Krylov and Polynomial Iterative Solvers Combined with Partial Spectral Factorization for SPD Linear Systems

Abstract

When solving the Symmetric Positive Definite (SPD) linear system Ax = b with the conjugate gradient method, the smallest eigenvalues in the matrix A often slow down the convergence. Consequently if the smallest eigenvalues in A could be somehow “removed”, the convergence may be improved. This observation is of importance even when a preconditioner is used, and some extra techniques might be investigated to improve furthermore the convergence rate of the conjugate gradient on the given preconditioned system. Several techniques have been proposed in the literature that either consist of updating the preconditioner or enforcing the conjugate gradient to work in the orthogonal complement of an invariant subspace associated with small eigenvalues. In this work, we compare the numerical efficiency, computational complexity, and sensitivity to the accuracy of the spectral information of the techniques presented in [1], [2] and [3]. A more detailed description of these approaches as well as other comparable techniques on a range of standard test problems is available in [4].