetex
Class: basic
Section: programming/specific
C-Name: writetex
Prototype: vss*
Help: writetex(filename,{str}*): appends the remaining arguments (same format as
 print) to filename, in TeX format.
Doc: as \kbd{write}, in \TeX\ format.
 %\syn{NO}

Function: zeta
Class: basic
Section: transcendental
C-Name: gzeta
Prototype: Gp
Help: zeta(s): Riemann zeta function at s with s a complex or a p-adic number.
Doc: For $s$ a complex number, Riemann's zeta
 function \sidx{Riemann zeta-function} $\zeta(s)=\sum_{n\ge1}n^{-s}$,
 computed using the \idx{Euler-Maclaurin} summation formula, except
 when $s$ is of type integer, in which case it is computed using
 Bernoulli numbers\sidx{Bernoulli numbers} for $s\le0$ or $s>0$ and
 even, and using modular forms for $s>0$ and odd.
 
 For $s$ a $p$-adic number, Kubota-Leopoldt zeta function at $s$, that
 is the unique continuous $p$-adic function on the $p$-adic integers
 that interpolates the values of $(1 - p^{-k}) \zeta(k)$ at negative
 integers $k$ such that $k \equiv 1 \pmod{p-1}$ (resp. $k$ is odd) if
 $p$ is odd (resp. $p = 2$).

Function: zetamult
Class: basic
Section: transcendental
C-Name: zetamult
Prototype: Gp
Help: zetamult(s): multiple zeta value at integral s = [s1,...,sd].
Doc: For $s$ a vector of positive integers such that $s[1] \geq 2$,
 returns the multiple zeta value (MZV)
 $$\zeta(s_1,\dots, s_k) = \sum_{n_1>\dots>n_k>0} n_1^{-s_1}\dots n_k^{-s_k}.$$
 \bprog
 ? zetamult([2,1]) - zeta(3) \\ Euler's identity
 %1 = 0.E-38
 @eprog

Function: zncharinduce
Class: basic
Section: number_theoretical
C-Name: zncharinduce
Prototype: GGG
Help: zncharinduce(G, chi, N): let G be idealstar(,q), let chi
 be a Dirichlet character mod q and let N be a multiple of q. Return
 the character modulo N induced by chi.
Doc: Let $G$ be attached to $(\Z/q\Z)^*$ (as per \kbd{G = idealstar(,q)})
 and let \kbd{chi} be a Dirichlet character on $(\Z/q\Z)^*$, given by
 
 \item a \typ{VEC}: a standard character on \kbd{bid.gen},
 
 \item a \typ{INT} or a \typ{COL}: a Conrey index in $(\Z/q\Z)^*$ or its
 Conrey logarithm;
 see \secref{se:dirichletchar} or \kbd{??character}.
 
 Let $N$ be a multiple of $q$, return the character modulo $N$ induced by
 \kbd{chi}. As usual for arithmetic functions, the new modulus $N$ can be
 given as a \typ{INT}, via a factorization matrix or a pair
 \kbd{[N, factor(N)]}, or by \kbd{idealstar(,N)}.
 
 \bprog
 ? G = idealstar(,4);
 ? chi = znconreylog(G,1); \\ trivial character mod 4
 ? zncharinduce(G, chi, 80)  \\ now mod 80
 %3 = [0, 0, 0]~
 ? zncharinduce(G, 1, 80)  \\ same using directly Conrey label
 %4 = [0, 0, 0]~
 ? G2 = idealstar(,80);
 ? zncharinduce(G, 1, G2)  \\ same
 %4 = [0, 0, 0]~
 
 ? chi = zncharinduce(G, 3, G2)  \\ induce the non-trivial character mod 4
 %5 = [1, 0, 0]~
 ? znconreyconductor(G2, chi, &chi0)
 %6 = [4, Mat([2, 2])]
 ? chi0
 %7 = [1]~
 @eprog\noindent Here is a larger example:
 \bprog
 ? G = idealstar(,126000);
 ? label = 1009;
 ? chi = znconreylog(G, label)
 %3 = [0, 0, 0, 14, 0]~
 ? N0 = znconreyconductor(G, label, &chi0)
 %4 = [125, Mat([5, 3])]
 ? chi0 \\ primitive character mod 5^3 attached to chi
 %5 = [14]~
 ? G0 = idealstar(,N0);
 ? zncharinduce(G0, chi0, G) \\ induce back
 %7 = [0, 0, 0, 14, 0]~
 ? znconreyexp(G, %)
 %8 = 1009
 @eprog

