Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1986"

A survey of matrix inverse eigenvalue problems

Abstract: In this paper, we present a survey of some recent results regarding direct methods for solving certain symmetric inverse eigenvalue problems. The problems we discuss in this paper are those of generating a symmetric matrix, either Jacobi, banded, or some variation thereof, given only some information on the eigenvalues of the matrix itself and some of its principal submatrices. Much of the motivation for the problems discussed in this paper came about from an interest in the inverse Sturm-Liouville problem. A preliminary version of this report was issued as a technical report of the Computer Science Department, University of Minnesota, TR 86-20, May 1986.

Efficient Eigenvalue and Frequency Response Methods Applied to Power System Small-Signal Stability Studies

Abstract: Frequency response and eigenvalue techniques are fundamental tools in the analysis of small signal stability of multimachine power systems. This paper describes two highly efficient algorithms which are expected to enhance the practical application of these techniques. One algorithm calculates exact eigenvalues and eigenvectors for a large power system, while the other produces the frequency response of the transfer functions between any two variables in the system. This paper also presents alternative computing procedures for the AESOPS eigenvalue estimation algorithm which are simpler and at least as efficient as those described in [1].

Coupled free oscillations of an aspherical, dissipative, rotating Earth: Galerkin theory

Abstract: Variational theory based on self-adjoint equations of motion cannot fully represent the interaction of the earth's seismic free oscillations in the presence of lateral structure, attenuation, and rotation. The more general Galerkin procedure can model correctly the frequencies and attenuation rates of hybrid oscillations. Implementation of either algorithm leads to a generalized matrix eigenvalue problem in which the potential and kinetic energy interactions are separated into distinct matrices. The interaction of the earth's seismic free oscillations due to aspherical structure, attenuation, and rotation is best treated as a matrix eigenvalue problem. The presence of attenuation causes the matrices to be non-Hermitian and requires the use of a general Galerkin procedure. Physical dispersion, represented as a logarithmic function in frequency, must be represented by a truncated Taylor series about a fiducial frequency in order to be incorporated in the Galerkin formalism in a numerically tractable manner. The earth's rotation introduces an interaction matrix distinct from the potential and kinetic energy matrices, leading to a quadratic eigenvalue problem. A simple approximation leads to an eigenvalue problem linear in squared frequency. Tests show that this approximation is accurate for calculations using modes of frequencies f ≳ 1 mHz, unless interaction across a wide frequency band is modeled. Hybrid oscillation particle motions are represented by matrix eigenvectors that can be significantly nonorthogonal. The degrees of freedom in the low-frequency seismic system remain distinct, since source excitation is calculated by using dual eigenvectors. Synthetic seismograms that are constructed from Galerkin coupling calculations without reference to this eigenvector nonorthogonality can be disastrously noncausal.

An error indicator for the generalized eigenvalue problem using the hierarchical finite element method

Abstract: The hierarchical concept applied to eigenvalue problems is considered. An error indicator is derived from the pertinent Rayleigh quotient. The indicator serves as an estimation ‘a posteriori’ of the relative change in an eigenvalue for a hierarchical refinement. A numerical example is included.

Multiprocessor Jacobi algorithms for dense symmetric eigenvalue and singular value decompositions

Abstract: Two parallel algorithms are presented based on Jacobi's method for real symmetric matrices to determine the complete eigensystem of a dense real symmetric matrix and the singular value decomposition of rectangular matrices on a multiprocessor. The intent is to study the advantages of using Jacobi and Jacobi-like schemes over new and existing EISPACK and LINPACK routines on an Alliant FX/8 computer system. For the dense symmetric eigenvalue problem, promising results are shown for small-order matrices. A ''one-sided'' Jacobi-like algorithm which produces the singular value decomposition of a rectangular matrix is shown to provide superior performance for rectangular matrices in which the number of rows is much larger than the number of columns. 17 refs., 9 figs., 5 tabs.

