python-numpy 1:1.6.1-6ubuntu1 / usr / share / pyshared / numpy / polynomial / legendre.py

"""
Objects for dealing with Legendre series.

This module provides a number of objects (mostly functions) useful for
dealing with Legendre series, including a `Legendre` class that
encapsulates the usual arithmetic operations.  (General information
on how this module represents and works with such polynomials is in the
docstring for its "parent" sub-package, `numpy.polynomial`).

Constants
---------
- `legdomain` -- Legendre series default domain, [-1,1].
- `legzero` -- Legendre series that evaluates identically to 0.
- `legone` -- Legendre series that evaluates identically to 1.
- `legx` -- Legendre series for the identity map, ``f(x) = x``.

Arithmetic
----------
- `legmulx` -- multiply a Legendre series in ``P_i(x)`` by ``x``.
- `legadd` -- add two Legendre series.
- `legsub` -- subtract one Legendre series from another.
- `legmul` -- multiply two Legendre series.
- `legdiv` -- divide one Legendre series by another.
- `legpow` -- raise a Legendre series to an positive integer power
- `legval` -- evaluate a Legendre series at given points.

Calculus
--------
- `legder` -- differentiate a Legendre series.
- `legint` -- integrate a Legendre series.

Misc Functions
--------------
- `legfromroots` -- create a Legendre series with specified roots.
- `legroots` -- find the roots of a Legendre series.
- `legvander` -- Vandermonde-like matrix for Legendre polynomials.
- `legfit` -- least-squares fit returning a Legendre series.
- `legtrim` -- trim leading coefficients from a Legendre series.
- `legline` -- Legendre series representing given straight line.
- `leg2poly` -- convert a Legendre series to a polynomial.
- `poly2leg` -- convert a polynomial to a Legendre series.

Classes
-------
- `Legendre` -- A Legendre series class.

See also
--------
`numpy.polynomial`

"""
from __future__ import division

__all__ = ['legzero', 'legone', 'legx', 'legdomain', 'legline',
        'legadd', 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow',
        'legval', 'legder', 'legint', 'leg2poly', 'poly2leg',
        'legfromroots', 'legvander', 'legfit', 'legtrim', 'legroots',
        'Legendre']

import numpy as np
import numpy.linalg as la
import polyutils as pu
import warnings
from polytemplate import polytemplate

legtrim = pu.trimcoef

def poly2leg(pol) :
    """
    Convert a polynomial to a Legendre series.

    Convert an array representing the coefficients of a polynomial (relative
    to the "standard" basis) ordered from lowest degree to highest, to an
    array of the coefficients of the equivalent Legendre series, ordered
    from lowest to highest degree.

    Parameters
    ----------
    pol : array_like
        1-d array containing the polynomial coefficients

    Returns
    -------
    cs : ndarray
        1-d array containing the coefficients of the equivalent Legendre
        series.

    See Also
    --------
    leg2poly

    Notes
    -----
    The easy way to do conversions between polynomial basis sets
    is to use the convert method of a class instance.

    Examples
    --------
    >>> from numpy import polynomial as P
    >>> p = P.Polynomial(np.arange(4))
    >>> p
    Polynomial([ 0.,  1.,  2.,  3.], [-1.,  1.])
    >>> c = P.Legendre(P.poly2leg(p.coef))
    >>> c
    Legendre([ 1.  ,  3.25,  1.  ,  0.75], [-1.,  1.])

    """
    [pol] = pu.as_series([pol])
    deg = len(pol) - 1
    res = 0
    for i in range(deg, -1, -1) :
        res = legadd(legmulx(res), pol[i])
    return res


def leg2poly(cs) :
    """
    Convert a Legendre series to a polynomial.

    Convert an array representing the coefficients of a Legendre series,
    ordered from lowest degree to highest, to an array of the coefficients
    of the equivalent polynomial (relative to the "standard" basis) ordered
    from lowest to highest degree.

    Parameters
    ----------
    cs : array_like
        1-d array containing the Legendre series coefficients, ordered
        from lowest order term to highest.

    Returns
    -------
    pol : ndarray
        1-d array containing the coefficients of the equivalent polynomial
        (relative to the "standard" basis) ordered from lowest order term
        to highest.

