On the power method in max algebra

The eigenvalue problem for an irreducible nonnegative matrix A = [a(ij)] in the max algebra system is A x x = lambda x, where (A x x)(i) = max(j)(a(ij)x(j)) and lambda turns out to be the maximum circuit geometric mean, mu(A). A power method algorithm is given to compute mu(A) and eigenvector x. The algorithm is developed by using results on the convergence of max powers of A, which are proved using nonnegative matrix theory. In contrast to an algorithm developed in [4], this new method works for any irreducible nonnegative A, and calculates eigenvectors in a simpler and more efficient way. Some asymptotic formulas relating mu(A), the spectral radius and norms are also given. (C) 1999]Elsevier Science Inc. All rights reserved.