Jacobi-type algorithms for eigenvalues on vector- and parallel computers

Algebra;Numerical Computation

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos--type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U+U^H. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank--structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank--structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration

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

Compression of unitary rank--structured matrices to CMV-like shape with an application to polynomial rootfinding

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos-type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U, i.e., of the matrix U+UH. By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank-structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank-structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration. © 2014 Elsevier Inc. All rights reserved

Structured Eigenvalue Problems

Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices

Algorithms and techniques for polynomial matrix decompositions

The concept of polynomial matrices is introduced and the potential application of polynomial matrix decompositions is discussed within the general context of multi-channel digital signal processing. A recently developed technique, known as the second order sequential rotation algorithm (SBR2), for performing the eigenvalue decomposition of a para-Hermitian polynomial matrix (PEVD) is presented. The potential benefit of using the SBR2 algorithm to impose strong decorrelation on the signals received by a broadband sensor array is demonstrated by means of a suitable numerical simulation. This demonstrates how the polynomial matrices produced as a result of the PEVD can be of unnecessarily high order. This is undesirable for many practical applications and slows down the iterative computational procedure. An effective truncation technique for controlling the growth in order of these polynomial matrices is proposed. Depending on the choice of truncation parameters, it provides an excellent compromise between reduced order polynomial matrix factors and accuracy of the resulting decomposition. This is demonstrated by means of a set of numerical simulations performed by applying the modified SBR2 algorithm with a variety of truncation parameters to a representative set of test matrices. Three new polynomial matrix decompositions are then introduced - one for implementing a polynomial matrix QR decomposition (PQRD) and two for implementing a polynomial matrix singular value decomposition (PSVD). Several variants of the PQRD algorithm (including polynomial order reduction) are proposed and compared by numerical simulation using an appropriate set of test matrices. The most effective variant w.r.t. computational speed, order of the polynomial matrix factors and accuracy of the resulting decomposition is identified. The PSVD can be computed using either the PEVD technique, based on the SBR2 algorithm, or the new algorithm proposed for implementing the PQRD. These two approaches are also compared by means of computer simulations which demonstrate that the method based on the PQRD is numerically superior. The potential application of the preferred PQRD and PSVD algorithms to multiple input multiple output (MIMO) communications for the purpose of counteracting both co-channel interference and inter-symbol interference (multi-channel equalisation) is demonstrated in terms of reduced bit error rate by means of representative computer simulations