Abstract: A type of tridiagonal pair is considered, said to be mild of q-Serre type.It is shown that these tridiagonal pairs induce the structure of a quantum affine algebra Uq( sl2)-module on their underlying vector space.This is done by presenting an explicit basis for the underlying vector space and describing the Uq( sl2)-action on that basis.

Abstract: We analyze the residuals of GMRES [Y Saad and M H Schultz, SIAM J Sci Statist Comput, 7 (1986), pp 856--859], when the method is applied to tridiagonal Toeplitz matrices We first derive formulas for the residuals as well as their norms when GMRES is applied to scaled Jordan blocks This problem has been studied previously by Ipsen [BIT, 40 (2000), pp 524--535] and Eiermann and Ernst [Private communication, 2002], but we formulate and prove our results in a different way We then extend the (lower) bidiagonal Jordan blocks to tridiagonal Toeplitz matrices and study extensions of our bidiagonal analysis to the tridiagonal case Intuitively, when a scaled Jordan block is extended to a tridiagonal Toeplitz matrix by a superdiagonal of small modulus (compared to the modulus of the subdiagonal), the GMRES residual norms for both matrices and the same initial residual should be close to each other We confirm and quantify this intuitive statement We also demonstrate principal difficulties of any GMRES convergence analysis which is based on eigenvector expansion of the initial residual when the eigenvector matrix is ill-conditioned Such analyses are complicated by a cancellation of possibly huge components due to close eigenvectors, which can prevent achieving well-justified conclusions

Abstract: A very efficient vectorized code is tailored to solve 3-D incompressible Navier-Stokes equations for mixed-convection flows in high streamwise aspect ratio channels. It is based on Goda's algorithm, second-order finite differences, an incremental factorization method of alternating direction implicit (ADI) type, spectral decomposition of the 1-D Laplace operators, and the tridiagonal matrix algorithm (TDMA). It is shown to be of second order in both space and time by a general method of determining code convergence orders and to have good performance on a NEC-SX5 supercomputer. It is validated through experiments of various Poiseuille-Rayleigh-Benard flows with steady longitudinal, unsteady transverse, and convectively unstable wavy rolls.

Abstract: The LBLT factorization of Bunch for solving linear systems involving a symmetric indefinite tridiagonal matrix T is a stable, efficient method. It computes a unit lower triangular matrix L and a block 1 × 1 and 2 × 2 matrix B such that T=LBLT. Choosing the pivot size requires knowing a priori the largest element σ of T in magnitude. In some applications, it is required to factor T as it is formed without necessarily knowing σ. In this paper, we present a modification of the Bunch algorithm that can satisfy this requirement. We demonstrate that this modification exhibits the same bound on the growth factor as the Bunch algorithm and is likewise normwise backward stable. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: In this paper, a new class of parallel Gaussian elimination algorithms is presented for the solution of tridiagonal matrix systems. The new algorithms, called ACER (alternating cyclic elimination and reduction), combine the advantages of the well known cyclic elimination algorithm (which is fast) and the cyclic reduction algorithms (which requires fewer operations). The ACER algorithms are developed with the unifying graph model.

Abstract: The development of the discrete fractional Fourier transform (DFRFT) necessitates having orthonormal eigenvectors for the DFT matrix, F. The objective of having the DFRFT approximate its continuous counterpart can be met if the eigenvectors of F approximate samples of the Hermite-Gaussian functions. Orthonormal Hermite-Gaussian-like eigenvectors for F are rigorously derived by a detailed analysis of an almost tridiagonal matrix, S, which commutes with F. By an appropriate similarity transformation, S is reduced to a 2/spl times/2 block diagonal form and the elements of the two exactly tridiagonal matrices forming the two diagonal blocks are explicitly derived in terms of the elements of matrix S.

Abstract: For MinRes and SymmLQ it is essential to compute a QR decomposition of a tridiagonal coefficient matrix gained in the Lanczos process. This QR decomposition is constructed by an update scheme applying in every step a single Givens rotation. Using complex Householder reflections we generalize this idea to block tridiagonal matrices that occur in generalizations of MinRes and SymmLQ to block methods for systems with multiple right-hand sides. Some implementation details are given, and we compare the method with an algorithm based on Givens rotations used in block QMR. Our approach generalizes to the QR decomposition of upper block Hessenberg matrices resulting from the block Arnoldi process and is applicable in block GMRes. Keywords— block Lanczos process, block Krylov space methods. I. The symmetric Lanczos algorithm In 1975 Christopher Paige and Michael Saunders [PAI 75] proposed two iterative Krylov subspace methods called MinRes and SymmLQ for solving sparse Hermitian indefinite linear systems

Abstract: When a matrix is reduced to Lanczos tridiagonal form, its matrix elements can be divided into an analytic smooth mean value and a fluctuating part. The next-neighbor spacing distribution P(s) and the spectral rigidity Delta _(3) are shown to be universal functions of the average value of the fluctuating part. It is explained why the behavior of these quantities suggested by random matrix theory is valid in far more general cases.

