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Finite difference discretization of the two-dimensional space-fractional diffusion equations derives a complicated linear system consisting of identity matrix and four scaled block-Toeplitz with Toeplitz block (BTTB) matrices resulted from the left and right Riemann–Liouville fractional derivatives in different directions. Incorporating with the diffusion coefficients and the symmetric parts of the BTTB matrices, we construct a diagonal and symmetric splitting (DSS) iteration method, which is demonstrated to be convergent conditionally when the considered space-fractional diffusion equations have sufficiently close diffusion coefficients. By further replacing the symmetric Toeplitz matrices involved in the BTTB matrices with τ matrices, an approximated DSS (ADSS) preconditioner based on two-dimensional fast sine transform is designed to accelerate the convergence rates of the Krylov subspace iteration methods. In this way, the total computational complexity of the ADSS-preconditioned GMRES method will be of O (n 2 log n) , where n 2 represents the dimension of the corresponding discrete linear system. In addition, theoretical analysis demonstrates that the eigenvalues of the ADSS-preconditioned matrix are weakly clustered around a complex disk centered at 1 with the radius less than 1. Numerical experiments show that the ADSS-preconditioned GMRES method is much more efficient than the other existing methods, and can show h -independent convergence behavior. [ABSTRACT FROM AUTHOR]

This paper considers the L 2 gain optimal problem for a class of discrete-time linear time-invariant with state-disturbance feedback controller and unknown system dynamic. Firstly, for a given stabilizing control policy, we establish the relation between the L 2 gain and a sequence of lower triangle Toeplitz matrices. Meanwhile, we show that the upper bound of optimal L 2 gain is proportional to the linear correlation degree between the input and disturbance matrices. Secondly, to overcome the obstacle arising from the unknown system dynamics, a data-based reinforcement learning scheme is developed for the optimal control policy by using linear matrix inequality technique and Q -learning with policy iteration. Under certain conditions, we prove that either the reinforcement learning process ends in a finite number of iterations, or the L 2 gain sequence is strictly monotonically convergent along the iteration axis provided that the disturbance data set can fully activate the closed-loop system. Finally, simulations are given to illustrate the effectiveness of our findings. [ABSTRACT FROM AUTHOR]

We propose a high precision algorithm for solving the Gelfand–Levitan–Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson's type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations. To the best of our knowledge, this is the first algorithm to solve this problem with an order of accuracy higher than the second. • Numerical solution of the Gelfand–Levitan–Marchenko equation. • Fast nonlinear Fourier transform algorithm. • Block Toeplitz Inner-Bordering method. • Gregory quadrature formulas. • Woodbury's formula for almost Toeplitz matrices. [ABSTRACT FROM AUTHOR]

This article deals with the limiting spectral distributions (LSD) of symmetric Toeplitz and Hankel matrices with dependent entries. For any fixed integer m ≥ 0 , we consider these n × n matrices with entries { Y j (m) / n ; j ∈ Z } , where Y j (m) = ∑ r = − m m X j + r and { X k } are i.i.d. random variables with mean zero and variance one. We provide explicit expressions for the LSDs. As a special case (m = 0), this article provides an alternate proof for the LSDs of these matrices when the entries are i.i.d. with mean zero and variance one. The method is based on the moment method. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm for solving quasi-pentadiagonal Toeplitz linear systems, that is based on Du et al. method (Du et al., 2014) and the Sherman–Morrison–Woodbury inversion formula. All algorithms have been implemented in Matlab and numerical experiments are given in order to illustrate the validity and efficiency of our algorithm. An application of the proposed algorithm to the Lax–Wendroff scheme is also considered. [ABSTRACT FROM AUTHOR]

Block Toeplitz matrices are a special class of matrices that exhibit reduced memory requirements and a reduced complexity of matrix-vector multiplications. We herein present an efficient computational approach to solve a sequence of block Toeplitz systems arising from a block Toeplitz system with multiple right-hand sides. Two different numerical schemes are implemented for the solution of the sequence of block Toeplitz systems based on global and block variants of the generalized minimal residual (GMRES) method. The performance of the schemes is assessed in terms of the wall clock time of the iterative solution process, the number of multiplications with the block Toeplitz system matrix and the peak memory usage. To demonstrate the method, two numerical examples are presented. In the first case study, aeroacoustic prediction of an airfoil in turbulent flow is examined, which requires multiple solutions of the wall pressure field beneath the turbulent boundary layer. The fluctuating pressure on the surface of the airfoil is synthesized in terms of uncorrelated wall plane waves, whereby each realization of the wall pressure field is an input to the acoustic solver based on the boundary element method (BEM). The total acoustic response from the airfoil in turbulent flow is then obtained from an ensemble average for the number of realizations considered. The number of realizations to yield a converged solution for the wall pressure field leads to a sequence of block Toeplitz systems. The second case study examines the nonlinear eigenvalue analysis of a sonic crystal barrier composed of locally resonant C-shaped sound-hard scatterers. The periodicity of the sound barrier leads to a block Toeplitz system matrix whereas the nonlinear eigenvalue problem requires the solution of sequences of linear systems. The combined technique to solve the sequences of block Toeplitz systems using the proposed variants of the GMRES is shown to yield a computationally efficient approach for flow noise prediction and nonlinear eigenvalue analysis. [ABSTRACT FROM AUTHOR]

