Perturbation analysis of the matrix equation \(X-A^*X^{-p}A=Q\)
The authors consider the matrix equation \(X-A^*X^{-p}A=Q\) with \(0<p\leq 1\), with \(X\), \(A\) and \(Q\) being \(n\times n\)-complex matrices, \(Q\) being positive definite. They prove (using a new method) existence and uniqueness of a positive definite solution \(X\). Except a generalization of known results for arbitrary \(p\in (0,1]\), a sharper perturbation bound and backward error of an approximation of this solution are evaluated. Explicit expressions of the condition number for the unique positive definite solution are obtained. The results are illustrated by numerical examples.
scientific article; zbMATH DE number 5603567