On Some Improved Harmonic Mean Newton-Like Methods for Solving Systems of Nonlinear Equations

Algorithms

Diyashvir Kreetee Rajiv Babajee³  Kalyanasundaram Madhu²  Jayakumar Jayaraman²  Alicia Cordero¹  Juan R. Torregrosa¹ 

[1] African Network for Policy Research & Advocacy for Sustainability (ANPRAS), Midlands, Curepipe 52501, Mauritius; E-Mail;Department of Mathematics, Pondicherry Engineering College, Pondicherry 605014, India; E-Mail:;African Network for Policy Research & Advocacy for Sustainability (ANPRAS), Midlands, Curepipe 52501, Mauritius; E-Mail:

关键词: system of nonlinear equation;    Newton’s method;    order of convergence;    point of attraction;   

DOI  :  10.3390/a8040895

来源: mdpi

【 摘 要 】

In this work, we have developed a fourth order Newton-like method based on harmonic mean and its multi-step version for solving system of nonlinear equations. The new fourth order method requires evaluation of one function and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each iteration. The proposed new scheme does not require the evaluation of second or higher order Fréchet derivatives and still reaches fourth order convergence. The multi-step version converges with order . We have proved that the root α is a point of attraction for a general iterative function, whereas the proposed new schemes also satisfy this result. Numerical experiments including an application to 1-D Bratu problem are given to illustrate the efficiency of the new methods. Also, the new methods are compared with some existing methods.

【 授权许可】

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