On the global convergence of a fast Halley’s family to solve nonlinear equations

Abstract

The purpose of this paper is to suggest an approach for increasing the convergence speed of Halley’s method to solve a non-linear equation. This approach is based on the second order Taylor polynomial and on Halley’s formula. By applying it a certain number of times, we obtain a new family of methods. The originality of this family is manifested in the fact that all its sequences are generated from one exceptional formula that depends on a natural integer parameter p. In addition, under certain conditions, the convergence speed of its sequences increases with p. The convergence analysis shows that the order of convergence of all proposed methods is three. A study on their global convergence is carried out. To illustrate the performance of this family, several numerical comparisons are made with other third and higher order methods.

Keywords

• Iterative methods
• Root finding
• Newton's method
• Order of convergence
• Third order method