A block preconditioned steepest descent method for symmetric eigenvalue problems

Abstract

To solve the symmetric eigenvalue problems, we propose a new preconditioning technique for the block steepest descent method based on a more accurate convergence estimate than the existing one and by employing the polynomial preconditioning technique designed originally for linear systems. Two classes of polynomial preconditioners are constructed under some mild and reasonable assumptions. Theoretical analysis shows that the group of eigenvalues with the polynomial preconditioners converge significantly faster than those with the standard preconditioner. Moreover, for the block preconditioned conjugate gradient method, the polynomial preconditioners can also be directly applied. Numerical examples further demonstrate the effectiveness and superiority of the polynomial preconditioners for both methods.