Tridiagonal Toeplitz Matrix Research Articles

Shell extremal eigenvalues of tridiagonal Toeplitz Matrices

The shell of a complex tridiagonal Toeplitz matrix is studied. Closed formulas for all quantities involved in its equation are presented. Necessary and sufficient conditions for a Toeplitz tridiagonal matrix to have shell extremal eigenvalues are given. Several, recently introduced, geometric quantities related to the shell are studied as measures of non-normality of these extremal eigenvalues of such matrices. These quantities are also proposed as measures of non-normality for the matrix itself.

Explicit potential function and fast algorithm for computing potentials in [formula omitted] conic surface resistor network

Resistor network research is of great importance, yet many resistor networks and their large-scale fast computations have not received sufficient attention. This paper proposes a new resistor network with idiosyncratic shape, i.e., a α×β conic surface (CS) resistor network that resembles the upper part of a three-dimensional Dirac function. Utilizing the Recursion Transform (RT-V) method of Tan, a recursive matrix equation model is constructed based on Kirchhoff’s law and nodal voltages, which contains the modified tridiagonal Toeplitz matrix. By using the orthogonal matrix transformation, the eigenvalues and eigenvectors of the modified tridiagonal Toeplitz are obtained. The discrete sine transform of the fourth type (DST−IV) is utilized to solve node voltages, while the explicit potential function is represented by the Chebyshev polynomials of the second kind. In addition, explicit potential functions for some special cases are provided, and the potential distribution is illustrated using dynamic three-dimensional graph. To achieve a rapid calculation of the potential, a fast algorithm based on the multiplication of DST-IV with a vector is proposed. In the end, analysis of computational efficiency for the explicit potential function and the fast algorithm are shown.

On a Nontrivial Situation Concerning the Pseudounitary Eigenvalues of a Positive Definite Matrix

Let Ip,q = Ip ⊕−Iq. Pseudounitary eigenvalues of a positive definite matrix A are the moduli of conventional eigenvalues of the matrix Ip,qA. They are invariants of pseudounitary *-congruences performed with A. For a fixed n = p+q, the sum of the squares σp,q of these numbers is a function of the parameter p. In general, its values for different p can differ very significantly. However, if A is the tridiagonal Toeplitz matrix with an entry a ≥ 2 on the principal diagonal and the entry −1 on the two neighboring diagonals, then σp,q has the same value for all p. This nontrivial fact is explained in the paper.

Analytical potential formulae and fast algorithm for a horn torus resistor network.

In this paper, a (u+1)×v horn torus resistor network with a special boundary is researched. According to Kirchhoff's law and the recursion-transform method, a model of the resistor network is established by the voltage V and a perturbed tridiagonal Toeplitz matrix. We obtain the exact potential formula of a horn torus resistor network. First, the orthogonal matrix transformation is constructed to obtain the eigenvalues and eigenvectors of this perturbed tridiagonal Toeplitz matrix; second, the solution of the node voltage is given by using the famous fifth kind of discrete sine transform (DST-V). We introduce Chebyshev polynomials to represent the exact potential formula. In addition, the equivalent resistance formulae in special cases are given and displayed by a three-dimensional dynamic view. Finally, a fast algorithm of computing potential is proposed by using the mathematical model, famous DST-V, and fast matrix-vector multiplication. The exact potential formula and the proposed fast algorithm realize large-scale fast and efficient operation for a (u+1)×v horn torus resistor network, respectively.

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.

Modeling One Dimensional Two-Cell Model with Tumor Interaction Using Krylov Subspace Methods

A brain tumor occurs when abnormal cells grow, sometimes very rapidly, into an abnormal mass of tissue. The tumor can infect normal tissue, so there is an interaction between healthy and infected cell. The aim of this paper is to propose some efficient and accurate numerical methods for the computational solution of one-dimensional continuous basic models for the growth and control of brain tumors. After computing the analytical solution, we construct approximations of the solution to the problem using a standard second order finite difference method for space discretization and the Crank-Nicolson method for time discretization. Then, we investigate the convergence behavior of Conjugate gradient and generalized minimum residual as Krylov subspace methods to solve the tridiagonal toeplitz matrix system derived.

