Jacobi - Davidson style QR and QZ algorithmsfor the partial reduction of matrix pencilsbyDiederik

The Jacobi-Davidson subspace iteration method ooers possibilities for solving a variety of eigen-problems. 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 eeects on the convergence of subspace methods. We will show how eeective 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 eecient computation of several (even multiple) eigenvalues, and the corresponding eigenvectors, near a user-speciied target value in the complex plane. 1 Introduction. In this paper we propose two iterative methods, one for computing solu