###### Options

# On recurrences converging to the wrong limit in finite precision and some new examples

Publikationstyp

Review Article

Publikationsdatum

2020-07-10

Sprache

English

Author

Institut

TORE-URI

Enthalten in

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.