MLPY/Lib/site-packages/numpy/polynomial/legendre.py

"""

==================================================

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

==================================================



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`).



Classes

-------

.. autosummary::

   :toctree: generated/



    Legendre



Constants

---------



.. autosummary::

   :toctree: generated/



   legdomain

   legzero

   legone

   legx



Arithmetic

----------



.. autosummary::

   :toctree: generated/



   legadd

   legsub

   legmulx

   legmul

   legdiv

   legpow

   legval

   legval2d

   legval3d

   leggrid2d

   leggrid3d



Calculus

--------



.. autosummary::

   :toctree: generated/



   legder

   legint



Misc Functions

--------------



.. autosummary::

   :toctree: generated/



   legfromroots

   legroots

   legvander

   legvander2d

   legvander3d

   leggauss

   legweight

   legcompanion

   legfit

   legtrim

   legline

   leg2poly

   poly2leg



See also

--------

numpy.polynomial



"""
import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index

from . import polyutils as pu
from ._polybase import ABCPolyBase

__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

    -------

    c : 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.], domain=[-1,  1], window=[-1,  1])

    >>> c = P.Legendre(P.legendre.poly2leg(p.coef))

    >>> c

    Legendre([ 1.  ,  3.25,  1.  ,  0.75], domain=[-1,  1], window=[-1,  1]) # may vary



    """
    [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(c):
    """

    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

    ----------

    c : 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

    --------

    >>> from numpy import polynomial as P

    >>> c = P.Legendre(range(4))

    >>> c

    Legendre([0., 1., 2., 3.], domain=[-1,  1], window=[-1,  1])

    >>> p = c.convert(kind=P.Polynomial)

    >>> p

    Polynomial([-1. , -3.5,  3. ,  7.5], domain=[-1.,  1.], window=[-1.,  1.])

    >>> P.leg2poly(range(4))

    array([-1. , -3.5,  3. ,  7.5])





    """
    from .polynomial import polyadd, polysub, polymulx

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

    --------

    numpy.polynomial.polynomial.polyline

    numpy.polynomial.chebyshev.chebline

    numpy.polynomial.laguerre.lagline

    numpy.polynomial.hermite.hermline

    numpy.polynomial.hermite_e.hermeline



    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 given roots.



    The function returns the coefficients of the polynomial



    .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),



    in Legendre form, where the `r_n` are the roots specified in `roots`.

    If a zero has multiplicity n, then it must appear in `roots` n times.

    For instance, if 2 is a root of multiplicity three and 3 is a root of

    multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The

    roots can appear in any order.



    If the returned coefficients are `c`, then



    .. math:: p(x) = c_0 + c_1 * L_1(x) + ... +  c_n * L_n(x)



    The coefficient of the last term is not generally 1 for monic

    polynomials in Legendre form.



    Parameters

    ----------

    roots : array_like

        Sequence containing the roots.



    Returns

    -------

    out : ndarray

        1-D array of coefficients.  If all roots are real then `out` is a

        real array, if some of the roots are complex, then `out` is complex

        even if all the coefficients in the result are real (see Examples

        below).



    See Also

    --------

    numpy.polynomial.polynomial.polyfromroots

    numpy.polynomial.chebyshev.chebfromroots

    numpy.polynomial.laguerre.lagfromroots

    numpy.polynomial.hermite.hermfromroots

    numpy.polynomial.hermite_e.hermefromroots



    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]) # may vary



    """
    return pu._fromroots(legline, legmul, roots)


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, legmulx, 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.])



    """
    return pu._add(c1, c2)


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, legmulx, 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.])



    """
    return pu._sub(c1, c2)


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



    Multiply the Legendre series `c` by x, where x is the independent

    variable.





    Parameters

    ----------

    c : array_like

        1-D array of Legendre series coefficients ordered from low to

        high.



    Returns

    -------

    out : ndarray

        Array representing the result of the multiplication.



