Abstract: A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described The overlap functions between the two dual basis are shown to satisfy a coupled system of recurrence relations and a set of discrete second-order $q-$difference equations which generalize the ones associated with the Askey-Wilson orthogonal polynomials with a discrete argument Normalizing the fundamental solution to unity, the hierarchy of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given

Abstract: We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the 'level' process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Abstract: The pivoting strategy of Bunch and Marcia for solving systems involving symmetric indefinite tridiagonal matrices uses two different methods for solving 2 × 2 systems when a 2 × 2 pivot is chosen. In this paper, we eliminate this need for two methods by adding another criterion for choosing a 1 × 1 pivot. We demonstrate that all the results from the Bunch and Marcia pivoting strategy still hold. Copyright © 2006 John Wiley & Sons, Ltd.

Abstract: We propose multisection for the multiple eigenvalues (MME) method for determining the eigenvalues of symmetric tridiagonal matrices. We also propose a method using runtime optimization, and show how to optimize its performance by dynamically selecting the implementation parameters. Performance results using a Hitachi SR8000 supercomputer with eight processors per node yield (1) up to 6.3x speedup over a conventional multisection method, and (2) up to 1.47x speedup over a statically optimized MME method.

Abstract: : Dielectric Barrier Discharge (DBD) type devices, when used as plasma actuators, have shown significant promise for use in many aeronautical applications. Experimentally, DBD actuator devices have been shown to induce motion in initially still air, and to cause re-attachment of air flow over a wing surface at a high angle of attack. This thesis explores the numerical simulation of the DBD device in both a lD and 2D environment. Using well established fluid equation techniques, along with the appropriate approximations for the regime under which these devices will be operating, computational results for various conditions and geometries are explored. In order to validate the code, results are compared to analytic or experimental data whenever possible, or matched with other similar numeric simulations to help establish the accuracy of the code. Solutions to Poisson's equation for the potential, electron and ion continuity equations, and the electron energy equation are solved semi-implicitly in a sequential manner. Each of the governing equations is solved by casting them into a tridiagonal grid, and using the computationally efficient Thomas algorithm to solve lD regions in a single iteration. The Scharfetter-Gummel flux discretization method is used to add stability to the code when transitioning from a field to diffusion dominated region or vice versa. Estimates for the ionization and recombination rates and for the transport coefficients of the background gas are calculated as a function of the local average electron energy, and updated for every calculation point in the domain on the completion of the solution to the electron energy equation.

Abstract: In this paper,the inverse of a tridiagonal matrix is investigated.By the LU and UL decompositions of a tridiagonal matrix and the special structure of the inverse matrix,an algorithm for inverting a tridiaognal matrix and the explicit expression of the elements of the inverse matrix are presented.The computing complexity and the computing time of this algorithm is lower than those of some existed algorithms for inverting a block tridiaognal matrix.

Abstract: The alternating-direction-implicit finite-difference time-domain method (ADI-FDTD) is considered as a very efficient algorithm. The key problem of the implementation of the ADI-FDTD method is to solve the tridiagonal matrix system. Two numerical algorithms for the tridiagonal matrix system are analyzed in detail in this paper. The interpretations for the stability of the algorithms are also given clearly

Abstract: The general expression of the l-th power (l ∈ N, l ≥ 2) for one type of tridiagonal matrices of order n = 2p (p ∈ N, p ≥ 2) with zeros in the first row is derived. Expressions of eigenvectors of the matrix and of the transforming matrix and its inverse are given, too.

Abstract: Computation of the eigenvalues of a symmetric tridiagonal matrix is a problem of great relevance. Many linear algebra libraries provide subroutines for solving it. But none of them is oriented to be executed in heterogeneous distributed memory multicomputers. In this work we focus on this kind of platforms. Two different load balancing schemes are presented and implemented. The experimental results show that only the algorithms that take into account the heterogeneity of the system when balancing the workload obtain optimum performance. This fact justifies the need of implementing specific load balancing techniques for heterogeneous parallel computers.

Abstract: In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix $$T$$ satisfies property A, one can easily construct a diagonal matrix $$\hat{D}$$ such that $$\hat{D}T\hat{D}$$ has the lowest condition number over all matrices $$DTD$$ , for any choice of diagonal matrix $$D$$ . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form.

Abstract: The four existing stable factorization methods for symmetric indefinite matrices suffer serious defects when applied to banded matrices. Partial pivoting (row or column exchanges) maintains a band structure in the reduced matrix and the factors, but destroys symmetry completely once an off-diagonal pivot is used. Two-by-two block pivoting and Gaussian reduction to tridiagonal (Aasen's algorithm) maintain symmetry at all times, but quickly destroy the band structure in the reduced matrices. Orthogonal reductions to tridiagonal maintain both symmetry and the band structure, but are too expensive for linear-equation solvers. We propose a new pivoting method, which we call snap-back pivoting. When applied to banded symmetric matrices, it maintains the band structure (like partial pivoting does), it keeps the reduced matrix symmetric (like 2-by-2 pivoting and reductions to tridiagonal), and it is fast. Snap-back pivoting reduces the matrix to a diagonal form using a sequence of elementary elimination steps, most of which are applied symmetrically from the left and from the right (but some are applied unsymmetrically). In snap-back pivoting, if the next diagonal element is too small, the next pivoting step might be unsymmetric, leading to asymmetry in the next row and column of the factors. But the reduced matrix snaps back to symmetry once the next step is completed.