Hankel/Toeplitz matrices and the static output feedback stabilization problem

The static output feedback (SOF) stabilization problem for general linear, continuous-time and discrete-time systems is discussed. A few novel necessary and sufficient conditions are proposed, and a modified SOF stabilization problem with performance is studied. For multiple-input single-output (or single-input multiple-output) systems the relation with a class of Hankel matrices, and their inverses, in the continuous-time case and with a class of Toeplitz matrices, in the discrete-time case, is established. These relationships are used to construct conceptual numerical algorithms. Finally, it is shown that, in the continuous-time case, the problem can be recast as a concave–convex programming problem. A few worked out examples illustrate the underlying theory.     

The Bruhat decomposition of finite Hankel matrices

The Bruhat decomposition of the space of n × n Hankel matrices is derived from the Bruhat decomposition of the general linear group. Some new results concerning the Hankel minors map H → (det H₁,…,det H_n) are obtained (det H_i denotes the i-th principal minor of H). The adherence order of the Bruhat decomposition is determined in the complex case.     

On the Berlekamp/Massey algorithm and counting singular Hankel matrices over a finite field

We derive an explicit count for the number of singular n×n$n\times n$ Hankel (Toeplitz) matrices whose entries range over a finite field with q$q$ elements by observing the execution of the Berlekamp/Massey algorithm on its elements. Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed. For example, the number of singular n×n$n\times n$ Toeplitz matrices with 0’s on the diagonal is q²ⁿ⁻³+qⁿ⁻¹−qⁿ⁻²$q^{2n-3}+q^{n-1}-q^{n-2}$.     

Efficiently solvable cases of quadratic assignment problem with generalized monotonic and incomplete anti-monge matrices

The paper describes new sufficient conditions that provide optimum to a quadratic assignment problem for a given substitution. The conditions separate out sets of matrices in the matrix space. The sets are formed by incomplete Anti-Monge matrices, which permit opposite ordering of rows and columns, and by additive monotone matrices, which are outside the class of Toeplitz matrices. The study was partially supported by INTAS (Project 03-51-5501). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 135–151, January–February 2007.     

Subspace identification of circulant systems

This article concerns the identification of a class of large scale systems called “circulant systems”. Circulant systems have a special property that allows them to be decomposed into simpler subsystems through a state transformation. This property has been used in literature for control design, and here we show how it can be used for system identification. The approach that is proposed here will both reduce the complexity of the problem as well as provide models which have a circulant structure that can be exploited for control design. A novel identification algorithm for circulant systems based on subspace identification is presented. The algorithm is then tested in simulation on an academic example of circulant system and on a realistic finite element model of a vibrating plate.     

Conditioning of infinite Hankel matrices of finite rank

Let H be an infinite Hankel matrix with h_i+j−2 as its (i,j)-entry, h_k=∑_l=1ⁿr_l z_l^k, k=0,1,…,|z_l|<1, and . We derive upper bounds for the 2-condition number of H as functions of n, r_l and z_l, which show that the Hankel matrix H becomes well conditioned whenever the z's are close to the unit circle but not extremely close to each other. Numerical results which illustrate the theory are provided.     

A Parallel Solver for Circulant Toeplitz Tridiagonal Systems on Hypercubes

Solving circulant Toeplitz tridiagonal systems arises in many engineering applications. This paper presents a fast parallel algorithm for solving this type of systems. The number of floating-point operations required in our algorithm is less than the previous parallel algorithm [cf. Kim and Lee (1990)] for solving the similar system. Specifically, an overlapping technique is proposed to reduce the communication steps required. In addition, an error analysis is given. The implementation of our algorithm on the nCUBE2/E with 16 processors has been carried out. The experimental results show that the speedup is almost linearly proportional to the number of processors.     

