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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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Extended Krylov Subspaces: Approximation of the Matrix Square Root and Related Functions SIAM. J. Matrix Anal. & Appl. 19, pp. 755-771 (17 pages) Online Publication Date: July 31, 2006
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We introduce an economical Gram--Schmidt orthogonalization on the extended Krylov subspace originated by actions of a symmetric matrix and its inverse. An error bound for a family of problems arising from the elliptic method of lines is derived. The bound shows that, for the same approximation quality, the diagonal variant of the extended subspaces requires about the square root of the dimension of the standard Krylov subspaces using only positive or negative matrix powers. An example of an application to the solution of a 2.5-D elliptic problem attests to the computational efficiency of the method for large-scale problems. |
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A New Sufficient Condition for the Uniqueness of Barabanov Norms 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. |
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Perturbation Analysis for Antitriangular Schur Decomposition 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. |
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Spectral and Inner-Outer Factorizations of Rational Matrices SIAM. J. Matrix Anal. & Appl. 10, pp. 1-17 (17 pages) Online Publication Date: July 17, 2006
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Spectral factorization and inner-outer factorization are basic techniques in treating many problems in electrical engineering. In this paper, the roblems of doing spectral and inner-outer factorizations via state-space methods are studied when the matrix to be factored is real-rational and surjective on the extended maginary axis. It is shown that our factorization problems can be reduced to solving a certain constrained Riccati equation, and that by examining some invariant ubspace of the associated Hamiltonian matrix there exists a unique solution to this equation. Finally, a state-space procedure to perform the factorization is proposed. |
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A Posteriori Error Analysis of Parameterized Linear Systems Using Spectral Methods SIAM. J. Matrix Anal. & Appl. 33, pp. 195-209 (15 pages) Online Publication Date: March 13, 2012
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We develop computable a posteriori error estimates for the pointwise evaluation of linear functionals of a solution to a parameterized linear system of equations. These error estimates are based on a variational analysis applied to polynomial spectral methods for forward and adjoint problems. We also use this error estimate to define an improved linear functional and we prove that this improved functional converges at a much faster rate than the original linear functional given a pointwise convergence assumption on the forward and adjoint solutions. The advantage of this method is that we are able to use low order spectral representations for the forward and adjoint systems to cheaply produce linear functionals with the accuracy of a higher order spectral representation. The method presented in this paper also applies to the case where only the convergence of the spectral approximation to the adjoint solution is guaranteed. We present numerical examples showing that the error in this improved functional is often orders of magnitude smaller. We also demonstrate that in higher dimensions, the computational cost required to achieve a given accuracy is much lower using the improved linear functional. |
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Stable Computation of the CS Decomposition: Simultaneous Bidiagonalization 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. |
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Solving Large-Scale Least Squares Semidefinite Programming by Alternating Direction Methods SIAM. J. Matrix Anal. & Appl. 32, pp. 136-152 (17 pages) Online Publication Date: February 08, 2011
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The well-known least squares semidefinite programming (LSSDP) problem seeks the nearest adjustment of a given symmetric matrix in the intersection of the cone of positive semidefinite matrices and a set of linear constraints, and it captures many applications in diversing fields. The task of solving large-scale LSSDP with many linear constraints, however, is numerically challenging. This paper mainly shows the applicability of the classical alternating direction method (ADM) for solving LSSDP and convinces the efficiency of the ADM approach. We compare the ADM approach with some other existing approaches numerically, and we show the superiority of ADM for solving large-scale LSSDP. |
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A Multilinear Singular Value Decomposition SIAM. J. Matrix Anal. & Appl. 21, pp. 1253-1278 (26 pages) Online Publication Date: July 31, 2006
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We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors. |
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SIAM. J. Matrix Anal. & Appl. 33, pp. 210-234 (25 pages) Online Publication Date: March 13, 2012
