Tridiagonal Matrix Research Articles

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.

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.

Order-preserving fuzzy transform for singular boundary value problems of polytropic gas flow and sewage diffusion

We describe a fuzzy transform method to analyze two-point singular and non-singular boundary value problems with high-order solution accuracies. This framework implements a fuzzy transform that approximates solutions with fourth-order accuracy at the interior mesh points. The fuzzy components and the triangular base function are locally arranged with a three-point linear combination of solution values. This yields a tri-diagonal Jacobian matrix that can be easily computed in a space-time efficient manner. Since a linear system relates solution values and fuzzy components, it is easy to obtain solution approximations via fuzzy components by a tri-diagonal matrix inversion. In addition to the numerical solution, it is easy to determine an approximate analytic solution using a cubic spline interpolating polynomial from the data available with fuzzy components. The error estimates for approximate analytical solutions and numerical solutions are obtained by integrated absolute error and maximum absolute errors. The new mechanism is analyzed for convergence using matrix theory. Several linear and nonlinear equations of practical importance related to sewage diffusion and polytropic gas flow models are simulated to corroborate the new scheme's utility and fourth-order convergence. Mathematics subject classification03B80, 65L10, 65L12

Decompositions of optimal averaged Gauss quadrature rules

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form uTf(A)v with a large matrix A∈RN×N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A; here u,v∈RN are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed.

A space-time multigrid method for poroelasticity equations with random hydraulic conductivity

In this work, we propose a space-time multigrid method for solving a finite difference discretization of the linear Biot’s model in two space dimensions considering random hydraulic conductivity. This method is an extension of the one developed by Franco et al. [1] and which was applied to the heat equation. Particularly for the poroelasticity problem, we need to use the Tri-Diagonal Matrix Algorithm (TDMA) for systems of equations with multiple variables (block TDMA) solver on the planes xt and yt with zebra-type wise-manner (the domain is divided into planes and solve each odd plane first and then each even plane independently [2]). On the planes xy, the F-cycle multigrid with fixed-stress smoother is improved by red-black relaxation. A fixed-stress smoother is compound solver, where the mechanical and flow parts are addressed by two distinctive solvers and symmetric Gauss-Seidel iteration. This is the basis of the proposed method which does not converge only with the standard line-in-time red-black solver [1]. This combination allows the code parallelization. The use of standard coarsening associated with line or plane smoothers in the strong coupling direction allows an application in real world problems and, in particular, the random hydraulic conductivity case. This would not be possible using the standard space-time method with semicoarsening in the strong coupling direction. The spatial and temporal approximations of the problem are performed, respectively, by using Central Difference Scheme (CDS) and implicit Euler methods. The robustness and excellent performance of the multigrid method, even with random hydraulic conductivity, are illustrated with numerical experiments through the average convergence factor.

Analysis of a Pre-Emptive Two-Priority Queuing System with Impatient Customers and Heterogeneous Servers

This paper presents a queuing system model that incorporates multiple priorities, multiple abandonments, and heterogeneous servers. Waiting for service easily leads to impatient behaviors. The impact of two kinds of impatient behaviors, balking and reneging, on queueing system performance is examined. The problem is formulated as continuous-time Markov chains. It also introduces a special state called the non-sojourn state to record the number of customers who abandon the system. The state transition rate matrix is transformed into a block tridiagonal matrix by appropriately setting the state numbers. A novel indicator called interstate transition frequency is proposed, which aids in distinguishing state transitions during the system evaluation process. Based on the interstate transition frequency, a set of indicators is derived to offer additional analytical perspectives for the queuing system. Finally, the proposed model is applied to an automobile repair shop to validate its effectiveness in practical scenarios.

Photoionization Cross Section for H@Cn${\text{H@C}}_{n}$ Implanted in Nonideal Classical Plasmas

AbstractIn this study, photoionization cross sections of guest hydrogen atom in endohedral fullerene () modeled by the Woods–Saxon confinement potential implanted in the nonideal classical plasma (NICP) under spherical confinement are reported for the first time in the related literature. The relevant wave equation is solved numerically via the tridiagonal matrix method and then the energy levels, bound and continuum wave functions are interpreted. Plasma effect is examined by considering plasma temperature and density and is evaluated in the photoionization process. Since the plasma modifies the discrete and continuum spectra by changing the potential energy of hydrogen atom, it closely affects the overlapping of the wave functions of ground state and continuum state. This effect has a distinct response on photoionization resonances. Using different values of endohedral confinement parameters, which means regarding different types of fullerenes, detailed analysis of energy levels, bound and continuum wave functions, and photoionization cross sections are provided by evaluating confinement width, depth, smoothing effect and distance from the spherical encompassement center. The photoionization process of implanted in the NICP (nonideal classical plasma) is highly sensitive to both plasma and endohedral confinement. At this sensitivity, Cooper resonance character is like an encoder for fullerene structure.