The shrinkage estimator of a high-dimensional covariance matrix relies on a preassigned target matrix during data processing. This paper provides an adaptive approach for selecting the optimal Toeplitz target matrix. We discover a sufficient and necessary condition for characterizing the two kinds of target matrices with the Toeplitz structure, and we propose an adaptive selection algorithm by measuring the similarity between the data and the Toeplitz structure. Numerical simulations and an empirical study on monetary funds verify the effectiveness of the selection approach. [ABSTRACT FROM AUTHOR]

Transmission zeros can be counted by using the Smith–McMillan form, pole/zero modules, or the dimension of the largest output-nulling invariant subspace. This article provides an alternative approach by showing that the number of transmission zeros of a multiple-input–multiple-output (MIMO) transfer function is given in terms of the defect of a block Toeplitz matrix and the defect of an augmented matrix consisting of an observability matrix and the block Toeplitz matrix. It is also shown that the number of infinite zeros is related to the defect of a block Toeplitz matrix. These results are illustrated with a numerical example. [ABSTRACT FROM AUTHOR]

In this paper, we address the identification of two-dimensional (2-D) spatial-temporal dynamical systems described by the Vector-AutoRegressive (VAR) form. The coefficient-matrices of the VAR model are parametrized as sums of Kronecker products. When the number of terms in the sum is small compared to the size of the matrices, such a Kronecker representation efficiently models large-scale VAR models. Estimating the coefficient matrices in least-squares sense gives rise to a bilinear estimation problem that is tackled using an Alternating Least Squares (ALS) algorithm. Regularization or parameter constraints on the coefficient-matrices allows to induce temporal network properties, such as stability, as well as spatial properties, such as sparsity or Toeplitz structure. Convergence of the regularized ALS is proved using fixed-point theory. A numerical example demonstrates the advantages of the new modeling paradigm. It leads to comparable variance of the prediction error with the unstructured least-squares estimation of VAR models. However, the number of parameters grows only linearly with respect to the number of nodes in the 2-D sensor network instead of quadratically in the case of fully unstructured coefficient-matrices. [ABSTRACT FROM AUTHOR]

We present faster algorithms for the residue multiplication modulo 521-bit Mersenne prime on 32- and 64-bit platforms by using Toeplitz matrix-vector product. The total arithmetic cost of our proposed algorithms is less than that of existing algorithms, with algorithms for 64- and 32-bit residue multiplication giving the best timing results on our test machine. The transition from 64- to 32-bit implementation is full of challenges because the number of limbs doubles and the limbs’ bitlengths are cut in half. Without using any intrinsics or SIMD/assembly instructions in our implementation on an Intel(R) Core i5 – 6402P CPU @ 2.80GHz, we find 136 and 550 cycles for our 64-and 32-bit residue multiplications, respectively. In addition, we implement constant-time variable- and fixed-base scalar multiplication for the standard NIST curve P-521 and Edwards curve E-521. [ABSTRACT FROM AUTHOR]

This note considers the identification of large-scale one-dimensional networks consisting of identical LTI dynamical systems. A subspace identification method is developed that only uses local input-output information and does not rely on knowledge about the local state interaction. The proposed identification method estimates the Markov parameters of a locally lifted system, following the state-space realization of a single subsystem. The Markov-parameter estimation is formulated as a rank minimization problem by exploiting the low-rank property and the two-layer Toeplitz structural property in the data equation, whereas the state-space realization of a single subsystem is formulated as a structured low-rank matrix-factorization problem. The effectiveness of the proposed identification method is demonstrated by simulation examples. [ABSTRACT FROM PUBLISHER]

This study presents special cases of inconsistent pairwise comparisons PC matrices. The analysis of the selected inconsistency indicators is provided. One of the compared inconsistencies is the popular eigenvalue-based consistency index (CI) which fails to follow axiomatization recently published by this journal. The other inconsistency, based on the exponentially invertible measure (also known as Koczkodaj's inconsistency indicator or Kii) follows the axiomatization. All studied special cases of PC matrices are Toeplitz matrices with only three different entries 1, x , and 1 / x . A circulant matrix has been used for pairwise comparisons to analyze inconsistency. Although this class of PC matrices may be perceived as restricted, it is general enough to cover all values of the eigenvalue-based inconsistency index from the lowest to the highest. Both exact mathematical expressions and estimations, where the exact expression was impossible to find, are provided. [ABSTRACT FROM AUTHOR]

