A Generalized Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems

Prof. Henk van der Vorst

Mathematics Institute
Utrecht University
Netherlands

Suppose we want to compute one or more eigenvalues and their corresponding eigenvectors of the n by n matrix A. Several iterative methods are available: Jacobi's method, the power method, the method of Lanczos, Arnoldi's method, and Davidson's method. The latter method has been reported as being quite successful, most notably in connection with certain symmetric problems in computational chemistry. The success of the method seems to depend quite heavily on (strong) diagonal dominance of A.

The method of Davidson is commonly seen as an extension to Lanczos' method, but as Saad points out, from the implementation point of view it is more related to Arnoldi's method. In spite of these relations the success of the method is not well understood. Some recent convergence results, as well as numerical experiments, are reported in Saad's book on eigenvalue problems and in a recent paper by Crouzeix, Philippe and Sadkane.

However, as we will show, Davidson's method has an interesting connection with an old and almost forgotten method of Jacobi. This leads to another view on the method of Davidson, that may help us to explain the behaviour of the method, and that may help to develop new algorithms for non-diagonally dominant and unsymmetric matrices as well.

The reported work is joint research with Gerard Sleijpen.