For a presentation of relevant theory for NHEPs, as well as pointers to literature, we refer to §2.5. In particular, we mention that the perturbation theory for these problems is delicate. One should keep in mind that a small norm of the residual for a computed eigenpair does not necessarily imply small errors in the computed values. For more information, see §2.5 and §7.13. Here we will give a brief introduction to those aspects that play a role in the selection of the algorithms. This will be followed by short characterizations of the different classes of methods, covered in this chapter.
In contrast to a Hermitian matrix, a non-Hermitian matrix does
not have an orthogonal set of eigenvectors; in other words,
a non-Hermitian matrix can in
general not be transformed by an orthogonal matrix to diagonal form
. Most non-Hermitian matrices can be transformed by
to diagonal form , but there exist matrices for which even
this is not possible. Such matrices can be viewed as limit cases for which
converges to a singular operator, and these matrices do not have a
complete set of eigenvectors; they are called defective. This reveals
a source for numerical instability: if is in some sense close to a
singular operator, then the transformation may be expected to be sensitive
to perturbations in . Such perturbations are introduced by the process
for the computation of and . Therefore, it is of great importance to
work with orthogonal, or close to orthogonal, transformations as often as
possible. It is well known that for any non-Hermitian matrix there exists
a unitary matrix that transforms it to upper triangular form
Reduction to Schur form can be done but it has its price. Let be the order of a non-Hermitian matrix . For , there is in general no finite algorithm that reduces to Schur form (if we exclude trivial cases, such as where is already upper triangular). The usual approach for matrices of modest order, say , is to reduce it first to upper Hessenberg form which can be done in a finite number of steps. Subsequently this upper Hessenberg matrix is reduced in an iterative manner (QR iterations) to Schur form. See §7.3 for more details and for pointers to software.
The main problem with this standard approach is that the computational costs are proportional to (hence the limitation to matrices of order of a few thousands at most) and it requires storage proportional to elements. In general it is not possible to exploit sparsity of the matrix in order to reduce the storage requirements. For these reasons alternatives have been proposed. These alternatives are all completely iterative, that is, there is no finite direct reduction step. They are usually only suitable for the computation of a handful of interesting eigenpairs. We will now briefly summarize the iterative methods, to be presented in this chapter.
The orthonormalization process leads to an by upper Hessenberg matrix (the approach is only practical for ) and this matrix can be interpreted as a suitably reduced form of the matrix restricted to the Krylov subspace. The eigenvalues of are approximations for some of the eigenvalues of and they can be computed by reducing to Schur form by the same standard method as discussed above for small dense matrices. An introduction to the Arnoldi method is given in §7.5. Clever restart techniques have been designed to keep memory requirements and computational overhead as low as possible: these techniques are known as implicitly restarted Arnoldi methods (IRAMs) and are discussed in §7.6. The computational overhead in the Arnoldi methods is not very suitable for parallelism, in particular not for some distributed memory computers. In order to create a larger ratio between computation and memory references block variants have been suggested. These block Arnoldi variants are discussed in §7.7.
A block version of the Lanczos method, and an implementation of it called ABLE, are presented in §7.9. ABLE is an adaptively blocked method designed to cure breakdown, and it attempts to eliminate the causes of slow convergence (such as a loss of biorthogonality, and clusters of nearby eigenvalues). Deflation within Krylov subspaces is nicely incorporated in yet another variant, the band Lanczos method, presented in §7.10. This is specially tailored for the reduced-order modeling of multiple-input and multiple-output linear dynamical systems of high dimensions.
A variant of the two-sided Lanczos method for complex symmetric systems is discussed in §7.11. With respect to the unsymmetric case, this variant reduces CPU time and computer storage by a factor of 2.
In Table 7.1 we present a summary of the various classes of methods. The table is not intended to replace further reading; it serves only as an indication of where to start and it may help the reader to follow a certain path in discovering which method serves which need best. The actual performance of each method depends, often critically, on many parameters, and one method that is best for one class of problems may encounter convergence problems for some particular matrices.
Explanation of Table 7.1