A Class of Spectral Two-Level Preconditioners

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

RAL and CERFACS

Abstract
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.