    See Also
    --------
    poly2leg

    Notes
    -----
    The easy way to do conversions between polynomial basis sets
    is to use the convert method of a class instance.

    Examples
    --------
    >>> c = P.Legendre(range(4))
    >>> c
    Legendre([ 0.,  1.,  2.,  3.], [-1.,  1.])
    >>> p = c.convert(kind=P.Polynomial)
    >>> p
    Polynomial([-1. , -3.5,  3. ,  7.5], [-1.,  1.])
    >>> P.leg2poly(range(4))
    array([-1. , -3.5,  3. ,  7.5])


    """
    from polynomial import polyadd, polysub, polymulx

    [cs] = pu.as_series([cs])
    n = len(cs)
    if n < 3:
        return cs
    else:
        c0 = cs[-2]
        c1 = cs[-1]
        # i is the current degree of c1
        for i in range(n - 1, 1, -1) :
            tmp = c0
            c0 = polysub(cs[i - 2], (c1*(i - 1))/i)
            c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i)
        return polyadd(c0, polymulx(c1))

#
# These are constant arrays are of integer type so as to be compatible
# with the widest range of other types, such as Decimal.
#

# Legendre
legdomain = np.array([-1,1])

# Legendre coefficients representing zero.
legzero = np.array([0])

# Legendre coefficients representing one.
legone = np.array([1])

# Legendre coefficients representing the identity x.
legx = np.array([0,1])


def legline(off, scl) :
    """
    Legendre series whose graph is a straight line.



    Parameters
    ----------
    off, scl : scalars
        The specified line is given by ``off + scl*x``.

    Returns
    -------
    y : ndarray
        This module's representation of the Legendre series for
        ``off + scl*x``.

    See Also
    --------
    polyline, chebline

    Examples
    --------
    >>> import numpy.polynomial.legendre as L
    >>> L.legline(3,2)
    array([3, 2])
    >>> L.legval(-3, L.legline(3,2)) # should be -3
    -3.0

    """
    if scl != 0 :
        return np.array([off,scl])
    else :
        return np.array([off])


def legfromroots(roots) :
    """
    Generate a Legendre series with the given roots.

    Return the array of coefficients for the P-series whose roots (a.k.a.
    "zeros") are given by *roots*.  The returned array of coefficients is
    ordered from lowest order "term" to highest, and zeros of multiplicity
    greater than one must be included in *roots* a number of times equal
    to their multiplicity (e.g., if `2` is a root of multiplicity three,
    then [2,2,2] must be in *roots*).

    Parameters
    ----------
    roots : array_like
        Sequence containing the roots.

    Returns
    -------
    out : ndarray
        1-d array of the Legendre series coefficients, ordered from low to
        high.  If all roots are real, ``out.dtype`` is a float type;
        otherwise, ``out.dtype`` is a complex type, even if all the
        coefficients in the result are real (see Examples below).

    See Also
    --------
    polyfromroots, chebfromroots

    Notes
    -----
    What is returned are the :math:`c_i` such that:

    .. math::

        \\sum_{i=0}^{n} c_i*P_i(x) = \\prod_{i=0}^{n} (x - roots[i])

    where ``n == len(roots)`` and :math:`P_i(x)` is the `i`-th Legendre
    (basis) polynomial over the domain `[-1,1]`.  Note that, unlike
    `polyfromroots`, due to the nature of the Legendre basis set, the
    above identity *does not* imply :math:`c_n = 1` identically (see
    Examples).

    Examples
    --------
    >>> import numpy.polynomial.legendre as L
    >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis
    array([ 0. , -0.4,  0. ,  0.4])
    >>> j = complex(0,1)
    >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis
    array([ 1.33333333+0.j,  0.00000000+0.j,  0.66666667+0.j])

    """
    if len(roots) == 0 :
        return np.ones(1)
    else :
        [roots] = pu.as_series([roots], trim=False)
        prd = np.array([1], dtype=roots.dtype)
        for r in roots:
            prd = legsub(legmulx(prd), r*prd)
        return prd


def legadd(c1, c2):
    """
    Add one Legendre series to another.

    Returns the sum of two Legendre series `c1` + `c2`.  The arguments
    are sequences of coefficients ordered from lowest order term to
    highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

    Parameters
    ----------
    c1, c2 : array_like
        1-d arrays of Legendre series coefficients ordered from low to
        high.

