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"""
A collection of functions to find the weights and abscissas for
Gaussian Quadrature.
These calculations are done by finding the eigenvalues of a
tridiagonal matrix whose entries are dependent on the coefficients
in the recursion formula for the orthogonal polynomials with the
corresponding weighting function over the interval.
Many recursion relations for orthogonal polynomials are given:
.. math::
a1n f_{n+1} (x) = (a2n + a3n x ) f_n (x) - a4n f_{n-1} (x)
The recursion relation of interest is
.. math::
P_{n+1} (x) = (x - A_n) P_n (x) - B_n P_{n-1} (x)
where :math:`P` has a different normalization than :math:`f`.
The coefficients can be found as:
.. math::
A_n = -a2n / a3n
\\qquad
B_n = ( a4n / a3n \\sqrt{h_n-1 / h_n})^2
where
.. math::
h_n = \\int_a^b w(x) f_n(x)^2
assume:
.. math::
P_0 (x) = 1
\\qquad
P_{-1} (x) == 0
For the mathematical background, see [golub.welsch-1969-mathcomp]_ and
[abramowitz.stegun-1965]_.
References
----------
.. [golub.welsch-1969-mathcomp]
Golub, Gene H, and John H Welsch. 1969. Calculation of Gauss
Quadrature Rules. *Mathematics of Computation* 23, 221-230+s1--s10.
.. [abramowitz.stegun-1965]
Abramowitz, Milton, and Irene A Stegun. (1965) *Handbook of
Mathematical Functions: with Formulas, Graphs, and Mathematical
Tables*. Gaithersburg, MD: National Bureau of Standards.
http://www.math.sfu.ca/~cbm/aands/
.. [townsend.trogdon.olver-2014]
Townsend, A. and Trogdon, T. and Olver, S. (2014)
*Fast computation of Gauss quadrature nodes and
weights on the whole real line*. :arXiv:`1410.5286`.
.. [townsend.trogdon.olver-2015]
Townsend, A. and Trogdon, T. and Olver, S. (2015)
*Fast computation of Gauss quadrature nodes and
weights on the whole real line*.
IMA Journal of Numerical Analysis
:doi:`10.1093/imanum/drv002`.
"""
#
# Author: Travis Oliphant 2000
# Updated Sep. 2003 (fixed bugs --- tested to be accurate)
# SciPy imports.
import numpy as np
from numpy import (exp, inf, pi, sqrt, floor, sin, cos, around,
hstack, arccos, arange)
from scipy import linalg
from scipy.special import airy
# Local imports.
from . import _ufuncs
from . import _ufuncs as cephes
_gam = cephes.gamma
from . import specfun
_polyfuns = ['legendre', 'chebyt', 'chebyu', 'chebyc', 'chebys',
'jacobi', 'laguerre', 'genlaguerre', 'hermite',
'hermitenorm', 'gegenbauer', 'sh_legendre', 'sh_chebyt',
'sh_chebyu', 'sh_jacobi']
# Correspondence between new and old names of root functions
_rootfuns_map = {'roots_legendre': 'p_roots',
'roots_chebyt': 't_roots',
'roots_chebyu': 'u_roots',
'roots_chebyc': 'c_roots',
'roots_chebys': 's_roots',
'roots_jacobi': 'j_roots',
'roots_laguerre': 'l_roots',
'roots_genlaguerre': 'la_roots',
'roots_hermite': 'h_roots',
'roots_hermitenorm': 'he_roots',
'roots_gegenbauer': 'cg_roots',
'roots_sh_legendre': 'ps_roots',
'roots_sh_chebyt': 'ts_roots',
'roots_sh_chebyu': 'us_roots',
'roots_sh_jacobi': 'js_roots'}
_evalfuns = ['eval_legendre', 'eval_chebyt', 'eval_chebyu',
'eval_chebyc', 'eval_chebys', 'eval_jacobi',
'eval_laguerre', 'eval_genlaguerre', 'eval_hermite',
'eval_hermitenorm', 'eval_gegenbauer',
'eval_sh_legendre', 'eval_sh_chebyt', 'eval_sh_chebyu',
'eval_sh_jacobi']
__all__ = _polyfuns + list(_rootfuns_map.keys()) + _evalfuns + ['poch', 'binom']
class orthopoly1d(np.poly1d):
def __init__(self, roots, weights=None, hn=1.0, kn=1.0, wfunc=None,
limits=None, monic=False, eval_func=None):
equiv_weights = [weights[k] / wfunc(roots[k]) for
k in range(len(roots))]
mu = sqrt(hn)
if monic:
evf = eval_func
if evf:
knn = kn
eval_func = lambda x: evf(x) / knn
mu = mu / abs(kn)
kn = 1.0
# compute coefficients from roots, then scale
poly = np.poly1d(roots, r=True)
np.poly1d.__init__(self, poly.coeffs * float(kn))
self.weights = np.array(list(zip(roots, weights, equiv_weights)))
self.weight_func = wfunc
self.limits = limits
self.normcoef = mu
# Note: eval_func will be discarded on arithmetic
self._eval_func = eval_func
def __call__(self, v):
if self._eval_func and not isinstance(v, np.poly1d):
return self._eval_func(v)
else:
return np.poly1d.__call__(self, v)
def _scale(self, p):
if p == 1.0:
return
self._coeffs *= p
evf = self._eval_func
if evf:
self._eval_func = lambda x: evf(x) * p
self.normcoef *= p
def _gen_roots_and_weights(n, mu0, an_func, bn_func, f, df, symmetrize, mu):
"""[x,w] = gen_roots_and_weights(n,an_func,sqrt_bn_func,mu)
Returns the roots (x) of an nth order orthogonal polynomial,
and weights (w) to use in appropriate Gaussian quadrature with that
orthogonal polynomial.
The polynomials have the recurrence relation
P_n+1(x) = (x - A_n) P_n(x) - B_n P_n-1(x)
an_func(n) should return A_n
sqrt_bn_func(n) should return sqrt(B_n)
mu ( = h_0 ) is the integral of the weight over the orthogonal
interval
"""
k = np.arange(n, dtype='d')
c = np.zeros((2, n))
c[0,1:] = bn_func(k[1:])
c[1,:] = an_func(k)
x = linalg.eigvals_banded(c, overwrite_a_band=True)
# improve roots by one application of Newton's method
y = f(n, x)
dy = df(n, x)
x -= y/dy
fm = f(n-1, x)
fm /= np.abs(fm).max()
dy /= np.abs(dy).max()
w = 1.0 / (fm * dy)
if symmetrize:
w = (w + w[::-1]) / 2
x = (x - x[::-1]) / 2
w *= mu0 / w.sum()
if mu:
return x, w, mu0
else:
return x, w
# Jacobi Polynomials 1 P^(alpha,beta)_n(x)
def roots_jacobi(n, alpha, beta, mu=False):
r"""Gauss-Jacobi quadrature.
