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 o f A, which are proved using nonnegative matrix theory. In contrast to an al gorithm developed in [4], this new method works for any irreducible nonnega tive A, and calculates eigenvectors in a simpler and more efficient way. So me asymptotic formulas relating mu(A), the spectral radius and norms are al so given. (C) 1999]Elsevier Science Inc. All rights reserved.