    Returns
    -------
    out : ndarray
        Array representing the Legendre series of their sum.

    See Also
    --------
    legsub, legmul, legdiv, legpow

    Notes
    -----
    Unlike multiplication, division, etc., the sum of two Legendre series
    is a Legendre series (without having to "reproject" the result onto
    the basis set) so addition, just like that of "standard" polynomials,
    is simply "component-wise."

    Examples
    --------
    >>> from numpy.polynomial import legendre as L
    >>> c1 = (1,2,3)
    >>> c2 = (3,2,1)
    >>> L.legadd(c1,c2)
    array([ 4.,  4.,  4.])

    """
    # c1, c2 are trimmed copies
    [c1, c2] = pu.as_series([c1, c2])
    if len(c1) > len(c2) :
        c1[:c2.size] += c2
        ret = c1
    else :
        c2[:c1.size] += c1
        ret = c2
    return pu.trimseq(ret)


def legsub(c1, c2):
    """
    Subtract one Legendre series from another.

    Returns the difference of two Legendre series `c1` - `c2`.  The
    sequences of coefficients are from lowest order term to highest, i.e.,
    [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

    Parameters
    ----------
    c1, c2 : array_like
        1-d arrays of Legendre series coefficients ordered from low to
        high.

    Returns
    -------
    out : ndarray
        Of Legendre series coefficients representing their difference.

    See Also
    --------
    legadd, legmul, legdiv, legpow

    Notes
    -----
    Unlike multiplication, division, etc., the difference of two Legendre
    series is a Legendre series (without having to "reproject" the result
    onto the basis set) so subtraction, just like that of "standard"
    polynomials, is simply "component-wise."

    Examples
    --------
    >>> from numpy.polynomial import legendre as L
    >>> c1 = (1,2,3)
    >>> c2 = (3,2,1)
    >>> L.legsub(c1,c2)
    array([-2.,  0.,  2.])
    >>> L.legsub(c2,c1) # -C.legsub(c1,c2)
    array([ 2.,  0., -2.])

    """
    # c1, c2 are trimmed copies
    [c1, c2] = pu.as_series([c1, c2])
    if len(c1) > len(c2) :
        c1[:c2.size] -= c2
        ret = c1
    else :
        c2 = -c2
        c2[:c1.size] += c1
        ret = c2
    return pu.trimseq(ret)


def legmulx(cs):
    """Multiply a Legendre series by x.

    Multiply the Legendre series `cs` by x, where x is the independent
    variable.


    Parameters
    ----------
    cs : array_like
        1-d array of Legendre series coefficients ordered from low to
        high.

    Returns
    -------
    out : ndarray
        Array representing the result of the multiplication.

    Notes
    -----
    The multiplication uses the recursion relationship for Legendre
    polynomials in the form

    .. math::

      xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1)

    """
    # cs is a trimmed copy
    [cs] = pu.as_series([cs])
    # The zero series needs special treatment
    if len(cs) == 1 and cs[0] == 0:
        return cs

    prd = np.empty(len(cs) + 1, dtype=cs.dtype)
    prd[0] = cs[0]*0
    prd[1] = cs[0]
    for i in range(1, len(cs)):
        j = i + 1
        k = i - 1
        s = i + j
        prd[j] = (cs[i]*j)/s
        prd[k] += (cs[i]*i)/s
    return prd


def legmul(c1, c2):
    """
    Multiply one Legendre series by another.

    Returns the product of two Legendre series `c1` * `c2`.  The arguments
    are sequences of coefficients, from lowest order "term" to highest,
    e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.

    Parameters
    ----------
    c1, c2 : array_like
        1-d arrays of Legendre series coefficients ordered from low to
        high.

    Returns
    -------
    out : ndarray
        Of Legendre series coefficients representing their product.

    See Also
    --------
    legadd, legsub, legdiv, legpow

    Notes
    -----
    In general, the (polynomial) product of two C-series results in terms
    that are not in the Legendre polynomial basis set.  Thus, to express
    the product as a Legendre series, it is necessary to "re-project" the
    product onto said basis set, which may produce "un-intuitive" (but
    correct) results; see Examples section below.

    Examples
    --------
    >>> from numpy.polynomial import legendre as L
    >>> c1 = (1,2,3)
    >>> c2 = (3,2)
    >>> P.legmul(c1,c2) # multiplication requires "reprojection"
    array([  4.33333333,  10.4       ,  11.66666667,   3.6       ])

    """
    # s1, s2 are trimmed copies
    [c1, c2] = pu.as_series([c1, c2])

    if len(c1) > len(c2):
        cs = c2
        xs = c1
    else:
        cs = c1
        xs = c2

    if len(cs) == 1:
        c0 = cs[0]*xs
        c1 = 0
    elif len(cs) == 2:
        c0 = cs[0]*xs
        c1 = cs[1]*xs
    else :
        nd = len(cs)
        c0 = cs[-2]*xs
        c1 = cs[-1]*xs
        for i in range(3, len(cs) + 1) :
            tmp = c0
            nd =  nd - 1
            c0 = legsub(cs[-i]*xs, (c1*(nd - 1))/nd)
            c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd)
    return legadd(c0, legmulx(c1))


def legdiv(c1, c2):
    """
    Divide one Legendre series by another.

    Returns the quotient-with-remainder of two Legendre series
    `c1` / `c2`.  The arguments are sequences of coefficients from lowest
    order "term" to highest, e.g., [1,2,3] represents the series
    ``P_0 + 2*P_1 + 3*P_2``.

    Parameters
    ----------
    c1, c2 : array_like
        1-D arrays of Legendre series coefficients ordered from low to
        high.

    Returns
    -------
    quo, rem : ndarrays
        Of Legendre series coefficients representing the quotient and
        remainder.

    See Also
    --------
    legadd, legsub, legmul, legpow

    Notes
    -----
    In general, the (polynomial) division of one Legendre series by another
    results in quotient and remainder terms that are not in the Legendre
    polynomial basis set.  Thus, to express these results as a Legendre
    series, it is necessary to "re-project" the results onto the Legendre
    basis set, which may produce "un-intuitive" (but correct) results; see
    Examples section below.

    Examples
    --------
    >>> from numpy.polynomial import legendre as L
    >>> c1 = (1,2,3)
    >>> c2 = (3,2,1)
    >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
    (array([ 3.]), array([-8., -4.]))
    >>> c2 = (0,1,2,3)
    >>> L.legdiv(c2,c1) # neither "intuitive"
    (array([-0.07407407,  1.66666667]), array([-1.03703704, -2.51851852]))

    """
    # c1, c2 are trimmed copies
    [c1, c2] = pu.as_series([c1, c2])
    if c2[-1] == 0 :
        raise ZeroDivisionError()

    lc1 = len(c1)
    lc2 = len(c2)
    if lc1 < lc2 :
        return c1[:1]*0, c1
    elif lc2 == 1 :
        return c1/c2[-1], c1[:1]*0
    else :
        quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype)
        rem = c1
        for i in range(lc1 - lc2, - 1, -1):
            p = legmul([0]*i + [1], c2)
            q = rem[-1]/p[-1]
            rem = rem[:-1] - q*p[:-1]
            quo[i] = q
        return quo, pu.trimseq(rem)


def legpow(cs, pow, maxpower=16) :
    """Raise a Legendre series to a power.

    Returns the Legendre series `cs` raised to the power `pow`. The
    arguement `cs` is a sequence of coefficients ordered from low to high.
    i.e., [1,2,3] is the series  ``P_0 + 2*P_1 + 3*P_2.``

    Parameters
    ----------
    cs : array_like
        1d array of Legendre series coefficients ordered from low to
        high.
    pow : integer
        Power to which the series will be raised
    maxpower : integer, optional
        Maximum power allowed. This is mainly to limit growth of the series
        to umanageable size. Default is 16

    Returns
    -------
    coef : ndarray
        Legendre series of power.

    See Also
    --------
    legadd, legsub, legmul, legdiv

    Examples
    --------

    """
    # cs is a trimmed copy
    [cs] = pu.as_series([cs])
    power = int(pow)
    if power != pow or power < 0 :
        raise ValueError("Power must be a non-negative integer.")
    elif maxpower is not None and power > maxpower :
        raise ValueError("Power is too large")
    elif power == 0 :
        return np.array([1], dtype=cs.dtype)
    elif power == 1 :
        return cs
    else :
        # This can be made more efficient by using powers of two
        # in the usual way.
        prd = cs
        for i in range(2, power + 1) :
            prd = legmul(prd, cs)
        return prd


def legder(cs, m=1, scl=1) :
    """
    Differentiate a Legendre series.