    See Also

    --------

    legadd, legmul, legmul, legdiv, legpow



    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)



    Examples

    --------

    >>> from numpy.polynomial import legendre as L

    >>> L.legmulx([1,2,3])

    array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary



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

    prd = np.empty(len(c) + 1, dtype=c.dtype)
    prd[0] = c[0]*0
    prd[1] = c[0]
    for i in range(1, len(c)):
        j = i + 1
        k = i - 1
        s = i + j
        prd[j] = (c[i]*j)/s
        prd[k] += (c[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, legmulx, 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 "reproject" the

    product onto said basis set, which may produce "unintuitive" (but

    correct) results; see Examples section below.



    Examples

    --------

    >>> from numpy.polynomial import legendre as L

    >>> c1 = (1,2,3)

    >>> c2 = (3,2)

    >>> L.legmul(c1,c2) # multiplication requires "reprojection"

    array([  4.33333333,  10.4       ,  11.66666667,   3.6       ]) # may vary



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

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

    if len(c) == 1:
        c0 = c[0]*xs
        c1 = 0
    elif len(c) == 2:
        c0 = c[0]*xs
        c1 = c[1]*xs
    else:
        nd = len(c)
        c0 = c[-2]*xs
        c1 = c[-1]*xs
        for i in range(3, len(c) + 1):
            tmp = c0
            nd = nd - 1
            c0 = legsub(c[-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, legmulx, 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 "reproject" the results onto the Legendre

    basis set, which may produce "unintuitive" (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])) # may vary



    """
    return pu._div(legmul, c1, c2)


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



    Returns the Legendre series `c` raised to the power `pow`. The

    argument `c` 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

    ----------

    c : array_like

        1-D 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 unmanageable size. Default is 16



    Returns

    -------

    coef : ndarray

        Legendre series of power.



    See Also

    --------

    legadd, legsub, legmulx, legmul, legdiv



    """
    return pu._pow(legmul, c, pow, maxpower)


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).



        .. versionadded:: 1.7.0



    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 "unintuitive," 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=True)
    if c.dtype.char in '?bBhHiIlLqQpP':
        c = c.astype(np.double)
    cnt = pu._deprecate_as_int(m, "the order of derivation")
    iaxis = pu._deprecate_as_int(axis, "the axis")
    if cnt < 0:
        raise ValueError("The order of derivation must be non-negative")
    iaxis = normalize_axis_index(iaxis, c.ndim)

    if cnt == 0:
        return c

    c = np.moveaxis(c, iaxis, 0)
    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.moveaxis(c, 0, iaxis)
    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 axis, 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 integral is taken. (Default: 0).



        .. versionadded:: 1.7.0



    Returns

    -------

    S : ndarray

        Legendre series coefficient array of the integral.



    Raises

    ------

    ValueError

        If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or

        ``np.ndim(scl) != 0``.



    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 "reprojected" onto the C-series basis set.  Thus, typically,

    the result of this function is "unintuitive," albeit correct; see

    Examples section below.



    Examples

    --------

    >>> from numpy.polynomial import legendre as L

    >>> c = (1,2,3)

    >>> L.legint(c)

    array([ 0.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary

    >>> L.legint(c, 3)

    array([  1.66666667e-02,  -1.78571429e-02,   4.76190476e-02, # may vary

             -1.73472348e-18,   1.90476190e-02,   9.52380952e-03])

    >>> L.legint(c, k=3)

     array([ 3.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary

    >>> L.legint(c, lbnd=-2)

    array([ 7.33333333,  0.4       ,  0.66666667,  0.6       ]) # may vary

    >>> L.legint(c, scl=2)

    array([ 0.66666667,  0.8       ,  1.33333333,  1.2       ]) # may vary



    """
    c = np.array(c, ndmin=1, copy=True)
    if c.dtype.char in '?bBhHiIlLqQpP':
        c = c.astype(np.double)
    if not np.iterable(k):
        k = [k]
    cnt = pu._deprecate_as_int(m, "the order of integration")
    iaxis = pu._deprecate_as_int(axis, "the axis")
    if cnt < 0:
        raise ValueError("The order of integration must be non-negative")
    if len(k) > cnt:
        raise ValueError("Too many integration constants")
    if np.ndim(lbnd) != 0:
        raise ValueError("lbnd must be a scalar.")
    if np.ndim(scl) != 0:
        raise ValueError("scl must be a scalar.")
    iaxis = normalize_axis_index(iaxis, c.ndim)

    if cnt == 0:
        return c

    c = np.moveaxis(c, iaxis, 0)
    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.moveaxis(c, 0, iaxis)
    return c


def legval(x, c, tensor=True):
    """

    Evaluate a Legendre series at points x.



