Note on the round-off errors in iterative processes
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- by J. Descloux PDF
- Math. Comp. 17 (1963), 18-27 Request permission
Abstract:
This paper discusses round-off errors in iterative processes for solving equations. Let ${x_{n + 1}} = {x_n} + F({x_n})$ be a scalar iterative converging process; the different values ${x_n}$ are represented in a computer with a certain precision; when ${x_n}$ is close to the limit, $F({x_n})$ is small and can perhaps be obtained easily with a higher absolute precision than ${x_n}$; consequently, the addition ${x_n} + F({x_n})$ will practically involve a rounding operation. Besides some general remarks, it will be shown that for a fixed-point computer an appropriate rounding method can provide a more accurate solution to the problem; analogous results are given in Appendix I for a floating-point computer; Appendix II deals with Aitken’s ${\delta ^2}$ process. The author is indebted to A. H. Taub for many suggestions and stimulating discussions.References
- Alston S. Householder, Principles of numerical analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059056
- Arnold Nordsieck, On numerical integration of ordinary differential equations, Math. Comp. 16 (1962), 22–49. MR 136519, DOI 10.1090/S0025-5718-1962-0136519-5 Jean Descloux, Remarks on the Round-Off Errors in Iterative Processes for Fixed-Point Computers. University of Illinois, Digital Computer Laboratory, Urbana, Illinois, Report No. 116, May, 1962. Jean Descloux, Remarks on Errors in First-Order Iterative Processes with Floating-Point Computers. University of Illinois, Digital Computer Laboratory, Urbana, Illinois, Report No. 113, March, 1962.
Additional Information
- © Copyright 1963 American Mathematical Society
- Journal: Math. Comp. 17 (1963), 18-27
- MSC: Primary 65.10; Secondary 65.50
- DOI: https://doi.org/10.1090/S0025-5718-1963-0152102-0
- MathSciNet review: 0152102