Alternative correction equations in the Jacobi-Davidson method (Q2760355)

The paper deals with variants of correction equations in the general framework of Jacobi-Davidson method for solving a nonsymmetric eigenvalue problem. A more restrictive correction equation for this method is proposed and its properties are investigated. The paper presents some theoretical considerations concerning relations of the standard and new techniques with respect to generated Krylov subspaces. The possible different behavior of the approaches is demonstrated by simple examples. The paper contains a set of results of a few numerical experiments. NEWLINENEWLINENEWLINEThe introduction and the subsequent section of the paper are devoted to an introduction into the Jacobi-Davidson method which is an iterative method for solving the nonsymmetric eigenvalue problem. Its relation to some Krylov space iterative solvers is explained. Starting from a description of the Rayleigh-Ritz procedure a derivation of the whole algorithm is given. Then a new and more restrictive correction equation for the search of new approximation in the space orthogonal to the current search space is introduced. A similarity of the resulting equations with the equations of the truncated RQ algorithm is mentioned. NEWLINENEWLINENEWLINELater, two simple examples which give some insight into the differences between the standard, restricted and intermediate forms of the correction equations are presented. NEWLINENEWLINENEWLINEThe problem of the exact solving of the correction equations is treated in the subsequent section. In particular, the relation between search vectors and Krylov subspaces is investigated. This section forms a theoretical core of the paper giving a theoretical insight into the search spaces induced by different correction equations. NEWLINENEWLINENEWLINEThe last section of the paper is devoted to numerical experiments. It is shown that both the standard and the restricted correction equations give comparable convergence rates in finite precision arithmetic provided the correction equations are solved very precisely. If the Krylov structure of the search space is seriously perturbed then the new restricted correction equation suffers more and its convergence rate in the experiments is significantly slower. In spite of the fact that the new restricted correction equation may have more favourable spectral properties which often corresponds to better solvability by linear solvers, this may not necessarily compensate the slower convergence of the algorithm.