COMPUTATION OF THE SPECTRAL RADIUS OF A
Nonnegative Irreducible Matrix · 1981
A class of methods for the computation of the maximal real eigenvalue and its associated eigenvector of a nonnegative irreducible matrix A is presented and the convergence of the methods is proved. These methods are based on the transformation of A to quasi-stochastic form by diagonal matrices where, in contrast to other classes of methods, the whole diagonal consists of numbers different from 1. A large number of special methods is described one of which is equivalent to the power method. A few numerical examples are given. 1. Introduction. The numerical methods treated in this paper make use of the well- known result of Perron and Frobenius (cf., e.g., Varga (15)) that a nonnegative irreducible matrix has a real simple eigenvalue equal to its spectral radius and an associated positive eigenvector which is the only positive one. The diagonal transformation methods of Brauer (1), (2), (3), Hall and Porsching (9), (10), Markham (12), Pham (13), (14) and Elsner (6), (7) utilize this fact in the following way: Let A be a nonnegative irreducible N x N-matrix, N E 1N, p its positive eigen- value equal to the spectral radius, and y an associated positive eigenvector. If