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We study iterative methods for finding the largest Hermitian positive definite solution of the matrix equation X + ∑ i = 1 m A i ∗ X − 1 A i = Q . Convergence rates of the basic fixed point iteration, inversion free variants of the basic fixed point iteration, and Stein iteration are considered. Some numerical examples are presented to illustrate the convergence behaviour of the various algorithms. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the positive definite solutions of nonlinear matrix equation X + A ⋆ X −δ A = Q, where δ ∈(0,1], which appears for the first time in [S.M. El-Sayed, A.C.M. Ran, On an iteration methods for solving a class of nonlinear matrix equations, SIAM J. Matrix Anal. Appl. 23 (2001) 632–645]. The necessary and sufficient conditions for the existence of a solution are derived. An iterative algorithm for obtaining the positive definite solutions of the equation is discussed. The error estimations are found. [Copyright &y& Elsevier]

Abstract: In this paper we investigate nonlinear matrix equations X + A*X −q A = Q and X − A*X −q A = Q where q ∈(0,1]. We derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations. We provide a sufficient condition for the equation X + A*X −q A = Q to have two different positive definite solutions and a sufficient condition for the equation X − A*X −q A = Q to have a unique positive definite solution. We also propose iterative methods for obtaining positive definite solutions for these equations. [Copyright &y& Elsevier]