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2012

Volume 33, Issue 1 (partial)


Stable Computation of the CS Decomposition: Simultaneous Bidiagonalization

Brian D. Sutton

SIAM. J. Matrix Anal. & Appl. 33, pp. 1-21 (21 pages)

Online Publication Date: January 05, 2012

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Since its discovery in 1977, the CS decomposition (CSD) has resisted computation, even though it is a sibling of the well-understood eigenvalue and singular value decompositions. Several algorithms have been developed for the reduced 2-by-1 form of the decomposition, but none have been extended to the complete 2-by-2 form of the decomposition in Stewart's original paper. In this article, we present an algorithm for simultaneously bidiagonalizing the four blocks of a unitary matrix partitioned into a 2-by-2 block structure. This serves as the first, direct phase of a two-stage algorithm for the CSD, much as Golub–Kahan–Reinsch bidiagonalization serves as the first stage in computing the singular value decomposition. Backward stability is proved.

dqds with Aggressive Early Deflation

Yuji Nakatsukasa, Kensuke Aishima, and Ichitaro Yamazaki

SIAM. J. Matrix Anal. & Appl. 33, pp. 22-51 (30 pages)

Online Publication Date: January 05, 2012

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The dqds algorithm computes all the singular values of an $n \times n$ bidiagonal matrix to high relative accuracy in $O(n^2)$ cost. Its efficient implementation is now available as a LAPACK subroutine and is the preferred algorithm for this purpose. In this paper we incorporate into dqds a technique called aggressive early deflation, which has been applied successfully to the Hessenberg QR algorithm. Extensive numerical experiments show that aggressive early deflation often reduces the dqds runtime significantly. In addition, our theoretical analysis suggests that with aggressive early deflation, the performance of dqds is largely independent of the shift strategy. We confirm through experiments that the zero-shift version is often as fast as the shifted version. We give a detailed error analysis to prove that with our proposed deflation strategy, dqds computes all the singular values to high relative accuracy.

Difference Filter Preconditioning for Large Covariance Matrices

Michael L. Stein, Jie Chen, and Mihai Anitescu

SIAM. J. Matrix Anal. & Appl. 33, pp. 52-72 (21 pages)

Online Publication Date: January 05, 2012

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In many statistical applications one must solve linear systems involving large, dense, and possibly irregularly structured covariance matrices. These matrices are often ill-conditioned; for example, the condition number increases at least linearly with respect to the size of the matrix when observations of a random process are obtained from a fixed domain. This paper discusses a preconditioning technique based on a differencing approach such that the preconditioned covariance matrix has a bounded condition number independent of the size of the matrix for some important process classes. When used in large scale simulations of random processes, significant improvement is observed for solving these linear systems with an iterative method.

Compact Fourier Analysis for Multigrid Methods based on Block Symbols

Thomas K. Huckle and Christos Kravvaritis

SIAM. J. Matrix Anal. & Appl. 33, pp. 73-96 (24 pages)

Online Publication Date: January 05, 2012

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The notion of compact Fourier analysis (CFA) is discussed. CFA allows description of multigrid (MG) in a nutshell and offers a clear overview on all MG components. The principal idea of CFA is to model the MG mechanisms by means of scalar generating functions and matrix functions (block symbols). The formalism of the CFA approach is presented by describing the symbols of the fine and coarse grid problems, the prolongation and restriction, the smoother, and the coarse grid correction, resp., smoothing corrections. CFA uses matrix functions and their features (e.g., product, inverse, adjugate, norm, spectral radius, eigenvectors, eigenvalues of multilevel $\omega$-circulant matrices), and scalar functions and their roots. This leads to an elementary description and allows for an easy analysis of MG algorithms. A first application is to utilize CFA for deriving MG as a direct solver, i.e., an MG cycle that will converge in just one iteration step. Necessary and sufficient conditions that have to be fulfilled by the MG components are given for obtaining MG functioning as a direct solver. Furthermore, new general and practical smoothers and transfer operators that lead to efficient MG methods are introduced. In addition, we study sparse approximations of the Galerkin coarse grid operator yielding efficient and practicable MG algorithms (approximately direct solvers). Numerical experiments demonstrate the theoretical results.

On Iterative Solution for Linear Complementarity Problem with an $H_{+}$-Matrix

A. Hadjidimos, M. Lapidakis, and M. Tzoumas

SIAM. J. Matrix Anal. & Appl. 33, pp. 97-110 (14 pages)

Online Publication Date: January 10, 2012

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The numerous applications of the linear complementarity problem (LCP) in, e.g., the solution of linear and convex quadratic programming, free boundary value problems of fluid mechanics, and moving boundary value problems of economics make its efficient numerical solution a very imperative and interesting area of research. For the solution of the LCP, many iterative methods have been proposed, especially, when the matrix of the problem is a real positive definite or an $H_{+}$-matrix. In this work we assume that the real matrix of the LCP is an $H_{+}$-matrix and solve it by using a new method, the scaled extrapolated block modulus algorithm, as well as an improved version of the very recently introduced modulus-based matrix splitting modified AOR iteration method. As is shown by numerical examples, the two new methods are very effective and competitive with each other. (A corrected PDF is attached to this article.)

Uni-mode and Partial Uniqueness Conditions for CANDECOMP/PARAFAC of Three-Way Arrays with Linearly Dependent Loadings

Xijing Guo, Sebastian Miron, David Brie, and Alwin Stegeman

SIAM. J. Matrix Anal. & Appl. 33, pp. 111-129 (19 pages)

Online Publication Date: January 13, 2012

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In this paper, three sufficient conditions are derived for the three-way CANDECOMP/PARAFAC (CP) model, which ensure uniqueness in one of the three modes (“uni-mode-uniqueness”). Based on these conditions, a partial uniqueness condition is proposed which allows collinear loadings in only one mode. We prove that if there is uniqueness in one mode, then the initial CP model can be uniquely decomposed in a sum of lower-rank tensors for which identifiability can be independently assessed. This condition is simpler and easier to check than other similar conditions existing in the specialized literature. These theoretical results are illustrated by numerical examples.

Verified Bounds for Least Squares Problems and Underdetermined Linear Systems

Siegfried M. Rump

SIAM. J. Matrix Anal. & Appl. 33, pp. 130-148 (19 pages)

Online Publication Date: January 13, 2012

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New algorithms are presented for computing verified error bounds for least squares problems and underdetermined linear systems. In contrast to previous approaches the new methods do not rely on normal equations and are applicable to sparse matrices. Computational results demonstrate that the new methods are faster than existing ones.

Block Tensor Unfoldings

Stefan Ragnarsson and Charles F. Van Loan

SIAM. J. Matrix Anal. & Appl. 33, pp. 149-169 (21 pages)

Online Publication Date: January 13, 2012

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Within the field of numerical multilinear algebra, block tensors are increasingly important. Accordingly, it is appropriate to develop an infrastructure that supports reasoning about block tensor computation. In this paper we establish concise notation that is suitable for the analysis and development of block tensor algorithms, prove several useful block tensor identities, and make precise the notion of a block tensor unfolding.
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