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ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis, pp. 358-369, 2020/07/09
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.
Keywords: 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