Tridiagonal Matrix Research Articles

Tridiagonal Weighting‐Based Interference Cancellation Method in High‐Mobility OFDM Systems

ABSTRACTIn orthogonal frequency division multiplexing (OFDM) systems, the time‐selective fading channel causes the signal Doppler expansion in high‐mobility scenarios, leading to intercarrier interference (ICI) in the received signals. People had employed a full‐dimensional weighting matrix method to reduce the Doppler‐assisted ICI, though it required a high implementation complexity. In order to reduce the ICI and produce lower complexities, this paper proposes to exploit the tridiagonal weighting matrix to suppress ICIs, in which an optimization problem is constructed to design the tridiagonal weighting matrix, and the nonzero coefficients within the tridiagonal matrix denote optimization variables. Moreover, the optimization objective exploits the system bit error rate (BER) instead of the conventional signal to interference plus noise ratio (SINR), which results in a highly nonlinear optimization problem. Therefore, an iterative solving approach is proposed according to the successive convex approximation (SCA). Simulations show that the proposed method can achieve a good balance between the complexity and BER degradation, especially in the scenario with moderate Doppler spreads.

Characteristic Equations of Chebyshev Polynomials of Third and Fourth Kinds and Their Generating Matrices

The main goal of the article is to obtain matrix representation for the third and fourth kinds of Chebyshev polynomials by using a tridiagonal matrix. We present a connection between the determinant of the tridiagonal matrix and the third and fourth kinds of Chebyshev polynomials. We also determine the characteristic equations for the third and fourth kinds of the Chebyshev polynomials up to degree three. We also prove some properties relating to matrix representation. We obtain a connection between the second, third kind and fourth kinds of Chebyshev polynomials and matrix power. It elaborates the theorem to validate through examples. The applications of the Chebyshev polynomials is also discussed. The practical application of the Chebyshev polynomials of the third kind in approximation theory is also detailed.

Numerical Study on the Impact of the Richardson Number on Convection in an Enclosure, with Moving Sidewalls for Three Aspect Ratios: A = 1, 2 and 0.5

Using mathematical models, this research investigation seeks to explore how varying aspect ratios and the Richardson number impact the mixing effectiveness, fluid flow velocity, and heat transfer characteristics in a rectangle-shaped space where cooling occurs along the sides and warm with a steady temperature exists throughout the 80% of the bottom section's center. Discretizing the governing equations through the finite volume approach, we solved them iteratively using the Tri-diagonal Matrix Algorithm, where pressure-velocity coupling was successfully implemented via the SIMPLER procedure. Innovative approaches for convection and diffusion terms involved employing a power-law difference scheme alongside a central differential scheme. In various simulations with specified parameters (Re = 100, Pr = 0.71), we visualized results in forms like streamlines, isothermal lines, and temperature/velocity profiles, along with local and average Nusselt numbers. We observe five unique structure formations during flow behavior by analyzing how Richardson numbers impact heat transmission within a specific geometric configuration (enclosure size A = 1). A change in aspect ratio leads to improved control over fluid movement while minimizing structural distortion. A subtle rise in heat transmission emerged as the average Nusselt number and Richardson number intersection strengthened force convection's domination while experiencing significant escalation under conditions conductive to natural convection. Decreasing the aspect ratio increased heat transfer efficiency when the Richardson number rose.

An efficient direct‐forcing immersed boundary method for flow around a pair of spheres

