path: root/numpy/polynomial/legendre.py

"""
Legendre Series (:mod: `numpy.polynomial.legendre`)
===================================================

.. currentmodule:: numpy.polynomial.polynomial

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
---------

.. autosummary::
   :toctree: generated/

   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
----------

.. autosummary::
   :toctree: generated/

   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.
   legval2d             evaluate a 2D Legendre series at given points.
   legval3d             evaluate a 3D Legendre series at given points.
   leggrid2d            evaluate a 2D Legendre series on a Cartesian product.
   leggrid3d            evaluate a 3D Legendre series on a Cartesian product.

Calculus
--------

.. autosummary::
   :toctree: generated/

   legder               differentiate a Legendre series.
   legint               integrate a Legendre series.

Misc Functions
--------------

.. autosummary::
   :toctree: generated/

   legfromroots          create a Legendre series with specified roots.
   legroots              find the roots of a Legendre series.
   legvander             Vandermonde-like matrix for Legendre polynomials.
   legvander2d           Vandermonde-like matrix for 2D power series.
   legvander3d           Vandermonde-like matrix for 3D power series.
   leggauss              Gauss-Legendre quadrature, points and weights.
   legweight             Legendre weight function.
   legcompanion          symmetrized companion matrix in Legendre form.
   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.polynomial
numpy.polynomial.chebyshev
numpy.polynomial.laguerre
numpy.polynomial.hermite
numpy.polynomial.hermite_e

"""
from __future__ import division

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

__all__ = ['legzero', 'legone', 'legx', 'legdomain', 'legline',
    'legadd', 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval',
    'legder', 'legint', 'leg2poly', 'poly2leg', 'legfromroots',
    'legvander', 'legfit', 'legtrim', 'legroots', 'Legendre','legval2d',
    'legval3d', 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d',
    'legcompanion', 'leggauss', 'legweight']

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(c, m=1, scl=1, axis=0) :
    """
    Differentiate a Legendre series.

    Returns the Legendre series coefficients `c` differentiated `m` times
    along `axis`.  At each iteration the result is multiplied by `scl` (the
    scaling factor is for use in a linear change of variable). The argument
    `c` is an array of coefficients from low to high degree along each
    axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
    while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
    2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
    ``y``.

    Parameters
    ----------
    c : array_like
        Array of Legendre series coefficients. If c is multidimensional the
        different axis correspond to different variables with the degree in
        each axis given by the corresponding index.
    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)
    axis : int, optional
        Axis over which the derivative is taken. (Default: 0).

    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
    >>> c = (1,2,3,4)
    >>> L.legder(c)
    array([  6.,   9.,  20.])
    >>> L.legder(c, 3)
    array([ 60.])
    >>> L.legder(c, scl=-1)
    array([ -6.,  -9., -20.])
    >>> L.legder(c, 2,-1)
    array([  9.,  60.])

    """
    c = np.array(c, ndmin=1, copy=1)
    if c.dtype.char in '?bBhHiIlLqQpP':
        c = c.astype(np.double)
    cnt, iaxis = [int(t) for t in [m, axis]]

    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")
    if iaxis != axis:
        raise ValueError("The axis must be integer")
    if not -c.ndim <= iaxis < c.ndim:
        raise ValueError("The axis is out of range")
    if iaxis < 0:
        iaxis += c.ndim

    if cnt == 0:
        return c

    c = np.rollaxis(c, iaxis)
    n = len(c)
    if cnt >= n:
        c = c[:1]*0
    else :
        for i in range(cnt):
            n = n - 1
            c *= scl
            der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
            for j in range(n, 2, -1):
                der[j - 1] = (2*j - 1)*c[j]
                c[j - 2] += c[j]
            if n > 1:
                der[1] = 3*c[2]
            der[0] = c[1]
            c = der
    c = np.rollaxis(c, 0, iaxis + 1)
    return c


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

    Returns the Legendre series coefficients `c` integrated `m` times from
    `lbnd` along `axis`. 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 `c` is an array of
    coefficients from low to high degree along each axix, e.g., [1,2,3]
    represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
    represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
    2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.

    Parameters
    ----------
    c : array_like
        Array of Legendre series coefficients. If c is multidimensional the
        different axis correspond to different variables with the degree in
        each axis given by the corresponding index.
    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)
    axis : int, optional
        Axis over which the derivative is taken. (Default: 0).

