Jacobi-Davidson style QR and QZ algorithms for the partial reduction of matrix pencils

Abstract

The Jacobi-Davidson subspace iteration method oers possibilities for solving a variety of eigenproblems. In practice one has to apply restarts because of memory limitations, in order to restrict computational overhead, and also if one wants to compute several eigenvalues. In general, restarting has negative eects on the convergence of subspace methods. We will show how eective restarts can be incorporated in the Jacobi-Davidson methods, very similar to the implicit restart procedure for the Arnoldi process. We will present two algorithms, JDQR for the standard eigenproblem, and JDQZ for the generalized eigenproblem, that are based on the iterative construction of the (generalized) partial Schur form with the Jacobi-Davidson approach. The algorithms are suitable for the ecient computation of several (even multiple) eigenvalues, and the corresponding eigenvectors, near a user-specied target value in the complex plane.