Nyström method

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:

$\underset{a}{\overset{b}{\int }}h(x)\mathrm{d}x\approx \sum _{k=1}^n{w}_{k}h(x_k)$

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

$f(x)=\lambda u(x)-\underset{a}{\overset{b}{\int }}K(x,x^{\prime })f(x^{\prime })\mathrm{d}{x}^{\prime }$,

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.