Abstract: The Darwin dynamical theory of diffraction for two beams yields a nonhomogeneous system of linear algebraic equations with a tridiagonal matrix. It is shown that different formulae of the two-beam Darwin theory can be obtained by a uniform view of the basic properties of tridiagonal matrices, their determinants (continuants) and their close relationship to continued fractions and difference equations. Some remarks concerning the relation of the Darwin theory in the three-beam case to tridiagonal block matrices are also presented.

Abstract: An algorithm to compute the eigenintervals of the generalized eigenvalue problem for tridiagonal symmetric matrices (A,B) with interval entries is developed. The algorithm is based on the Sturm Sequences and requires twice as much computational effort as for point matrices. The algorithm is applied to the problem of computing the natural frequencies of a vibrating beam where the manufacturing tolerance is taken into consideration.

Abstract: The dynamics of relativistic atomic wave functions evolving under the influence of intense laser pulses is used as an example of a general class of applications employing the alternating direction implicit method. The method requires the solution of many tridiagonal systems of linear equations. A range of parallel algorithms for this setting are analyzed with respect to their scalability on large parallel machines.

Abstract: The approximability of the resolvent of an operator induced by a band matrix by the resolvents of its finite-dimensional sections is studied. For bounded perturbations of self-adjoint matrices a positive result is obtained. The convergence domain of the sequence of resolvents can be described in this case in terms of matrices involved in the representation. This result is applied to tridiagonal complex matrices to establish conditions for the convergence of Chebyshev continued fractions on sets in the complex domain. In the particular case of compact perturbations this result is improved and a connection between the poles of the limit function and the eigenvalues of the tridiagonal matrix is established.

Abstract: This paper relates disconjugacy of linear Hamiltonian difference systems (LHdS) (and hence positive definiteness of certain discrete quadratic functionals) to positive definiteness of some block tridiagonal matrices associated with these systems and functionals. As a special case of a Hamiltonian system, Sturm--Liouville difference equations are considered, and analogous results are obtained for these important objects.

Abstract: The associated Hermite, Laguerre, Jacobi, and Bessel polynomials appear naturally when Bochner's problem [5] of characterizing orthogonal polynomials satisfying a second-order differential equation is extended to doubly infinite tridiagonal matrices. We obtain an explicit expression for the spectral matrix measure corresponding to the associated doubly infinite tridiagonal matrix in the Jacobi case. We show that, in an appropriate basis of \"bispectral\" functions, the spectral matrix can be put into a nice diagonal form, restoring the simplicity of the familiar orthogonality relations satisfied by the Jacobi polynomials.

Abstract: For a given real or complex polynomial p of degree n we modify the Euclidean algorithm to find a general tridiagonal matrix representation T of the monic version of p and then use the tridiagonal DQR eigenvalue algorithm on T in order to find all roots ofp with their multiplicities in O(n 2) operations and 0(n) storage. We include details of the implementation and comparisons with several, standard and recent, essentially 0(n 3) polynomial root finders.

Abstract: Componentwise error analysis for a parallel partitioning algorithm for tridiagonal systems is presented. Bounds on the equivalent perturbations are obtained, depending on three constants. Then bounds on the forward error are presented as well, depending on two types of condition numbers. Estimates on the first constant come directly from the roundoff error analysis of the tridiagonal Gaussian elimination [N. Higham, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 521--530]. In the present paper, the second and third constants are bounded for some special classes of matrices, i.e., diagonally dominant (row or column), symmetric positive definite, M-matrices, and totally nonnegative. One of the features of the analysis is that the exact forward and backward errors are bounded, not just their first order approximations, with respect to the machine precision. In all the bounds, the linear terms are given separately to show that the terms of higher order are small enough.

Abstract: In this study we propose a novel method to parallelize high-order compact numerical algorithms for the solution of three-dimensional PDEs in a space-time domain. For this numerical integration most of the computer time is spent in computation of spatial derivatives at each stage of the Runge-Kutta temporal update. The most efficient direct method to compute spatial derivatives on a serial computer is a version of Gaussian elimination for narrow linear banded systems known as the Thomas algorithm. In a straightforward pipelined implementation of the Thomas algorithm processors are idle due to the forward and backward recurrences of the Thomas algorithm. To utilize processors during this time, we propose to use them for either non-local data independent computations, solving lines in the next spatial direction, or local data-dependent computations by the Runge-Kutta method. To achieve this goal, control of processor communication and computations by a static schedule is adopted. Thus, our parallel code is driven by a communication and computation schedule instead of the usual ``creative programming" approach. The obtained parallelization speed-up of the novel algorithm is about twice as much as that for the standard pipelined algorithm and close to that for the explicit DRP algorithm.

Abstract: We show that the stability of Gaussian elimination with partial pivoting relates to the well definition of the reduced triangular systems. We develop refined perturbation bounds that generalize Skeel bounds to the case of ill conditioned systems. We finally develop reliable algorithms for solving general bidiagonal systems of linear equations with applications to the fast and stable solution of tridiagonal systems.

