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 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 equations and 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 [ citation needed]operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non-singular problems.
Standard quadrature methods seek to represent an integral as a weighed sum in the following manner:
where are the weights of the quadrature rule, and points are the abscissas.
Applying this to the inhomogeneous Fredholm equation of the second kind
results in
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 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 equations and 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 [ citation needed]operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non-singular problems.
Standard quadrature methods seek to represent an integral as a weighed sum in the following manner:
where are the weights of the quadrature rule, and points are the abscissas.
Applying this to the inhomogeneous Fredholm equation of the second kind
results in