Function: zncharisodd
Class: basic
Section: number_theoretical
C-Name: zncharisodd
Prototype: lGG
Help: zncharisodd(G, chi): let G be idealstar(,N), let chi
 be a Dirichlet character mod N, return 1 if and only if chi(-1) = -1
 and 0 otherwise.
Doc: Let $G$ be attached to $(\Z/N\Z)^*$ (as per \kbd{G = idealstar(,N)})
 and let \kbd{chi} be a Dirichlet character on $(\Z/N\Z)^*$, given by
 
 \item a \typ{VEC}: a standard character on \kbd{bid.gen},
 
 \item a \typ{INT} or a \typ{COL}: a Conrey index in $(\Z/q\Z)^*$ or its
 Conrey logarithm;
 see \secref{se:dirichletchar} or \kbd{??character}.
 
 Return $1$ if and only if \kbd{chi}$(-1) = -1$ and $0$ otherwise.
 
 \bprog
 ? G = idealstar(,8);
 ? zncharisodd(G, 1)  \\ trivial character
 %2 = 0
 ? zncharisodd(G, 3)
 %3 = 1
 ? chareval(G, 3, -1)
 %4 = 1/2
 @eprog

Function: znchartokronecker
Class: basic
Section: number_theoretical
C-Name: znchartokronecker
Prototype: GGD0,L,
Help: znchartokronecker(G, chi, {flag=0}): let G be idealstar(,N), let chi
 be a Dirichlet character mod N, return the discriminant D if chi is
 real equal to the Kronecker symbol (D/.) and 0 otherwise. If flag
 is set, return the fundamental discriminant attached to the corresponding
 primitive character.
Doc: Let $G$ be attached to $(\Z/N\Z)^*$ (as per \kbd{G = idealstar(,N)})
 and let \kbd{chi} be a Dirichlet character on $(\Z/N\Z)^*$, given by
 
 \item a \typ{VEC}: a standard character on \kbd{bid.gen},
 
 \item a \typ{INT} or a \typ{COL}: a Conrey index in $(\Z/q\Z)^*$ or its
 Conrey logarithm;
 see \secref{se:dirichletchar} or \kbd{??character}.
 
 If $\fl = 0$, return the discriminant $D$ if \kbd{chi} is real equal to the
 Kronecker symbol $(D/.)$ and $0$ otherwise. The discriminant $D$ is
 fundamental if and only if \kbd{chi} is primitive.
 
 If $\fl = 1$, return the fundamental discriminant attached to the
 corresponding primitive character.
 
 \bprog
 ? G = idealstar(,8); CHARS = [1,3,5,7]; \\ Conrey labels
 ? apply(t->znchartokronecker(G,t), CHARS)
 %2 = [4, -8, 8, -4]
 ? apply(t->znchartokronecker(G,t,1), CHARS)
 %3 = [1, -8, 8, -4]
 @eprog

Function: znconreychar
Class: basic
Section: number_theoretical
C-Name: znconreychar
Prototype: GG
Help: znconreychar(bid,m): Dirichlet character attached to m in (Z/qZ)*
 in Conrey's notation, where bid is idealstar(,q).
Doc: Given a \var{bid} attached to $(\Z/q\Z)^*$ (as per
 \kbd{bid = idealstar(,q)}), this function returns the Dirichlet character
 attached to $m \in (\Z/q\Z)^*$ via Conrey's logarithm, which
 establishes a ``canonical'' bijection between $(\Z/q\Z)^*$ and its dual.
 
 Let $q = \prod_p p^{e_p}$ be the factorization of $q$ into distinct primes.
 For all odd  $p$ with $e_p > 0$, let $g_p$ be the element in $(\Z/q\Z)^*$
 which is
 
 \item congruent to $1$ mod $q/p^{e_p}$,
 
 \item congruent mod $p^{e_p}$ to the smallest integer whose order
 is $\phi(p^{e_p})$.
 
 For $p = 2$, we let $g_4$ (if $2^{e_2} \geq 4$) and $g_8$ (if furthermore
 ($2^{e_2} \geq 8$) be the elements in $(\Z/q\Z)^*$ which
 are
 
 \item congruent to $1$ mod $q/2^{e_2}$,
 
 \item $g_4 = -1 \mod 2^{e_2}$,
 
 \item $g_8 = 5 \mod 2^{e_2}$.
 
