On the semigroup property for some structured iterations Abstract   Nonlinear matrix equations play a crucial role in science and engineering problems. However, solutions of nonlinear matrix equations cannot, in general, be given analytically. One standard way of solving nonlinear matrix equations is to apply the fixedpoint iteration with usually only the linear convergence rate. To advance the existing methods, we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order. We realize our way by starting with examples of solving the scalar equations and, also, connect this method with some wellknown equations including, but not limited to, the Stein matrix equation, the generalized eigenvalue problem, the generalized nonlinear matrix equation, the discretetime algebraic Riccati equations to express the capacity of this method.

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.