Nyström method

In mathematics numerical analysis, the Nyström method[1] 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 discrete intervals; quadrature or numerical integration determines the weights and locations of representative points for the integral.

This article is about the method of solving integral equations. For the low-rank matrix approximation, see Low-rank matrix approximations § Nyström approximation.

The problem becomes a system of linear equations with equations and 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 operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non-singular problems.

Discretization of the integral

Standard quadrature methods seek to represent an integral as a weighed sum in the following manner:

where are the weights of the quadrature rule, and points are the abscissas.

Example

Applying this to the inhomogeneous Fredholm equation of the second kind

,

results in

.

See also

References
  1. Nyström, Evert Johannes (1930). "Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben". Acta Mathematica. 54 (1): 185–204. doi:10.1007/BF02547521.

Bibliography
  • Leonard M. Delves & Joan E. Walsh (eds): Numerical Solution of Integral Equations, Clarendon, Oxford, 1974.
  • Hans-Jürgen Reinhardt: Analysis of Approximation Methods for Differential and Integral Equations, Springer, New York, 1985.
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