Hostname: page-component-89b8bd64d-ktprf Total loading time: 0 Render date: 2026-05-07T13:14:12.598Z Has data issue: false hasContentIssue false
  • English
  • Français

High order algorithms for calculating roots

Published online by Cambridge University Press:  17 October 2016

Ulrich Abel
Ulrich Abel*
Affiliation:
Technische Hochschule Mittelhessen, Department MND, Wilhelm-Leuschner-Straße 13, 61169 Friedberg, Germany e-mail: Ulrich.Abel@mnd.thm.de
Get access

Extract

In a recent Note [1] Michael D. Hirschhorn presented high order algorithms for calculating numerically square roots and cube roots. In particular, he obtained the method

(1)

with , where the convergence is of tenth order:

We recall his idea in the case of an arbitrary square root with a > 0. Let p ⩾ 2 be a fixed integer. Our starting point is the relation

Choose

Information

Type
Articles
Information
The Mathematical Gazette , Volume 100 , Issue 549 , November 2016 , pp. 420 - 428
Copyright
Copyright © Mathematical Association 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Hirschhorn, Michael D., Tenth order algorithms for calculating and , Math. Gaz. 95 (November 2011) pp. 526–528.Google Scholar
2. Schröder, E., Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen 2 (1870) pp. 317–365.CrossRefGoogle Scholar
3. Householder, Alston S., Principles of Numerical Analysis, McGraw-Hill, New York-London (1953).Google Scholar
4. Shidkow, N. P. and Beresin, I. S., Numerische Methoden 2, VEB Deutscher Verlag der Wissenschaften, Berlin (1971).Google Scholar
5. Walther, Richard, Konstruktion und Analyse von Iterationsmethoden höherer Ordnung (Bachelor thesis), Technische Hochschule Mittelhessen, Friedberg (2011).Google Scholar
6. Traub, J. F., Comparison of iterative methods for the calculation of n th roots, Commun. ACM 4 (1961) pp. 143–145.CrossRefGoogle Scholar
7. Traub, J. F., Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, New Jersey (1964).Google Scholar
8. Halley, E., A new, exact, and easy method of finding the roots of any equations generally and that without any previous reduction (Latin), [English translation: Philos. Trans. Roy. Soc. Abridged. Vol. III, London, pp. 640-649 (1809)], Trans. Roy. Soc. London 18 (1694) pp. 136–148.Google Scholar
9. Uspensky, J. V., Note on the computation of roots, Amer. Math. Monthly 34 (1927) pp. 130–134.Google Scholar
10. Dunkel, O., Discussions: A note on the computation of arithmetic roots, Amer. Math. Monthly 34 (1927) pp. 366–368.Google Scholar
11. Barlow, P., New mathematical tables, London (1814) p. 259.Google Scholar