Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1987"
Book•
01 Jan 1987
TL;DR: This book discusses iterative projection methods for solving Eigenproblems, and some of the techniques used to solve these problems came from the literature on Hermitian Eigenvalue.
Abstract: List of symbols and acronyms List of iterative algorithm templates List of direct algorithms List of figures List of tables 1: Introduction 2: A brief tour of Eigenproblems 3: An introduction to iterative projection methods 4: Hermitian Eigenvalue problems 5: Generalized Hermitian Eigenvalue problems 6: Singular Value Decomposition 7: Non-Hermitian Eigenvalue problems 8: Generalized Non-Hermitian Eigenvalue problems 9: Nonlinear Eigenvalue problems 10: Common issues 11: Preconditioning techniques Appendix: of things not treated Bibliography Index .
1,418 citations
TL;DR: In this paper, the eigenvalue problem of the Schrodinger operator with the magnetic field on a compact Riemannian manifold was considered and a representation of the least eigen value by the variational formula was given.
81 citations
TL;DR: In this article, the authors considered a symmetric partial algebraic eigenvalue problem and analyzed the convergence rates of several methods of solving it by means of a preconditioner.
Abstract: In this paper we consider a symmetric partial algebraic eigenvalue problem. In Section 1 we present several estimates for the rates of convergence of some classical algorithms of vector iterations. Estimates of the accuracy of the Rayleigh-Ritz method and of the subspace iterations are considered in Section 2. The convergence rates of several methods of solving generalized eigenvalue problems by means of a preconditioner are analysed in Section 3. Finally, Section 4 deals with the Temple-Lehmann two-sided estimates for eigenvalues. The paper constitutes a systematic review of recent results mainly due to the author. Consider in a Euclidean space H the problem of computing p maximal eigenvalues A! > · · · > λρ and the corresponding eigenvectors for a generalized eigenvalue problem Mu = XLu, M = M*, L = L*>0 (0.1) (to simplify the notation, all the eigenvalues λί > ··· > λρ are taken to be simple). To calculate the minimal eigenvalues of (0.1), we need only to replace Μ by — Μ throughout. In computational practice the problem is traditionally tackled by implicit reduction of the generalized eigenproblem (0.1) to the ordinary one, i.e. ΑΗ = λΐΛ, A = A* (0.2) where, for example, A = L\"M in the space HL equipped with the scalar product (Λ *)*. = (£·,*). In Section 1 we present several estimates for the convergence rates of some classical methods of vector iterations for problem (0.2) with ρ = 1. In Section 2 we consider some estimates for the accuracy of the Rayleigh-Ritz method and of the subspace iterations for problem (0.2) with ρ > 1. In Section 3 we investigate the convergence rates of several methods of solving eigenvalue problem (0.1) using a preconditioner Β = Β* > 0 such that the system Bu = f can be efficiently solved, and the ratio δ, δ = δ0/δΐ9 0<δ0Β*ζΙιζδ1Β (0.3) is as close to 1 as possible. Finally, Section 4 deals with the Temple-Lehmann approach to obtaining two-sided estimates for the eigenvalues of (0.1). The results of this paper can be useful in evaluating the efficiency of various iterative techniques for solving problems of type (0.1), which can occur from the finite difference or finite element discretization of differential eigenvalue problems. Our choice of Originally published in Russian in Numerical Methods and Mathematical Modelling, Transactions of the Department of Numerical Mathematics of the USSR Academy of Sciences, Moscow, 1986.
67 citations
TL;DR: In this article, the problem of eigenvalue assignment for a class of linear time-varying multivariable systems is considered using matrix operators and canonical transformations, and a simple algorithm for the design of the state feedback is provided.
Abstract: The problem of eigenvalue assignments for a class of linear time-varying multivariable systems is considered. Using matrix operators and canonical transformations, it is shown that a time-varying system that is 'lexicography-fixedly controllable' can be made via state feedback to be equivalent to a time-invariant system whose eigenvalues are arbitrarily assignable. A simple algorithm for the design of the state feedback is provided.
58 citations
TL;DR: The homotopy continuation method is used to find all the isolated zeros of the polynomial system which corresponds to the eigenpairs of the generalized eigenvalue problem.
