Gausskvadraturák

Gausskvadraturák, also known as Gaussian quadrature, are numerical integration methods used to approximate definite integrals. The core idea is to select specific points, called nodes, within the integration interval and assign weights to them, such that a weighted sum of the function's values at these nodes approximates the integral with high accuracy. Unlike Newton-Cotes formulas which use equally spaced points, Gaussian quadrature chooses nodes and weights optimally to achieve the highest possible degree of precision for a given number of points.

The general form of a Gaussian quadrature rule is a sum of the form $\sum_{i=1}^n w_i f(x_i)$, where $x_i$ are the nodes and $w_i$ are the corresponding weights. The key advantage is that for $n$ nodes, Gaussian quadrature can integrate polynomials of degree up to $2n-1$ exactly. This means that if the function being integrated is a polynomial of degree $2n-1$ or less, the approximation will be exact. For functions that can be approximated well by polynomials, Gaussian quadrature provides a very efficient and accurate method.

The choice of nodes and weights depends on the specific type of Gaussian quadrature. The most common is Gauss-Legendre quadrature, which uses Legendre polynomials and integrates over the interval $[-1, 1]$. Other variants exist for different intervals and weight functions, such as Gauss-Hermite quadrature for the interval $(-\infty, \infty)$ with a weight function $e^{-x^2}$, and Gauss-Laguerre quadrature for the interval $[0, \infty)$ with a weight function $e^{-x}$. The nodes for these methods are typically the roots of orthogonal polynomials.