On generalized one-step derivative-free iterative family for evaluating multiple roots

Abstract

In this study, we propose a family of iterative procedures with no deriva-tives for calculating multiple roots of one-variable nonlinear equations. We also present an iterative technique to approximate the multiplicity of the roots. The new class is optimal since it fits the Kung–Traub hypothesis and has second-order convergence. Derivative-free methods for calculating mul-tiple roots are rarely found in literature, especially in the case of one-step methods, which are the simplest ones in terms of their structure. Moreover, this new family contains almost all the existing single-step derivative-free iterative schemes as its special cases, with an additional degree of freedom. Several results are used to confirm its theoretical order of convergence. Through the complex discrete dynamics analysis, the stability of the sug-gested class is illustrated, and the most stable methods are found. Several test problems are included to check the performance of the proposed meth-ods, whether the multiplicity of the roots is estimated or known, comparing the numerical results with those obtained by other methods.

Keywords

• nonlinear equations
• derivative–free iterative method
• multiple roots
• order of convergence
• stability