/usr/share/axiom-20170501/src/algebra/LPOLY.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 | )abbrev domain LPOLY LiePolynomial
++ Author: Michel Petitot (petitot@lifl.fr).
++ Date Created: 91
++ Date Last Updated: 7 Juillet 92
++ Fix History: compilation v 2.1 le 13 dec 98
++ References:
++ Free Lie Algebras by C. Reutenauer (Oxford science publications).
++ Description:
++ This type supports Lie polynomials in Lyndon basis
++ see Free Lie Algebras by C. Reutenauer
++ (Oxford science publications).
LiePolynomial(VarSet,R) : SIG == CODE where
VarSet : OrderedSet
R : CommutativeRing
MAGMA ==> Magma(VarSet)
LWORD ==> LyndonWord(VarSet)
WORD ==> OrderedFreeMonoid(VarSet)
XDPOLY ==> XDistributedPolynomial(VarSet,R)
XRPOLY ==> XRecursivePolynomial(VarSet,R)
NNI ==> NonNegativeInteger
RN ==> Fraction Integer
EX ==> OutputForm
TERM ==> Record(k: LWORD, c: R)
SIG ==> Join(FreeLieAlgebra(VarSet,R), FreeModuleCat(R,LWORD)) with
LiePolyIfCan : XDPOLY -> Union($, "failed")
++ \axiom{LiePolyIfCan(p)} returns \axiom{p} in Lyndon basis
++ if \axiom{p} is a Lie polynomial, otherwise \axiom{"failed"}
++ is returned.
construct : (LWORD, LWORD) -> $
++ \axiom{construct(x,y)} returns the Lie bracket \axiom{[x,y]}.
construct : (LWORD, $) -> $
++ \axiom{construct(x,y)} returns the Lie bracket \axiom{[x,y]}.
construct : ($, LWORD) -> $
++ \axiom{construct(x,y)} returns the Lie bracket \axiom{[x,y]}.
CODE ==> FreeModule1(R, LWORD) add
import(TERM)
--representation
Rep := List TERM
-- fonctions locales
cr1 : (LWORD, $ ) -> $
cr2 : ($, LWORD ) -> $
crw : (LWORD, LWORD) -> $ -- crochet de 2 mots de Lyndon
DPoly: LWORD -> XDPOLY
lquo1: (XRPOLY , LWORD) -> XRPOLY
lyndon: (LWORD, LWORD) -> $
makeLyndon: (LWORD, LWORD) -> LWORD
rquo1: (XRPOLY , LWORD) -> XRPOLY
RPoly: LWORD -> XRPOLY
eval1: (LWORD, VarSet, $) -> $ -- 08/03/98
eval2: (LWORD, List VarSet, List $) -> $ -- 08/03/98
-- Evaluation
eval1(lw,v,nv) == -- 08/03/98
not member?(v, varList(lw)$LWORD) => LiePoly lw
(s := retractIfCan(lw)$LWORD) case VarSet =>
if (s::VarSet) = v then nv else LiePoly lw
l: LWORD := left lw
r: LWORD := right lw
construct(eval1(l,v,nv), eval1(r,v,nv))
eval2(lw,lv,lnv) == -- 08/03/98
p: Integer
(s := retractIfCan(lw)$LWORD) case VarSet =>
p := position(s::VarSet, lv)$List(VarSet)
if p=0 then lw::$ else elt(lnv,p)$List($)
l: LWORD := left lw
r: LWORD := right lw
construct(eval2(l,lv,lnv), eval2(r,lv,lnv))
eval(p:$, v: VarSet, nv: $): $ == -- 08/03/98
+/ [t.c * eval1(t.k, v, nv) for t in p]
eval(p:$, lv: List(VarSet), lnv: List($)): $ == -- 08/03/98
+/ [t.c * eval2(t.k, lv, lnv) for t in p]
lquo1(p,lw) ==
constant? p => 0$XRPOLY
retractable? lw => lquo(p, retract lw)$XRPOLY
lquo1(lquo1(p, left lw),right lw) - lquo1(lquo1(p, right lw),left lw)
rquo1(p,lw) ==
constant? p => 0$XRPOLY
retractable? lw => rquo(p, retract lw)$XRPOLY
rquo1(rquo1(p, left lw),right lw) - rquo1(rquo1(p, right lw),left lw)
coef(p, lp) == coef(p, lp::XRPOLY)$XRPOLY
lquo(p, lp) ==
lp = 0 => 0$XRPOLY
+/ [t.c * lquo1(p,t.k) for t in lp]
rquo(p, lp) ==
lp = 0 => 0$XRPOLY
+/ [t.c * rquo1(p,t.k) for t in lp]
LiePolyIfCan p == -- inefficace a cause de la rep. de XDPOLY
not quasiRegular? p => "failed"
p1: XDPOLY := p ; r:$ := 0
while p1 ^= 0 repeat
t: Record(k:WORD, c:R) := mindegTerm p1
w: WORD := t.k; coef:R := t.c
(l := lyndonIfCan(w)$LWORD) case "failed" => return "failed"
lp:$ := coef * LiePoly(l::LWORD)
r := r + lp
p1 := p1 - lp::XDPOLY
r
--definitions locales
makeLyndon(u,v) == (u::MAGMA * v::MAGMA) pretend LWORD
crw(u,v) == -- u et v sont des mots de Lyndon
u = v => 0
lexico(u,v) => lyndon(u,v)
- lyndon (v,u)
lyndon(u,v) == -- u et v sont des mots de Lyndon tq u < v
retractable? u => monom(makeLyndon(u,v),1)
u1: LWORD := left u
u2: LWORD := right u
lexico(u2,v) => cr1(u1, lyndon(u2,v)) + cr2(lyndon(u1,v), u2)
monom(makeLyndon(u,v),1)
cr1 (l, p) ==
+/[t.c * crw(l, t.k) for t in p]
cr2 (p, l) ==
+/[t.c * crw(t.k, l) for t in p]
DPoly w ==
retractable? w => retract(w) :: XDPOLY
l:XDPOLY := DPoly left w
r:XDPOLY := DPoly right w
l*r - r*l
RPoly w ==
retractable? w => retract(w) :: XRPOLY
l:XRPOLY := RPoly left w
r:XRPOLY := RPoly right w
l*r - r*l
-- definitions
coerce(v:VarSet) == monom(v::LWORD , 1)
construct(x:$ , y:$):$ ==
+/[t.c * cr1(t.k, y) for t in x]
construct(l:LWORD , p:$):$ == cr1(l,p)
construct(p:$ , l:LWORD):$ == cr2(p,l)
construct(u:LWORD , v:LWORD):$ == crw(u,v)
coerce(p:$):XDPOLY ==
+/ [t.c * DPoly(t.k) for t in p]
coerce(p:$):XRPOLY ==
+/ [t.c * RPoly(t.k) for t in p]
LiePoly(l) == monom(l,1)
varList p ==
le : List VarSet := "setUnion"/[varList(t.k)$LWORD for t in p]
sort(le)$List(VarSet)
mirror p ==
[[t.k, (odd? length t.k => t.c; -t.c)]$TERM for t in p]
trunc(p, n) ==
degree(p) > n => trunc( reductum p , n)
p
degree p ==
null p => 0
length( p.first.k)$LWORD
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