New Algorithms for Solving Singular Linear System

Abstract

The DFOM method is an iterative method for computing the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax = b, where A ϵ ℂ n × n is a singular and in general non-Hermitian matrix that has an arbitrary index. This method is generally used with restarting. But the restarting often slows down the convergence and DFOM often stagnates. We show that adding some approximate error vectors or approximate eigenvectors (corresponding to a few of the smallest eigenvalues) to the Krylov subspace can improve the convergence just like the method proposed by R. Morgan in [8]. We derive the implementation of these methods and present some numerical examples to show the advantages of these methods.