Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1993"
TL;DR: This expository paper explores the relationships among a number of algorithms for solving eigenvalue problems, including the power method, subspace iteration, the $QR$ algorithm, and the Arnoldi and symmetric Lanczos algorithms.
Abstract: This expository paper explores the relationships among a number of algorithms for solving eigenvalue problems, including the power method, subspace iteration, the $QR$ algorithm, and the Arnoldi and symmetric Lanczos algorithms. The symmetric Lanczos algorithm is shown to be identical to the three-term recursion (Stieltjes procedure) for computing orthogonal polynomials with respect to a measure on the real line. The connection between measures on the line and symmetric tridiagonal (Jacobi) matrices is investigated. If such a matrix is transformed by a step of the $QR$ algorithm, there is a corresponding transformation in the measure. The tridiagonal matrices are also exploited for the construction of Gaussian quadrature formulas for measures on the line. The developments on the real line are replicated with suitable modifications on the unit circle via Lanczos-like procedures for unitary operators. The best-known procedure of this type is the recursion of Szego for computing orthogonal polynomials on the...
155 citations
TL;DR: In this article, the dependence of Schrodinger equation eigenvalue problems on the coupling-constant parameters in the potential was considered and it was shown that unless great care is taken, analytic continuation in these parameters can lead to surprising and paradoxical conclusions.
127 citations
TL;DR: A convergence property of Horst’s method by forming it as a generalization of the so-called power method is then proved and a closed form on the cardinality of solutions for the multivariate eigenvalue problem is first proved.
Abstract: Multivariate eigenvalue problems for symmetric and positive definite matrices arise from multivariate statistics theory where coefficients are to be determined so that the resulting linear combinations of sets of random variables are maximally correlated. By using the method of Lagrange multipliers such an optimization problem can be reduced to the multivariate eigenvalue problem. For over 30 years an iterative method proposed by Horst [Psychometrika, 26 (1961), pp. 129–149J has been used for solving the multivariate eigenvalue problem. Yet the theory of convergence has never been complete. The number of solutions to the multivariate eigenvalue problem also remains unknown. This paper contains two new results. By using the degree theory, a closed form on the cardinality of solutions for the multivariate eigenvalue problem is first proved. A convergence property of Horst’s method by forming it as a generalization of the so-called power method is then proved. The discussion leads to new formulations of nume...
76 citations
TL;DR: Two nonmodal algorithms are described, and it is shown mathematically that under some mild assumptions on the damping matrix, the feedback matrix obtained by each algorithm indeed stabilizes a closed-loop system.
59 citations
43 citations
TL;DR: An approach of employing the Lanczos two-sided recursion to solve the quadratic eigenvalue problem without the need to use the method of matrix augmentation traditionally used to cast the problem as a linear eigen value problem of order 2n is presented.
41 citations
TL;DR: In this paper, a review of boundary element eigenvalue analysis focusing on the Helmholtz equation in terms of a scalar-valued function is presented, and the most popular approach using domain cells for domain integration and some transformation methods, such as the dual reciprocity method (DRM) and the multiple reciprocity methods (MRM), are discussed.
Abstract: Recent developments in boundary element eigenvalue analysis are reviewed, focusing on the Helmholtz equation in terms of a scalar-valued function. The problem is of fundamental importance in the framework of the sophisticated boundary element scheme, in general devised for the non-homogeneous differential equation. The most popular approach using domain cells for domain integration and some transformation methods, such as the dual reciprocity method (DRM) and the multiple reciprocity method (MRM), are discussed. Two key issues are the solution without domain integration and the standard routine eigenvalue search in contrast to the conventional domain cell discretization and the direct eigenvalue search using distribution of the magnitude of the determinant, which are the marked item of the boundary element method for numerical efficiency and are preffered over other methods.
37 citations
TL;DR: A new algorithm on minimizing the largest eigenvalue over an affine family of symmetric matrices is proposed and under certain assumptions it is shown that, if started close enough to the minimizer, the proposed algorithm converges to x ∗ quadratically.
28 citations
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01 Jan 1993
TL;DR: In this article, asymptotic boundary conditions for uniform convergence of generalized eigenfunction expansions are given by recursion, expansion theorems are established by an analytical study of the asymptic behaviour of Greens function.
Abstract: This work deals with nonself-adjoint lambda-nonlinear boundary eigenvalue problems for ordinary differential equations. Asymptotic boundary conditions for uniform convergence of generalized eigenfunction expansions are given by recursion, expansion theorems are established by an analytical study of the asymptotic behaviour of Greens function. The theory is illustrated by various examples from technical mechanics.