Computing The Complex Eigenvalue Spectrum for Resistive Magnetohydrodynamics

Abstract: The spectrum of resistive MHD is evaluted by applying the Galerkin method in conjunction with finite elements. This leads to the general eigenvalue problem Ax = λBx. where A is a general non-Hermitian and B a symmetric positive-definite matrix. As this is a stiff problem, large matrix dimensions evolve. The QR algorithm can only be applied for a coarse grid. The fine grids necessary are treated by applying inverse vector iteration. Specific eigenvalue curves in the complex plane are obtained. By applying a continuation procedure it is possible by inverse vector iteration to map out in succession complete branches of the spectrum, e.g. all resistive Alfven modes, for matrix dimensions of up to 3.742.

The Impact of Parallel Architectures on The Solution of Eigenvalue Problems

Abstract: This paper presents a short survey of recent work on parallel implementations of Numerical Linear Algebra algorithms with emphasis on those relating to the solution of the symmetric eigenvalue problem on loosely coupled multiprocessor architectures. The vital operations in the formulation of most eigenvalue algorithms are matrix vector multiplication, matrix transposition, and linear system solution. Their implementations on several representative multiprocessor systems will be described, as well as parallel implementations of the following classes of eigenvalue methods : QR, bisection, divide-and-conquer, and Lanczos algorithm.

Sturmian eigenvalue equations with a Bessel function basis

Abstract: A non‐self‐adjoint Sturmian eigenvalue equation of the form Av=f, encountered in quantum scattering theory, is solved as a complex general matrix eigenvalue problem. The matrix form is obtained on expansion of the solution in a discrete set of spherical Sturmian–Bessel functions of complex argument. This set of basis functions gives better convergence behavior for both the eigenvalues and eigenfunctions when compared to the results of a Chebyshev polynomial method reported previously.

Sensitivity analysis and approximation methods for general eigenvalue problems

Abstract: Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy Proper eigenvector normalization is discussed An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity Important numerical aspects of this method are also discussed To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated Linear and quadratic approximations are based directly on the Taylor series Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem Approximation methods based on trace theorem give high accuracy without needing any derivatives Operation counts for the computation of the approximations are given General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought

Exact solution of a nonlinear eigenvalue problem.

Abstract: We show that Shastry's exact solution of a nonlinear eigenvalue problem in one dimension can be recovered by a method which is familiar in the theory of nonlinear ordinary differential equations.

The symmetric eigenvalue problem on a multiprocessor

Abstract: This paper is a survey of optimal methods for solving the symmetric eigenvalue problem on a multiprocessor. Although the inherent parallelism of the Jacobi method is well known, the convergence is not insured and the total number of operations is larger than that of the methods which transform the full matrix into a tridiagonal matrix. In this paper we are concerned with the methods which deal with the tridiagonalized eigenvalue systems. We review the sequential methods for solving symmetric eigenvalue problems by tridiagonal reduction using the appropriate routines from EISPACK. We also discuss the different ways of exploiting parallelism, examples of linear recurrence and orthogonalization are presented. Section 4 and Section 5 deal with the two algorithms, SESUPD, a parallel version of TQL2 in EISPACK, and TREPS, a parallel version of BISECT + TINVIT, respectively.

On The Implementation Of A Fully Parallel Algorithm For The Symmetric Eigenvalue Problem

Abstract: In this paper we present a parallel algorithm for the symmetric algebraic eigenvalue problem. The algorithm is based upon a divide and conquer scheme suggested by Cuppen for computing the eigensystem of a symmetric tridiagonal matrix. We extend this idea to obtain a parallel algorithm that retains a number of active parallel processes that is greater than or equal to the initial number throughout the course of the computation. Computation of the eigensystem of the tridiagonal matrix is reviewed. Also, brief analysis of the numerical properties and sensitivity to round off error is presented to indicate where numerical difficulties may occur. We show how to explicitly overlap the initial reduction to tridiagonal form with the parallel computation of the eigensystem of the tridiagonal matrix. The algorithm is therefore able to exploit parallelism at all levels of the computation and is well suited to a variety of architectures. Computational results have been presented in [4] for several machines. These results are very encouraging with respect to both accuracy and speedup. A surprising result is that the parallel algorithm for the tridiagonal case, even when run in serial mode, can be significantly faster than the previously best sequential algorithm on large problems, and is effective on moderate size problems when run in serial mode.