Compute the sample points and weights for Gauss-Jacobi
quadrature. The sample points are the roots of the nth degree
Jacobi polynomial, :math:`P^{\alpha, \beta}_n(x)`. These sample
points and weights correctly integrate polynomials of degree
:math:`2n - 1` or less over the interval :math:`[-1, 1]` with
weight function :math:`w(x) = (1 - x)^{\alpha} (1 +
x)^{\beta}`. See 22.2.1 in [AS]_ for details.
Parameters
----------
n : int
quadrature order
alpha : float
alpha must be > -1
beta : float
beta must be > -1
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
m = int(n)
if n < 1 or n != m:
raise ValueError("n must be a positive integer.")
if alpha <= -1 or beta <= -1:
raise ValueError("alpha and beta must be greater than -1.")
if alpha == 0.0 and beta == 0.0:
return roots_legendre(m, mu)
if alpha == beta:
return roots_gegenbauer(m, alpha+0.5, mu)
mu0 = 2.0**(alpha+beta+1)*cephes.beta(alpha+1, beta+1)
a = alpha
b = beta
if a + b == 0.0:
an_func = lambda k: np.where(k == 0, (b-a)/(2+a+b), 0.0)
else:
an_func = lambda k: np.where(k == 0, (b-a)/(2+a+b),
(b*b - a*a) / ((2.0*k+a+b)*(2.0*k+a+b+2)))
bn_func = lambda k: 2.0 / (2.0*k+a+b)*np.sqrt((k+a)*(k+b) / (2*k+a+b+1)) \
* np.where(k == 1, 1.0, np.sqrt(k*(k+a+b) / (2.0*k+a+b-1)))
f = lambda n, x: cephes.eval_jacobi(n, a, b, x)
df = lambda n, x: 0.5 * (n + a + b + 1) \
* cephes.eval_jacobi(n-1, a+1, b+1, x)
return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, False, mu)
def jacobi(n, alpha, beta, monic=False):
r"""Jacobi polynomial.
Defined to be the solution of
.. math::
(1 - x^2)\frac{d^2}{dx^2}P_n^{(\alpha, \beta)}
+ (\beta - \alpha - (\alpha + \beta + 2)x)
\frac{d}{dx}P_n^{(\alpha, \beta)}
+ n(n + \alpha + \beta + 1)P_n^{(\alpha, \beta)} = 0
for :math:`\alpha, \beta > -1`; :math:`P_n^{(\alpha, \beta)}` is a
polynomial of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
alpha : float
Parameter, must be greater than -1.
beta : float
Parameter, must be greater than -1.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
P : orthopoly1d
Jacobi polynomial.
Notes
-----
For fixed :math:`\alpha, \beta`, the polynomials
:math:`P_n^{(\alpha, \beta)}` are orthogonal over :math:`[-1, 1]`
with weight function :math:`(1 - x)^\alpha(1 + x)^\beta`.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
wfunc = lambda x: (1 - x)**alpha * (1 + x)**beta
if n == 0:
return orthopoly1d([], [], 1.0, 1.0, wfunc, (-1, 1), monic,
eval_func=np.ones_like)
x, w, mu = roots_jacobi(n, alpha, beta, mu=True)
ab1 = alpha + beta + 1.0
hn = 2**ab1 / (2 * n + ab1) * _gam(n + alpha + 1)
hn *= _gam(n + beta + 1.0) / _gam(n + 1) / _gam(n + ab1)
kn = _gam(2 * n + ab1) / 2.0**n / _gam(n + 1) / _gam(n + ab1)
# here kn = coefficient on x^n term
p = orthopoly1d(x, w, hn, kn, wfunc, (-1, 1), monic,
lambda x: eval_jacobi(n, alpha, beta, x))
return p
# Jacobi Polynomials shifted G_n(p,q,x)
def roots_sh_jacobi(n, p1, q1, mu=False):
"""Gauss-Jacobi (shifted) quadrature.
Compute the sample points and weights for Gauss-Jacobi (shifted)
quadrature. The sample points are the roots of the nth degree
shifted Jacobi polynomial, :math:`G^{p,q}_n(x)`. These sample
points and weights correctly integrate polynomials of degree
:math:`2n - 1` or less over the interval :math:`[0, 1]` with
weight function :math:`w(x) = (1 - x)^{p-q} x^{q-1}`. See 22.2.2
in [AS]_ for details.
Parameters
----------
n : int
quadrature order
p1 : float
(p1 - q1) must be > -1
q1 : float
q1 must be > 0
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
if (p1-q1) <= -1 or q1 <= 0:
raise ValueError("(p - q) must be greater than -1, and q must be greater than 0.")
x, w, m = roots_jacobi(n, p1-q1, q1-1, True)
x = (x + 1) / 2
scale = 2.0**p1
w /= scale
m /= scale
if mu:
return x, w, m
else:
return x, w
def sh_jacobi(n, p, q, monic=False):
r"""Shifted Jacobi polynomial.
Defined by
.. math::
G_n^{(p, q)}(x)
= \binom{2n + p - 1}{n}^{-1}P_n^{(p - q, q - 1)}(2x - 1),
where :math:`P_n^{(\cdot, \cdot)}` is the nth Jacobi polynomial.
Parameters
----------
n : int
Degree of the polynomial.
p : float
Parameter, must have :math:`p > q - 1`.
q : float
Parameter, must be greater than 0.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
G : orthopoly1d
Shifted Jacobi polynomial.
Notes
-----
For fixed :math:`p, q`, the polynomials :math:`G_n^{(p, q)}` are
orthogonal over :math:`[0, 1]` with weight function :math:`(1 -
x)^{p - q}x^{q - 1}`.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
wfunc = lambda x: (1.0 - x)**(p - q) * (x)**(q - 1.)
if n == 0:
return orthopoly1d([], [], 1.0, 1.0, wfunc, (-1, 1), monic,
eval_func=np.ones_like)
n1 = n
x, w = roots_sh_jacobi(n1, p, q)
hn = _gam(n + 1) * _gam(n + q) * _gam(n + p) * _gam(n + p - q + 1)
hn /= (2 * n + p) * (_gam(2 * n + p)**2)
# kn = 1.0 in standard form so monic is redundant. Kept for compatibility.
kn = 1.0
pp = orthopoly1d(x, w, hn, kn, wfunc=wfunc, limits=(0, 1), monic=monic,
eval_func=lambda x: eval_sh_jacobi(n, p, q, x))
return pp
# Generalized Laguerre L^(alpha)_n(x)
def roots_genlaguerre(n, alpha, mu=False):
r"""Gauss-generalized Laguerre quadrature.
Compute the sample points and weights for Gauss-generalized
Laguerre quadrature. The sample points are the roots of the nth
degree generalized Laguerre polynomial, :math:`L^{\alpha}_n(x)`.
These sample points and weights correctly integrate polynomials of
degree :math:`2n - 1` or less over the interval :math:`[0,
\infty]` with weight function :math:`w(x) = x^{\alpha}
e^{-x}`. See 22.3.9 in [AS]_ for details.
