This paper describes implementations of Cuppen's method, bisection, and multisection for the computation of all eigenvalues and eigenvectors of a real symmetric tridiagonal matrix on a distributed-memory hypercube multiprocessor. Numerical results and timings for Intel's iPSC-1 are presented. Cuppen's method is found to be the numerically most accurate of three methods, while bisection with inverse iteration is observed experimentally to be the fastest method.

Solving the symmetric tridiagonal eigenvalues problem on the hypercube

In the last twenty years the field of Markov-modulated fluid queues has received considerable attention. In these models a fluid reservoir receives and/or releases fluid at rates which depend on the actual state of a background Markov chain. In the first chapter of this thesis we give a short introduction on how the stationary distribution for such a model is usually found, as well as a literature overview on Markov-modulated and related uid queues. The rest of the thesis is concerned with �nding stationary distributions for some types of fluid models that have received little or no attention until now. The two main contributions are the following.
1. We focus on models in which the state space of the regulating Markov process is infinitely large, either denumerable or not. Regarding the first type, we mainly look into regulating processes that are of birth-death type. We present procedures to find the stationary distribution, using the theory of orthogonal polynomials. In the nondenumerable case, we look into simple systems of fluid queues, in which one fluid queue regulates the behaviour of another (one example being a fluid tandem queue).
2. We look into models in which the state of the fluid reservoir in quences the behaviour of the regulating process, so that the latter does not constitute a Markov process. We call suchlike systems feedback fluid queues, to emphasize the two-way dependence between fluid reservoir and regulating process.

/pdf/markov-modulated-and-feedback-fluid-queues-1iunes5rbv.pdf

Markov-modulated and feedback fluid queues

LetH?? n xn be a unitary upper Hessenberg matrix whose eigenvalues, and possibly also eigenvectors, are to be determined. We describe how this eigenproblem can be solved by a divide and conquer method, in which the matrixH is split into two smaller unitary upper Hessenberg matricesH 1 andH 2 by a rank-one modification ofH. The eigenproblems forH 1 andH 2 can be solved independently, and the solutions of these smaller eigenproblems define a rational function, whose zeros on the unit circle are the eigenvalues ofH. The eigenvector ofH can be determined from the eigenvalues ofH and the eigenvectors ofH 1 andH 2. The outlined splitting of unitary upper Hessenberg matrices into smaller such matrices is carried out recursively. This gives rise to a divide and conquer method that is suitable for implementation on a parallel computer.

WhenH?? n xn is orthogonal, the divide and conquer scheme simplifies and is described separately. Our interest in the orthogonal eigenproblem stems from applications in signal processing. Numerical examples for the orthogonal eigenproblem conclude the paper.

A divide and conquer method for unitary and orthogonal eigenproblems

We present a FORTRAN implementation of a divide-and-conquer method for computing the spectral resolution of a unitary upper Hessenberg matrix H. Any such matrix H of order n, normalized so that its subdiagonal elements are nonnegative, can be written as a product of n-1* Givens matrices and a diagonal matrix. This representation, which we refer to as the Schur parametric form of H, arises naturally in applications such as in signal processing and in the computation of Gauss-Szego quadrature rules. Our programs utilize the Schur parametrization to compute the spectral decomposition of H without explicitly forming the elements of H. If only the eigenvalues and first components of the eigenvectors are desired, as in the applications mentioned above, the algorithm requires only O(n2) arithmetic operations. Experimental results presented indicate that the algorithm is reliable and competitive with the general QR algorithm applied to this problem. Moreover, the algorithm can be easily adapted for parallel implementation.—Authors' Abstract

/pdf/an-implementation-of-a-divide-and-conquer-algorithm-for-the-1rgz5gv3ss.pdf

An implementation of a divide and conquer algorithm for the unitary eigen problem

Surprisingly simple corollaries from the Courant-Fischer minimax characterization theorem enable us to devise a very effective algorithm for the evaluation of a set S interleaving the set E of the eigenvalues of a real symmetric tridiagonal matrix Tn (as well a.s a point that splits E into two subsets of comparable cardinalities). As a result, we dramatically decrease the previous record upper estimates for the parallel complexity of the eigenvalue computation for a symmetric tridiagonal matrix (which is a major computational problem in linear algebra); our new upper bounds are within polylogarithmic factors from the information lower bounds. The algorithm can be extended to approximating to the zeros of a polynomial that ha-s only real zeros (aa an alternative to the algorithm of [BOT]).

Parallel complexity of tridiagonal symmetric eigenvalue problem

The Symmetric Tridiagonal Eigenproblem has been the topic of some recent work. Many methods have been advanced for the computation of the eigenvalues of such a matrix. In this paper, we present a divide-and-conquer approach to the computation of the eigenvalues of a symmetric tridiagonal matrix via the evaluation of the characteristic polynomial. The problem of evaluation of the characteristic polynomial is partitioned into smaller parts which are solved and these solutions are then combined to form the solution to the original problem. We give the update equations for the characteristic polynomial and certain auxiliary polynomials used in the computation. Furthermore, this set of recursions can be implemented on a regulartree structure. If the concurrency exhibited by this algorithm is exploited, it can be shown that thetime for computation of all the eigenvalues becomesO(nlogn) instead ofO(n2) as is the case for the approach where the order is increased by only one at every step. We address the numerical problems associated with the use of the characteristic polynomial and present a numerically stable technique for the eigenvalue computation.

Eigenvalues of a symmetric tridiagonal matrix: A divide-and-conquer approach

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Eigenvalues of a symmetric tridiagonal matrix: A divide-and-conquer approach