A parallel algorithm for computing the group inverse via Perron complementation

Abstract

Abstract.A parallel algorithm is presented for computing the group inverse of a singular M-matrix of the form A = I − T , where T ∈ R n×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 the algorithm is analyzed, which shows that the algorithm saves a significant amount of computation over the direct computation of the group inverse of A.

Perron Complementation on Linear Systems Involving M-Matrices

“…Such an eigen problem is indeed a special case of a linear system which involves an M-matrix. The Perron complementation technique was later applied to the computation of mean first passage times for Markov chains [8] and group generalized inverses of certain Q-matrices [21] 1 However, the Perron complementation over linear systems featuring M-matrices has yet to be addressed. In this paper, we show how Perron complementation emerges naturally on a linear system involving an M-matrix through some regular splitting.…”