Pole-Swapping Algorithms for the Eigenvalue Problem

The front matter includes the Title page, Author page, Copyright page, Dedication, Table of Contents, and Preface.

The symbols ℝ and ℂ denote the fields of real and complex numbers, respectively. All matrices in this book will have entries in ℂ. Now and then we will restrict our attention to matrices with real entries. The symbol ℂ^n×m denotes the set of n × m matrices with complex entries, and ℂⁿ denotes the set of column vectors with n complex entries (so ℂ = ℂⁿ× 1). We will also use the analogous symbols ℝ^n×m and ℝⁿ.

The pole-swapping algorithms that are the subject of this book make use of processes that are exactly the same as those used to interchange two adjacent eigenvalues or blocks of eigenvalues in a triangular pencil. We will therefore study this problem in detail. We wade into the subject gradually by considering the single-matrix case first.

3.1 ■ Krylov subspaces

In Chapter 1 we suggested a general approach to solving eigenvalue problems: reduce the matrix to a condensed form (e.g., Hessenberg), then iterate to get the eigenvalues. At each step we do a similarity transformation. This approach is feasible for small- to medium-sized problems with n not exceeding a few thousand. For larger problems, other methods are required. The Krylov subspace methods that are discussed in this chapter are among the most important.⁶

⁶ Krylov subspaces have other uses as well. For example, they play a role in the study of Francis's and other bulgechasing algorithms in [69], and are crucial to the convergence theory of Francis's algorithm laid out in [70, 71]. In the present book, in our discussions of the convergence of pole-swapping algorithms, we introduce rational Krylov subspaces. See Section 4.7.

This chapter draws heavily from [21] and [25].

The rational Krylov algorithm produces Hessenberg pairs. In this chapter we will introduce pole-swapping algorithms that compute the eigenvalues/vectors of such a pair. As we mentioned in Chapter 2, pole swapping turns out to be a matter of interchanging blocks of eigenvalues. In this chapter we look at the simplest case, in which the blocks are 1 × 1. Larger blocks will be discussed in Chapter 7.

Up to this point we have focused on the generalized eigenvalue problem for a matrix pencil A - λB . It is natural to ask whether pole-swapping methods can be used to solve the standard eigenvalue problem for a single matrix A ∈ ℂ^n×n. This is a special case of the generalized eigenvalue problem, where the pencil is A - λI. Let us assume that A has already been reduced to Hessenberg form, so the pair (A, I) is a Hessenberg pair. If we now apply the basic algorithm from Chapter 4 to this pair, the identity matrix will not be preserved, as the transformations are unitary equivalences but not similarities. However the transformed matrix will have some useful properties. For one, it will remain upper Hessenberg. Moreover it will be unitary because I is unitary and so are all of the transformations. Thus we will have to deal with Hessenberg pairs (A, U ) for which U is unitary. This may look like a losing strategy, since it forces us to work with two matrices instead of one. It turns out not to be a loser because unitary Hessenberg matrices can be stored in a compact factored form requiring only O (n ) storage and can be manipulated almost for free.

We now return for a second look at the Krylov processes that we introduced in Chapter 3.

6.1 ■ Implicitly restarted Arnoldi process

The implicitly restarted Arnoldi process was invented by Sorensen [57] and is also described in [47, 70], for example. The description given here is a bit different from that in [70], but it is equivalent.

In Chapter 4 we developed algorithms that swap poles one pair at a time. Swapping a pair of poles is the same as swapping a pair of eigenvalues of the pole pencil, which can be done by methods described in Chapter 2. In that chapter we also showed how to swap whole blocks of eigenvalues, and in this chapter we consider algorithms that swap the poles in blocks. A primary reference is [58].

An obvious motivation for this is the case of real pencils. Some of the eigenvalues may be complex, but these occur in complex conjugate pairs. To find complex eigenvalues we will need to use complex shifts, but they can also be taken in conjugate pairs, which can be housed in real 2×2 blocks. This allows for an algorithm that stays entirely in real arithmetic by swapping 2×2 blocks.

The back matter includes Bibliography, Index, and backcover.