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.