A Class of Spectral Two-Level Preconditioners

B. Carpentieri, I.S. Duff, L. Giraud, J.C. Rioual


When solving the linear system Ax= b with a Krylov method,the smallest eigenv alues of the matrix A often slow down the convergence. In the SPD case, this is clearly highlighted by the bound on the rate of convergence ofthe Conjugate Gradient method (CG). From this bound it can be said that enlarging the smallest eigenvalues would improve the convergence rate of CG. Consequently if the smallest eigenvalues of A could be somehow "removed" the convergence of CG will be improved. Similarly for unsymmetric systems arguments exist to explain the bad effect of the smallest eigenv alues on the rate of convergence ofthe unsymmetric Krylov solver. We first present our techniques for unsymmetric linear systems and then derive a variant for symmetric and SPD matrices.