Abstract: Eigenvectors of the tridiagonal matrices of Sylvester type are explicitly determined. These are closely related to orthogonal polynomials named after Krawtchouk, (dual) Hahn and Racah as well as to q-Racah polynomials.

Abstract: The problem of generating a matrix A with specified eigenpairs, where A is a tridiagonal symmetric matrix, is presented. A general expression of such a matrix is provided, and the set of such matrices is denoted by S"E. Moreover, the corresponding least-squares problem under spectral constraint is considered when the set S"E is empty, and the corresponding solution set is denoted by S"L. The best approximation problem associated with S"E(S"L) is discussed, that is: to find the nearest matrix A@^ in S"E(S"L) to a given matrix. The existence and uniqueness of the best approximation are proved and the expression of this nearest matrix is provided. At the same time, we also discuss similar problems when A is a tridiagonal bisymmetric matrix.

Abstract: We present a matrix factorization called WZ factorization for the solution of symmetric tridiagonal linear systems. When combined with partitioning scheme, it renders a divide and conquer algorithm. Existence theorems are presented and backward error analysis is given. A variant of WZ factorization called WDZ factorization is also presented. Both WZ and WDZ algorithms for parallel solution of large tridiagonal symmetric positive definite linear systems are implemented on parallel machine with MPI as inter node communication.

Abstract: A mathematical model is chosen and a finite-difference algorithm is developed to describe the drying of agro products in a batch fluidized bed. The mathematical model is discretized according to the implicit scheme for two particle geometries: sphere and chip. The system of algebraic equations is solved using the tridiagonal matrix algorithm (TDMA). Stability and convergence of the numerical algorithm are analyzed by variation of the time step and mesh size, finding computer time savings when using nonuniform space steps and boundary conditions are discretized using three nodal positions. Obtained results are compared with numerical results from the literature and experimental data of potato chips drying, concluding that the chosen model describes the phenomena correctly.

Abstract: A method for the inversion of block tridiagonal matrices encountered in electronic structure calculations is developed, with the goal of efficiently determining the matrices involved in the Fisher–Lee relation for the calculation of electron transmission coefficients. The new method leads to faster transmission calculations compared to traditional methods, as well as freedom in choosing alternate Green’s function matrix blocks for transmission calculations. The new method also lends itself to calculation of the tridiagonal part of the Green’s function matrix. The effect of inaccuracies in the electrode self-energies on the transmission coefficient is analyzed and reveals that the new algorithm is potentially more stable towards such inaccuracies. 2007 Elsevier Inc. All rights reserved. PACS: 71.15. m; 02.70. c

Abstract: Two algorithms for computing the inverse factors of general tridiagonal and pentadiagonal matrices are obtained. Then, these algorithms are used for computing a block ILU preconditioner for the block tridiagonal linear system of equations. Some numerical results are given to show the robustness and efficiency of the preconditioner. The performance of the proposed preconditioner is compared with a recently proposed method.

Abstract: The problem of constructing a hyperbolic interpolation spline can be formulated as a differential multipoint boundary value problem. Its discretization yields a linear system with a five-diagonal matrix, which may be ill-conditioned for unequally spaced data. It is shown that this system can be split into diagonally dominant tridiagonal systems, which are solved without computing hyperbolic functions and admit effective parallelization.

Abstract: We show that certain integral positive definite symmetric tridiagonal matrices of determinant $n$ are in one to one correspondence with elements of $(\mathbb Z/n\mathbb Z)^*$ We study some properties of this correspondence In a somewhat unrelated second part we discuss a construction which associates a sequence of integral polytopes to every integral symmetric matrix

Abstract: In this note we show how to improve some recent upper and lower bounds for the elements of the inverse of diagonally dominant tridiagonal matrices. In particular, a technique described by (R. Peluso, and T. Politi, Some improvements on two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Lin. Alg. Appl. Vol. 330 (2001) 1-14), is used to obtain better bounds for the diagonal elements.