27 citations
TL;DR: In this article, the analysis and design of rotor bearing systems with gyroscopic effects are discussed, where the original problem of this nonproportionally damped mechanism is transformed into the state space form so that the transformed problem is similar to the eigenvalue problem for an undamped system.
Abstract: The analysis and design of rotor-bearing systems with gyroscopic effects are discussed in this paper. The original problem of this nonproportionally damped mechanism is transformed into the state space form so that the transformed problem is similar to the eigenvalue problem for an undamped system. It can be easily solved by widely available eigensolvers and the eigenvalue sensitivities needed for the design optimization can also be obtained conveniently using the transformed form. The sequential linear programming technique is employed to solve the design optimization problem. The easy implementation of this proposed approach with a general purpose finite element program is shown in the solution algorithm.
24 citations
TL;DR: Cuppen's divide-and-conquer method for solving the symmetric tridiagonal eigenvalue problem has been shown to be very efficient on shared memory multiprocessor architectures.
Abstract: Cuppen’s divide-and-conquer method for solving the symmetric tridiagonal eigenvalue problem has been shown to be very efficient on shared memory multiprocessor architectures.In this paper, some error analysis issues concerning this method are resolved. The method is shown to be stable and a slightly different stopping criterion for finding the zeroes of the spectral function is suggested.These error analysis results extend to general update methods for the symmetric eigenvalue problem. That is, good backward error bounds are obtained for methods to find the eigenvalues and eigenvectors of $A + \rho ww^T $, given those of A. These results can also be used to analyze a new fast method for finding the eigenvalues of banded, symmetric Toeplitz matrices.
TL;DR: In this article, sufficient conditions of eigenvalue clustering for interval matrices are presented, which can be applied to both continuous and discrete-time dynamic interval systems, and three examples are given to illustrate the applicability of the results.
Abstract: Various sufficient conditions of eigenvalue clustering for interval matrices are presented. The proposed sufficient conditions guarantee that all eigenvalues of the interval matrices lie inside various specified regions in the complex plane. The derived theorems can be applied to both continuous- and discrete-time dynamic interval systems. The dynamical characteristics of a linear system are influenced by the eigenvalue locations of the system. Therefore, by the analysis of eigenvalue clustering, we can understand more properties about interval dynamic systems such as stability margin, performance robustness and so on. Three examples are given to illustrate the applicability of the results.
TL;DR: In this article, the authors considered the inverse eigenvalue problem for an arbitrary square matrix and for arbitrary additive perturbations belonging to a matrix Lie algebra, motivated by several examples arising in linear systems theory.
Abstract: Motivated by several examples arising in linear systems theory, the problem considered here is the inverse eigenvalue problem for an arbitrary square matrix and for arbitrary additive perturbations belonging to a matrix Lie algebra. For an algebraically closed field with characteristic zero, the main theorem gives necessary and sufficient conditions for the positive solution of the corresponding additive inverse eigenvalue problem. There are, of course, several antecedents of this result in the literature involving special coordinate systems and special representations of particular matrix Lie algebras, most notable among these being the result due to S. Friedland on the inverse eigenvalue problem for diagonal perturbations.
TL;DR: In this article, sufficient conditions of robust eigenvalue clustering for perturbed systems are presented, where all the eigenvalues of the nominal system are located in a specified region of the complex plane.
Abstract: Sufficient conditions of robust eigenvalue clustering for perturbed systems are presented. If all the eigenvalues of the nominal system are located in a specified region of the complex plane then the proposed sufficient conditions guarantee that all the eigenvalues of the perturbed system remain inside the same region. The characteristics of a linear dynamical system are influenced by the eigenvalue locations. Therefore, through the study of eigenvalue clustering, more features about perturbed systems, such as stability margin, performance robustness and so on, can be investigated. It is emphasized that the proposed perturbation bounds for robust eigenvalue clustering in a specified region can be applied to both continuous- and discrete-time systems. Three illustrative examples are given to show the applicability of the proposed theorems.
TL;DR: In this article, the authors presented an algorithm for the optimization of the maximum eigenvalue of a symmetric definite pencil depending affinely on a vector of parameters using a hybrid approach, combining a scheme based on the method of centers, developed by Boyd and El Ghaoui [Linear Algebra Appl., 188 (1993), pp.~63--112], with a new quadratically convergent local scheme.