Parameters
----------
n : int
quadrature order
alpha : float
alpha must be > -1
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
m = int(n)
if n < 1 or n != m:
raise ValueError("n must be a positive integer.")
if alpha < -1:
raise ValueError("alpha must be greater than -1.")
mu0 = cephes.gamma(alpha + 1)
if m == 1:
x = np.array([alpha+1.0], 'd')
w = np.array([mu0], 'd')
if mu:
return x, w, mu0
else:
return x, w
an_func = lambda k: 2 * k + alpha + 1
bn_func = lambda k: -np.sqrt(k * (k + alpha))
f = lambda n, x: cephes.eval_genlaguerre(n, alpha, x)
df = lambda n, x: (n*cephes.eval_genlaguerre(n, alpha, x)
- (n + alpha)*cephes.eval_genlaguerre(n-1, alpha, x))/x
return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, False, mu)
def genlaguerre(n, alpha, monic=False):
r"""Generalized (associated) Laguerre polynomial.
Defined to be the solution of
.. math::
x\frac{d^2}{dx^2}L_n^{(\alpha)}
+ (\alpha + 1 - x)\frac{d}{dx}L_n^{(\alpha)}
+ nL_n^{(\alpha)} = 0,
where :math:`\alpha > -1`; :math:`L_n^{(\alpha)}` is a polynomial
of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
alpha : float
Parameter, must be greater than -1.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
L : orthopoly1d
Generalized Laguerre polynomial.
Notes
-----
For fixed :math:`\alpha`, the polynomials :math:`L_n^{(\alpha)}`
are orthogonal over :math:`[0, \infty)` with weight function
:math:`e^{-x}x^\alpha`.
The Laguerre polynomials are the special case where :math:`\alpha
= 0`.
See Also
--------
laguerre : Laguerre polynomial.
"""
if alpha <= -1:
raise ValueError("alpha must be > -1")
if n < 0:
raise ValueError("n must be nonnegative.")
if n == 0:
n1 = n + 1
else:
n1 = n
x, w = roots_genlaguerre(n1, alpha)
wfunc = lambda x: exp(-x) * x**alpha
if n == 0:
x, w = [], []
hn = _gam(n + alpha + 1) / _gam(n + 1)
kn = (-1)**n / _gam(n + 1)
p = orthopoly1d(x, w, hn, kn, wfunc, (0, inf), monic,
lambda x: eval_genlaguerre(n, alpha, x))
return p
# Laguerre L_n(x)
def roots_laguerre(n, mu=False):
r"""Gauss-Laguerre quadrature.
Compute the sample points and weights for Gauss-Laguerre
quadrature. The sample points are the roots of the nth degree
Laguerre polynomial, :math:`L_n(x)`. These sample points and
weights correctly integrate polynomials of degree :math:`2n - 1`
or less over the interval :math:`[0, \infty]` with weight function
:math:`w(x) = e^{-x}`. See 22.2.13 in [AS]_ for details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
numpy.polynomial.laguerre.laggauss
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
return roots_genlaguerre(n, 0.0, mu=mu)
def laguerre(n, monic=False):
r"""Laguerre polynomial.
Defined to be the solution of
.. math::
x\frac{d^2}{dx^2}L_n + (1 - x)\frac{d}{dx}L_n + nL_n = 0;
:math:`L_n` is a polynomial of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
L : orthopoly1d
Laguerre Polynomial.
Notes
-----
The polynomials :math:`L_n` are orthogonal over :math:`[0,
\infty)` with weight function :math:`e^{-x}`.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
if n == 0:
n1 = n + 1
else:
n1 = n
x, w = roots_laguerre(n1)
if n == 0:
x, w = [], []
hn = 1.0
kn = (-1)**n / _gam(n + 1)
p = orthopoly1d(x, w, hn, kn, lambda x: exp(-x), (0, inf), monic,
lambda x: eval_laguerre(n, x))
return p
# Hermite 1 H_n(x)
def roots_hermite(n, mu=False):
r"""Gauss-Hermite (physicist's) quadrature.
Compute the sample points and weights for Gauss-Hermite
quadrature. The sample points are the roots of the nth degree
Hermite polynomial, :math:`H_n(x)`. These sample points and
weights correctly integrate polynomials of degree :math:`2n - 1`
or less over the interval :math:`[-\infty, \infty]` with weight
function :math:`w(x) = e^{-x^2}`. See 22.2.14 in [AS]_ for
details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
Notes
-----
For small n up to 150 a modified version of the Golub-Welsch
algorithm is used. Nodes are computed from the eigenvalue
problem and improved by one step of a Newton iteration.
The weights are computed from the well-known analytical formula.
For n larger than 150 an optimal asymptotic algorithm is applied
which computes nodes and weights in a numerically stable manner.
The algorithm has linear runtime making computation for very
large n (several thousand or more) feasible.
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
numpy.polynomial.hermite.hermgauss
roots_hermitenorm
References
----------
.. [townsend.trogdon.olver-2014]
Townsend, A. and Trogdon, T. and Olver, S. (2014)
*Fast computation of Gauss quadrature nodes and
weights on the whole real line*. :arXiv:`1410.5286`.
.. [townsend.trogdon.olver-2015]
Townsend, A. and Trogdon, T. and Olver, S. (2015)
*Fast computation of Gauss quadrature nodes and
weights on the whole real line*.
IMA Journal of Numerical Analysis
:doi:`10.1093/imanum/drv002`.
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
m = int(n)
if n < 1 or n != m:
raise ValueError("n must be a positive integer.")
mu0 = np.sqrt(np.pi)
if n <= 150:
an_func = lambda k: 0.0*k
bn_func = lambda k: np.sqrt(k/2.0)
f = cephes.eval_hermite
df = lambda n, x: 2.0 * n * cephes.eval_hermite(n-1, x)
return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, True, mu)
else:
nodes, weights = _roots_hermite_asy(m)
if mu:
return nodes, weights, mu0
else:
return nodes, weights
def _compute_tauk(n, k, maxit=5):
"""Helper function for Tricomi initial guesses
For details, see formula 3.1 in lemma 3.1 in the
original paper.
Parameters
----------
n : int
Quadrature order
k : ndarray of type int
Index of roots :math:`\tau_k` to compute
maxit : int
Number of Newton maxit performed, the default
value of 5 is sufficient.
Returns
-------
tauk : ndarray
Roots of equation 3.1
See Also
--------
initial_nodes_a
roots_hermite_asy
"""
a = n % 2 - 0.5
c = (4.0*floor(n/2.0) - 4.0*k + 3.0)*pi / (4.0*floor(n/2.0) + 2.0*a + 2.0)
f = lambda x: x - sin(x) - c
df = lambda x: 1.0 - cos(x)
xi = 0.5*pi
for i in range(maxit):
xi = xi - f(xi)/df(xi)
return xi
def _initial_nodes_a(n, k):
r"""Tricomi initial guesses
Computes an initial approximation to the square of the `k`-th
(positive) root :math:`x_k` of the Hermite polynomial :math:`H_n`
of order :math:`n`. The formula is the one from lemma 3.1 in the
original paper. The guesses are accurate except in the region
near :math:`\sqrt{2n + 1}`.
