Kvadratuurisäännöt

Kvadratuurisäännöt, or quadrature rules, are numerical integration techniques used to approximate definite integrals of functions. The term originates from the Latin word “quadratura,” meaning “to make a square,” and reflects the method’s goal of estimating areas under curves. These rules are essential tools in applied mathematics, engineering, physics, and computational science for evaluating integrals that cannot be solved analytically or for which an analytical solution is too cumbersome.

Quadrature rules are based on weighted sums of function values at specific sample points within the integration interval. The general form is

∫ₐᵇ f(x)dx ≈ Σᵢ wᵢ f(xᵢ),

where the xᵢ are nodes and the wᵢ are weights. Different choices of nodes and weights give rise to markedly different methods, each with advantages concerning accuracy, computational cost, and applicability to particular classes of functions.

Common kvadratuurisäännöt include the trapezoidal rule and Simpson’s rule, which employ equally spaced nodes and are easy to implement but may require many subintervals for high accuracy on oscillatory or sharply varying functions. Gaussian quadrature uses non‑equally spaced nodes and optimal weights chosen from orthogonal polynomials (Legendre, Chebyshev, etc.) to achieve maximal accuracy for polynomials of a given degree with fewer nodes. Extensions such as Clenshaw–Curtis quadrature and adaptive algorithms allow efficient handling of singularities and multidimensional integrals.

The effectiveness of a quadrature rule is measured by its error term, which depends on the function’s smoothness and the rule’s polynomial degree of exactness. In practice, numerical analysts select an appropriate kvadratuurisääntö by balancing precision requirements against computational resources, and often combine several rules within composite or adaptive schemes to recover robustness while retaining efficiency.