Solvability Theory and Iteration Method for One Self-Adjoint Polynomial Matrix Equation

Zhigang Jia,Sitao Ling,Minghui Wang,Meixiang Zhao

doi:10.1155/2014/681605

Abstract

The solvability theory of an important self-adjoint polynomial matrix equation is presented, including the boundary of its Hermitian positive definite (HPD) solution and some sufficient conditions under which the (unique or maximal) HPD solution exists. The algebraic perturbation analysis is also given with respect to the perturbation of coefficient matrices. An efficient general iterative algorithm for the maximal or unique HPD solution is designed and tested by numerical experiments.

Highlights

In this paper, we consider the following self-adjoint polynomial matrix equation: Xs − A∗XtA = Q, (1)where s, t are positive integers, A, Q ∈ Cn×n, and Q > 0
If A is stable, discrete-time algebraic Lyapunov equation (DALE) has a unique Hermitian positive definite (HPD) solution. Such strong relation between the spectral property of A and the solvability theory is owned by (1), which can be considered as a nonlinear DALE if s ≠ 1 or t ≠ 1
A general iteration method for (1) given in this paper can be seen as a new algorithm for the algebraic Riccati equation (2), setting s = 2 and t = 1

Summary

Introduction

If A is stable (with respect to the unit circle), DALE has a unique Hermitian positive definite (HPD) solution. Such strong relation between the spectral property of A and the solvability theory is owned by (1), which can be considered as a nonlinear DALE if s ≠ 1 or t ≠ 1. A general iteration method for (1) given in this paper can be seen as a new algorithm for the algebraic Riccati equation (2), setting s = 2 and t = 1. Nonlinear matrix equations, X − A∗XqA = Q (see, e.g., [18, 19]), are equivalence models of Ys − A∗YtA = Q and Y = X1/s, where s, t are positive integers and q = t/s. Suppose that X and Y are Hermitian matrices; we write X ≥ Y(X > Y) if X − Y is positively semidefinite (definite) and denote the matrices set {X | X − αI ≥ 0 and βI − X ≥ 0} by [αI, βI]

Solvability of Self-Adjoint Polynomial Matrix Equation

Algebraic Perturbation Analysis

Algorithm and Numerical Experiments

Conclusion