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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Note on the round-off errors in iterative processes


Author: J. Descloux
Journal: Math. Comp. 17 (1963), 18-27
MSC: Primary 65.10; Secondary 65.50
MathSciNet review: 0152102
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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.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1963-0152102-0
PII: S 0025-5718(1963)0152102-0
Article copyright: © Copyright 1963 American Mathematical Society