Computational Physics Lectures: Numerical integration, from Newton-Cotes quadrature to Gaussian quadrature

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Integration points and weights with orthogonal polynomials

To understand how the weights and the mesh points are generated, we define first a polynomial of degree $2N-1$ (since we have $2N$ variables at hand, the mesh points and weights for $N$ points). This polynomial can be represented through polynomial division by

$\begin{equation*} P_{2N-1}(x)=L_N(x)P_{N-1}(x)+Q_{N-1}(x), \end{equation*}$ where

$P_{N-1}(x)$ and

$Q_{N-1}(x)$ are some polynomials of degree

$N-1$ or less. The function

$L_N(x)$ is a Legendre polynomial of order

$N$ .

Recall that we wanted to approximate an arbitrary function $f(x)$ with a polynomial $P_{2N-1}$ in order to evaluate

$\begin{equation*} \int_{-1}^1f(x)dx\approx \int_{-1}^1P_{2N-1}(x)dx. \end{equation*}$