Chapter 16 - Gauss Quadrature and Multi-dimensional Integrals

Abstract

Gauss quadrature rules are designed so that an N-point quadrature rule will exactly integrate a polynomial of degree 2N−1 or lower. This is done by picking the N weights and N evaluation points (i.e., abscissas) to integrate the 2N terms in a degree 2N−1 polynomial. In particular we cover Gauss–Legendre quadrature formulas finite domain. On smooth functions, Gauss quadrature can converge exponentially to the correct answer. On functions with singularities, the convergence is slower. Multidimensional integrals are computed by repeatedly applying 1-D quadrature rules. In the exercises Gauss–Hermite and Gauss–Lobatto quadrature rules are discussed.