In this paper, by applying the real representations of complex matrices, the particular structure of the real representations and the Moore–Penrose generalized inverse, we obtain the explicit expression of the minimal norm least squares Hermitian solution of the complex matrix equation A X B + C X D = E . And we also derive the minimal norm least squares Hermitian solution of the complex matrix equation A X B = E . Our proposed formulas only involve real matrices, and therefore are more effective and portable than those reported in Yuan and Liao (2014). The corresponding algorithms only perform real arithmetic which also consider the particular structure of the real representations of complex matrices. Two numerical examples are provided to demonstrate the effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm, that is based on computing approximate greatest common divisors (GCD) of polynomials, for solving the problem of blind image deconvolution. Specifically, we design a specialized algorithm for computing the GCD of bivariate polynomials corresponding to z -transforms of blurred images to recover the original image. The new algorithm is based on the fast GCD algorithm for univariate polynomials in which the successive transformation matrices are upper triangular Toeplitz matrices. The complexity of our algorithm is O ( n 2 log ( n ) ) where the size of blurred images is n × n . All algorithms have been implemented in Matlab and experimental results with synthetically blurred images are included to illustrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

A method for the rigorous computation of the electromagnetic scattering of large dielectric volumes is presented. One goal is to simplify the analysis of large dielectric targets with translational symmetries taken advantage of their Toeplitz symmetry. Then, the matrix-fill stage of the Method of Moments is efficiently obtained because the number of coupling terms to compute is reduced. The Multilevel Fast Multipole Method is applied to solve the problem. Structured meshes are obtained efficiently to approximate the dielectric volumes. The regular mesh grid is achieved by using parallelepipeds whose centres have been identified as internal to the target. The ray casting algorithm is used to classify the parallelepiped centres. It may become a bottleneck when too many points are evaluated in volumes defined by parametric surfaces, so a hierarchical algorithm is proposed to minimize the number of evaluations. Measurements and analytical results are included for validation purposes. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to develop and analyze two fast and conservative algorithms for the fractional sine-Gordon equation. The numerical schemes are derived by using the second-and fourth-order difference method in space and the implicit midpoint rule in time to approximate an equivalent system obtained via the energy quadratic method. In addition, the conservation, existence and uniqueness, and convergence of the two schemes are investigated. Based on the properties of the Toeplitz matrix, a fast algorithm is given in the calculation for the proposed schemes. Numerical experiments verify the theoretical analysis of the schemes, showing their efficiency and excellent behavior. • We propose the Hamiltonian formulation of the 2D fractional sine-Gordon equation. • Two linearly implicit energy-preserving schemes are developed for the 2D fractional sine-Gordon equation. • An optimal l ∞ -error estimate for the proposed scheme is established without any restriction on the grid ratio. • A fast algorithm based on the fast Fourier transformation technique is given. [ABSTRACT FROM AUTHOR]

This technical note presents a new proof of the circle criterion for multivariate, discrete-time systems with time-varying feedback nonlinearities. A new proof for the multivariate Tsypkin criterion for time-invariant monotonic feedback nonlinearities is derived as well. Both integrator- and non-integrator systems are considered. The proofs are direct in the sense that they do not resort to any existing result in systems theory, such as Lyapunov theory, passivity theory or the small-gain theorem. Instead, the proofs refer to the asymptotic properties of block-Toeplitz matrices. One major advantage of the new proof is that it elegantly handles integrator systems without resorting to loop transformation/pole shifting techniques. Additionally, less conservative stability bounds are derived by making stronger assumptions on the sector bound conditions on the feedback nonlinearities. In particular, it is exemplified how this technique relaxes stability conditions of (i) a model predictive control (MPC) rule and (ii) an integrator system. [ABSTRACT FROM PUBLISHER]

In array processing, mutual coupling between antenna elements is inevitable, which has an adverse effect on the estimation of parameters. Meanwhile, the expensive computational requirements are also not conducive to the real-time processing. In this paper, a new Hermitian Toeplitz matrix reconstruction algorithm suitable for distributed sparse arrays with contiguous difference co-array is proposed. The computational complexity of the new algorithm is quantitatively analyzed, which is much lower than that of the spatial smoothing algorithm. Then, a new array called distributed super nested arrays (DSNA) is proposed to reduce the mutual coupling between antenna elements. In contrast to super nested arrays (SNA) and nested arrays (NA), the proposed array can further improve the accuracy of direction-of-arrival (DOA) estimation by using the multiscale estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm. In high-frequency surface wave radar (HFSWR), the proposed array and algorithm can effectively reduce the mutual coupling in a wide frequency range, improve DOA estimation performance, significantly increase the number of detectable source signals, and reduce computational complexity. Numerical simulations demonstrate the superiority of DSNA using the proposed Hermitian Toeplitz matrix reconstruction algorithm. [ABSTRACT FROM AUTHOR]

