A QMR-Type Algorithm for Drazin-Inverse Solution of Singular Nonsymmetric Linear Systems

Alireza Ataei

doi:10.4236/alamt.2016.64011

Abstract

In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.

Highlights

IntroductionWhere A ∈ N×N is a singular matrix and ind ( A) is arbitrary. Here ind ( A) , the index of A is the size of the largest Jordan block corresponding to the zero eigenvalue of A
Open AccessConsider the linear system Ax = b, (1)where A ∈ N×N is a singular matrix and ind ( A) is arbitrary
We propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax = b where A ∈ N×N is a singular and in general non-hermitian matrix that has an arbitrary index

Summary

Introduction

Where A ∈ N×N is a singular matrix and ind ( A) is arbitrary. Here ind ( A) , the index of A is the size of the largest Jordan block corresponding to the zero eigenvalue of A. He presented several Krylov subspace methods of Arnoldi, DGCR and Lancoze types. In [13] [14], Sidi has continued to drive two Krylov subspace methods to compute ADb. One is DGMRES method, which is the implementation of the DGCR method for singular systems which is analogues to GMRES for non-singular systems. The Drazin-Quasi-minimal residual algorithm (DQMR hereafter) is another implementation of the projection method for singular linear systems is analogues to Lanczos algorithm for non-singular systems. We design DQMR when we set ind ( A) = 0 throughout, DQMR reduces to QMR In this sense, DQMR is an extension of QMR that archives the Drazin-inverse solution of singular systems.

Some Basic Theorem and Projection Methods for ADb

DQMR Algorithm

Numerical Examples

Conclusion