Gaussquadrature

Gaussquadrature, also known as Gaussian quadrature, is a family of numerical methods for approximating definite integrals. The core idea is to replace the integral with a weighted sum of function evaluations at specific points, called nodes. Unlike simpler methods like the trapezoidal rule or Simpson's rule, which use equally spaced nodes, Gaussian quadrature selects the nodes and weights optimally to achieve the highest possible degree of accuracy for a given number of function evaluations.

The general form of a Gaussian quadrature formula for integrating a function f(x) over the interval [a, b] is given by the sum of w_i * f(x_i), where x_i are the nodes and w_i are the corresponding weights. The key insight is that for a fixed number of nodes, say n, Gaussian quadrature can achieve an exact result for polynomials of degree up to 2n-1. This is a significant advantage over methods with fixed node positions.

The nodes and weights for Gaussian quadrature are typically derived from orthogonal polynomials. For integration over the interval [-1, 1], the nodes are the roots of Legendre polynomials, and the weights are determined based on these roots. For other intervals, a change of variables is used to transform the integral to the standard [-1, 1] interval. Different types of Gaussian quadrature exist, named after the orthogonal polynomials they employ, such as Gauss-Legendre, Gauss-Hermite, Gauss-Laguerre, and Gauss-Chebyshev quadrature, each suited for integrating functions with different weight functions or over different domains. These methods are widely used in various scientific and engineering fields for numerical integration.