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