Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1972"

Large Eigenvalue Problems in Dynamic Analysis

Abstract: An effective solution technique is presented to calculate the p lowest eigenvalues and corresponding vectors in the problem KΦ = ω²MΦ, when the order and bandwidth of the matrices is large. The eigenvalue problem is solved directly without a transformation to the standard form. The mass matrix, M, may be diagonal with zero elements as in a lumped mass analysis or may be banded as in a consistent mass formulation. The algorithm establishes q starting vectors, q > p , from the elements in M and K and iterates with all vectors simultaneously. This iteration is described as a subspace iteration, where best eigenvalue and eigenvector approximations can be calculated in each iteration. Operation counts are given which show the cost effectiveness of the algorithm when the bandwidth of the system is large. A program is described to solve the eigenvalue problem when the system has practically any order and bandwidth. Two example analyses are presented.

Note on matrices with a very ill-conditioned eigenproblem

Abstract: Gives a bound for the distance of a matrix having an ill-conditioned eigenvalue problem from a matrix having a multiple eigenvalue which is generally sharper than that which has been published hitherto.

Lusternik-Schnirelman theory and non-linear eigenvalue problems

Abstract: In this paper we employ the ideas of Lusternik and Schnirelman [14] to establish the existence of infinitely many distinct eigenfunctions for problem (1.1). This problem has already attracted considerable interest [3-7, 9-11, 13, 14, 18, 19]. All these investigations have been based on variants of the so-called Lusternik-Schnirelman theory. In the early 1930's Lusternik and Schnirelman developed a theory of critical points for differentiable functions on finite-dimensional Riemannian manifolds. One of the principal tools for establishing the existence of "intermediate" critical points (i.e. of critical points not belonging to absolute maxima or minima) is the same as in the Morse theory, namely the deformation of the manifold along gradient lines. The application of this theory to infinite-dimensional eigenvalue problems of the form (1.1) which arise in connection with differential and integral equations require the generalization of the Lusternik-Schnirelman theory to infinite-dimensional manifolds. This extension has been made by Schwartz [16, 17] for Riemannian manifolds modelled on Hilbert spaces and by R. S. Palais [15] for Finsler manifolds modelled on arbitrary Banach spaces. These generalizations are based on a fundamental compactness assumption, the so-called Palais-Smale Condition. In applying this general Lusternik-Schnirelman theory to the eigenvalue problem (1.1) one is faced with two technical difficulties. First, one

An Algorithm for the Ill-Conditioned Generalized Eigenvalue Problem

Abstract: The spectrum of $Ax - \lambda Bx = 0$ consists of stable and unstable eigenvalues, which undergo, respectively, small and large changes in response to small changes in A and B. The algorithm isolates and accurately computes the eigenspace associated with the stable eigenvalues.

Derivation of weighting matrices towards satisfying eigenvalue requirements

Abstract: The present work is concerned with the problem of designing appropriate weighting matrices for the performance index of the linear regulator optimization problem, guch that the resulting optimal policy satisfies given eigenvalue requirements. The text provides an algorithm for deriving a diagonal state-weighting matrix according to these eigenvalue requirements, the latter often being better defined than the requirements for the stato-weighting matrix. The solution given employs matrix differential calculus and a static gradient minimization sub-routine to minimize an eigenvalue error criterion, thus facilitating optimal control within desired eigenvalue requirements.

Solution of eigenvalue problems by Sturm sequence method.

Abstract: A generalized eigenvalue algorithm is presented herein along with the complete listing of the associated computer program, which may be conveniently utilized for the efficient solution of certain broad classes of eigenvalue problems. Extensive applications of the procedure are envisaged in the analysis of many important engineering problems, such as stability and natural frequency analysis of practical discrete structural systems, idealized by the finite element technique. The procedure based on the Sturm sequence method is accurate and fast, possessing several significant advantages over other known methods of such analysis. Numerical results are also presented for two representative structural engineering problems.

