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

Contents

Orthogonal polynomials, Legendre

It is common to choose the normalization condition $$ \begin{equation*} L_N(1)=1. \end{equation*} $$ With these equations we can determine a Legendre polynomial of arbitrary order with input polynomials of order $ N-1 $ and $ N-2 $.

As an example, consider the determination of $ L_0 $, $ L_1 $ and $ L_2 $. We have that $$ \begin{equation*} L_0(x) = c, \end{equation*} $$ with $ c $ a constant. Using the normalization equation $ L_0(1)=1 $ we get that $$ \begin{equation*} L_0(x) = 1. \end{equation*} $$