Volume 52, pp. 571-575, 2020.

Addendum to “On recurrences converging to the wrong limit in finite precision and some new examples”

Siegfried M. Rump

Abstract

In a recent paper [Electron. Trans. Numer. Anal, 52 (2020), pp. 358–369], we analyzed Muller's famous 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 gave necessary and sufficient conditions for such recurrences to produce only nonzero iterates. In the above-mentioned paper, an example was given where only finitely many terms of the recurrence over $\mathbb{R}$ are well defined, but floating-point evaluation indicates convergence to the attracting fixed point. The input data of that example, however, are not representable in binary floating-point, and the question was posed whether such examples exist with binary representable data. This note answers that question in the affirmative.

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Key words

recurrences, rounding errors, IEEE-754, exactly representable data, bfloat, half precision (binary16), single precision (binary32), double precision (binary64)

AMS subject classifications

65G50, 11B37

Links to the cited ETNA articles

[5]	Vol. 52 (2020), pp. 358-369 Siegfried M. Rump: On recurrences converging to the wrong limit in finite precision and some new examples

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