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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 Language NIntegrate function uses Gauss-Kronrod quadrature for one-dimensional integrals.

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References

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 SSSR 154, 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.