"""
==================================================
Laguerre Series (:mod:`numpy.polynomial.laguerre`)
==================================================

This module provides a number of objects (mostly functions) useful for
dealing with Laguerre series, including a `Laguerre` 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/

   Laguerre

Constants
---------
.. autosummary::
   :toctree: generated/

   lagdomain
   lagzero
   lagone
   lagx

Arithmetic
----------
.. autosummary::
   :toctree: generated/

   lagadd
   lagsub
   lagmulx
   lagmul
   lagdiv
   lagpow
   lagval
   lagval2d
   lagval3d
   laggrid2d
   laggrid3d

Calculus
--------
.. autosummary::
   :toctree: generated/

   lagder
   lagint

Misc Functions
--------------
.. autosummary::
   :toctree: generated/

   lagfromroots
   lagroots
   lagvander
   lagvander2d
   lagvander3d
   laggauss
   lagweight
   lagcompanion
   lagfit
   lagtrim
   lagline
   lag2poly
   poly2lag

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__ = [
    'lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline', 'lagadd',
    'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow', 'lagval', 'lagder',
    'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', 'lagvander',
    'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d', 'lagval3d',
    'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d', 'lagcompanion',
    'laggauss', 'lagweight']

lagtrim = pu.trimcoef


def poly2lag(pol):
    """
    poly2lag(pol)

    Convert a polynomial to a Laguerre 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 Laguerre 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 Laguerre
        series.

    See Also
    --------
    lag2poly

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

    Examples
    --------
    >>> from numpy.polynomial.laguerre import poly2lag
    >>> poly2lag(np.arange(4))
    array([ 23., -63.,  58., -18.])

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


def lag2poly(c):
    """
    Convert a Laguerre series to a polynomial.

    Convert an array representing the coefficients of a Laguerre 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 Laguerre 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
    --------
    poly2lag

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

    Examples
    --------
    >>> from numpy.polynomial.laguerre import lag2poly
    >>> lag2poly([ 23., -63.,  58., -18.])
    array([0., 1., 2., 3.])

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

    [c] = pu.as_series([c])
    n = len(c)
    if n == 1:
        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, polysub((2*i - 1)*c1, polymulx(c1))/i)
        return polyadd(c0, polysub(c1, 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.
#

# Laguerre
lagdomain = np.array([0, 1])

# Laguerre coefficients representing zero.
lagzero = np.array([0])

# Laguerre coefficients representing one.
lagone = np.array([1])

# Laguerre coefficients representing the identity x.
lagx = np.array([1, -1])


def lagline(off, scl):
    """
    Laguerre 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 Laguerre series for
        ``off + scl*x``.

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

    Examples
    --------
    >>> from numpy.polynomial.laguerre import lagline, lagval
    >>> lagval(0,lagline(3, 2))
    3.0
    >>> lagval(1,lagline(3, 2))
    5.0

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


def lagfromroots(roots):
    """
    Generate a Laguerre 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 Laguerre 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 Laguerre 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
    --------
    polyfromroots, legfromroots, chebfromroots, hermfromroots, hermefromroots

    Examples
    --------
    >>> from numpy.polynomial.laguerre import lagfromroots, lagval
    >>> coef = lagfromroots((-1, 0, 1))
    >>> lagval((-1, 0, 1), coef)
    array([0.,  0.,  0.])
    >>> coef = lagfromroots((-1j, 1j))
    >>> lagval((-1j, 1j), coef)
    array([0.+0.j, 0.+0.j])

    """
    return pu._fromroots(lagline, lagmul, roots)


def lagadd(c1, c2):
    """
    Add one Laguerre series to another.

    Returns the sum of two Laguerre 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 Laguerre series coefficients ordered from low to
        high.

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

    See Also
    --------
    lagsub, lagmulx, lagmul, lagdiv, lagpow

    Notes
    -----
    Unlike multiplication, division, etc., the sum of two Laguerre series
    is a Laguerre 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.laguerre import lagadd
    >>> lagadd([1, 2, 3], [1, 2, 3, 4])
    array([2.,  4.,  6.,  4.])


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


def lagsub(c1, c2):
    """
    Subtract one Laguerre series from another.