Recursive Algorithms for Solving a Class of Nonlinear Matrix Equations with Applications to Certain Sensitivity Optimization Problems

TLDR: This paper is concerned with solving a class of nonlinear algebraic matrix equations and two recursive algorithms are proposed in terms of matrix difference equations and are studied, and a locally exponential convergence property is proved for one of them.

Abstract: This paper is concerned with solving a class of nonlinear algebraic matrix equations. Two recursive algorithms are proposed in terms of matrix difference equations and are studied. A set of initial values is characterized, from which the convergence of the algorithms can be guaranteed. Based on the general results, several effective algorithms are presented to compute $L^2$-sensitivity optimal realizations, as well as Euclidean norm balancing realizations, of a given linear system. A locally exponential convergence property is proved for one of them. As is shown by simulation in this paper, these algorithms prove to be far more practical for digital computer implementation than the gradient flows previously proposed.