Abstract: This paper presents the mathematical model of the thermal process from thermal power plant to aquatic environment of the reservoir-cooler, which is located in the Pavlodar region, 17 Km to the north-east of Ekibastuz town. The thermal process in reservoir-cooler with different hydrometeorological conditions is considered, which is solved by three-dimensional Navier-Stokes equations and temperature equation for an incompressible flow in a stratified medium. A numerical method based on the projection method, divides the problem into three stages. At the first stage, it is assumed that the transfer of momentum occurs only by convection and diffusion. Intermediate velocity field is solved by fractional steps method. At the second stage, three-dimensional Poisson equation is solved by the Fourier method in combination with tridiagonal matrix method (Thomas algorithm). Finally, at the third stage, it is expected that the transfer is only due to the pressure gradient. Numerical method determines the basic laws of the hydrothermal processes that qualitatively and quantitatively are approximated depending on different hydrometeorological conditions.

Abstract: The 1D active magnetic regenerator model developed at the University of Applied Sciences of Western Switzerland is described. The system of two partial differential equations is discretized with the finite differences method backward in time and solved with the tridiagonal matrix algorithm. New features, not found in the literature in 1D models, are thermal losses in the regenerator, parasitic heat exchange, and the calculation of the AMR cycle output power in steady state. The model is implemented in MATLAB and it has a graphical user interface.

Abstract: Numeric algorithms for solving the linear systems of tridiagonal type have already existed. The well-known Thomas algorithm is an example of such algorithms. The current paper is mainly devoted to constructing symbolic algorithms for solving tridiagonal linear systems of equations via transformations. The new symbolic algorithms remove the cases where the numeric algorithms fail. The computational cost of these algorithms is given. MAPLE procedures based on these algorithms are presented. Some illustrative examples are given.

Abstract: In the current paper, we present a novel symbolic algorithm for solving periodic tridiagonal linear systems without imposing any restrictive conditions. The computational cost of the algorithm is less than or almost equal to those of three well-known algorithms given by Chawla and Khazal (Int. J. Comput. Math. 79(4):473–484, 2002) and by El-Mikkawy (Appl. Math. Comput. 161:691–696, 2005), respectively. In addition, the solution of periodic anti-tridiagonal linear systems is also discussed. Two numerical experiments are provided in order to illustrate the performance and effectiveness of our algorithm. All of the experiments were performed on a computer with aid of programs written in MATLAB.

Abstract: In this article, a fourth-order compact alternating direction implicit (ADI) method, based on the clasical Douglas-Gunn ADI method combined with Richardson's extrapolation technique, is proposed for solving 3-D unsteady convection-diffusion equations with discontinuous coefficients. To achieve fourth-order temporal accuracy, a correction term that ensures the realization of extrapolation and reduces the splitting error is introduced. The scheme has unconditional stability and can be solved by the Thomas algorithm. Numerical experiments, including continuous/discontinuous coefficients cases, are conducted to verify the robustness and the high accuracy of this new method.

Abstract: Tridiagonal matrices are considered which are totally nonnegative, i. e., all their mi- nors are nonnegative. The largest amount is given by which the single entries of such a matrix can be perturbed without losing the property of total nonnegativity.

Abstract: In this paper, we settle Higham’s conjecture for the LU factorization of diagonally dominant tridiagonal matrices. We establish a strong componentwise perturbation bound for the solution of a diagonally dominant tridiagonal linear system, independent of the traditional condition number of the coefficient matrix. We then accurately and efficiently solve the linear system by the GTH-like algorithm without pivoting, as suggested by the perturbation result.

Abstract: A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.

Abstract: Heat conduction in multilayered films with the Neumann (or insulated) boundary condition is often encountered in engineering applications, such as laser process in a gold thin-layer padding on a chromium thin-layer for micromachining and patterning. Predicting the temperature distribution in a multilayered thin film is essential for precision of laser process. This article presents an accurate finite difference (FD) scheme for solving heat conduction in a double-layered thin film with the Neumann boundary condition. In particular, the heat conduction equation is discretized using a fourth-order accurate compact FD method in space coupled with the Crank–Nicolson method in time, where the Neumann boundary condition and the interfacial condition are approximated using a third-order accurate compact FD method. The overall scheme is proved to be convergent and hence unconditionally stable. Furthermore, the overall scheme can be written into a tridiagonal linear system so that the Thomas algorithm can be easily used. Numerical errors and convergence rates of the solution are tested by an example. Numerical results coincide with the theoretical analysis. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1291–1314, 2014

