On the Iterative Method for the System of Nonlinear Matrix Equations

Abstract

The positive definite solutions for the system of nonlinear matrix equations + * − = , + * − = are considered, where n, m are two positive integers and A, B are nonsingular complex matrices. Some sufficient conditions for the existence of positive definite solutions for the system are derived. Under some conditions, an iterative algorithm for computing the positive definite solutions for the system is proposed. Also, the estimation of the error is obtained. 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].…”

The Inversion-Free Iterative Methods for Solving the Nonlinear Matrix EquationX+AHX-1A+BHHuang1, Ma2 2013Abstract and Applied Analysis5010View full textAdd to dashboardCite

“…The nonlinear matrix equation (2) has many applications in nano research, control theory, dynamic programming, statistics, ladder networks, stochastic filtering, and so forth (see [1][2][3][4][5][6][7]). The special case (2) has been widely studied by some authors (see [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]). Different iterative methods for computing the positive definite solutions of (2) have been proposed, for example, the fixed-point iteration (see [15]), structure-preserving doubling algorithm (see [7,16]), and some inversion-free iterations (see [17,20,23,27]).…”