Abstract: A parallel solver for general tridiagonal irreducible systems is described.Solver based on Spike framework and Givens-QR with occasional low-rank modification.Modifications handle singularities exposed by QR in blocks of the parallel partition.The GPU implementation has similar performance to existing methods.Method returns accurate results when current GPU tridiagonal solvers fail. g-Spike, a parallel algorithm for solving general nonsymmetric tridiagonal systems for the GPU, and its CUDA implementation are described. The solver is based on the Spike framework, applying Givens rotations and QR factorization without pivoting. It also implements a low-rank modification strategy to compute the Spike DS decomposition even when the partitioning defines singular submatrices along the diagonal. The method is also used to solve the reduced system resulting from the Spike partitioning. Numerical experiments with problems of high order indicate that g-Spike is competitive in runtime with existing GPU methods, and can provide acceptable results when other methods cannot be applied or fail.

Abstract: Two symbolic algorithms for inverting k -tridiagonal matrices have been recently found by El-Mikkawy and Atlan (2014, 2015). These two algorithms are mainly based on the Doolittle LU factorization of the k -tridiagonal matrix. In the current paper, we present a new explicit analytic expression for the inverses of general tridiagonal matrices at first. By using a block diagonalization technique, we then relate k -tridiagonal matrix inversion to tridiagonal matrix inversion. Meanwhile, an efficient algorithm is derived for computing the inverses of nonsingular k -tridiagonal matrices with the help of any algorithm for computing the inverses of tridiagonal matrices. Three examples are given in order to illustrate the performance and efficiency of the proposed algorithms.

Abstract: In this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layers. A fitting factor is introduced and the model equation is discretized by a finite difference scheme on an uniform mesh. Thomas algorithm is used to solve the tri-diagonal system. The stability of algorithm is investigated. It is shown that proposed technique provides first order accuracy independent of perturbation parameter. Linear and nonlinear problems are solved by proposed method and numerical results are presented to illustrate the present technique.

Abstract: In this paper, we give some relations in terms of k- Balancing number which generalize some well known results concerning the relation between the determinant and Chebyshev polynomials which is due to tridiagonal matrix B(n)(k). Also for the other tridiagonal matrix W(n)(k); we deduce the cofactor matrix of it then we nd another relations for k- Balancing number.

Abstract: Summary We study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off-diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner. Copyright © 2014 John Wiley & Sons, Ltd.

Abstract: An optimized implementation of a block tridiagonal solver based on the block cyclic reduction (BCR) algorithm is introduced and its portability to graphics processing units (GPUs) is explored. The computations are performed on the NVIDIA GTX480 GPU. The results are compared with those obtained on a single core of Intel Core i7-920 (2.67 GHz) in terms of calculation runtime. The BCR linear solver achieves the maximum speedup of 5.84x with block size of 32 over the CPU Thomas algorithm in double precision. The proposed BCR solver is applied to discontinuous Galerkin (DG) simulations on structured grids via alternating direction implicit (ADI) scheme. The GPU performance of the entire computational fluid dynamics (CFD) code is studied for different compressible inviscid flow test cases. For a general mesh with quadrilateral elements, the ADI-DG solver achieves the maximum total speedup of 7.45x for the piecewise quadratic solution over the CPU platform in double precision.

Abstract: While dealing in [1] with the supersymmetry of a tridiagonal Hamiltonian H, we have proved that its partner Hamiltonian H(+) also have a tridiagonal matrix representation in the same basis and that the polynomials associated with the eigenstates expansion of H(+) are precisely the kernel polynomials of those associated with H. This formalism is here applied to the case of the Morse oscillator which may have a finite discrete energy spectrum in addition to a continuous one. This completes the treatment of tridiagonal Hamiltonians with pure continuous energy spectrum, a pure discrete one, or a spectrum of mixed discrete and continous parts.

Abstract: In the current paper, the authors present a symbolic algorithm for solving doubly bordered k-tridiagonal linear system having n equations and n unknowns. The proposed algorithm is derived by using partition together with UL factorization. The cost of the algorithm is O(n). The algorithm is implemented using the computer algebra system, MAPLE. Some illustrative examples are given.

Abstract: In this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer at left (or right) end of the domain. A fitting factor is introduced and the model equation is discretized by a finite difference scheme on an uniform mesh. Thomas algorithm is used to solve the tri-diagonal system. The stability of the algorithm is investigated. It is shown that proposed technique provides first order accuracy independent of the perturbation parameter. Several linear and nonlinear problems are solved by the proposed method and numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.