34 citations
Journal Article•
TL;DR: In this paper, the conditions générales d'utilisation (http://www.numdam.org/conditions) are defined, i.e., toute utilisation commerciale ou impression systématique is constitutive d'une infraction pénale.
Abstract: L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
26 citations
TL;DR: In this paper, the problem of maximizing the minimum eigenvalue of a selfadjoint matrix operator is considered and the necessary conditions for a local maximum of the eigen value of arbitrary multiplicity p with an isoperimetric limit are obtained.
26 citations
TL;DR: In this article, a compressible stability analysis computer code is developed for the boundary layer profiles of an LFC swept wing, which uses a matrix finite-difference method for local eigenvalue solution when a good guess for the eigen value is available.
Abstract: A compressible stability analysis computer code is developed. The code uses a matrix finite-difference method for local eigenvalue solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used initial-value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed that may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black-box stability code. Sample stability calculations are presented for the boundary layer profiles of an LFC swept wing.
23 citations
TL;DR: In this article, it is shown that the Sturm-Liouville eigenvalue problems with singularities in the interior of the interval of definition of the differential equation can be represented by self-adjoint, unbounded differential operators in a suitable chosen Hilbert function space.
Abstract: This paper is concerned with Sturm-Liouville eigenvalue problems with singularities in the interior of the interval of definition of the differential equation. Such circumstances arise in mathematical models of certain physical problems. It is shown that the eigenvalue problems can be represented by self-adjoint, unbounded differential operators in a suitable chosen Hilbert function space. Numerical values for the eigenvalues can be obtained using the SLEIGN computer programme.
19 citations
TL;DR: This work discusses the use of recent new formulations of the Tau method for the numerical approximation of differential eigenvalue problems where the spectral parameter appears nonlinearly and formulated by using a standard technique easy to implement in a computer.
TL;DR: The Lanczos approach is generally faster and allows an out-of-core implementation for extremely large problems, and some of the advantages of the Lanczos algorithm are demonstrated.
15 Aug 1987
TL;DR: In this article, a new approach for the complex resonance eigenvalue problem mixing the optical potential model with a perturbational treatment is investigated, illustrated by the study of two model potentials using the collocation approximation.
Abstract: A new approach for the complex resonance eigenvalue problem mixing the optical potential model with a perturbational treatment is investigated. The method is illustrated by the study of two model potentials using the collocation approximation. It is shown that resonance eigenvalues can be calculated with a small amount of computer time, i.e. making only a few diagonalizations of the hamiltonian matrix and no overlap integral calculation.
TL;DR: An application of the so-called Hill determinant method to the bound-state eigenvalue problem in the elementary quarkonium potential V(r)=-a/r+br is described, proved and illustrated for a few examples in this paper.
Abstract: An application of the so-called Hill determinant method to the bound-state eigenvalue problem in the elementary quarkonium potential V(r)=-a/r+br is described, proved and illustrated for a few examples An improvement of the method, which is based on an extended continued-fraction formulation of the eigenvalue condition, is also proposed
Proceedings Article•
01 Dec 1987
TL;DR: An inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace, and the rate of convergence of the method is established, and a technique for improving it is discussed.
Abstract: An algorithm for computing a few of the smallest (or largest) eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem Ax = ABx is presented. The matrices A and B are assumed to be symmetric, and haphazardly sparse, with B being positive definite. The problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace. The rate of convergence of the method is established, and a technique for improving it is discussed. Numerical experiments and comparisons with other methods are presented.
TL;DR: In this paper, the eigenvalue problems of singularly perturbed linear ODEs are studied and a local stability analysis of the time-dependent semiconductor device equations via an eigen value problem is performed, and approximate eigenvalues in the case of a symmetric diode in the equilibrium state are computed.
Abstract: This paper is concerned with eigenvalue problems of singularly perturbed linear ordinary differential equations. A common way to treat such problems is to derive an approximating eigenvalue problem by the use of matched asymptotic expansions.It is shown that under appropriate assumptions a domain in the complex plane can be identified, in which the eigenvalues of the approximating problem are isolated and that the eigenvalues and invariant subspaces of the singularly perturbed problem converge to these eigenvalues and corresponding invariant subspaces of the approximating eigenvalue problem as the perturbation parameter tends to zero.As an application a local stability analysis of the time-dependent semiconductor device equations via an eigenvalue problem is performed, and approximations of the eigenvalues in the case of a symmetric diode in the equilibrium state are computed.