Abstract: An algorithm is presented for the optimization of the maximum eigenvalue of a symmetric definite pencil depending affinely on a vector of parameters. The algorithm uses a hybrid approach, combining a scheme based on the method of centers, developed by Boyd and El Ghaoui [Linear Algebra Appl., 188 (1993), pp.~63--112], with a new quadratically convergent local scheme. A convenient expression for the generalized gradient of the maximum eigenvalue of the pencil is also given, expressed in terms of a dual matrix. The algorithm computes the dual matrix that establishes the optimality of the computed solution.
TL;DR: In this article, the eigenvalue of the three-dimensional Helmholtz equation was determined efficiently by extending the previously developed method for the two-dimensional problem by using higher order fundamental solutions for the Laplace equation; yielding polynomial coefficient matrices in terms of unknown wavenumber.
TL;DR: In this paper, it was shown that the maximal eigenvalue of a class of (n * n)-dimensional real symmetric interval matrices, say A, coincides with the maximum eigen value of a single vertex matrix whose entries are the right endpoint of its intervals.
Abstract: It is proved that the maximal eigenvalue of a class of (n*n)-dimensional real symmetric interval matrices, say A, coincides with the maximal eigenvalue of a single vertex matrix whose entries are the right endpoint of its intervals. The elements of the interval matrix A are intervals whose right endpoint is not smaller than the absolute value of the left endpoint. As a corollary, a necessary and sufficient condition for A to be Hurwitz-namely, that the above-mentioned vertex matrix is Hurwitz-is obtained. Furthermore, the Hurwitz stability of A implies the Hurwitz stability of the general interval matrix whose entries are allowed to vary in the intervals of A. >
TL;DR: An efficient numerical algorithm is developed to solve the quadratic eigenvalue problems arising in the dynamic analysis of damped structural systems, which takes full advantage of the sparseness and symmetry of the system matrices and requires no complex arithmetic, making it very economical for use in solving large problems.
Abstract: An efficient numerical algorithm is developed to solve the quadratic eigenvalue problems arising in the dynamic analysis of damped structural systems. The algorithm can even be applied to structural systems with non-symmetric matrices. The algorithm is based on the use of Arnoldi's method to generate a Krylov subspace of trial vectors, which is then used to reduce a large eigenvalue problem to a much smaller one. The reduced eigenvalue problem is solved and the solutions are used to construct approximate solutions to the original large system. In the process, the algorithm takes full advantage of the sparseness and symmetry of the system matrices and requires no complex arithmetic, therefore, making it very economical for use in solving large problems. The numerical results from test examples are presented to demonstrate that a large fraction of the approximate solutions calculated are very accurate, indicating that the algorithm is highly effective for extracting a number of vibration modes for a large dynamic system, whether it is lightly or heavily damped.
01 Jan 1993
TL;DR: In this paper, it was shown that many well known algorithms in numerical analysis can be reformulated in a useful and intrinsic way, by making use of basic ideas in symplectic geometry.
Abstract: It turns out that many well known algorithms in numerical analysis can be reformulated in a useful and intrinsic way, by making use of basic ideas in symplectic geometry. Actually, from a modern perspective, it is surprising to see how many examples of important techniques in the theory of dynamical systems with special symmetries were known to numerical analysts. The methods of symplectic geometry enable us to extend our understanding of the algorithms, by providing a conceptual setup in which to interpret a number of algebraic procedures. This text is an attempt to illustrate the above claims.
01 Jan 1993
TL;DR: In this paper, the authors describe the first distributed memory implementation of the split-merge algorithm, an eigenvalue solver for symmetric tridiagonal matrices that uses Laguerre's iteration and exploits the separation property in order to create independent subtasks.
Abstract: Abstract Both massively parallel computers and clusters of workstations are considered promising platforms for numerical scientific computing. This paper describes the first distributed-memory implementation of the split-merge algorithm, an eigenvalue solver for symmetric tridiagonal matrices that uses Laguerre's iteration and exploits the separation property in order to create independent subtasks. Implementations of the split-merge algorithm on both an nCUBE-2 hypercube and a cluster of Sun Spare-10 workstations are described, with emphasis on load balancing, communication overhead, and interaction with other user processes. A performance study demonstrates the advantage of the new algorithm over a parallelization of the well-known bisection algorithm. A comparison of the performance of the nCUBE-2 and cluster implementations supports the claim that workstation clusters offer a cost-effective alternative to massively parallel computers for certain scientific applications.
