Optimal quadrature formulas using generalized inverses. I. General theory and minimum variance formulas

Duris, C. S.

doi:10.1090/S0025-5718-1971-0295567-0

Optimal quadrature formulas using generalized inverses. I. General theory and minimum variance formulas
HTML articles powered by AMS MathViewer

by C. S. Duris PDF

Math. Comp. 25 (1971), 495-504 Request permission

Abstract:

This paper is the first of two papers concerning the derivation of optimal quadrature formulas. In Part I, we develop results concerning generalized inverses and use these results to derive some minimum variance quadrature formulas. The formulas are obtained by inverting appropriate systems of numerical differentiation formulas. The second paper, Part II, will use the same results concerning generalized inverses to derive Sard “best” quadrature formulas.

References

C. S. Duris, A simplex sufficiency condition for quadrature formulas, Math. Comp. 20 (1966), 68–78. MR 196934, DOI 10.1090/S0025-5718-1966-0196934-4
Vladimir Ivanovich Krylov, Approximate calculation of integrals, The Macmillan Company, New York-London, 1962, 1962. Translated by Arthur H. Stroud. MR 0144464
Philip J. Davis and Philip Rabinowitz, Numerical integration, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1967. MR 0211604
Arthur Sard, Linear approximation, American Mathematical Society, Providence, R.I., 1963. MR 0158203, DOI 10.1090/surv/009
Arthur Sard, Smoothest approximation formulas, Ann. Math. Statistics 20 (1949), 612–615. MR 36278, DOI 10.1214/aoms/1177729956
T. N. E. Greville, Some applications of the pseudoinverse of a matrix, SIAM Rev. 2 (1960), 15–22. MR 110185, DOI 10.1137/1002004
Charles A. Rohde, Some results on generalized inverses, SIAM Rev. 8 (1966), 201–205. MR 210722, DOI 10.1137/1008040
R. P. Tewarson, On some representations of generalized inverses, SIAM Rev. 11 (1969), 272–276. MR 245584, DOI 10.1137/1011045
Richard J. Hanson and Charles L. Lawson, Extensions and applications of the Householder algorithm for solving linear least squares problems, Math. Comp. 23 (1969), 787–812. MR 258258, DOI 10.1090/S0025-5718-1969-0258258-9