Iterative Algorithm for Solving a System of Nonlinear Matrix Equations

Abstract

We discuss the positive definite solutions for the system of nonlinear matrix equations X − A * Y −n A I and Y − B * X −m B I, where n, m are two positive integers. Some properties of solutions are studied, and the necessary and sufficient conditions for the existence of positive definite solutions are given. An iterative algorithm for obtaining positive definite solutions of the system is proposed. Moreover, the error estimations are found. Finally, some numerical examples are given to show the efficiency of the proposed iterative algorithm.

Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations

“…(2), Ω 1 (X) � Ω 2 (X) � X, Q 1 (X) � ± X − p 1 , P 1 (X) � ± X − p 2 , and Q 2 (X) � Q 2 (X) � O, Q � I and p 1 , p 2 ∈ I + [9,10] For (1) and (2) Ω 1 (X) � Ω 2 (X) � X, P 1 (X) � X − p 1 , Q 1 (X) � X − p 2 , and Q 2 (X) � Q 2 (X) � O, Q � I and p 1 , p 2 ∈ (0, 1] [11] For different types of applications of the Riccati equation, one can check [4,12,13].…”

Positive Definite Solutions for the System of Nonlinear Matrix Equations <i>X</i> + <i>A<sup>*</sup>Y<sup>-n</sup>A</i> = <i>I</i>, <i>Y</i> + <i>B<sup>*</sup>X<sup>-m</sup>B</i> = <i>I</i>

“…of algebraic discrete-type Riccati equations appears in many applications [9]- [12]. Czornik and Swierniak [10] have studied the lower bounds for eigenvalues and matrix lower bound of a solution for the special case of the System: ( ) (1.4) have been studied in some papers [14] [15].…”

On the Iterative Method for the System of Nonlinear Matrix Equations

“…Linear and nonlinear matrix equations have been widely used for solving many problems in several areas such as control theory, optimal control, optimization control, stability theory, communication system, dynamic programming, signal processing, and stochastic filtering and statistics, [1][2][3]. Many authors studied the existence of solutions for several classes of the matrix equations (see, e.g., [4][5][6][7][8][9][10][11][12][13][14]), in particular, Lyapunov matrix equation [15], Sylvester matrix equations [11,14], algebraic Riccati equations [3], some special case of linear and nonlinear matrix equations [16][17][18][19][20][21], and coupled matrix equations [22][23][24].…”

“…The efficient numerical solutions for some special case of the system (2) have been extensively studied by several authors [4][5][6][7][8][9][10][22][23][24][25][26]. For example, Mukaidani [22] proposed a new algorithm for solving crosscoupled sign-indefinite algebraic Riccati equations for weakly coupled large-scale systems, while in [4,5] Al-Dubiban has studied special cases of Sys. (2) by obtained sufficient conditions for the existence of positive definite solutions for the systems and proposed iterative algorithms to calculate the solutions.…”