Computing the Smallest Eigenvalue of an M-Matrix

A computation of the smallest eigenvalue and the corresponding eigenvector of an irreducible nonsingular M-matrix A is considered. It is shown that if the entries of A are known with high relative accuracy, the smallest eigenvalue and each component of the corresponding eigenvector will be determined to high relative accuracy. A known inverse iteration algorithm with new stopping criterion is presented to compute them. Under certain assumptions, the algorithm will have a small componentwise backward error, which is consistent with the perturbation results.