Detection and estimation of feeble signals in noise is of great importance in practice. In this paper, we use the properties of eigen-space and eigen-spectrum of symmetric and Toeplitz covariance matrix, to address the problem of detecting/estimating very low signal-to-noise ratio (SNR) sinusoidal signals. We show that as the input signal vector length increases, the sign of the dominant eigenvalue of flipped covariance matrix changes with the same frequency as the input sinusoidal signal. Based on this fact, we introduce two algorithms to detect weak sinusoidal signal buried in additive white Gaussian noise. These algorithms have similarities with the bordered-autocorrelation-method Karhunen-Loeve transform (BAM-KLT) method, which is previously proposed in the literature for weak signal detection. However, the BAM-KLT has some limitations and bottlenecks in computational burden, performance and implementation. Moreover, its exact analysis in the discrete domain is missed in the literature. In this paper, we provide the detailed analysis of discrete BAM-KLT. We also show analytically that if the discrete BAM-KLT is applied to flipped covariance matrix instead of covariance matrix itself, its limitations and bottlenecks can be overcome. This idea is the basis of the proposed algorithms. The algorithms are studied analytically and evaluated through numerical simulation. The results show that the proposed methods effectively increase the capability of existing methods, in terms of detection/estimation performance, root-mean-square frequency error, contrast, as well as the number of detectable signal sources. [ABSTRACT FROM AUTHOR]

We obtain a formula for the determinant of a block Toeplitz matrix associated with a quadratic fermionic chain with complex coupling. Such couplings break reflection symmetry and/or charge conjugation symmetry. We then apply this formula to compute the Renyi entropy of a partial observation to a subsystem consisting of contiguous sites in the limit of large size. The present work generalizes similar results due to Its, Jin, and Korepin [Fields Institute Communications, Universality and Renormalization, Vol. 50, p. 151, 2007] and Its, Mezzadri, and Mo [Commun. Math. Phys. 284, 117 (2008)]. A striking feature of our formula for the entanglement entropy is the appearance of a term scaling with the logarithm of the size. This logarithmic behavior originates from certain discontinuities in the symbol of the block Toeplitz matrix. Equipped with this formula we analyze the entanglement entropy of a Dzyaloshinsk-Moriya spin chain and a Kitaev fermionic chain with long-range pairing. [ABSTRACT FROM AUTHOR]

The paper establishes weak convergence in C [ 0 , 1 ] of normalized stochastic processes, generated by Toeplitz type quadratic functionals of a continuous time Gaussian stationary process, exhibiting long-range dependence. Both central and non-central functional limit theorems are obtained. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe a Matlab toolbox, called μ -diff, for modeling and numerically solving two-dimensional complex multiple scattering by a large collection of circular cylinders. The approximation methods in μ -diff are based on the Fourier series expansions of the four basic integral operators arising in scattering theory. Based on these expressions, an efficient spectrally accurate finite-dimensional solution of multiple scattering problems can be simply obtained for complex media even when many scatterers are considered as well as large frequencies. The solution of the global linear system to solve can use either direct solvers or preconditioned iterative Krylov subspace solvers for block Toeplitz matrices. Based on this approach, this paper explains how the code is built and organized. Some complete numerical examples of applications (direct and inverse scattering) are provided to show that μ -diff is a flexible, efficient and robust toolbox for solving some complex multiple scattering problems. Program summary Program title: μ -diff Catalogue identifier: AEWH_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEWH_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 13,718 No. of bytes in distributed program, including test data, etc.: 1,466,301 Distribution format: tar.gz Programming language: Matlab. Computer: PC, Mac. Operating system: Windows, Mac OS, Linux. Has the code been vectorized or parallelized?: Yes RAM: 2000 Megabytes Classification: 4.6, 10, 18. Nature of problem: Modeling and simulation of two-dimensional multiple wave scattering by large clusters of circular cylinders for any frequency. The program is well-designed to manage highly accurate solutions for deterministic or random media, with various boundary conditions and physics properties of the scatterers. Pre- and post-processing facilities are designed specifically for these problems. Solution method: We use spectral Fourier approximation schemes and direct or iterative Krylov subspace methods. Running time: From a few seconds for simple problems to a few minutes for more complex situations on a medium computer. [ABSTRACT FROM AUTHOR]

