Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1982"

A Trace Minimization Algorithm for the Generalized Eigenvalue Problem

Abstract: An algorithm for computing a few of the smallest (or largest) eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem $Ax = \lambda Bx$ 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.

Analytical selection of masters for the reduced eigenvalue problem

Abstract: Masters are defined as the degrees-of-freedom that are retained in the reduced eigenvalue problem. Various qualitative guidelines to select masters are published in the literature, but it is difficult to apply them to complex structures. In this paper a computational algorithm to select masters for complex structures is presented. This algorithm is based on a guideline14 which assures that the associated Guyan reduction process is valid. This algorithm eliminates one degree-of-freedom at a time satisfying the guideline, and preserves lower frequencies in the reduced eigenvalue problem. The algorithm presented in this paper is used to select masters for four different structural models. The natural frequencies of the associated reduced eigenvalue problems are calculated and compared with those calculated from the full eigenvalue problems.

On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems

Abstract: We present an algorithmic method for obtaining an hereditary symmetry (the generalized squared‐eigenfunction operator) from a given isospectral eigenvalue problem. This method is applied to the n×n eigenvalue problem considered by Ablowitz and Haberman and to the eigenvalue problem considered by Alonso. The relevant Hamiltonian formulations are also determined. Finally, an alternative method is presented in the case two evolution equations are related by a Miura type transformation and their Hamiltonian formulations are known.

On estimating the largest eigenvalue with the Lanczos algorithm

Abstract: The Lanczos algorithm applied to a positive definite matrix produces good approximations to the eigenvalues at the extreme ends of the spectrum after a few iterations. In this note we utilize this behavior and develop a simple algorithm which computes the largest eigenvalue. The algorithm is especially economical if the order of the matrix is large and the accuracy requirements are low. The phenomenon of misconvergence is discussed. Some simple extensions of the algorithm are also indicated. Finally, some numerical examples and a comparison with the power method are given.

On the eigenvalue spectrum of the susceptibility matrix for random spin systems

Abstract: The eigenvalues and eigenvectors of the inverse of the susceptibility matrix are discussed for random spin systems. At a phase transition precipitated by the vanishing of the smallest eigenvalue it is shown that the corresponding eigenvector must be extended, and that the density of states must vanish at zero eigenvalue. Above the transition temperature, generalised Griffiths singularities are associated with the existence of localised states with arbitrarily small eigenvalues. For infinite spin dimensionality the eigenvalue problem is equivalent to an Anderson problem with correlated diagonal and off-diagonal disorder.

A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems

Abstract: The theory of Rayleigh functionals for non-linear eigenvalue problems T(λ) u = 0 is extended to cases where the functional is defined only on a proper subset. The theory applies to problems which do not satisfy an overdamping condition and yields a minimax characterization of eigenvalues. Applications to damped free vibrations of an elastic body are discussed.

Design with Several Eigenvalue Constraints by Finite Elements and Linear Programming

Abstract: A finite element discretization, combined with a powerful numerical eigenvalue procedure, has proved to be a unified approach to eigenvalue analysis of elastic solids. Treating the sensitivity analysis as an integrated part of this approach, one obtains gradients of the eigenvalues without any new eigenvalue analysis. This forms the necessary information for an optimal redesign which is formulated as a linear programming problem. By a sequence of optimal redesigns, one then obtains a solution to the problem of optimal design or a solution to an inverse eigenvalue problem. Taking as an example the vibration of Timoshenko beams, we focus on the gradient functions, on the dependence of slenderness, and on the inherent problem of local optima.

Systolic Arrays For Eigenvalue Computation

Abstract: A machine architecture for computing the eigenvalues and eigenvectors of an Hermitian matrix is presented. Two systolic arrays are used, one for reducing full matrices to band matrices, the second for performing QR iteration on band matrices. A one-parameter family of systems, parameterized by the bandwidth of the reduced matrix, is available. This allows a tradeoff of processors for execution time.

A Systolic Architecture for Almost Linear-Time Solution of the Symmetric Eigenvalue Problem

Abstract: An algorithm is presented for computing the eigenvalues and eigenvectors of an n x n real symmetric matrix. The algorithm is essentially a Jacobi method implemented on a two-dimensional systolic array of $O(n^{2})$ processors with nearest-neighbor communication between processors. The speedup over the serial Jacobi method is $\Theta(n^{2})$, so the algorithm converges to working accuracy in time $O(nS))$, where $S$ is the number of sweeps (typically $S \leq 10)$. Key Words and Phrases: Eigenvalue decomposition, real symmetric matrices, Hermitian matrices, Jacobi method, linear-time computation, systolic arrays, VLSI, real-time computation.