    Returns the series `cs` differentiated `m` times.  At each iteration the
    result is multiplied by `scl` (the scaling factor is for use in a linear
    change of variable).  The argument `cs` is the sequence of coefficients
    from lowest order "term" to highest, e.g., [1,2,3] represents the series
    ``P_0 + 2*P_1 + 3*P_2``.

    Parameters
    ----------
    cs : array_like
        1-D array of Legendre series coefficients ordered from low to high.
    m : int, optional
        Number of derivatives taken, must be non-negative. (Default: 1)
    scl : scalar, optional
        Each differentiation is multiplied by `scl`.  The end result is
        multiplication by ``scl**m``.  This is for use in a linear change of
        variable. (Default: 1)

    Returns
    -------
    der : ndarray
        Legendre series of the derivative.

    See Also
    --------
    legint

    Notes
    -----
    In general, the result of differentiating a Legendre series does not
    resemble the same operation on a power series. Thus the result of this
    function may be "un-intuitive," albeit correct; see Examples section
    below.

    Examples
    --------
    >>> from numpy.polynomial import legendre as L
    >>> cs = (1,2,3,4)
    >>> L.legder(cs)
    array([  6.,   9.,  20.])
    >>> L.legder(cs,3)
    array([ 60.])
    >>> L.legder(cs,scl=-1)
    array([ -6.,  -9., -20.])
    >>> L.legder(cs,2,-1)
    array([  9.,  60.])

    """
    cnt = int(m)

    if cnt != m:
        raise ValueError, "The order of derivation must be integer"
    if cnt < 0 :
        raise ValueError, "The order of derivation must be non-negative"

    # cs is a trimmed copy
    [cs] = pu.as_series([cs])
    if cnt == 0:
        return cs
    elif cnt >= len(cs):
        return cs[:1]*0
    else :
        for i in range(cnt):
            n = len(cs) - 1
            cs *= scl
            der = np.empty(n, dtype=cs.dtype)
            for j in range(n, 0, -1):
                der[j - 1] = (2*j - 1)*cs[j]
                cs[j - 2] += cs[j]
            cs = der
        return cs


def legint(cs, m=1, k=[], lbnd=0, scl=1):
    """
    Integrate a Legendre series.

    Returns a Legendre series that is the Legendre series `cs`, integrated
    `m` times from `lbnd` to `x`.  At each iteration the resulting series
    is **multiplied** by `scl` and an integration constant, `k`, is added.
    The scaling factor is for use in a linear change of variable.  ("Buyer
    beware": note that, depending on what one is doing, one may want `scl`
    to be the reciprocal of what one might expect; for more information,
    see the Notes section below.)  The argument `cs` is a sequence of
    coefficients, from lowest order Legendre series "term" to highest,
    e.g., [1,2,3] represents the series :math:`P_0(x) + 2P_1(x) + 3P_2(x)`.

    Parameters
    ----------
    cs : array_like
        1-d array of Legendre series coefficients, ordered from low to high.
    m : int, optional
        Order of integration, must be positive. (Default: 1)
    k : {[], list, scalar}, optional
        Integration constant(s).  The value of the first integral at
        ``lbnd`` is the first value in the list, the value of the second
        integral at ``lbnd`` is the second value, etc.  If ``k == []`` (the
        default), all constants are set to zero.  If ``m == 1``, a single
        scalar can be given instead of a list.
    lbnd : scalar, optional
        The lower bound of the integral. (Default: 0)
    scl : scalar, optional
        Following each integration the result is *multiplied* by `scl`
        before the integration constant is added. (Default: 1)

    Returns
    -------
    S : ndarray
        Legendre series coefficients of the integral.

    Raises
    ------
    ValueError
        If ``m < 0``, ``len(k) > m``, ``np.isscalar(lbnd) == False``, or
        ``np.isscalar(scl) == False``.

