An error analysis for numerical multiple integration. III

Barnhill, Robert E.; Nielson, Gregory M.

doi:10.1090/S0025-5718-1970-0275665-7

An error analysis for numerical multiple integration. III
HTML articles powered by AMS MathViewer

by Robert E. Barnhill and Gregory M. Nielson PDF

Math. Comp. 24 (1970), 301-314 Request permission

Abstract:

Asymptotic results of two types are given for minimum norm cubatures, defined on a certain Hubert space of analytic functions. Cubatures with a high degree of precision are realized as the limits of minimum norm cubatures, and an algebraic approach for the determination of efficient rules is derived. Numerical methods and examples are also included.

References

J. Albrecht and L. Collatz, Zur numerischen Auswertung mehrdimensionaler Integrale, Z. Angew. Math. Mech. 38 (1958), 1–15 (German, with English, French and Russian summaries). MR 93912, DOI 10.1002/zamm.19580380102
Robert E. Barnhill, An error analysis for numerical multiple integration. I, Math. Comp. 22 (1968), 98–109. MR 226852, DOI 10.1090/S0025-5718-1968-0226852-6
Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1963. MR 0157156
Walter Gautschi, Construction of Gauss-Christoffel quadrature formulas, Math. Comp. 22 (1968), 251–270. MR 228171, DOI 10.1090/S0025-5718-1968-0228171-0

Blending-Function Methods of Bivariate and Multivariate Interpolation and Approximation

Preston C. Hammer, Numerical evaluation of multiple integrals, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Publication of the Mathematics Research Center, U.S. Army, the University of Wisconsin, no. 1, University of Wisconsin Press, Madison, Wis., 1959, pp. 99–115. Edited by R. E. Langer. MR 0100355
P. M. Hirsch, Evaluation of orthogonal polynomials and relationship to evaluating multiple integrals, Math. Comp. 22 (1968), 280–285. MR 226855, DOI 10.1090/S0025-5718-1968-0226855-1
Johann Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 (1948), 286–300 (German). MR 33206, DOI 10.1007/BF01525334
Arthur Sard, Linear approximation, American Mathematical Society, Providence, R.I., 1963. MR 0158203
A. H. Stroud, Integration formulas and orthogonal polynomials, SIAM J. Numer. Anal. 4 (1967), 381–389. MR 228180, DOI 10.1137/0704034
D. D. Stancu, The remainder of certain linear approximation formulas in two variables, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 1 (1964), 137–163. MR 177240

Approximate Calculation of Multiple Integrals

Arthur H. Stroud, Quadrature methods for functions of more than one variable, Ann. New York Acad. Sci. 86 (1960), 776–791 (1960). MR 119417
G. W. Tyler, Numerical integration of functions of several variables, Canad. J. Math. 5 (1953), 393–412. MR 56379, DOI 10.4153/cjm-1953-044-1

Applications of Functional Analysis to Optimal Approximation for Analytic Functions

Nonlinear Approximations in the ${l_2}$ Norm