Sensitivity of the Nonlinear Matrix Equation Xp = A plus M(B plus X-1)-1 M*

The paper deals with the sensitivity of the nonlinear matrix equation F(X, Q) := X-p - A - M(B + X-1)(-1) M* = 0, for a positive integer p >= 1. Applying the local and the nonlocal perturbation analysis, based on the techniques of Frechet derivatives, the method of Lyapunov majorants and Schauder fixed point principle, local and nonlocal perturbation bounds are derived. The local bound is a first order perturbation bound for the error in the solution X. It is asymptotically valid for sufficiently small perturbations in the data. The formulated nonlocal perturbation bound involves the local bound, as well as terms of second order of the perturbations in the data. The non-local bound is valid for perturbations included in a given a priori prescribed domain. This inclusion guarantees the existence of an unique solution to the perturbed equation in a neighborhood of the unperturbed solution. Numerical examples illustrate the effectiveness of the perturbation bounds proposed.