Perturbation Analysis of a Nonlinear Matrix Equation Arising in Tree-Like Stochastic Processes

Abstract

The solution, obtained in the environment of finite precision machine arithmetic must be allays accompanied by an analysis of the conditioning of the problem solved. The perturbation analysis derives measures for the sensitivity of the solution to perturbations in the matrix coefficients. Motivated by these, in order to ascertain the accuracy of an iteratively calculated solution to a nonlinear matrix equation arising in Tree-like stochastic processes, in this paper norm-wise, mixed and component-wise condition numbers, as well as local perturbation bounds are formulated and norm-wise non-local residual bounds are derived using the methods of nonlinear perturbation analysis (Frechet derivatives, Lyapunov majorants, fixed-point principles). The residual bounds are formulated in terms of the computed approximate solution to the equation and can be used as a stop criteria of the iterations, when solving the considered nonlinear matrix equation by a numerically stable iterative algorithm.