MLPY/Lib/site-packages/sympy/polys/orthopolys.py

"""Efficient functions for generating orthogonal polynomials."""
from sympy.core.symbol import Dummy
from sympy.polys.densearith import (dup_mul, dup_mul_ground,
    dup_lshift, dup_sub, dup_add, dup_sub_term, dup_sub_ground, dup_sqr)
from sympy.polys.domains import ZZ, QQ
from sympy.polys.polytools import named_poly
from sympy.utilities import public

def dup_jacobi(n, a, b, K):
    """Low-level implementation of Jacobi polynomials."""
    if n < 1:
        return [K.one]
    m2, m1 = [K.one], [(a+b)/K(2) + K.one, (a-b)/K(2)]
    for i in range(2, n+1):
        den = K(i)*(a + b + i)*(a + b + K(2)*i - K(2))
        f0 = (a + b + K(2)*i - K.one) * (a*a - b*b) / (K(2)*den)
        f1 = (a + b + K(2)*i - K.one) * (a + b + K(2)*i - K(2)) * (a + b + K(2)*i) / (K(2)*den)
        f2 = (a + i - K.one)*(b + i - K.one)*(a + b + K(2)*i) / den
        p0 = dup_mul_ground(m1, f0, K)
        p1 = dup_mul_ground(dup_lshift(m1, 1, K), f1, K)
        p2 = dup_mul_ground(m2, f2, K)
        m2, m1 = m1, dup_sub(dup_add(p0, p1, K), p2, K)
    return m1

@public
def jacobi_poly(n, a, b, x=None, polys=False):
    r"""Generates the Jacobi polynomial `P_n^{(a,b)}(x)`.

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    a
        Lower limit of minimal domain for the list of coefficients.
    b
        Upper limit of minimal domain for the list of coefficients.
    x : optional
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.
    """
    return named_poly(n, dup_jacobi, None, "Jacobi polynomial", (x, a, b), polys)

def dup_gegenbauer(n, a, K):
    """Low-level implementation of Gegenbauer polynomials."""
    if n < 1:
        return [K.one]
    m2, m1 = [K.one], [K(2)*a, K.zero]
    for i in range(2, n+1):
        p1 = dup_mul_ground(dup_lshift(m1, 1, K), K(2)*(a-K.one)/K(i) + K(2), K)
        p2 = dup_mul_ground(m2, K(2)*(a-K.one)/K(i) + K.one, K)
        m2, m1 = m1, dup_sub(p1, p2, K)
    return m1

def gegenbauer_poly(n, a, x=None, polys=False):
    r"""Generates the Gegenbauer polynomial `C_n^{(a)}(x)`.

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    x : optional
    a
        Decides minimal domain for the list of coefficients.
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.
    """
    return named_poly(n, dup_gegenbauer, None, "Gegenbauer polynomial", (x, a), polys)

def dup_chebyshevt(n, K):
    """Low-level implementation of Chebyshev polynomials of the first kind."""
    if n < 1:
        return [K.one]
    # When n is small, it is faster to directly calculate the recurrence relation.
    if n < 64: # The threshold serves as a heuristic
        return _dup_chebyshevt_rec(n, K)
    return _dup_chebyshevt_prod(n, K)

def _dup_chebyshevt_rec(n, K):
    r""" Chebyshev polynomials of the first kind using recurrence.

    Explanation
    ===========

    Chebyshev polynomials of the first kind are defined by the recurrence
    relation:

    .. math::
        T_0(x) &= 1\\
        T_1(x) &= x\\
        T_n(x) &= 2xT_{n-1}(x) - T_{n-2}(x)

    This function calculates the Chebyshev polynomial of the first kind using
    the above recurrence relation.

    Parameters
    ==========

    n : int
        n is a nonnegative integer.
    K : domain

    """
    m2, m1 = [K.one], [K.one, K.zero]
    for _ in range(n - 1):
        m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2), K), m2, K)
    return m1

def _dup_chebyshevt_prod(n, K):
    r""" Chebyshev polynomials of the first kind using recursive products.

    Explanation
    ===========

    Computes Chebyshev polynomials of the first kind using

    .. math::
        T_{2n}(x) &= 2T_n^2(x) - 1\\
        T_{2n+1}(x) &= 2T_{n+1}(x)T_n(x) - x

    This is faster than ``_dup_chebyshevt_rec`` for large ``n``.

    Parameters
    ==========

    n : int
        n is a nonnegative integer.
    K : domain

    """
    m2, m1 = [K.one, K.zero], [K(2), K.zero, -K.one]
    for i in bin(n)[3:]:
        c = dup_sub_term(dup_mul_ground(dup_mul(m1, m2, K), K(2), K), K.one, 1, K)
        if  i  == '1':
            m2, m1 = c, dup_sub_ground(dup_mul_ground(dup_sqr(m1, K), K(2), K), K.one, K)
        else:
            m2, m1 = dup_sub_ground(dup_mul_ground(dup_sqr(m2, K), K(2), K), K.one, K), c
    return m2

def dup_chebyshevu(n, K):
    """Low-level implementation of Chebyshev polynomials of the second kind."""
    if n < 1:
        return [K.one]
    m2, m1 = [K.one], [K(2), K.zero]
    for i in range(2, n+1):
        m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2), K), m2, K)
    return m1

@public
def chebyshevt_poly(n, x=None, polys=False):
    r"""Generates the Chebyshev polynomial of the first kind `T_n(x)`.