Channel Time-Variation Suppression With Optimized Receive Beamforming for High-Mobility OFDM Downlink Transmissions

A channel time variation suppression scheme based on beamforming is proposed for high-mobility orthogonal frequency division multiplexing (OFDM) downlink transmissions. The residual channel time variation after Doppler shift compensation by finite resolution receive beamforming can be measured by the Doppler spread. The interchannel interference (ICI) due to the time-varying channel is then analyzed and the relation between the signal-to-interference ratio (SIR) and Doppler spread is given. Based on the derived Doppler spread and ICI, a beamforming network optimization scheme is proposed to reduce the channel time variation. A closed-form solution is first obtained to minimize the average ICI, which exploits the structure of a tridiagonal Toeplitz matrix. Next, an optimization problem is formulated to minimize the maximum Doppler spread to further verify the impact of average ICI on channel time variation. The sequential parametric convex approximation (SPCA) algorithm is exploited to solve the non-convex min-max problem. Numerical results verify the theoretical analysis.

Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation

We present a quantum algorithm to solve systems of linear equations of the form Ax = b , where A is a tridiagonal Toeplitz matrix and b results from discretizing an analytic function, with a circuit complexity of O (1/√ε, poly (log κ, log N )), where N denotes the number of equations, ε is the accuracy, and κ the condition number. The repeat-until-success algorithm has to be run O (κ/(1-ε)) times to succeed, leveraging amplitude amplification, and needs to be sampled O (1/ε 2 ) times. Thus, the algorithm achieves an exponential improvement with respect to N over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation, and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with 1/√ε circuits that can be run in parallel each with circuit complexity 1/√ ε instead of 1/ε. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.

DETERMINAN SUATU MATRIKS TOEPLITZ K-TRIDIAGONAL MENGGUNAKAN METODE REDUKSI BARIS DAN EKSPANSI KOFAKTOR

This paper discusses the determinants of a k-tridiagonal Toeplitz matrix using row reduction and cofactor expansion methods. The analysis was carried out recursively from the general form of the determinant of the tridiagonal Toeplitz matrix, the determinant of the 2-tridiagonal Toeplitz matrix, and the determinant of the 3-tridiagonal Toeplitz matrix. In the end, the general form of the determinant of the k-tridiagonal Toeplitz matrix is obtained.

INTERESTING DETERMINANTS AND INVERSES OF SKEW LOEPLITZ AND FOEPLITZ MATRICES

<abstract> In this paper, we show that there is an intimate relationship between Toeplitz matrix, tridiagonal Toeplitz matrix, the Fibonacci number, the Lucas number, and the Golden Ratio. We introduce skew Loeplitz and skew Foeplitz matrices and derive their determinants and inverses by construction. Specifically, the determiant of $ n\times n $ skew Loeplitz matrix can be expressed by the $ (n+1) $st Fibonacci number. The inverse of skew Loeplitz matrix is sparse and can be expressed by the $ n $th and $ (n+1) $st Fibonacci numbers. Similarly, the determinant of $ n\times n $ skew Foeplitz matrix also can be expressed by the $ (n+1) $st Lucas number. The inverse of skew Foeplitz matrix can be expressed by only seven elements with each element being the explicit expression of the Lucas or Fibonacci numbers. We also calculate the determinants and inverses of skew Lankel and skew Fankel matrices. </abstract>

