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"""
Evaluate orthogonal polynomial values using recurrence relations
or by calling special functions.
References
----------
.. [AMS55] Abramowitz & Stegun, Section 22.5.
.. [MH] Mason & Handscombe, Chebyshev Polynomials, CRC Press (2003).
"""
#
# Copyright (C) 2009 Pauli Virtanen
# Distributed under the same license as Scipy.
#
#------------------------------------------------------------------------------
# Direct evaluation of polynomials
#------------------------------------------------------------------------------
cdef extern from "math.h":
double sqrt(double x) nogil
cdef double eval_poly_chebyt(long k, double x) nogil:
# Use Chebyshev T recurrence directly, see [MH]
cdef long m
cdef double b2, b1, b0
b2 = 0
b1 = -1
b0 = 0
x = 2*x
for m in range(k+1):
b2 = b1
b1 = b0
b0 = x*b1 - b2
return (b0 - b2)/2.0
#------------------------------------------------------------------------------
# Ufunc boilerplate
#------------------------------------------------------------------------------
cdef extern from "numpy/arrayobject.h":
void import_array()
ctypedef int npy_intp
cdef enum NPY_TYPES:
NPY_LONG
NPY_DOUBLE
cdef extern from "numpy/ufuncobject.h":
void import_ufunc()
ctypedef void (*PyUFuncGenericFunction)(char**, npy_intp*, npy_intp*, void*)
object PyUFunc_FromFuncAndData(PyUFuncGenericFunction* func, void** data,
char* types, int ntypes, int nin, int nout,
int identity, char* name, char* doc, int c)
cdef void _loop_id_d(char **args, npy_intp *dimensions, npy_intp *steps,
void *func) nogil:
cdef int i
cdef double x
cdef char *ip1=args[0], *ip2=args[1], *op=args[2]
for i in range(0, dimensions[0]):
(<double*>op)[0] = (<double(*)(long,double) nogil>func)(
(<long*>ip1)[0], (<double*>ip2)[0])
ip1 += steps[0]; ip2 += steps[1]; op += steps[2]
cdef char _id_d_types[3]
cdef PyUFuncGenericFunction _id_d_funcs[1]
_id_d_types[0] = NPY_LONG
_id_d_types[1] = NPY_DOUBLE
_id_d_types[2] = NPY_DOUBLE
_id_d_funcs[0] = _loop_id_d
import_array()
import_ufunc()
#--
cdef void *chebyt_data[1]
chebyt_data[0] = <void*>eval_poly_chebyt
_eval_chebyt = PyUFunc_FromFuncAndData(_id_d_funcs, chebyt_data,
_id_d_types, 1, 2, 1, 0, "", "", 0)
#------------------------------------------------------------------------------
# Actual evaluation functions
#------------------------------------------------------------------------------
import numpy as np
from scipy.special._cephes import gamma, hyp2f1, hyp1f1, gammaln
from numpy import exp
def binom(n, k):
"""Binomial coefficient"""
return np.exp(gammaln(1+n) - gammaln(1+k) - gammaln(1+n-k))
def eval_jacobi(n, alpha, beta, x, out=None):
"""Evaluate Jacobi polynomial at a point."""
d = binom(n+alpha, n)
a = -n
b = n + alpha + beta + 1
c = alpha + 1
g = (1-x)/2.0
return hyp2f1(a, b, c, g) * d
def eval_sh_jacobi(n, p, q, x, out=None):
"""Evaluate shifted Jacobi polynomial at a point."""
factor = np.exp(gammaln(1+n) + gammaln(n+p) - gammaln(2*n+p))
return factor * eval_jacobi(n, p-q, q-1, 2*x-1)
def eval_gegenbauer(n, alpha, x, out=None):
"""Evaluate Gegenbauer polynomial at a point."""
d = gamma(n+2*alpha)/gamma(1+n)/gamma(2*alpha)
a = -n
b = n + 2*alpha
c = alpha + 0.5
g = (1-x)/2.0
return hyp2f1(a, b, c, g) * d
def eval_chebyt(n, x, out=None):
"""
Evaluate Chebyshev T polynomial at a point.
This routine is numerically stable for `x` in ``[-1, 1]`` at least
up to order ``10000``.
"""
return _eval_chebyt(n, x, out)
def eval_chebyu(n, x, out=None):
"""Evaluate Chebyshev U polynomial at a point."""
d = n+1
a = -n
b = n+2
c = 1.5
g = (1-x)/2.0
return hyp2f1(a, b, c, g) * d
def eval_chebys(n, x, out=None):
"""Evaluate Chebyshev S polynomial at a point."""
return eval_chebyu(n, x/2, out=out)
def eval_chebyc(n, x, out=None):
"""Evaluate Chebyshev C polynomial at a point."""
return 2*eval_chebyt(n, x/2.0, out)
def eval_sh_chebyt(n, x, out=None):
"""Evaluate shifted Chebyshev T polynomial at a point."""
return eval_chebyt(n, 2*x-1, out=out)
def eval_sh_chebyu(n, x, out=None):
"""Evaluate shifted Chebyshev U polynomial at a point."""
return eval_chebyu(n, 2*x-1, out=out)
def eval_legendre(n, x, out=None):
"""Evaluate Legendre polynomial at a point."""
d = 1
a = -n
b = n+1
c = 1
g = (1-x)/2.0
return hyp2f1(a, b, c, g) * d
def eval_sh_legendre(n, x, out=None):
"""Evaluate shifted Legendre polynomial at a point."""
return eval_legendre(n, 2*x-1, out=out)
def eval_genlaguerre(n, alpha, x, out=None):
"""Evaluate generalized Laguerre polynomial at a point."""
d = binom(n+alpha, n)
a = -n
b = alpha + 1
g = x
return hyp1f1(a, b, g) * d
def eval_laguerre(n, x, out=None):
"""Evaluate Laguerre polynomial at a point."""
return eval_genlaguerre(n, 0., x, out=out)
def eval_hermite(n, x, out=None):
"""Evaluate Hermite polynomial at a point."""
n, x = np.broadcast_arrays(n, x)
n, x = np.atleast_1d(n, x)
if out is None:
out = np.zeros_like(0*n + 0*x)
if (n % 1 != 0).any():
raise ValueError("Order must be integer")
even = (n % 2 == 0)
m = n[even]/2
out[even] = ((-1)**m * 2**(2*m) * gamma(1+m)
* eval_genlaguerre(m, -0.5, x[even]**2))
m = (n[~even]-1)/2
out[~even] = ((-1)**m * 2**(2*m+1) * gamma(1+m)
* x[~even] * eval_genlaguerre(m, 0.5, x[~even]**2))
return out
def eval_hermitenorm(n, x, out=None):
"""Evaluate normalized Hermite polynomial at a point."""
return eval_hermite(n, x/sqrt(2)) * 2**(-n/2.0)
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