In this paper we propose an expression for the entanglement entropy of several intervals in a stationary state of a free, translational invariant Hamiltonian in a fermionic chain. We check numerically the accuracy of our proposal and conjecture a formula for the asymptotic behavior of principal submatrices of a Toeplitz matrix. [ABSTRACT FROM AUTHOR]

This paper introduces a special nonlinear generalized coprime array, namely circular partial coprime array (CPCA). The antennas of CPCA are distributed on a circle, but after the antenna positions of CPCA are orthogonally projected to a certain baseline, the antenna spacing satisfies the coprime spacing characteristic of linear coprime array (LCA). Because the CPCA signal cannot be equivalent to the virtual difference coarray signal of LCA after vectoring the covariance matrix of incoming signals, the existing direction-of-arrival (DOA) estimation ideas of LCA model will be invalid in CPCA model. To make full use of the coprime information of CPCA model, the coprime interpolation method which can transform the steering manifolds of CPCA into the steering manifolds of expected LCA (ELCA) will be proposed. To alleviate the negative impact of interpolation error caused by coprime interpolation method on DOA estimation accuracy of CPCA model, a sparse Hermitian Toeplitz matrix (HTM) related to the spatial smoothed matrix will be recovered by the nuclear norm optimization model, where recovering the sparse part of HTM is equivalent to fitting the virtual hole data. Therefore, the recovered HTM can improve the virtual degree of freedom (DOF) of ELCA, and the high-precise DOA estimation can be obtained by combining the recovered HTM and MUSIC spectrum. In addition, the influence of inevitable interpolation error on the CPCA signal model is quantitatively analysed. Finally, simulation experiments verify the effectiveness and compatibility of the coprime interpolation method, and show the estimation performance advantages of the proposed method in CPCA model. [ABSTRACT FROM AUTHOR]

The rational covariance extension problem to determine a rational spectral density given a finite number of covariance lags can be seen as a matrix completion problem to construct an infinite-dimensional positive-definite Toeplitz matrix the northwest corner of which is given. The circulant rational covariance extension problem considered in this paper is a modification of this problem to partial stochastic realization of periodic stationary process, which are better represented on the discrete unit circle \BBZ 2N rather than on the discrete real line \BBZ . The corresponding matrix completion problem then amounts to completing a finite-dimensional Toeplitz matrix that is circulant. Another important motivation for this problem is that it provides a natural approximation, involving only computations based on the fast Fourier transform, for the ordinary rational covariance extension problem, potentially leading to an efficient numerical procedure for the latter. The circulant rational covariance extension problem is an inverse problem with infinitely many solutions in general, each corresponding to a bilateral ARMA representation of the underlying periodic process. In this paper, we present a complete smooth parameterization of all solutions and convex optimization procedures for determining them. A procedure to determine which solution that best matches additional data in the form of logarithmic moments is also presented. [ABSTRACT FROM PUBLISHER]

We address the problem of fast system analysis and controller synthesis for heterogeneous subsystems interconnected in a loop. Such distributed systems have state-space realizations with a special matrix structure, which we show to be a generalization of the Sequentially Semi-Separable (SSS) structure employed in recent research on systems interconnected in a Cartesian array. By extending the \cal O(N) structure-preserving SSS arithmetic to the matrices induced by this circular type of interconnection, we introduce a new cyclic SSS matrix structure and arithmetic which leads to fast and efficient procedures for linear computational complexity optimal distributed controller synthesis for arbitrarily heterogeneous subsystems connected in a loop. In the homogeneous case, where all subsystems are identical, the computational complexity reduces to \cal O(1) and an interesting relationship with the infinite case is demonstrated. CSSS matrices can be non-Toeplitz (as compared to circulant matrices), a great expansion of domain, but when restricted to be Toeplitz (and thus circulant), the arithmetic reduces to \cal O(1), a huge cost-savings. The procedures are demonstrated on two computational examples, using a freely available MATLAB toolbox implementation of these algorithms. [ABSTRACT FROM AUTHOR]

Abstract: The problem of scattering of plane monochromatic elastic waves on an isolated heterogeneous inclusion of arbitrary shape is considered. Volume integral equations for elastic displacements in heterogeneous media are used for reducing this problem to the region occupied by the inclusion. Discretization of this equation is carried out by Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problem are calculated in explicit analytical forms. For regular grids of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and the matrix–vector products can be calculated by the Fast Fourier Transform technique. The latter strongly accelerates the process of the iterative solution of the discretized problem. Elastic displacements and differential cross-sections of a homogeneous spherical inclusion are calculated for longitudinal and transversal incident waves of various wave lengths. The numerical results are compared with exact solutions. The displacement fields and differential cross-sections of a cylindrical inclusion are calculated for incident fields of different directions with respect to the cylinder axis. [Copyright &y& Elsevier]

