Gausskvadraturát

Gausskvadraturát, also known as Gaussian quadrature, is a numerical method for approximating definite integrals. It's a technique that evaluates a weighted sum of function values at specific points, known as nodes, within the integration interval. The key innovation of Gaussian quadrature lies in its ability to choose both the nodes and the weights optimally to achieve a higher degree of accuracy for a given number of function evaluations compared to simpler methods like the trapezoidal rule or Simpson's rule.

The method is derived by assuming that the integral of a polynomial can be exactly represented by a weighted sum of function values at certain points. By carefully selecting these points and weights, Gaussian quadrature can exactly integrate polynomials of degree up to 2n-1, where n is the number of nodes used. This efficiency makes it a valuable tool in various fields, including physics, engineering, and computer graphics, where accurate integration is crucial for solving complex problems. The specific nodes and weights are typically derived from orthogonal polynomials, such as Legendre polynomials for integrals over the interval [-1, 1]. For integrals over different intervals, a transformation is applied.