Simultaneous iterations algorithm for general eigenvalue problems on parallel processors

Abstract: The method of simultaneous iteration with shift is extended to extraction of m-eigenpairs of a general eigenvalue problem of large order n in a parallel processing environment. The algorithm combines the power method and the Jacobi technique, and reduces to performing four basic operations. Parallel implementation of the algorithm is discussed in detail. The analysis accounts for computation and communication costs, and utilizes a parallel processing architecture of the ensemble type. Expressions for the computational efficiency and speedup are defined as a function of the problem and hardware parameters. Selected representative problems exhibit efficiencies ranging from 60 to 98 percent.

Addendum to Eigenvalue Method for Solving Transient Heat Conduction Problems

Abstract: Computer time comparisons were performed between the eigenvalue method, the Crank-Nicolson method, and the fully implicit method for computing transient heat conduction in a two-and a three-dimensional problem. It was found that a large number of nodal points required excessive computer time for computation of the complete sets of eigenvalues and eigenvectors, as needed by the eigenvalue method.

Effective methods for solving eigenvalue problems with fourth-order elliptic operators

Abstract: A generalized eigenvalue problem with an elliptic fourth-order operator in a domain with a piecewise-smooth boundary is approximated by a finite element scheme. An efficient iterative method is suggested for evaluating the smallest eigenvalues of the constructed generalized algebraic eigenvalue problem, and an estimate is found for the complexity of the iterative method. The results are illustrated by using as examples three main types of problems arising in the theory of elastic plates. This paper is connected with the author's previous publications [6-14] and is devoted to effective numerical procedures of finding the smallest eigenvalue λί of eigenvalue problems with linear elliptic fourth-order operators in a bounded domain Ω on the plane with a piecewise-smooth boundary Γ. Such problems can be written in the operator form Lw = AMw, 0<λί<λ2<·(0.1) with bounded symmetric linear operators L and M in Hubert space H, where L and M are self-adjoint and positive definite; L~M is a compact operator. The Hubert space//can be of the form//! χ H2 χ ··· χ //p,where//risasubspaceof the Sobolev space W\\(£l) if 1 < r ̂ k^ ^ p, and a subspace of W\\(ty i f k 1 < p and r > k^ (see [3,4,24]). The elements of H are denoted by w = {w{,..., wp}. Let the eigenspace corresponding to λί consist of functions w such that WreW + (Q), y>0, l^r^p (0.2) where wr is the rth component of any eigenfunction w corresponding to m(r) = 2 if I k1. The main result of the paper is the construction, under proper conditions on the types of the boundary value problems, of some variants of the finite element method (FEM) and iterative methods (IM) for finding the smallest eigenvalues of the arising algebraic problems such that an approximation of λ± with 0(e)-accuracy may be obtained at the computational cost of W(s) arithmetic operations, where W(s) = 0(s-?\\lns\\\\ 1=12. (0.3) A generalization is possible if the condition Μ > 0 is not satisfied and λ1 is either the smallest positive or the largest negative eigenvalue. For such problems, the estimates (0.3) are still valid. The admissible boundary conditions can be of a rather general nature. If, for example, either the equation Δνν> = λ\\ν or Δνν = — ΑΔνν is considered, then the boundary conditions can be any of three main types known in the theory of elastic plates that correspond to the cases of clamped, simply supported, and free plates.

On the limit behavior of the largest eigenvalue of an elliptic operator with a small parameter

Abstract: The asymptotic behavior of the largest eigenvalue of a differential operator with a small parameter is studied.Figures: 1. Bibliography: 10 titles.

Parallel solution of eigenvalue problems in acoustics on the distributed array processor (DAP)

Abstract: The paper deals with the parallel solution of the partial differential equation -Δu=u in an 2− or 3-dimensional domain on the Distributed Arrary Processor (DAP). We want to find the eigenfunctions corresponding to the two or three smallest eigenvalues. A discretisation leads to a system of linear equations Ax=λ Bx, A,B ∈ R nxn ,x e R n , A and B large and sparse.