Some High-Order Iterative Methods for Finding All the Real Zeros

Fazlollah Soleymani

Abstract

This paper concerns the numerical solution of one variable  nonlinear equations. Some four-point four-step iterative schemes are given. The new methods are attained by a simple but powerful approximation of the first derivative ofthe function in the fourth step of our cycle, where the first three steps are any of the optimal derivative-involved eighth-order methods. Analytical proof of the main theorem is given to clarify the fourteenth-order convergence. The extension of one high-order method for multiple zeros will be given as well. A hybrid algorithm has also been proposed to extract all the real zeros of nonlinear functions in a given interval. Finally, we furnish numerical comparisons to attest the theoretical results and the fast rate of convergence.