Parameters
----------
n : int
Quadrature order
k : ndarray of type int
Index of roots to compute
Returns
-------
xksq : ndarray
Square of the approximate roots
See Also
--------
initial_nodes
roots_hermite_asy
"""
tauk = _compute_tauk(n, k)
sigk = cos(0.5*tauk)**2
a = n % 2 - 0.5
nu = 4.0*floor(n/2.0) + 2.0*a + 2.0
# Initial approximation of Hermite roots (square)
xksq = nu*sigk - 1.0/(3.0*nu) * (5.0/(4.0*(1.0-sigk)**2) - 1.0/(1.0-sigk) - 0.25)
return xksq
def _initial_nodes_b(n, k):
r"""Gatteschi initial guesses
Computes an initial approximation to the square of the kth
(positive) root :math:`x_k` of the Hermite polynomial :math:`H_n`
of order :math:`n`. The formula is the one from lemma 3.2 in the
original paper. The guesses are accurate in the region just
below :math:`\sqrt{2n + 1}`.
Parameters
----------
n : int
Quadrature order
k : ndarray of type int
Index of roots to compute
Returns
-------
xksq : ndarray
Square of the approximate root
See Also
--------
initial_nodes
roots_hermite_asy
"""
a = n % 2 - 0.5
nu = 4.0*floor(n/2.0) + 2.0*a + 2.0
# Airy roots by approximation
ak = specfun.airyzo(k.max(), 1)[0][::-1]
# Initial approximation of Hermite roots (square)
xksq = (nu +
2.0**(2.0/3.0) * ak * nu**(1.0/3.0) +
1.0/5.0 * 2.0**(4.0/3.0) * ak**2 * nu**(-1.0/3.0) +
(9.0/140.0 - 12.0/175.0 * ak**3) * nu**(-1.0) +
(16.0/1575.0 * ak + 92.0/7875.0 * ak**4) * 2.0**(2.0/3.0) * nu**(-5.0/3.0) -
(15152.0/3031875.0 * ak**5 + 1088.0/121275.0 * ak**2) * 2.0**(1.0/3.0) * nu**(-7.0/3.0))
return xksq
def _initial_nodes(n):
"""Initial guesses for the Hermite roots
Computes an initial approximation to the non-negative
roots :math:`x_k` of the Hermite polynomial :math:`H_n`
of order :math:`n`. The Tricomi and Gatteschi initial
guesses are used in the region where they are accurate.
Parameters
----------
n : int
Quadrature order
Returns
-------
xk : ndarray
Approximate roots
See Also
--------
roots_hermite_asy
"""
# Turnover point
# linear polynomial fit to error of 10, 25, 40, ..., 1000 point rules
fit = 0.49082003*n - 4.37859653
turnover = around(fit).astype(int)
# Compute all approximations
ia = arange(1, int(floor(n*0.5)+1))
ib = ia[::-1]
xasq = _initial_nodes_a(n, ia[:turnover+1])
xbsq = _initial_nodes_b(n, ib[turnover+1:])
# Combine
iv = sqrt(hstack([xasq, xbsq]))
# Central node is always zero
if n % 2 == 1:
iv = hstack([0.0, iv])
return iv
def _pbcf(n, theta):
r"""Asymptotic series expansion of parabolic cylinder function
The implementation is based on sections 3.2 and 3.3 from the
original paper. Compared to the published version this code
adds one more term to the asymptotic series. The detailed
formulas can be found at [parabolic-asymptotics]_. The evaluation
is done in a transformed variable :math:`\theta := \arccos(t)`
where :math:`t := x / \mu` and :math:`\mu := \sqrt{2n + 1}`.
Parameters
----------
n : int
Quadrature order
theta : ndarray
Transformed position variable
Returns
-------
U : ndarray
Value of the parabolic cylinder function :math:`U(a, \theta)`.
Ud : ndarray
Value of the derivative :math:`U^{\prime}(a, \theta)` of
the parabolic cylinder function.
See Also
--------
roots_hermite_asy
References
----------
.. [parabolic-asymptotics]
https://dlmf.nist.gov/12.10#vii
"""
st = sin(theta)
ct = cos(theta)
# https://dlmf.nist.gov/12.10#vii
mu = 2.0*n + 1.0
# https://dlmf.nist.gov/12.10#E23
eta = 0.5*theta - 0.5*st*ct
# https://dlmf.nist.gov/12.10#E39
zeta = -(3.0*eta/2.0) ** (2.0/3.0)
# https://dlmf.nist.gov/12.10#E40
phi = (-zeta / st**2) ** (0.25)
# Coefficients
# https://dlmf.nist.gov/12.10#E43
a0 = 1.0
a1 = 0.10416666666666666667
a2 = 0.08355034722222222222
a3 = 0.12822657455632716049
a4 = 0.29184902646414046425
a5 = 0.88162726744375765242
b0 = 1.0
b1 = -0.14583333333333333333
b2 = -0.09874131944444444444
b3 = -0.14331205391589506173
b4 = -0.31722720267841354810
b5 = -0.94242914795712024914
# Polynomials
# https://dlmf.nist.gov/12.10#E9
# https://dlmf.nist.gov/12.10#E10
ctp = ct ** arange(16).reshape((-1,1))
u0 = 1.0
u1 = (1.0*ctp[3,:] - 6.0*ct) / 24.0
u2 = (-9.0*ctp[4,:] + 249.0*ctp[2,:] + 145.0) / 1152.0
u3 = (-4042.0*ctp[9,:] + 18189.0*ctp[7,:] - 28287.0*ctp[5,:] - 151995.0*ctp[3,:] - 259290.0*ct) / 414720.0
u4 = (72756.0*ctp[10,:] - 321339.0*ctp[8,:] - 154982.0*ctp[6,:] + 50938215.0*ctp[4,:] + 122602962.0*ctp[2,:] + 12773113.0) / 39813120.0
u5 = (82393456.0*ctp[15,:] - 617950920.0*ctp[13,:] + 1994971575.0*ctp[11,:] - 3630137104.0*ctp[9,:] + 4433574213.0*ctp[7,:]
- 37370295816.0*ctp[5,:] - 119582875013.0*ctp[3,:] - 34009066266.0*ct) / 6688604160.0
v0 = 1.0
v1 = (1.0*ctp[3,:] + 6.0*ct) / 24.0
v2 = (15.0*ctp[4,:] - 327.0*ctp[2,:] - 143.0) / 1152.0
v3 = (-4042.0*ctp[9,:] + 18189.0*ctp[7,:] - 36387.0*ctp[5,:] + 238425.0*ctp[3,:] + 259290.0*ct) / 414720.0
v4 = (-121260.0*ctp[10,:] + 551733.0*ctp[8,:] - 151958.0*ctp[6,:] - 57484425.0*ctp[4,:] - 132752238.0*ctp[2,:] - 12118727) / 39813120.0
v5 = (82393456.0*ctp[15,:] - 617950920.0*ctp[13,:] + 2025529095.0*ctp[11,:] - 3750839308.0*ctp[9,:] + 3832454253.0*ctp[7,:]
+ 35213253348.0*ctp[5,:] + 130919230435.0*ctp[3,:] + 34009066266*ct) / 6688604160.0
# Airy Evaluation (Bi and Bip unused)
Ai, Aip, Bi, Bip = airy(mu**(4.0/6.0) * zeta)
# Prefactor for U
P = 2.0*sqrt(pi) * mu**(1.0/6.0) * phi
