Band Matrix Research Articles

On Spectral Properties of Compact Toeplitz Operators on Bergman Space with Logarithmically Decaying Symbol and Applications to Banded Matrices

Abstract Let $L^2({{\mathbb{D}}})$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a({{\mathbb{D}}})$ be the Bergman space, that is, the (closed) subspace of analytic functions in $L^2({{\mathbb{D}}})$. $P_+$ stays for the orthogonal projection going from $L^2({{\mathbb{D}}})$ to $L^2_a({{\mathbb{D}}})$. For a function $\varphi \in L^\infty ({{\mathbb{D}}})$, the Toeplitz operator $T_\varphi : L^2_a({{\mathbb{D}}})\to L^2_a({{\mathbb{D}}})$ is defined as $$\begin{align*} & T_\varphi f=P_+\varphi f, \quad f\in L^2_a({{\mathbb{D}}}). \end{align*}$$The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is, $$\begin{align*} & \varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0, \end{align*}$$where $z=re^{i\theta }$ and $\varphi _1$ is a continuous (or piece-wise continuous) function on the unit circle. The result is applied to the spectral analysis of banded (including Jacobi) matrices.

Generalized finite difference method for electroelastic analysis of three-dimensional piezoelectric structures

This short communication makes the first attempt to apply the generalized finite difference method (GFDM), a newly-developed meshless collocation method, for the numerical solutions of three-dimensional (3D) piezoelectric problems. In the present method, the entire computational domain is divided into a set of overlapping subdomains in which the local Taylor series expansion and moving-least square approximation are applied to construct the local systems of linear equations. By satisfying the coupled mechanical and electrical governing equations, a sparse and banded stiffness matrix can be established which makes the method very attractive for large-scale engineering simulations. Preliminary numerical experiments are presented to demonstrate the applicability and accuracy of the present method, where the results obtained are compared with the analytical solutions with very good agreement.

Solution of the symmetric band partial inverse eigenvalue problem for the damped mass spring system

ABSTRACT The structured partial quadratic inverse eigenvalue problem (SPQIEP) is to construct the structured quadratic matrix polynomial using the partial eigendata. The structures arising in physical applications include symmetry, band (tridiagonal, diagonal, pentagonal) etc. The construction of the structured matrix polynomial is the most difficult aspect of this problem and the research on structured inverse eigenvalue problem is rare. In this paper, the symmetric band partial quadratic inverse eigenvalue problem (SBPQIEP) for the damped mass spring system is considered. This problem concerns in finding the symmetric band matrices , and C with bandwidth p from m ( ) prescribed eigenpairs so that the corresponding quadratic matrix polynomial has the given eigenpairs as its eigenvalues and eigenvectors. In general, SBPQIEP is very hard to solve due to the additional band structure constraint. We propose a novel method, based on the matrix-vectorization and Kronecker product of matrices for solving this problem. Furthermore, explicit expressions for general solutions are presented. Numerical experiments on a spring mass problem are presented to illustrate the applicability and the practical usefulness of the proposed method.

Eigenvalue Clusters of Large Tetradiagonal Toeplitz Matrices

Toeplitz matrices are typically non-Hermitian and hence they evade the well-elaborated machinery one can employ in the Hermitian case. In a pioneering paper of 1960, Palle Schmidt and Frank Spitzer showed that the eigenvalues of large banded Toeplitz matrices cluster along a certain limiting set which is the union of finitely many closed analytic arcs. Finding this limiting set nevertheless remains a challenge. We here present an algorithm in the spirit of Richard Beam and Robert Warming that reduces testing O(N^2) points in the plane for membership in the limiting set by testing only O(N) points along a one-dimensional curve. For tetradiagonal Toeplitz matrices, we describe all types of the limiting sets, we classify their exceptional points, and we establish asymptotic formulas for the analytic arcs near their endpoints.

Multistage Linear Gauss Pseudospectral Method for Piecewise Continuous Nonlinear Optimal Control Problems

This article aims at proposing a multistage linear Gauss pseudospectral method (MS-LGPM) for solving the piecewise continuous nonlinear optimal control problem (OCP) with interior-point constraints and terminal constraints. First, the first-order necessary conditions for the multistage nonlinear OCP are derived, and a typical multipoint boundary value problem is obtained. Second, the original problem can be solved iteratively by integral prediction and quasi-linearization. Then Lagrange interpolation polynomials are used to approximate the state, control, and costate variables, to transfer those differential equations into a set of linear algebraic equations. Therefore, the control update coming closer to the optimal solution can be derived in an analytical manner. Additionally, those linear algebraic equations are formulated in regular banded matrix forms to make the code as concise as possible. Finally, the proposed method is applied to the midcourse guidance design for a dual-pulse missile to evaluate its performance. Simulation results show that the proposed method performs well in providing the optimal control with high accuracy and computational efficiency. Furthermore, a comparison with other typical guidance method and Monte Carlo simulations are also provided. Results demonstrate, even with some random uncertainties, the proposed method still has strong robustness and superior performance.

