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

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Integration points and weights with orthogonal polynomials

If we identify the weights with

$2(L^{-1})_{0i}$ , where the points

$x_i$ are the zeros of

$L_N$ , we have an integration formula of the type

$\begin{equation*} \int_{-1}^1P_{2N-1}(x)dx=\sum_{i=0}^{N-1}\omega_iP_{2N-1}(x_i) \end{equation*}$ and if our function

$f(x)$ can be approximated by a polynomial

$P$ of degree

$2N-1$ , we have finally that

$\begin{equation*} \int_{-1}^1f(x)dx\approx \int_{-1}^1P_{2N-1}(x)dx=\sum_{i=0}^{N-1}\omega_iP_{2N-1}(x_i) . \end{equation*}$ In summary, the mesh points

$x_i$ are defined by the zeros of an orthogonal polynomial of degree

$N$ , that is

$L_N$ , while the weights are given by

$2(L^{-1})_{0i}$ .