ELA, Volume 13, pp. 131-145, April 2005, abstract.

A Parallel Algorithm For Computing The Group 
Inverse Via Perron Complementation

Michael Neumann and Jianhong Xu

A parallel algorithm is presented for computing 
the group inverse of a singular M-matrix of the
form A = I-T, where T is a real n-by-n matrix 
that 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.