Fast solvers for tridiagonal Toeplitz linear systems

Let A be a tridiagonal Toeplitz matrix denoted by $$A = {\text {Tritoep}} (\beta , \alpha , \gamma )$$ . The matrix A is said to be: strictly diagonally dominant if $$|\alpha |>|\beta |+|\gamma |$$ , weakly diagonally dominant if $$|\alpha |\ge |\beta |+|\gamma |$$ , subdiagonally dominant if $$|\beta |\ge |\alpha |+|\gamma |$$ , and superdiagonally dominant if $$|\gamma |\ge |\alpha |+|\beta |$$ . In this paper, we consider the solution of a tridiagonal Toeplitz system $$A\mathbf {x}= \mathbf {b}$$ , where A is subdiagonally dominant, superdiagonally dominant, or weakly diagonally dominant, respectively. We first consider the case of A being subdiagonally dominant. We transform A into a block $$2\times 2$$ matrix by an elementary transformation and then solve such a linear system using the block LU factorization. Compared with the LU factorization method with pivoting, our algorithm takes less flops, and needs less memory storage and data transmission. In particular, our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency. Then, we deal with superdiagonally dominant and weakly diagonally dominant cases, respectively. Numerical experiments are finally given to illustrate the effectiveness of our algorithms.

An explicit formula for the smoother weights of the Hodrick–Prescott filter

Abstract By applying the Sherman–Morrison–Woodbury (SMW) formula and a discrete cosine transformation matrix, De Jong and Sakarya [De Jong, R. M., and N. Sakarya. 2016. “The Econometrics of the Hodrick–Prescott Filter.” Review of Economics and Statistics 98 (2): 310–317] recently derived an explicit formula for the smoother weights of the Hodrick–Prescott filter. More recently, by applying the SMW formula and the spectral decomposition of a symmetric tridiagonal Toeplitz matrix, Cornea-Madeira [Cornea-Madeira, A. 2017. “The Explicit Formula for the Hodrick–Prescott Filter in Finite Sample.” Review of Economics and Statistics 99: 314–318] provided a simpler formula. This paper provides an alternative simpler formula for it and explains the reason why our approach leads to a simpler formula.

Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

SummaryIt is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a0 and on the two first off‐diagonals the constants a1(lower) and a−1(upper), which are all complex values, there exist closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. For example, for the 1D discrete Laplacian, this triple is (a0,a1,a−1)=(2,−1,−1). In the first part of this article, we consider a tridiagonal Toeplitz matrix of the same form (a0,aω,a−ω), but where the two off‐diagonals are positioned ω steps from the main diagonal instead of only one. We show that its eigenvalues and eigenvectors can also be identified in closed form and that interesting connections with the standard Toeplitz symbol are identified. Furthermore, as numerical evidences clearly suggest, it turns out that the eigenvalue behavior of a general banded symmetric Toeplitz matrix with real entries can be described qualitatively in terms of the symmetrically sparse tridiagonal case with real a0, aω=a−ω, ω=2,3,…, and also quantitatively in terms of those having monotone symbols. A discussion on the use of such results and on possible extensions complements the paper.

The envelope of tridiagonal Toeplitz matrices and block-shift matrices

The envelope of a square complex matrix is a spectrum encompassing region in the complex plane. It is contained in and is akin to the numerical range in the sense that the envelope is obtained as an infinite intersection of unbounded regions contiguous to cubic curves, rather than half-planes. In this article, the geometry and properties of the envelopes of special matrices are examined. In particular, symmetries of the envelope of a tridiagonal Toeplitz matrix are obtained, and the envelopes of block-shift matrices, Jordan blocks and 2×2 matrices are explicitly characterized.

Role of quantum coherence in shaping the line shape of an exciton interacting with a spatially and temporally correlated bath.

