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 C-1 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 C-1 functions with the same properties of convergence as the method proposed by Costabile, Gualtieri and Serra (1999) but with order 1 + root2 = 2.41 for C-3 functions. Compared with the methods of order 1 + root2 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 + rootn(2) + 4)/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.