Difference sequence spaces based on Lucas band matrix and modulus function

In this work, we define some difference sequence spaces based upon Lucas band matrix and associated with the idea of sequence of modulus functions and then establish that aforementioned spaces are BK-spaces of non absolute type.We further investigate that our spaces are linearly isomorphic to the space l(p), where $$l(p)=\{x=(x_{k}) \in w:\sum _{k}|x_{k}|^{p_{k}}<\infty \}$$ . Moreover, we demonstrate inclusion relationship between newly defined spaces as well as establish certain geometrical properties.

A survey on the spectra of the difference operators over the Banach space c

Recently, due to its numerous applications, the spectra of the bounded operators over Banach spaces have been studied extensively. This work aims to collect some of the investigations on the spectra of difference operators or matrices on the Banach space c in the literature and provide a foundation for related problems. To the best of our investigations, the problem has been solved over the sequence space c so far up to maximum order 2. In the present work, the fine spectra of the difference operator $$\Delta ^m, m\in \mathbb {N}$$ on c have been computed. The generalized difference operator $$\Delta ^m$$ on the Banach space c is defined by $$(\Delta ^mx)_k= \sum _{i=0}^m(-1)^i\left( {\begin{array}{c}m\\ i\end{array}}\right) x_{k-i},\;k=0,1,2,3,\dots $$ with $$ x_{k} = 0$$ for $$k<0$$ . Indeed, the operator $$\Delta ^m$$ is represented by an $$(m+1)$$ -th band matrix which generalizes several difference operators such as $$\Delta ,\Delta ^2,B(r,s)$$ and B(r, s, t) etc, under different limiting conditions. Initially, we provide some essential background results on the linearity and boundedness of the backward difference operator $$\Delta ^m$$ . Finally, the sets for the spectrum and fine spectra such as the point spectrum, the continuous spectrum and the residual spectrum of the defined operator on the space c have been computed. The geometrical interpretation for the spectral subdivisions of the above difference operator is also incorporated.

Localized method of fundamental solutions for two- and three-dimensional transient convection-diffusion-reaction equations

This paper investigates the use of the localized method of fundamental solutions (LMFS) for the numerical solution of general transient convection-diffusion-reaction equation in both two-(2D) and three-dimensional (3D) materials. The method is developed as a generalization of the author's earlier work on Laplace's equation to transient convection-diffusion-reaction equation. The popular Crank-Nicolson (CN) time-stepping technology is adopted to perform the temporal simulations. The LMFS approach is then introduced for solving the resulting inhomogeneous boundary value problems, where a pseudo-spectral Chebyshev collocation scheme (CCS) is employed for the approximation of the corresponding particular solutions. As compared with the classical MFS and boundary element method (BEM), the present CN-CCS-LMFS approach produces sparse and banded stiffness matrix which makes the method possible to perform large-scale dynamic simulations. Several benchmark numerical examples are presented to demonstrate the efficiency and feasibility of the present method.

On certain difference operators and their explicitly inverses

In recent years, a number of extensive applications of difference operators through sequence spaces have been developed. The most crucial application is being used in the study of functional analysis, operator theory and matrix theory. In this context, the present article makes an attempt to provide a survey on various difference operators and unify them by introducing two m+ 1-th sequential band matrices. The purpose of this work is also to extend the determination of their inverses and derive an adaptive recursive free formula for matrix inversions. We provide two relevant formulas for inversion of m + 1-th sequential lower and upper band matrices. Subsequently, the idea is being applied to develop a new explicitly formula for matrix inversion.

