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

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] S. Haber, ``A modified Monte-Carlo quadrature,'' Math. Comp., v. 20, 1966, pp. 361-368. MR 35 #1178. MR 0210285 (35:1178)
[2] S. Haber, ``A modified Monte-Carlo quadrature. II,'' Math. Comp., v. 21, 1967, pp. 388-397. MR 0234606 (38:2922)
[3] N. S. Bahvalov, ``On approximate calculation of multiple integrals,'' Vestnik Moscow Univ., v. 4, 1959, pp. 3-18. (Russian) MR 0115275 (22:6077)
[4] P. J. Davis, ``A construction of nonnegative approximate quadratures,'' Math. Comp., v. 21, 1967, pp. 578-582. MR 36 #5584. MR 0222534 (36:5584)
[5] A. H. Stroud, ``Quadrature methods for functions of more than one variable,'' Ann. New York Acad. Sci., v. 86, I960, pp. 776-791. MR 22 #10179. MR 0119417 (22:10179)
[6] J. M. Cook, ``Rational formulae for the production of a spherically symmetric probability distribution,'' MTAC, v. 11, 1957, pp. 81-82. MR 19, 466. MR 690630 (19:466g)
[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] J. Williamson, ``Note on Hadamard's determinant theorem,'' Bull. Amer. Math. Soc., v. 53, 1947, pp. 608-613. MR 8, 559. MR 0020538 (8:559g)
[11] E. Spence, ``A new class of Hadamard matrices,'' Glasgow Math. J., v. 8, 1967, pp. 59-62. MR 35 #1496. MR 0210610 (35:1496)
[12] V. I. Krylov, Approximate Calculation of Integrals, Fizmatgiz, Moscow, 1959; English transl., Macmillan, New York, 1962. MR 22 #2002; MR 26 #2008. MR 0144464 (26:2008)
[13] S. L. Sobolev, ``On the order of convergence of cubature formulae,'' Dokl. Akad. Nauk SSSR, v. 162, 1965, pp. 1005-1008 = Soviet Math. Dokl., v. 6, 1965, pp. 808-811. MR 31 #3776. MR 0179528 (31:3776)
[14] S. Haber, ``A combination of Monte-Carlo and classical methods for evaluating multiple integrals,'' Bull. Amer. Math. Soc., v. 74, 1968, pp. 683-686. MR 0243718 (39:5039)
[15] R. E. Barnhill, ``An error analysis for numerical multiple integration. II,'' Math. Comp., v. 22, 1968, pp. 286-292. MR 37 #6027. MR 0230465 (37:6027)