# Terms for U
# https://dlmf.nist.gov/12.10#E42
phip = phi ** arange(6, 31, 6).reshape((-1,1))
A0 = b0*u0
A1 = (b2*u0 + phip[0,:]*b1*u1 + phip[1,:]*b0*u2) / zeta**3
A2 = (b4*u0 + phip[0,:]*b3*u1 + phip[1,:]*b2*u2 + phip[2,:]*b1*u3 + phip[3,:]*b0*u4) / zeta**6
B0 = -(a1*u0 + phip[0,:]*a0*u1) / zeta**2
B1 = -(a3*u0 + phip[0,:]*a2*u1 + phip[1,:]*a1*u2 + phip[2,:]*a0*u3) / zeta**5
B2 = -(a5*u0 + phip[0,:]*a4*u1 + phip[1,:]*a3*u2 + phip[2,:]*a2*u3 + phip[3,:]*a1*u4 + phip[4,:]*a0*u5) / zeta**8
# U
# https://dlmf.nist.gov/12.10#E35
U = P * (Ai * (A0 + A1/mu**2.0 + A2/mu**4.0) +
Aip * (B0 + B1/mu**2.0 + B2/mu**4.0) / mu**(8.0/6.0))
# Prefactor for derivative of U
Pd = sqrt(2.0*pi) * mu**(2.0/6.0) / phi
# Terms for derivative of U
# https://dlmf.nist.gov/12.10#E46
C0 = -(b1*v0 + phip[0,:]*b0*v1) / zeta
C1 = -(b3*v0 + phip[0,:]*b2*v1 + phip[1,:]*b1*v2 + phip[2,:]*b0*v3) / zeta**4
C2 = -(b5*v0 + phip[0,:]*b4*v1 + phip[1,:]*b3*v2 + phip[2,:]*b2*v3 + phip[3,:]*b1*v4 + phip[4,:]*b0*v5) / zeta**7
D0 = a0*v0
D1 = (a2*v0 + phip[0,:]*a1*v1 + phip[1,:]*a0*v2) / zeta**3
D2 = (a4*v0 + phip[0,:]*a3*v1 + phip[1,:]*a2*v2 + phip[2,:]*a1*v3 + phip[3,:]*a0*v4) / zeta**6
# Derivative of U
# https://dlmf.nist.gov/12.10#E36
Ud = Pd * (Ai * (C0 + C1/mu**2.0 + C2/mu**4.0) / mu**(4.0/6.0) +
Aip * (D0 + D1/mu**2.0 + D2/mu**4.0))
return U, Ud
def _newton(n, x_initial, maxit=5):
"""Newton iteration for polishing the asymptotic approximation
to the zeros of the Hermite polynomials.
Parameters
----------
n : int
Quadrature order
x_initial : ndarray
Initial guesses for the roots
maxit : int
Maximal number of Newton iterations.
The default 5 is sufficient, usually
only one or two steps are needed.
Returns
-------
nodes : ndarray
Quadrature nodes
weights : ndarray
Quadrature weights
See Also
--------
roots_hermite_asy
"""
# Variable transformation
mu = sqrt(2.0*n + 1.0)
t = x_initial / mu
theta = arccos(t)
# Newton iteration
for i in range(maxit):
u, ud = _pbcf(n, theta)
dtheta = u / (sqrt(2.0) * mu * sin(theta) * ud)
theta = theta + dtheta
if max(abs(dtheta)) < 1e-14:
break
# Undo variable transformation
x = mu * cos(theta)
# Central node is always zero
if n % 2 == 1:
x[0] = 0.0
# Compute weights
w = exp(-x**2) / (2.0*ud**2)
return x, w
def _roots_hermite_asy(n):
r"""Gauss-Hermite (physicist's) quadrature for large n.
Computes the sample points and weights for Gauss-Hermite quadrature.
The sample points are the roots of the nth degree Hermite polynomial,
:math:`H_n(x)`. These sample points and weights correctly integrate
polynomials of degree :math:`2n - 1` or less over the interval
:math:`[-\infty, \infty]` with weight function :math:`f(x) = e^{-x^2}`.
This method relies on asymptotic expansions which work best for n > 150.
The algorithm has linear runtime making computation for very large n
feasible.
Parameters
----------
n : int
quadrature order
Returns
-------
nodes : ndarray
Quadrature nodes
weights : ndarray
Quadrature weights
See Also
--------
roots_hermite
References
----------
.. [townsend.trogdon.olver-2014]
Townsend, A. and Trogdon, T. and Olver, S. (2014)
*Fast computation of Gauss quadrature nodes and
weights on the whole real line*. :arXiv:`1410.5286`.
.. [townsend.trogdon.olver-2015]
Townsend, A. and Trogdon, T. and Olver, S. (2015)
*Fast computation of Gauss quadrature nodes and
weights on the whole real line*.
IMA Journal of Numerical Analysis
:doi:`10.1093/imanum/drv002`.
"""
iv = _initial_nodes(n)
nodes, weights = _newton(n, iv)
# Combine with negative parts
if n % 2 == 0:
nodes = hstack([-nodes[::-1], nodes])
weights = hstack([weights[::-1], weights])
else:
nodes = hstack([-nodes[-1:0:-1], nodes])
weights = hstack([weights[-1:0:-1], weights])
# Scale weights
weights *= sqrt(pi) / sum(weights)
return nodes, weights
def hermite(n, monic=False):
r"""Physicist's Hermite polynomial.
Defined by
.. math::
H_n(x) = (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2};
:math:`H_n` is a polynomial of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
H : orthopoly1d
Hermite polynomial.
Notes
-----
The polynomials :math:`H_n` are orthogonal over :math:`(-\infty,
\infty)` with weight function :math:`e^{-x^2}`.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
if n == 0:
n1 = n + 1
else:
n1 = n
x, w = roots_hermite(n1)
wfunc = lambda x: exp(-x * x)
if n == 0:
x, w = [], []
hn = 2**n * _gam(n + 1) * sqrt(pi)
kn = 2**n
p = orthopoly1d(x, w, hn, kn, wfunc, (-inf, inf), monic,
lambda x: eval_hermite(n, x))
return p
# Hermite 2 He_n(x)
def roots_hermitenorm(n, mu=False):
r"""Gauss-Hermite (statistician's) quadrature.
Compute the sample points and weights for Gauss-Hermite
quadrature. The sample points are the roots of the nth degree
Hermite polynomial, :math:`He_n(x)`. These sample points and
weights correctly integrate polynomials of degree :math:`2n - 1`
or less over the interval :math:`[-\infty, \infty]` with weight
function :math:`w(x) = e^{-x^2/2}`. See 22.2.15 in [AS]_ for more
details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
Notes
-----
For small n up to 150 a modified version of the Golub-Welsch
algorithm is used. Nodes are computed from the eigenvalue
problem and improved by one step of a Newton iteration.
