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Mathematics of Computation

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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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Article copyright: © Copyright 1963 American Mathematical Society