An error analysis for numerical multiple integration. III

Authors:
Robert E. Barnhill and Gregory M. Nielson

Journal:
Math. Comp. **24** (1970), 301-314

MSC:
Primary 65.55

DOI:
https://doi.org/10.1090/S0025-5718-1970-0275665-7

MathSciNet review:
0275665

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**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**0093912**, https://doi.org/10.1002/zamm.19580380102**[2]**Robert E. Barnhill,*An error analysis for numerical multiple integration. I*, Math. Comp.**22**(1968), 98–109. MR**0226852**, https://doi.org/10.1090/S0025-5718-1968-0226852-6**[3]**Philip J. Davis,*Interpolation and approximation*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. MR**0157156****[4]**Walter Gautschi,*Construction of Gauss-Christoffel quadrature formulas*, Math. Comp.**22**(1968), 251–270. MR**0228171**, https://doi.org/10.1090/S0025-5718-1968-0228171-0**[5]**W. J. Gordon,*Blending-Function Methods of Bivariate and Multivariate Interpolation and Approximation*. General Motors Research Report GMR-834, Warren, Michigan, 1968.**[6]**Preston C. Hammer,*Numerical evaluation of multiple integrals*, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Edited by R. E. Langer. Publication no. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin, The University of Wisconsin Press, Madison, 1959, pp. 99–115. MR**0100355****[7]**P. M. Hirsch,*Evaluation of orthogonal polynomials and relationship to evaluating multiple integrals*, Math. Comp.**22**(1968), 280–285. MR**0226855**, https://doi.org/10.1090/S0025-5718-1968-0226855-1**[8]**Johann Radon,*Zur mechanischen Kubatur*, Monatsh. Math.**52**(1948), 286–300 (German). MR**0033206**, https://doi.org/10.1007/BF01525334**[9]**Arthur Sard,*Linear approximation*, American Mathematical Society, Providence, R.I., 1963. MR**0158203****[10]**A. H. Stroud,*Integration formulas and orthogonal polynomials*, SIAM J. Numer. Anal.**4**(1967), 381–389. MR**0228180**, https://doi.org/10.1137/0704034**[11]**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**0177240****[12]**A. H. Stroud,*Approximate Calculation of Multiple Integrals*. (Manuscript.)**[13]**Arthur H. Stroud,*Quadrature methods for functions of more than one variable*, Ann. New York Acad. Sci.**86**(1960), 776–791 (1960). MR**0119417****[14]**G. W. Tyler,*Numerical integration of functions of several variables*, Canadian J. Math.**5**(1953), 393–412. MR**0056379****[15]**R. A. Valentin,*Applications of Functional Analysis to Optimal Approximation for Analytic Functions*, Ph.D. Thesis, Division of Applied Math., Brown Univ., Providence, R. I., 1965.**[16]**G. M. Nielson,*Nonlinear Approximations in the Norm*, M.S. Thesis, Department of Mathematics, Univ. of Utah, Salt Lake City, Utah, 1968.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0275665-7

Keywords:
Cubature,
numerical integration,
approximate integration,
error bounds

Article copyright:
© Copyright 1970
American Mathematical Society