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DEFINITE INTEGRATION- Numerical techniques

Numerical integration is the approximate computation of an integral using numerical technique. The numerical computation of an integral is sometimes called quadrature. The word quadrature has incompatible meanings. Integration by quadrature either means solving an integral analytically (i.e., symbolically in terms of known functions), or solving an integral numerically. The word "quadrature" means numerical computation of a univariate integral, and "cubature" means numerical computation of a multiple integral.
Numerical integration methods can generally be described as combining evaluation of the integrand to get an approximation of the integral. The integrand is evaluated at a finite set of points called integration points and a weighted sum of these values is used to approximate the integral. The integration points and weights depend on the specific method used and the accuracy required from the approximation.
There are a wide range of methods available for numerical integration. The most straightforward numerical integration technique uses the Newton-Cotes formulas (also called quadrature formulas), which approximate a function tabulated at a sequence of regularly spaced intervals by various degree polynomials. If the endpoints are tabulated, then the 2- and 3-point formulas are called the trapezoidal rule and Simpson's rule, respectively.
Modern numerical integration methods based on information theory have been developed to simulate information systems such as computer controlled systems, communication systems, and control systems since in these cases; the classical methods (which are based on approximation theory) are not as efficient.

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