A family of Three-point Methods of Eighth-order for Finding Multiple Roots of Nonlinear Equations

Abstract

In this paper, two new three-point eighth-order iterative methods for solving nonlinear equations are constructed.� It is proved that these methods have the convergence order of eight requiring only four function evaluations per iteration.� In fact, we have obtained the optimal order of convergence which supports the Kung and Traub conjecture. Kung and Traub conjectured that the multipoint iteration methods, without memory based on \(n\) evaluations, could achieve optimal convergence order \(2^{n-1}\).� Thus, we present new iterative methods which agree with the Kung and Traub conjecture for \(n=4\) Numerical comparisons are included to demonstrate exceptional convergence speed of the proposed methods using only a few function evaluations.