Abstract: We provide a new representation for the inverse of block tridiagonal and banded matrices. The new representation is shown to be numerically stable over a variety of block tridiagonal matrices, in addition of being more computationally efficient than the previously proposed techniques. We provide two algorithms for commonly encountered problems that illustrate the usefulness of the results.

Abstract: We consider the following inverse eigenvalue problem: to construct a symmetrical tridiagonal matrix from the minimal and maximal eigenvalues of all its leading principal submatrices. We give a necessary and sufficient condition for the existence of such a matrix and for the existence of a nonnegative symmetrical tridiagonal matrix. Our results are constructive, in the sense that they generate an algorithmic procedure to construct the matrix.

Abstract: Graphical Processing Units (GPUs) potentially promise widespread and inexpensive high performance computation. However, architectural limitations (only some operations and memory access patterns can be performed quickly, partial support for IEEE floating point arithmetic) make it necessary to change existing algorithms to attain high performance and correctness. Here we show how to make the bisection algorithm for eigenvalues of symmetric tridiagonal matrices (sstebz from LAPACK) run both fast and correctly on an ATI Radeon X1900 GPU. Our fastest algorithm takes up to 156! less time than Intel's Math Kernel Library version of sstebz running on the CPU, but does so by doing many redundant floating point operations compared to the CPU version. We use an automatic tuning procedure analogous to ATLAS or PHiPAC to decide the optimal redundancy. Correctness despite partial IEEE floating point semantics required explicitly adding 0 in the inner loop. The problems and solutions discussed here are of interest on other GPU architectures. 1 Motivation and Objectives Modern graphics processors (GPUs) are data parallel architectures that can run general-purpose computations in single precision (so far) at high computational rates. They are capable of achieving 110 GFLOPS in matrix-matrix multiplication [Segal and Peercy 2006] and show 30-40x speedups compared to the recent Intel Xeon processors in computationally intensive applications such as Black-Scholes option pricing [McCool et al. 2006] and gas dynamics solvers [Hagen et al. 2007]. It is tempting to exploit this computational power in solving other common numerical problems. In this work we consider an implementation of another widely used linear algebra routine — the bisection algorithm for finding the eigenvalues of symmetric tridiagonal matrices. A numerically robust, vectorized implementation of this algorithm in single precision is available in LAPACK’s sstebz routine [Anderson et al. 1999]. Our goal is to port the vectorized segments of the code to the GPU. In order to increase the utilization of the parallel resources, we use the Multi-section with Multiple Eigenvalues method used previously by Katagiri et al. [2006]. For the purpose of this study we restrict our attention to finding all eigenvalues of the matrix. The extension to finding a subset of the eigenvalues as done in LAPACK’s sstebz routine, is straightforward. 2 The Bisection Algorithm

Abstract: The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for reaction-diffusion systems (three equations) with a tridiagonal matrix of diffusion coefficients and with nonhomogeneous boundary conditions after the work of Kouachi (2004) on the system of reaction diffusion with a full 2-square matrix. Our techniques are based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth.

Abstract: The present invention provides an eigenvalue decomposition apparatus that can perform processing in parallel at high speed and high accuracy. The eigenvalue decomposition apparatus comprises a matrix dividing portion 14 that repeatedly divides a symmetric tridiagonal matrix T into two symmetric tridiagonal matrices, an eigenvalue decomposition portion 15 that performs eigenvalue decomposition on the symmetric tridiagonal matrix after the division, an eigenvalue computing portion 17 that repeatedly computes eigenvalues of the symmetric tridiagonal matrix that is the division origin and matrix elements of the symmetric tridiagonal matrix that is the division origin, based on eigenvalues and matrix elements obtained by eigenvalue decomposition performed by the eigenvalue decomposition portion 15, the matrix elements being part of elements of orthogonal matrices constituted by eigenvectors, until an eigenvalue of the symmetric tridiagonal matrix T is computed, and an eigenvector computing portion 19 that computes an eigenvector of the symmetric tridiagonal matrix T based on the symmetric tridiagonal matrix T and the eigenvalue thereof using twisted factorization.

Abstract: OF DISSERTATION

Abstract: In this paper, we consider shifted tridiagonal matrices. We prove that the standard algorithm to compute the LU factorization in this situation is mixed forward-backward stable and, therefore, componentwise forward stable. Moreover, we give a formula to compute the corresponding condition number in O(n) flops.