Abstract: The paper establishes error orders for integral limit approximations to the traces of products of Toeplitz matrices generated by integrable real symmetric functions defined on the unit circle. These approximations and the corresponding error bounds are of importance in the statistical analysis of discrete-time stationary processes: asymptotic distributions and large deviations of Toeplitz type random quadratic forms, estimation of the spectral parameters and functionals, etc. [Copyright &y& Elsevier]

Abstract: We study two specific symmetric random block Toeplitz (of dimension ) matrices, where the blocks (of size ) are (i) matrices with i.i.d. entries and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after scaling by ) when (a) is fixed and (b) is fixed and (c) and go to simultaneously. Further, the LSDs obtained in (a) and (b) coincide with those in (c) when or respectively tends to infinity. This limit in (c) is the semicircle law in Case (i). In Case (ii), the limit is related to the limit of the random symmetric Toeplitz matrix as obtained by and . [Copyright &y& Elsevier]

Structured covariances occurring in spectral analysis, filtering and identification need to be estimated from a finite observation record. The corresponding sample covariance usually fails to possess the required structure. This is the case, for instance, in the Byrnes–Georgiou–Lindquist THREE-like tunable, high-resolution spectral estimators. There, the output covariance \Sigma of a linear filter is needed to initialize the spectral estimation technique. The sample covariance estimate \mathhat \Sigma , however, is usually not compatible with the filter. In this paper, we present a new, systematic way to overcome this difficulty. The new estimate \Sigma \circ is obtained by solving an ancillary problem with an entropic-type criterion. Extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances. [ABSTRACT FROM AUTHOR]

Abstract: This paper can be thought of as a remark of , where the authors studied the eigenvalue distribution of random block Toeplitz band matrices with given block order . In this paper, we will give explicit density functions of when the bandwidth grows slowly. In fact, these densities are exactly the normalized one-point correlation functions of Gaussian unitary ensemble (GUE for short). The series can be seen as a transition from the standard normal distribution to semicircle distribution. We also show a similar relationship between GOE and block Toeplitz band matrices with symmetric blocks. [Copyright &y& Elsevier]

Abstract: The work is devoted to the calculation of static elastic fields in 3D-composite materials consisting of a homogeneous host medium (matrix) and an array of isolated heterogeneous inclusions. A self-consistent effective field method allows reducing this problem to the problem for a typical cell of the composite that contains a finite number of the inclusions. The volume integral equations for strain and stress fields in a heterogeneous medium are used. Discretization of these equations is performed by the radial Gaussian functions centered at a system of approximating nodes. Such functions allow calculating the elements of the matrix of the discretized problem in explicit analytical form. For a regular grid of approximating nodes, the matrix of the discretized problem has the Toeplitz properties, and matrix–vector products with such matrices may be calculated by the fast fourier transform technique. The latter accelerates significantly the iterative procedure. First, the method is applied to the calculation of elastic fields in a homogeneous medium with a spherical heterogeneous inclusion and then, to composites with periodic and random sets of spherical inclusions. Simple cubic and FCC lattices of the inclusions which material is stiffer or softer than the material of the matrix are considered. The calculations are performed for cells that contain various numbers of the inclusions, and the predicted effective constants of the composites are compared with the numerical solutions of other authors. Finally, a composite material with a random set of spherical inclusions is considered. It is shown that the consideration of a composite cell that contains a dozen of randomly distributed inclusions allows predicting the composite effective elastic constants with sufficient accuracy. [Copyright &y& Elsevier]

Abstract: In the work, a numerical method for calculation of electro and thermo static fields in matrix composite materials is considered. Such materials consist of a regular or random set of isolated inclusions embedded in a homogeneous background medium (matrix). The proposed method is based on fast calculation of fields in a homogeneous medium containing a finite number of isolated inclusions. By the solution of this problem, the volume integral equations for the fields in heterogeneous media are used. Discretization of these equations is carried out by Gaussian approximating functions that allow calculating the elements of the matrix of the discretized problem in explicit analytical forms. If the grid of approximating nodes is regular, the matrix of the discretized problem proves to have the Toeplitz structure, and the matrix–vector product with such matrices can be calculated by the Fast Fourier Transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. In the case of an infinite medium containing a homogeneous in space random set of inclusions, our approach combines a self-consistent effective field method with the numerical solution of the conductivity problem for a typical cell. The method allows constructing detailed static (electric or temperature) fields in the composites with inclusions of arbitrary shapes and calculating effective conductivity coefficients of the composites. Results are given for 2D and 3D-composites and compared with the existing exact and numerical solutions. [ABSTRACT FROM AUTHOR]