Linear algebra for tensor problems

By a tensor problem in general, we mean one where all the data on input and output are given (exactly or approximately) in tensor formats, the number of data representation parameters being much smaller than the total amount of data. For such problems, it is natural to seek for algorithms working with data only in tensor formats maintaining the same small number of representation parameters—by the price of all results of computation to be contaminated by approximation (recompression) to occur in each operation. Since approximation time is crucial and depends on tensor formats in use, in this paper we discuss which are best suitable to make recompression inexpensive and reliable. We present fast recompression procedures with sublinear complexity with respect to the size of data and propose methods for basic linear algebra operations with all matrix operands in the Tucker format, mostly through calls to highly optimized level-3 BLAS/LAPACK routines. We show that for three-dimensional tensors the canonical format can be avoided without any loss of efficiency. Numerical illustrations are given for approximate matrix inversion via proposed recompression techniques.      

基于低秩矩阵估计的暗光图像增强模型

在暗光或逆光拍照时,获得的图像常常出现过暗或光照分布不均的现象,导致图像视觉质量较差.基于Retinex模型的暗光增强模型能实现有效地光照增强.但此类暗光增强模型也存在一些问题,即待处理图像中暗光区域的可视度虽然得到了有效改善,但其中隐藏的噪声也被放大和凸显,依旧影响了增强结果的视觉质量.为解决这一问题,构建了基于低秩...     

Symmetric subresultants and applications

Schur’s transforms of a polynomial are used to count its roots in the unit disk. These are generalized then by introducing the sequence of symmetric subresultants of two polynomials. Although they do have a determinantal definition, we show that they satisfy a structure theorem which allows us to compute them with a type of Euclidean division. As a consequence, a fast algorithm based on a dichotomic process and FFT is designed.     

Parallel computation of the eigenvalues of symmetric Toeplitz matrices through iterative methods

This paper presents a new procedure to compute many or all of the eigenvalues and eigenvectors of symmetric Toeplitz matrices. The key to this algorithm is the use of the “Shift–and–Invert” technique applied with iterative methods, which allows the computation of the eigenvalues close to a given real number (the “shift”). Given an interval containing all the desired eigenvalues, this large interval can be divided in small intervals. Then, the “Shift–and–Invert” version of an iterative method (Lanczos method, in this paper) can be applied to each subinterval. Since the extraction of the eigenvalues of each subinterval is independent from the other subintervals, this method is highly suitable for implementation in parallel computers. This technique has been adapted to symmetric Toeplitz problems, using the symmetry exploiting Lanczos process proposed by Voss [H. Voss, A symmetry exploiting Lanczos method for symmetric Toeplitz matrices, Numerical Algorithms 25 (2000) 377–385] and using fast solvers for the Toeplitz linear systems that must be solved in each Lanczos iteration. The method compares favourably with ScaLAPACK routines, specially when not all the spectrum must be computed.     

A new version of Kovarik’s approximate orthogonalization algorithm without matrix inversion

In 1970 Kovarik proposed approximate orthogonalization algorithms. One of them (algorithm B) has quadratic convergence but requires at each iteration the inversion of a matrix of similar dimension to the initial one. An attempt to overcome this difficulty was made by replacing the inverse with a finite Neumann series expansion involving the original matrix and its adjoint. Unfortunately, this new algorithm loses the quadratic convergence and requires a large number of terms in the Neumann series which results in a dramatic increase in the computational effort per iteration. In this paper we propose a much simpler algorithm which, by using only the first two terms in a different series expansion, gives us the desired result with linear convergence. Systematic numerical experiments for collocation and Toeplitz matrices are also described.     

A new measure of consistency for positive reciprocal matrices

The analytic hierarchy process (AHP) provides a decision maker with a way of examining the consistency of entries in a pairwise comparison matrix and the hierarchy as a whole through the consistency ratio measure. It has always seemed to us that this commonly used measure could be improved upon. The purpose of this paper is to present an alternative consistency measure and demonstrate how it might be applied in different types of matrices.     

