Perturbation analysis of the maximal solution of the matrix equation \(X+A^*X^{-1}A=P\). II
The nonlinear matrix equation \(X+A^*X^{-1}A=P\) is considered where \(A\), \(P\) are \(n\times n\) complex matrices with \(P\) Hermitian positive definite and \(A^*\) the conjugate transpose of \(A\). One is often interested in Hermitian positive definite solutions \(X\). The inequality \(X\geq Y\) means that \(X-Y\) is positive semidefinite. One defines the maximal and minimal solutions \(X_L\) and \(X_S\) such that \(X_S\leq X\leq X_L\) for any Hermitian positive definite solution \(X\). The paper is the second part of the paper by the second author [Linear Algebra Appl. 336, 61-70 (2001; Zbl 0992.15013)] and uses the techniques developed by the first author [Linear Algebra Appl. 259, 183-208 (1997; Zbl 0883.93041); ibid. 350, 237-261 (2002; Zbl 1003.15004)]. A perturbation bound (sharper than the one found in the first part) for the maximal solution to the matrix equation is derived, explicit expressions of the condition number for the maximal solution are obtained, and the backward error of an approximate solution to the maximal solution is evaluated. The results are illustrated by numerical examples.
scientific article; zbMATH DE number 1880076