Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem

To compute the smallest eigenvalues and associated eigenvectors of a real symmetric matrix, we consider the Jacobi–Davidson method with inner preconditioned conjugate gradient iterations for the arising linear systems. We show that the coefficient matrix of these systems is indeed positive definite with the smallest eigenvalue bounded away from zero. We also establish a relation between the residual norm reduction in these inner linear systems and the convergence of the outer process towards the desired eigenpair. From a theoretical point of view, this allows to prove the optimality of the method, in the sense that solving the eigenproblem implies only a moderate overhead compared with solving a linear system. From a practical point of view, this allows to set up a stopping strategy for the inner iterations that minimizes this overhead by exiting precisely at the moment where further progress would be useless with respect to the convergence of the outer process. These results are numerically illustrated on some model example. Direct comparison with some other eigensolvers is also provided. Copyright © 2001 John Wiley & Sons, Ltd.