Abstract: This paper explores a seemingly counter-intuitive idea: the possibility of accelerating the solution of certain linear equations by adding even more equations to the problem. The basic insight is to trade-off problem size by problem structure. We test this idea on Toeplitz equations, in which case the expense of a larger set of equations easily leads to circulant structure. The idea leads to a very simple iterative algorithm, which works for a certain class of Toeplitz matrices, each iteration requiring only two circular convolutions. In the symmetric definite case, numerical experiments show that the method can compete with the preconditioned conjugate gradient method (PCG), which achieves performance. Because the iteration does not converge for all Toeplitz matrices, we give necessary and sufficient conditions to ensure convergence (for not necessarily symmetric matrices), and suggest an efficient convergence test. In the positive definite case we determine the value of the free parameter of the circulant that leads to the fastest convergence, as well as the corresponding value for the spectral radius of the iteration matrix. Although the usefulness of the proposed iteration is limited in the case of ill-conditioned matrices, we believe that the results show that the problem size/problem structure trade-off deserves further study. [Copyright &y& Elsevier]

In factor analysis, which is used for example in econometrics, by definition the number of latent variables has to exceed the number of factor variables. The associated transfer function matrix has more rows than columns, and when the factor variables are independent zero mean white noise sequences and the transfer function matrix is stable, then the output spectrum is singular. While a related paper focusses on the properties of such a nonsquare transfer function matrix, in this paper, we explore a number of properties of the spectral matrix and associated covariance sequence. In particular, a zero free minimum degree spectral factor can be computed with a finite number of rational calculations from the spectrum (in contrast to typical spectral factor calculations), assuming the spectrum fulfills a generic condition. Application of the result to Kalman filtering is indicated, and presentation of the results is also achieved using finite block Toeplitz matrices with entries obtained from the covariance of the latent variable vector. [ABSTRACT FROM AUTHOR]

Abstract: The problem of computing an approximate solution of an overdetermined system of linear equations is considered. The usual approach to the problem is least squares, in which the 2-norm of the residual is minimized. This produces the minimum variance unbiased estimator of the solution when the errors in the observations are independent and normally distributed with mean 0 and constant variance. It is well known, however, that the least squares solution is not robust if outliers occur, i.e., if some of the observations are contaminated by large error. In this case, alternate approaches have been proposed which judge the size of the residual in a way that is less sensitive to these components. These include the Huber M-function, the Talwar function, the logistic function, the Fair function, and the norm. New algorithms are proposed to compute the solution to these problems efficiently, in particular, when the matrix A has small displacement rank. Matrices with small displacement rank include matrices that are Toeplitz, block-Toeplitz, block-Toeplitz with Toeplitz blocks, Toeplitz plus Hankel, and a variety of other forms. For exposition, only Toeplitz matrices are considered, but the ideas apply to all matrices with small displacement rank. Algorithms are also presented to compute the solution efficiently when a regularization term is included to handle the case when the matrix of the coefficients is ill-conditioned or rank-deficient. The techniques are illustrated on a problem of FIR system identification. [Copyright &y& Elsevier]

When a covariance matrix with a Toeplitz structure is written as the sum of a singular one and a positive scalar multiple of the identity, the singular summand corresponds to the covariance of a purely deterministic component of a time-series whereas the identity corresponds to white noise—this is the Carathéodory—Fejér—Pisarenko (CFP) decomposition. In the present paper we study multivariable analogs for block-Toeplitz matrices as well as for matrices with the structure of state-covariances of finite-dimensional linear systems (which include block-Toeplitz ones). To this end, we develop theory which addresses questions of existence, uniqueness and realization of multivariable power spectra, possibly having deterministic components. We characterize state-covariances which admit only a deterministic input power spectrum, and we explain how to realize multivariable power spectra which are consistent with singular state covariances via decomposing the contribution of the singular part. We then show that multivariable decomposition of a state-covariance in accordance with a ‘deterministic component + white noise’ hypothesis for the input does not exist in general. We finally reinterpret the CFP-dictum and consider replacing the ‘scalar multiple of the identity’ by a covariance of maximal trace which is admissible as a summand. The summand can be either (block-)diagonal corresponding to white noise or have a ‘short-range correlation structure’ correponding to a moving average component. The trace represents the maximal variance/energy that can be accounted for by a process at the input (e.g., noise) with the aforementioned structure, and this maximal solution can be computed via convex optimization. The decomposition of covariances and spectra according to the range of their time-domain correlations is an alternative to the CFP-dictum with potentially great practical significance. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of synchronization errors and their effects on feedback loops that are closed over communication networks. It is assumed that the feedback loop consists of two discrete-time systems, the clock frequency ratio of which has a special rational form, and is close to one. A Toeplitz matrix approach is taken to model the input/output relationship of the arising feedback systems. Based on this representation, the effect of synchronization errors on stability of the feedback systems is analyzed. It is shown that stability of the synchronized feedback loop can not guarantee stability of the loop in the presence of synchronization errors. Necessary and sufficient stability conditions are derived. [ABSTRACT FROM AUTHOR]

