The Jacobi-Davidson Method for Eigenvalue and Generalized Eigenvalue Problems

We consider variants of Davidson's method for the iterative computation of one or more eigenvalues and their corresponding eigenvectors of an n n matrix A. The original Davidson method 3], for real normal matrices A, may be viewed as an accelerated Gauss-Jacobi method, and the success of the method seems to depend quite heavily on diagonal dominance of A 3, 4, 17]. In the hope to enlarge the scope of the method, one has investigated the eeect of replacing the diagonal preconditioner by other, more general, ones. Recent convergence results as well as numerical experiments with these generalized Davidson methods are reported in 2, 8, 9, 10]. Unfortunately , preconditioners that represent the eigenproblem well may lead to slow convergence or even to stagnation. In 13] an explanation for this unsatisfactory situation is given. Further exploiting ideas of Jacobi 7] has led to a new robust method with superior convergence properties, for non-diagonally dominant, non-normal, complex, matrices as well: this Jacobi-Davidson 13] method takes advantage of the quality of the preconditioner and is not sensitive to the eeects of rounding errors. Moreover, the Jacobi-Davidson method usually converges faster with better preconditioners. The eeciency per step of this new method is comparable to a step of the Davidson method if the same way of preconditioning is used. The Jacobi-Davidson method will be discussed and its relation with Davidson's method. A convenient way of incorporating preconditioners will be given. We will argue that the new method can be viewed as an improvement on methods such as Davidson's, Olson's (cf. 11]), Arnoldi's, and (accelerated inexact) Shift-and-Invert. Most of the material in these part can be found in 13]. We will give numerical examples from Quantum Chemistry 16] in which Davidson's method was the preferred one before. The basic scheme of the Jacobi-Davidson method, given in 13], needs some modiication for determining a portion of the spectrum with associated eigenvectors. Our approach 14, 6] is based on deeation: by projecting on the orthogonal complement of the subspace spanned by detected eigenvectors