Abstract: A new kind of prism element with a triangular base is presented for discretization of multi-scale layered structures within the discontinuous Galerkin time-domain framework. Mixed-order curl-conforming vector basis functions are used in the triangular bases of the prismatic element. The height of the prism adopts spectral basis functions based on Gauss–Lobatto–Legendre polynomials, with an arbitrary order of interpolation. This method combines the flexibility of triangles with the accuracy of spectral elements for layered structures. Eigenvalues obtained show better results than traditional finite elements using tetrahedrons and hexahedrons. For transient analysis, the implicit Crank–Nicholson method is implemented for sequential sub-domains. This kind of arrangement of sub-domains produces a block tridiagonal linear system, thus allowing a block Thomas algorithm to solve the system efficiently. A package-to-chip example shows the efficacy of this method.

Abstract: In this paper we derive an explicit formula for the numerical range of (non-self-adjoint) tridiagonal random operators. As a corollary we obtain that the numerical range of such an operator is always the convex hull of its spectrum, this (surprisingly) holding whether or not the random operator is normal. Furthermore, we introduce a method to compute numerical ranges of (not necessarily random) tridiagonal operators that is based on the Schur test. In a somewhat combinatorial approach we use this method to compute the numerical range of the square of the (generalized) Feinberg-Zee random hopping matrix to obtain an improved upper bound to the spectrum. In particular, we show that the spectrum of the Feinberg-Zee random hopping matrix is not convex.

Abstract: An optimized parallel algorithm is proposed to solve the problem occurred in the process of complicated backward substitution of cyclic reduction during solving tridiagonal linear systems. Adopting a hybrid parallel model, this algorithm combines the cyclic reduction method and the partition method. This hybrid algorithm has simple backward substitution on parallel computers comparing with the cyclic reduction method. In this paper, the operation count and execution time are obtained to evaluate and make comparison for these methods. On the basis of results of these measured parameters, the hybrid algorithm using the hybrid approach with a multi-threading implementation achieves better efficiency than the other parallel methods, i.e., the cyclic reduction and the partition methods. Among them, the cyclic reduction method is previously found to be the fastest algorithm in many ways for solutions. In particular, the approach involved in this paper has the least scalar operation count and the shortest execution time on multi-core computer when the size of an equation is large enough. The hybrid parallel algorithm improves the performance of the cyclic reduction and partition methods by 30% and 20% respectively.

Abstract: A circulant tridiagonal system is a special type of Toeplitz system that appears in a variety of problems in scientific computation. In this paper we give a formula for the inverse of a symmetric circulant tridiagonal matrix as a product of a circulant matrix and its transpose, and discuss the utility of this approach for solving the associated system.

Abstract: Block tridiagonal systems of linear equations arise in a wide variety of scientific and engineering applications. Recursive doubling algorithm is a well-known prefix computation-based numerical algorithm that requires O(M3(N/P + logP)) work to compute the solution of a block tridiagonal system with N block rows and block size M on P processors. In real-world applications, solutions of tridiagonal systems are most often sought with multiple, often hundreds and thousands, of different right hand sides but with the same tridiagonal matrix. Here, we show that a recursive doubling algorithm is sub-optimal when computing solutions of block tridiagonal systems with multiple right hand sides and present a novel algorithm, called the accelerated recursive doubling algorithm, that delivers O(R) improvement when solving block tridiagonal systems with R distinct right hand sides. Since R is typically ~ 102--104, this improvement translates to very significant speedups in practice. Detailed complexity analyses of the new algorithm with empirical confirmation of runtime improvements are presented. To the best of our knowledge, this algorithm has not been reported before in the literature.

Abstract: This paper presents the mathematical model of the thermal power plant in cooling pond under different hydrometeorological conditions, which is solved by three dimensional Navier - Stokes equations and temperature equation for an incompressible fluid in a stratified medium. A numerical method based on the projection method, which divides the problem into three stages. At the first stage it is assumed that the transfer of momentum occurs only by convection and diffusion. Intermediate velocity field is solved by method of fractional steps. At the second stage, three-dimensional Poisson equation is solved by the Fourier method in combination with tridiagonal matrix method (Thomas algorithm). Finally at the third stage it is expected that the transfer is only due to the pressure gradient. To increase the order of approximation compact scheme was used. Then qualitatively and quantitatively approximate the basic laws of the hydrothermal processes depending on different hydrometeorological conditions are determined.

