Eberlein measure and mechanical quadrature formulae II. Numerical results

Sarma, V. L. N.; Stroud, A. H.

doi:10.1090/S0025-5718-1969-0258282-6

Eberlein measure and mechanical quadrature formulae II. Numerical results
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by V. L. N. Sarma and A. H. Stroud PDF

Math. Comp. 23 (1969), 781-784 Request permission

Abstract:

In a previous paper it was shown how a probability measure (Eberlein measure) on the closed unit ball of the sequence space, ${l_1}$, can be used to find the variance ${\sigma ^2}$ of the error functional for a quadrature formula for the $k$-dimensional cube, regarded as a random variable. Here we give values of $\sigma$ for some specific formulae.

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

Messenger of Math.

G. M. Ewing, Questions, Discussions, and Notes: On Approximate Cubature, Amer. Math. Monthly 48 (1941), no. 2, 134–136. MR 1525068, DOI 10.2307/2303604
Preston C. Hammer and Arthur H. Stroud, Numerical evaluation of multiple integrals. II, Math. Tables Aids Comput. 12 (1958), 272–280. MR 102176, DOI 10.1090/S0025-5718-1958-0102176-6
Preston C. Hammer and A. Wayne Wymore, Numerical evaluation of multiple integrals. I, Math. Tables Aids Comput. 11 (1957), 59–67. MR 87220, DOI 10.1090/S0025-5718-1957-0087220-6
John Lamperti, Probability. A survey of the mathematical theory, W. A. Benjamin, Inc., New York-Amsterdam, 1966. MR 0206996

Proc. Cambridge Philos. Soc.

D. Mustard, J. N. Lyness, and J. M. Blatt, Numerical quadrature in $n$ dimensions, Comput. J. 6 (1963/64), 75–87. MR 158539, DOI 10.1093/comjnl/6.1.75
I. P. Mysovskih, On the construction of cubature formulae for the simplest regions, Ž. Vyčisl. Mat i Mat. Fiz. 4 (1964), 3–14 (Russian). MR 160334
Johann Radon, Zur mechanischen Kubatur, Monatsh. Math. 52 (1948), 286–300 (German). MR 33206, DOI 10.1007/BF01525334
Michael Sadowsky, A formula for approxiamte computation of a triple integral, Amer. Math. Monthly 47 (1940), 539–543. MR 2496, DOI 10.2307/2303834
V. L. N. Sarma, Eberlein measure and mechanical quadrature formulae. I. Basic theory, Math. Comp. 22 (1968), 607–616. MR 253559, DOI 10.1090/S0025-5718-1968-0253559-1
A. H. Stroud, Some fifth degree integration formulas for symmetric regions. II, Numer. Math. 9 (1966/67), 460–468. MR 218017, DOI 10.1007/BF02162160
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