ELA, Volume 8, pp. 83-93, June 2001, abstract.

Additional results on index splittings for Drazin 
inverse solutions of singular linear systems
                
Yimin Wei and Hebing Wu

Given an n-by-n singular matrix A of index k, an 
index splitting of A is one of the form A = U-V, 
where R(U) = R(A^k) and N(U) = N(A^k). This splitting, 
introduced by the first author, generalizes the proper 
splitting proposed by Berman and Plemmons. Regarding 
singular systems Au = f, the first author has shown that 
the iterations u^(i+1) = U^# Vu^(i) + U^# f converge 
to A^D f, the Drazin inverse solution to the system, 
if and only if the spectral radius of U^# V is less 
than one. The aim of this paper is to further study 
index splittings in order to extend some previous results 
by replacing the Moore-Penrose inverse A^+ and A^{-1} 
with the Drazin inverse A^D. The characteristics of the 
Drazin inverse solution A^D f are established. Some 
criteria are given for comparing convergence rates of 
U_i^# V_i, where A = U_1-V_1 = U_2-V_2. Results of Collatz, 
Marek and Szyld on monotone-type iterations are extended. 
A characterization of the iteration matrix of an index 
splitting is also presented.