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    x : optional
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.
    """
    return named_poly(n, dup_chebyshevt, ZZ,
            "Chebyshev polynomial of the first kind", (x,), polys)

@public
def chebyshevu_poly(n, x=None, polys=False):
    r"""Generates the Chebyshev polynomial of the second kind `U_n(x)`.

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    x : optional
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.
    """
    return named_poly(n, dup_chebyshevu, ZZ,
            "Chebyshev polynomial of the second kind", (x,), polys)

def dup_hermite(n, K):
    """Low-level implementation of Hermite polynomials."""
    if n < 1:
        return [K.one]
    m2, m1 = [K.one], [K(2), K.zero]
    for i in range(2, n+1):
        a = dup_lshift(m1, 1, K)
        b = dup_mul_ground(m2, K(i-1), K)
        m2, m1 = m1, dup_mul_ground(dup_sub(a, b, K), K(2), K)
    return m1

def dup_hermite_prob(n, K):
    """Low-level implementation of probabilist's Hermite polynomials."""
    if n < 1:
        return [K.one]
    m2, m1 = [K.one], [K.one, K.zero]
    for i in range(2, n+1):
        a = dup_lshift(m1, 1, K)
        b = dup_mul_ground(m2, K(i-1), K)
        m2, m1 = m1, dup_sub(a, b, K)
    return m1

@public
def hermite_poly(n, x=None, polys=False):
    r"""Generates the Hermite polynomial `H_n(x)`.

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    x : optional
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.
    """
    return named_poly(n, dup_hermite, ZZ, "Hermite polynomial", (x,), polys)

@public
def hermite_prob_poly(n, x=None, polys=False):
    r"""Generates the probabilist's Hermite polynomial `He_n(x)`.

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    x : optional
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.
    """
    return named_poly(n, dup_hermite_prob, ZZ,
            "probabilist's Hermite polynomial", (x,), polys)

def dup_legendre(n, K):
    """Low-level implementation of Legendre polynomials."""
    if n < 1:
        return [K.one]
    m2, m1 = [K.one], [K.one, K.zero]
    for i in range(2, n+1):
        a = dup_mul_ground(dup_lshift(m1, 1, K), K(2*i-1, i), K)
        b = dup_mul_ground(m2, K(i-1, i), K)
        m2, m1 = m1, dup_sub(a, b, K)
    return m1

@public
def legendre_poly(n, x=None, polys=False):
    r"""Generates the Legendre polynomial `P_n(x)`.

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    x : optional
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.
    """
    return named_poly(n, dup_legendre, QQ, "Legendre polynomial", (x,), polys)

def dup_laguerre(n, alpha, K):
    """Low-level implementation of Laguerre polynomials."""
    m2, m1 = [K.zero], [K.one]
    for i in range(1, n+1):
        a = dup_mul(m1, [-K.one/K(i), (alpha-K.one)/K(i) + K(2)], K)
        b = dup_mul_ground(m2, (alpha-K.one)/K(i) + K.one, K)
        m2, m1 = m1, dup_sub(a, b, K)
    return m1

@public
def laguerre_poly(n, x=None, alpha=0, polys=False):
    r"""Generates the Laguerre polynomial `L_n^{(\alpha)}(x)`.

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    x : optional
    alpha : optional
        Decides minimal domain for the list of coefficients.
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.
    """
    return named_poly(n, dup_laguerre, None, "Laguerre polynomial", (x, alpha), polys)

def dup_spherical_bessel_fn(n, K):
    """Low-level implementation of fn(n, x)."""
    if n < 1:
        return [K.one, K.zero]
    m2, m1 = [K.one], [K.one, K.zero]
    for i in range(2, n+1):
        m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(2*i-1), K), m2, K)
    return dup_lshift(m1, 1, K)

def dup_spherical_bessel_fn_minus(n, K):
    """Low-level implementation of fn(-n, x)."""
    m2, m1 = [K.one, K.zero], [K.zero]
    for i in range(2, n+1):
        m2, m1 = m1, dup_sub(dup_mul_ground(dup_lshift(m1, 1, K), K(3-2*i), K), m2, K)
    return m1

def spherical_bessel_fn(n, x=None, polys=False):
    """
    Coefficients for the spherical Bessel functions.

    These are only needed in the jn() function.

    The coefficients are calculated from:

    fn(0, z) = 1/z
    fn(1, z) = 1/z**2
    fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)

    Parameters
    ==========

    n : int
        Degree of the polynomial.
    x : optional
    polys : bool, optional
        If True, return a Poly, otherwise (default) return an expression.

    Examples
    ========

    >>> from sympy.polys.orthopolys import spherical_bessel_fn as fn
    >>> from sympy import Symbol
    >>> z = Symbol("z")
    >>> fn(1, z)
    z**(-2)
    >>> fn(2, z)
    -1/z + 3/z**3
    >>> fn(3, z)
    -6/z**2 + 15/z**4
    >>> fn(4, z)
    1/z - 45/z**3 + 105/z**5

    """
    if x is None:
        x = Dummy("x")
    f = dup_spherical_bessel_fn_minus if n < 0 else dup_spherical_bessel_fn
    return named_poly(abs(n), f, ZZ, "", (QQ(1)/x,), polys)