AbstractThe numerical study of flow around a pair of spheres and a square array of spheres is investigated by using a direct‐forcing immersed boundary method. Using high resolution three‐dimensional computations, we analyzed the flow around several configurations: a sphere, a pair of spheres in a tandem arrangement with center‐to‐center streamwise ratio L/D ranging from 1 to 6, and a square array with 9 spheres in a uniform arrangement. In the latter case, we explore the ratio of array diameter (DG) to sphere diameter (D) at 4, 5, 6 and 7. The center‐to‐center streamwise and transverse pitch is the same, varied from L/D = 1.5, 2, 2.5 to 3, and they were arranged in a square periodic array to allow uniform distribution within the array. Based on the effective direct‐forcing immersed boundary projection method, the fractional time marching methodology is applied for solving four field variables involving three velocities and one pressure component. The pressure Poisson equation is advanced in space by using the fast Fourier transform (FFT) and a tridiagonal matrix algorithm (TDMA), effectively solving for the diagonally dominant tridiagonal matrix equations. A direct‐forcing immersed boundary method is involved to treat the interfacial terms by adding the appropriate sources as force function at the boundary, separating the phases. Geometries featuring the stationary solid obstacles in the flow are embedded in the Cartesian grid with special discretizations near the embedded boundary using a discrete Dirac delta function to ensure the accuracy of the solution in the cut cells. An important characteristic of flow over the multiple spheres is devised by comparing with the drag and lift coefficients, as well as vortex shedding.

1D drift-kinetic numerical model based on semi-implicit particle-in-cell method

The paper presents a new one-dimensional drift-kinetic electrostatic model based on the particle-in-cell method and capable of simulating the processes of plasma heating and confinement in mirror traps. The most of particle and energy losses in these traps occur along the magnetic field lines. The key role in limiting these losses is played by the ambipolar electric potential which creates a potential barrier for electrons and significantly reduces the heat flux that, without this barrier, would go to wall due to the classical electron thermal conductivity. However, modeling the formation of such a potential on real spatial and temporal scales of experiments is a challenging problem, since it requires a detailed description of not only ion, but also electron kinetics. In this work, we propose to solve the problem of taking into account electron kinetic effects on the time scale of plasma confinement in a mirror trap using the particle-in-cell method adapted to the approximate drift-kinetic equations of plasma motion. Unlike other electrostatic particle-in-cell models, which use fully implicit schemes to solve the nonlinear system of Vlasov-Poisson and Vlasov-Ampere equations, we propose a semi-implicit approach. By analogy with the Energy Conserving Semi-Implicit Method (ECSIM), it allows for precise conservation of energy and reduces the procedure for finding the electric field to inverting a tridiagonal matrix without multiple nonlinear iterations. Such a model will be useful for simulating not only collisional losses of hot plasma in fusion experiments, but also for studying the features of creating cold starting plasma in mirror traps using plasma or electron guns.

Tridiagonal and single-pair matrices and the inverse sum of two single-pair matrices

A novel factorization for the sum of two single-pair matrices is established as product of lower-triangular, tridiagonal, and upper-triangular matrices, leading to semi-closed-form formulas for tridiagonal matrix inversion. Subsequent factorizations are established, leading to semi-closed-form formulas for the inverse sum of two single-pair matrices. An application to derive the symbolic inverse of a particular Gram matrix is presented, and the numerical stability of the formulas is studied.

Optimizing Resource Allocation in M/M/1/N Queues with Feedback, Discouraged Arrivals, and Reneging for Enhanced Service Delivery

This article presents a novel computational approach for analyzing M/M/1/N queues with feedback, discouraged arrivals, and reneging, under the first-come, first-served (FCFS) discipline. We calculate explicit transient state probabilities and represent results using symmetric tridiagonal matrix eigenvalues. Through numerical simulations, we validate our method, providing practical insights for optimizing resource allocation. Our study contributes to both theory and application, advancing queueing theory and aiding decision-makers in improving service quality and resource management.