    Returns
    -------
    S : ndarray
        Legendre series coefficient array 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
    >>> c = (1,2,3)
    >>> L.legint(cs)
    array([ 0.33333333,  0.4       ,  0.66666667,  0.6       ])
    >>> L.legint(c, 3)
    array([  1.66666667e-02,  -1.78571429e-02,   4.76190476e-02,
            -1.73472348e-18,   1.90476190e-02,   9.52380952e-03])
    >>> L.legint(c, k=3)
    array([ 3.33333333,  0.4       ,  0.66666667,  0.6       ])
    >>> L.legint(c, lbnd=-2)
    array([ 7.33333333,  0.4       ,  0.66666667,  0.6       ])
    >>> L.legint(c, scl=2)
    array([ 0.66666667,  0.8       ,  1.33333333,  1.2       ])

    """
    c = np.array(c, ndmin=1, copy=1)
    if c.dtype.char in '?bBhHiIlLqQpP':
        c = c.astype(np.double)
    if not np.iterable(k):
        k = [k]
    cnt, iaxis = [int(t) for t in [m, axis]]

    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")
    if iaxis != axis:
        raise ValueError("The axis must be integer")
    if not -c.ndim <= iaxis < c.ndim:
        raise ValueError("The axis is out of range")
    if iaxis < 0:
        iaxis += c.ndim

    if cnt == 0:
        return c

    c = np.rollaxis(c, iaxis)
    k = list(k) + [0]*(cnt - len(k))
    for i in range(cnt) :
        n = len(c)
        c *= scl
        if n == 1 and np.all(c[0] == 0):
            c[0] += k[i]
        else:
            tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
            tmp[0] = c[0]*0
            tmp[1] = c[0]
            if n > 1:
                tmp[2] = c[1]/3
            for j in range(2, n):
                t = c[j]/(2*j + 1)
                tmp[j + 1] = t
                tmp[j - 1] -= t
            tmp[0] += k[i] - legval(lbnd, tmp)
            c = tmp
    c = np.rollaxis(c, 0, iaxis + 1)
    return c


def legval(x, c, tensor=True):
    """ Evaluate a Legendre series.

    If `c` is of length ``n + 1``, this function returns the value:

    ``p(x) = c[0]*L_0(x) + c[1]*L_1(x) + ... + c[n]*L_n(x)``

    If `x` is a sequence or array and `c` is 1 dimensional, then ``p(x)``
    will have the same shape as `x`.  If `x` is a algebra_like object that
    supports multiplication and addition with itself and the values in `c`,
    then an object of the same type is returned.

    In the case where c is multidimensional, the shape of the result
    depends on the value of `tensor`. If tensor is true the shape of the
    return will be ``c.shape[1:] + x.shape``, where the shape of a scalar
    is the empty tuple. If tensor is false the shape is ``c.shape[1:]`` if
    `x` is broadcast compatible with that.

    If there are trailing zeros in the coefficients they still take part in
    the evaluation, so they should be avoided if efficiency is a concern.

    Parameters
    ----------
    x : array_like, algebra_like
        If x is a list or tuple, it is converted to an ndarray. Otherwise
        it is left unchanged and if it isn't an ndarray it is treated as a
        scalar. In either case, `x` or any element of an ndarray must
        support addition and multiplication with itself and the elements of
        `c`.
    c : array_like
        Array of coefficients ordered so that the coefficients for terms of
        degree n are contained in ``c[n]``. If `c` is multidimesional the
        remaining indices enumerate multiple Legendre series. In the two
        dimensional case the coefficients may be thought of as stored in
        the columns of `c`.
    tensor : boolean, optional
        If true, the coefficient array shape is extended with ones on the
        right, one for each dimension of `x`. Scalars are treated as having
        dimension 0 for this action. The effect is that every column of
        coefficients in `c` is evaluated for every value in `x`. If False,
        the `x` are broadcast over the columns of `c` in the usual way.
        This gives some flexibility in evaluations in the multidimensional
        case. The default value it ``True``.

    Returns
    -------
    values : ndarray, algebra_like
        The shape of the return value is described above.

    See Also
    --------
    legval2d, leggrid2d, legval3d, leggrid3d

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

    Examples
    --------

    """
    c = np.array(c, ndmin=1, copy=0)
    if c.dtype.char in '?bBhHiIlLqQpP':
        c = c.astype(np.double)
    if isinstance(x, (tuple, list)):
        x = np.asarray(x)
    if isinstance(x, np.ndarray) and tensor:
        c = c.reshape(c.shape + (1,)*x.ndim)

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


def legval2d(x, y, c):
    """
    Evaluate 2D Legendre series at points (x,y).