Theoretical analysis of unsteady buoyant turbulent heat and mass transport from a vertical plate using LRN k-ϵ model

The contemporary analysis has been developed to investigate the transient natural convective combined heat and mass transport of a turbulent flow adjacent to the plate by utilizing the low Reynolds number (LRN) k-ϵ model. The research problem on turbulent flow is assumed to be two dimensional and viscous incompressible. The governing turbulent equations such as continuity, momentum, energy, concentration, turbulent kinetic energy (TKE) and dissipation rate of TKE are considered as per the flow geometry. Due to the non-availability of analytical or direct numerical techniques, the corresponding highly nonlinear PDE’s are solved by adopting the standard implicit type of finite difference method namely Crank–Nicolson scheme. Also, this scheme is unconditionally stable and helps to keep the governing turbulent flow equations in discretized form and are solved via tridiagonal matrix algorithm. The simulated results for several control parameters including turbulent Reynolds number are graphically displayed and also analyzed the flow profiles. To understand the physical phenomenon of turbulent flow, the authors shown the turbulent behavior of average momentum, energy and mass transfer rates. It has been noted that, the average temperature and concentration fields boosted with rising values of . Also, TKE and dissipation rate of TKE profiles increase with enhancing turbulent values. In addition, the simulated turbulent flow results from the LRN k-ϵ model are compared with usual laminar flows as a special case and found to be in respectable agreement.

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.

Ising analogs of quantum spin chains with multispin interactions

A new family of free fermionic quantum spin chains with multispin interactions was recently introduced. Here we show that it is possible to build standard quantum Ising chains---but with inhomogeneous couplings---which have the same spectra as the novel spin chains with multispin interactions. The Ising models are obtained by associating an antisymmetric tridiagonal matrix to the polynomials that characterize the quasienergies of the system via a modified Euclidean algorithm. For the simplest nontrivial case, corresponding to the Fendley model, the phase diagram of the inhomogeneous Ising model is investigated numerically. It is characterized by gapped phases separated by critical lines with order-disorder transitions depending on the parity of the total number of energy density operators in the Hamiltonian.

LEFT AND RIGHT EIGENVECTORS OF A VARIANT OF THE SYLVESTER–KAC MATRIX

AbstractAs an extension of Sylvester’s matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form.

The second, third harmonic generations and nonlinear optical rectification of the Mathieu quantum dot with the external electric, magnetic and laser field

In this study, the nonlinear optical properties of the InxGa1−xAs/GaAs Mathieu quantum dot (MQD) are investigated for the first time, focusing on the nonlinear optical rectification (NOR), second harmonic generation (SHG), and third harmonic generation (THG). The effects of external fields such as the electric field, magnetic field, and laser radiation field on the nonlinear optical properties of MQD are examined, along with structural parameters such as quantum dot depth and width determined by the indium concentration. The motivation of the study is to explain the NOR, SHG, and THG characteristics of MQD in response to changes in external fields and structural factors. To investigate the effects of the laser field, the time-dependent part of the laser field is transferred to the potential energy term of the wave equation using the Kramers–Henneberger (KH) and dipole approximations, creating a laser-dressed potential. Then, the effective potential wave equation is solved using the tridiagonal matrix method. The effects of external fields and structural parameters of MQD on the NOR, SHG, and THG coefficients are discussed in detail. The optimality of the structure for device design and applications considering MQD is revealed, and alternative parameter analysis is conducted for this optimality.

The bilateral birth–death chain generated by the associated Jacobi polynomials

AbstractWe give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three‐term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one‐step transition probability matrix of a discrete‐time bilateral birth–death chain with state space on . We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.

EFFECTS OF MAGNETIC PARTICLES DIAMETER AND PARTICLE SPACING ON BIOMAGNETIC FLOW AND HEAT TRANSFER OVER A LINEAR/NONLINEAR STRETCHED CYLINDER IN THE PRESENCE OF MAGNETIC DIPOLE

Magnetic particles are essential in materials science, biomedical, bioengineering, heat exchangers due to their exceptional thermal conductivity and unique properties. This work aims to model and analyze the biomagnetic fluid flow and heat transfer, namely the flow of blood with magnetic particles (Fe3O[Formula: see text] induced by stretching cylinder with linear and nonlinear stretching velocities. Additionally, this study investigates the impact of particles diameter and their spacing under the influence of ferrohydrodynamics (FHD) principle. The collection of partial differential equations is transformed using similarity transformations to produce the theoretically stated ordinary differential system. An efficient numerical technique, which is further based on common finite difference method with central differencing, a tridiagonal matrix manipulation and an iterative procedure are used to solve the problem numerically. The major goal of this extensive study is to enhance heat transformation under the influence of numerous parameters. There have been numerous displays of the velocity profile, temperature distribution, local skin friction factor and rate of heat transfer in terms of the appearing physical parameters. It is observed that variation in velocity and temperature distributions is the cause of increasing the ferromagnetic interaction parameter and the size of magnetic particles. The enhancement of particle diameter causes an increment in the skin friction while the rate of heat transfer declines. For verifying purposes, a comparison is also shown with previously published scientific work and found to possess suitable accuracy.

