Symmetric Toeplitz Matrix Research Articles

The determinant and a factorization of a Toeplitz matrix with some type of Horadam numbers entries

Abstract In this note, we define a symmetric Toeplitz matrix whose entries are from a particular Horadam sequence. We provide the determinant, the inverse and a factorization for such matrices and their special cases. These matrices are always nonsingular and their inverses are pentadiagonal matrices.

A smoothing spline algorithm to interpolate and predict the eigenvalues of matrices extracted from the sequence of preconditioned banded symmetric Toeplitz matrices

<abstract><p>Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about $ \Omega $ and $ \Phi $, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices $ T_{n}^{-1}(\Phi) T_n(\Omega) $, given a monotonic ratio $ r = \Omega/\Phi $. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices.</p></abstract>

Matrix splitting preconditioning based on sine transform for solving two-dimensional space-fractional diffusion equations

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(n2logn), where n2 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.

Equality of MNK and Aitken estimations of the parameter of the linear regression model, when the covariance matrix of the deviations is a symmetric Toeplitz matrix of the general form

Equality of MNK and Aitken estimations of the parameter of the linear regression model, when the covariance matrix of the deviations is a symmetric Toeplitz matrix of the general form

Pairs of Symmetric and Skew-Symmetric Toeplitz Matrices with Identical Squares

Certain sets consisting of pairs of symmetric and skew-symmetric Toeplitz matrices with identical squares are described.

A Graduated Non-Convexity Technique for Dealing Large Point Spread Functions

This paper focuses on reducing the computational cost of a GNC Algorithm for deblurring images when dealing with full symmetric Toeplitz block matrices composed of Toeplitz blocks. Such a case is widespread in real cases when the PSF has a vast range. The analysis in this paper centers around the class of gamma matrices, which can perform vector multiplications quickly. The paper presents a theoretical and experimental analysis of how γ-matrices can accurately approximate symmetric Toeplitz matrices. The proposed approach involves adding a minimization step for a new approximation of the energy function to the GNC technique. Specifically, we replace the Toeplitz matrices found in the blocks of the blur operator with γ-matrices in this approximation. The experimental results demonstrate that the new GNC algorithm proposed in this paper reduces computation time by over 20% compared with its previous version. The image reconstruction quality, however, remains unchanged.

Toeplitz Determinants for a Certain Family of Analytic Functions Endowed with Borel Distribution

In this work, we derive coefficient bounds for the symmetric Toeplitz matrices T2(2), T2(3), T3(1), and T3(2), which are the known first four determinants for a new family of analytic functions with Borel distribution series in the open unit disk U. Further, some special cases of results obtained are also pointed.

The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix

This paper is concerned with the distance of a symmetric tridiagonal Toeplitz matrix $T$ to the manifold of similarly structured singular matrices, and with determining the closest matrix to $T$ in this manifold. Explicit formulas are presented, exploiting the analysis of the sensitivity of the spectrum of $T$ with respect to structure-preserving perturbations of its entries.

Algorithms for solving a class of real quasi-symmetric Toeplitz linear systems and its applications

<abstract><p>In this paper, fast numerical methods for solving the real quasi-symmetric Toeplitz linear system are studied in two stages. First, based on an order-reduction algorithm and the factorization of Toeplitz matrix inversion, a sequence of linear systems with a constant symmetric Toeplitz matrix are solved. Second, two new fast algorithms are employed to solve the real quasi-symmetric Toeplitz linear system. Furthermore, we show a fast algorithm for quasi-symmetric Toeplitz matrix-vector multiplication. In addition, the stability analysis of the splitting symmetric Toeplitz inversion is discussed. In mathematical or engineering problems, the proposed algorithms are extraordinarily effective for solving a sequence of linear systems with a constant symmetric Toeplitz matrix. Fast matrix-vector multiplication and a quasi-symmetric Toeplitz linear solver are proven to be suitable for image encryption and decryption.</p></abstract>

ПАРАЛЕЛЬНЕ ВИЗНАЧЕННЯ СПЕКТРУ СИМЕТРИЧНИХ ТЕПЛИЦЕВИХ МАТРИЦЬ З ПЕРЕТВОРЕННЯМИ ЛЕВІНСОНА-ДАРБІНА

Purpose. The paper investigates modern approaches to mathematical modeling of random fields using correlation matrices of superhigh dimensions. Methodology. Techniques for constructing surrogate models focused on reducing the high dimension of stochastic input spaces have been considered. A comparative analysis of modern numerical methods for searching for eigenvalues of filled matrices has been carried out with an assessment of the possibilities of parallelizing the computation process. To determine the range of the distribution spectrum of eigenvalues of symmetric Toeplitz matrices and subsequent localization of eigenvalues, the use of Gershgorin circles has been proposed. As test matrices, we used the generation of random vectors of arbitrary dimensions, followed by the formation of symmetric Toeplitz matrices on such vectors. Results. The studies have been carried out both for serial and parallel implementations of computational processes. When testing, the main tasks were aimed at conducting a comparative analysis of the accuracy of the solutions obtained, at estimating the coefficient of the ratio of the total number of partitioning problems to those that led to obtaining an eigenvalue. Scientific novelty. The use of a parallel procedure for separating and refining all matrix eigenvalues using a priority search for the smallest(s), largest(s) of eigenvalues has been justified. A software application has been developed for separating and refining all eigenvalues of symmetric Toeplitz filled matrices, which are formed in the course of mathematical modeling of random fields. Practical significance. The proposed software product can be used in numerical simulators for the implementation of mathematical models of groundwater flows and physicochemical processes of dissolution in deep aquifers.