Indoor Direct Positioning With Imperfect Massive MIMO Array Using Measured Near-Field Channels

In this article, we focus on indoor direct positioning using a practical massive multiple-input and multiple-output (MIMO) array with imperfections. First, the imperfect array needs to be calibrated in an anechoic chamber from the view of array signal processing. This brings inconveniences due to the large size of massive MIMO systems. To solve this problem, we propose onsite calibration, where a banded calibration matrix, calibrating both phase-gain error and mutual coupling, can be estimated by performing least-squares on the measure near-field channels, i.e., the site survey data. The feasibility of this calibration method can be explained by that the non-line-of-sight (NLOS) channel is able to be modeled using Rayleigh fading; i.e., the sum of the multipath components can be treated as Gaussian noise. Then, to estimate the user position, we propose a new two-stage direct positioning scheme. In the first stage, the coarse position is obtained via global search-based conventional beamforming (CBF) using a global calibration matrix. In the second stage, fine positioning is achieved via local search-based multiple signal classification (MUSIC) using a local calibration matrix. The proposed method is computationally efficient, insensitive to model order selection, free of dense site survey, and robust to partial line-of-sight (LOS) blocking. Finally, the proposed method is verified by practical measurements from a distributed array (DA), uniform linear array (ULA), and uniform rectangular array (URA), and compared with a deep learning positioning-method. The results show that the DA with the proposed method performs the best, achieving sub-centimeter accuracy (typically below 5 mm).

Stress analysis of elastic bi-materials by using the localized method of fundamental solutions

&lt;abstract&gt; &lt;p&gt;The localized method of fundamental solutions belongs to the family of meshless collocation methods and now has been successfully tried for many kinds of engineering problems. In the method, the whole computational domain is divided into a set of overlapping local subdomains where the classical method of fundamental solutions and the moving least square method are applied. The method produces sparse and banded stiffness matrix which makes it possible to perform large-scale simulations on a desktop computer. In this paper, we document the first attempt to apply the method for the stress analysis of two-dimensional elastic bi-materials. The multi-domain technique is employed to handle the non-homogeneity of the bi-materials. Along the interface of the bi-material, the displacement continuity and traction equilibrium conditions are applied. Several representative numerical examples are presented and discussed to illustrate the accuracy and efficiency of the present approach.&lt;/p&gt; &lt;/abstract&gt;

Bayesian estimation of the stochastic volatility model with double exponential jumps

This paper generalizes the stochastic volatility model to allow for the double exponential jumps. To derive the jumps and time-varying volatility in returns, we implement an efficient Markov chain Monte Carlo approach based on the band and sparse matrix algorithms used in Chan and Hsiao (SSRN Electron J., 2013, https://doi.org/10.2139/ssrn.2359838 ) to estimate this model. We illustrate the the methodology using the daily data for the Shanghai Composite Index, Hangseng Index, Nikkei 225 Index and Kospi Index. We find that the stochastic volatility model with double exponential jumps provide better fitness in sample period.

Quintic Spline functions and Fredholm integral equation

A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we first formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.

Parallel gaussian elimination of symmetric positive definite band matrices for shared-memory multicore architectures

This study presents a new parallel Gaussian elimination approach for symmetric positive definite band systems. For each task, the appropriate start time and adequate processor are determined. Unnecessary dependencies between tasks are eliminated. Simultaneously, all processors perform their associated tasks with precedence constraints under consideration. Our main goal is to obtain a high degree of parallelism by balancing the load of processors and reducing the total idle and parallel execution times. The theoretical lower bounds for parallel execution time and number of processors required to execute the precedence graph at an optimal time are also computed. The validity of our investigation is confirmed by carrying out several experiments on a shared-memory multicore architecture using OpenMP. Practical results prove the efficiency of the proposed method.

ECG Baseline Estimation and Denoising With Group Sparse Regularization

Baseline wander (BW) and electrocardiogram (ECG) noise reduction play an important role in ECG data analysis and disease diagnosis. This article introduces a sparse optimization method, which takes into account the group sparse characteristics of the signal, and combines low-pass filter to denoise the ECG signal and estimate the baseline. Derived from the classic total variation (TV) denoising method, a denoising method considering the structural characteristics of ECG signals is proposed. This method uses a band matrix to represent the sparse optimization problem, and adopts majorization-minimization (MM) algorithm to optimize the solution of the convergence problem. Through data comparison and detailed analysis, we first compares the method with two TV denoising methods. Then, the proposed method is validated in the MIT-BIH arrhythmia database of ECG signals, and compared with nonlocal means (NLM) and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) methods. The simulation experiment results show that the proposed algorithm has lower root mean square error (RMSE) and higher signal-to-noise ratio improvement (SNR_imp).