PaScaL_TDMA 2.0: A multi-GPU-based library for solving massive tridiagonal systems

PaScaL_TDMA 2.0: A multi-GPU-based library for solving massive tridiagonal systems

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.

PaScaL_TCS: A versatile solver for large-scale turbulent convective heat transfer problems with temperature-dependent fluid properties

PaScaL_TCS is an efficient and scalable solver for large-scale direct numerical simulations of natural convective flow that considers temperature-dependent fluid properties. For increased scalability on a massive-scale distributed memory system, the solver decomposes the computational domain into a cubic sub-domain. The numerical procedure is based on the monolithic projection method with staggered time discretization [1], which is an implicit and non-iterative solver for wall-bounded domains. The PaScaL_TDMA library is used to solve batched tridiagonal systems, which are partitioned according to domain decomposition. For parallel fast Fourier transform in a Fourier Poisson solver for pressure, two transpose schemes with different communicator sizes are proposed and compared according to the number of cores. An explicit intermediate aggregation scheme for MPI-IO is suggested to reduce the number of processes that simultaneously take part in parallel IO. The overall implementation was evaluated using the Nurion cluster system of the Korea Institute of Science and Technology Information with detailed performance profiling. With the efficient communication, the results showed good strong and weak scalability up to 131,072 cores. The file IO performance improved with the suggested MPI-IO scheme using explicit aggregation, especially when many processes are involved in parallel IO. The solver was verified by conducting large-scale differentially heated vertical convection flow simulations under the Oberbeck–Boussinesq (OB) approximation. The capability of the solver to capture and quantitatively describe the non-OB effect was demonstrated via glycerol Rayleigh–Bénard convection flow simulations. Program summaryProgram Title: PaScaL_TCSCPC Library link to program files:https://doi.org/10.17632/3b2bysrcxm.1Developer's repository link:https://github.com/MPMC-LabLicensing provisions: MITProgramming language: Fortran90Nature of problem: Solving the incompressible Navier-Stokes equations for natural convective flow considering the non-Oberbeck–Boussinesq effect, which considers the temperature dependence of fluids properties, in three-dimensional channel domain.Solution method: PaScaL_TCS uses the monolithic projection-based method with staggered time discretization (MPM-STD) [1]. It employs the Crank-Nicolson scheme along with staggered time which evaluates scalar and vector variables at the (n+1/2) and (n+1) time level, respectively. PaScaL_TDMA [2] is used to solve multiple tridiagonal matrix systems for decoupled energy and momentum equations in a distributed memory system. The Poisson equation solver for the pressure-correction scheme is based on Fourier diagonalization. MPI_Alltoallw is used for parallel FFT. PaScaL_TCS supports a single-file-based parallel IO for efficient data generation in large-scale computing.

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.

A novel high-dimensional and multi-physics modeling approach of proton exchange membrane fuel cell for real-time simulation

Fuel cell model provides state observation and quantitative analysis for Hardware-In-the-Loop (HIL) test platform. For a high-dimensional multi-physics model, the observed distribution of physical properties is meaningful for fuel cell stack optimization, manufacturing process and model-based control development. However, traditional modeling approach is difficult to consider both the calculation accuracy and efficiency. To solve this problem, this paper proposes a high-dimensional multi-physics real-time modeling approach for fuel cells based on partial differential equations (PDEs)-tridiagonal matrix equation transformation (TMET) and symmetric successive overrelaxation (SSOR) algorithm. In the proposed method, a generalized framework of TMET and a novel staggered grid technique is developed. In this case, both the convection and diffusion phenomena can be accurately described. For different fuel cells or parameters change, supplementary model modification or development is not required, only the global variables need to be re-assigned. Moreover, SSOR with bi-directional iteration mechanism is used to increase the probability of computational convergence and thus improve the calculation efficiency. Compared with commercial COMSOL model, by using the proposed modeling approach, the mean absolute percentage error (MAPE) of the simulated polarization output characteristics is 1.77%, the MAPE of different distribution characteristics can remain within 10%, and the calculation time can be shortened from hours even days to seconds.