 Then the $g_p$ (and the extra $g_4$ and $g_8$ if $2^{e_2}\geq 2$) are
 independent
 generators of $(\Z/q\Z)^*$, i.e. every $m$ in $(\Z/q\Z)^*$ can be written
 uniquely as $\prod_p g_p^{m_p}$, where $m_p$ is defined modulo the order
 $o_p$ of $g_p$
 and $p \in S_q$, the set of prime divisors of $q$ together with $4$
 if $4 \mid q$ and $8$ if $8 \mid q$. Note that the $g_p$ are in general
 \emph{not} SNF
 generators as produced by \kbd{znstar} or \kbd{idealstar} whenever
 $\omega(q) \geq 2$, although their number is the same. They however allow
 to handle the finite abelian group $(\Z/q\Z)^*$ in a fast and elegant
 way. (Which unfortunately does not generalize to ray class groups or Hecke
 characters.)
 
 The Conrey logarithm of $m$ is the vector $(m_p)_{p\in S_q}$, obtained
 via \tet{znconreylog}. The Conrey character $\chi_q(m,\cdot)$  attached to
 $m$ mod $q$ maps
 each $g_p$, $p\in S_q$ to $e(m_p / o_p)$, where $e(x) = \exp(2i\pi x)$.
 This function returns the Conrey character expressed in the standard PARI
 way in terms of the SNF generators \kbd{bid.gen}.
 
 \misctitle{Note} It is useless to include the generators
 in the \var{bid}, except for debugging purposes: they are well defined from
 elementary matrix operations and Chinese remaindering, their explicit value
 as elements in $(\Z/q\Z)^*$ is never used.
 
 \bprog
 ? G = idealstar(,8,2); /*add generators for debugging:*/
 ? G.cyc
 %2 = [2, 2]  \\ Z/2 x Z/2
 ? G.gen
 %3 = [7, 3]
 ? znconreychar(G,1)  \\ 1 is always the trivial character
 %4 = [0, 0]
 ? znconreychar(G,2)  \\ 2 is not coprime to 8 !!!
   ***   at top-level: znconreychar(G,2)
   ***                 ^-----------------
   *** znconreychar: elements not coprime in Zideallog:
     2
     8
   ***   Break loop: type 'break' to go back to GP prompt
 break>
 
 ? znconreychar(G,3)
 %5 = [0, 1]
 ? znconreychar(G,5)
 %6 = [1, 1]
 ? znconreychar(G,7)
 %7 = [1, 0]
 @eprog\noindent We indeed get all 4 characters of $(\Z/8\Z)^*$.
 
 For convenience, we allow to input the \emph{Conrey logarithm} of $m$
 instead of $m$:
 \bprog
 ? G = idealstar(,55);
 ? znconreychar(G,7)
 %2 = [7, 0]
 ? znconreychar(G, znconreylog(G,7))
 %3 = [7, 0]
 @eprog

Function: znconreyconductor
Class: basic
Section: number_theoretical
C-Name: znconreyconductor
Prototype: GGD&
Help: znconreyconductor(bid,chi, {&chi0}): let bid be idealstar(,q) and chi
 be a Dirichlet character on (Z/qZ)* given by its Conrey logarithm. Return
 the conductor of chi, and set chi0 to (the Conrey logarithm of) the
 attached primitive character. If chi0 != chi, return the conductor
 and its factorization.
Doc: Let \var{bid} be attached to $(\Z/q\Z)^*$ (as per
 \kbd{bid = idealstar(,q)}) and \kbd{chi} be a Dirichlet character on
 $(\Z/q\Z)^*$, given by
 
 \item a \typ{VEC}: a standard character on \kbd{bid.gen},
 
 \item a \typ{INT} or a \typ{COL}: a Conrey index in $(\Z/q\Z)^*$ or its
 Conrey logarithm;
 see \secref{se:dirichletchar} or \kbd{??character}.
 
 Return the conductor of \kbd{chi}, as the \typ{INT} \kbd{bid.mod}
 if \kbd{chi} is primitive, and as a pair \kbd{[N, faN]} (with \kbd{faN} the
 factorization of $N$) otherwise.
 
