Rounding error analysis in solving M-matrix linear systems of block Hessenberg form

The problem of solving large M-matrix linear systems with sparse coefficient matrix in block Hessenberg form is here addressed. In previous work of the authors a divide-and-conquer strategy was proposed and a backward error analysis of the resulting algorithm was presented showing its effectiveness for the solution of computational problems of queueing theory and Markov chains. In particular, it was shown that for block Hessenberg M-matrices the algorithm is weakly backward stable in the sense that the computed solution is the exact solution of a nearby linear system, where the norm of the perturbation is proportional to the condition number of the coefficient matrix. In this note a better error estimate is given by showing that for block Hessenberg M-matrices the algorithm is even backward stable.