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

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Integration points and weights with orthogonal polynomials

Instead of an integration problem we need now to define the coefficient

$\alpha_0$ . Since we know the values of

$Q_{N-1}$ at the zeros of

$L_N$ , we may rewrite Eq. (15) as

$\begin{equation} Q_{N-1}(x_k)=\sum_{i=0}^{N-1}\alpha_iL_i(x_k)=\sum_{i=0}^{N-1}\alpha_iL_{ik} \hspace{1cm} k=0,1,\dots, N-1. \tag{16} \end{equation}$ Since the Legendre polynomials are linearly independent of each other, none of the columns in the matrix

$L_{ik}$ are linear combinations of the others.