A Family of Newton Type Iterative Methods for Solving Nonlinear Equations

Algorithms

Xiaofeng Wang³  Yuping Qin¹  Weiyi Qian³  Sheng Zhang³  Xiaodong Fan³  Alicia Cordero²  Juan R. Torregrosa² 

[1] College of Engineering, Bohai University, Jinzhou 121013, China; E-Mai:;School of Mathematics and Physics, Bohai University, Jinzhou 121013, China; E-Mais;School of Mathematics and Physics, Bohai University, Jinzhou 121013, China; E-Mais:

关键词: multipoint iterative methods;    nonlinear equations;    R-order convergence;    root-finding methods;   

DOI  :  10.3390/a8030786

来源: mdpi

【 摘 要 】

In this paper, a general family of n-point Newton type iterative methods for solving nonlinear equations is constructed by using direct Hermite interpolation. The order of convergence of the new n-point iterative methods without memory isrequiring the evaluations of n functions and one first-order derivative in per full iteration, which implies that this family is optimal according to Kung and Traub’s conjecture (1974). Its error equations and asymptotic convergence constants are obtained. The n-point iterative methods with memory are obtained by using a self-accelerating parameter, which achieve much faster convergence than the corresponding n-point methods without memory. The increase of convergence order is attained without any additional calculations so that the n-point Newton type iterative methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.