 If \kbd{chi0} is present, set it to the Conrey logarithm of the attached
 primitive character.
 
 \bprog
 ? G = idealstar(,126000);
 ? znconreyconductor(G,11)   \\ primitive
 %2 = 126000
 ? znconreyconductor(G,1)    \\ trivial character, not primitive!
 %3 = [1, matrix(0,2)]
 ? N0 = znconreyconductor(G,1009, &chi0) \\ character mod 5^3
 %4 = [125, Mat([5, 3])]
 ? chi0
 %5 = [14]~
 ? G0 = idealstar(,N0);      \\ format [N,factor(N)] accepted
 ? znconreyexp(G0, chi0)
 %7 = 9
 ? znconreyconductor(G0, chi0) \\ now primitive, as expected
 %8 = 125
 @eprog\noindent The group \kbd{G0} is not computed as part of
 \kbd{znconreyconductor} because it needs to be computed only once per
 conductor, not once per character.

Function: znconreyexp
Class: basic
Section: number_theoretical
C-Name: znconreyexp
Prototype: GG
Help: znconreyexp(bid, chi): Conrey exponential attached to bid =
 idealstar(,q). Returns the element m in (Z/qZ)^* attached to the character
 chi on bid: znconreylog(bid, m) = chi.
Doc: Given a \var{bid} attached to $(\Z/q\Z)^*$ (as per
 \kbd{bid = idealstar(,q)}), this function returns the Conrey exponential of
 the character \var{chi}: it returns the integer
 $m \in (\Z/q\Z)^*$ such that \kbd{znconreylog(\var{bid}, $m$)} is \var{chi}.
 
 The character \var{chi} is given either as a
 
 \item \typ{VEC}: in terms of the generators \kbd{\var{bid}.gen};
 
 \item \typ{COL}: a Conrey logarithm.
 
 \bprog
 ? G = idealstar(,126000)
 ? znconreylog(G,1)
 %2 = [0, 0, 0, 0, 0]~
 ? znconreyexp(G,%)
 %3 = 1
 ? G.cyc \\ SNF generators
 %4 = [300, 12, 2, 2, 2]
 ? chi = [100, 1, 0, 1, 0]; \\ some random character on SNF generators
 ? znconreylog(G, chi)  \\ in terms of Conrey generators
 %6 = [0, 3, 3, 0, 2]~
 ? znconreyexp(G, %)  \\ apply to a Conrey log
 %7 = 18251
 ? znconreyexp(G, chi) \\ ... or a char on SNF generators
 %8 = 18251
 ? znconreychar(G,%)
 %9 = [100, 1, 0, 1, 0]
 @eprog

Function: znconreylog
Class: basic
Section: number_theoretical
C-Name: znconreylog
Prototype: GG
Help: znconreylog(bid,m): Conrey logarithm attached to m in (Z/qZ)*,
 where bid is idealstar(,q).
Doc: Given a \var{bid} attached to $(\Z/q\Z)^*$ (as per
 \kbd{bid = idealstar(,q)}), this function returns the Conrey logarithm of
 $m \in (\Z/q\Z)^*$.
 
 Let $q = \prod_p p^{e_p}$ be the factorization of $q$ into distinct primes,
 where we assume $e_2 = 0$ or $e_2 \geq 2$. (If $e_2 = 1$, we can ignore $2$
 from the factorization, as if we replaced $q$ by $q/2$, since $(\Z/q\Z)^*
 \sim (\Z/(q/2)\Z)^*$.)
 
 For all odd  $p$ with $e_p > 0$, let $g_p$ be the element in $(\Z/q\Z)^*$
 which is
 
 \item congruent to $1$ mod $q/p^{e_p}$,
 
 \item congruent mod $p^{e_p}$ to the smallest integer whose order
 is $\phi(p^{e_p})$ for $p$ odd,
 
 For $p = 2$, we let $g_4$ (if $2^{e_2} \geq 4$) and $g_8$ (if furthermore
 ($2^{e_2} \geq 8$) be the elements in $(\Z/q\Z)^*$ which
 are
 
 \item congruent to $1$ mod $q/2^{e_2}$,
 
 \item $g_4 = -1 \mod 2^{e_2}$,
 
 \item $g_8 = 5 \mod 2^{e_2}$.
 
