Abstract: The solving of tridiagonal systems is one of the most computationally expensive parts in many applications, so that multiple studies have explored the use of NVIDIA GPUs to accelerate such computation. However, these studies have mainly focused on using parallel algorithms to compute such systems, which can efficiently exploit the shared memory and are able to saturate the GPUs capacity with a low number of systems, presenting a poor scalability when dealing with a relatively high number of systems. We propose a new implementation (cuThomasBatch) based on the Thomas algorithm. To achieve a good scalability using this approach is necessary to carry out a transformation in the way that the inputs are stored in memory to exploit coalescence (contiguous threads access to contiguous memory locations). The results given in this study proves that the implementation carried out in this work is able to beat the reference code when dealing with a relatively large number of Tridiagonal systems (2,000–256,000), being closed to \(3{\times }\) (in double precision) and \(4{\times }\) (in single precision) faster using one Kepler NVIDIA GPU.

Abstract: A Crank-Nicolson-type difference scheme is presented for the spatial variable coefficient subdiffusion equation with Riemann-Liouville fractional derivative. The truncation errors in temporal and spatial directions are analyzed rigorously. At each time level, it results in a linear system in which the coefficient matrix is tridiagonal and strictly diagonally dominant, so it can be solved by the Thomas algorithm. The unconditional stability and convergence of the scheme are proved in the discrete $L_{2}$ norm by the energy method. The convergence order is $\min \{2-\frac{\alpha}{2}, 1+\alpha \}$ in the temporal direction and two in the spatial one. Finally, numerical examples are presented to verify the efficiency of our method.

Abstract: In this paper, we proposed a numerical integration technique with exponential integrating factor for the solution of singularly perturbed differential-difference equations with negative shift, namely the delay differential equation, with layer behaviour. First, the negative shift in the differentiated term is approximated by Taylor's series, provided the shift is of $o(\varepsilon )$. Subsequently, the delay differential equation is replaced by an asymptotically equivalent first order neutral type delay differential equation. An exponential integrating factor is introduced into the first order delay equation. Then Trapezoidal rule, along with linear interpolation, has been employed to get a three term recurrence relation. The resulting tri-diagonal system is solved by Thomas algorithm. The proposed technique is implemented on model examples, for different values of delay parameter, $\delta $ and perturbation parameter, $\varepsilon $. Maximum absolute errors are tabulated and compared to validate the technique. Convergence of the proposed method has also been discussed.

Abstract: We present the perturbation and backward error analyses of the partitioned LU factorization for block tridiagonal matrices. In addition, we consider the bounds of perturbations for the partitioned LU factorization for block-tridiagonal linear systems. Finally, numerical examples are given to verify the obtained results.

Abstract: This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices. Under quite general assumptions, we prove that the traces are approximately normally distributed. A Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth–death Markov kernel, the random birth–death Q matrix and the $$\beta $$ -Hermite ensemble. Furthermore, under an independent-and-identically-distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.

Abstract: The paper addresses the problem of computational efficiency of the pipe-flow model used in leak detection and identification systems. Analysis of the model brings attention to its specific structure, where all matrices are sparse. With certain rearrangements, the model can be reduced to a set of equations with tridiagonal matrices. Such equations can be solved using the Thomas algorithm. This method provides almost the same values of the state vector and maintains stability for the same discretization grid, while the computational overhead is vastly reduced.

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

Abstract: Let T be a tridiagonal operator on which has strict row and column dominant property except for some finite number of rows and columns. This matrix is shown to be invertible under certain conditions. This result is also extended to double infinite tridiagonal matrices. Further, a general theorem is proved for solving an operator equation using its finite-dimensional truncations, where T is a double infinite tridiagonal operator. Finally, it is also shown that these results can be applied in order to obtain a stable set of sampling for a shift-invariant space.

Abstract: In the paper, the authors find a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

Abstract: Summary Computing the extremal eigenvalue bounds of interval matrices is non-deterministic polynomial-time (NP)-hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.

