Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2008"

The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods

Abstract: This book presents the first in-depth, complete, and unified theoretical discussion of the two most important classes of algorithms for solving matrix eigenvalue problems: QR-like algorithms for dense problems and Krylov subspace methods for sparse problems. The author discusses the theory of the generic GR algorithm, including special cases (for example, QR, SR, HR), and the development of Krylov subspace methods. Also addressed are a generic Krylov process and the Arnoldi and various Lanczos algorithms, which are obtained as special cases. The chapter on product eigenvalue problems provides further unification, showing that the generalized eigenvalue problem, the singular value decomposition problem, and other product eigenvalue problems can all be viewed as standard eigenvalue problems. The author provides theoretical and computational exercises in which the student is guided, step by step, to the results. Some of the exercises refer to a collection of MATLAB programs compiled by the author that are available on a Web site that supplements the book. Audience: Readers of this book are expected to be familiar with the basic ideas of linear algebra and to have had some experience with matrix computations. This book is intended for graduate students in numerical linear algebra. It will also be useful as a reference for researchers in the area and for users of eigenvalue codes who seek a better understanding of the methods they are using. Contents: Preface; Chapter 1: Preliminary Material; Chapter 2: Basic Theory of Eigensystems; Chapter 3: Elimination; Chapter 4: Iteration; Chapter 5: Convergence; Chapter 6: The Generalized Eigenvalue Problem; Chapter 7: Inside the Bulge; Chapter 8: Product Eigenvalue Problems; Chapter 9: Krylov Subspace Methods; Bibliography; Index.

Convergence and optimal complexity of adaptive finite element eigenvalue computations

Abstract: In this paper, an adaptive finite element method for elliptic eigenvalue problems is studied. Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved. The analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper.

Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems.

Abstract: In this paper we provide new fast and simple solutions to two important minimal problems in computer vision, the five-point relative pose problem and the six-point focal length problem. We show that these two problems can easily be formulated as polynomial eigenvalue problems of degree three and two and solved using standard efficient numerical algorithms. Our solutions are somewhat more stable than state-of-the-art solutions by Nister and Stewenius and are in some sense more straightforward and easier to implement since polynomial eigenvalue problems are well studied with many efficient and robust algorithms available. The quality of the solvers is demonstrated in experiments 1.

On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm

Abstract: This paper is devoted to the eigenvalue complementarity problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex (Queiroz et al., Math. Comput. 73, 1849–1863, 2004). We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigenvalue by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projected gradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP.

On Slow-Fading MIMO Systems With Nonseparable Correlation

Abstract: In a frequency-selective slow-fading channel in a multiple-input multiple-output (MIMO) system, the channel matrix is of the form of a block matrix. A method is proposed to calculate the limit of the eigenvalue distribution of block matrices if the size of the blocks tends to infinity. Asymptotic eigenvalue distribution of is also calculated, where the entries of are jointly Gaussian, with a correlation of the form (where is fixed and does not increase with the size of the matrix). An operator-valued free probability approach is used to achieve this goal. Using this method, a system of equations is derived, which can be solved numerically to compute the desired eigenvalue distribution.

Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers - eScholarship

Abstract: PERFORMANCE AND ACCURACY OF LAPACK’S SYMMETRIC TRIDIAGONAL EIGENSOLVERS JAMES W DEMMEL † OSNI A MARQUES ‡ BERESFORD N PARLETT † AND CHRISTOF V OMEL ‡ Abstract We compare four algorithms from the latest LAPACK 31 release for computing eigenpairs of a symmetric tridiagonal matrix These include QR iteration, bisection and inverse iteration (BI), the Divide-and-Conquer method (DC), and the method of Multiple Relatively Robust Representations (MR) Our evaluation considers speed and accuracy when computing all eigenpairs, and additionally subset computations Using a variety of carefully selected test problems, our study includes a variety of today’s computer architectures Our conclusions can be summarized as follows (1) DC and MR are generally much faster than QR and BI on large matrices (2) MR almost always does the fewest floating point operations, but at a lower MFlop rate than all the other algorithms (3) The exact performance of MR and DC strongly depends on the matrix at hand (4) DC and QR are the most accurate algorithms with observed accuracy O( ne) The accuracy of BI and MR is generally O(ne) (5) MR is preferable to BI for subset computations Key words LAPACK, symmetric eigenvalue problem, inverse iteration, Divide & Conquer, QR algorithm, MRRR algorithm, accuracy, performance, benchmark AMS subject classifications 15A18, 15A23 1 Introduction One goal of the latest 31 release [25] of LAPACK [1] is to pro- duce the fastest possible symmetric eigensolvers subject to the constraint of delivering small residuals and orthogonal eigenvectors For an input matrix A that may be dense or banded, one standard approach is the conversion to tridiagonal form T , then the eigenvalues and eigenvectors of T are found, and last the eigenvectors of T transformed to eigenvectors of A Depending on the situation, all the eigenpairs or just some of them may be de- sired LAPACK, for some algorithms, allows selection by eigenvalue indices (‘find λ i , λ i+1 , λ j , where λ 1 ≤ λ 2 ≤ · · · ≤ λ n are all the eigenvalues in increasing order’, and their eigenvectors) or by an interval (‘find all the eigenvalues in [a, b] and their eigenvectors’) This paper analyzes the performance and accuracy of four algorithms: 1 QR iteration, in LAPACK’s driver STEV (QR for short), 2 Bisection and Inverse Iteration, in STEVX (BI for short), 3 Divide and Conquer, in STEVD (DC for short), 4 Multiple Relatively Robust Representations, in STEVR (MR for short) Section 2 gives a brief description of these algorithms with references For a representative picture of each algorithm’s capacities, we developed an ex- tensive set of test matrices [7], broken into two classes: (1) ‘practical matrices’ based on reducing matrices from a variety of practical applications to tridiagonal form, and generating some other tridiagonals with similar spectra, and (2) synthetic ‘testing ma- trices’ chosen to have extreme distributions of eigenvalues or other properties designed to exercise one or more of the algorithms, see Section 31 for details The timing and † Mathematics Department and Computer Science Division, University of California, Berkeley, CA 94720, USA {demmel@cs,parlett@math}berkeleyedu ‡ Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA {oamarques,cvoemel}@lblgov

