A New Iterative Method for the Computation of the Solutions of Nonlinear Equations

Recently, by Costabile, Gualtieri and Serra (1999), an iterative method was presented for the computation of zeros of C1 functions. This method combines the assured convergence of the bisection-like algorithms with a superlinear convergence speed which characterizes Newton-like methods. The order of the method and the cost per iteration is exactly equivalent to the Newton method. In this paper we present a new iterative method for the computation of the zeros of C1 functions with the same properties of convergence as the method proposed by Costabile, Gualtieri and Serra (1999) but with order 1+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sqrt 2 $$ \end{document} ≐2.41 for C3 functions. Compared with the methods of order 1+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sqrt 2 $$ \end{document} presented by Traub (1964), our methods ensure global convergence. Then we consider a generalization of this procedure which gives a class of methods of order (n+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sqrt {n^2 + 4} $$ \end{document})/2, where n is the degree of the approximating polynomial, with one-point iteration functions with memory. Finally a number of numerical tests are performed. The numerical results seem to show that, at least on a set of problems, the new methods work better than the methods proposed, and, therefore, than both the Newton and Alefeld and Potra (1992) methods.