    See Also
    --------
    legder

    Notes
    -----
    Note that the result of each integration is *multiplied* by `scl`.
    Why is this important to note?  Say one is making a linear change of
    variable :math:`u = ax + b` in an integral relative to `x`.  Then
    :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a`
    - perhaps not what one would have first thought.

    Also note that, in general, the result of integrating a C-series needs
    to be "re-projected" onto the C-series basis set.  Thus, typically,
    the result of this function is "un-intuitive," albeit correct; see
    Examples section below.

    Examples
    --------
    >>> from numpy.polynomial import legendre as L
    >>> cs = (1,2,3)
    >>> L.legint(cs)
    array([ 0.33333333,  0.4       ,  0.66666667,  0.6       ])
    >>> L.legint(cs,3)
    array([  1.66666667e-02,  -1.78571429e-02,   4.76190476e-02,
            -1.73472348e-18,   1.90476190e-02,   9.52380952e-03])
    >>> L.legint(cs, k=3)
    array([ 3.33333333,  0.4       ,  0.66666667,  0.6       ])
    >>> L.legint(cs, lbnd=-2)
    array([ 7.33333333,  0.4       ,  0.66666667,  0.6       ])
    >>> L.legint(cs, scl=2)
    array([ 0.66666667,  0.8       ,  1.33333333,  1.2       ])

    """
    cnt = int(m)
    if np.isscalar(k) :
        k = [k]

    if cnt != m:
        raise ValueError, "The order of integration must be integer"
    if cnt < 0 :
        raise ValueError, "The order of integration must be non-negative"
    if len(k) > cnt :
        raise ValueError, "Too many integration constants"

    # cs is a trimmed copy
    [cs] = pu.as_series([cs])
    if cnt == 0:
        return cs

    k = list(k) + [0]*(cnt - len(k))
    for i in range(cnt) :
        n = len(cs)
        cs *= scl
        if n == 1 and cs[0] == 0:
            cs[0] += k[i]
        else:
            tmp = np.empty(n + 1, dtype=cs.dtype)
            tmp[0] = cs[0]*0
            tmp[1] = cs[0]
            for j in range(1, n):
                t = cs[j]/(2*j + 1)
                tmp[j + 1] = t
                tmp[j - 1] -= t
            tmp[0] += k[i] - legval(lbnd, tmp)
            cs = tmp
    return cs


def legval(x, cs):
    """Evaluate a Legendre series.

    If `cs` is of length `n`, this function returns :

    ``p(x) = cs[0]*P_0(x) + cs[1]*P_1(x) + ... + cs[n-1]*P_{n-1}(x)``

    If x is a sequence or array then p(x) will have the same shape as x.
    If r is a ring_like object that supports multiplication and addition
    by the values in `cs`, then an object of the same type is returned.

    Parameters
    ----------
    x : array_like, ring_like
        Array of numbers or objects that support multiplication and
        addition with themselves and with the elements of `cs`.
    cs : array_like
        1-d array of Legendre coefficients ordered from low to high.

    Returns
    -------
    values : ndarray, ring_like
        If the return is an ndarray then it has the same shape as `x`.

    See Also
    --------
    legfit

    Notes
    -----
    The evaluation uses Clenshaw recursion, aka synthetic division.

    Examples
    --------

    """
    # cs is a trimmed copy
    [cs] = pu.as_series([cs])
    if isinstance(x, tuple) or isinstance(x, list) :
        x = np.asarray(x)

    if len(cs) == 1 :
        c0 = cs[0]
        c1 = 0
    elif len(cs) == 2 :
        c0 = cs[0]
        c1 = cs[1]
    else :
        nd = len(cs)
        c0 = cs[-2]
        c1 = cs[-1]
        for i in range(3, len(cs) + 1) :
            tmp = c0
            nd =  nd - 1
            c0 = cs[-i] - (c1*(nd - 1))/nd
            c1 = tmp + (c1*x*(2*nd - 1))/nd
    return c0 + c1*x


def legvander(x, deg) :
    """Vandermonde matrix of given degree.

    Returns the Vandermonde matrix of degree `deg` and sample points `x`.
    This isn't a true Vandermonde matrix because `x` can be an arbitrary
    ndarray and the Legendre polynomials aren't powers. If ``V`` is the
    returned matrix and `x` is a 2d array, then the elements of ``V`` are
    ``V[i,j,k] = P_k(x[i,j])``, where ``P_k`` is the Legendre polynomial
    of degree ``k``.