Abstract: In this work we present flow simulations in laminar and turbulent regime within a representative elementary volume of a simplified porous media by solving the Navier–Stokes equations and a Low-Re turbulence $$k{-}\epsilon$$ model. Numerical solution was achieved with an implementation of the SIMPLE algorithm for pressure velocity coupling of variables, and the solution of the tridiagonal systems of algebraic equations was accomplished by a parallelized ADI scheme based on the Thomas algorithm. Implementation of the numerical solution was done with an in-house C code which combined OMP and CUDA technologies for computations based on CPU and GPU, respectively. Exponential structured grids were employed in the wall vicinity to capture the turbulence behavior. Results indicate that similar profiles for velocity, pressure, turbulent kinetic energy and its dissipation were found. Several CUDA grids were tested and their performances measured over two GPUs: GTX 680 and GTX TITAN. Considerable speedup was achieved by the GPUs over the CPU schemes even without the use of the device shared memory which was not explored due to the nature of the algorithm.

Abstract: For the further development of semiconductor heterostructures and devices based on it we have to know precisely the basic parameters such as energy band discontinuities, free carriers distribution, the magnitude of the accumulated charge, etc. Physical description of systems with reduced dimensionality gives the system of Schrodinger and Poisson equations that are too complicated for solving analytically and is usually solved by numerical methods. In this regard, in the paper questions of numerical calculation and energy band offsets determination on the example of quantum well by solving self-consistently Schrodinger and Poisson are considered. The determination of the Schrodinger equation is implemented by the “shooting” algorithm, and the determination of the Poisson equation using Thomas algorithm, in connection with the speed of operation and optimum accuracy of the solution.

Abstract: The following study introduces an algorithm for numerically solving a coupled system of differential equations that discretize to produce a banded pentadiagonal matrix with tridiagonal submatrices on either side. The article then proceeds to illustrate a step-by-step procedure to solve a two-equation system in a coupled manner. A generalization of this method to multiequation system follows. The proposed algorithm is highly efficient as it fully exploits the banded nature of the matrix. Finally, the solution of the two-equation k − ω and the four-equation turbulence models in plane channel flows by the current method serves as a demonstration exercise and wraps up the paper.

Abstract: In this short note, we present a novel method for computing exact lower and upper bounds of eigenvalues of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and avoids any assumptions. Our construction explicitly yields those matrices for which particular lower and upper bounds are attained.

Abstract: This paper proposes a method for speeding up the estimation of the absolute value of largest eigenvalue of an asymmetric tridiagonal matrix based on Power method. An error analysis shows that the proposed method provide errors no greater than the usual Power method. The proposed method involves the computation of the tridiagonal matrix square under analysis, which is performed through a proposed fast algorithm specially tailored for tridiagonal matrices. We perform numerical simulations on Matlab® platform showing the reliability of the method and the claimed speedup using Sylvester-Kac test matrix.

Abstract: Physical phenomena with critical blow-up regimes simulated by the 3D nonlinear diffusion equation in a spherical shell are studied. For solving the model numerically, the original differential operator is split along the radial coordinate, as well as an original technique of using two coordinate maps for solving the 2D subproblem on the sphere is involved. This results in 1D finite difference subproblems with simple periodic boundary conditions in the latitudinal and longitudinal directions that lead to unconditionally stable implicit second-order finite difference schemes. A band structure of the resulting matrices allows applying fast direct (non-iterative) linear solvers using the Sherman-Morrison formula and Thomas algorithm. The developed method is tested in several numerical experiments. Our tests demonstrate that the model allows simulating different regimes of blow-up in a 3D complex domain. In particular, heat localisation is shown to lead to the breakup of the medium into individual fragments followed by the formation and development of self-organising patterns, which may have promising applications in thermonuclear fusion, nonlinear inelastic deformation and fracture of loaded solids and media and other areas.

Abstract: When solving the linear inviscid shallow water equations with variable depth in one dimension using finite differences, a tridiagonal system of equations must be solved. Here we present an approach, which is more efficient than the commonly used numerical method, to solve this tridiagonal system of equations using a recursion formula. We illustrate this approach with an example in which we solve for a rectangular channel to find the resonance modes. Our numerical solution agrees very well with the analytical solution. This new method is easy to use and understand by undergraduate students, so it can be implemented in undergraduate courses such as Numerical Methods, Lineal Algebra or Differential Equations.

Abstract: Unbounded solutions (critical blow-up regimes) simulated by the 3D nonlinear diffusion equation in a spherical shell are studied. The coordinate splitting of the differential operator coupled with two spherical coordinate maps makes it possible to use periodic boundary conditions in the latitudinal and longitudinal directions and employ the computationally efficient Sherman-Morrison formula and Thomas algorithm. The resulting finite difference method is direct, with implicit and unconditionally stable schemes of second-order approximation in all the variables. Numerical tests demonstrate that it allows simulating different blow-up regimes in complex computational domains.

