"""
==============================================================
Hermite Series, "Physicists" (:mod:`numpy.polynomial.hermite`)
==============================================================

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

   Hermite

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

   hermdomain
   hermzero
   hermone
   hermx

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

   hermadd
   hermsub
   hermmulx
   hermmul
   hermdiv
   hermpow
   hermval
   hermval2d
   hermval3d
   hermgrid2d
   hermgrid3d

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

   hermder
   hermint

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

   hermfromroots
   hermroots
   hermvander
   hermvander2d
   hermvander3d
   hermgauss
   hermweight
   hermcompanion
   hermfit
   hermtrim
   hermline
   herm2poly
   poly2herm

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__ = [
    'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd',
    'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval',
    'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots',
    'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite',
    'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d',
    'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight']

hermtrim = pu.trimcoef


def poly2herm(pol):
    """
    poly2herm(pol)

    Convert a polynomial to a Hermite 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 Hermite 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 Hermite
        series.

    See Also
    --------
    herm2poly

    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.hermite import poly2herm
    >>> poly2herm(np.arange(4))
    array([1.   ,  2.75 ,  0.5  ,  0.375])

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


def herm2poly(c):
    """
    Convert a Hermite series to a polynomial.

    Convert an array representing the coefficients of a Hermite 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 Hermite 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
    --------
    poly2herm

    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.hermite import herm2poly
    >>> herm2poly([ 1.   ,  2.75 ,  0.5  ,  0.375])
    array([0., 1., 2., 3.])

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

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

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

# Hermite
hermdomain = np.array([-1, 1])

# Hermite coefficients representing zero.
hermzero = np.array([0])

# Hermite coefficients representing one.
hermone = np.array([1])

# Hermite coefficients representing the identity x.
hermx = np.array([0, 1/2])


def hermline(off, scl):
    """
    Hermite 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 Hermite series for
        ``off + scl*x``.

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

    Examples
    --------
    >>> from numpy.polynomial.hermite import hermline, hermval
    >>> hermval(0,hermline(3, 2))
    3.0
    >>> hermval(1,hermline(3, 2))
    5.0

    """
    if scl != 0:
        return np.array([off, scl/2])
    else:
        return np.array([off])


def hermfromroots(roots):
    """
    Generate a Hermite 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 Hermite 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 * H_1(x) + ... +  c_n * H_n(x)

    The coefficient of the last term is not generally 1 for monic
    polynomials in Hermite 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, lagfromroots, chebfromroots, hermefromroots

    Examples
    --------
    >>> from numpy.polynomial.hermite import hermfromroots, hermval
    >>> coef = hermfromroots((-1, 0, 1))
    >>> hermval((-1, 0, 1), coef)
    array([0.,  0.,  0.])
    >>> coef = hermfromroots((-1j, 1j))
    >>> hermval((-1j, 1j), coef)
    array([0.+0.j, 0.+0.j])

    """
    return pu._fromroots(hermline, hermmul, roots)


def hermadd(c1, c2):
    """
    Add one Hermite series to another.

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

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

    See Also
    --------
    hermsub, hermmulx, hermmul, hermdiv, hermpow

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

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


def hermsub(c1, c2):