Jacobi-Davidson Style QR and QZ Algorithms for the Reduction of Matrix Pencils

Recently the Jacobi–Davidson subspace iteration method has been introduced as a new powerful technique for solving a variety of eigenproblems. In this paper we will further exploit this method and enhance it with several techniques so that practical and accurate algorithms are obtained. We will present two algorithms, JDQZ for the generalized eigenproblem and JDQR for the standard eigenproblem, that are based on the iterative construction of a (generalized) partial Schur form. The algorithms are suitable for the efficient computation of several (even multiple) eigenvalues and the corresponding eigenvectors near a user-specified 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 problems. We will show how effective restarts can be incorporated in the Jacobi–Davidson methods, very similar to the implicit restart procedure for 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 wellchosen numerical experiments.