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.