The intravarietal relationship among different accessions of Dolichos biflorus was checked based on protein profiling using SDS-PAGE.

In this study, the phylogenetic relationship within the selected Eleven Indian (Dolichos biflorus (horse gram) varieties was analyzed for total soluble seed protein. Twenty-five bands were documented through SDS PAGE based on 100 seed weight of each variety and were studied for genetic diversity. Jaccard’s similarity matrix was acquired and used in UPGMA cluster analysis based on the polymorphism generated by the presence (1) or absence (0) of protein bands. Thus, the dendrogram showed four major groups that corre-spond to an earlier study on polymorphisms of 11 accessions of Indian Dolichos. Signifi-cant correspondence between the clustering pattern and the pedigree was observed; thus, a high genetic diversity could be kept within the Dolichos varieties. A similarity matrix among the targeted genotypes and phylogenetic analysis is considered a unique feature in the present work. Therefore, the current investigation was carried out to analyse Protein diversity of unexplored Dolichos genotypes at the molecular level, construct a dendro-gram based on similarity band matrix and generate efficiency in genetic divergence analy-sis among Dolichos. This study underlines the importance of using genetic diversity based in the Dolichos breeding program.

Discrete spectra for some complex infinite band matrices

Under suitable assumptions the eigenvalues for an unbounded discrete operator \(A\) in \(l_2\), given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let \[\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},\] where \({\rm Lim}_{n\to \infty} \lambda_n\) is the set of all limit points of the sequence \((\lambda_n)\) and \(A_n\) is a finite dimensional orthogonal truncation of \(A\). The aim of this article is to provide the conditions that are sufficient for the relations \(\sigma(A) \subset \Lambda(A)\) or \(\Lambda (A) \subset \sigma (A)\) to be satisfied for the band operator \(A\).

An Unobserved Components Model of Total Factor Productivity and the Relative Price of Investment

This paper applies the common stochastic trends representation approach to the time series of total factor productivity and the relative price of investment to investigate the relationship between neutral technology and investment-specific technology. The permanent and transitory movements in both series are estimated efficiently via MCMC methods using band matrix algorithms. The results indicate that total factor productivity and the relative price of investment are, each, well-represented by an integrated process of order one. In addition, their time series share a common trend component that we interpret as reflecting changes in General Purpose Technology. These results suggest that (1) the traditional view of assuming that neutral technology and investment-specific technology follow independent processes is not supported by the features of the time series and (2) advances in information and communication technologies are general purpose technological progress that drive the trend in aggregate TFP in the United States.

Almost convergent sequence spaces derived by the domain of quadruple band matrix

Abstract In this work, we construct the sequence spaces f(Q(r, s, t, u)), f 0(Q(r, s, t, u)) and fs(Q(r, s, t, u)), where Q(r, s, t, u) is quadruple band matrix which generalizes the matrices Δ3, B(r, s, t), Δ2, B(r, s) and Δ, where Δ3, B(r, s, t), Δ2, B(r, s) and Δ are called third order difference, triple band, second order difference, double band and difference matrix, respectively. Also, we prove that these spaces are BK-spaces and are linearly isomorphic to the sequence spaces f, f 0 and fs, respectively. Moreover, we give the Schauder basis and β, γ-duals of those spaces. Lastly, we characterize some matrix classes related to those spaces.

Sequence spaces derived by the triple band generalized Fibonacci difference operator

In this article we introduce the generalized Fibonacci difference operator mathsf{F}(mathsf{B}) by the composition of a Fibonacci band matrix F and a triple band matrix mathsf{B}(x,y,z) and study the spaces ell _{k}( mathsf{F}(mathsf{B})) and ell _{infty }(mathsf{F}(mathsf{B})). We exhibit certain topological properties, construct a Schauder basis and determine the Köthe–Toeplitz duals of the new spaces. Furthermore, we characterize certain classes of matrix mappings from the spaces ell _{k}(mathsf{F}(mathsf{B})) and ell _{infty }(mathsf{F}(mathsf{B})) to space mathsf{Y}in {ell _{infty },c_{0},c,ell _{1},cs_{0},cs,bs} and obtain the necessary and sufficient condition for a matrix operator to be compact from the spaces ell _{k}(mathsf{F}(mathsf{B})) and ell _{infty }(mathsf{F}(mathsf{B})) to mathsf{Y}in { ell _{infty }, c, c_{0}, ell _{1},cs_{0},cs,bs} using the Hausdorff measure of non-compactness.