    This function returns the values:

    ``p(x,y) = \sum_{i,j} c[i,j] * L_i(x) * L_j(y)``

    Parameters
    ----------
    x,y : array_like, algebra_like
        The two dimensional Legendre seres is evaluated at the points
        ``(x,y)``, where `x` and `y` must have the same shape.  If `x` or
        `y` is a list or tuple, it is first converted to an ndarray.
        Otherwise it is left unchanged and if it isn't an ndarray it is
        treated as a scalar.  See `legval` for explanation of algebra_like.
    c : array_like
        Array of coefficients ordered so that the coefficients for terms of
        degree i,j are contained in ``c[i,j]``. If `c` has dimension
        greater than 2 the remaining indices enumerate multiple sets of
        coefficients.

    Returns
    -------
    values : ndarray, algebra_like
        The values of the two dimensional Legendre series at points formed
        from pairs of corresponding values from `x` and `y`.

    See Also
    --------
    legval, leggrid2d, legval3d, leggrid3d

    """
    try:
        x, y = np.array((x, y), copy=0)
    except:
        raise ValueError('x, y are incompatible')

    c = legval(x, c)
    c = legval(y, c, tensor=False)
    return c


def leggrid2d(x, y, c):
    """
    Evaluate 2D Legendre series on the Cartesion product of x,y.

    This function returns the values:

    ``p(a,b) = \sum_{i,j} c[i,j] * L_i(a) * L_j(b)``

    where the points ``(a,b)`` consist of all pairs of points formed by
    taking ``a`` from `x` and ``b`` from `y`. The resulting points form a
    grid with `x` in the first dimension and `y` in the second.

    Parameters
    ----------
    x,y : array_like, algebra_like
        The two dimensional Legendre series is evaluated at the points in
        the Cartesian product of `x` and `y`.  If `x` or `y` is a list or
        tuple, it is first converted to an ndarray, Otherwise it is left
        unchanged and if it isn't an ndarray it is treated as a scalar. See
        `legval` for explanation of algebra_like.
    c : array_like
        Array of coefficients ordered so that the coefficients for terms of
        degree i,j are contained in ``c[i,j]``. If `c` has dimension
        greater than 2 the remaining indices enumerate multiple sets of
        coefficients.

    Returns
    -------
    values : ndarray, algebra_like
        The values of the two dimensional Legendre series at points in the
        Cartesion product of `x` and `y`.

    See Also
    --------
    legval, legval2d, legval3d, leggrid3d

    """
    c = legval(x, c)
    c = legval(y, c)
    return c


def legval3d(x, y, z, c):
    """
    Evaluate 3D Legendre series at points (x,y,z).

    This function returns the values:

    ``p(x,y,z) = \sum_{i,j,k} c[i,j,k] * L_i(x) * L_j(y) * L_k(z)``

    Parameters
    ----------
    x,y,z : array_like, algebra_like
        The three dimensional Legendre seres is evaluated at the points
        ``(x,y,z)``, where `x`, `y`, and `z` must have the same shape.  If
        any of `x`, `y`, or `z` is a list or tuple, it is first converted
        to an ndarray. Otherwise it is left unchanged and if it isn't an
        ndarray it is treated as a scalar.  See `legval` for explanation of
        algebra_like.
    c : array_like
        Array of coefficients ordered so that the coefficients for terms of
        degree i,j are contained in ``c[i,j]``. If `c` has dimension
        greater than 2 the remaining indices enumerate multiple sets of
        coefficients.

    Returns
    -------
    values : ndarray, algebra_like
        The values of the three dimensional Legendre series at points formed
        from triples of corresponding values from `x`, `y`, and `z`.

    See Also
    --------
    legval, legval2d, leggrid2d, leggrid3d

    """
    try:
        x, y, z = np.array((x, y, z), copy=0)
    except:
        raise ValueError('x, y, z are incompatible')

    c = legval(x, c)
    c = legval(y, c, tensor=False)
    c = legval(z, c, tensor=False)
    return c


def leggrid3d(x, y, z, c):
    """
    Evaluate 3D Legendre series on the Cartesian product of x,y,z.

    This function returns the values:

    ``p(a,b,c) = \sum_{i,j,k} c[i,j,k] * L_i(a) * L_j(b) * L_k(c)``

    where the points ``(a,b,c)`` consist of all triples formed by taking
    ``a`` from `x`, ``b`` from `y`, and ``c`` from `z`. The resulting
    points form a grid with `x` in the first dimension, `y` in the second,
    and `z` in the third.

