Diederik R. Fokkema

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 well-chosen numerical experiments.

Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems

In this paper we will show how the Jacobi-Davidson iterative method can be used to solve generalized eigenproblems. Similar ideas as for the standard eigenproblem are used, but the projections, that are required to reduce the given problem to a small manageable size, need more attention. We show that by proper choices for the projection operators quadratic convergence can be achieved. The advantage of our approach is that none of the involved operators needs to be inverted. It turns out that similar projections can be used for the iterative approximation of selected eigenvalues and eigenvectors of polynomial eigenvalue equations. This approach has already been used with great success for the solution of quadratic eigenproblems associated with acoustic problems.

BiCGstab(l) and other hybrid Bi-CG methods

It is well-known that Bi-CG can be adapted so that the operations withAT can be avoided, and hybrid methods can be constructed in which it is attempted to further improve the convergence behaviour. Examples of this are CGS, Bi-CGSTAB, and the more general BiCGstab(l) method. In this paper it is shown that BiCGstab(l) can be implemented in different ways. Each of the suggested approaches has its own advantages and disadvantages. Our implementations allow for combinations of Bi-CG with arbitrary polynomial methods. The choice for a specific implementation can also be made for reasons of numerical stability. This aspect receives much attention. Various effects have been illustrated by numerical examples.

Generalized conjugate gradient squared

The Conjugate Gradient Squared (CGS) is an iterative method for solving nonsymmetric linear systems of equations. However, during the iteration large residual norms may appear, which may lead to inaccurate approximate solutions or may even deteriorate the convergence rate. Instead of squaring the Bi-CG polynomial as in CGS, we propose to consider products of two nearby Bi-CG polynomials which leads to generalized CGS methods, of which CGS is just a particular case. This approach allows the construction of methods that converge less irregularly than CGS and that improve on other convergence properties as well. Here, we are interested in a property that got less attention in literature: we concentrate on retaining the excellent approximation qualities of CGS with respect to components of the solution in the direction of eigenvectors associated with extreme eigenvalues. This property seems to be important in connection with Newtons scheme for nonlinear equations: our numerical experiments show that the number of Newton steps may decrease significantly when using a generalized CGS method as linear solver for the Newton correction equations.

Further Improvements in Nonsymmetric Hybrid Iterative Methods

In the past few years new methods have been proposed that can be seen as combinations of standard Krylov subspace methods, such as Bi—CG and GM-RES. Such hybrid schemes include CGS, BiCGSTAB, QMRS, TFQMR, and the nested GMRESR method. These methods have been successful in solving relevant sparse nonsymmetric linear systems, but there is still a need for further improvements. In this paper we will highlight some of the recent advancements in the search for effective iterative solvers.