A computation of the smallest eigenvalue and the corresponding eigenvector of an irreducible nonsingular \(\text{M}\)-matrix \(A\) is considered. Section 2 introduces some lemmas for M-matrices. Sections 3 and 4 discuss perturbation theory for the eigenvalues and for each component of the corresponding eigenvector. Section 5 presents an algorithm for computing the smallest eigenvalue and the corresponding eigenvector and gives the componentwise backward error. A numerical example illustrates the stability of the proposed algorithm in the last section.