The weights are computed from the well-known analytical formula.
For n larger than 150 an optimal asymptotic algorithm is used
which computes nodes and weights in a numerical stable manner.
The algorithm has linear runtime making computation for very
large n (several thousand or more) feasible.
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
numpy.polynomial.hermite_e.hermegauss
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
m = int(n)
if n < 1 or n != m:
raise ValueError("n must be a positive integer.")
mu0 = np.sqrt(2.0*np.pi)
if n <= 150:
an_func = lambda k: 0.0*k
bn_func = lambda k: np.sqrt(k)
f = cephes.eval_hermitenorm
df = lambda n, x: n * cephes.eval_hermitenorm(n-1, x)
return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, True, mu)
else:
nodes, weights = _roots_hermite_asy(m)
# Transform
nodes *= sqrt(2)
weights *= sqrt(2)
if mu:
return nodes, weights, mu0
else:
return nodes, weights
def hermitenorm(n, monic=False):
r"""Normalized (probabilist's) Hermite polynomial.
Defined by
.. math::
He_n(x) = (-1)^ne^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2};
:math:`He_n` is a polynomial of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
He : orthopoly1d
Hermite polynomial.
Notes
-----
The polynomials :math:`He_n` are orthogonal over :math:`(-\infty,
\infty)` with weight function :math:`e^{-x^2/2}`.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
if n == 0:
n1 = n + 1
else:
n1 = n
x, w = roots_hermitenorm(n1)
wfunc = lambda x: exp(-x * x / 2.0)
if n == 0:
x, w = [], []
hn = sqrt(2 * pi) * _gam(n + 1)
kn = 1.0
p = orthopoly1d(x, w, hn, kn, wfunc=wfunc, limits=(-inf, inf), monic=monic,
eval_func=lambda x: eval_hermitenorm(n, x))
return p
# The remainder of the polynomials can be derived from the ones above.
# Ultraspherical (Gegenbauer) C^(alpha)_n(x)
def roots_gegenbauer(n, alpha, mu=False):
r"""Gauss-Gegenbauer quadrature.
Compute the sample points and weights for Gauss-Gegenbauer
quadrature. The sample points are the roots of the nth degree
Gegenbauer polynomial, :math:`C^{\alpha}_n(x)`. These sample
points and weights correctly integrate polynomials of degree
:math:`2n - 1` or less over the interval :math:`[-1, 1]` with
weight function :math:`w(x) = (1 - x^2)^{\alpha - 1/2}`. See
22.2.3 in [AS]_ for more details.
Parameters
----------
n : int
quadrature order
alpha : float
alpha must be > -0.5
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
m = int(n)
if n < 1 or n != m:
raise ValueError("n must be a positive integer.")
if alpha < -0.5:
raise ValueError("alpha must be greater than -0.5.")
elif alpha == 0.0:
# C(n,0,x) == 0 uniformly, however, as alpha->0, C(n,alpha,x)->T(n,x)
# strictly, we should just error out here, since the roots are not
# really defined, but we used to return something useful, so let's
# keep doing so.
return roots_chebyt(n, mu)
mu0 = np.sqrt(np.pi) * cephes.gamma(alpha + 0.5) / cephes.gamma(alpha + 1)
an_func = lambda k: 0.0 * k
bn_func = lambda k: np.sqrt(k * (k + 2 * alpha - 1)
/ (4 * (k + alpha) * (k + alpha - 1)))
f = lambda n, x: cephes.eval_gegenbauer(n, alpha, x)
df = lambda n, x: (-n*x*cephes.eval_gegenbauer(n, alpha, x)
+ (n + 2*alpha - 1)*cephes.eval_gegenbauer(n-1, alpha, x))/(1-x**2)
return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, True, mu)
def gegenbauer(n, alpha, monic=False):
r"""Gegenbauer (ultraspherical) polynomial.
Defined to be the solution of
.. math::
(1 - x^2)\frac{d^2}{dx^2}C_n^{(\alpha)}
- (2\alpha + 1)x\frac{d}{dx}C_n^{(\alpha)}
+ n(n + 2\alpha)C_n^{(\alpha)} = 0
for :math:`\alpha > -1/2`; :math:`C_n^{(\alpha)}` is a polynomial
of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
C : orthopoly1d
Gegenbauer polynomial.
Notes
-----
The polynomials :math:`C_n^{(\alpha)}` are orthogonal over
:math:`[-1,1]` with weight function :math:`(1 - x^2)^{(\alpha -
1/2)}`.
"""
base = jacobi(n, alpha - 0.5, alpha - 0.5, monic=monic)
if monic:
return base
# Abrahmowitz and Stegan 22.5.20
factor = (_gam(2*alpha + n) * _gam(alpha + 0.5) /
_gam(2*alpha) / _gam(alpha + 0.5 + n))
base._scale(factor)
base.__dict__['_eval_func'] = lambda x: eval_gegenbauer(float(n), alpha, x)
return base
# Chebyshev of the first kind: T_n(x) =
# n! sqrt(pi) / _gam(n+1./2)* P^(-1/2,-1/2)_n(x)
# Computed anew.
def roots_chebyt(n, mu=False):
r"""Gauss-Chebyshev (first kind) quadrature.
Computes the sample points and weights for Gauss-Chebyshev
quadrature. The sample points are the roots of the nth degree
Chebyshev polynomial of the first kind, :math:`T_n(x)`. These
sample points and weights correctly integrate polynomials of
degree :math:`2n - 1` or less over the interval :math:`[-1, 1]`
with weight function :math:`w(x) = 1/\sqrt{1 - x^2}`. See 22.2.4
in [AS]_ for more details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
numpy.polynomial.chebyshev.chebgauss
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
m = int(n)
if n < 1 or n != m:
raise ValueError('n must be a positive integer.')
x = _ufuncs._sinpi(np.arange(-m + 1, m, 2) / (2*m))
w = np.full_like(x, pi/m)
if mu:
return x, w, pi
else:
return x, w
def chebyt(n, monic=False):
r"""Chebyshev polynomial of the first kind.
Defined to be the solution of
.. math::
(1 - x^2)\frac{d^2}{dx^2}T_n - x\frac{d}{dx}T_n + n^2T_n = 0;
:math:`T_n` is a polynomial of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
T : orthopoly1d
Chebyshev polynomial of the first kind.
Notes
-----
The polynomials :math:`T_n` are orthogonal over :math:`[-1, 1]`
with weight function :math:`(1 - x^2)^{-1/2}`.
See Also
--------
chebyu : Chebyshev polynomial of the second kind.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
wfunc = lambda x: 1.0 / sqrt(1 - x * x)
if n == 0:
return orthopoly1d([], [], pi, 1.0, wfunc, (-1, 1), monic,
lambda x: eval_chebyt(n, x))
n1 = n
x, w, mu = roots_chebyt(n1, mu=True)
hn = pi / 2
kn = 2**(n - 1)
p = orthopoly1d(x, w, hn, kn, wfunc, (-1, 1), monic,
lambda x: eval_chebyt(n, x))
return p
# Chebyshev of the second kind
# U_n(x) = (n+1)! sqrt(pi) / (2*_gam(n+3./2)) * P^(1/2,1/2)_n(x)
def roots_chebyu(n, mu=False):
r"""Gauss-Chebyshev (second kind) quadrature.