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We introduce a generalized Rayleigh-quotient $\rho_A$ on the direct product of Grassmannians $\mathrm{Gr}({\bf m},{\bf n})$ enabling a unified approach to well-known optimization tasks from different areas of numerical linear algebra, such as best low-rank approximations of tensors (data compression), geometric measures of entanglement (quantum computing), and subspace clustering (image processing). We compute the Riemannian gradient of $\rho_A$, characterize its critical points, and prove that they are generically nondegenerated. Moreover, we derive an explicit necessary condition for the nondegeneracy of the Hessian. Finally, we present two intrinsic methods for optimizing $\rho_A$—a Newton-like and a conjugated gradient—and compare our algorithms tailored to the above-mentioned applications with established ones from the literature. |
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On the Design of Deterministic Matrices for Fast Recovery of Fourier Compressible Functions SIAM. J. Matrix Anal. & Appl. 33, pp. 263-289 (27 pages) Online Publication Date: March 27, 2012
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We present a general class of compressed sensing matrices which are then demonstrated to have associated sublinear-time sparse approximation algorithms. We then develop methods for constructing specialized matrices from this class which are sparse when multiplied with a discrete Fourier transform matrix. Ultimately, these considerations improve previous sampling requirements for deterministic sparse Fourier transform methods. |
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On a Nonlinear Matrix Equation Arising in Nano Research SIAM. J. Matrix Anal. & Appl. 33, pp. 235-262 (28 pages) Online Publication Date: March 15, 2012
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The matrix equation $X+A^{\top}X^{-1}A=Q$ arises in Green's function calculations in nano research, where $A$ is a real square matrix and $Q$ is a real symmetric matrix dependent on a parameter and is usually indefinite. In practice one is mainly interested in those values of the parameter for which the matrix equation has no stabilizing solutions. The solution of interest in this case is a special weakly stabilizing complex symmetric solution $X_*$, which is the limit of the unique stabilizing solution $X_{\eta}$ of the perturbed equation $X+A^{\top}X^{-1}A=Q+i\eta I$, as $\eta\to 0^+$. It has been shown that a doubling algorithm can be used to compute $X_{\eta}$ efficiently even for very small values of $\eta$, thus providing good approximations to $X_*$. It has been observed by nano scientists that a modified fixed-point method can sometimes be quite useful, particularly for computing $X_{\eta}$ for many different values of the parameter. We provide a rigorous analysis of this modified fixed-point method and its variant and of their generalizations. We also show that the imaginary part $X_I$ of the matrix $X_*$ is positive semidefinite and we determine the rank of $X_I$ in terms of the number of unimodular eigenvalues of the quadratic pencil $\lambda^2 A^{\top}-\lambda Q+A$. Finally we present a new structure-preserving algorithm that is applied directly on the equation $X+A^{\top}X^{-1}A=Q$. In doing so, we work with real arithmetic most of the time. |
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Verified Bounds for Least Squares Problems and Underdetermined Linear Systems 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. |
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Alternating-directional Doubling Algorithm for $M$-Matrix Algebraic Riccati Equations SIAM. J. Matrix Anal. & Appl. 33, pp. 170-194 (25 pages) Online Publication Date: March 08, 2012
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A new doubling algorithm—the alternating-directional doubling algorithm (ADDA)—is developed for computing the unique minimal nonnegative solution of an $M$-matrix algebraic Riccati equation (MARE). It is argued by both theoretical analysis and numerical experiments that ADDA is always faster than two existing doubling algorithms: SDA of Guo, Lin, and Xu (Numer. Math., 103 (2006), pp. 393–412) and SDA-ss of Bini, Meini, and Poloni (Numer. Math., 116 (2010), pp. 553–578) for the same purpose. Also demonstrated is that all three methods are capable of delivering minimal nonnegative solutions with entrywise relative accuracies as warranted by the defining coefficient matrices of a MARE. The three doubling algorithms, differing only in their initial setups, correspond to three special cases of the general bilinear (also called Möbius) transformation. It is explained that ADDA is the best among all possible doubling algorithms resulted from all bilinear transformations. |
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Disjunctive Programming and a Hierarchy of Relaxations for Discrete Optimization Problems SIAM. J. on Algebraic and Discrete Methods 6, pp. 466-486 (21 pages) Online Publication Date: August 02, 2006
