Adaptive polynomial preconditioning for the conjugate gradient algorithm

Abstract

For the parallel conjugate gradient algorithm polynomial preconditioners are more suitable than the more common incomplete Cholesky preconditioner. In this paper we examine the Chebyshev polynomial preconditioner. This preconditioner is based on an interval which approximately contains the eigenvalues of the matrix. If we know the extreme eigenvalues of the matrix then the preconditioner based on this interval minimises the condition number of the preconditioned matrix. Unfortunately this does not minimise the number of conjugate gradient iterations. We propose an adaptive procedure to find the interval which gives optimal rate of convergence. We demonstrate the success of this adaptive procedure on three matrices from the Harwell-Boeing collection.