TL;DR: In this article, the eigenvalue interlacing property of Z-matrices and Hermitian matrices is investigated, i.e., the two smallest real eigenvalues of a matrix are interlaced by any principal submatrix of order one less.
TL;DR: Some iterative methods for eigenvalue problems and systems of linear equations involving large matrices are reviewed in this paper, where applications to the computation of molecular eigenstates and properties are presented and discussed.
Abstract: Some iterative methods for eigenvalue problems and systems of linear equations involving large matrices are reviewed. Applications to the computation of molecular eigenstates and properties are presented and discussed.
TL;DR: In this paper, a Chebyshev polynomial method was used to study the spectral properties of a non-self-adjoint Sturmian eigenvalue equation encountered in Quantum Scattering theory.
01 Jan 1987
TL;DR: In this article, a completely parallel algorithm for the symmetric eigenvalue problem AX = Lambda X is presented, in which a large number of operations can be programmed to be performed concurrently on a computer with multiple central processing units.
Abstract: A completely parallel algorithm for the symmetric eigenproblem AX = Lambda BX is outlined. The algorithm is parallel in the sense that the numerical operations do not occur in a fixed sequence. Therefore, a large number of operations can be programmed to be performed concurrently on a computer with multiple central processing units. The standard symmetric eigenvalue problem AX = Lambda X has the property that the n eigenvalues of the principal submatrix of A of order n are separated by the (n-1) eignvalues of the principal submatrix of order (n-1). The separation property delineated n intervals containing one eigenvalue. Each eigenvalue and corresponding eigenvector can be computed independently. The n eigenproblem calculations can be divided among multiple processing units.
TL;DR: In this paper, a straightforward application of the Pi Theorem will reveal the presence of an eigenvalue condition and will also provide some information about the form of the eigen value condition.
Abstract: The Pi Theorem of dimensional analysis says that the solution to any problem can be expressed using just dimensionless combinations of the relevant quantities. The illustrations given show that in some problems a straightforward application of the Pi Theorem will reveal the presence of an eigenvalue condition and will also provide some information about the form of the eigenvalue condition. This can happen even when the investigator is not specifically seeking the eigenvalue condition. These same illustrations make it clear why the presence of homogeneous rather than nonhomogeneous boundary conditions can result in an eigenvalue condition being present rather than absent.
01 Jun 1987
TL;DR: In this article, the set E⊂ℝ n should be defined as (3) F 0 (υ)=υ+x should read F(υ)-x.
Abstract: In the paper referred to above (see [1]), the following corrections should be made. (1) Hypothesis 2.2 should read: (2) The set E⊂ℝ n should be defined as (3) F 0 (υ)=υ+x should read F (υ)=υ–x. (4) The unique solution of F 0 (υ)=0 is υ= x (not υ= –x as printed). (5) Page 345, line 14:
01 Jan 1987
TL;DR: By using the expansions of piecewise-linear polynomial functions, the Sturm-Liouville eigenvalue problem can be dealt with as a set of linear algebraic equations as mentioned in this paper.
Abstract: By using the expansions of piecewise-linear polynomial functions, the Sturm-Liouville eigenvalue problem can be dealt with as a set of linear algebraic equations. Owing to the available recursive algorithm, these linear algebraic equations can be solved by straightforward substitution. An iterative improvement is used for finding eigenvalues in order to speed up the convergence. The usefulness of this method is demonstrated by an example and the results are satisfactory.
01 Jan 1987
TL;DR: In this article, the authors discuss the use of control canonical forms to help construct feedback controls solving the eigenvalue specification problem for a class of infinite-dimensional linear control systems.
Abstract: In this paper we discuss the use of control canonical forms to help construct feedback controls solving the eigenvalue specification problem for a class of infinite dimensional linear control systems.
TL;DR: In this paper, an efficient method of calculating angular basis functions for the expansion of scattering wave functions for molecular collisions is proposed, in which the perturbed rotor eigenvalue differential equation is replaced by a finite difference equation which can be solved step by step.