Condition Numbers and Backward Error of a Matrix Polynomial Equation Arising in Stochastic Models

摘要

We consider a matrix polynomial equation (MPE) A(n)X(n) + A(n-1)X(n-1) + ... + A(0) = 0, where A(n), A(n-1),... ,A(0) is an element of R-mxm are the coefficient matrices, and X is an element of R-mxm is the unknown matrix. A sufficient condition for the existence of the minimal nonnegative solution is derived, where minimal means that any other solution is componentwise no less than the minimal one. The explicit expressions of normwise, mixed and componentwise condition numbers of the matrix polynomial equation are obtained. A backward error of the approximated minimal nonnegative solution is defined and evaluated. Some numerical examples are given to show the sharpness of the three kinds of condition numbers.