Abstract: First, the Cartesian and generalized coordinate systems and the coordinate transformation are introduced. We also discuss the method of virtual boundaries and the need to introduce a forcing term to represent the geometry. Next, we present the formulation for low Mach number, valid for most cases of reactive flows. Then the large-eddy simulations formulation is discussed as an improvement of Reynolds averaged Navier-Stokes (RANS) formulation, which is less expensive than direct numerical simulations (DNS) formulation. Subsequently, for a reactive flow model, the equations of momentum, energy, enthalpy, and chemical species are written as a general equation, which is approximated by methods of finite difference, finite volume, and finite element, to be integrated by Runge-Kutta methods. After that, approximations of order 3 and 4 are given, as well as some compact schemes of order of approximation 6. Then, we discuss some of the main methods used in the flow solution such as Gauss-Seidel, simplified Runge-Kutta, tridiagonal matrix algorithm (TDMA), Newton, strongly modified implicit procedure (MSI), and LU-SSOR, which is an LU decomposition with the introduction of dissipation. Then, we indicate some methods for solving stiff systems of equations, such as Newton’s method and Rosenbrock’s method, which can be seen as a combination of the methods of Newton and Runge-Kutta. After that, the principal boundary conditions, such as permeable and impermeable wall, symmetry and cut, far field and periodic are given, which are common in jet diffusion flames, and in reactive flows in porous media. Finally, some techniques for the acceleration of convergence as local time-stepping, residual smoothing, and the multigrid technique are introduced. Moreover, some numerical implementation details and the analysis of uncertainties for the solution of reactive flows is discussed.

Abstract: Chain models can be represented by a tridiagonal matrix with periodic entries along its diagonals. Eigenmodes of open chains are represented by spectra of such tridiagonal $k$-Toeplitz matrices, where $k$ represents length of the repeated unit, allowing for a maximum of $k$ distinct types of elements in the chain. We present an analysis that allows for generality in $k$ and values in $\mathbb{C}$ representing elements of the chain, including non-Hermitian systems. Numerical results of spectra of some special $k$-Toeplitz matrices are presented as a motivation. This is followed by analysis of a general tridiagonal $k$-Toeplitz matrix of increasing dimensions, beginning with 3-term recurrence relations between their characteristic polynomials involving a $k^{th}$ order coefficient polynomial, with the variables and coefficients in $\mathbb{C}$. The existence of limiting zeros for these polynomials and their convergence are established, and the conditioned $k^{th}$ order coefficient polynomial is shown to provide a continuous support for the limiting spectra representing modes of the chain. This analysis also includes the at most $2k$ eigenvalues outside this continuous set. It is shown that this continuous support can as well be derived using Widom's conditional theorems (and its recent extensions) for the existence of limiting spectra for block-Toeplitz operators, except in special cases. Numerical examples are used to graphically demonstrate theorems. As an addendum, we derive expressions for $O(k)$ computation of the determinant of tridiagonal $k$-Toeplitz matrices of any dimension.

Abstract: In this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer. A fitting factor is introduced and the model equation is discretized by a finite difference scheme on an uniform mesh. Thomas algorithm is used to solve the tri-diagonal system. The stability of the algorithm is investigated. It is shown that the proposed technique is of first order accurate and the error constant is independent of the perturbation parameter. Several problems are solved and numerical results are presented to support the theoretical error bounds established.

Abstract: Tridiagonal matrices are considered for which the main diagonal consists of zeroes, the sup-diagonal of all ones, and the entries on the sub-diagonal form a geometric progression. The criterion for the numerical range of such matrices to have line segments on its boundary is established, and the number and orientation of these segments is described.

Abstract: In this work, we present a stable parallel algorithm based on WZ factorization for solving diagonally dominant tridiagonal linear system of algebraic equations, using divide and conquer approach. Existence results are given and the backward error analysis of the method is presented. Numerical stability of the algorithm is proved. The given parallel algorithm for diagonally dominant tridiagonal linear systems is compared with the Truncated SPIKE version of the SPIKE algorithm [12].

Abstract: The decomposition method which makes the parallel solution of the block-tridiagonal matrix systems possible is presented. The performance of the method is analytically estimated based on the number of elementary multiplicative operations for its parallel and serial parts. The computational speedup with respect to the conventional sequential Thomas algorithm is assessed for various types of the application of the method. It is observed that the maximum of the analytical speedup for a given number of blocks on the diagonal is achieved at some finite number of parallel processors. The values of the parameters required to reach the maximum computational speedup are obtained. The benchmark calculations show a good agreement of analytical estimations of the computational speedup and practically achieved results. The application of the method is illustrated by employing the decomposition method to the matrix system originated from a boundary value problem for the two-dimensional integro-differential Faddeev equations. The block-tridiagonal structure of the matrix arises from the proper discretization scheme including the finite-differences over the first coordinate and spline approximation over the second one. The application of the decomposition method for parallelization of solving the matrix system reduces the overall time of calculation up to 10 times.

Abstract: In this paper, we give some relations in terms of k Balancing number which generalize some well known results concerning the relation between the determinant and Chebyshev polynomials which is due to tridiagonal matrix B(n)(k). Also for the other tridiagonal matrix W(n)(k); we deduce the cofactor matrix of it then we nd another relations for k Balancing number.