 Then the $g_p$ (and the extra $g_4$ and $g_8$ if $2^{e_2}\geq 2$) are
 independent
 generators of $\Z/q\Z^*$, i.e. every $m$ in $(\Z/q\Z)^*$ can be written
 uniquely as $\prod_p g_p^{m_p}$, where $m_p$ is defined modulo the
 order $o_p$ of $g_p$
 and $p \in S_q$, the set of prime divisors of $q$ together with $4$
 if $4 \mid q$ and $8$ if $8 \mid q$.
 Note that the $g_p$ are in general \emph{not} SNF
 generators as produced by \kbd{znstar} or \kbd{idealstar} whenever
 $\omega(q) \geq 2$, although their number is the same. They however allow
 to handle the finite abelian group $(\Z/q\Z)^*$ in a fast and elegant
 way. (Which unfortunately does not generalize to ray class groups or Hecke
 characters.)
 
 The Conrey logarithm of $m$ is the vector $(m_p)_{p\in S_q}$. The inverse
 function \tet{znconreyexp} recovers the Conrey label $m$ from a character.
 
 \bprog
 ? G = idealstar(,126000);
 ? znconreylog(G,1)
 %2 = [0, 0, 0, 0, 0]~
 ? znconreyexp(G, %)
 %3 = 1
 ? znconreylog(G,2)  \\ 2 is not coprime to modulus !!!
   ***   at top-level: znconreylog(G,2)
   ***                 ^-----------------
   *** znconreylog: elements not coprime in Zideallog:
     2
     126000
   ***   Break loop: type 'break' to go back to GP prompt
 break>
 ? znconreylog(G,11) \\ wrt. Conrey generators
 %4 = [0, 3, 1, 76, 4]~
 ? log11 = ideallog(,11,G)   \\ wrt. SNF generators
 %5 = [178, 3, -75, 1, 0]~
 @eprog\noindent
 
 For convenience, we allow to input the ordinary discrete log of $m$,
 $\kbd{ideallog(,m,bid)}$, which allows to convert discrete logs
 from \kbd{bid.gen} generators to Conrey generators.
 \bprog
 ? znconreylog(G, log11)
 %7 = [0, 3, 1, 76, 4]~
 @eprog\noindent We also allow a character (\typ{VEC}) on \kbd{bid.gen} and
 return its representation on the Conrey generators.
 \bprog
 ? G.cyc
 %8 = [300, 12, 2, 2, 2]
 ? chi = [10,1,0,1,1];
 ? znconreylog(G, chi)
 %10 = [1, 3, 3, 10, 2]~
 ? n = znconreyexp(G, chi)
 %11 = 84149
 ? znconreychar(G, n)
 %12 = [10, 1, 0, 1, 1]
 @eprog

Function: zncoppersmith
Class: basic
Section: number_theoretical
C-Name: zncoppersmith
Prototype: GGGDG
Help: zncoppersmith(P, N, X, {B=N}): finds all integers x
 with |x| <= X such that  gcd(N, P(x)) >= B. X should be smaller than
 exp((log B)^2 / (deg(P) log N)).
Doc: $N$ being an integer and $P\in \Z[X]$, finds all integers $x$ with
 $|x| \leq X$ such that
 $$\gcd(N, P(x)) \geq B,$$
 using \idx{Coppersmith}'s algorithm (a famous application of the \idx{LLL}
 algorithm). $X$ must be smaller than $\exp(\log^2 B / (\deg(P) \log N))$:
 for $B = N$, this means $X < N^{1/\deg(P)}$. Some $x$ larger than $X$ may
 be returned if you are very lucky. The smaller $B$ (or the larger $X$), the
 slower the routine will be. The strength of Coppersmith method is the
 ability to find roots modulo a general \emph{composite} $N$: if $N$ is a prime
 or a prime power, \tet{polrootsmod} or \tet{polrootspadic} will be much
 faster.
 
 We shall now present two simple applications. The first one is
 finding non-trivial factors of $N$, given some partial information on the
 factors; in that case $B$ must obviously be smaller than the largest
 non-trivial divisor of $N$.
 \bprog
 setrand(1); \\ to make the example reproducible
 interval = [10^30, 10^31];
 p = randomprime(interval);
 q = randomprime(interval); N = p*q;
 p0 = p % 10^20; \\ assume we know 1) p > 10^29, 2) the last 19 digits of p
 L = zncoppersmith(10^19*x + p0, N, 10^12, 10^29)
 
 \\ result in 10ms.
 %6 = [738281386540]
 ? gcd(L[1] * 10^19 + p0, N) == p
 %7 = 1
 @eprog\noindent and we recovered $p$, faster than by trying all
 possibilities $ < 10^{12}$.
 
