A multishift, multipole rational QZ method with aggressive early deflation

The rational QZ method generalizes the QZ method by implicitly supporting rational subspace iteration. In this paper we extend the rational QZ method by introducing shifts and poles of higher multiplicity in the Hessenberg pencil, which is a pencil consisting of two Hessenberg matrices. The result is a multishift, multipole iteration on block Hessenberg pencils which allows one to stick to real arithmetic for a real input pencil. In combination with optimally packed shifts and aggressive early deflation as an advanced deflation technique we obtain an efficient method for the dense generalized eigenvalue problem. In the numerical experiments we compare the results with state-of-the-art routines for the generalized eigenvalue problem and show that we are competitive in terms of speed and accuracy

On pole-swapping algorithms for the eigenvalue problem

Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms

Multishift variants of the QZ algorithm with aggressive early deflation

New variants of the QZ algorithm for solving the generalized eigenvalue problem are proposed. An extension of the small-bulge multishift QR algorithm is developed, which chases chains of many small bulges instead of only one bulge in each QZ iteration. This allows the effective use of level 3 BLAS operations, which in turn can provide efficient utilization of high performance computing systems with deep memory hierarchies. Moreover, an extension of the aggressive early deflation strategy is proposed, which can identify and de. ate converged eigenvalues long before classic deflation strategies would. Consequently, the number of overall QZ iterations needed until convergence is considerably reduced. As a third ingredient, we reconsider the deflation of infinite eigenvalues and present a new deflation algorithm, which is particularly effective in the presence of a large number of infinite eigenvalues. Combining all these developments, our implementation significantly improves existing implementations of the QZ algorithm. This is demonstrated by numerical experiments with random matrix pairs as well as with matrix pairs arising from various applications

A CMV--based eigensolver for companion matrices

In this paper we present a novel matrix method for polynomial rootfinding. By exploiting the properties of the QR eigenvalue algorithm applied to a suitable CMV-like form of a companion matrix we design a fast and computationally simple structured QR iteration.Comment: 14 pages, 4 figure

A rational QZ method

We propose a rational QZ method for the solution of the dense, unsymmetric generalized eigenvalue problem. This generalization of the classical QZ method operates implicitly on a Hessenberg, Hessenberg pencil instead of on a Hessenberg, triangular pencil. Whereas the QZ method performs nested subspace iteration driven by a polynomial, the rational QZ method allows for nested subspace iteration driven by a rational function, this creates the additional freedom of selecting poles. In this article we study Hessenberg, Hessenberg pencils, link them to rational Krylov subspaces, propose a direct reduction method to such a pencil, and introduce the implicit rational QZ step. The link with rational Krylov subspaces allows us to prove essential uniqueness (implicit Q theorem) of the rational QZ iterates as well as convergence of the proposed method. In the proofs, we operate directly on the pencil instead of rephrasing it all in terms of a single matrix. Numerical experiments are included to illustrate competitiveness in terms of speed and accuracy with the classical approach. Two other types of experiments exemplify new possibilities. First we illustrate that good pole selection can be used to deflate the original problem during the reduction phase, and second we use the rational QZ method to implicitly filter a rational Krylov subspace in an iterative method

A parallel Schur method for solving continuous-time algebraic Riccati equations

Numerical algorithms for solving the continuous-time algebraic Riccati matrix equation on a distributed memory parallel computer are considered. In particular, it is shown that the Schur method, based on computing the stable invariant subspace of a Hamiltonian matrix, can be parallelized in an efficient and scalable way. Our implementation employs the state-of-the-art library ScaLAPACK as well as recently developed parallel methods for reordering the eigenvalues in a real Schur form. Some experimental results are presented, confirming the scalability of our implementation and comparing it with an existing implementation of the matrix sign iteration from the PLiCOC library

Implicit QR algorithms for palindromic and even eigenvalue problems

In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques. © 2008 Springer Science+Business Media, LLC