Abstract: In this note a comparison of the odd/even and stride of 3 reduction algorithms is presented for a periodic tridiagonal linear system.

Abstract: This paper gives a parallel algorithm,PPD algorithm, for the solution of diagonally dominant tridiagonal linear systems. Its computation complexity is about as same as the best sequential algorithm and its communication complexity is a constant. But now the computation and communication complexity of the best parallel algorithm are about 17 n and log P . It implements PPD algorithm on a MPP supercomputer. The results show that speedup improves linearly and the efficiency of our method is up to 90%.

Abstract: Recently, a generalized SOR method with multiple relaxation parameters were considered for solving a linear system of equations and it was shown that if a pair of parameter values is computed from the pivots of the Gaussian elimination applied to the system, then the spectral radius of the iterative matrix is reduced to zero. A proper choice of orderings and starting vectors for the iteration were also proposed.

Abstract: In this article, a fully scalable parallel algorithm is presented for solving symmetric tridiagonal eigenvalue problems using the quasi-Laguerre's method. The algorithm is implemented using Parallel Virtual Machine and is tested on a variety of matrices with a load balancing scheme. Test results show that the algorithm has high parallel efficiency. Compared with other existing algorithms, our algorithm seems to be the best for distributed memory parallel architecture.

Abstract: The main results of a componentwise error analysis for a parallel partitioning algorithm [4] in the cases of banded and tridiagonal linear systems are presented. It is shown that for some special classes of matrices, i.e. diagonally dominant (row or column), symmetric positive definite, and M-matrices, the algorithm is numericaly stable.

Abstract: A two-dimensional mathematical model was built to describe the melting process of cylindrical basalt particle bed in a crucible. The melting processes with respect to the factors of thermal boundary conditions and particle sizes of basalt were simulated by using the numerical method (FDM). The governing equations were discretized in tridiagonal matrix form and were solved by using the tridiagonal matrix algorithm (TDMA) as well as the alternative direction implicit (ADI) solver. The temperature distribution, the moving law of the two dimensional phase-change boundaries, the thermal current distribution were given through the numerical simulation.

Abstract: In this note, we present a numerical scheme for finding an approximate solution of an equation which can be viewed as a model for spatial diffusion of age-dependent biological populations. Discretization of the model yields a linear system with a block tridiagonal matrix. Our main concern will be discussion of stability for this scheme by examining the eigenvalues of the block tridiagonal matrix. Numerical results are presented.

Abstract: with cd ^ 0, other entries a^ being zero. Shin gives explicit formulas for eigenpairs ofmatrix (1) when the order n is odd, and implicit formulas, for n even.The purpose of this note is to draw attention to a paper [3], with which Shin wasobviously unfamiliar. In [3], the class of so-called tridiagonal 2-Toeplitz matrices isstudied. These are tridiagonal matrices that satisfy the relation(2) a

Abstract: It is well known that the inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is given by two sequences of real numbers, {ui} and {vi}, such that ci,j = u i vj for $i \leq j$. A similar result holds for nonsymmetric matrices A. There the inverse can be described by four sequences {ui},{vi}, {xi},$ and {vi} with u ivi = xiyi. Here we characterize certain properties of A, i.e., being an M-matrix or positive definite, in terms of the ui, vi,xi, and yi. We also establish a relation of zero row sums and zero column sums of A and pairwise constant ui,vi, xi, and yi. Moreover, we consider decay rates for the entries of the inverse of tridiagonal and block tridiagonal (banded) matrices. For diagonally dominant matrices we show that the entries of the inverse strictly decay along a row or column. We give a sharp decay result for tridiagonal irreducible M-matrices and tridiagonal positive definite matrices. We also give a decay rate for arbitrary banded M-matrices.

Abstract: We describe a parallel algorithm for finding the eigenvalues and eigenvectors of a dense symmetric matrix, with an emphasis on the dense linear algebra operations. We follow the traditional three-step process: reduce to tridiagonal form, solve the tridiagonal problem, then backtransform the result. Since the different steps have different algorithmic characteristics, this problem serves as a perfect vehicle for exploring some issues associated with parallel linear algebra calculations. In particular, we examine the effects of matrix distribution and blocking on the computational performance of tridiagonalization and backtransformation. Through experiments on an Intel Paragon, we demonstrate that block storage of the matrix is not necessary for a highly efficient block algorithm. The performance of our approach compares very favorably with that of the corresponding ScaLAPACK routines.

Abstract: We construct a Lyapunov function for tridiagonal competitive-cooperative systems. The same function is a Lyapunov function for Kolmogorov tridiagonal systems, which are defined on a closed positive orthant in Rn . We show that all bounded orbits converge to the set of equilibria. Moreover, we show that there can be no heteroclinic cycles on the boundary of the first orthant, extending the results of H. I. Freedman and H. L. Smith [Differential Equations Dynam. Systems, 3 (1995), pp. 367--382].