Improved bisection eigenvalue method for band symmetric Toeplitz matrices

We apply a general bisection eigenvalue algorithm, developed forHermitian matrices with quasiseparable representations, to the particular caseof real band symmetric Toeplitz matrices. We show that every band symmetricToeplitz matrix $T_q$ with bandwidth $q$ admits the representation$T_q=A_q+H_q$, where the eigendata of $A_q$ are obtained explicitly andthe matrix $H_q$ has nonzero entries only in two diagonal blocksof size $(q-1)\times (q-1)$. Based on this representation, one obtainsan interlacing property of the eigenvalues of the matrix $T_q$ and the knowneigenvalues of the matrix $A_q$. This allows us to essentially improve the performance of thebisection eigenvalue algorithm. We also present an algorithm tocompute the corresponding eigenvectors.

Patterned random matrices: deviations from universality

We investigate the level spacing distribution for three ensembles of real symmetric matrices having additional structural constraint to reduce the number of independent entries to only in contrast to the for a real symmetric matrix of size n × n. We derive all the results analytically exactly for the matrices and show that spacing distribution display a range of behaviour based on the structural constraint. The spacing distribution of the ensemble of reverse circulant matrices with additional zeros is found to fall slower than exponential for larger spacing while that of symmetric circulant matrices has poisson spacings. The palindromic symmetric Toeplitz matrices on the other hand show level repulsion but the distribution is significantly different from Wigner. The behaviour of spacings for all the three ensembles clearly show the departure from universal result of Wigner distribution for real symmetric matrices. The deviation from universality continues in large n cases as well, which we study numerically.

Entries of the inverses of large positive definite Toeplitz matrices

This is an expository paper embarking on the asymptotic behaviorof the entries of the inverses of positive definite symmetric Toeplitzmatrices as the matrix dimension goes to infinity. We consider the behaviorof the entries in neighborhoods of the four corners as well as thedensity of the distribution of the entries over all of the inverse matrix.

Symmetric Toeplitz Matrices for a New Family of Prestarlike Functions

By making use of prestarlike functions, we introduce in this paper a certain family of normalized holomorphic functions defined in the open unit disk, and we establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices T2(2), T2(3), T3(2) and T3(1) for the functions belonging to this family. We also mention some known and new results that follow as special cases of our results.

Certain New Subclass of Multivalent Q-Starlike Functions Associated with Q-Symmetric Calculus

In our present investigation, we extend the idea of q-symmetric derivative operators to multivalent functions and then define a new subclass of multivalent q-starlike functions. For this newly defined function class, we discuss some useful properties of multivalent functions, such as the Hankel determinant, symmetric Toeplitz matrices, the Fekete–Szego problem, and upper bounds of the functional ap+1−μap+12 and investigate some new lemmas for our main results. In addition, we consider the q-Bernardi integral operator along with q-symmetric calculus and discuss some applications of our main results.

Spectral Analysis for Preconditioning of Multi-Dimensional Riesz Fractional Diffusion Equations

In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which will generate symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient method converges linearly. The proposed method can be extended to multi-level Toeplitz matrices generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to demonstrate our new theoretical results in the literature and show the convergence performance of the proposed preconditioner that is better than other existing preconditioners.

Empirical Bayes identification of stationary processes and approximation of Toeplitz spectra

We study Statistical Consistency of an approximate subspace identification procedure for the infinite dimensional a posteriori model of a frequency estimation problem in an Empirical Bayesian framework. By first imposing a natural uniform prior probability density on the unknown frequencies, the estimation of the hyperparameters of the a priori distribution can be accomplished by a sequence of subspace identification techniques. These techniques exploit the special structure of the covariance matrix of the a posteriori process which is discovered by making a connection with classical results on energy concentration in deterministic signal processing. The convergence of the spectrum of the subspace estimates to the (nonrational) spectral density of the a posteriori process has an analytic counterpart in the approximation of symmetric positive definite Toeplitz matrices by submatrices of finite rank. This is proven by a weak-sense convergence theorem for Toeplitz spectra.

On Pairs of Symmetric Toeplitz Matrices Whose Squares Are Identical

On Pairs of Symmetric Toeplitz Matrices Whose Squares Are Identical

An upper bound on the minimum rank of a symmetric Toeplitz matrix completion problem

We consider a symmetric Toeplitz matrix completion problem, of which the matrix possesses special row and column structures. It has wide applications in diverse areas and is well-known to be computationally NP-hard. This note presents an upper bound on the objective of minimizing the rank of the symmetric Toeplitz matrix in the completion problem based on conclusions from the trigonometric moment problem and the semi-infinite problem. We prove that the upper bound is less than twice the number of the active constraints of the associated semi-infinite problem. Moreover, it is less than twice the number of linear constraints of the problem.

On the rational approximation of Markov functions, with applications to the computation of Markov functions of Toeplitz matrices

We investigate the problem of approximating the matrix function f(A) by r(A), with f a Markov function, r a rational interpolant of f, and A a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error 1 − r/f on the spectral interval of A. By minimizing this upper bound over all interpolation points, we obtain a new, simple, and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant r. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating r(A), where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for r following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of r leading to a small error, even in presence of finite precision arithmetic.