Gauss-Kronrod quadrature in an adaptive Gaussian quadrature method for numerical integration in which error is estimation based
on evaluation at special points known as "Kronrod points." By suitably
picking these points, abscissas from previous iterations can be reused as part of
the new set of points, whereas usual Gaussian
quadrature would require recomputation of all abscissas at each iteration. This
is particularly important when some specified degree of accuracy is needed but the
number of points needed to achieve this accuracy is not known ahead of time. Kronrod
(1964) showed how to pick Kronrod points optimally from Legendre-Gauss
quadrature, and Patterson (1968, 1969) showed how to compute continued extensions
of this kind (Press et al. 1992, p. 154).
With Method -> Automatic, the Wolfram LanguageNIntegrate
function uses Gauss-Kronrod quadrature for one-dimensional integrals.
Calvetti, D.; Golub, G. H.; Gragg, W. B. and Reichel, L. "Computation of Gauss-Kronrod Quadrature Rules." Math. Comput.69,
1035-1052, 2000.Calvetti, D.; Golub, G. H.; Gragg, W. B. and
Reichel, L. "Computation of Gauss-Kronrod Quadrature Rules." Stanford University
Scientific Computing/Computational Mathematics Report SCCM-98-09. http://www-sccm.stanford.edu/pub/sccm/sccm98-09.ps.gz.Kronrod,
A. S. [Russian]. Doklady Akad. Nauk SSSR154, 283-286, 1964.Patterson,
T. N. L. Math. Comput.22, 847-856 and C1-C11, 1968.Patterson,
T. N. L. Math. Comput.23, 892, 1969.Piessens,
R.; de Doncker, E.; Uberhuber, C. W.; and Kahaner, D. K. QUADPACK:
A Subroutine Package for Automatic Integration. New York: Springer-Verlag,
1983.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.;
and Vetterling, W. T. Numerical
Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, p. 154, 1992.Ueberhuber, C. W.
Numerical
Computation 2: Methods, Software, and Analysis. Berlin: Springer-Verlag,
pp. 105-106, 1997.
