Integral Approximation Task

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A Integral Approximation Task is a numerical approximation task that requires the production of an approximate function to a function's integral.



References

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2015

  • (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/numerical_integration Retrieved:2015-6-24.
    • In numerical analysis, numerical integration constitutes 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. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals. Numerical integration over more than one dimension is sometimes incorrectly described as cubature, since the meaning of quadrature is understood for higher-dimensional integration as well.

      The basic problem in numerical integration is to compute an approximate solution to a definite integral : [math] \int_a^b\! f(x)\, dx [/math] to a given degree of accuracy. If f(x) 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.

2014

  • (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Numerical_integration#Reasons_for_numerical_integration Retrieved:2014-4-26.
    • There are several reasons for carrying out numerical integration. The integrand f(x) may be known only at certain points, such as obtained by sampling. Some embedded systems and other computer applications may need numerical integration for this reason.

      A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative that is an elementary function. An example of such an integrand is f(x) = exp(−x2), the antiderivative of which (the error function, times a constant) cannot be written in elementary form. It may be possible to find an antiderivative symbolically, but it may be easier to compute a numerical approximation than to compute the antiderivative. That may be the case if the antiderivative is given as an infinite series or product, or if its evaluation requires a special function that is not available.