Abstract
We present a local convergence analysis for a family of Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.
Highlights
In this study we are concerned with the problem of approximating a locally unique solution x∗ of equation F (x) = 0, (1)where F : D ⊆ S → S is a nonlinear function, D is a convex subset of S and S is R or C
The semi-local convergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls [3, 5, 20, 21, 22, 24, 26]
We study the local convergence of third order Steffensen-type method defined for each n = 0, 1, 2, · · · by yn
Summary
In this study we are concerned with the problem of approximating a locally unique solution x∗ of equation We study the local convergence of third order Steffensen-type method defined for each n = 0, 1, 2, · · · by yn Method (2) was studied in [18] under hypotheses reaching upto the fourth derivative of function F . The local convergence of the preceding methods has been shown under hypotheses up to the fourth derivative (or even higher).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
