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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
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.
Received: 10.03.2016
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
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https://www.mathnet.ru/eng/sjvm636 https://www.mathnet.ru/eng/sjvm/v20/i1/p59
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Abstract page: | 162 | Full-text PDF : | 83 | References: | 25 | First page: | 2 |
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