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Algorithms 2015, 8(4), 1111-1120; doi:10.3390/a8041111

An Optimal Biparametric Multipoint Family and Its Self-Acceleration with Memory for Solving Nonlinear Equations

College of Sciences, North China University of Technology, Beijing 100144, China
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Academic Editor: Alicia Cordero
Received: 8 October 2015 / Revised: 22 November 2015 / Accepted: 24 November 2015 / Published: 1 December 2015
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
View Full-Text   |   Download PDF [210 KB, uploaded 1 December 2015]

Abstract

In this paper, a family of Steffensen-type methods of optimal order of convergence with two parameters is constructed by direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub (J. Assoc. Comput. Math. 1974, 21, 634–651) that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2m-1 . Furthermore, the family of Steffensen-type methods of super convergence is suggested by using arithmetic expressions for the parameters with memory but no additional new evaluation of the function. Their error equations, asymptotic convergence constants and convergence orders are obtained. Finally, they are compared with related root-finding methods in the numerical examples. View Full-Text
Keywords: nonlinear equation; Newton’s method; Steffensen’s method; derivative free; optimal convergence; super convergence nonlinear equation; Newton’s method; Steffensen’s method; derivative free; optimal convergence; super convergence
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Zheng, Q.; Zhao, X.; Liu, Y. An Optimal Biparametric Multipoint Family and Its Self-Acceleration with Memory for Solving Nonlinear Equations. Algorithms 2015, 8, 1111-1120.

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