Error Analysis in Homography Estimation by First Order Approximation Tools: A General Technique

This paper shows how to analytically calculate the statistical properties of the errors in estimated parameters. The basic tools to achieve this aim include first order approximation/perturbation techniques, such as matrix perturbation theory and Taylor Series. This analysis applies for a general class of parameter estimation problems that can be abstracted as a linear (or linearized) homogeneous equation. Of course there may be many reasons why one might which to have such estimates. Here, we concentrate on the situation where one might use the estimated parameters to carry out some further statistical fitting or (optimal) refinement. In order to make the problem concrete, we take homography estimation as a specific problem. In particular, we show how the derived statistical errors in the homography coefficients, allow improved approaches to refining these coefficients through subspace constrained homography estimation (Chen and Suter in Int. J. Comput. Vis. 2008). Indeed, having derived the statistical properties of the errors in the homography coefficients, before subspace constrained refinement, we do two things: we verify the correctness through statistical simulations but we also show how to use the knowledge of the errors to improve the subspace based refinement stage. Comparison with the straightforward subspace refinement approach (without taking into account the statistical properties of the homography coefficients) shows that our statistical characterization of these errors is both correct and useful.

Methods of computing weighted pseudoinverse matrices with singular weights

Three iterative processes are constructed and investigated for computing weighted pseudoinverse matrices with singular weights and ML-weighted pseudoinverse matrices. Two of them are based on the decompositions of the weighted pseudoinverse matrix with singular weights into matrix power series, and the third is a generalization of the Schulz method to nonsingular square matrices. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 150–169, September–October, 1999.     

Analytic inversion for a general case of certain classes of matrices

The explicit inverse and determinant of a class of matrices are presented. The class under consideration is defined by 4n ? 2 parameters, analytic expressions of which form the elements of the upper Hessenberg type inverse. These analytic expressions enable recursion formulae to be obtained, which reduce the arithmetic operations to O(n ²). The Hadamard product of two specific structures' matrices provides the class presented, special cases of which are already known classes of test matrices.     

Semi-tensor product of matrices and its application to Morgen's problem

This paper proposes a new matrix product, namely, semi-tensor product. It is a general-ization of the conventional matrix product. Meanwhile, it is also closely related to Kronecker (tensor) product of matrices. The purpose of introducing this product is twofold: (i) treat multi-dimensional da-ta; (ii) treat nonlinear problems in a linear way. Then the computer and numerical methods can be easily used for solving nonlinear problems. Properties and formulas are deduced. As an application, the Morgan's problem for control systems is formulated as a numerically solvable problem.     

Operational matrices of Bernstein polynomials and their applications

The Bernstein polynomials (B-polynomials) operational matrices of integration P, differentiation D and product ? are derived. A general procedure of forming these matrices are given. These matrices can be used to solve problems such as calculus of variations, differential equations, optimal control and integral equations. Illustrative examples are included to demonstrate the validity and applicability of the operational matrices.     

A new preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

In this paper, based on the preconditioners presented by Cao [A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices, Journal of Computational and Applied Mathematics 238(15) (2013), pp. 109–115], we introduce and study a new augmentation block preconditioners for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and its minimal polynomial. Finally, numerical examples show that the eigenvalue distribution with presented preconditioner has the same spectral clustering with preconditioners in the literature when choosing the optimal parameters and the preconditioner in this paper and in the literature improve the convergence of BICGSTAB and GMRES iteration efficiently when they are applied to the preconditioned BICGSTAB and GMRES to solve the Stokes equation and two-dimensional time-harmonic Maxwell equations by choosing different parameters.     

Determination of the number of unambiguous bit matrices

A computationally convenient formula is derived that allows the calculation of the number of unambiguous bit matrices of orderm × n. The result shows that the compression of a large binary matrix by using unambiguous bit matrices is generally impractical unless additional information about the original matrix is available.The authors would like to express their thanks to the referees for their criticism and comments. The support of the National Research Council of Canada and of the University Council for Research of St. Francis Xavier University is gratefully acknowledged.