Nyström method

Short description: Method of solving integral equations

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

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 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(n3) [citation needed]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.

## Contents

* 1 Discretization of the integral
* 2 Example
* 3 See also
* 4 References
* 5 Bibliography

## Discretization of the integral

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

∫abh(x)dx≈∑k=1nwkh(xk)

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

## Example

Applying this to the inhomogeneous Fredholm equation of the second kind

f(x)=λu(x)−∫abK(x,x′)f(x′)dx′,

results in

f(x)≈λu(x)−∑k=1nwkK(x,xk)f(xk).

## See also

* Boundary element method

## 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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