On Iterative Algorithm and Perturbation Analysis for the Nonlinear Matrix Equation

Abstract

In this study, an iterative algorithm is proposed to solve the nonlinear matrix equation $X+A^{*}{e}^{X}A=I_{n}$. Explicit expressions for mixed and componentwise condition numbers with their upper bounds are derived to measure the sensitivity of the considered nonlinear matrix equation. Comparative analysis for the derived condition numbers and the proposed algorithm are presented. The proposed iterative algorithm reduces the number of iterations significantly when incorporated with exact line searches. Componentwise condition number seems more reliable to detect the sensitivity of the considered equation than mixed condition number as validated by numerical examples.