A new optimal family of three-step methods for efficient finding of a simple root of a nonlinear equation

Abstract

This study presents a new efficient family of eighth order methods for finding the simple root of nonlinear equation. The new family consists of three steps: the Newton's step, any optimal fourth order iteration scheme and the simply structured third step which improves the convergence order up to at least eight, and ensures the efficiency index 1.6818. For several relevant numerical test functions, the numerical performances confirm the theoretical results.