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



    .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)



    The parameter `x` is converted to an array only if it is a tuple or a

    list, otherwise it is treated as a scalar. In either case, either `x`

    or its elements must support multiplication and addition both with

    themselves and with the elements of `c`.



    If `c` is a 1-D array, then `p(x)` will have the same shape as `x`.  If

    `c` is multidimensional, then the shape of the result depends on the

    value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +

    x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that

    scalars have shape (,).



    Trailing zeros in the coefficients will be used in the evaluation, so

    they should be avoided if efficiency is a concern.



    Parameters

    ----------

    x : array_like, compatible object

        If `x` is a list or tuple, it is converted to an ndarray, otherwise

        it is left unchanged and treated as a scalar. In either case, `x`

        or its elements must support addition and multiplication with

        with themselves and with 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 multidimensional the

        remaining indices enumerate multiple polynomials. In the two

        dimensional case the coefficients may be thought of as stored in

        the columns of `c`.

    tensor : boolean, optional

        If True, the shape of the coefficient array is extended with ones

        on the right, one for each dimension of `x`. Scalars have dimension 0

        for this action. The result is that every column of coefficients in

        `c` is evaluated for every element of `x`. If False, `x` is broadcast

        over the columns of `c` for the evaluation.  This keyword is useful

        when `c` is multidimensional. The default value is True.



        .. versionadded:: 1.7.0



    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.



    """
    c = np.array(c, ndmin=1, copy=False)
    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 a 2-D Legendre series at points (x, y).



    This function returns the values:



    .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)



    The parameters `x` and `y` are converted to arrays only if they are

    tuples or a lists, otherwise they are treated as a scalars and they

    must have the same shape after conversion. In either case, either `x`

    and `y` or their elements must support multiplication and addition both

    with themselves and with the elements of `c`.



    If `c` is a 1-D array a one is implicitly appended to its shape to make

    it 2-D. The shape of the result will be c.shape[2:] + x.shape.



    Parameters

    ----------

    x, y : array_like, compatible objects

        The two dimensional series 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.

    c : array_like

        Array of coefficients ordered so that the coefficient of the term

        of multi-degree i,j is contained in ``c[i,j]``. If `c` has

        dimension greater than two the remaining indices enumerate multiple

        sets of coefficients.



    Returns

    -------

    values : ndarray, compatible object

        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



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._valnd(legval, c, x, y)


def leggrid2d(x, y, c):
    """

    Evaluate a 2-D Legendre series on the Cartesian product of x and y.



    This function returns the values:



    .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)



    where the points `(a, b)` consist of all pairs 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.



    The parameters `x` and `y` are converted to arrays only if they are

    tuples or a lists, otherwise they are treated as a scalars. In either

    case, either `x` and `y` or their elements must support multiplication

    and addition both with themselves and with the elements of `c`.



    If `c` has fewer than two dimensions, ones are implicitly appended to

    its shape to make it 2-D. The shape of the result will be c.shape[2:] +

    x.shape + y.shape.



    Parameters

    ----------

    x, y : array_like, compatible objects

        The two dimensional 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.

    c : array_like

        Array of coefficients ordered so that the coefficient of the term of

        multi-degree i,j is contained in `c[i,j]`. If `c` has dimension

        greater than two the remaining indices enumerate multiple sets of

        coefficients.



    Returns

    -------

    values : ndarray, compatible object

        The values of the two dimensional Chebyshev series at points in the

        Cartesian product of `x` and `y`.



    See Also

    --------

    legval, legval2d, legval3d, leggrid3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._gridnd(legval, c, x, y)


def legval3d(x, y, z, c):
    """

    Evaluate a 3-D Legendre series at points (x, y, z).



