A Convex Combination Approach for Mean-Based Variants of Newton’s Method

Abstract

Several authors have designed variants of Newton’s method for solving nonlinear equations by using different means. This technique involves a symmetry in the corresponding fixed-point operator. In this paper, some known results about mean-based variants of Newton’s method (MBN) are re-analyzed from the point of view of convex combinations. A new test is developed to study the order of convergence of general MBN. Furthermore, a generalization of the Lehmer mean is proposed and discussed. Numerical tests are provided to support the theoretical results obtained and to compare the different methods employed. Some dynamical planes of the analyzed methods on several equations are presented, revealing the great difference between the MBN when it comes to determining the set of starting points that ensure convergence and observing their symmetry in the complex plane.

Keywords

• nonlinear equations
• iterative methods
• general means
• basin of attraction