Stage–discharge prediction in the multi-stage ice-covered compound channel

The multi-stage compound channel, which is a common pattern in natural alluvial rivers and the regulation projects of urban rivers, inevitably freezes in winter when it is situated in cold northern areas with high latitudes. Given that ascertaining the stage–discharge relationship for rivers is the foundation for the development of flood control schemes and water resources management, this study concentrates on proposing an analytical model for predicting the stage–discharge curves of multi-stage ice-covered compound channels. In deducing the analytical model, the cross section of the channel is first segmented into several homogeneous subregions that can be grouped into seven categories according to the geometric characteristics. Through analyzing the momentum transfer between adjacent subregions, the force balance equation for each subregion is then established to get the bulk mean velocity for the corresponding subregion, thereby obtaining the discharge by solving a tridiagonal matrix. Subsequently, measurements from two-stage and three-stage ice-covered compound channel experiments and three sets of experimental data from the literature are used to validate the performance of the proposed model. Good agreement between the predictions and the measured data suggests that the deduced model can accurately estimate the discharge for the multi-stage ice-covered compound channels when the flow depth is given. Finally, sensitivity analysis indicates that Manning's roughness coefficient of the channel bed has a more pronounced impact on the stage–discharge relationship than that of the ice cover. Moreover, when compared to the two-stage ice-covered compound channel, the multi-stage ice-covered compound channel offers greater potential for water resource utilization.

Tridiagonal matrix decomposition for Hamiltonian simulation on a quantum computer

Tridiagonal matrix decomposition for Hamiltonian simulation on a quantum computer

Parallel Algorithms for Numerical Solution of Spatially Three-Dimensional Diffusion-Convection Equations in Coastal Systems Based on Splitting Schemes

Introduction. To prevent the occurrence and mitigate the consequences of hazardous and catastrophic phenomena associated with sediment transport in natural systems, it is necessary to develop operational and scientifically justified forecasts, identify critical states at which the emergence of emergency situations is possible. For these purposes, it is necessary to create an accurate and efficient toolkit, including algorithms for numerical solution of a model problem that takes into account the specifics of natural systems. In this work, parallel algorithms for numerical solution of a spatially three-dimensional diffusion-convection problem of sediment are presented, which allow a significant reduction in computation time (by more than 4 times) compared to calculations conducted using a sequential algorithm.Materials and Methods. For the parallel solution of the spatially three-dimensional diffusion-convection problem, an implicit splitting scheme is constructed, in which the original continuous problem is replaced by a chain of two-dimensional and one-dimensional problems. The splitting schemes proposed in the work are physically justified and take into account the specifics of coastal marine systems, for which the influence of micro-turbulent diffusion and advective transport of substances are comparable, and the Peclet number does not exceed unity when approximating real problems. For the parallel numerical implementation, a method of decomposing the grid domain into two families of vertical planes parallel to the coordinate planes Oxz and Oyz, combined with the Seidel method for solving two-dimensional grid problems in horizontal planes and the tridiagonal matrix algorithm when solving one-dimensional three-point problems in the vertical direction, is used. Within the framework of the parallel computing software implementation, a parallel algorithm is presented that implements the diffusion-convection problem on a computing system using MPI technology.Results. A comparative analysis of parallel and sequential algorithms is obtained using a model problem.Discussion and Conclusions. The developed software allows its practical use for solving specific hydrophysical problems, including as part of a software complex.

Radiative dynamics of laser-driven Li@C n embedded in quantum plasma

This work considers a guest Lithium atom (Li@C n ) in an endohedral fullerene, embedded in a quantum plasma modeled by the more general exponential cosine screened Coulomb (MGECSC) potential, under the influence of a spherical confinement and laser radiation field. The system is examined in nonrelativistic form and the related wave equation is solved using the tridiagonal matrix method (TMM), thus obtaining the discrete-continuum spectrum and related wave functions. The numerical values of the relevant parameters in this process are physically accessible values. The effects of the plasma, laser field and endohedral cavity on the photoionization cross section (PCS) are analysed in detail. The shielding effect of the plasma medium and the pulsating effect of the laser field modify the effective potential energy of the system, affecting the localizations of the 2s and continuum states, causing various overlapping cases. Considering different values of the endohedral encompassement parameters, which means that different types of fullerenes are taken into account, overlapping cases occur for different spectra and wave functions. Scrutinising these overlappings, the confinement and Cooper resonances of the PCSs are analysed. This analysis provides many details for the radiative dynamics of an artificial system Li@C n . The relevant ranges and critical values of plasma, laser field, and endohedral encapsulation parameters in the formation process of these resonances and PCSs are explained, as well as the cross-section curves, resonance positions, effective photoelectron energy range, and general PCS behavior, which can be important for potential experiments in addition to other theoretical investigations.

