Stochastic quadrature formulas

Haber, Seymour

doi:10.1090/S0025-5718-1969-0260139-1

Stochastic quadrature formulas

Author: Seymour Haber
Journal: Math. Comp. 23 (1969), 751-764
MSC: Primary 65.15
DOI: https://doi.org/10.1090/S0025-5718-1969-0260139-1
Corrigendum: Math. Comp. 24 (1970), 1001.
Corrigendum: Math. Comp. 24 (1970), 1001.
MathSciNet review: 0260139
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A class of formulas for the numerical evaluation of multiple integrals is described, which combines features of the Monte-Carlo and the classical methods. For certain classes of functions--defined by smoothness conditions--these formulas provide the fastest possible rate of convergence to the integral. Asymptotic error estimates are derived, and a method is described for obtaining good a posteriori error bounds when using these formulas. Equal-coefficients formulas of this class, of degrees up to 3, are constructed.

[1] Seymour Haber, A modified Monte-Carlo quadrature, Math. Comp. 20 (1966), 361–368. MR 0210285, https://doi.org/10.1090/S0025-5718-1966-0210285-0
[2] Seymour Haber, A modified Monte-Carlo quadrature. II, Math. Comp. 21 (1967), 388–397. MR 0234606, https://doi.org/10.1090/S0025-5718-1967-0234606-9
[3] N. S. Bahvalov, Approximate computation of multiple integrals, Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fiz. Him. 1959 (1959), no. 4, 3–18 (Russian). MR 0115275
[4] Philip J. Davis, A construction of nonnegative approximate quadratures, Math. Comp. 21 (1967), 578–582. MR 0222534, https://doi.org/10.1090/S0025-5718-1967-0222534-4
[5] Arthur H. Stroud, Quadrature methods for functions of more than one variable, Ann. New York Acad. Sci. 86 (1960), 776–791 (1960). MR 0119417
[6] J. M. Cook, Rational formulae for the production of a spherically symmetric probability distribution, Math. Tables Aids Comput. 11 (1957), 81–82. MR 690630, https://doi.org/10.1090/S0025-5718-1957-0690630-7
[7] M. E. Muller, ``A note on a method for generating points uniformly on -dimensional spheres,'' Comm. ACM, v. 2, 1959, pp. 19-20.
[8] J. Hadamard, ``Resolution d'une question relative aux determinants,'' Bull. Sci. Math., (2), v. 17, 1893, pp. 240-246.
[9] R. E. A. C. Paley, ``On orthogonal matrices,'' J. Math, and Phys., v. 12, 1933, pp. 311-320.
[10] John Williamson, Note on Hadamard’s determinant theorem, Bull. Amer. Math. Soc. 53 (1947), 608–613. MR 0020538, https://doi.org/10.1090/S0002-9904-1947-08853-4
[11] E. Spence, A new class of Hadamard matrices, Glasgow Math. J. 8 (1967), 59–62. MR 0210610, https://doi.org/10.1017/S0017089500000094
[12] Vladimir Ivanovich Krylov, Approximate calculation of integrals, Translated by Arthur H. Stroud, The Macmillan Co., New York-London, 1962, 1962. MR 0144464
[13] S. L. Sobolev, On the order of convergence of cubature formulae, Dokl. Akad. Nauk SSSR 162 (1965), 1005–1008 (Russian). MR 0179528
[14] Seymour Haber, A combination of Monte Carlo and classical methods for evaluating multiple integrals, Bull. Amer. Math. Soc. 74 (1968), 683–686. MR 0243718, https://doi.org/10.1090/S0002-9904-1968-11993-7
[15] Robert E. Barnhill, An error analysis for numerical multiple integration. II, Math. Comp. 22 (1968), 286–292. MR 0230465, https://doi.org/10.1090/S0025-5718-1968-0230465-X

Bibliography