JACOBI-DAVIDSON STYLE QR AND QZ ALGORITHMS FOR THE REDUCTION OF MATRIX PENCILS

Recently the Jacobi-Davidson subspace iteration method has been introd uced as a new powerful technique for solving a variety of eigenproblem s. In this paper we will further exploit this method and enhance it wi th several techniques so that practical and accurate algorithms are ob tained. We will present two algorithms, JDQZ for the generalized eigen problem and JDQR for the standard eigenproblem, that are based on the iterative construction of a (generalized) partial Schur form. The algo rithms are suitable for the efficient computation of several (even mul tiple) eigenvalues and the corresponding eigenvectors near a user-spec ified target value in the complex plane. An attractive property of our algorithms is that explicit inversion of operators is avoided, which makes them potentially attractive for very large sparse matrix problem s. We will show how effective restarts can be incorporated in the Jaco bi-Davidson methods, very similar to the implicit restart procedure fo r the Arnoldi process. Then we will discuss the use of preconditioning , and, finally, we will illustrate the behavior of our algorithms by a number of well-chosen numerical experiments.