    This function returns the values:



    .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)



    The parameters `x`, `y`, and `z` are converted to arrays only if

    they are tuples or a lists, otherwise they are treated as a scalars and

    they must have the same shape after conversion. In either case, either

    `x`, `y`, and `z` or their elements must support multiplication and

    addition both with themselves and with the elements of `c`.



    If `c` has fewer than 3 dimensions, ones are implicitly appended to its

    shape to make it 3-D. The shape of the result will be c.shape[3:] +

    x.shape.



    Parameters

    ----------

    x, y, z : array_like, compatible object

        The three dimensional series 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.

    c : array_like

        Array of coefficients ordered so that the coefficient of the term of

        multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension

        greater than 3 the remaining indices enumerate multiple sets of

        coefficients.



    Returns

    -------

    values : ndarray, compatible object

        The values of the multidimensional polynomial on points formed with

        triples of corresponding values from `x`, `y`, and `z`.



    See Also

    --------

    legval, legval2d, leggrid2d, leggrid3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._valnd(legval, c, x, y, z)


def leggrid3d(x, y, z, c):
    """

    Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z.



    This function returns the values:



    .. math:: 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.



    The parameters `x`, `y`, and `z` are converted to arrays only if they

    are tuples or a lists, otherwise they are treated as a scalars. In

    either case, either `x`, `y`, and `z` or their elements must support

    multiplication and addition both with themselves and with the elements

    of `c`.



    If `c` has fewer than three dimensions, ones are implicitly appended to

    its shape to make it 3-D. The shape of the result will be c.shape[3:] +

    x.shape + y.shape + z.shape.



    Parameters

    ----------

    x, y, z : array_like, compatible objects

        The three dimensional series is evaluated at the points in the

        Cartesian product of `x`, `y`, and `z`.  If `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.

    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 two the remaining indices enumerate multiple sets of

        coefficients.



    Returns

    -------

    values : ndarray, compatible object

        The values of the two dimensional polynomial at points in the Cartesian

        product of `x` and `y`.



    See Also

    --------

    legval, legval2d, leggrid2d, legval3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._gridnd(legval, c, x, y, z)


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



    Returns the pseudo-Vandermonde matrix of degree `deg` and sample points

    `x`. The pseudo-Vandermonde matrix is defined by



    .. math:: V[..., i] = L_i(x)



    where `0 <= i <= deg`. The leading indices of `V` index the elements of

    `x` and the last index is the degree of the Legendre polynomial.



    If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the

    array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and

    ``legval(x, c)`` are the same up to roundoff. This equivalence is

    useful both for least squares fitting and for the evaluation of a large

    number of Legendre series of the same degree and sample points.



    Parameters

    ----------

    x : array_like

        Array of points. The dtype is converted to float64 or complex128

        depending on whether any of the elements are complex. If `x` is

        scalar it is converted to a 1-D array.

    deg : int

        Degree of the resulting matrix.



    Returns

    -------

    vander : ndarray

        The pseudo-Vandermonde matrix. The shape of the returned matrix is

        ``x.shape + (deg + 1,)``, where The last index is the degree of the

        corresponding Legendre polynomial.  The dtype will be the same as

        the converted `x`.



    """
    ideg = pu._deprecate_as_int(deg, "deg")
    if ideg < 0:
        raise ValueError("deg must be non-negative")

    x = np.array(x, copy=False, ndmin=1) + 0.0
    dims = (ideg + 1,) + x.shape
    dtyp = x.dtype
    v = np.empty(dims, dtype=dtyp)
    # 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.moveaxis(v, 0, -1)


def legvander2d(x, y, deg):
    """Pseudo-Vandermonde matrix of given degrees.



    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample

    points `(x, y)`. The pseudo-Vandermonde matrix is defined by



    .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),



    where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of

    `V` index the points `(x, y)` and the last index encodes the degrees of

    the Legendre polynomials.



    If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`

    correspond to the elements of a 2-D coefficient array `c` of shape

    (xdeg + 1, ydeg + 1) in the order



    .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...



    and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same

    up to roundoff. This equivalence is useful both for least squares

    fitting and for the evaluation of a large number of 2-D Legendre

    series of the same degrees and sample points.



