Higher-order families of multiple root finding methods suitable for non-convergent cases and their dynamics

• Published in: Mathematical modelling and analysis Vol. 24; no. 3; pp. 422 - 444
• Main Authors: Behl, Ramandeep, Kanwar, Vinay, Kim, Young Ik
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 06.06.2019
• Subjects: basins of attraction; Convergence; Convergence (Mathematics); Divergence; Family; Iteration (Mathematics); Iterative methods; Mathematical research; Methods; multiple roots; Nonlinear theories; Parameter modification; Rall’s method; Schröder’s method; super-Halley’s method; India; Rall’s method; Schröder’s method; super-Halley’s method; basins of attraction; multiple roots
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2019.026

In this paper, we present many new one-parameter families of classical Rall’s method (modified Newton’s method), Schröder’s method, Halley’s method and super-Halley method for the first time which will converge even though the guess is far away from the desired root or the derivative is small in the vicinity of the root and have the same error equations as those of their original methods respectively, for multiple roots. Further, we also propose an optimal family of iterative methods of fourth-order convergence and converging to a required root in a stable manner without divergence, oscillation or jumping problems. All the methods considered here are found to be more effective than the similar robust methods available in the literature. In their dynamical study, it has been observed that the proposed methods have equal or better stability and robustness as compared to the other methods.