    Parameters
    ----------
    x,y,z : array_like, algebra_like
        The three dimensional Legendre seres is evaluated at the points
        in the cartesian product of `x`, `y`, and `z`
        ``(x,y,z)``, where `x` and `y` must have the same shape.  If `x` or
        `y` is a list or tuple, it is first converted to an ndarray,
        otherwise it is left unchanged and treated as a scalar.  See
        `legval` for explanation of algebra_like.
    c : array_like
        Array of coefficients ordered so that the coefficients for terms of
        degree i,j are contained in ``c[i,j]``. If `c` has dimension
        greater than 2 the remaining indices enumerate multiple sets of
        coefficients.

    Returns
    -------
    values : ndarray, algebra_like
        The values of the three dimensional Legendre series at points formed
        from triples of corresponding values from `x`, `y`, and `z`.

    See Also
    --------
    legval, legval2d, leggrid2d, legval3d

    """
    c = legval(x, c)
    c = legval(y, c)
    c = legval(z, c)
    return c


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 legvander2d(x, y, deg) :
    """Pseudo Vandermonde matrix of given degree.

    Returns the pseudo Vandermonde matrix for 2D Legendre series in `x` and
    `y`. The sample point coordinates must all have the same shape after
    conversion to arrays and the dtype will be converted to either float64
    or complex128 depending on whether any of `x` or 'y' are complex.  The
    maximum degrees of the 2D Legendre series in each variable are specified in
    the list `deg` in the form ``[xdeg, ydeg]``. The return array has the
    shape ``x.shape + (order,)`` if `x`, and `y` are arrays or ``(1, order)
    if they are scalars. Here order is the number of elements in a
    flattened coefficient array of original shape ``(xdeg + 1, ydeg + 1)``.
    The flattening is done so that the resulting pseudo Vandermonde array
    can be easily used in least squares fits.

    Parameters
    ----------
    x,y : array_like
        Arrays of point coordinates, each of the same shape.
    deg : list
        List of maximum degrees of the form [x_deg, y_deg].

    Returns
    -------
    vander2d : ndarray
        The shape of the returned matrix is described above.

    See Also
    --------
    legvander, legvander3d. legval2d, legval3d

    """
    ideg = [int(d) for d in deg]
    is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)]
    if is_valid != [1, 1]:
        raise ValueError("degrees must be non-negative integers")
    degx, degy = ideg
    x, y = np.array((x, y), copy=0) + 0.0

    vx = legvander(x, degx)
    vy = legvander(y, degy)
    v = vx[..., None]*vy[..., None, :]
    return v.reshape(v.shape[:-2] + (-1,))


def legvander3d(x, y, z, deg) :
    """Psuedo Vandermonde matrix of given degree.

    Returns the pseudo Vandermonde matrix for 3D Legendre series in `x`, `y`,
    or `z`. The sample point coordinates must all have the same shape after
    conversion to arrays and the dtype will be converted to either float64
    or complex128 depending on whether any of `x`, `y`, or 'z' are complex.
    The maximum degrees of the 3D Legendre series in each variable are
    specified in the list `deg` in the form ``[xdeg, ydeg, zdeg]``. The
    return array has the shape ``x.shape + (order,)`` if `x`, `y`, and `z`
    are arrays or ``(1, order) if they are scalars. Here order is the
    number of elements in a flattened coefficient array of original shape
    ``(xdeg + 1, ydeg + 1, zdeg + 1)``.  The flattening is done so that the
    resulting pseudo Vandermonde array can be easily used in least squares
    fits.

    Parameters
    ----------
    x,y,z : array_like
        Arrays of point coordinates, each of the same shape.
    deg : list
        List of maximum degrees of the form [x_deg, y_deg, z_deg].

    Returns
    -------
    vander3d : ndarray
        The shape of the returned matrix is described above.

    See Also
    --------
    legvander, legvander3d. legval2d, legval3d

    """
    ideg = [int(d) for d in deg]
    is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)]
    if is_valid != [1, 1, 1]:
        raise ValueError("degrees must be non-negative integers")
    degx, degy, degz = ideg
    x, y, z = np.array((x, y, z), copy=0) + 0.0

    vx = legvander(x, degx)
    vy = legvander(y, degy)
    vz = legvander(z, degz)
    v = vx[..., None, None]*vy[..., None, :, None]*vz[..., None, None, :]
    return v.reshape(v.shape[:-3] + (-1,))


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 legcompanion(cs):
    """Return the scaled companion matrix of cs.