    Parameters

    ----------

    x, y : array_like

        Arrays of point coordinates, all of the same shape. The dtypes

        will be converted to either float64 or complex128 depending on

        whether any of the elements are complex. Scalars are converted to

        1-D arrays.

    deg : list of ints

        List of maximum degrees of the form [x_deg, y_deg].



    Returns

    -------

    vander2d : ndarray

        The shape of the returned matrix is ``x.shape + (order,)``, where

        :math:`order = (deg[0]+1)*(deg[1]+1)`.  The dtype will be the same

        as the converted `x` and `y`.



    See Also

    --------

    legvander, legvander3d, legval2d, legval3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._vander_nd_flat((legvander, legvander), (x, y), deg)


def legvander3d(x, y, z, deg):
    """Pseudo-Vandermonde matrix of given degrees.



    Returns the pseudo-Vandermonde matrix of degrees `deg` and sample

    points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,

    then The pseudo-Vandermonde matrix is defined by



    .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),



    where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`.  The leading

    indices of `V` index the points `(x, y, z)` and the last index encodes

    the degrees of the Legendre polynomials.



    If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns

    of `V` correspond to the elements of a 3-D coefficient array `c` of

    shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order



    .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...



    and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the

    same up to roundoff. This equivalence is useful both for least squares

    fitting and for the evaluation of a large number of 3-D Legendre

    series of the same degrees and sample points.



    Parameters

    ----------

    x, y, z : array_like

        Arrays of point coordinates, all of the same shape. The dtypes will

        be converted to either float64 or complex128 depending on whether

        any of the elements are complex. Scalars are converted to 1-D

        arrays.

    deg : list of ints

        List of maximum degrees of the form [x_deg, y_deg, z_deg].



    Returns

    -------

    vander3d : ndarray

        The shape of the returned matrix is ``x.shape + (order,)``, where

        :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`.  The dtype will

        be the same as the converted `x`, `y`, and `z`.



    See Also

    --------

    legvander, legvander3d, legval2d, legval3d



    Notes

    -----



    .. versionadded:: 1.7.0



    """
    return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg)


def legfit(x, y, deg, rcond=None, full=False, w=None):
    """

    Least squares fit of Legendre series to data.



    Return the coefficients of a Legendre series of degree `deg` that is the

    least squares fit to the data values `y` given at points `x`. If `y` is

    1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple

    fits are done, one for each column of `y`, and the resulting

    coefficients are stored in the corresponding columns of a 2-D return.

    The fitted polynomial(s) are in the form



    .. math::  p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),



    where `n` is `deg`.



    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 or 1-D array_like

        Degree(s) of the fitting polynomials. If `deg` is a single integer

        all terms up to and including the `deg`'th term are included in the

        fit. For NumPy versions >= 1.11.0 a list of integers specifying the

        degrees of the terms to include may be used instead.

    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.



        .. versionadded:: 1.5.0



    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`. If `deg` is specified as a list, coefficients for

        terms not included in the fit are set equal to zero in the

        returned `coef`.



    [residuals, rank, singular_values, rcond] : list

        These values are only returned if `full` = True



        resid -- sum of squared residuals of the least squares fit

        rank -- the numerical rank of the scaled Vandermonde matrix

        sv -- singular values of the scaled Vandermonde matrix

        rcond -- value of `rcond`.



        For more details, see `numpy.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', np.RankWarning)



    See Also

    --------

    numpy.polynomial.polynomial.polyfit

    numpy.polynomial.chebyshev.chebfit

    numpy.polynomial.laguerre.lagfit

    numpy.polynomial.hermite.hermfit

    numpy.polynomial.hermite_e.hermefit

    legval : Evaluates a Legendre series.

    legvander : Vandermonde matrix of Legendre series.

    legweight : Legendre weight function (= 1).

    numpy.linalg.lstsq : Computes a least-squares fit from the matrix.

    scipy.interpolate.UnivariateSpline : Computes spline fits.



