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Nyström method

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

∫ a b h ( x ) d x ≈ ∑ k = 1 n w k h ( x k ) {\displaystyle \int _{a}^{b}h(x)\;\mathrm {d} x\approx \sum _{k=1}^{n}w_{k}h(x_{k})}

where w k {\displaystyle w_{k}} are the weights of the quadrature rule, and points x k {\displaystyle x_{k}} are the abscissas.

Example

Applying this to the inhomogeneous Fredholm equation of the second kind

f ( x ) = λ u ( x ) − ∫ a b K ( x , x ′ ) f ( x ′ ) d x ′ {\displaystyle f(x)=\lambda u(x)-\int _{a}^{b}K(x,x')f(x')\;\mathrm {d} x'} ,

results in

f ( x ) ≈ λ u ( x ) − ∑ k = 1 n w k K ( x , x k ) f ( x k ) {\displaystyle f(x)\approx \lambda u(x)-\sum _{k=1}^{n}w_{k}K(x,x_{k})f(x_{k})} .

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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