On the Convergence of the Self-Consistent Field Iteration for a Class of Nonlinear Eigenvalue Problems

Abstract: We investigate the convergence of the self-consistent field (SCF) iteration used to solve a class of nonlinear eigenvalue problems. We show that for the class of problems considered, the SCF iteration produces a sequence of approximate solutions that contain two convergent subsequences. These subsequences may converge to two different limit points, neither of which is the solution to the nonlinear eigenvalue problem. We identify the condition under which the SCF iteration becomes a contractive fixed point iteration that guarantees its convergence. This condition is characterized by an upper bound placed on a parameter that weighs the contribution from the nonlinear component of the eigenvalue problem. We derive such a bound for the general case as well as for a special case in which the dimension of the problem is $2$.

Constant-sign and sign-changing solutions for nonlinear eigenvalue problems

Abstract: We prove the existence of multiple constant-sign and sign-changing solutions for a nonlinear elliptic eigenvalue problem under Dirichlet boundary condition involving the p -Laplacian. More precisely, we establish the existence of a positive solution, of a negative solution, and of a nontrivial sign-changing solution when the eigenvalue parameter λ is greater than the second eigenvalue λ 2 of the negative p -Laplacian, extending results by Ambrosetti–Lupo, Ambrosetti–Mancini, and Struwe. Our approach relies on a combined use of variational and topological tools (such as, e.g., critical points, Mountain-Pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the p -Laplacian) and comparison arguments for nonlinear differential inequalities. In particular, the existence of extremal nontrivial constant-sign solutions plays an important role in the proof of sign-changing solutions.

Block Krylov–Schur method for large symmetric eigenvalue problems

Abstract: Stewart’s Krylov–Schur algorithm offers two advantages over Sorensen’s implicitly restarted Arnoldi (IRA) algorithm. The first is ease of deflation of converged Ritz vectors, the second is the avoidance of the potential forward instability of the QR algorithm. In this paper we develop a block version of the Krylov–Schur algorithm for symmetric eigenproblems. Details of this block algorithm are discussed, including how to handle rank deficient cases and how to use varying block sizes. Numerical results on the efficiency of the block Krylov–Schur method are reported.

A Fast Adaptive Algorithm for the Generalized Symmetric Eigenvalue Problem

Abstract: In this letter, we propose a new adaptive algorithm for the generalized symmetric eigenvalue problem, which can extract the principal and minor generalized eigenvectors, as well as their corresponding subspaces, at a low computational cost. A comparison with other adaptive algorithms from the literature, including the batch generalized singular value decomposition (GSVD) technique, is also given to show the superiority of the proposed algorithm in terms of convergence performance and computational complexity.

Random matrix ensembles of time-lagged correlation matrices: derivation of eigenvalue spectra and analysis of financial time-series

Abstract: We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as real, asymmetric random matrices where the time-shift superimposes some structure. We demonstrate that, for large matrices, the associated eigenvalue spectrum is circular symmetric in the complex plane. This fact allows us to exactly compute the eigenvalue density via an inverse Abel-transform of the density of the symmetrized problem. We demonstrate the validity of this approach numerically. Theoretical findings are then compared with eigenvalue densities obtained from actual high-frequency (5 min) data of the S&P 500 and the observed deviations are discussed. We identify various non-trivial, non-random patterns and find asymmetric dependencies associated with eigenvalues departing strongly from the Gaussian prediction in the imaginary part. For the same time-series, with the market contribution removed, we observe strong ...

Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbations

Abstract: In this paper we address the problem of efficiently computing all the eigenvalues of a large N×N Hermitian matrix modified by a possibly non Hermitian perturbation of low rank. Previously proposed fast adaptations of the QR algorithm are considerably simplified by performing a preliminary transformation of the matrix by similarity into an upper Hessenberg form. The transformed matrix can be specified by a small set of parameters which are easily updated during the QR process. The resulting structured QR iteration can be carried out in linear time using linear memory storage. Moreover, it is proved to be backward stable. Numerical experiments show that the novel algorithm outperforms available implementations of the Hessenberg QR algorithm already for small values of N.

Harmonic and refined Rayleigh-Ritz for the polynomial eigenvalue problem

Abstract: After reviewing the harmonic Rayleigh–Ritz approach for the standard and generalized eigenvalue problem, we discuss several extraction processes for subspace methods for the polynomial eigenvalue problem. We generalize the harmonic and refined Rayleigh–Ritz approaches which lead to new approaches to extract promising approximate eigenpairs from a search space. We give theoretical as well as numerical results of the methods. In addition, we study the convergence of the Jacobi–Davidson method for polynomial eigenvalue problems with exact and inexact linear solves and discuss several algorithmic details. Copyright © 2008 John Wiley & Sons, Ltd.

Eigenvalue Separation in Some Random Matrix Models

Abstract: The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the size of the matrices are fixed and c goes to infinity, and higher rank analogues of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions, and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogues an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.

Harmonic Rayleigh-Ritz extraction for the multiparameter eigenvalue problem

Abstract: We study harmonic and refined extraction methods for the multiparameter eigenvalue problem. These techniques are generalizations of their counterparts for the standard and generalized eigenvalue problem. The methods aim to approximate interior eigenpairs, generally more accurately than the standard extraction does. We study their properties and give Saad-type theorems. The processes can be combined with any subspace expansion approach, for instance a Jacobi-Davidson type technique, to form a subspace method for multiparameter eigenproblems of high dimension.

Jacobi Correction Equation, Line Search, and Conjugate Gradients in Hermitian Eigenvalue Computation I: Computing an Extreme Eigenvalue

Abstract: This paper is concerned with the convergence properties of iterative algorithms of conjugate gradient type applied to the computation of an extreme eigenvalue of a Hermitian operator via the optimization of the Rayleigh quotient functional. The algorithms in focus employ the line search for the extremum of the Rayleigh quotient functional in the direction that is a linear combination of the gradient and the previous search direction. An asymptotically explicit equation for the reduction in the eigenvalue error after the line search step as a function of the search direction is derived and applied to the analysis of the local convergence properties (i.e., the error reduction after an iteration) of the algorithms in question. The local (stepwise) asymptotic equivalence of various conjugate gradient algorithms and the generalized Davidson method is proved, and a new convergence estimate for conjugate gradient iterations is derived, showing the reduction in the eigenvalue error after any two consecutive iterations. The paper's analysis extensively employs remarkable properties of the operator of the so-called Jacobi orthogonal complement correction equation.

Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods

Abstract: In this paper, we analyze the biharmonic eigenvalue problem by two nonconforming finite elements, Q and E Q. We obtain full order convergence rate of the eigenvalue approximations for the biharmonic eigenvalue problem based on asymptotic error expansions for these two nonconforming finite elements. Using the technique of eigenvalue error expansion, the technique of integral identities, and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Eigenvalues of the antiplane-shear crack problem for a power-law material

Abstract: This paper discusses the problem of finding the eigenvalue spectrum in determining the stress and strain fields at the tip of an antiplane-shear crack in a power-law material. It is shown that the perturbation method provides an analytical dependence of the eigenvalue on the material nonlinearity parameter and the eigenvalue of the linear problem. Thus, it is possible to find the entire spectrum of eigenvalues and not only the eigenvalue of the Hutchinson-Rice-Rosengren problem.

Eigenvalue Characterization and Computation for the Laplacian on General 2-D Domains

Abstract: In this paper, we address the problem of determining and efficiently computing an approximation to the eigenvalues of the negative Laplacian − ▵ on a general domain Ω ⊂ ℝ2 subject to homogeneous Dirichlet or Neumann boundary conditions. The basic idea is to look for eigenfunctions as the superposition of generalized eigenfunctions of the corresponding free space operator, in the spirit of the classical method of particular solutions (MPS). The main novelties of the proposed approach are the possibility of targeting each eigenvalue independently without the need for extensive scanning of the positive real axis and the use of small matrices. This is made possible by iterative inclusion of more basis functions in the expansions and a projection idea that transforms the minimization problem associated with MPS and its variants into a relatively simple zero-finding problem, even for expansions with very few basis functions.