Computation Of Singular Values Research Articles

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

Accurate computations with collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis

A fast and accurate algorithm to compute the bidiagonal decomposition of collocation matrices of the Lupaş-type (p,q)-analogue of the Bernstein basis is presented. The error analysis of the algorithm and the perturbation theory for the bidiagonal decomposition are also included. Starting from this bidiagonal decomposition, the accurate and efficient solution of several linear algebra problems involving these matrices is addressed: linear system solving, eigenvalue and singular value computation, and computation of the inverse and the Moore-Penrose inverse. The numerical experiments carried out show the good behaviour of the algorithm.

Modular Computation of Restoration Entropy for Networks of Systems: A Dissipativity Approach

The problem of state estimation based on information received over a finite bit rate channel gives rise to the study of minimal bit rate above which state can be estimated with any desired accuracy. In the past few years, researchers have studied the minimal average bit rate which is sufficient enough for state estimation such that the estimation error stays within a given factor of its initial value. The notion of restoration entropy characterizes this type of bit rate. Recent results proposed numerical schemes to estimate restoration entropy by the computation of singular values of the linearized systems. Such schemes are either complex to implement or suffer severely from computational complexity and the size of the state dimension. In this paper, we describe a modular approach to compute an upper bound of the restoration entropy of a large network by decomposing the network to an interconnection of smaller subsystems. Then, we formulate a distributed optimization problem which is solved for each subsystem separately and then their optimization results are composed to get an upper bound of the restoration entropy for the overall network. We illustrate the effectiveness of our results by two examples.

Skeleton-Based Human Action Recognition through Third-Order Tensor Representation and Spatio-Temporal Analysis

Given the broad range of applications from video surveillance to human–computer interaction, human action learning and recognition analysis based on 3D skeleton data are currently a popular area of research. In this paper, we propose a method for action recognition using depth sensors and representing the skeleton time series sequences as higher-order sparse structure tensors to exploit the dependencies among skeleton joints and to overcome the limitations of methods that use joint coordinates as input signals. To this end, we estimate their decompositions based on randomized subspace iteration that enables the computation of singular values and vectors of large sparse matrices with high accuracy. Specifically, we attempt to extract different feature representations containing spatio-temporal complementary information and extracting the mode-n singular values with regards to the correlations of skeleton joints. Then, the extracted features are combined using discriminant correlation analysis, and a neural network is used to recognize the action patterns. The experimental results presented use three widely used action datasets and confirm the great potential of the proposed action learning and recognition method.

English

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R m×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A −1) and Tr(A −2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A −1) and Tr(A −2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A −1)}2/Tr(A −2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.

Special Section: 2016 Copper Mountain Conference

The Fourteenth Copper Mountain Conference on Iterative Methods was held March 20--25, 2016, in Copper Mountain, Colorado. The meeting featured more than 170 presentations covering a broad spectrum of topics in scientific computing. Highlighted topics for the 2016 conference included iterative linear algebraic techniques in data mining, optimization, inverse problems, nonlinear solvers, methods for large-scale eigenvalue and singular value computations, implementation of iterative solvers on emerging architectures, uncertainty quantification, robust and scalable iterative solution of coupled multi-physics problems, model reduction, and methods based on low-rank approximations. Besides data mining, applications to energy, climate, imaging, fluids and electromagnetics were especially encouraged. Several speakers addressed discretization techniques for partial differential equations from the point of view of their impact on the design and performance of iterative solvers. As is customary for Copper Mountain, ...

On Singular Values Related to DAEs in Kronecker Canonical Form

The index and the structural properties of differential algebraic equations (DAEs) are often determined by rank considerations of the derivative array. Since the Kronecker canonical form is a well-understood standard form that permits deep insight into the properties of DAEs, in this contribution we undertake an analysis of the singular values of this specific derivative array. To this end, the special structure of the obtained block matrices is pointed out, such that some formulas for the computation and estimation of eigenvalues and singular values can be applied. Actually, we explore the relationship between the spectra of particular block tridiagonal matrices and some perturbed Jacobi matrices.

Computation of extremum singular values and the strong H-infinity norm of SISO time-delay systems

We consider the computation of H-infinity norms for Single-Input–Single-Output (SISO) time-delay systems, which are described by delay differential algebraic equations. Unlike the iterative level set methods in the literature, we present a novel numerical method to compute the H-infinity norm. This method requires solving one eigenvalue problem of at most twice the size of the eigenvalue problem in every iteration of a level set method, but in practice often considerably lower. We first show that the computation of extrema of the transfer function can be turned into the computation of the imaginary axis zeros of a transcendental function. We compute these zeros by a predictor–corrector type algorithm. It is known that the H-infinity norm of delay differential algebraic systems, which can model both retarded and neutral type systems, might be sensitive with respect to arbitrarily small delay perturbations. This recently led to the concept of strong H-infinity norms, which explicitly take into account such small delay perturbations. We present a direct numerical method to compute the strong H-infinity norm of SISO time-delay systems. Our algorithm is applicable to the closed-loop system of interconnections (series, parallel, feedback, junctions) of time-delay systems and/or controllers.

