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2012

Volume 33, Issue 2 (partial)


Candecomp/Parafac: From Diverging Components to a Decomposition in Block Terms

Alwin Stegeman

SIAM. J. Matrix Anal. & Appl. 33, pp. 291-316 (26 pages)

Online Publication Date: April 17, 2012

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Fitting an $R$-component Candecomp/Parafac (CP) decomposition to a multiway array or higher-order tensor ${\cal Z}$ is equivalent to finding a best rank-$R$ approximation of ${\cal Z}$. Such a best rank-$R$ approximation may not exist due to the fact that the set of multiway arrays with rank at most $R$ is not closed. In this case, trying to compute the approximation results in diverging CP components. We present an approach to avoid diverging components for real $I\times J\times K$ arrays with $R\le\min(I,J,K)$. We show that a CP decomposition $({\bf A},{\bf B},{\bf C})$ featuring diverging components can be rewritten as a decomposition in block terms, where each block term corresponds to a group of diverging components. Moreover, we show that if the diverging components occur in groups of two or three, then the limiting boundary point ${\cal X}$ (i.e., the limit of the sequence of CP updates) can be obtained by fitting an appropriate constrained Tucker3 model to ${\cal Z}$, using the block term decomposition of $({\bf A},{\bf B},{\bf C})$ as initial values. Our results are demonstrated by means of numerical experiments.

A New Sufficient Condition for the Uniqueness of Barabanov Norms

Ian D. Morris

SIAM. J. Matrix Anal. & Appl. 33, pp. 317-324 (8 pages)

Online Publication Date: April 17, 2012

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The joint spectral radius of a bounded set of $d \times d$ real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. Under quite mild conditions such a set of matrices admits an associated vector norm, called a Barabanov norm, which can be used to characterize those sequences of matrices which achieve this maximum rate of exponential growth. In this note we continue an earlier investigation into the problem of determining when the Barabanov norm associated to such a set of matrices is unique. We give a new sufficient condition for this uniqueness and provide some examples in which our condition applies. We also give a theoretical application which shows that the property of having a unique Barabanov norm can in some cases be highly sensitive to small perturbations of the set of matrices.

Perturbation Analysis for Antitriangular Schur Decomposition

Xiao Shan Chen, Wen Li, and Michael K. Ng

SIAM. J. Matrix Anal. & Appl. 33, pp. 325-335 (11 pages)

Online Publication Date: April 17, 2012

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Let $Z$ be an $n\times n$ complex matrix. A decomposition $Z=\overline{U}M U^H$ is called an antitriangular Schur decomposition of $Z$ if $U$ is an $n\times n$ unitary matrix and $M$ is an $n\times n$ antitriangular matrix. The antitriangular Schur decomposition is a useful tool for solving palindromic eigenvalue problems. However, there is no perturbation result for an antitriangular Schur decomposition in the literature. The main contribution of this paper is to give a perturbation bound of such decomposition and show that the bound depends inversely on $f(M):= \min_{\| X_N \|_F = 1} \| (\mbox{Aup}(MX_L-\overline{X}_U M), \mbox{Aup}(M^TX_L -\overline{X}_U M^T)) \|_F$, where $X_L$ and $X_U$ are the strictly lower triangular and upper triangular parts of $X, X_N=X_L+X_U,$ and $\mbox{Aup}(Y)$ denotes the strictly upper antitriangular part of Y. The quantity $\sqrt{2}/f(M)$ can be used to characterize the condition number of the decomposition, i.e., when $\sqrt{2}/f(M)$ is large (or small), the decomposition problem is ill-conditioned (or well-conditioned). Numerical examples are presented to illustrate the theoretical result.

The Quasi-Kronecker Form For Matrix Pencils

Thomas Berger and Stephan Trenn

SIAM. J. Matrix Anal. & Appl. 33, pp. 336-368 (33 pages)

Online Publication Date: May 03, 2012

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We study singular matrix pencils and show that the so-called Wong sequences yield a quasi-Kronecker form. This form decouples the matrix pencil into an underdetermined part, a regular part, and an overdetermined part. This decoupling is sufficient to fully characterize the solution behavior of the differential-algebraic equations associated with the matrix pencil. Furthermore, we show that the minimal indices of the pencil can be determined with only the Wong sequences and that the Kronecker canonical form is a simple corollary of our result; hence, in passing, we also provide a new proof for the Kronecker canonical form. The results are illustrated with an example given by a simple electrical circuit.

Result Verification for the Real Quadratic Eigenvalue Problem

Friederike Voos

SIAM. J. Matrix Anal. & Appl. 33, pp. 369-387 (19 pages)

Online Publication Date: May 10, 2012

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A method for verifying and enclosing real simple eigenvalues and associated eigenvectors of the real quadratic eigenvalue problem is presented. This method uses appropriate eigenpair approximations. Improved bounds for the eigenpair can be calculated iteratively. The paper closes with numerical examples.

On the Complexity of Some Hierarchical Structured Matrix Algorithms

Jianlin Xia

SIAM. J. Matrix Anal. & Appl. 33, pp. 388-410 (23 pages)

Online Publication Date: May 17, 2012

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In recent years, hierarchical structured matrices have been widely used in fast solutions of integral equations, PDEs, structured matrix (such as Toeplitz) problems, companion eigenproblems, etc. In this paper, we systematically study the complexity of some hierarchical structured matrix algorithms, in terms of hierarchically semiseparable (HSS) matrices. Several important aspects are considered. We perform detailed complexity analysis for some typical HSS algorithms, with the aid of certain graph techniques. This analysis helps us provide some significant improvements to classical HSS methods. One improvement is to propose more efficient HSS construction and solution algorithms. Another improvement is a recompression procedure which reorthonormalizes some HSS generators and converts noncompact HSS forms to compact ones. A precise theoretical justification of the compactness is also given. The third improvement is to relax the rank requirement in HSS operations. Unlike many classical HSS methods where the appropriate off-diagonal (numerical) ranks are often required to be bounded, we allow the ranks to increase. Certain general rank patterns are proposed, so that similar performance can be achieved with the maximum rank unbounded. These improvements significantly enhance both the efficiency and the applicability of HSS methods. Numerical examples from some applications are included to support the analysis.
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