Generalized Eigenvalue Problems with Positive Semi-Definite Matrices.

Abstract: In a recent paper in this journal McDonald, Torii, and Nishisato show that generalized eigenvalue problems in which both matrices are singular can sometimes be solved by reducing them to similar problems of smaller order. In this paper a more extensive analysis of such problems is used to sharpen and clarify the results of McDonald, Torii, and Nishisato. Possible extensions are also indicated. The relevant mathematical literature is reviewed briefly.

Continuation algorithms for the eigenvalue problems

Abstract: Three algorithms for the solution of the eigenvalue problem for a continuously parameterized family of sparse matrices are presented; a continuousLU (orLR) algorithm, a continuousQR algorithm, and a continuous Hessenberg algorithm Each of the three algorithms may be implemented recursively and the sparsity of the given matrices is preserved throughout the numerical process

Computation Of Minimum Eigenvalue Of Toeplitz Matrix By Levinson Algorithm

Abstract: This paper considers the computation of the minimum eigenvalue of a symmetric Toeplitz matrix via the Levinson algorithm. By exploiting the relationship between the minimum eigen-value and the residues obtained in the Levinson algorithm, a fast iterative procedure is established to successively estimate the minimum eigenvalue. Although the computational complexity analysis is yet inconclusive, we have found that the approximation of the minimum eigenvalue has an important application in high resolution spectrum estimation problems. Based on simulation results for such an application, some improvements are observed in both the computing speed as well as accuracy of estimates.© (1982) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Correction of Sturm-Liouville eigenvalue estimates

Abstract: The error in the Sturm-Liouville eigenvalue estimates obtained by replacing the coefficient function with a piecewise constant interpolate is not uniform. In this paper we present a method for correcting these estimates to obtain a uniform approximation of all

Solution of linear gyroscopic systems

Abstract: HE solution of gyroscopic system equations requires solution of an eigenvalue problem which by present methods is accomplished via a transformation to a standard non-gyroscopic eigenvalue problem on two symmetric matrices with its associated computational advantages. However this is done at the expense of doubling the dimension and simultaneously doubling the multiplicities of the eigenfrequencies. In this paper a method is presented which, for most practical problems eliminates these difficulties. The method takes advantage of special phase relationships satisfied in many practical systems.

Discretization of multiparameter eigenvalue problems

Abstract: Although multiparameter eigenvalue problems, as for example Mathieu's differential equation, have been known for a long time, so far no work has been done on the numerical treatment of these problems. So in this paper we extend the spectral theory for one parameter (cf. [7, II, VII]) to multiparameter eigenvalue problmes, formulate in the framework of discrete approximation a convergent numerical treatment, establish algebraic bifurcation equations for the intersection points of the eigenvalue curves and illustrate this with some numerical examples.

On the eigenvalue problem of 't Hooft

Abstract: We determine the asymptotic distribution of the eigenvalues in ′ t Hoofts eigenvalue problem in two-dimensional quantum chromodynamics. We formulate the problem as an eigenvalue problem for a singular pseudodifferential operator and use systematically its basic invariance properties.

Numerical Method for Nonlinear Eigenvalue Problems

Abstract: A new efficient numerical scheme is presented for analyzing nonlinear boundary value and eigenvalue problems. The method consists of combining the shooting method and Newton's iteration scheme. The method is applied to study the nonlinear interaction of strong laser light with plasma. By the numerical analysis of the structure resonance in laser plasma interaction, multi-valued refraction coefficient as a function of incident laser intensity, and associated anomalous transmission and absorption phenomena are obtained.

A fast solver for nonlinear eigenvalue problems

Abstract: A numerical method recently proposed by the author is shown to be a very efficient and robust method for the solution of a class of discrete nonlinear eigenvalue problems. In particular it is applied to follow the relevant and the spurious solution curves. Numerical results show that also in the neighbourhood of turning or bifurcation points the work required is considerably less than for usual continuation procedures and that a larger steplength may be chosen. A corresponding multi-grid method is used for following spurious solution branches.