The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration.
It 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.
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.
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).
These extra points are the zeros of Stieltjes polynomials.
The difference between a Gauss quadrature rule and its Kronrod extension are often used as an estimate of the approximation error.
are scaled to the interval as follows: Patterson (1968) showed how to find further extensions of this type, Piessens & Branders (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.