Finding the Weights in Gaussian Quadrature

Copyright (C) 2020 Andreas Kloeckner

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Here's a utility routine to do biseciton, to use below:

Set the number of nodes:

Node selection: Plain Gauss

Gauss nodes are the roots of the $n$th Legendre polynomial $P_n$:

Node Selection: Gauss-Lobatto

Gauss-Lobatto nodes are (except for the endpoints) the roots of $P_{n-1}'$:

(See here or here for a formula for $P_n'$.)

Node Selection: Gauss-Radau

For Gauss-Radau (with the left endpoint included), the nodes are the roots of the following function:

Finding the weights

Use method of undetermined coefficients to find the interpolatory quadrature rule for nodes:

Now compare the approximate integrals of monomials from our rule with the true answers: