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Sibirskii Zhurnal Vychislitel'noi Matematiki, 2017, Volume 20, Number 1, Pages 59–75
DOI: https://doi.org/10.15372/SJNM20170106
(Mi sjvm636)
 

Semilocal convergence of a continuation method in Banach spaces

M. Prashanth, S. Motsa

Department of Mathematics, Statistics and Computer science, University of Kawazulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa
References:
Abstract: This paper is concerned with the semilocal convergence of a continuation method between two third-order iterative methods, namely, Halley's method and the convex acceleration of Newton's method, also known as super-Halley's method. This convergence analysis is discussed using a recurrence relations approach. This approach simplifies the analysis and leads to improved results. The convergence is established under the assumption that the second Fréchet derivative satisfies the Lipschitz continuity condition. An existence-uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter $\alpha\in[0,1]$. Two numerical examples are worked out to demonstrate the efficiency of our approach. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences [15], we observed that our analysis gives better results. Further, we observed that for particular values of $\alpha$ our analysis reduces to Halley's method ($\alpha=0$) and convex acceleration of Newton's method ($\alpha=1$), respectively, with improved results.
Key words: Halley's method, convex acceleration of Newton's method, continuation method, Banach space, Lipschitz condition, Fréchet derivative.
Funding agency Grant number
University of Kawazulu-Natal, Pietermaritzburg, South Africa
Received: 10.03.2016
English version:
Numerical Analysis and Applications, 2017, Volume 10, Issue 1, Pages 47–62
DOI: https://doi.org/10.1134/S1995423917010062
Bibliographic databases:
Document Type: Article
MSC: 6505, 65H99
Language: Russian
Citation: M. Prashanth, S. Motsa, “Semilocal convergence of a continuation method in Banach spaces”, Sib. Zh. Vychisl. Mat., 20:1 (2017), 59–75; Num. Anal. Appl., 10:1 (2017), 47–62
Citation in format AMSBIB
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\paper Semilocal convergence of a~continuation method in Banach spaces
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\pages 59--75
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