"""
orthopoly.py - A suite of functions for generating orthogonal polynomials
and quadrature rules.
Copyright (c) 2014 Greg von Winckel
All rights reserved.
Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:
The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
Last updated on Wed Jan 1 14:29:25 MST 2014
Modified by David A. Ham (david.ham@imperial.ac.uk), 2016
"""
import numpy as np
from functools import reduce
from math import gamma
[docs]
def gauss(alpha, beta):
"""
Compute the Gauss nodes and weights from the recursion
coefficients associated with a set of orthogonal polynomials
Inputs:
alpha - recursion coefficients
beta - recursion coefficients
Outputs:
x - quadrature nodes
w - quadrature weights
Adapted from the MATLAB code by Walter Gautschi
http://www.cs.purdue.edu/archives/2002/wxg/codes/gauss.m
"""
from numpy.linalg import eigh
A = np.diag(np.sqrt(beta)[1:], 1) + np.diag(alpha)
x, V = eigh(A, "U")
w = beta[0] * np.real(np.power(V[0, :], 2))
return x, w
[docs]
def lobatto(alpha, beta, xl1, xl2):
"""
Compute the Lobatto nodes and weights with the preassigned
nodea xl1,xl2
Inputs:
alpha - recursion coefficients
beta - recursion coefficients
xl1 - assigned node location
xl2 - assigned node location
Outputs:
x - quadrature nodes
w - quadrature weights
Based on the section 7 of the paper
"Some modified matrix eigenvalue problems"
by Gene Golub, SIAM Review Vol 15, No. 2, April 1973, pp.318--334
"""
from numpy.linalg import solve
n = len(alpha) - 1
en = np.zeros(n)
en[-1] = 1
A1 = np.vstack((np.sqrt(beta), alpha - xl1))
J1 = np.diag(A1[0, 1:-1], 1) + np.diag(A1[1, 1:]) + np.diag(A1[0, 1:-1], -1)
A2 = np.vstack((np.sqrt(beta), alpha - xl2))
J2 = np.diag(A2[0, 1:-1], 1) + np.diag(A2[1, 1:]) + np.diag(A2[0, 1:-1], -1)
g1 = solve(J1, en)
g2 = solve(J2, en)
C = np.array(((1, -g1[-1]), (1, -g2[-1])))
xl = np.array((xl1, xl2))
ab = solve(C, xl)
alphal = alpha
alphal[-1] = ab[0]
betal = beta
betal[-1] = ab[1]
x, w = gauss(alphal, betal)
return x, w
[docs]
def rec_jacobi(N, a, b):
"""
Generate the recursion coefficients alpha_k, beta_k
P_{k+1}(x) = (x-alpha_k)*P_{k}(x) - beta_k P_{k-1}(x)
for the Jacobi polynomials which are orthogonal on [-1,1]
with respect to the weight w(x)=[(1-x)^a]*[(1+x)^b]
Inputs:
N - polynomial order
a - weight parameter
b - weight parameter
Outputs:
alpha - recursion coefficients
beta - recursion coefficients
Adapted from the MATLAB code by Dirk Laurie and Walter Gautschi
http://www.cs.purdue.edu/archives/2002/wxg/codes/r_jacobi.m
"""
nu = (b - a) / float(a + b + 2)
mu = 2 ** (a + b + 1) * gamma(a + 1) * gamma(b + 1) / gamma(a + b + 2)
if N == 1:
alpha = nu
beta = mu
else:
n = np.arange(1.0, N)
nab = 2 * n + a + b
alpha = np.hstack((nu, (b ** 2 - a ** 2) / (nab * (nab + 2))))
n = n[1:]
nab = nab[1:]
B1 = 4 * (a + 1) * (b + 1) / float((a + b + 2) ** 2 * (a + b + 3))
B = 4 * (n + a) * (n + b) * n * (n + a + b) / \
(nab ** 2 * (nab + 1) * (nab - 1))
beta = np.hstack((mu, B1, B))
return alpha, beta
[docs]
def rec_jacobi01(N, a, b):
"""
Generate the recursion coefficients alpha_k, beta_k
for the Jacobi polynomials which are orthogonal on [0,1]
See rec_jacobi for the recursion coefficients on [-1,1]
Inputs:
N - polynomial order
a - weight parameter
b - weight parameter
Outputs:
alpha - recursion coefficients
beta - recursion coefficients
Adapted from the MATLAB implementation:
https://www.cs.purdue.edu/archives/2002/wxg/codes/r_jacobi01.m
"""
if a <= -1 or b <= -1:
raise ValueError('''Jacobi coefficients are defined only
for alpha,beta > -1''')
if not isinstance(N, int):
raise TypeError('N must be an integer')
if N < 1:
raise ValueError('N must be at least 1')
c, d = rec_jacobi(N, a, b)
alpha = (1 + c) / 2
beta = d / 4
beta[0] = d[0] / 2 ** (a + b + 1)
return alpha, beta
[docs]
def polyval(alpha, beta, x):
"""
Evaluate polynomials on x given the recursion coefficients alpha and beta
"""
N = len(alpha)
m = len(x)
P = np.zeros((m, N + 1))
P[:, 0] = 1
P[:, 1] = (x - alpha[0]) * P[:, 0]
for k in range(1, N):
P[:, k + 1] = (x - alpha[k]) * P[:, k] - beta[k] * P[:, k - 1]
return P
[docs]
def jacobi(N, a, b, x, NOPT=1):
"""
JACOBI computes the Jacobi polynomials which are orthogonal on [-1,1]