Features based on intrinsic mode functions for classification of EMG signals

In this paper, the features based on Intrinsic Mode Functions (IMFs) for classification of EMG signals are presented. The EMD method decomposes EMG signals into a set of narrow-band components known as IMFs. The features, namely mean frequency estimation and singular value computation, extracted from IMFs are exploited for classification of EMG signals. These parameters are used as an input to Least Squares Support Vector Machine (LS-SVM) with Radial Basis Function (RBF) for automatic classification of EMG signals. The classification accuracy for classification of normal and abnormal EMG signals obtained by the proposed method is 99.03% with RBF kernel of LS-SVM. The experimental results are presented to show the effectiveness of the proposed method for classification of normal and abnormal EMG signals (myopathy and neuropathy).

On Efficiently Computing the Eigenvalues of Limited-Memory Quasi-Newton Matrices

In this paper, we consider the problem of efficiently computing the eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact formulation. In addition, we produce a compact formula for quasi-Newton matrices generated by any member of the Broyden convex class of updates. Our proposed method makes use of efficient updates to the QR factorization that substantially reduce the cost of computing the eigenvalues after the quasi-Newton matrix is updated. Numerical experiments suggest that the proposed method is able to compute eigenvalues to high accuracy. Applications for this work include modified quasi-Newton methods and trust-region methods for large-scale optimization, the efficient computation of condition numbers and singular values, and sensitivity analysis.

A shift strategy for superquadratic convergence in the dqds algorithm for singular values

A new shift strategy is proposed for the differential quotient difference with shifts (dqds) algorithm for the computation of singular values of bidiagonal matrices. While maintaining global convergence, the proposed shift realizes asymptotic superquadratic convergence of the dqds algorithm.

On best rank one approximation of tensors

SUMMARYToday, compact and reduced data representations using low rank data approximation are common to represent high‐dimensional data sets in many application areas as for example, genomics, multimedia, quantum chemistry, social networks, or visualization. To produce such low rank data representations, the input data is typically approximated by the so‐called alternating least squares (ALS) algorithms. However, not all of these ALS algorithms are guaranteed to converge. To address this issue, we suggest a new algorithm for the computation of a best rank one approximation of tensors, called alternating singular value decomposition. This method is based on the computation of maximal singular values and the corresponding singular vectors of matrices. We also introduce a modification for this method and the ALS method, which ensures that alternating iterations will always converge to a semi‐maximal point (a critical point in several vector variables is semi‐maximal if it is maximal with respect to each vector variable, while other vector variables are kept fixed). We present several numerical examples that illustrate the computational performance of the new method in comparison to the ALS method. Copyright © 2013 John Wiley & Sons, Ltd.

Accurate computations with totally positive Bernstein-Vandermonde matrices

The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein-Vandermonde matrix is considered. Bernstein-Vandermonde ma- trices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials of degree less than or equal to n the Bernstein basis instead of the monomial basis. The approach in this paper is based on the computation of the bidiagonal factorization of a totally positive Bernstein-Vandermonde matrix or of its inverse. The explicit expressions obtained for the determinants involved in the process make the algorithm both fast and accurate. The error analysis of this algorithm for computing this bidiagonal factorization and the perturbation theory for the bidiagonal factorization of totally positive Bernstein-Vandermonde matrices are also carried out. Several applications of the computation with this type of matrices are also pointed out.

Automated Detection of Influential Patents Using Singular Values

Centrality measures such as degree centrality have been utilized to identify influential and important patents in a citation network. However, no existing centrality measures take into consideration information from the change of the similarity matrix. This paper presents a new centrality measure based on the change of a node similarity matrix. The proposed approach gives more intuitive understanding of the finding of the influential nodes. The present study starts off with the assumption that the change of matrix that may result from removing a given node would assess the importance of the node since each node make a contribution to the given similarity matrix between nodes. The various matrix norms using the singular values such as nuclear norm which is the sum of all singular values, are used for calculating the contribution of a given node to a node similarity matrix. In other words, we can obtain the change of matrix norms for a given node after we calculate the singular values for the case of the nonexistence and the case of existence of the node. Then, the node resulting in the largest change (i.e., decrease) of matrix norms can be considered as the most important node. Computation of singular values can be computationally intensive when the similarity matrix size is large. Therefore, the singular value update technique is also developed for the case of the network with large nodes. We compare the performance of our proposed approach with other widely used centrality measures using U.S. patents data in the area of information and security. Experimental results show that our proposed approach is competitive or even performs better compared to existing approaches.

