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:


 * $$\int_a^b h (x) \;\mathrm d x \approx \sum_{k=1}^n w_k h (x_k)$$

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


 * $$f (x) = \lambda u (x) - \int_a^b K (x, x') f (x') \;\mathrm d x'$$,

results in


 * $$f (x) \approx \lambda u (x) - \sum_{k=1}^n w_k K (x, x_k) f (x_k)$$.