An a priori estimate for the truncation error of a continued fraction expansion to the Gaussian error function

Abstract

--Zusammenfassung An a priori Estimate for the Truncation Error of a Continued Fraction Expansion to the Gaussian ErrorFunction. The truncation error for a continued fraction to the Gaussian error function is estimated. The precision of the obtained bounds is verified by comparison with the exact values. The related precision as well as the number of needed iterations are discussed in several ways.AMS Subject Classifications. Primary: 33A20; secondary: 30A04, 3A22, 41A20, 41A21,65D20, 65G05. Key words." Gaussian error function, continued fraction expansion, truncation error. Eine a priori Absch~itzung des Abbruchfehlers einer Kettenbruchentwicklung zum Gaul~sehenFehlerintegral. Der Abbruchfehler ffir eine Kettenbruchentwicklung zurn Gaugschen Fehlerintegral wird beidseitig abgesch~tzt. Die Giite der Absch~itzungen wird durch Vergleich mit den exakten Werten geprtift. Die zugeh6rige erreichbare Genauigkeit sowie die Anzahl der ben6tigten Iterationen werden in verschiedener Weise diskutiert.

Numeration and discrete dynamical systems

“…Transducers also allow one to deduce the continued fraction expansion of an homographic image from the continued fraction expansion of the original number (see [87]). The analytic theory of continued fractions has now proved its efficiency for the evaluation of functions (see the classic references [55,97] and also [17,28] for a priori truncation error estimates for continued fraction representations). Lastly, based on the work of Gosper [47], exact real computer arithmetic with continued fractions has also been thoroughly investigated (see [95], and also [60,66,73]).…”