Abstract: Beginning with the method of whole path iterative ray-tracing and according to the positive definiteness of the coefficient matrix of the systems of linear equations, a symmetry block tridiagonal matrix was decomposed into the product of block bidiagonal triangular matrix and its transpose by means of Cholesky decomposition. Then an algorithm for solving systems of block bidiagonal triangular linear equations was given, which is not necessary to treat with the zero elements out of banded systems. A fast algorithm for solving the systems of symmetry block tridiagonal linear equations was deduced, which can quicken the speed of ray-tracing. Finally, the simulation based on this algorithm for ray-tracing in three dimensional media was carried out. Meanwhile, the segmentally-iterative ray-tracing method and banded method for solving the systems of block tridiagonal linear equations were compared in the same model mentioned above. The convergence condition was assumed that the L-2 norm summation for mk, 1 and mk, 2 in the whole ray path was limited in 10−6. And the calculating speeds of these methods were compared. The results show that the calculating speed of this algorithm is faster than that of conventional method and the calculated results are accurate enough. In addition, its precision can be controlled according to the requirement of ray-tracing

Abstract: The aim of this paper is to show that a kind of boundary value problem for second-order ordinary differential equations which reduces to the problem of solving tridiagonal systems of linear equations can be efficiently solved on modern multicore computer architectures. A new method for solving such tridiagonal systems of linear equations, based on recently developed algorithms for solving linear recurrence systems with constant coefficients, can be easily vectorized and parallelized. Further improvements can be achieved when novel data formats for dense matrices are used.

Abstract: We discuss the exact solvability of a lattice model with a flow within the matrix product ansatz to interacting many-body systems from the point of view of a tridiagonal boundary algebra. It reveals deep symmetry properties of the multiparticle process which allow for the exact solution in the stationary state and puts the description of the dynamics into perspective.

Abstract: Determination of effective diffusivity of a substrate in biocatalyst particles is a key requirement in modeling heterogeneous reactions. The diffusivity is mainly controlled by the molecular-diffusion characteristics of the substrate, the tortuousness of the diffusion path within the biocatalyst, and the void fraction of particle volume available for diffusion. A general numerical model describing reactions in a biocatalyst particle following zero-order, first-order, and Michaelis-Menten kinetics was developed. Finite volume method was used to discretize the nonlinear second-order differential equation, and tridiagonal matrix algorithm was applied to solve the algebraic equation iteratively after discretization. Computer codes were written for calculating the diffusivity under specified boundary conditions. The numerical solution of the diffusion-reaction equations was validated against the experimental data from literatures. The model was further calibrated with experimental data obtained from fungal pellet experiments and then verified using additional data. The results show that the inverse methodology developed in this study was capable of predicting diffusivity in biocatalyst particles. Based on the predicted diffusivity, oxygen consumption by an individual pellet was simulated, oxygen consumptions by small versus large pellets were compared, and the effect of reaction rate on oxygen consumption was evaluated. © 2008 American Institute of Chemical Engineers AIChE J, 2008

Abstract: It appears that large scale calculations in particle physics often require to solve systems of linear equations with rational number coefficients exactly. If classical Gaussian elimination is applied to a dense system, the time needed to solve such a system grows exponentially in the size of the system. In this tutorial paper, we present a standard technique from computer algebra that avoids this exponential growth: homomorphic images. Using this technique, big dense linear systems can be solved in a much more reasonable time than using Gaussian elimination over the rationals.

Abstract: The eigenvalue problem for a certain tridiagonal matrix with complex coefficients is considered The eigenvalues and eigenvectors are shown to be expressible in terms of solutions of a certain scalar trigonometric equation Explicit solutions of this equation are obtained for several special cases, and further analysis of this equation in several other cases provides information about the distribution of eigenvalues

Abstract: We give explicit expressions for the eigenvalues and eigenvectors of some tridiagonal matrices with non-constant diagonal entries. Our techniques are based on the theory of recurrent sequences.

Abstract: Publisher Summary This chapter presents some of the basic computational techniques that can be employed to solve the governing equations of fluid dynamics. The first stage of obtaining the computational solution involves the conversion of the governing equations into a system of algebraic equations. This is usually known as the discretization stage. It discusses some of the discretization tools, such as the finite- difference and finite-volume methods, which form the foundation of understanding the basic features of discretization. Both of these methods are abundant in many CFD applications. The second stage involves numerically solving the system of algebraic equations, which can be achieved by either the direct methods or iterative methods. Basic direct methods such as the Gaussian elimination and the Thomas algorithm are discussed. Simple iterative methods such as the point-by-point Jacobi and Gauss-Siedel methods are also described. Nevertheless, CFD problems are generally multidimensional and comprise a large system of equations to be solved. Efficient iterative methods such as the ADI or Stone's SIP are often applied to solve such a system of equations. To further enhance the convergence of the computational solution, precondition conjugate gradient methods or multigrid methods are employed to accelerate the iteration process. Finally, this chapter discusses the assessment of convergence. In practice, the algebraic equations that result from the discretization process yield the flow solution at each nodal point on a finite-grid layout.