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We discuss a new conceptual framework for the convexification of discrete optimization problems, and a general technique for obtaining approximations to the convex hull of the feasible set. The concepts come from disjunctive programming and the key tool is a description of the convex hull of a union of polyhedra in terms of a higher dimensional polyhedron. Although this description was known for several years, only recently was it shown by Jeroslow and Lowe to yield improved representations of discrete optimization problems. We express the feasible set of a discrete optimization problem as the intersection (conjunction) of unions of polyhedra, and define an operation that takes one such expression into another, equivalent one, with fewer conjuncts. We then introduce a class of relaxations based on replacing each conjunct (union of polyhedra) by its convex hull. The strength of the relaxations increases as the number of conjuncts decreases, and the class of relaxations forms a hierarchy that spans the spectrum between the common linear programming relaxation, and the convex hull of the feasible set itself. Instances where this approach has advantages include critical path problems in disjunctive graphs, network synthesis problems, certain fixed charge network flow problems, etc. We illustrate the approach on the first of these problems, which is a model for machine sequencing. |
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A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices SIAM. J. Matrix Anal. & Appl. 13, pp. 707-728 (22 pages) Online Publication Date: July 17, 2006
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In this paper some results are reviewed concerning the characterization of inverses of symmetric tridiagonal and block tridiagonal matrices as well as results concerning the decay of the elements of the inverses. These results are obtained by relating the elements of inverses to elements of the Cholesky decompositions of these matrices. This gives explicit formulas for the elements of the inverse and gives rise to stable algorithms to compute them. These expressions also lead to bounds for the decay of the elements of the inverse for problems arising from discretization schemes. |
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Hierarchical Singular Value Decomposition of Tensors SIAM. J. Matrix Anal. & Appl. 31, pp. 2029-2054 (26 pages) Online Publication Date: May 26, 2010
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We define the hierarchical singular value decomposition (SVD) for tensors of order $d\geq2$. This hierarchical SVD has properties like the matrix SVD (and collapses to the SVD in $d=2$), and we prove these. In particular, one can find low rank (almost) best approximations in a hierarchical format ($\mathcal{H}$-Tucker) which requires only $\mathcal{O}((d-1)k^3+dnk)$ parameters, where $d$ is the order of the tensor, $n$ the size of the modes, and $k$ the (hierarchical) rank. The $\mathcal{H}$-Tucker format is a specialization of the Tucker format and it contains as a special case all (canonical) rank $k$ tensors. Based on this new concept of a hierarchical SVD we present algorithms for hierarchical tensor calculations allowing for a rigorous error analysis. The complexity of the truncation (finding lower rank approximations to hierarchical rank $k$ tensors) is in $\mathcal{O}((d-1)k^4+dnk^2)$ and the attainable accuracy is just 2–3 digits less than machine precision. |
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A Dual Approach to Semidefinite Least-Squares Problems SIAM. J. Matrix Anal. & Appl. 26, pp. 272-284 (13 pages) Online Publication Date: July 31, 2006
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In this paper, we study the projection onto the intersection of an affine subspace and a convex set and provide a particular treatment for the cone of positive semidefinite matrices. Among applications of this problem is the calibration of covariance matrices. We propose a Lagrangian dualization of this least-squares problem, which leads us to a convex differentiable dual problem. We propose to solve the latter problem with a quasi-Newton algorithm. We assess this approach with numerical experiments which show that fairly large problems can be solved efficiently. |
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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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Candecomp/Parafac: From Diverging Components to a Decomposition in Block Terms 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. |
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SIAM. J. Matrix Anal. & Appl. 32, pp. 1079-1094 (16 pages) Online Publication Date: October 04, 2011
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We derive some properties of complex Hessenberg matrices and use the relevant normal matrix cases to examine the lengths of Ritz vectors in the rounding error analysis of the Lanczos tridiagonalization process. This question is important for the computational use of the process and has already been studied for the real symmetric matrix case, but because of its intricate and unedifying nature, part of the theory was never submitted to scientific journals. We develop a new and more palatable theory which also applies to Lanczos processes adapted to any form of normal matrix with collinear eigenvalues such as a Hermitian or skew-Hermitian matrix. The nonnormal matrix properties are intended to help in the analysis of the unsymmetric Lanczos process. |
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