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| {{Context|date=October 2009}}
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| In [[numerical analysis]], the '''Nyström method'''<ref>{{cite journal|last=Nyström|first=Evert Johannes|title=Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben|journal=Acta Mathematica|year=1930|volume=54|issue=1|pages=185–204|doi=10.1007/BF02547521}}</ref> 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>n</math> discrete intervals, quadrature or [[numerical integration]] determines the weights and locations of representative points for the integral. The discrete problem to be solved is now a [[system of linear equations]] with n equations and n unknowns.
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| From the n solved points the function value at other points is interpolated consistent with the chosen quadrature method.
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| Depending on the original problem and the choice of quadrature the problem may be ill-conditioned.
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| Since the linear equations require <math>O(n^3)</math> operations to solve, hence high-order quadrature rules perform better because low-order quadrature rules require large <math>n</math> for a given accuracy. [[Gaussian quadrature]] is normally a good choice for smooth, non-singular problems.
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| == Discretization of the integral ==
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| :<math>\int_a^b h (x) \;\mathrm d x \approx \sum_{k=1}^n w_k h (x_k)</math>
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| :where <math>w_k</math> are the weights of the quadrature rule, and points <math>x_k</math> are the abscissas. | |
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| == Example ==
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| Applying this to the inhomogeneous [[Fredholm equation]] of the second kind
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| :<math>f (x) = \lambda u (x) - \int_a^b K (x, x') f (x') \;\mathrm d x'</math>,
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| results in
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| :<math>f (x) \approx \lambda u (x) - \sum_{k=1}^n w_k K (x, x_k) f (x_k)</math>.
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| == References ==
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| {{reflist}}
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| * Leonard M. Delves & Joan Walsh (eds): ''Numerical Solution of Integral Equations'', Clarendon, Oxford, 1974.
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| * Hans-Jürgen Reinhardt: ''Analysis of Approximation Methods for Differential and Integral Equations'', Springer, New York, 1985.
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| {{DEFAULTSORT:Nystrom method}}
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| [[Category:Integral equations]]
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| [[Category:Numerical analysis]]
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| [[Category:Numerical integration (quadrature)]]
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| {{mathapplied-stub}}
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