Semilocal convergence of a continuation method with Holder continuous second derivative in Banach spaces

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:236

Article

Gupta, D. K.¹  Prashanth, M.¹ 

[1] Indian Inst Technol, Dept Math, Kharagpur 721302, W Bengal, India

关键词: Continuation method;    Lipschitz continuous;    Holder continuous;    Cubic convergence;    Recurrence relations;    Majorizing sequence;   

DOI  :  10.1016/j.cam.2012.02.015

来源: Elsevier

【 摘 要 】

In this paper, the semilocal convergence of a continuation method combining the Chebyshev method and the convex acceleration of Newton's method used for solving nonlinear equations in Banach spaces is established by using recurrence relations under the assumption that the second Frechet derivative satisfies the Holder continuity condition. This condition is mild and works for problems in which the second Frechet derivative fails to satisfy Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. The existence and uniqueness regions along with a closed form of the error bounds in terms of a real parameter alpha is an element of [0, 1] for the solution x* is given. Two numerical examples are worked out to demonstrate the efficacy of our approach. On comparing the existence and uniqueness regions for the solution obtained by our analysis with those obtained by using majorizing sequences under Holder continuity condition on F '', it is found that our analysis gives improved results. Further, we have observed that for particular values of the alpha, our analysis reduces to those for the Chebyshev method (alpha = 0) and the convex acceleration of Newton's method (alpha = 1) respectively with improved results. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

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