Kubo's fluctuation theory of line shape forms the backbone of our understanding of optical and vibrational line shapes, through such concepts as static heterogeneity and motional narrowing. However, the theory does not properly address the effects of quantum coherences on optical line shape, especially in extended systems where a large number of eigenstates are present. In this work, we study the line shape of an exciton in a one-dimensional lattice consisting of regularly placed and equally separated optical two level systems. We consider both linear array and cyclic ring systems of different sizes. Detailed analytical calculations of line shape have been carried out by using Kubo's stochastic Liouville equation (SLE). We make use of the observation that in the site representation, the Hamiltonian of our system with constant off-diagonal coupling J is a tridiagonal Toeplitz matrix (TDTM) whose eigenvalues and eigenfunctions are known analytically. This identification is particularly useful for long chains where the eigenvalues of TDTM help understanding crossover between static and fast modulation limits. We summarize the new results as follows. (i) In the slow modulation limit when the bath correlation time is large, the effects of spatial correlation are not negligible. Here the line shape is broadened and the number of peaks increases beyond the ones obtained from TDTM (constant off-diagonal coupling element J and no fluctuation). (ii) However, in the fast modulation limit when the bath correlation time is small, the spatial correlation is less important. In this limit, the line shape shows motional narrowing with peaks at the values predicted by TDTM (constant J and no fluctuation). (iii) Importantly, we find that the line shape can capture that quantum coherence affects in the two limits differently. (iv) In addition to linear chains of two level systems, we also consider a cyclic tetramer. The cyclic polymers can be designed for experimental verification. (v) We also build a connection between line shape and population transfer dynamics. In the fast modulation limit, both the line shape and the population relaxation, for both correlated and uncorrelated bath, show similar behavior. However, in slow modulation limit, they show profoundly different behavior. (vi) This study explains the unique role of the rate of fluctuation (inverse of the bath correlation time) in the sustenance and propagation of coherence. We also examine the effects of off-diagonal fluctuation in spectral line shape. Finally, we use Tanimura-Kubo formalism to derive a set of coupled equations to include temperature effects (partly neglected in the SLE employed here) and effects of vibrational mode in energy transfer dynamics.

On decomposition of k-tridiagonal ℓ-Toeplitz matrices and its applications

Abstract We consider a k-tridiagonal ℓ-Toeplitz matrix as one of generalizations of a tridiagonal Toeplitz matrix. In the present paper, we provide a decomposition of the matrix under a certain condition. By the decomposition, the matrix is easily analyzed since one only needs to analyze the small matrix obtained from the decomposition. Using the decomposition, eigenpairs and arbitrary integer powers of the matrix are easily shown as applications.

Nonlinear Schrödinger equations with multiple-well potential

We consider the stationary solutions for a class of Schrödinger equations with a N-well potential and a nonlinear perturbation. By means of semiclassical techniques we prove that the dominant term of the ground state solutions is described by a N-dimensional Hamiltonian system, where the coupling term among the coordinates is a tridiagonal Toeplitz matrix. In particular, in the limit of large focusing nonlinearity we prove that the ground state stationary solutions consist of N wavefunctions localized on a single well. Furthermore, we consider in detail the case of N=4 wells, where we show the occurrence of spontaneous symmetry-breaking bifurcation effect.

Approximate Analysis of Power Offset over Spatially Correlated MIMO Channels

Power offset is zero-order term in the capacity versus signal-to-noise ratio curve. In this paper, approximate analysis of power offset is presented to describe MIMO system with uniform linear antenna arrays of fixed length. It is assumed that the number of receive antenna is larger than that of transmit antenna. Spatially Correlated MIMO Channel is approximated by tri-diagonal toeplitz matrix. The determinant of tri-diagonal toeplitz matrix, which is fitted by elementary curve, is one of the key factors related to power offset. Based on the curve fitting, the determinant of tri-diagonal toeplitz matrix is mathematically tractable. Consequently, the expression of local extreme points can be derived to optimize power offset. The simulation results show that approximation above is accurate in local extreme points of power offset. The proposed expression of local extreme points is helpful to approach optimal power offset.

Explicit inverses of Toeplitz and associated matrices

We discuss Toeplitz and associated matrices which have simple explicit expressions for their inverses. We first review existing results and generalize these where possible, including matrices with hyperbolic and trigonometric elements. In Section 2 we discuss and generalize the Fiedler matrix. In Section 3 we give an analytic procedure for inverting any band Toeplitz matrix, in Section 4 we invert a tridiagonal Toeplitz matrix with modified corner elements.

On the finite difference approximation to the convection–diffusion equation

In this paper, the system of ordinary differential equations arisen from discretizing the convection–diffusion equation with respect to the space variable to compute its approximate solution along a time level is considered. This system involves in computing e k A y for some vector y, where k is the time step-size and A is a large tridiagonal Toeplitz matrix. The common ways to compute an approximate solution of the convection–diffusion equation are based on replacing e k A y by an its approximation. We give an explicit expression for the exact value of e k A y and then the numerical results of our method with that of some well-known methods are compared.