The Gauss–Kronrod quadrature formula is an
adaptive method for
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral.
The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...
. It is a variant of
Gaussian quadrature
In numerical analysis, an -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for .
Th ...
, 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
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
.
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
In numerical analysis, an -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for .
Th ...
:
where ''w''
''i'', are the
weights and ''x''
''i'' are 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
points to an
-point rule in such a way that the resulting rule is exact for polynomials of degree less than or equal to
(; the corresponding Gauss rule is of order
). 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 . Because the Gauss points are incorporated into the Kronrod points, a total of only 15 function evaluations are needed.
:
The integral is then estimated by the Kronrod rule
and the error can be estimated as
.
For an arbitrary interval