Quadratisointi

Quadratisointi, also known as cubature, is a numerical method used to approximate definite integrals, particularly in higher dimensions. It is a generalization of quadrature, which is used for one-dimensional integrals. Quadratisointi aims to estimate the value of a multivariable integral over a given region by evaluating the integrand at a specific set of points within that region and combining these values using a weighted sum.

The core idea behind quadratisointi is to construct a formula that accurately approximates the integral of a polynomial of a certain degree. If the integrand is a polynomial of that degree or less, the quadratisointi formula will yield the exact value of the integral. For more complex integrands, the formula provides an approximation, and the accuracy depends on the degree of the polynomial that the formula can exactly integrate and the behavior of the integrand.

Various quadratisointi methods exist, differing in the choice of evaluation points (nodes) and the corresponding weights. Common methods include Newton-Cotes type rules adapted for higher dimensions, Gaussian quadratisointi, and adaptive quadratisointi techniques. Gaussian quadratisointi is often favored for its efficiency and ability to achieve high accuracy with a relatively small number of function evaluations by choosing optimal node locations. Adaptive quadratisointi refines the approximation by subdividing the integration domain in regions where the integrand varies rapidly.

Quadratisointi finds applications in many scientific and engineering fields, including computational physics, fluid dynamics, finance, and machine learning, where analytical solutions to integrals are often impossible to obtain.