 The second application is an attack on RSA with low exponent, when the
 message $x$ is short and the padding $P$ is known to the attacker. We use
 the same RSA modulus $N$ as in the first example:
 \bprog
 setrand(1);
 P = random(N);    \\ known padding
 e = 3;            \\ small public encryption exponent
 X = floor(N^0.3); \\ N^(1/e - epsilon)
 x0 = random(X);   \\ unknown short message
 C = lift( (Mod(x0,N) + P)^e ); \\ known ciphertext, with padding P
 zncoppersmith((P + x)^3 - C, N, X)
 
 \\ result in 244ms.
 %14 = [2679982004001230401]
 
 ? %[1] == x0
 %15 = 1
 @eprog\noindent
 We guessed an integer of the order of $10^{18}$, almost instantly.

Function: znlog
Class: basic
Section: number_theoretical
C-Name: znlog0
Prototype: GGDG
Help: znlog(x,g,{o}): return the discrete logarithm of x in
 (Z/nZ)* in base g. If present, o represents the multiplicative
 order of g. Return [] if no solution exist.
Doc: This functions allows two distinct modes of operation depending
 on $g$:
 
 \item if $g$ is the output of \tet{znstar} (with initialization),
 we compute the discrete logarithm of $x$ with respect to the generators
 contained in the structure. See \tet{ideallog} for details.
 
 \item else $g$ is an explicit element in $(\Z/N\Z)^*$, we compute the
 discrete logarithm of $x$ in $(\Z/N\Z)^*$ in base $g$. The rest of this
 entry describes the latter possibility.
 
 The result is $[]$ when $x$ is not a power of $g$, though the function may
 also enter an infinite loop in this case.
 
 If present, $o$ represents the multiplicative order of $g$, see
 \secref{se:DLfun}; the preferred format for this parameter is
 \kbd{[ord, factor(ord)]}, where \kbd{ord} is the order of $g$.
 This provides a definite speedup when the discrete log problem is simple:
 \bprog
 ? p = nextprime(10^4); g = znprimroot(p); o = [p-1, factor(p-1)];
 ? for(i=1,10^4, znlog(i, g, o))
 time = 205 ms.
 ? for(i=1,10^4, znlog(i, g))
 time = 244 ms. \\ a little slower
 @eprog
 
 The result is undefined if $g$ is not invertible mod $N$ or if the supplied
 order is incorrect.
 
 This function uses
 
 \item a combination of generic discrete log algorithms (see below).
 
 \item in $(\Z/N\Z)^*$ when $N$ is prime: a linear sieve index calculus
 method, suitable for $N < 10^{50}$, say, is used for large prime divisors of
 the order.
 
 The generic discrete log algorithms are:
 
 \item Pohlig-Hellman algorithm, to reduce to groups of prime order $q$,
 where $q | p-1$ and $p$ is an odd prime divisor of $N$,
 
 \item Shanks baby-step/giant-step ($q < 2^{32}$ is small),
 
 \item Pollard rho method ($q > 2^{32}$).
 
 The latter two algorithms require $O(\sqrt{q})$ operations in the group on
 average, hence will not be able to treat cases where $q > 10^{30}$, say.
 In addition, Pollard rho is not able to handle the case where there are no
 solutions: it will enter an infinite loop.
 \bprog
 ? g = znprimroot(101)
 %1 = Mod(2,101)
 ? znlog(5, g)
 %2 = 24
 ? g^24
 %3 = Mod(5, 101)
 
 ? G = znprimroot(2 * 101^10)
 %4 = Mod(110462212541120451003, 220924425082240902002)
 ? znlog(5, G)
 %5 = 76210072736547066624
 ? G^% == 5
 %6 = 1
 ? N = 2^4*3^2*5^3*7^4*11; g = Mod(13, N); znlog(g^110, g)
 %7 = 110
 ? znlog(6, Mod(2,3))  \\ no solution
 %8 = []
 @eprog\noindent For convenience, $g$ is also allowed to be a $p$-adic number:
 \bprog
 ? g = 3+O(5^10); znlog(2, g)
 %1 = 1015243
 ? g^%
 %2 = 2 + O(5^10)
 @eprog
Variant: The function
 \fun{GEN}{znlog}{GEN x, GEN g, GEN o} is also available

