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. 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.

These formulas are named after Alexander Kronrod, who invented them in the 1960s, and Carl Friedrich Gauss.

## Description

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

${\displaystyle \int _{a}^{b}f(x)\,dx.}$

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

${\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i}),}$

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 ${\displaystyle n+1}$ points to an ${\displaystyle n}$-point rule in such a way that the resulting rule is of order ${\displaystyle 3n+1}$ (Laurie (1997, p. 1133); the corresponding Gauss rule is of order ${\displaystyle 2n-1}$). These extra points are the zeros of Stieltjes polynomials. This allows for computing higher-order estimates while reusing 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 (Kahaner, Moler & Nash 1989, §5.5). Because the Gauss points are incorporated into the Kronrod points, a total of only 15 function evaluations are needed.

(G7, K15) on [−1,1]
Gauss nodesWeights
±0.94910 79123 427590.12948 49661 68870
±0.74153 11855 993940.27970 53914 89277
±0.40584 51513 773970.38183 00505 05119
0.00000 00000 000000.41795 91836 73469
Kronrod nodesWeights
±0.99145 53711 208130.02293 53220 10529
±0.94910 79123 427590.06309 20926 29979
±0.86486 44233 597690.10479 00103 22250
±0.74153 11855 993940.14065 32597 15525
±0.58608 72354 676910.16900 47266 39267
±0.40584 51513 773970.19035 05780 64785
±0.20778 49550 078980.20443 29400 75298
0.00000 00000 000000.20948 21410 84728

The integral is then estimated by the Kronrod rule ${\displaystyle K15}$ and the error can be estimated as ${\displaystyle |G7-K15|}$.

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.[1]

## Implementations

Routines for Gauss–Kronrod quadrature are provided by the QUADPACK library, the GNU Scientific Library, the NAG Numerical Libraries, R,[2] and the C++ library Boost.[3]