Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:330

Article

Kumar, Abhimanyu¹  Gupta, D. K.¹  Martinez, Eulalia²  Singh, Sukhjit³ 

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

[2] Univ Politecn Valencia, Inst Univ Matemat Multidisciplinar, E-46022 Valencia, Spain

[3] Natl Inst Technol Hamirpur, Dept Math, Hamirpur 1777005, India

关键词: Semilocal convergence;    Double step Secant method;    Divided differences;    Majorizing sequences;    Error bounds;    Efficiency index;   

DOI  :  10.1016/j.cam.2017.02.042

来源: Elsevier

【 摘 要 】

The semilocal convergence of double step Secant method to approximate a locally unique solution of a nonlinear equation is described in Banach space setting. Majorizing sequences are used under the assumption that the first-order divided differences of the involved operator satisfy the weaker Lipschitz and the center-Lipschitz continuity conditions. A theorem is established for the existence-uniqueness region along with the estimation of error bounds for the solution. Our work improves the results derived in Ren and Argyros (2015) in more stringent Lipschitz and center Lipschitz conditions and gives finer majorizing sequences. Also, an example is worked out where the conditions of Ren and Argyros (2015) fail but our's work. Numerical examples including nonlinear elliptic differential equations and integral equations are worked out. It is found that our conditions enlarge the convergence domain of the solution. Finally, taking a nonlinear system of in equations, the Efficiency Index (EI) and the Computational Efficiency Index (CEI) of double step Secant method are computed and its comparison with respect to other similar existing iterative methods are summarized in the tabular forms. (C) 2017 Elsevier B.V. All rights reserved.

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