Abstract: We have recently developed a vectorized Thomas solver for quasi-block tridiagonal linear algebraic equation systems using Streaming SIMD Extensions (SSE) and Advanced Vector Extensions (AVX) in operations on dense blocks [D. Barnaś and L. K. Bieniasz, Int. J. Comput. Meth., accepted]. The acceleration caused by vectorization was observed for large block sizes, but was less satisfactory for small blocks. In this communication we report on another version of the solver, optimized for small blocks of size up to four rows and/or columns.We have recently developed a vectorized Thomas solver for quasi-block tridiagonal linear algebraic equation systems using Streaming SIMD Extensions (SSE) and Advanced Vector Extensions (AVX) in operations on dense blocks [D. Barnaś and L. K. Bieniasz, Int. J. Comput. Meth., accepted]. The acceleration caused by vectorization was observed for large block sizes, but was less satisfactory for small blocks. In this communication we report on another version of the solver, optimized for small blocks of size up to four rows and/or columns.

Abstract: This paper aims at extending our previous work on the solution of the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). In the approach, we expand the spinor wavefunction in terms of suitable square integrable basis functions that support a tridiagonal matrix representation of the wave operator. This will transform the problem from solving a system of coupled first order differential equations to solving an algebraic three-term recursion relation for the expansion coefficients of the wavefunction. In some cases, solutions to this recursion relation can be related to well-known classes of orthogonal polynomials whereas in other situations solutions represent new class of polynomials. In this work, we will discuss various solvable potentials that obey the tridiagonal representation requirement with special emphasis on simple cases with spin-symmetric and pseudospin-symmetric potential couplings. We conclude by mentioning some potential applications in graphene.

Abstract: This work presents a simplified steady state one dimensional heat transfer model for stabilized premixed flames in porous inert media. Two energy conservation equations describe the heat transfer process in solid and fluid regions of a porous burner. The thermophysical properties are considered constant and a plug flow is adopted. The stabilized premixed flame acts as a heat source in a specified section of the domain. The energy conservation equations are discretized by the finite volume method, using upwind scheme on the convective terms and central difference scheme on the diffusive terms. The linear systems of algebraic equations are solved by Tridiagonal Matrix Algorithm (TDMA). The results are compared with experimental and theoretical data. The effects of the porosity, Peclet number and thermal conductivity ratio between the solid and the fluid on temperature fields are depicted. Furthermore, the results reveal that the model is able to represent superadiabatic flames and the heat recirculation process in the porous burner.

Abstract: In this work, we discuss two articles which Jonas Rimas published in 2005. We found that they have some interesting properties of a special type of tridiagonal matrices. These properties serve the purpose of computing the eigenvalues and eigenvectors of this type of matrices, and consequently in computing positive integer powers of such matrices.

Abstract: The aim of this work is to find exact solutions of the Dirac equation in 1+1 space-time beyond the already known class. We consider exact spin (and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus (and minus) the vector potential. We also include pseudo-scalar potentials in the interaction. The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis, which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric. This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.

Abstract: In this work, four numerical time-splitting methods are proposed for the (1 + 1)-dimensional nonlinear Dirac equation. All of these methods (or schemes) are proved to satisfy the charge conservation in the discrete level. To enhance the computation efficiency, the block Thomas algorithm is adopted. Numerical experiments are given to test the accuracy order for these schemes, to simulate numerically the binary collision including two standing waves and two moving solitons, meanwhile, the dynamic properties for the nonlinear Dirac equation are discussed. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2017

Abstract: This paper aims at extending our previous work on the solution of the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). In the approach, we expand the spinor wavefunction in terms of suitable square integrable basis functions that support a tridiagonal matrix representation of the wave operator. This will transform the problem from solving a system of coupled first order differential equations to solving an algebraic three-term recursion relation for the expansion coefficients of the wavefunction. In some cases, solutions to this recursion relation can be related to well-known classes of orthogonal polynomials whereas in other situations solutions represent new class of polynomials. In this work, we will discuss various solvable potentials that obey the tridiagonal representation requirement with special emphasis on simple cases with spin-symmetric and pseudospin-symmetric potential couplings. We conclude by mentioning some potential applications in graphene.