A GMRES(m) method with two stage deflated preconditioners

The gmres(m) method is often used to compute Krylov subspace solutions of large sparse linear systems of equations. Morgan developed a new procedure that deflates the smallest eigenvalues and improves the eigenvalue distribution. Several preconditioning techniques have been explored in numerous research papers. In particular, the deflated GMRES proposed by Erhel and others replaces the smallest eigenvalues of the original coefficient matrix of the linear system with the largest modulus of the eigenvalues. We explore a new deflated gmres which uses a two stage deflation technique. Further, the results of the numerical experiments for test matrices are tabulated to illustrate that our approach is effective in solving a wide range of problems.