Computing the Weights in Newton-Cotes Rules

Copyright (C) 2020 Andreas Kloeckner

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In [5]:
import numpy as np

import numpy.linalg as la

We start by choosing our quadrature nodes, the maximum degree which will be exact, as well as the interval $(a,b)$ on which we integrate:

In [6]:
nodes = [0, 1]

#nodes = [0, 0.5, 1]

#nodes = [3, 3.5, 4]

#nodes = [0, 1, 2]

#nodes = np.linspace(0, 1, 12)



max_degree = len(nodes)-1



a = nodes[0]

b = nodes[-1]

Next, we compute the transpose of the Vandermonde matrix $V^T$ and the integrals $\int_a^b x^i$ as rhs:

In [13]:
nodes = np.array(nodes)

powers = np.arange(max_degree+1)



Vt = nodes ** powers.reshape(-1, 1)



rhs = 1/(powers+1) * (b**(powers+1) - a**(powers+1))



if len(nodes) <= 4:

    print(Vt)

[[1 1]
[0 1]]

Set up the linear system for the weights:

$$ \begin{align*} \alpha_0 x_0^0 + \cdots + \alpha_{n-1} x_{n-1}^{0} &= \int_a^b x^0\\ \vdots &= \vdots \\ \alpha_0 x_0^{n-1} + \cdots + \alpha_{n-1} x_{n-1}^{n-1} &= \int_a^b x^{n-1} \end{align*} $$
In [9]:
#clear

weights = la.solve(Vt, rhs)



print(weights)

[ 0.5  0.5]

Now we test our quadrature rule by integrating the monomials $\int_a^b x^i dx$ and comparing quadrature results to the true answers:

In [12]:
for i in range(len(nodes) + 1):

    approx = weights @ nodes**i

    true = 1/(i+1)*(b**(i+1) - a**(i+1))

    

    print("Error at degree %d: %g" % (i, approx-true))

Error at degree 0: 0
Error at degree 1: 0
Error at degree 2: 0.166667
In [24]: