On a posteriori error estimates

Miel, George

doi:10.1090/S0025-5718-1977-0426418-4

On a posteriori error estimates
HTML articles powered by AMS MathViewer

by George Miel PDF

Math. Comp. 31 (1977), 204-213 Request permission

Abstract:

Consider a sequence $\{ {x_n}\} _{n = 0}^\infty$ in a normed space X converging to some ${x^\ast } \in X$. It is shown that the sequence satisfies a condition of the type \[ \left \| {{x^\ast } - {x_n}} \right \| \leqslant \alpha \left \| {{x_n} - {x_{n - 1}}} \right \|\] for some constant $\alpha$ and every $n \geqslant 1$, if the associated null sequence $\{ {e_n}\} _{n = 0}^\infty ,{e_n} = {x^\ast } - {x_n}$ , is uniformly decreasing in norm or if it is alternating with respect to any ordering whose cone of positive elements is acute.

References

Abraham Berman and Robert J. Plemmons, Cones and iterative methods for best least squares solutions of linear systems, SIAM J. Numer. Anal. 11 (1974), 145–154. MR 348984, DOI 10.1137/0711015
G. Blanch, Numerical evaluation of continued fractions, SIAM Rev. 6 (1964), 383–421. MR 171374, DOI 10.1137/1006092
G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, Chelsea Publishing Co., New York, 1959. 6th ed. MR 0121327
Lothar Collatz, Functional analysis and numerical mathematics, Academic Press, New York-London, 1966. Translated from the German by Hansjörg Oser. MR 0205126
Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. [Ginn and Co.], New York-Toronto-London, 1963. MR 0157156
William B. Gragg, Truncation error bounds for $g$-fractions, Numer. Math. 11 (1968), 370–379. MR 228158, DOI 10.1007/BF02161885
W. B. Gragg, Truncation error bounds for $\pi$-fractions, Bull. Amer. Math. Soc. 76 (1970), 1091–1094. MR 262723, DOI 10.1090/S0002-9904-1970-12574-5
Peter Henrici and Pia Pfluger, Truncation error estimates for Stieltjes fractions, Numer. Math. 9 (1966), 120–138. MR 212991, DOI 10.1007/BF02166031
Graham Jameson, Ordered linear spaces, Lecture Notes in Mathematics, Vol. 141, Springer-Verlag, Berlin-New York, 1970. MR 0438077
Thomas H. Jefferson, Truncation error estimates for $T$-fractions, SIAM J. Numer. Anal. 6 (1969), 359–364. MR 260147, DOI 10.1137/0706033
William B. Jones and W. J. Thron, A posteriori bounds for the truncation error of continued fractions, SIAM J. Numer. Anal. 8 (1971), 693–705. MR 295536, DOI 10.1137/0708063

Theory and Application of Infinite Series

Pierre-Jean Laurent, Approximation et optimisation, Collection Enseignement des Sciences, No. 13, Hermann, Paris, 1972 (French). MR 0467080
E. P. Merkes, On truncation errors for continued fraction computations, SIAM J. Numer. Anal. 3 (1966), 486–496. MR 202283, DOI 10.1137/0703042
George Miel, On a posteriori error estimates, Math. Comp. 31 (1977), no. 137, 204–213. MR 426418, DOI 10.1090/S0025-5718-1977-0426418-4
Alexandre Ostrowski, Les estimations des erreurs a posteriori dans les procédés itératifs, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A275–A278 (French). MR 305629
A. M. Ostrowski, Solution of equations in Euclidean and Banach spaces, Pure and Applied Mathematics, Vol. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Third edition of Solution of equations and systems of equations. MR 0359306
A. M. Ostrowski, A posteriori error estimates in iterative procedures, SIAM J. Numer. Anal. 10 (1973), 290–298. MR 339443, DOI 10.1137/0710028
J. H. Rowland and G. J. Miel, Exit criteria for Newton-Cotes quadrature rules, SIAM J. Numer. Anal. 14 (1977), no. 6, 1145–1150. MR 458814, DOI 10.1137/0714079
J. H. Rowland and Y. L. Varol, Exit criteria for Simpson’s compound rule, Math. Comp. 26 (1972), 699–703. MR 341823, DOI 10.1090/S0025-5718-1972-0341823-8
Arthur Sard, Integral representations of remainders, Duke Math. J. 15 (1948), 333–345. MR 26757
Daniel Shanks, A logarithm algorithm, Math. Tables Aids Comput. 8 (1954), 60–64. MR 61464, DOI 10.1090/S0025-5718-1954-0061464-9

Exit Criteria for Some Numerical Algorithms