Nyström method

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 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 $O(n^{3})$ 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.

Discretization of the integral

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

$\int _{a}^{b}h(x)\;\mathrm {d} x\approx \sum _{k=1}^{n}w_{k}h(x_{k})$

where $w_{k}$ are the weights of the quadrature rule, and points $x_{k}$ are the abscissas.

Example

Applying this to the inhomogeneous Fredholm equation of the second kind

$f(x)=\lambda u(x)-\int _{a}^{b}K(x,x')f(x')\;\mathrm {d} x'$,

results in

$f(x)\approx \lambda u(x)-\sum _{k=1}^{n}w_{k}K(x,x_{k})f(x_{k})$.

See also

• Boundary element method

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.