This note deals with generating a series of random matrices uniformly and independently from contractive lower triangular block Toeplitz (LTBT) matrices. All the contractive LTBT matrices are parameterized nonlinearly but recursively in a closed-form by a sequence of contractive unstructured matrices. Moreover, the conditional probability density function (PDF) is obtained in a closed-form for all the matrix parameters, as well as the squares of their singular values. It has been made clear that the conditional PDFs of the matrix parameters are unitarily invariant. A procedure is developed to accurately produce the required random matrix samples. [ABSTRACT FROM AUTHOR]

• A three-level linearized finite difference scheme is presented. • The unique solvability and boundedness of solution of the difference scheme are analyzed by the discrete energy method. • The difference scheme is unconditionally stable and second-order accurate in time and space under l ∞-norm. • The difference scheme (i.e. a linear system) can be solved effciently by complex conjugate gradient method In this paper, we propose a three-level linearized finite difference scheme for the high-dimensional nonlinear Ginzburg–Landau equation with fractional Laplacian. The Crank–Nicolson scheme is used for time discretization, and the fractional Laplacian is discretized by the fractional centered difference scheme. The proposed difference scheme (i.e. a linear system) can be solved efficiently by fast Fourier transform (FFT) and complex conjugate gradient method, since the coefficient matrix is a multi-level block Toeplitz matrix. Furthermore, we analyze the unique solvability and boundedness of solution of the difference scheme by the discrete energy method. It is also proved that the difference scheme is unconditionally stable and second-order accurate in time and space with respect to l ∞ -norm. Finally, several numerical examples are provided to validate the theoretical results. [ABSTRACT FROM AUTHOR]

• We propose a new fourth-order dissipation-preserving difference scheme for space fractional nonlinear damped wave equations. • Our proposed fourth-order scheme can generate the full toeplitz matrix, it is convenient for us to speed up the matrix-vector multiplication by fast Fourier transform. • We obtain the L 2 and H α /2 norm estimates, which are discussed for two-dimensional (high-dimensional) space fractional nonlinear damped wave equations. • We obtain the convergence analysis of fully-discrete scheme without any restriction on step-size ratio. • Our proposed scheme performs more robust than some existing schemes for conserving dissipation-preserving law. In this paper, we numerically investigate the space fractional nonlinear damped wave equation. We construct a novel high-accuracy dissipation-preserving finite difference scheme by using the new fourth-order fractional central difference operator. Thanks to the toeplitz-like differentiation matrix, we further raise the computation efficiency of the proposed scheme by fast Fourier transform. Moreover, we obtain the error estimate of our proposed scheme in L 2 and H α /2 (1 < α ≤ 2) norm, respectively. Finally, we verify the discrete dissipation-preserving law and convergence of the proposed scheme by one- and two-dimensional numerical experiments in long-time observation. [ABSTRACT FROM AUTHOR]

We discuss the process convergence of the time dependent fluctuations of linear eigenvalue statistics of band Toeplitz matrices with independent Brownian motion entries, as the dimension of matrix goes to infinity. Here the band width b n of the n × n Toeplitz matrix goes to infinity with n at the rate of o (n). [ABSTRACT FROM AUTHOR]

Traditional blind restoration model for super-resolution image based on neural network is easy to fall into the problem of local minimum, so a blind restoration model for super-resolution image based on chaotic neural network is constructed. Firstly, the noise of degraded image is removed by using wavelet transform to reduce the influence of the noise of degraded image on the result of blind restoration. Then a simplified chaotic neural network model is constructed by introducing transient chaos and time-varying increment into chaotic neural network. The gray value of image is taken as the input of the network. Two Toeplitz matrices are generated by point spread function and Laplace operator. The connection weights and bias inputs of chaotic neural network are calculated by generating matrix. Iterative degradation of image gray is updated continuously according to the connection weights and bias outputs. Degree value is used to judge whether the allowable error value of network convergence meets the requirement, and when it meets the requirement, the blind restoration of super-resolution image will be output. The experimental results show that the mean values of blind restoration time, image sharpness and energy consumption are 9.273 ms, 99.045% and 118.524 J, respectively. The restoration performance of the model is good, and the blind super-resolution image has the highest similarity with the original image. [ABSTRACT FROM AUTHOR]