In mathematics

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

, the Nyström method or quadrature method seeks the

numerical solution Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

of an

integral equation In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...

by replacing the integral with a representative weighted sum. The continuous problem is broken into

n

discrete intervals; quadrature or

numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...

determines the weights and locations of representative points for the integral. The problem becomes a

system of linear equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three ...

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 In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more ...

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_^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 In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to sol ...

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_^n w_k K (x, x_k) f (x_k)

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. {{DEFAULTSORT:Nystrom method Integral equations Numerical analysis Numerical integration (quadrature)

Discretization of the integral

Example

See also

References

Bibliography