An optimal scheme for multiple roots of nonlinear equations with eighth-order convergence

Abstract

We have a good number of eighth-order iterative methods for simple zeros of nonlinear equations in the available literature. But, unfortunately, we don’t have a single iterative method of eighth-order for multiple zeros with known or unknown multiplicity. Some scholars from the worldwide have tried to present optimal or non-optimal multipoint eighth-order iteration functions. But, unfortunately, none of them get success in this direction and attained maximum sixth-order convergence in the case of multiple zeros with known multiplicity m. Motivated and inspired by this fact, we propose an optimal scheme with eighth-order convergence based on weight function approach. In addition, an extensive convergence study is discussed in order to demonstrate the eighth-order convergence of the present scheme. Moreover, we also show the applicability of our scheme on some real life and academic problems. These problems illustrate that our methods are more efficient among the available multiple root finding techniques.