A modified Chebyshev–Halley‐type iterative family with memory for solving nonlinear equations and its stability analysis

Abstract

In this paper, we propose a new iterative scheme with memory for solving nonlinear equations numerically in order to achieve higher order of convergence in comparison to the cubically convergent Chebyshev–Halley‐type method. Several modifications on Chebyshev–Halley‐type methods without memory have been considered in order to extend it to the scheme with memory. We have used self accelerating parameter in order to attain acceleration of convergence speed which is estimated from the current and previous iterations using divided differences. Therefore, the order of convergence increases from 3 to 3.30 without any further functional evaluation. We study the complex and real dynamics of the proposed family. The parameter spaces and dynamical planes are presented. From the parameter spaces, we can detect different members of the proposed family that have good and bad convergence properties. From the dynamical planes, we can analyze the stability of the proposed family in terms of different values of the parameter involved. This study aids in determining the family members with stable behavior which in turn are suitable for practical problems. Numerical examples and comparisons with some of the existing methods are included to confirm the theoretical results. Furthermore, basins of attraction are included to describe a clear picture of the convergence of the proposed as well as some of the existing methods.