Gauss–Kronrod quadrature formula

In numerical mathematics, the Gauss–Kronrod quadrature formula is a method for numerical integration (calculating approximate values of integrals). Gauss–Kronrod quadrature is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation. It is an example of what is called a nested quadrature rule: for the same set of function evaluation points, it has two quadrature rules, one higher-order and one lower-order (the latter called an "embedded" rule), and the difference between these two approximations is used to estimate the calculational error of the integration.

These formulas are named after Alexander Kronrod, who invented them in the 1960s, and Carl Friedrich Gauss. Gauss–Kronrod quadrature is used in the QUADPACK library, the NAG Numerical Libraries and Matlab.

Description

The problem in numerical integration is to approximate definite integrals of the form

:$int\_a^b\; f(x),dx.$

Such integrals can be approximated, for example, by "n"-point Gaussian quadrature

:$int\_a^b\; f(x),dx\; approx\; sum\_\{i=1\}^n\; w\_i\; f(x\_i).$

where "w"_"i", "x"_"i" are the weights and points at which to evaluate the function "f"("x").

If the interval ["a", "b"] is subdivided, the Gauss evaluation points of the new subintervals never coincide with the previous evaluation points (except at zero for odd numbers), and thus the integrand must be evaluated at every point. Gauss–Kronrod formulas are extensions of Gauss quadrature formulas generated by adding $n+1$ points to an $n$-point rule in such a way that the resulting rule is of order $3n+1$. This allows for computing higher-order estimates while re-using the function values of a lower-order estimate. The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error.

Example

A popular example combines a 7-point Gauss rule with a 15-point Kronrod rule Harv|Kahaner|Moler|Nash|1989|loc=§5.5. Because the Gauss points are incorporated into the Kronrod points, a total of only 15 function evaluations yields both a quadrature estimate and an error estimate.

:

harvtxt|Patterson|1968 showed how to find further extensions of this type.

References

*
* (Authorized translation from the Russian)
* (Reference guide for QUADPACK)
*. Erratum in "Math. Comput." 23: 892.

External links

* QUADPACK (part of SLATEC): description [http://www.netlib.org/slatec/src/qpdoc.f] , source code [http://www.netlib.org/slatec/src] . QUADPACK is a collection of algorithms, in Fortran, for numerical integration based on Gauss-Kronrod rules. SLATEC (at Netlib) is a large public domain library for numerical computing.