Nyström method

{{About|the method of solving integral equations|the low-rank matrix approximation|Low-rank matrix approximations#Nyström approximation}}

In mathematics numerical analysis, the Nyström method{{cite journal|last=Nyström|first=Evert Johannes|title=Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben |journal=Acta Mathematica |year=1930|volume=54|issue=1|pages=185–204|doi=10.1007/BF02547521|doi-access=free}} or quadrature method seeks the numerical solution of an integral equation by replacing the integral with a representative weighted sum. The continuous problem is broken into $n$ discrete intervals; quadrature or numerical integration determines the weights and locations of representative points for the integral.

The problem becomes a system of linear equations with $n$ equations and $n$ unknowns, and the underlying function is implicitly represented by an interpolation using the chosen quadrature rule. This discrete problem may be ill-conditioned, depending on the original problem and the chosen quadrature rule.

Since the linear equations require $O(n^3)$ {{fact|reason=Inversion by Strassen algorithm is ~n^2.3|date=May 2021}}operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large $n$ for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non-singular problems.