Abstract: An algorithm for computing the inverse of a general tridiagonal matrix is introduced. This algorithm is obtained by factoring this matrix into the product of two bidiagonal matrices using Crout’s LU factorization, one upper and one lower bidiagonal. A simple recurrence relation is used to generate a sequence of numbers, this sequence is then used to fill in the matrices L, U, L −1, U −1 and consequently the required inverse.

Abstract: Differential qd (dqd) algorithm with shifts is probably the fastest known algorithm which computes eigenvalues of symmetric tridiagonal matrices with high relative accuracy. In this paper we will construct a similar algorithm for computing eigenvalues of skew-symmetric matrices, which is based on implicit usage of both the QR and the symplectic QR factorizations. If we apply this algorithm to tridiagonal skew-symmetric matrices, we obtain the skew-symmetric dqd algorithm. This algorithm also enjoys high relative stability. However, incorporation of shifts is much harder then in the symmetric case, and yet to be implemented. Finally, the standard algorithm for computing the eigenvalues of tridiagonal skew-symmetric matrices can also be interpreted in the context of the skew-symmetric dqd algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Abstract: An efficient implicit dynamic finite element method (FEM) for elastic 3D objects with uniform cross-sections was developed. In this method, the finite element mesh is generated in such a way that the object to be analysed is at first sliced into layers with the same thickness along its generatrix and then each layer is discretized into finite elements of the same pattern. This way of discretization makes the mass, viscosity, and stiffness matrices into the repetitive block tridiagonal matrices. The repetitive block tridiagonal matrix has the characteristic, that the sequence of matrices which appears in the Gaussian elimination for the repetitive block tridiagonal matrix is a rapid convergent sequence. The process of the Gaussian elimination can be terminated when the sequence converges. The rest of the sequence is not necessary to be stored. The present method can save the computational time and memory by utilising this characteristic of the repetitive block tridiagonal matrix. A few examples of analyses including whole Hopkinson-bar analysis were performed to demonstrate the effectiveness of the present method. The present method is applicable not only to the elasto-dynamics but also to many other problems, such as thermal problems, electrical problems, and plastic problems without geometric non-linearity. Copyright © 2005 John Wiley & Sons, Ltd.

Abstract: The inverse of a class of block tridiagonal matrices is investigated With the LU decomposition of the block tridiagonal matrix,an explicit expression of the block inverse elements is obtained A relation between the inverse elements is found,and a new algorithm for inverting a block tridiagonal matrix is established The computing complexity and computing time of this algorithm is lower than that of existed algorithms

Abstract: A method for computing orthogonal URV/ULV decompositions of block tridiagonal (or banded) matrices is presented. The method discussed transforms the matrix into structured triangular form and has several attractive properties: The block tridiagonal structure is fully exploited; high data locality is achieved, which is important for high efficiency on modern computer systems; very little fill-in occurs, which leads to no or very low memory overhead; and in most practical situations observed the transformed matrix has very favorable numerical properties. Two variants of this method are introduced and compared.

Abstract: The objective of the present work is to study the solution of quaternion block quasi-tridiagonalsystems. Kershaw and Rozsa and Romani have proposed a method for calculating matrix inverses using second kind Chebyshev polynomials. This method has been generalized later to block tridiagonal matrices with quaternionic entries by Costa and Serodio. In the present work, we make use of this method to solve block quasi-tridiagonal systems of the same kind. The computational effort to obtain the solution is evaluated, and future more efficient strategies are proposed.

Abstract: This work examines the relation between Gaussian elimination and the conjugate directions algorithm [Hestenes and Steifel, 1952]. Analysis is extended to the case where the sequence of the conjugated vectors is modified, which is shown to result in reordering of the solution vector. Based on these analyses an algorithm is described which combines Gaussian elimination with a look-ahead algorithm. The purpose of the algorithm is to employ Gaussian elimination on a system of smaller order and to use this solution to approximate the solution of the original system. The algorithm was tested on a range of linear systems and performed well when the components in the solution vector varied by large magnitude.

Abstract: We obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k−Toeplitz matrix A. The proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of Chebyshev polynomials of the second kind. We also compute the characteristic polynomial of A which enables us to state some conditions for the existence of A−1. Our results also extend known results for the case when the residue mod k of the order of A is equal to 0 or k−1 (Numer. Math., 10 (1967), pp. 153–161.).

Abstract: Tridiagonal matrices appear frequently in mathematical models In this note, we derive the eigenvalues and the corresponding eigenvectors of several tridiagonal matrices by the method of symbolic calculus in (1)