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). The difference between these two approximations is used to estimate the calculational error of the integration.
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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 GNU Scientific Library, the NAG Numerical Libraries and R.
Description
The problem in numerical integration is to approximate definite integrals of the form
Such integrals can be approximated, for example, by n-point Gaussian quadrature
where wi, xi 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 the midpoint for odd numbers of evaluation points), and thus the integrand must be evaluated at every point. Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by adding
Example
A popular example combines a 7-point Gauss rule with a 15-point Kronrod rule (Kahaner, Moler & Nash 1989, §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.
The recommended error estimate is
Patterson (1968) showed how to find further extensions of this type, Piessens (1974) and Monegato (1978) proposed improved algorithms, and finally the most efficient algorithm was proposed by Laurie (1997). Quadruple precision (34 decimal digits) coefficients for (G7, K15), (G10, K21), (G15,K31), (G20,K41) and others are computed and tabulated.