On the Ratio L12/L21 For n = 2 Eigenvalue Problems: An Empirical Constraint For The Calculation of Force Constants (Extended L Matrix Approximation)

Abstract: Abstract The variation of the ratio L12/L21 (where Lij refers to the element of the L matrix) as a function of the mass coupling term T=G12/|G|1/2 has been studied for molecules of the type XY4(Td), XY3(D3h) and XY2C2v) using the exact force field data. In the case of the XY4 and XY3 type molecules, the ratio L12/L21 is nearly independent of the mass coupling and in the case of the XY2 type, it shows a nearly linear dependence on T. The force constants have been evaluated using an empirically determined constraint for L12/L21. The values are generally in good agreement with the exact ones. In all cases except VCl4 the exact interaction term of the F matrix is in agreement with the sign rule of Müller et al.

The convergence of the lowest eigenvalue of a real nearly-symmetric matrix

Abstract: The lowest eigenvalue of a real nearly-symmetric matrix is expressed as a perturbation series in terms of the eigenvalues of the symmetric part and the matrix elements of the skew-symmetric part. It is shown that the resulting series is closely related to the perturbation series for the lowest eigenvalue of a related hermitian matrix. This enables the behaviour of the lowest eigenvalue of a nearly symmetric matrix as the dimension of the matrix is increased to be deduced from the behaviour of the lowest eigenvalue of a hermitian matrix. This is of considerable importance as the behaviour of the lowest eigenvalue of a hermitian matrix as the dimension of the matrix is increased can be much more readily established. A possible application to Boys' transcorrelated method of calculating atomic and molecular energies is suggested.

Eigenvalue bounds in linear inviscid stability theory

Abstract: New eigenvalue bounds are derived for the linear stability of inviscid parallel flows, both for homogeneous and for stratified fluids. The usefulness of these bounds, as compared with that of previous results, is assessed for several examples. For homogeneous fluids the new upper bounds for the imaginary part c i of the complex phase velocity are sometimes better than previous criteria. For both homogeneous and stratified flows, the new upper bounds for the wave-number α of neutrally stable disturbances improve on previous results, giving values within 10% of the known exact solution in several cases.

On the Reduction of the Generalized RPA Eigenvalue Problem

Abstract: The 2n‐dimensional eigenvalue problem, which arises when the random phase approximation (RPA) matrix is not real, is reduced to an n‐dimensional eigenvalue problem. Some properties of the reduced eigenvalue problem are studied. A numerical example is considered for illustrative purposes.

On the bifurcation theory of semilinear elliptic eigenvalue problems

Abstract: A standard "bootstrap" method is used to show that the bifurcation problem for the semilinear eigenvalue problem Au + Af(x, u) = O in Q, ulan = O, where f(x, O)-O, and (a/au)f(x, 0) > 0, and when formulated in terms of weak solutions, is a local problem, i.e. independent of the behavior of f for large u. A principle of linearization for this problem is proved under mild differentiability conditions onf. 1. This note is concerned with the bifurcation problem for the semilinear elliptic eigenvalue problem (1) -Au = A(P(x)u + g(x, u)), x E Q, ulan = 0. Here Q is a bounded region in RN (N > 2) for which the Dirichlet problem is solvable, and g, defined for small u, is odd and monotone in u. For recent contributions to bifurcation theory for nonlinear elliptic eigenvalue problems and for additional references see [5]. The object here is twofold, namely to show that in the variational treatment of the bifurcation problem for (1) a polynomial growth condition on g, as usually required, is unnecessary and using this fact and results of [3] and/or [6], to derive bifurcation theorems for (1). For simplicity we have restricted the linear operator on the left in (1) to be the Laplacian. Without difficulty the results obtained below can be extended to the larger class of real linear formally selfadjoint operators considered in [2]. The machinery for doing this is set up in [2]. 2. The approach here to the eigenvalue problem (1) is through the study of the integral equation (2) u(x) = iAf G(x, t)f(t, u(t)) dt, Received by the editors September 21, 1970 and, in revised form, April 9, 1971. AMS 1970 subject classiflcations. Primary 35G30, 35J20, 35J25.

On the Existence of Relations among the Eigenvalues of Difference Equations

Abstract: By translating the eigenvalue problem of a difference equation to one of abstract operators, we give a method of finding relations among the eigenvalues of the difference equations.