Augmented And Deflated RRGMRES Method For Singular Linear Systems

GMRES is a popular iterative method for solving large non-symmetric non-singular linear systems of equations, which is a Krylov subspace iterative method. When GMRES method is applied to a singular system, we should consider how to deal with the inconsistent system. The RRGMRES was designed as a modification of GMRES.To improve the effectiveness and efficiency, an augmented RRGMRES was pro-posed, where the Krylov subspace is expanded carefully according to the property of the discussed problem. In addition, deflation is also a technique to improve the RRGMRES, both of the techniques can be applied to either nonsingular or singular systems. In our work, we propose the combination of augmentation and deflation for RRGMRES to solve singular linear systems. The base of the augmented part is chosen carefully so that the reduced system has the same residual as the original system. We have derived the conditions for the designed augmented space the-oretically. Finally some numerical experiments illustrate that our method has its advantage over the other two methods in improving the accuracy of the approximate solution.The contribution of our work is that, in terms of improving the accuracy of the approximate solution, we successfully carry out the combination of augmentation and deflation of RRGMRES for a singular system, which was originally introduced to GMRES for nonsingular systems.