Tridiagonal Interval Matrix: Exploring New Perspectives and Application

Tridiagonal interval matrices are relevant in diverse applications, especially in dealing with parameter estimation, optimization and circuit analysis uncertainties. This research paper aims to improve the computational efficiency of obtaining the inverse of a general tridiagonal interval matrix. This matrix is pivotal in electric circuit analysis. We achieve this by employing interval arithmetic operations in the LU decomposition process, enabling effective handling of circuit parameter uncertainties. This approach generates an inverse interval matrix that addresses uncertainties in circuit analyses.

Entropy generation analysis of a ternary hybrid nanofluid (Au-CuO-GO/blood) containing gyrotactic microorganisms in bifurcated artery

PurposeThe purpose of this paper is to study haemodynamic flow characteristics and entropy analysis in a bifurcated artery system subjected to stenosis, magnetohydrodynamic (MHD) flow and aneurysm conditions. The findings of this study offer significant insights into the intricate interplay encompassing electro-osmosis, MHD flow, microorganisms, Joule heating and the ternary hybrid nanofluid.Design/methodology/approachThe governing equations are first non-dimensionalised, and subsequently, a coordinate transformation is used to regularise the irregular boundaries. The discretisation of the governing equations is accomplished by using the Crank–Nicolson scheme. Furthermore, the tri-diagonal matrix algorithm is applied to solve the resulting matrix arising from the discretisation.FindingsThe investigation reveals that the velocity profile experiences enhancement with an increase in the Debye–Hückel parameter, whereas the magnetic field parameter exhibits the opposite effect, reducing the velocity profile. A comparative study demonstrates the velocity distribution in Au-CuO hybrid nanofluid and Au-CuO-GO ternary hybrid nanofluid. The results indicate a notable enhancement in velocity for the ternary hybrid nanofluid compared to the hybrid nanofluids. Moreover, an increase in the Brinkmann number results in an augmentation in entropy generation.Originality/valueThis study investigates the flow characteristics and entropy analysis in a bifurcated artery system subjected to stenosis, MHD flow and aneurysm conditions. The governing equations are non-dimensionalised, and a coordinate transformation is applied to regularise the irregular boundaries. The Crank–Nicolson scheme is used to model blood flow in the presence of a ternary hybrid nanofluid (Au-CuO-GO/blood) within the arterial domain. The findings shed light on the complex interactions involving stenosis, MHD flow, aneurysms, Joule heating and the ternary hybrid nanofluid. The results indicate a decrease in the wall shear stress (WSS) profile with increasing stenosis size. The MHD effects are observed to influence the velocity distribution, as the velocity profile exhibits a declining nature with an increase in the Hartmann number. In addition, entropy generation increases with an enhancement in the Brinkmann number. This research contributes to understanding fluid dynamics and heat transfer mechanisms in bifurcated arteries, providing valuable insights for diagnosing and treating cardiovascular diseases.

The Generalized Hermite Spectral Method Combined with Laplace Transform for Solution of the Time Fractional Fokker–Planck Equation

The paper proposes a method to solve the time fractional Fokker–Planck equation using the generalized Hermite spectral method and Laplace transform. By the Laplace transform, the ordinary differential equation about the orthogonal expansion coefficient is transformed into a linear algebraic equation system, with the coefficient matrix being a tridiagonal lower triangular matrix. The orthogonal expansion coefficients of frequency domain components are obtained through this transformation. The approximate solution at any time is then obtained by using the numerical inverse Laplace transform with the Talbot algorithm. Numerical experiments have been carried out to demonstrate the high accuracy and efficiency of the proposed method.

