New methods for computing the Drazin-inverse solution of singular linear systems

The DGMRES method is an iterative method for computing the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b,Ax=b, where A∈Cn×nA∈Cn×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 DGMRES often stagnates. Based on the LGMRES and GMRES-E methods, we present two new techniques for accelerating the convergence of restarted DGMRES by adding some approximate error vectors or approximate eigenvectors (corresponding to a few of the smallest eigenvalues) to the Krylov subspace. We derive the implementation of these methods and present some numerical examples to show the advantages of these methods.