Finding the Weights in Gaussian Quadrature¶
Copyright (C) 2020 Andreas Kloeckner
MIT License
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.In [2]:
import numpy as np
import numpy.linalg as la
import scipy.special as sps
import matplotlib.pyplot as pt
Here's a utility routine to do biseciton, to use below:
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def bisection(f, a, b, tol=1e-14):
assert np.sign(f(a)) != np.sign(f(b))
while b-a > tol:
m = a + (b-a)/2
fm = f(m)
if np.sign(f(a)) != np.sign(fm):
b = m
else:
a = m
return m
Set the number of nodes:
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n = 5
Node selection: Plain Gauss¶
Gauss nodes are the roots of the $n$th Legendre polynomial $P_n$:
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nodes = sps.legendre(n).weights[:, 0]
mesh = np.linspace(-1, 1, 300)
pt.plot(mesh, sps.eval_legendre(n, mesh))
pt.plot(nodes, 0*nodes, "o")
Out[5]:
Node Selection: Gauss-Lobatto¶
In [27]:
def eval_legendre_deriv(n, x):
return (
(x*sps.eval_legendre(n, x) - sps.eval_legendre(n-1, x))
/
((x**2-1)/n))
brackets = sps.legendre(n-1).weights[:, 0]
nodes = np.zeros(n)
nodes[0] = -1
nodes[-1] = 1
from functools import partial
# Use the fact that the roots of P_{n-1} bracket the roots of P_{n-1}':
for i in range(n-2):
nodes[i+1] = bisection(
partial(eval_legendre_deriv, n-1),
brackets[i], brackets[i+1])
mesh = np.linspace(-1, 1, 300)
pt.plot(mesh, eval_legendre_deriv(n-1, mesh))
pt.plot(nodes, 0*nodes, "o")
Out[27]:
Node Selection: Gauss-Radau¶
For Gauss-Radau (with the left endpoint included), the nodes are the roots of the following function:
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def radau_func(n, x):
return (
(sps.eval_legendre(n-1, x) + sps.eval_legendre(n, x))
/
(1+x))
nodes = None
# Root finding left as an exercise for the reader. :)
mesh = np.linspace(-1, 1, 300)
pt.plot(mesh, radau_func(n, mesh))
Out[31]:
Finding the weights¶
Use method of undetermined coefficients to find the interpolatory quadrature rule for nodes:
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max_degree = len(nodes) - 1
powers = np.arange(max_degree+1)
Vt = nodes ** powers.reshape(-1, 1)
a, b = -1, 1
rhs = 1/(powers+1) * (b**(powers+1) - a**(powers+1))
weights = la.solve(Vt, rhs)
Now compare the approximate integrals of monomials from our rule with the true answers:
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for i in range(2*n + 1):
approx = weights @ nodes**i
true = 1/(i+1)*(1. - (-1)**(i+1))
print("Error at degree %d: %g" % (i, approx-true))
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