Inhomogeneous isotropic quantum spin chain associated with the difference Lamé equation

Abstract The spectrum and orthogonal eigenbasis are computed of a tridiagonal matrix encoding a finite-dimensional reduction of the difference Lamé equation at the single-gap integral value of the coupling parameter. This entails the exact solution, in terms of single-gap difference Lamé wave functions, for the spectral problem of a corresponding open inhomogeneous isotropic $XY$ chain with coupling constants built from elliptic integers.

On the eigenvalue-separation properties of real tridiagonal matrices

In this paper, we give a simple sufficient condition for the eigenvalue-separation properties of real tridiagonal matrices T. This result is much more than the statement that the pertinent eigenvalues are distinct. Its derivation is based on recurrence formulae satisfied by the polynomials made up by the minors of the characteristic polynomial det(xE-T) that are proven to form a Sturm sequence. This is a new result, and it proves the simple spectrum property of a symmetric tridiagonal matrix studied in Grünbaum's paper. Two numerical examples underpin the theoretical findings. The style of the paper is expository in order to address a large readership.

On a Determinantal Formula for Derangement Numbers

The aim of this note is to provide succinct proofs for a recent formula of the derangement numbers in terms of the determinant of a tridiagonal matrix.

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.

Long-time adaptive coherent detection of small targets in sea clutter by fast inversion algorithm of block tridiagonal speckle covariance matrices

Low-velocity small target detection in maritime surveillance radars is always a challenging task. Low signal-to-clutter ratio requires long-time coherent integration to obtain enough gain of target returns. However, long-time coherent integration encounters insufficient secondary data due to the spatial inhomogeneity of sea clutter. In this paper, considering the decorrelation time of speckle component of sea clutter short up to a dozen of milliseconds, the spherical invariant random vector (SIRV) model with block tridiagonal speckle covariance matrix and the inverse Gamma distributed texture is proposed to model sea clutter sequences in several tenths of a second. In this model, a long-time adaptive generalized likelihood ratio test with linear threshold detector (GLRT-LTD) is constructed. Owing to the block tridiagonal structure of speckle covariance matrices, the adaptive detection requires much less reference cells for speckle covariance matrix estimation and much lower computational cost for its inversion. The proposed detector is verified by an X-band high-resolution island-based radar data with an unmanned aerial vehicle (UAV) as test target. The experimental result shows that it obtains competitive detection performance in comparison with the state-of-the-art detectors.

Computational analysis of entropy generation in three-dimensional mixed convection flow with thermal dissipation and variable thermal conductivity

Entropy generation in steady three-dimensional thermal flow is investigated under the coupled influence of viscous dissipation and mixed convection. The stretching sheet with an exponential velocity profile induces the flow. In this examination, we presume that the fluid is incompressible and possesses a Newtonian behavior and temperature-dependent thermal conductivity. Using the appropriate classical similarity variables reduces the dimensional nonlinear dynamical system of governing equations to a dimensionless nonlinear dynamical set of ordinary differential equations. It’s important to note that the selection of an exponentially varying surface temperature is deliberate, as it enables precise similarity transformations to be established in situations involving both viscous dissipation and buoyancy forces. The simplified finite difference approach (SFDM) is used to solve the acquired nonlinear dynamical set of ordinary differential equations. The tridiagonal matrix approach is used to solve the resulting system of algebraic equations. In a particular situation, the present numerical simulation is numerically checked with the previously available results and observed a good agreement. The effects of physical parameters on velocity, temperature, surface stresses and heat tranfer rate, entropy generation, and the irreversibility parameter are investigated. The entropy generation rate exhibits a decreasing trend with respect to the mixed convection parameter in the vicinity of the stretching surface. Entropy reduction can be achieved by employing a working fluid with a low Prandtl number and by minimizing the fluid friction.