    Notes

    -----

    The solution is the coefficients of the Legendre series `p` that

    minimizes the sum of the weighted squared errors



    .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,



    where :math:`w_j` are the weights. This problem is solved by setting up

    as the (typically) overdetermined matrix equation



    .. math:: V(x) * c = w * y,



    where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the

    coefficients to be solved for, `w` are the weights, and `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

    coefficient 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",

           https://en.wikipedia.org/wiki/Curve_fitting



    Examples

    --------



    """
    return pu._fit(legvander, x, y, deg, rcond, full, w)


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



    The basis polynomials are scaled so that the companion matrix is

    symmetric when `c` is an Legendre basis polynomial. This provides

    better eigenvalue estimates than the unscaled case and for basis

    polynomials the eigenvalues are guaranteed to be real if

    `numpy.linalg.eigvalsh` is used to obtain them.



    Parameters

    ----------

    c : 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).



    Notes

    -----



    .. versionadded:: 1.7.0



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

    n = len(c) - 1
    mat = np.zeros((n, n), dtype=c.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] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
    return mat


def legroots(c):
    """

    Compute the roots of a Legendre series.



    Return the roots (a.k.a. "zeros") of the polynomial



    .. math:: p(x) = \\sum_i c[i] * L_i(x).



    Parameters

    ----------

    c : 1-D array_like

        1-D array of coefficients.



    Returns

    -------

    out : ndarray

        Array of the roots of the series. If all the roots are real,

        then `out` is also real, otherwise it is complex.



    See Also

    --------

    numpy.polynomial.polynomial.polyroots

    numpy.polynomial.chebyshev.chebroots

    numpy.polynomial.laguerre.lagroots

    numpy.polynomial.hermite.hermroots

    numpy.polynomial.hermite_e.hermeroots



    Notes

    -----

    The root estimates are obtained as the eigenvalues of the companion

    matrix, Roots far from the origin of the complex plane may have large

    errors due to the numerical instability of the 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 origin 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, all real roots

    array([-0.85099543, -0.11407192,  0.51506735]) # may vary



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

    # rotated companion matrix reduces error
    m = legcompanion(c)[::-1,::-1]
    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 :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with

    the weight function :math:`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

    -----



    .. versionadded:: 1.7.0



    The results have only been tested up to degree 100, higher degrees may

    be problematic. The weights are determined by using the fact that



    .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))



    where :math:`c` is a constant independent of :math:`k` and :math:`x_k`

    is the k'th root of :math:`L_n`, and then scaling the results to get

    the right value when integrating 1.



    """
    ideg = pu._deprecate_as_int(deg, "deg")
    if ideg <= 0:
        raise ValueError("deg must be a positive 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.eigvalsh(m)

    # 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 is :math:`1` and the interval of integration is

    :math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not

    normalized, with respect to this weight function.



    Parameters

    ----------

    x : array_like

       Values at which the weight function will be computed.



    Returns

    -------

    w : ndarray

       The weight function at `x`.



    Notes

    -----



    .. versionadded:: 1.7.0



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

#
# Legendre series class
#

class Legendre(ABCPolyBase):
    """A Legendre series class.



    The Legendre class provides the standard Python numerical methods

    '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the

    attributes and methods listed in the `ABCPolyBase` documentation.



    Parameters

    ----------

    coef : array_like

        Legendre coefficients in order of increasing degree, i.e.,

        ``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``.

    domain : (2,) array_like, optional

        Domain to use. The interval ``[domain[0], domain[1]]`` is mapped

        to the interval ``[window[0], window[1]]`` by shifting and scaling.

        The default value is [-1, 1].

    window : (2,) array_like, optional

        Window, see `domain` for its use. The default value is [-1, 1].



        .. versionadded:: 1.6.0



    """
    # Virtual Functions
    _add = staticmethod(legadd)
    _sub = staticmethod(legsub)
    _mul = staticmethod(legmul)
    _div = staticmethod(legdiv)
    _pow = staticmethod(legpow)
    _val = staticmethod(legval)
    _int = staticmethod(legint)
    _der = staticmethod(legder)
    _fit = staticmethod(legfit)
    _line = staticmethod(legline)
    _roots = staticmethod(legroots)
    _fromroots = staticmethod(legfromroots)

    # Virtual properties
    domain = np.array(legdomain)
    window = np.array(legdomain)
    basis_name = 'P'