Development of Optimal Eighth Order Derivative-Free Methods for Multiple Roots of Nonlinear Equations

Symmetry

IoannisK. Argyros¹  JanakRaj Sharma²  Sunil Kumar² 

[1] Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA;Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur 148106, India;

关键词: nonlinear equations;    multiple roots;    derivative-free method;    optimal convergence;   

DOI  :  10.3390/sym11060766

来源: DOAJ

【 摘 要 】

A number of higher order iterative methods with derivative evaluations are developed in literature for computing multiple zeros. However, higher order methods without derivative for multiple zeros are difficult to obtain and hence such methods are rare in literature. Motivated by this fact, we present a family of eighth order derivative-free methods for computing multiple zeros. Per iteration the methods require only four function evaluations, therefore, these are optimal in the sense of Kung-Traub conjecture. Stability of the proposed class is demonstrated by means of using a graphical tool, namely, basins of attraction. Boundaries of the basins are fractal like shapes through which basins are symmetric. Applicability of the methods is demonstrated on different nonlinear functions which illustrates the efficient convergence behavior. Comparison of the numerical results shows that the new derivative-free methods are good competitors to the existing optimal eighth-order techniques which require derivative evaluations.

【 授权许可】

Unknown