Function: znorder
Class: basic
Section: number_theoretical
C-Name: znorder
Prototype: GDG
Help: znorder(x,{o}): order of the integermod x in (Z/nZ)*.
 Optional o represents a multiple of the order of the element.
Description: 
 (gen):int             order($1)
 (gen,):int            order($1)
 (gen,int):int         znorder($1, $2)
Doc: $x$ must be an integer mod $n$, and the
 result is the order of $x$ in the multiplicative group $(\Z/n\Z)^*$. Returns
 an error if $x$ is not invertible.
 The parameter o, if present, represents a non-zero
 multiple of the order of $x$, see \secref{se:DLfun}; the preferred format for
 this parameter is \kbd{[ord, factor(ord)]}, where \kbd{ord = eulerphi(n)}
 is the cardinality of the group.
Variant: Also available is \fun{GEN}{order}{GEN x}.

Function: znprimroot
Class: basic
Section: number_theoretical
C-Name: znprimroot
Prototype: G
Help: znprimroot(n): returns a primitive root of n when it exists.
Doc: returns a primitive root (generator) of $(\Z/n\Z)^*$, whenever this
 latter group is cyclic ($n = 4$ or $n = 2p^k$ or $n = p^k$, where $p$ is an
 odd prime and $k \geq 0$). If the group is not cyclic, the result is
 undefined. If $n$ is a prime power, then the smallest positive primitive
 root is returned. This may not be true for $n = 2p^k$, $p$ odd.
 
 Note that this function requires factoring $p-1$ for $p$ as above,
 in order to determine the exact order of elements in
 $(\Z/n\Z)^*$: this is likely to be costly if $p$ is large.

Function: znstar
Class: basic
Section: number_theoretical
C-Name: znstar0
Prototype: GD0,L,
Help: znstar(n,{flag=0}): 3-component vector v = [no,cyc,gen], giving the
 structure of the abelian group (Z/nZ)^*;
 no is the order (i.e. eulerphi(n)), cyc is a vector of cyclic components,
 and gen is a vector giving the corresponding generators.
Doc: gives the structure of the multiplicative group $(\Z/n\Z)^*$.
 The output $G$ depends on the value of \fl:
 
 \item $\fl = 0$ (default), an abelian group structure $[h,d,g]$,
 where $h = \phi(n)$ is the order (\kbd{G.no}), $d$ (\kbd{G.cyc})
 is a $k$-component row-vector $d$ of integers $d_i$ such that $d_i>1$,
 $d_i \mid d_{i-1}$ for $i \ge 2$ and
 $$ (\Z/n\Z)^* \simeq \prod_{i=1}^k (\Z/d_i\Z), $$
 and $g$ (\kbd{G.gen}) is a $k$-component row vector giving generators of
 the image of the cyclic groups $\Z/d_i\Z$.
 
 \item $\fl = 1$ the result is a \kbd{bid} structure without generators
 (which are well defined but not explicitly computed, which saves time);
 this allows computing discrite logarithms using \tet{znlog} (also in the
 non-cyclic case!).
 
 \item $\fl = 2$ same as $\fl = 1$ with generators.
 
 \bprog
 ? G = znstar(40)
 %1 = [16, [4, 2, 2], [Mod(17, 40), Mod(21, 40), Mod(11, 40)]]
 ? G.no   \\ eulerphi(40)
 %2 = 16
 ? G.cyc  \\ cycle structure
 %3 = [4, 2, 2]
 ? G.gen  \\ generators for the cyclic components
 %4 = [Mod(17, 40), Mod(21, 40), Mod(11, 40)]
 ? apply(znorder, G.gen)
 %5 = [4, 2, 2]
 @eprog\noindent According to the above definitions, \kbd{znstar(0)} is
 \kbd{[2, [2], [-1]]}, corresponding to $\Z^*$.
Variant: Instead the above hardcoded numerical flags, one should rather use
 \fun{GEN}{ZNstar}{GEN N, long flag}, where \kbd{flag} is
 an or-ed combination of \tet{nf_GEN} (include generators) and \tet{nf_INIT}
 (return a full \kbd{bid}, not a group), possibly $0$. This offers
 one more combination: no gen and no init.