    The basis polynomials are scaled so that the companion matrix is
    symmetric when `cs` represents a single Legendre polynomial. This
    provides better eigenvalue estimates than the unscaled case and in the
    single polynomial case the eigenvalues are guaranteed to be real if
    `numpy.linalg.eigvalsh` is used to obtain them.

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

    Returns
    -------
    mat : ndarray
        Scaled companion matrix of dimensions (deg, deg).

    """
    # cs is a trimmed copy
    [cs] = pu.as_series([cs])
    if len(cs) < 2:
        raise ValueError('Series must have maximum degree of at least 1.')
    if len(cs) == 2:
        return np.array(-cs[0]/cs[1])

    n = len(cs) - 1
    mat = np.zeros((n, n), dtype=cs.dtype)
    scl = 1./np.sqrt(2*np.arange(n) + 1)
    top = mat.reshape(-1)[1::n+1]
    bot = mat.reshape(-1)[n::n+1]
    top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
    bot[...] = top
    mat[:,-1] -= (cs[:-1]/cs[-1])*(scl/scl[-1])*(n/(2*n - 1))
    return mat


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

    Returns 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.
    maxiter : int, optional
        Maximum number of iterations of Newton to use in refining the
        roots.

    Returns
    -------
    out : ndarray
        Sorted 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
    -----
    The root estimates are obtained as the eigenvalues of the companion
    matrix, Roots far from the real interval [-1, 1] in the complex plane
    may have large errors due to the numerical instability of the Lengendre
    series for such values. Roots with multiplicity greater than 1 will
    also show larger errors as the value of the series near such points is
    relatively insensitive to errors in the roots. Isolated roots near the
    interval [-1, 1] can be improved by a few iterations of Newton's
    method.

    The Legendre series basis polynomials aren't powers of ``x`` so the
    results of this function may seem unintuitive.

    Examples
    --------
    >>> import numpy.polynomial.legendre as leg
    >>> leg.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) < 2:
        return np.array([], dtype=cs.dtype)
    if len(cs) == 2:
        return np.array([-cs[0]/cs[1]])

    m = legcompanion(cs)
    r = la.eigvals(m)
    r.sort()
    return r


def leggauss(deg):
    """Gauss Legendre quadrature.

    Computes the sample points and weights for Gauss-Legendre quadrature.
    These sample points and weights will correctly integrate polynomials of
    degree ``2*deg - 1`` or less over the interval ``[-1, 1]`` with the
    weight function ``f(x) = 1``.

    Parameters
    ----------
    deg : int
        Number of sample points and weights. It must be >= 1.

    Returns
    -------
    x : ndarray
        1-D ndarray containing the sample points.
    y : ndarray
        1-D ndarray containing the weights.

    Notes
    -----
    The results have only been tested up to degree 100. Higher degrees may
    be problematic. The weights are determined by using the fact that

          w = c / (L'_n(x_k) * L_{n-1}(x_k))

    where ``c`` is a constant independent of ``k`` and ``x_k`` is the k'th
    root of ``L_n``, and then scaling the results to get the right value
    when integrating 1.

    """
    ideg = int(deg)
    if ideg != deg or ideg < 1:
        raise ValueError("deg must be a non-negative integer")

    # first approximation of roots. We use the fact that the companion
    # matrix is symmetric in this case in order to obtain better zeros.
    c = np.array([0]*deg + [1])
    m = legcompanion(c)
    x = la.eigvals(m)
    x.sort()

    # improve roots by one application of Newton
    dy = legval(x, c)
    df = legval(x, legder(c))
    x -= dy/df

    # compute the weights. We scale the factor to avoid possible numerical
    # overflow.
    fm = legval(x, c[1:])
    fm /= np.abs(fm).max()
    df /= np.abs(df).max()
    w = 1/(fm * df)

    # for Legendre we can also symmetrize
    w = (w + w[::-1])/2
    x = (x - x[::-1])/2

    # scale w to get the right value
    w *= 2. / w.sum()

    return x, w


def legweight(x):
    """Weight function of the Legendre polynomials.

    The weight function for which the Legendre polynomials are orthogonal.
    In this case the weight function is simply one. Note that the Legendre
    polynomials are not normalized.

    Parameters
    ----------
    x : array_like
       Values at which the weight function will be computed.

    Returns
    -------
    w : ndarray
       The weight function at `x`.

    """
    w = x*0.0 + 1.0
    return w

#
# Legendre series class
#

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