Computes the sample points and weights for Gauss-Chebyshev
quadrature. The sample points are the roots of the nth degree
Chebyshev polynomial of the second kind, :math:`U_n(x)`. These
sample points and weights correctly integrate polynomials of
degree :math:`2n - 1` or less over the interval :math:`[-1, 1]`
with weight function :math:`w(x) = \sqrt{1 - x^2}`. See 22.2.5 in
[AS]_ for details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
m = int(n)
if n < 1 or n != m:
raise ValueError('n must be a positive integer.')
t = np.arange(m, 0, -1) * pi / (m + 1)
x = np.cos(t)
w = pi * np.sin(t)**2 / (m + 1)
if mu:
return x, w, pi / 2
else:
return x, w
def chebyu(n, monic=False):
r"""Chebyshev polynomial of the second kind.
Defined to be the solution of
.. math::
(1 - x^2)\frac{d^2}{dx^2}U_n - 3x\frac{d}{dx}U_n
+ n(n + 2)U_n = 0;
:math:`U_n` is a polynomial of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
U : orthopoly1d
Chebyshev polynomial of the second kind.
Notes
-----
The polynomials :math:`U_n` are orthogonal over :math:`[-1, 1]`
with weight function :math:`(1 - x^2)^{1/2}`.
See Also
--------
chebyt : Chebyshev polynomial of the first kind.
"""
base = jacobi(n, 0.5, 0.5, monic=monic)
if monic:
return base
factor = sqrt(pi) / 2.0 * _gam(n + 2) / _gam(n + 1.5)
base._scale(factor)
return base
# Chebyshev of the first kind C_n(x)
def roots_chebyc(n, mu=False):
r"""Gauss-Chebyshev (first kind) quadrature.
Compute the sample points and weights for Gauss-Chebyshev
quadrature. The sample points are the roots of the nth degree
Chebyshev polynomial of the first kind, :math:`C_n(x)`. These
sample points and weights correctly integrate polynomials of
degree :math:`2n - 1` or less over the interval :math:`[-2, 2]`
with weight function :math:`w(x) = 1 / \sqrt{1 - (x/2)^2}`. See
22.2.6 in [AS]_ for more details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
x, w, m = roots_chebyt(n, True)
x *= 2
w *= 2
m *= 2
if mu:
return x, w, m
else:
return x, w
def chebyc(n, monic=False):
r"""Chebyshev polynomial of the first kind on :math:`[-2, 2]`.
Defined as :math:`C_n(x) = 2T_n(x/2)`, where :math:`T_n` is the
nth Chebychev polynomial of the first kind.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
C : orthopoly1d
Chebyshev polynomial of the first kind on :math:`[-2, 2]`.
Notes
-----
The polynomials :math:`C_n(x)` are orthogonal over :math:`[-2, 2]`
with weight function :math:`1/\sqrt{1 - (x/2)^2}`.
See Also
--------
chebyt : Chebyshev polynomial of the first kind.
References
----------
.. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions"
Section 22. National Bureau of Standards, 1972.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
if n == 0:
n1 = n + 1
else:
n1 = n
x, w = roots_chebyc(n1)
if n == 0:
x, w = [], []
hn = 4 * pi * ((n == 0) + 1)
kn = 1.0
p = orthopoly1d(x, w, hn, kn,
wfunc=lambda x: 1.0 / sqrt(1 - x * x / 4.0),
limits=(-2, 2), monic=monic)
if not monic:
p._scale(2.0 / p(2))
p.__dict__['_eval_func'] = lambda x: eval_chebyc(n, x)
return p
# Chebyshev of the second kind S_n(x)
def roots_chebys(n, mu=False):
r"""Gauss-Chebyshev (second kind) quadrature.
Compute the sample points and weights for Gauss-Chebyshev
quadrature. The sample points are the roots of the nth degree
Chebyshev polynomial of the second kind, :math:`S_n(x)`. These
sample points and weights correctly integrate polynomials of
degree :math:`2n - 1` or less over the interval :math:`[-2, 2]`
with weight function :math:`w(x) = \sqrt{1 - (x/2)^2}`. See 22.2.7
in [AS]_ for more details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
x, w, m = roots_chebyu(n, True)
x *= 2
w *= 2
m *= 2
if mu:
return x, w, m
else:
return x, w
def chebys(n, monic=False):
r"""Chebyshev polynomial of the second kind on :math:`[-2, 2]`.
Defined as :math:`S_n(x) = U_n(x/2)` where :math:`U_n` is the
nth Chebychev polynomial of the second kind.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
S : orthopoly1d
Chebyshev polynomial of the second kind on :math:`[-2, 2]`.
Notes
-----
The polynomials :math:`S_n(x)` are orthogonal over :math:`[-2, 2]`
with weight function :math:`\sqrt{1 - (x/2)}^2`.
See Also
--------
chebyu : Chebyshev polynomial of the second kind
References
----------
.. [1] Abramowitz and Stegun, "Handbook of Mathematical Functions"
Section 22. National Bureau of Standards, 1972.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
if n == 0:
n1 = n + 1
else:
n1 = n
x, w = roots_chebys(n1)
if n == 0:
x, w = [], []
hn = pi
kn = 1.0
p = orthopoly1d(x, w, hn, kn,
wfunc=lambda x: sqrt(1 - x * x / 4.0),
limits=(-2, 2), monic=monic)
if not monic:
factor = (n + 1.0) / p(2)
p._scale(factor)
p.__dict__['_eval_func'] = lambda x: eval_chebys(n, x)
return p
# Shifted Chebyshev of the first kind T^*_n(x)
def roots_sh_chebyt(n, mu=False):
r"""Gauss-Chebyshev (first kind, shifted) quadrature.
Compute the sample points and weights for Gauss-Chebyshev
quadrature. The sample points are the roots of the nth degree
shifted Chebyshev polynomial of the first kind, :math:`T_n(x)`.
These sample points and weights correctly integrate polynomials of
degree :math:`2n - 1` or less over the interval :math:`[0, 1]`
with weight function :math:`w(x) = 1/\sqrt{x - x^2}`. See 22.2.8
in [AS]_ for more details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
xw = roots_chebyt(n, mu)
return ((xw[0] + 1) / 2,) + xw[1:]
def sh_chebyt(n, monic=False):
r"""Shifted Chebyshev polynomial of the first kind.
Defined as :math:`T^*_n(x) = T_n(2x - 1)` for :math:`T_n` the nth
Chebyshev polynomial of the first kind.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
T : orthopoly1d
Shifted Chebyshev polynomial of the first kind.
Notes
-----
The polynomials :math:`T^*_n` are orthogonal over :math:`[0, 1]`
with weight function :math:`(x - x^2)^{-1/2}`.
