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On recurrences converging to the wrong limit in finite precision and some new examples
Publikationstyp
Review Article
Date Issued
2020-07-10
Sprache
English
Author(s)
Institut
TORE-URI
Start Page
358
End Page
369
Citation
Electronic Transactions on Numerical Analysis (52): 358-369 (2020)
Publisher DOI
Scopus ID
Publisher
Kent State Univ.
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.
Subjects
Bfloat
Different precisions
Double precision (binary64)
Extended precision (binary128)
Half precision (binary16)
IEEE-754
Multiple precision
Pisot sequence
Recurrences
Rounding errors
Single precision (binary32)
Skolem problem
DDC Class
004: Informatik
510: Mathematik