Note on the round-off errors in iterative processes

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.