A parallel algorithm is presented for computing the group inverse of a singular M–matrix of the form A = I − T , where T ∈ Rn×n is irreducible and stochastic. The algorithm is constructed in the spirit of Meyer’s Perron complementation approach to computing the Perron vector of an irreducible nonnegative matrix. The asymptotic number of multiplication operations that is necessary to implement t...

We investigate (0, 1)-matrices which are totally nonnegative and therefore which have all of their eigenvalues equal to nonnegative real numbers. Such matrices are characterized by four forbidden submatrices (of orders 2 and 3). We show that the maximum number of 0s in an irreducible (0, 1)-matrix of order n is (n − 1) and characterize those matrices with this number of 0s. We also show that th...

For monotone linear differential systems with periodic coefficients, the (first) Floquet eigenvalue measures the growth rate of the system. We define an appropriate arithmetico-geometric time average of the coefficients for which we can prove that the Perron eigenvalue is smaller than the Floquet eigenvalue. We apply this method to Partial Differential Equations, and we use it for an age-struct...

In this paper, we apply the Perron-Frobenius theory for non-negative matrices to the analysis of variance asymptotics for simulations of finite state Markov chain to which importance sampling is applied. The results show that we can typically expect the variance to grow (at least) exponentially rapidly in the length of the time horizon simulated. The exponential rate constant is determined by t...