A Bi-CG type iterative method for Drazin-inverse solution of singular inconsistent nonsymmetric linear systems of arbitrary index

Summary

Summary: Consider the linear system Ax = b, where b is a vector in C N , A 2 C N$\Theta N$ is a singular matrix, and ind (A) = a is arbitrary. Here ind $(\Delta )$ denotes the index of a matrix. The Drazininverse 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 $\Gamma a$ steps. A detailed error analysis for this method is provided and the results are illustrated with suitable numerical examples.

Mathematics Subject Classification

15A06, 15A09, 65F10, 65F50

Keywords/Phrases

singular linear systems, drazin-inverse solution, Krylov subspace methods, Lanczos method, bi-conjugate gradient algorithm