Abstract: [PROBLEMS] An eigenvalue decomposing device capable of performing parallel processing at high speed with high accuracy [MEANS FOR SOLVING PROBLEMS] An eigenvalue decomposing device comprises a matrix decomposing section (14) for repeating division of a symmetric tridiagonal matrix (T) into two symmetric tridiagonal matrices, an eigenvalue decomposing section (15) for performing eigenvalue decomposition of the divided symmetric tridiagonal matrices, an eigenvalue calculating section (17) for repeating calculation of the eigenvalue of the symmetric tridiagonal matrix before the division and the matrix elements of the symmetric tridiagonal matrix before the division from the eigenvalue subjected to the eigenvalue decomposition by the eigenvalue decomposition section (15) and matrix elements which are a part of the elements of an orthogonal matrix composed of eigenvectors until the eigenvalue of the symmetric tridiagonal matrix (T) is calculated, and an eigenvector calculating section (19) for calculating the eigenvectors of the symmetric tridiagonal matrix (T) by the twist decomposition form the symmetric tridiagonal matrix T and its eigenvalue

Abstract: A method and algorithm to determine if a tridiagonal Toeplitz matrix is well-condition or weakly well-condition is presented

Abstract: We consider the following inverse eigenvalue problem: to construct a symmetrical tridiagonal matrix from the minimal and maximal eigenvalues of all its leading principal submatrices. We give a necessary and sufficient condition for the existence of such a matrix and for the existence of a nonnegative symmetrical tridiagonal matrix. Our results are constructive, in the sense that they generate an algorithmic procedure to construct the matrix. c 2007 Elsevier Ltd. All rights reserved.

Abstract: A scalable parallel algorithm of block tridiagonal systems for solving the initial boundary value problem of 3D-parabolic equation with the Dirichlet boundary condition is discussed A parallel degree of the difference scheme is proposed for showing its intrinsic parallelism of the difference scheme The relation between the parallel degree and the performance of the parallel algorithm is investigated The method proposed in this paper has been implemented on the super computer "ZiQiang 3000" of Shanghai University, and the numerical results match closely with theoretical analysis With the given accuracy, the line speedup is obtained, and the parallel implementation efficiency over 90% is reached

Abstract: We use variable transformation from the real line to finite or semi-infinite spaces where we expand the regular solution of the 1D time-independent Schrodinger equation in terms of square integrable bases. We also require that the basis support an infinite tridiagonal matrix representation of the wave operator. By this requirement, we deduce a class of solvable potentials along with their corresponding bound states and stationary wavefunctions expressed as infinite series in terms of these bases. This approach allows for simultaneous treatment of the discrete (bound states) as well as the continuous (scattering states) spectrum on the same footing. The problem translates into finding solutions of the resulting three-term recursion relation for the expansion coefficients of the wavefunction. These are written in terms of orthogonal polynomials, some of which are modified versions of known polynomials. The examples given, which are not exhaustive, illustrate the power of this approach in dealing with 1D quantum problems.

Abstract: In this paper, the main objective is to introduce a new modified approach for deriving of implicit Runge-Kutta methods by construction a tridiagonal implicit matrices of unknown coefficients.

Abstract: The level spacing distribution of general, non-normal, Gaussian random 2D matrices is derived. In particular, tridiagonal matrices have no level repulsion and show a halfsided Gaussian distribution. General non-normal matrices show strong level repulsion. The repulsion exponent is 2 − 0log.

Abstract: The problem of computing the distance in the Frobenius norm of a given real irreducible tridiagonal matrix T to the algebraic variety of real normal irreducible tridiag onal matrices is solved. Simple formulas for computing the distance and a normal tridiagonal matrix at this distance are presented. The special case of tridiagonal Toeplitz matrices also is considered.

Abstract: This is the first in a series of articles in which we study the rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation. Here, we compute the bound states energy spectrum by diagonalizing the finite dimensional Hamiltonian matrix of H2, LiH, HCl and CO molecules for arbitrary angular momentum. The calculation was performed using the J-matrix basis that supports a tridiagonal matrix representation for the reference Hamiltonian. Our results for these diatomic molecules have been compared with available numerical data satisfactorily. The proposed method is handy, very efficient, and it enhances accuracy by combining analytic power with a convergent and stable numerical technique.

Abstract: In the matrix-product states approach to interacting multiparticle systems the stationary probability distribution is expressed as a matrix-product state with respect to a quadratic algebra determined by the dynamics of the process. The states involved in the matrix elements are determined by the boundary conditions. This reflects the intriguing feature of open systems that the bulk behaviour in the steady state strongly depends on the boundary rates. Led by the importance of the boundary conditions we consider the boundary operators as generators of a tridiagonal algebra whose irreducible modules are the Askey–Wilson polynomials. The matrices of the matrix-product ansatz obey the tridiagonal algebraic relations as well for particular values of the structure constants. This suggests the formulation of the steady-state properties in terms of noncommutative matrices generating a tridiagonal Askey–Wilson algebra. The previously known representations, both infinite dimensional and finite dimensional ones, are recovered within the tridiagonal framework.