Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-28T01:23:13.635Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the convergence of Halley's method for solving operator equations

Published online by Cambridge University Press:  17 February 2009

Shiming Zheng  and
Desmond Robbie
Shiming Zheng
Affiliation:
Department of Mathematics, Hangzhou University, Hangzhou 310028, China.
Desmond Robbie
Affiliation:
Department of Mathematics, University of Melbourne, Parkville 3052, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Halley's method is a famous iteration for solving nonlinear equations. Some Kantorovich-like theorems have been given. The purpose of this note is to relax the region conditions and give another Kantorovich-like theorem for operator equations.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 1 , July 1995 , pp. 16 - 25
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Alefeld, G., “On the convergence of Halley's method”, Amer. Math. Monthly 88 (1981) 530–536.CrossRefGoogle Scholar
[2]Brown, G. N., “On Halley's variation of Newton's method”, Amer. Math. Monthly 84 (1977) 726–728.CrossRefGoogle Scholar
[3]Chen, D., Argyros, I. K. and Qian, Q. S., “A note on the Halley method in Banach spaces”, Appl. Math, and Comp. 58 (1993) 215–224.CrossRefGoogle Scholar
[4]Cuyt, A. and Rall, L. B., “Computational implementation of the multivariate Halley method for solving nonlinear systems of equations”, ACM Trans. on Math. Softw. 11 (1985) 20–36.CrossRefGoogle Scholar
[5]Davies, M. and Dawson, B., “On the global convergence of Halley's iteration formula”, Numer.Mart. 24 (1975) 133–135.CrossRefGoogle Scholar
[6]Halley, E., “Methodus nova, accurata & facilis inveniendi radices aequationum quarumcumque generaliter, sine praevia reductione”, Philos. Trans. Roy. Soc. London 18 (1694) 136–148.CrossRefGoogle Scholar
[7]Salehov, G. S. and Mertvetsova, M. A., “On the convergence of some iterative processes”, Izv. Kazansk. Phil. ANSSSR, Ser. Phiz-Matem. i Techn. 5 (1954) 77–108, (in Russian).Google Scholar
[8]Smale, S., “Newton's method estimates from data at one point”, in Proc. of a conf. in Honor of Gail Young, (Laramie, New York, 1986), 1–16.Google Scholar
[9]Zheng, S., “The convergence of Halley's method and its optimal error bounds”, J. Hangzhou University 9 (1982) 285–289, (in Chinese).Google Scholar
[10]Zheng, S., “Point estimates for Halley's iteration”, Acta Math. Appl. Sinica 14 (1991) 376–383, (in Chinese).Google Scholar