with respect to the weight w(x)=[(1-x)^a]*[(1+x)^b] and evaluate them
on the given grid up to P_N(x). Setting NOPT=2 returns the
L2-normalized polynomials
"""
m = len(x)
P = np.zeros((m, N + 1))
apb = a + b
a1 = a - 1
b1 = b - 1
c = apb * (a - b)
P[:, 0] = 1
if N > 0:
P[:, 1] = 0.5 * (a - b + (apb + 2) * x)
if N > 1:
for k in range(2, N + 1):
k2 = 2 * k
g = k2 + apb
g1 = g - 1
g2 = g - 2
d = 2.0 * (k + a1) * (k + b1) * g
P[:, k] = (g1 * (c + g2 * g * x) * P[:, k - 1] -
d * P[:, k - 2]) / (k2 * (k + apb) * g2)
if NOPT == 2:
k = np.arange(N + 1)
pnorm = 2 ** (apb + 1) * gamma(k + a + 1) * gamma(k + b + 1) / \
((2 * k + a + b + 1) * (gamma(k + 1) * gamma(k + a + b + 1)))
P *= 1 / np.sqrt(pnorm)
return P
[docs]
def jacobiD(N, a, b, x, NOPT=1):
"""
JACOBID computes the first derivatives of the normalized Jacobi
polynomials which are orthogonal on [-1,1] with respect
to the weight w(x)=[(1-x)^a]*[(1+x)^b] and evaluate them
on the given grid up to P_N(x). Setting NOPT=2 returns
the derivatives of the L2-normalized polynomials
"""
z = np.zeros((len(x), 1))
if N == 0:
Px = z
else:
Px = 0.5 * np.hstack((z, jacobi(N - 1, a + 1, b + 1, x, NOPT) *
((a + b + 2 + np.arange(N)))))
return Px
[docs]
def mm_log(N, a):
"""
MM_LOG Modified moments for a logarithmic weight function.
The call mm=MM_LOG(n,a) computes the first n modified moments of the
logarithmic weight function w(t)=t^a log(1/t) on [0,1] relative to
shifted Legendre polynomials.
REFERENCE: Walter Gautschi,``On the preceding paper `A Legendre
polynomial integral' by James L. Blue'',
Math. Comp. 33 (1979), 742-743.
Adapted from the MATLAB implementation:
https://www.cs.purdue.edu/archives/2002/wxg/codes/mm_log.m
"""
if a <= -1:
raise ValueError('Parameter a must be greater than -1')
prod = lambda z: reduce(lambda x, y: x * y, z, 1)
mm = np.zeros(N)
c = 1
for n in range(N):
if isinstance(a, int) and a < n:
p = range(n - a, n + a + 2)
mm[n] = (-1) ** (n - a) / prod(p)
mm[n] *= gamma(a + 1) ** 2
else:
if n == 0:
mm[0] = 1 / (a + 1) ** 2
else:
k = np.arange(1, n + 1)
s = 1 / (a + 1 + k) - 1 / (a + 1 - k)
p = (a + 1 - k) / (a + 1 + k)
mm[n] = (1 / (a + 1) + sum(s)) * prod(p) / (a + 1)
mm[n] *= c
c *= 0.5 * (n + 1) / (2 * n + 1)
return mm
[docs]
def mod_chebyshev(N, mom, alpham, betam):
"""
Calcuate the recursion coefficients for the orthogonal polynomials
which are are orthogonal with respect to a weight function which is
represented in terms of its modifed moments which are obtained by
integrating the monic polynomials against the weight function.
References
----------
John C. Wheeler, "Modified moments and Gaussian quadratures"
Rocky Mountain Journal of Mathematics, Vol. 4, Num. 2 (1974), 287--296
Walter Gautschi, "Orthogonal Polynomials (in Matlab)
Journal of Computational and Applied Mathematics, Vol. 178 (2005) 215--234
Adapted from the MATLAB implementation:
https://www.cs.purdue.edu/archives/2002/wxg/codes/chebyshev.m
"""
if not isinstance(N, int):
raise TypeError('N must be an integer')
if N < 1:
raise ValueError('N must be at least 1')
N = min(N, int(len(mom) / 2))
alpha = np.zeros(N)
beta = np.zeros(N)
normsq = np.zeros(N)
sig = np.zeros((N + 1, 2 * N))
alpha[0] = alpham[0] + mom[1] / mom[0]
beta[0] = mom[0]
sig[1, :] = mom
for n in range(2, N + 1):
for m in range(n - 1, 2 * N - n + 1):
sig[n, m] = sig[n - 1, m + 1] - (alpha[n - 2] - alpham[m]) * sig[n - 1, m] - \
beta[n - 2] * sig[n - 2, m] + betam[m] * sig[n - 1, m - 1]
alpha[n - 1] = alpham[n - 1] + sig[n, n] / sig[n, n - 1] - sig[n - 1, n - 1] / \
sig[n - 1, n - 2]
beta[n - 1] = sig[n, n - 1] / sig[n - 1, n - 2]
normsq = np.diagonal(sig, -1)
return alpha, beta, normsq
[docs]
def rec_jaclog(N, a):
"""
Generate the recursion coefficients alpha_k, beta_k
P_{k+1}(x) = (x-alpha_k)*P_{k}(x) - beta_k P_{k-1}(x)
for the monic polynomials which are orthogonal on [0,1]
with respect to the weight w(x)=x^a*log(1/x)
Inputs:
N - polynomial order
a - weight parameter
Outputs:
alpha - recursion coefficients
beta - recursion coefficients
Adated from the MATLAB code:
https://www.cs.purdue.edu/archives/2002/wxg/codes/r_jaclog.m
"""
alphaj, betaj = rec_jacobi01(2 * N, 0, 0)
mom = mm_log(2 * N, a)
alpha, beta, _ = mod_chebyshev(N, mom, alphaj, betaj)
return alpha, beta