Robust Performance Analysis of Uncertain LTI Systems: Dual LMI Approach and Verifications for Exactness

This paper addresses robust performance analysis problems of linear time-invariant (LTI) systems affected by real parametric uncertainties. These problems, known also as a special class of structured singular value computation problems, are inherently intractable (NP-hard problems). As such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions, where linear matrix inequality (LMI) plays an important. Nevertheless, since LMI-based conditions are expected to be conservative in general, it is often the case that we cannot conclude anything if the LMI at hand turns out to be infeasible. This motivates us to consider the dual of the LMI and examine the structure of the dual solution. By pursuing this direction, in this paper, we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained. In particular, a set of uncertain parameters that violates the specified performance can be computed. These results come from block-moment matrix structure of the dual variable, which is consistent with the recent results on polynomial optimization. This particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained.

Subspace Gap Residuals for Rayleigh–Ritz Approximations

Large-scale eigenvalue and singular value computations are usually based on extracting information from a compression of the matrix to suitably chosen low dimensional subspaces. This paper introduces new a posteriori relative error bounds based on a residual expressed using the largest principal angle (gap) between relevant subspaces. The eigenvector approximations are estimated using subspace gaps and relative separation of the eigenvalues.

Computing the distance to uncontrollability via LMIs: Lower bound computation with exactness verification

In this paper, we consider the problem of computing the distance to uncontrollability (DTUC) of a given controllable pair A ∈ C n × n and B ∈ C n × m . It is known that this problem is equivalent to computing the minimum of the smallest singular value of [ A − z I B ] over z ∈ C . With this fact, Gu et al. proposed an algorithm that correctly estimates the DTUC at a computation cost O ( n 4 ) . From the viewpoints of linear control system theory, on the other hand, this problem can be regarded as a special case of the structured singular value computation problems and thus it is expected that we can establish an alternative LMI-based algorithm. In fact, this paper first shows that we can compute a lower bound of the DTUC by simply applying the existing techniques to solve robust LMIs. Moreover, we show via convex duality theory that this lower bound can be characterized by a very concise dual SDP. In particular, this dual SDP enables us to derive a condition on the dual variable under which the computed lower bound surely coincides with the exact DTUC. On the other hand, in the second part of the paper, we consider the problem of computing the similarity transformation matrix T that maximizes the lower bound of the DTUC of ( T − 1 A T , T − 1 B ) . We clarify that this problem can be reduced to a generalized eigenvalue problem and thus solved efficiently. In view of the correlation between the DTUC and the numerical difficulties of the associated pole placement problem, this computation of the transformation matrix would lead to an effective and efficient conditioning of the pole placement problem for the pair ( A , B ) .

Recent additions to SLEPc, the Scalable Library for Eigenvalue Problem computations

AbstractSLEPc is a software library for the solution of large, sparse eigenvalue problems on parallel computers. We have recently extended its functionality by incorporating a Krylov‐Schur eigensolver and a module for singular value computation. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Accurate computation of singular values in terms of shifted integrable schemes

A new scheme with a shift of origin for computing singular values σk is presented. A shift θ is introduced into the recurrence relation defined by the discrete integrable Lotka-Volterra system with variable step-size. A suitable shift strategy is given so that the singular value computation becomes numerically stable. It is proved that variables in the new scheme converge to σ 2 - Σ θ2. A comparison of the zero-shift and the nonzero-shift routines is drawn. With respect to both the computational time and the numerical accuracy, it is shown that the nonzero-shift routine is more accurate and faster than a credible LAPACK routine for singular values at least in four different types of test matrices.

A cubic polynomial system with seven limit cycles at infinity

In this paper, we study bifurcation of limit cycles at infinity for a class of cubic polynomial system with no singular points at infinity, in which the problem for bifurcation of limit cycles from infinity be transferred into that from the origin. By computation of singular point values, the conditions of the origin (correspondingly, infinity) to be a center and the highest degree fine focus are derived. Consequently, we construct a cubic system which can bifurcate seven limit cycles from infinity when let normal parameters be suitable values. The positions of these limit cycles without constructing Poincaré cycle fields can be pointed out exactly.