Abstract: In this paper , the quintic B-spline collocation scheme is employed to approximate numerical solution of the KdV-like Rosenau equation . This scheme is based on the Crank-Nicolson formulation for time integration and quintic B-spline functions for space integration . The unconditional stability of the present method is proved using Von- Neumann approach . Since we do not know the exact solution of the nonlinear KdV-like Rosenau equation , a comparison between the numerical solutions on a coarse mesh and those on a refine mesh is made to show the efficiency of discussed method.

Abstract: In this paper, we discuss the solution of singularly perturbed differential-difference equations exhibiting dual layer using the higher order finite differences. First, the second order singularly perturbed differential-difference equations is replaced by an asymptotically equivalent second order singular perturbed ordinary differential equation. Then, fourth order stable finite difference scheme is applied to get a three term recurrence relation which is easily solved by Thomas algorithm. Some numerical examples have been solved to validate the computational efficiency of the proposed numerical scheme. To analyze the effect of the parameters on the solution, the numerical solution has also been plotted using graphs. The error bound and convergence of the method have also been established.

Abstract: A fitted-stable central difference method is presented for solving singularly perturbed two point boundary value problems with the boundary layer at one end (left or right) of the interval. A fitting factor is introduced in second order stable central difference scheme (SCD Method) and its value is obtained using the theory of singular perturbations. Thomas Algorithm (also known as Discrete Invariant Imbedding Algorithm) is used to solve the resulting tri-diagonal system. To validate the applicability of the method, some linear and non-linear examples have been solved for different values of the perturbation parameter. The numerical results are tabulated and compared with exact solutions. The error bound and convergence of the proposed method has also been established. From the results, it is observed that the present method approximates the exact solution very well. Key words : Singular perturbation problems, stable, central differences, fitted methods

Abstract: In the present paper numerical discusses a theoretical model for instability phenomenon in double phase flow through homogeneous porous medium. Relation between relative permeability and saturation has been considered based on earlier experiment. A governing nonlinear partial differential equation is solved by collocation method with cubic B-splines. To obtain the scheme of the equation the nonlinear term is approximated by Taylor series which leads to tridiagonal system and has been solved by well-known Thomas Algorithm. The Numerical solution is obtained by using MATLAB coding.

Abstract: The Generalized Minimal Residual method (GMRES) is often used to solve a large and spars system Ax=b. This paper establishes error bound for residuals of GMRES on solving a normal tridiagonal Toeplitz linear system. this problem has been studied previously by Li [R.-C. Li, Convergence of CG and GMRES on a tridiagonal Toeplitz linear system, BIT 47 (3) (2007) 577-599.], for two special right-hand sides. Also, Li and Zhang [R.-C. Li, W. Zhang, The rate of convergence of GMRES on a tridiagonal Toeplitz linear system, Numer . Math. 112 (2009) 267-293.] for non-symmetric matrix $A$, presented upper bound for GMRES residuals. But in this paper we establish the upper bound on normal tridiagonal Toeplitz linear systems for another special right-hand sides.

Abstract: Traditionally, solving the parameter identification inverse problems of partial differential equations encountered many difficulties and insufficiency. In this paper, we propose an improved GEP (Gene Expression Programming) to identify the parameters in the reverse problems of partial differential equations based on the self-adaption, self-organization and self-learning characters of GEP. This algorithm simulates a parametric function itself of a partial differential equation directly through the observed values by fully taking into account inverse results caused by noises of a measured value. Modeling is unnecessary to add regularization in the modeling process aiming at special problems again. The experiment results show that the algorithm has good noise-immunity. In case there is no noise or noise is very low, the identified parametric function is almost the same as the original accurate value; when noise is very high, good results can still be obtained, which successfully realizes automation of the parameter modeling process for partial differential equations.

Abstract: In this paper, the cubic B-spline collocation scheme is implemented to find numerical solution of diffusion convection problem of chemical engineering. The scheme is based on the Crank–Nicolson formulation for time integration and cubic B-spline functions for space integration. The numerical results are found to be in good agreement with the exact solutions. Results are also shown graphically and are compared with results given in the literature. Keywords: Diffusion; Cubic B-spline; Collocation; Tridiagonal system; Thomas algorithm.

Abstract: We discuss the convergence rate of the QR algorithm with Wilkinson’s shift for tridiagonal symmetric eigenvalue problems It is well known that the convergence rate is theoretically at least quadratic, and practically better than cubic for most matrices In an effort to derive the convergence rate, the limiting patterns of some lower right submatrices have been intensively investigated In this paper, we first describe a new limiting pattern of the lower right 3-by-3 submatrix with a concrete example, and then prove that the convergence rate of this new pattern is strictly cubic In addition, we stress that our analysis identifies three classes of the limiting patterns of the tridiagonal QR algorithm with Wilkinson’s shift