"""
base = sh_jacobi(n, 0.0, 0.5, monic=monic)
if monic:
return base
if n > 0:
factor = 4**n / 2.0
else:
factor = 1.0
base._scale(factor)
return base
# Shifted Chebyshev of the second kind U^*_n(x)
def roots_sh_chebyu(n, mu=False):
r"""Gauss-Chebyshev (second kind, shifted) quadrature.
Computes the sample points and weights for Gauss-Chebyshev
quadrature. The sample points are the roots of the nth degree
shifted Chebyshev polynomial of the second kind, :math:`U_n(x)`.
These sample points and weights correctly integrate polynomials of
degree :math:`2n - 1` or less over the interval :math:`[0, 1]`
with weight function :math:`w(x) = \sqrt{x - x^2}`. See 22.2.9 in
[AS]_ for more details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
x, w, m = roots_chebyu(n, True)
x = (x + 1) / 2
m_us = cephes.beta(1.5, 1.5)
w *= m_us / m
if mu:
return x, w, m_us
else:
return x, w
def sh_chebyu(n, monic=False):
r"""Shifted Chebyshev polynomial of the second kind.
Defined as :math:`U^*_n(x) = U_n(2x - 1)` for :math:`U_n` the nth
Chebyshev polynomial of the second kind.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
U : orthopoly1d
Shifted Chebyshev polynomial of the second kind.
Notes
-----
The polynomials :math:`U^*_n` are orthogonal over :math:`[0, 1]`
with weight function :math:`(x - x^2)^{1/2}`.
"""
base = sh_jacobi(n, 2.0, 1.5, monic=monic)
if monic:
return base
factor = 4**n
base._scale(factor)
return base
# Legendre
def roots_legendre(n, mu=False):
r"""Gauss-Legendre quadrature.
Compute the sample points and weights for Gauss-Legendre
quadrature. The sample points are the roots of the nth degree
Legendre polynomial :math:`P_n(x)`. These sample points and
weights correctly integrate polynomials of degree :math:`2n - 1`
or less over the interval :math:`[-1, 1]` with weight function
:math:`w(x) = 1.0`. See 2.2.10 in [AS]_ for more details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
numpy.polynomial.legendre.leggauss
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
m = int(n)
if n < 1 or n != m:
raise ValueError("n must be a positive integer.")
mu0 = 2.0
an_func = lambda k: 0.0 * k
bn_func = lambda k: k * np.sqrt(1.0 / (4 * k * k - 1))
f = cephes.eval_legendre
df = lambda n, x: (-n*x*cephes.eval_legendre(n, x)
+ n*cephes.eval_legendre(n-1, x))/(1-x**2)
return _gen_roots_and_weights(m, mu0, an_func, bn_func, f, df, True, mu)
def legendre(n, monic=False):
r"""Legendre polynomial.
Defined to be the solution of
.. math::
\frac{d}{dx}\left[(1 - x^2)\frac{d}{dx}P_n(x)\right]
+ n(n + 1)P_n(x) = 0;
:math:`P_n(x)` is a polynomial of degree :math:`n`.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
P : orthopoly1d
Legendre polynomial.
Notes
-----
The polynomials :math:`P_n` are orthogonal over :math:`[-1, 1]`
with weight function 1.
Examples
--------
Generate the 3rd-order Legendre polynomial 1/2*(5x^3 + 0x^2 - 3x + 0):
>>> from scipy.special import legendre
>>> legendre(3)
poly1d([ 2.5, 0. , -1.5, 0. ])
"""
if n < 0:
raise ValueError("n must be nonnegative.")
if n == 0:
n1 = n + 1
else:
n1 = n
x, w = roots_legendre(n1)
if n == 0:
x, w = [], []
hn = 2.0 / (2 * n + 1)
kn = _gam(2 * n + 1) / _gam(n + 1)**2 / 2.0**n
p = orthopoly1d(x, w, hn, kn, wfunc=lambda x: 1.0, limits=(-1, 1),
monic=monic, eval_func=lambda x: eval_legendre(n, x))
return p
# Shifted Legendre P^*_n(x)
def roots_sh_legendre(n, mu=False):
r"""Gauss-Legendre (shifted) quadrature.
Compute the sample points and weights for Gauss-Legendre
quadrature. The sample points are the roots of the nth degree
shifted Legendre polynomial :math:`P^*_n(x)`. These sample points
and weights correctly integrate polynomials of degree :math:`2n -
1` or less over the interval :math:`[0, 1]` with weight function
:math:`w(x) = 1.0`. See 2.2.11 in [AS]_ for details.
Parameters
----------
n : int
quadrature order
mu : bool, optional
If True, return the sum of the weights, optional.
Returns
-------
x : ndarray
Sample points
w : ndarray
Weights
mu : float
Sum of the weights
See Also
--------
scipy.integrate.quadrature
scipy.integrate.fixed_quad
References
----------
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover, 1972.
"""
x, w = roots_legendre(n)
x = (x + 1) / 2
w /= 2
if mu:
return x, w, 1.0
else:
return x, w
def sh_legendre(n, monic=False):
r"""Shifted Legendre polynomial.
Defined as :math:`P^*_n(x) = P_n(2x - 1)` for :math:`P_n` the nth
Legendre polynomial.
Parameters
----------
n : int
Degree of the polynomial.
monic : bool, optional
If `True`, scale the leading coefficient to be 1. Default is
`False`.
Returns
-------
P : orthopoly1d
Shifted Legendre polynomial.
Notes
-----
The polynomials :math:`P^*_n` are orthogonal over :math:`[0, 1]`
with weight function 1.
"""
if n < 0:
raise ValueError("n must be nonnegative.")
wfunc = lambda x: 0.0 * x + 1.0
if n == 0:
return orthopoly1d([], [], 1.0, 1.0, wfunc, (0, 1), monic,
lambda x: eval_sh_legendre(n, x))
x, w = roots_sh_legendre(n)
hn = 1.0 / (2 * n + 1.0)
kn = _gam(2 * n + 1) / _gam(n + 1)**2
p = orthopoly1d(x, w, hn, kn, wfunc, limits=(0, 1), monic=monic,
eval_func=lambda x: eval_sh_legendre(n, x))
return p
# -----------------------------------------------------------------------------
# Code for backwards compatibility
# -----------------------------------------------------------------------------
# Import functions in case someone is still calling the orthogonal
# module directly. (They shouldn't be; it's not in the public API).
poch = cephes.poch
# eval_chebyu, eval_sh_chebyt and eval_sh_chebyu: These functions are not
# used in orthogonal.py, they are not in _rootfuns_map, but their names
# do appear in _evalfuns, so they must be kept.
from ._ufuncs import (binom, eval_jacobi, eval_sh_jacobi, eval_gegenbauer,
eval_chebyt, eval_chebyu, eval_chebys, eval_chebyc,
eval_sh_chebyt, eval_sh_chebyu, eval_legendre,
eval_sh_legendre, eval_genlaguerre, eval_laguerre,
eval_hermite, eval_hermitenorm)
# Make the old root function names an alias for the new ones
_modattrs = globals()
for newfun, oldfun in _rootfuns_map.items():
_modattrs[oldfun] = _modattrs[newfun]
__all__.append(oldfun)