    Parameters
    ----------
    x : array_like
        Array of points. The values are converted to double or complex
        doubles. If x is scalar it is converted to a 1D array.
    deg : integer
        Degree of the resulting matrix.

    Returns
    -------
    vander : Vandermonde matrix.
        The shape of the returned matrix is ``x.shape + (deg+1,)``. The last
        index is the degree.

    """
    ideg = int(deg)
    if ideg != deg:
        raise ValueError("deg must be integer")
    if ideg < 0:
        raise ValueError("deg must be non-negative")

    x = np.array(x, copy=0, ndmin=1) + 0.0
    v = np.empty((ideg + 1,) + x.shape, dtype=x.dtype)
    # Use forward recursion to generate the entries. This is not as accurate
    # as reverse recursion in this application but it is more efficient.
    v[0] = x*0 + 1
    if ideg > 0 :
        v[1] = x
        for i in range(2, ideg + 1) :
            v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i
    return np.rollaxis(v, 0, v.ndim)


def legfit(x, y, deg, rcond=None, full=False, w=None):
    """
    Least squares fit of Legendre series to data.

    Fit a Legendre series ``p(x) = p[0] * P_{0}(x) + ... + p[deg] *
    P_{deg}(x)`` of degree `deg` to points `(x, y)`. Returns a vector of
    coefficients `p` that minimises the squared error.

    Parameters
    ----------
    x : array_like, shape (M,)
        x-coordinates of the M sample points ``(x[i], y[i])``.
    y : array_like, shape (M,) or (M, K)
        y-coordinates of the sample points. Several data sets of sample
        points sharing the same x-coordinates can be fitted at once by
        passing in a 2D-array that contains one dataset per column.
    deg : int
        Degree of the fitting polynomial
    rcond : float, optional
        Relative condition number of the fit. Singular values smaller than
        this relative to the largest singular value will be ignored. The
        default value is len(x)*eps, where eps is the relative precision of
        the float type, about 2e-16 in most cases.
    full : bool, optional
        Switch determining nature of return value. When it is False (the
        default) just the coefficients are returned, when True diagnostic
        information from the singular value decomposition is also returned.
    w : array_like, shape (`M`,), optional
        Weights. If not None, the contribution of each point
        ``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the
        weights are chosen so that the errors of the products ``w[i]*y[i]``
        all have the same variance.  The default value is None.

    Returns
    -------
    coef : ndarray, shape (M,) or (M, K)
        Legendre coefficients ordered from low to high. If `y` was 2-D,
        the coefficients for the data in column k  of `y` are in column
        `k`.

    [residuals, rank, singular_values, rcond] : present when `full` = True
        Residuals of the least-squares fit, the effective rank of the
        scaled Vandermonde matrix and its singular values, and the
        specified value of `rcond`. For more details, see `linalg.lstsq`.

    Warns
    -----
    RankWarning
        The rank of the coefficient matrix in the least-squares fit is
        deficient. The warning is only raised if `full` = False.  The
        warnings can be turned off by

        >>> import warnings
        >>> warnings.simplefilter('ignore', RankWarning)

    See Also
    --------
    legval : Evaluates a Legendre series.
    legvander : Vandermonde matrix of Legendre series.
    polyfit : least squares fit using polynomials.
    chebfit : least squares fit using Chebyshev series.
    linalg.lstsq : Computes a least-squares fit from the matrix.
    scipy.interpolate.UnivariateSpline : Computes spline fits.

    Notes
    -----
    The solution are the coefficients ``c[i]`` of the Legendre series
    ``P(x)`` that minimizes the squared error

    ``E = \\sum_j |y_j - P(x_j)|^2``.

    This problem is solved by setting up as the overdetermined matrix
    equation

    ``V(x)*c = y``,

    where ``V`` is the Vandermonde matrix of `x`, the elements of ``c`` are
    the coefficients to be solved for, and the elements of `y` are the
    observed values.  This equation is then solved using the singular value
    decomposition of ``V``.

    If some of the singular values of ``V`` are so small that they are
    neglected, then a `RankWarning` will be issued. This means that the
    coeficient values may be poorly determined. Using a lower order fit
    will usually get rid of the warning.  The `rcond` parameter can also be
    set to a value smaller than its default, but the resulting fit may be
    spurious and have large contributions from roundoff error.

