ELA, Volume 6, pp. 72-94, August 2000, abstract.
 
A Bi-CG Type Iterative Method for Drazin-Inverse Solution of 
Singular Inconsistent Nonsymmetric Linear Systems of Arbitrary Index

Avram Sidi and Vladimir Kluzner

Consider the linear system Ax=b, where b is a vector in C^N,
A in C^{N times N} is a singular matrix, and ind(A)=a is arbitrary.
Here ind(.) denotes the index of a matrix. The Drazin-inverse solution 
of this system is defined to be the vector A^{D}b, where the matrix 
A^{D} is the Drazin inverse of A. The Drazin-inverse solution of 
singular linear systems has been considered recently by the first 
author within the context of extrapolation methods, when ind(A) is
arbitrary. It has also been considered within the context of Krylov 
subspace methods, when A is real symmetric (hence ind(A)=1 necessarily). 
In addition, semi-iterative methods have been developed for the cases 
in which ind(A)=1 and ind(A)>1, assuming that the spectrum of A is real 
nonnegative. The purpose of the present work is to develop a Bi-CG type 
Krylov subspace method suitable for the general case in which A is not 
necessarily real symmetric, its index is arbitrary, and its spectrum
is not necessarily real. The method that is  developed can be 
implemented via a 4-term recursion relation independently of the size 
of ind(A) and produces A^{D}b in at most N-a steps.  A detailed error 
analysis for this method is provided and the results are illustrated 
with suitable numerical examples.