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In this paper we study a special case of linearly perturbed discretetime algebraic Riccati equation. We give some sufficient conditions for the existence of a positive definite solution of the considered equation. We propose a basic fixed point iteration and its inversion free variant for finding a positive definite solution. Moreover, by specially choosing the initial value in the basic fixed point iteration we prove that it converges to the largest solution. The theoretical results are illustrated by numerical examples. [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]