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  :18,72,functions. This method combines the assured convergence of the bisectionlike algorithms with a superlinear convergence speed which characterizes Newtonlike 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  functions with the same properties of convergence as the method proposed by Costabile, Gualtieri and Serra (1999) but with order                                                                            :167,270,388,2                  :169,272,392,xe2x89x902.41 for  :18,72,functions. :188,Compared with the methods of order                                                                            :167,270,388,2                  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                                                                            :314,n2  4                 )2, where n is the degree of the approximating polynomial, with onepoint 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.

Iterative acceleration methods with secondorder time accuracy for nonlinear diffusion equations For a twolayer finite difference scheme with secondorder time accuracy of nonlinear diffusion equations, we present three iterative solving algorithms, including Picard, PicardNewton and derivativefree PicardNewton iterations. The main purpose of this paper is to solve the twolayer scheme efficiently and accurately, and give strict theoretical proofs of the convergence and efficiency of the three iterative methods. For the three iterative methods, two strategies are adopted to offer iterative initial evaluations, one is to use the value at the previous time step with firstorder accuracy, the other is an extrapolation with secondorder accuracy. With an induction reasoning technique, we prove that the solutions of the three iterative methods all converge to the exact solution of the diffusion problem with secondorder accuracy both in space and time after two nonlinear iteration steps, even if the firstorder initial value is used. It is also proved that the solutions of Picard iteration converge linearly to the solution of the discrete scheme, while PicardNewton and derivativefree PicardNewton iterative solutions converge with a quadratic speed. Moreover, no difference occurs in convergent speed with the two different initial values. Finally, some numerical tests are presented to verify our theoretical results, which show that compared with Picard iteration, PicardNewton and derivativefree PicardNewton iterations are more efficient for solving nonlinear problems.