Abstract: In this paper, we derive the general expression of the sth (s in N if n is odd; s in Z if n is even) power of some tridiagonal matrices.

Abstract: Gaussian elimination is used in special linear groups to solve the word problem. In this paper, we extend Gaussian elimination to unitary groups. These algorithms have an application in building a public-key cryptosystem, we demonstrate that.

Abstract: The design and implementation of the Thomas algorithm optimised for hardware acceleration on an FPGA is presented. The hardware based algorithm combined with custom data flow and low level parallelism available in an FPGA reduces the overall complexity from 8N down to 5N arithmetic operations, and combined with a data streaming interface reduces memory overheads to only 2 N-length vectors per N-tridiagonal system to be solved. The Thomas Core developed allows for multiple tridiagonal systems to be solved in parallel, giving potential use for solving multiple implicit finite difference schemes or accelerating higher dimensional alternating-direction-implicit schemes used in financial derivatives pricing. This paper also discusses the limitations arising from the fixed-point arithmetic used in the design and how the resultant rounding errors can be controlled to meet a specified tolerance level.

Abstract: The security of many cryptographic systems is based on the difficulty of solving systems of quadratic multivariate polynomial equations. XL family which stands for eXtended Linearization is a traditional type of algorithm for solving systems of multivariate polynomial equations over finite fields. The main idea of these algorithms is to extend the initial system by monomials multiplications and use linear algebra techniques to solve the system. The overall complexity of XL family is essentially determined by the workload of Gaussian elimination step. In this paper, we use the structured Gaussian elimination and parallel fast Gaussian elimination to reduce the complexity of XL family over GF2. Because of the sparsity of the extended system, structured Gaussian elimination is applied to reduce the dimensions of the extended system. Thus, a great reduction of the storage can be obtained. By slightly modifying the logic of ordinarily Gaussian elimination, we parallelly compute the reduced system. Theory analysis indicates that the complexity of our improved XL, XL_SP, is λT2, which is far lower than 7/64T2.8, the complexity of XL, where T is the number of monomials and λ is the ratio of the dimensions of the reduced system to the extended system. Further experiments to Hidden Field Equation cryptosystems show that about 90% reduction of the extended system is achieved by using structured Gaussian elimination, which means λ is about 1/10 in our experiments. Copyright © 2013 John Wiley & Sons, Ltd.

Abstract: In this paper, we grasp an inverse eigenvalue problem which constructs a tridiagonal matrix with specified multiple eigenvalues, from the viewpoint of the quotient difference (qd) recursion formula. We also prove that the characteristic and the minimal polynomials of a constructed tridiagonal matrix are equal to each other. As an application of the qd formula, we present a procedure for getting a tridiagonal matrix with specified multiple eigenvalues. Examples are given through providing with four tridiagonal matrices with specified multiple eigenvalues.

Abstract: An interesting semi-analytic solution is given for the Helmholtz equation. This solution is obtained from a rigorous discussion of the regularity and the inversion of the tridiagonal symmetric matrix. Then, applications are given, showing very good accuracy. This work provides also the analytical inverse of the skew-symmetric tridiagonal matrix.

Abstract: We present a design and implementation of the Thomas algorithm optimized for hardware acceleration on an FPGA, the Thomas Core. The hardware-based algorithm combined with the custom data flow and low level parallelism available in an FPGA reduces the overall complexity from 8N down to 5N serial arithmetic operations, and almost halves the overall latency by parallelizing the two costly divisions. Combining this with a data streaming interface, we reduce memory overheads to 2 N-length vectors per N-tridiagonal system to be solved. The Thomas Core allows for multiple independent tridiagonal systems to be continuously solved in parallel, providing an efficient and scalable accelerator for many numerical computations. Finally we present applications for derivatives pricing problems using implicit finite difference schemes on an FPGA accelerated system and we investigate the use and limitations of fixed-point arithmetic in our algorithm.

Abstract: It is known that certain tridiagonal matrices have exact eigenvalues and eigenvectors There are sixteen documented tridiagonal matrix families, from the discretization of the one-dimensional Helmholtz equation that possess such properties Extended members of these matrices share a same set of eigenvectors making them commutative with respective to matrix multiplication We may therefore construct, in a fairly straightforward way, exact closed-form solutions of certain tridiagonal generalized matrix eigenvalue problems

Abstract: The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for m-component reaction-diffusion systems with a tridiagonal symmetric toeplitz matrix of diffusion coefficients and with nonhomogeneous boundary conditions. The proposed technique is based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth.