Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1990"

Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation

Abstract: A starting point Formal problems in linear algebra The singular-value decomposition and its use to solve least-squares problems Handling larger problems Some comments on the formation of the cross-product matrix ATA Linear equations-a direct approach The Choleski decomposition The symmetric positive definite matrix again The algebraic eigenvalue generalized problem Real symmetric matrices The generalized symmetric matrix eigenvalue problem Optimization and nonlinear equations One-dimensional problems Direct search methods Descent to a minimum I-variable metric algorithms Descent to a minimum II-conjugate gradients Minimizing a nonlinear sum of squares Leftovers The conjugate gradients method applied to problems in linear algebra Appendices Bibliography Index

Classical Integrable Systems Generated Through Nonlinearization of Eigenvalue Problems

Abstract: It is a challenge for us to look for new finite-dimensional completely integrable systems. H. Flaschka [l] pointed out an important principle to obtain finite-dimensional integrable systems by constraining infinite-dimensional integrable systems on finite-dimensional invariant subset.

Stability For a Multidimensional Inverse Spectral Theorem

Abstract: We consider the eigenvalue problem in Ω Where Ω is a bounded domain in Rd with smooth boundary,a nd q is a bounded, measurable function on Ω The eigenvalue problem has discrete spectrum; we denote by and a nondecreasing sequence of eigenvalue and corresponding (orthonormal) eigenfunctions. It is known ([N–S–U]) that knowledge of the eigenvalues and the boundary values of the normal derivatives of the corresponding eigenfunctions is sufficient to uniquely determine a coefficient, q.

Parallel complexity of tridiagonal symmetric eigenvalue problem

Abstract: 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]).

Bit-level systolic algorithm for the symmetric eigenvalue problem

Abstract: An arithmetic algorithm is presented which speeds up the parallel Jacobi method for the eigen-decomposition of real symmetric matrices. After analyzing the elementary mathematical operations in the Jacobi method (i.e. the evaluation and application of Jacobi rotations), the author devises arithmetic algorithms that effect these mathematical operations with few primitive operations (i.e. few shifts and adds) and enable the most efficient use of the parallel hardware. The matrices to which the plane Jacobi rotations are applied are decomposed into even and odd parts, enabling the application of the rotations from a single side and thus removing some sequentiality from the original method. The rotations are evaluated and applied in a fully concurrent fashion with the help of an implicit CORDIC algorithm. In addition, the CORDIC algorithm can perform rotations with variable resolution, which lead to a significant reduction in the total computation time. >

Estimator eigenvalue placement in positive real control

Abstract: A numerical technique is presented to place the eigenvalues of the estimator in a positive real control environment. The estimator is based on a reduced-order model of the full system. The technique involves posing the problem as a constrained optimization problem. Two formulations are presented for solving the optimization problem that differ in their objective functions and in the way constraints are handled. Examples are given illustrating the pole placement for estimators based on an arbitrary second-order model, and on fourthand sixth-order models of the DRAPER I tetrahedral truss structure.

A comparison of some robust eigenvalue assignment techniques

Abstract: Three techniques for robust eigenvalue assignment are presented. The first is well known and is based on iteratively assigning the closed-loop eigenvectors so as to be maximally orthogonal to one another. The second has been recently presented by the authors and is an improvement of the first which gives better results for problems where complex-conjugate eigenvalue pairs are to be assigned. The final method is new and is founded on the iterative replacement of the current closed-loop eigenvector matrix with a new matrix which is the projection of the columns of the nearest orthogonal matrix into the allowable eigenvector subspaces. Some numerical examples are given which are used to illustrate the improved results obtained using the second technique in place of the first and to compare these with the performance of the last algorithm which is based on an alternative approach.

Algorithms for Solving Inverse Eigenvalue Problems for Sturm-Liouville Equations

Abstract: In the present paper we will consider the inverse eigenvalue problem for a Sturm-Liouville equation in so called impedance form, $$\left( {{p^2}\left( x \right)u\prime \left( x \right)} \right)\prime + {w^2}{p^2}\left( x \right)u\left( x \right) = 0, 0 \leqslant x \leqslant L, p\left( x \right)> 0 $$ (1) and with the following boundary conditions and nations for the eigenfrequencies.

Regular eigenvalue problem with eigenparameter contained in the equation and the boundary conditions

Abstract: The purpose of this paper is to establish the expansion theorem for a regular right-definite eigenvalue problem for the Laplace operator in Rn, (n > 2) with an eigenvalue parameter % contained in the equation and the Robin boundary conditions on two "parts" of a smooth boundary of a simply connected bounded domain.

Quadrature and extrapolation for the variable coefficient elliptic eigenvalue problem

Abstract: For the variable coefficient elliptic eigenvalue problem on a smooth domain or aconvex polygonal domain,a numerical quadrature scheme over triangles is used for computingthe coefficient of the resulting linear finite element system.The effect of numerical integrationis studied.The corresponding discrete eigenvalue with linear finite elements is shown to admitasymptotic error expansions for certain classes of“uniform”meshes.Hence,the Richardsonextrapolation increases the accuracy of the scheme from second to fourth order.

An interactive method for the eigenvalue problem for matrices

Abstract: The algorithm described in this article uses Householder reflections to obtain the largest eigenvalue of a full positive definite matrix. The algorithm can be used to obtain all the eigenvalues of a symmetric matrix and may be suitable for a parallel processing machine. We also consider the differential equations analogue of this method and prove the convergence of that method together with the convergence rates. Finally some numerical examples are given. This algorithm is very suitable to determine the eigenvalues in an interactive environment. Unlike the QL algorithm the full matrix is operated upon at each stage of the iterative process, hence one could use the APL programming language to write a very brief code to implement this program.

Parallel normreducing transformations for the algebraic eigenvalue problem

Abstract: This article presents a unified approach for parallel normreducing methods for the algebraic eigenproblem. The so-called Euclidean parameters presents the problem, to minimize the Frobenius norm of the transform matrix, in a simple form. The use of appropriate preprocessing unitary transforms together with an appropriate pivot strategy leads to convergence to formality.

Solution to the 2-D eigenvalue assignment problem

Abstract: A method for eigenvalue assignment by state feedback for 2-D systems described by the Roesser model is proposed. An algorithm for the choice of the state feedback matrix has been provided and illustrated by a numerical example

An overview of parallel algorithms for the singular value and symmetric eigenvalue problems

Abstract: Solving dense symmetric eigenvalue problems and computing singular value decompositions continue to be two of the most dominating tasks in numerous scientific applications. With the advent of multiprocessor computer systems, the design of efficient parallel algorithms to determine these solutions becomes of paramount importance. In this paper, we discuss two fundamental approaches for determining the eigenvalues and eigenvectors of dense symmetric matrices on machines such as the Alliant FX/8 and CRAY X-MP. One approach capitalizes upon the inherent parallelism offered by Jacobi methods, while the other relies upon an efficient reduction to tridiagonal form via Householder's transformations followed by a multisectioning technique to obtain the eigenvalues and eigenvectors of the corresponding symmetric tridiagonal matrix. For the singular value decomposition, we discuss an efficient method for rectangular matrices in which the number of rows is substantially larger or smaller than the number of columns. This scheme performs an initial orthogonal factorization using block Householder transformation, followed by a parallel one-sided Jacobi method to obtain the singular values and singular vectors of the resulting upper-triangular matrix. Exceptional performance for this SVD scheme is demonstrated for tall matrices of full or deficient rank having clustered or multiple singular values. A hybrid method that combines one- and two-sided Jacobi schemes is also discussed. Performance results for each of the above algorithms on the Alliant FX/8 and CRAY X-MP computer systems will be presented with particular emphasis given to speedups obtained over such classical EISPACK algorithms.

Parallel Jacobi algorithms for the algebraic eigenvalue problem

Abstract: The standard method for computing the eigenvalues of a general dense matrix on a traditional sequential computer is the QR algorithm. With the advent of parallel computers, a variety of parallel eigenvalue algorithms have been proposed. One approach has been to recognize the inherent parallelism in the Jacobi method, which was the standard algorithm for the problem before the invention of the QR algorithm. This thesis investigates some Jacobi-like algorithms for solving eigenvalue problems on parallel computers. First considered is the (easier) case of Hermitian matrices. The issue of convergence of the cyclic Jacobi method for the Hermitian eigenvalue problem has never been conclusively settled. New theoretical results are developed that prove global convergence of a large class of parallel Jacobi algorithms for the Hermitian eigen-problem. Next the eigenvalue problem for general matrices is considered. A new parallel Jacobi-like algorithm for general matrices developed which promises to be very competitive on massively parallel computers. It is proven that this new algorithm converges quadratically. Experimental results are presented that indicate that the algorithm can be expected to take $O(n{\rm log}\sp2 n)$ time using $O(n\sp2)$ processors. Finally an implementation of this algorithm on the massively parallel Connection Machine is described and performance results are presented.

Localization of the spectrum of the eigenvalue problem nonlinear in the spectral parameter

Abstract: An approach which enables one to localize the spectrum of some classes of eigenvalue problems nonlinear in the spectral parameters is proposed and proved. The application of the proposed approach to some test examples is described.

An estimate of the rate of convergence of Weinstein's method in the eigenvalue problem for an L-shaped membrane

Abstract: General estimates for the rate of convergence of Weinstein's method of intermediate problems, obtained previously by the author, are applied particularly to the case of the eigenvalue problem for the L-shaped membrane attached at its edges.

Numerical algorithms for eigenvalue assignment problem via observer matrix equations

Abstract: A unified approach to the development of several algorithms for the well-known eigenvalue and canonical form assignment problems is presented via observer matrix equations. Furthermore, a relationship between these algorithms and the Q-R factorization method is explicitly stated. This relationship allows the algorithms to be implemented in a more numerically robust way. The algorithms are discussed, and several research problems are outlined. >

A parallel algorithm for the symmetric eigenvalue problem

Abstract: A description is given of the authors' experiences in implementing a divide-and-conquer-type parallel algorithm due to J.J. Cuppen (1981) on the Alliant multivector processor. An analysis of the algorithm and its implementation on the Alliant multivector processor are presented. To promote portability of parallel algorithms across various architectures, J.J. Dongarra and D.C. Sorenson (1987) recently developed a tool called SCHEDULE. Results obtained with and without SCHEDULE are compared. >

Some remarks on the eigenvalue spectrum of a large symmetric Wigner random-sign matrix

Abstract: The replica method is used to calculate the averaged eigenvalue spectrum as N → ∞, of the ensemble of Wigner random-sign real symmetric N × N matrices. Results are presented for the cases where the individual matrix elements have a mean value of zero and also where the mean value of the individual matrix elements has a finite nonzero value. It is shown that the replica method provides a straightforward framework within which it is possible to verify the Wigner conjecture that any reasonably well-behaved distribution of matrix elements must lead to the well-known semicircular averaged eigenvalue spectrum of the Gaussian orthogonal ensemble of random matrices. Some numerical simulations of the averaged eigenvalue spectrum of these random-sign matrices are presented and they lend support to the prediction that if the individual matrix elements have sufficiently large a mean value, then a single eigenvalue will split off from the main semicircular band of eigenvalues.