Siegfried M. Rump
S. 358 - 369
doi:
10.1553/etna_vol52s358
Verlag der Österreichischen Akademie der Wissenschaften
doi:
10.1553/etna_vol52s358
Abstract:
In 1989, Jean-Michel Muller gave a famous example of a recurrence where, for particular initial values, the iteration over real numbers converges to a repellent fixed point, whereas finite precision arithmetic produces a different result, the attracting fixed point. We analyze recurrences in that spirit and remove a gap in previous arguments in the literature, that is, the recursion must be well defined. The latter is known as the Skolem problem. We identify initial values producing a limit equal to the repellent fixed point, show that in every ε-neighborhood of such initial values the recurrence is not well defined, and characterize initial values for which the recurrence is well defined. We give some new examples in that spirit. For example, the correct real result, i.e., the repellent fixed point, may be correctly computed in bfloat, half, single, double, formerly extended precision (80 bit format), binary128 as well as many formats of much higher precision. Rounding errors may be beneficial by introducing some regularizing effect.
recurrences, rounding errors, IEEE-754, different precisions, bfloat, half precision (binary16), single precision (binary32), double precision (binary64), extended precision (binary128), multiple precision, Skolem problem, Pisot sequence
Published Online:
2020/07/09 11:59:00
Object Identifier:
0xc1aa5576 0x003ba8e8
Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613.
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