    Fits using Legendre series are usually better conditioned than fits
    using power series, but much can depend on the distribution of the
    sample points and the smoothness of the data. If the quality of the fit
    is inadequate splines may be a good alternative.

    References
    ----------
    .. [1] Wikipedia, "Curve fitting",
           http://en.wikipedia.org/wiki/Curve_fitting

    Examples
    --------

    """
    order = int(deg) + 1
    x = np.asarray(x) + 0.0
    y = np.asarray(y) + 0.0

    # check arguments.
    if deg < 0 :
        raise ValueError, "expected deg >= 0"
    if x.ndim != 1:
        raise TypeError, "expected 1D vector for x"
    if x.size == 0:
        raise TypeError, "expected non-empty vector for x"
    if y.ndim < 1 or y.ndim > 2 :
        raise TypeError, "expected 1D or 2D array for y"
    if len(x) != len(y):
        raise TypeError, "expected x and y to have same length"

    # set up the least squares matrices
    lhs = legvander(x, deg)
    rhs = y
    if w is not None:
        w = np.asarray(w) + 0.0
        if w.ndim != 1:
            raise TypeError, "expected 1D vector for w"
        if len(x) != len(w):
            raise TypeError, "expected x and w to have same length"
        # apply weights
        if rhs.ndim == 2:
            lhs *= w[:, np.newaxis]
            rhs *= w[:, np.newaxis]
        else:
            lhs *= w[:, np.newaxis]
            rhs *= w

    # set rcond
    if rcond is None :
        rcond = len(x)*np.finfo(x.dtype).eps

    # scale the design matrix and solve the least squares equation
    scl = np.sqrt((lhs*lhs).sum(0))
    c, resids, rank, s = la.lstsq(lhs/scl, rhs, rcond)
    c = (c.T/scl).T

    # warn on rank reduction
    if rank != order and not full:
        msg = "The fit may be poorly conditioned"
        warnings.warn(msg, pu.RankWarning)

    if full :
        return c, [resids, rank, s, rcond]
    else :
        return c


def legroots(cs):
    """
    Compute the roots of a Legendre series.

    Return the roots (a.k.a "zeros") of the Legendre series represented by
    `cs`, which is the sequence of coefficients from lowest order "term"
    to highest, e.g., [1,2,3] is the series ``L_0 + 2*L_1 + 3*L_2``.

    Parameters
    ----------
    cs : array_like
        1-d array of Legendre series coefficients ordered from low to high.

    Returns
    -------
    out : ndarray
        Array of the roots.  If all the roots are real, then so is the
        dtype of ``out``; otherwise, ``out``'s dtype is complex.

    See Also
    --------
    polyroots
    chebroots

    Notes
    -----
    Algorithm(s) used:

    Remember: because the Legendre series basis set is different from the
    "standard" basis set, the results of this function *may* not be what
    one is expecting.

    Examples
    --------
    >>> import numpy.polynomial as P
    >>> P.polyroots((1, 2, 3, 4)) # 4x^3 + 3x^2 + 2x + 1 has two complex roots
    array([-0.60582959+0.j        , -0.07208521-0.63832674j,
           -0.07208521+0.63832674j])
    >>> P.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0 has only real roots
    array([-0.85099543, -0.11407192,  0.51506735])

    """
    # cs is a trimmed copy
    [cs] = pu.as_series([cs])
    if len(cs) <= 1 :
        return np.array([], dtype=cs.dtype)
    if len(cs) == 2 :
        return np.array([-cs[0]/cs[1]])

    n = len(cs) - 1
    cs /= cs[-1]
    cmat = np.zeros((n,n), dtype=cs.dtype)
    cmat[1, 0] = 1
    for i in range(1, n):
        tmp = 2*i + 1
        cmat[i - 1, i] = i/tmp
        if i != n - 1:
            cmat[i + 1, i] = (i + 1)/tmp
        else:
            cmat[:, i] -= cs[:-1]*(i + 1)/tmp
    roots = la.eigvals(cmat)
    roots.sort()
    return roots


#
# Legendre series class
#

exec polytemplate.substitute(name='Legendre', nick='leg', domain='[-1,1]')