Parameter modified versions of preconditioning and iterative inner product free refinement methods for two-by-two block matrices

Abstract

A special two-by-two block matrix form arises in many important applications. Extending earlier results it is shown that parameter modified versions of a very efficient preconditioner does not improve its rate of convergence. This holds also for iterative refinement methods corresponding to a few fixed steps of the Chebyshev accelerated method. The parameter version can improve the defect-correction method but the convergence of this method is slower than an iterative refinement method with an optimal parameter. The paper includes also a discussion of how one can save computer elapsed times by avoiding use of global inner products such as by use of a Chebyshev accelerated method instead of a Krylov subspace method. Since accurate and even sharp eigenvalue bounds are available, the Chebyshev iteration method converges as fast as the Krylov subspace method.