01 Jan 1993
TL;DR: Newton methods for general large sparse eigenvalue problems satisfying the overdamping condition are examined and how to use sparse iterative solvers more eeectively in them are shown.
Abstract: We present a new Chebyshev{Arnoldi algorithm for nding the lowest energy eigen-functions of an elliptic operator. The algorithm, which is essentially the same for symmetric , nonsymmetric, and complex nonhermitian matrices, is adapted to a speciic problem by two subroutines which encapsulate the problem{speciic deenition of energy, plus the discretization and matrix{vector multiply routines. We adapt the algorithm to two important problems, the self{consistent Schrr odinger{Poisson model of quantum{eeect devices, and the vector Helmholtz equation for a dielectric waveguide, addressing other important physical, numerical and computational issues as they arise. An asymptotic convergence estimate is derived which shows the Chebyshev{Arnoldi algorithm to be superior to Chebyshev{preconditioned subspace iteration. We also examine Newton methods for general large sparse eigenvalue problems satisfying the overdamping condition and show how to use sparse iterative solvers more eeectively in them. Table Page I Occupancy levels of the rst four wavefunctions for applied potentials V 1 = V 2 = 0:2V , with the exchange{correlation potential V xc (n QW) Closeup of the conduction band potential energy V , the quantum electron density n QW , and the rst two wavefunctions for applied potentials V 1 = V 2 = 0:2V , with the exchange{correlation potential V xc (n QW) Closeup of the conduction band potential energy V , the quantum electron density n QW , and the rst two wavefunctions for applied potentials V 1 = 0:7V and V 2 = 0:2V , with the exchange{correlation potential V xc
TL;DR: In this paper, a perturbation bound for the Jacobi matrix inverse eigenvalue problem is given, which is the same as the bound given in this paper, but with a different solution.
Abstract: In this paper, we give a perturbation bound for the solution of the Jacobi matrix inverse eigenvalue problem.
TL;DR: In this paper, a class of operator equations in an abstract Hilbert space and the L2-theory of certain nonlinear Schrodinger equations which can be viewed as special cases of (i) are discussed.
Abstract: IN THIS PAPER we study: (i) a class of operator equations in an abstract Hilbert space; and (ii) the L2-theory of certain nonlinear Schrodinger equations which can be viewed as special cases of (i). In order to describe the type of abstract nonlinear eigenvalue problems to be discussed, consider a real Hilbert space H with scalar product (* , *) and norm II.11 and let S be a (not necessarily bounded) positive self-adjoint linear operator in li. We write S in the form
TL;DR: A lower bound is given, based on geometrical data, for the first eigenvalue to the problem in a domain D, such that u = 0 on aD and 0.
Abstract: In this paper we give a lower bound, based on geometrical data, for the first eigenvalue to the problem in a domain D, such that u = 0 on aD and 0
01 Jan 1993
TL;DR: The theme of this chapter is that techniques from dynamical systems can be applied to the study of certain problems in numerical analysis and this chapter will focus on the particular numerical analysis problem of approximating the eigenvalues of a real matrix.
Abstract: In recent years, there has been a great deal of interdisciplinary work between numerical analysis and dynamical systems. The theme of this chapter is that techniques from dynamical systems can be applied to the study of certain problems in numerical analysis. We will focus on the particular numerical analysis problem of approximating the eigenvalues of a real matrix. The discussion is from the point of view of dynamical systems and assumes a basic knowledge of dynamical systems [D2] and general topology [M]. A substantial portion of the chapter consists of motivating and defining the relevant numerical algorithms and no background in numerical analysis is required.
TL;DR: Hybrid codes that combine elements of the QR and LR algorithms are described and, for certain choices of parameters, the hybrid codes possess the same high reliability as the QR algorithm and are significantly faster.
Abstract: Hybrid codes that combine elements of the QR and LR algorithms are described. The codes can calculate the eigenvalues and, optionally, eigenvectors of real, nonsymmetric matrices. Extensive tests are presented as evidence that, for certain choices of parameters, the hybrid codes possess the same high reliability as the QR algorithm and are significantly faster. The greatest success has been achieved with the codes that calculate eigenvalues only. These can do the task in 15% to 50% less time than the QR algorithm.
TL;DR: Three algorithms to solve the sparse symmetric generalized eigenvalue problem on vector and parallel computers are compared: reduction to a standard problem, use of Sturm sequences, and Lanczos method.