|
Adaptive Quadrature
Adaptive quadrature is a numerical integration method in which the integral of a function f(x) is approximated using static quadrature rules on adaptively refined subintervals of the region of integration. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well behaved" integrands, but are also effective for "badly behaved" integrands for which traditional algorithms may fail. General scheme Adaptive quadrature follows the general scheme 1. procedure integrate ( f, a, b, τ ) 2. Q \approx \int_a^b f(x)\,\mathrmx 3. \varepsilon \approx \left, Q - \int_a^b f(x)\,\mathrmx\ 4. if ''ε'' > ''τ'' then 5. m = (a + b) / 2 6. Q = integrate(f, a, m, τ/2) + integrate(f, m, b, τ/2) 7. endif 8. return Q An approximation Q to the integral of f(x) over the interval ,b/math> is computed (line 2), as well as an error estimate \varepsilon (line 3). If the estimated error is larger than the required tol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure ('' quadrature'' or ''squaring''), as in the quadrature of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
