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 [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ n }[/math] equations and [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ O(n^3) }[/math]^{[citation needed]}operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large [math]\displaystyle{ n }[/math] 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:

[math]\displaystyle{ \int_a^b h (x) \;\mathrm d x \approx \sum_{k=1}^n w_k h (x_k) }[/math]

where [math]\displaystyle{ w_k }[/math] are the weights of the quadrature rule, and points [math]\displaystyle{ x_k }[/math] are the abscissas.

Example

Applying this to the inhomogeneous Fredholm equation of the second kind

[math]\displaystyle{ f (x) = \lambda u (x) - \int_a^b K (x, x') f (x') \;\mathrm d x' }[/math],

results in

[math]\displaystyle{ f (x) \approx \lambda u (x) - \sum_{k=1}^n w_k K (x, x_k) f (x_k) }[/math].

See also

• Boundary element method

References

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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