Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2006"
TL;DR: In this article, the p -Laplace operator subject to different kinds of boundary conditions on a bounded domain is studied and the existence of a non-decreasing sequence of nonnegative eigenvalues is shown.
Abstract: We study nonlinear eigenvalue problems for the p -Laplace operator subject to different kinds of boundary conditions on a bounded domain. Using the Ljusternik–Schnirelman principle, we show the existence of a nondecreasing sequence of nonnegative eigenvalues. We prove the simplicity and isolation of the principal eigenvalue and give a characterization for the second eigenvalue.
262 citations
TL;DR: In this article, the authors studied the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices.
Abstract: We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries Mij,i≤j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix Open image in new window and its rank. This rank is also allowed to increase with N in some restricted way.
203 citations
TL;DR: A family of eigenvalue inequalities for the product of a Hermitian matrix and a positive-semidefinite matrix is presented and the theorem contains or extends some existing results on trace and eigenvalues.
Abstract: We present a family of eigenvalue inequalities for the product of a Hermitian matrix and a positive-semidefinite matrix. Our theorem contains or extends some existing results on trace and eigenvalues
93 citations
Journal Article•
TL;DR: In this paper, the notion of first eigenvalue for fully nonlinear operators which are nonvariational but homogeneous was introduced, and the importance of the spectrum of a linear operator was minimized.
Abstract: In this paper we introduce the notion of first eigenvalue for fully nonlinear operators which are nonvariational but homogeneous. It is unnecessary to emphasize the importance of knowing the spectrum of a linear operator. When the operator is a uniformly elliptic operator of second order Lu = tr(A(x)D2u) associated with a Dirichlet problem in a bounded domain Ω the spectrum is a point spectrum bounded from below and the first eigenvalue λ is paramount. It is well known that λ is positive and it satisfies: • There exists a positive function φ satisfying { Lφ + λφ = 0 in Ω φ = 0 on ∂Ω.
83 citations
TL;DR: New accurate eigenvalue bounds for symmetric matrices of saddle point form are derived and applied for both unpreconditioned and preconditionsed versions of the matrices to enable a better understanding of how preconditioners should be chosen.
Abstract: Eigenvalue bounds for saddle point matrices on symmetric or, more generally, nonsymmetric form are derived and applied for preconditioned versions of the matrices. The preconditioners enable efficient iterative solution of the corresponding linear systems.
73 citations
TL;DR: The Ritz--Galerkin method for symmetric eigenvalue problems is analyzed and what is apparently the first truly a priori error estimates that show the levels of the eigen value errors depending on approximability of eigenfunctions in the corresponding eigenspace are proved.
Abstract: We analyze the Ritz--Galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates. For a simple eigenvalue, we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants. For a multiple eigenvalue we prove, in addition, what is apparently the first truly a priori error estimates that show the levels of the eigenvalue errors depending on approximability of eigenfunctions in the corresponding eigenspace. These estimates reflect a known phenomenon that different eigenfunctions in the corresponding eigenspace may have different approximabilities, thus resulting in different levels of errors for the approximate eigenvalues. For clustered eigenvalues, we derive eigenvalue error bounds that do not depend on the width of the cluster. Our results are readily applicable to the classical Ritz method for compact symmetric integral operators and to finite element method eigenvalue approximation for symmetric positive definite differential operators.
72 citations
TL;DR: This paper investigates the effect of structure-preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems, and shows that under reasonable assumptions on the scalar product, the structured and unstructured eigen value condition numbers are equal for structures in Jordan algebras.
Abstract: This paper investigates the effect of structure-preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo-symmetric, persymmetric, skew-symmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number.
70 citations
TL;DR: This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigen value problems of large order with a monotone dependence on the spectral parameter: preconditionsed simple iteration method, preconditioned steepest descent method, and precONDitioned conjugate gradient method.
66 citations
TL;DR: The local averaging technique is introduced to solve a class of symmetric eigenvalue problems and its efficiency and reliability are proved by both the theory and numerical experiments structured meshes as well as irregular meshes.
Abstract: The local averaging technique has become a popular tool in adaptive finite element methods for solving partial differential boundary value problems since it provides efficient a posteriori error estimates by a simple postprocessing. In this paper, the technique is introduced to solve a class of symmetric eigenvalue problems. Its efficiency and reliability are proved by both the theory and numerical experiments structured meshes as well as irregular meshes.
61 citations
TL;DR: By means of eigenvalue error expansion and integral expansion techniques, this paper proposed and analyzed the stream function-vorticity-pressure method for the eigen value problem associated with the Stokes equations on the unit square.
Abstract: By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally, numerical experiments are reported.
48 citations
TL;DR: Differences between the BR and the QR algorithms with performance comparison in terms of CPU time based on stopping criteria and storage requirement suggest that the BR algorithm is a more efficient algorithm for large-scale power system small signal stability eigenanalysis.
Abstract: The BR algorithm is a novel and efficient method to find all eigenvalues of upper Hessenberg matrices and has never been applied to eigenanalysis for power system small signal stability. This paper analyzes differences between the BR and the QR algorithms with performance comparison in terms of CPU time based on stopping criteria and storage requirement. The BR algorithm utilizes accelerating strategies to improve its performance when computing eigenvalues of narrowly banded, nearly tridiagonal upper Hessenberg matrices. These strategies significantly reduce the computation time at a reasonable level of precision. Compared with the QR algorithm, the BR algorithm requires fewer iteration steps and less storage space without depriving of appropriate precision in solving eigenvalue problems of large-scale power systems. Numerical examples demonstrate the efficiency of the BR algorithm in pursuing eigenanalysis tasks of 39-, 68-, 115-, 300-, and 600-bus systems. Experiment results suggest that the BR algorithm is a more efficient algorithm for large-scale power system small signal stability eigenanalysis
TL;DR: The methods are iterative in nature and utilize alternating projection ideas, and the main computational component of each iteration is an eigenvalue-eigenvector decomposition, while for the other algorithm, it is a Schur matrix decomposition.
Abstract: Presented here are two related numerical methods, one for the inverse eigenvalue problem for nonnegative or stochastic matrices and another for the inverse eigenvalue problem for symmetric nonnegative matrices. The methods are iterative in nature and utilize alternating projection ideas. For the algorithm for the symmetric problem, the main computational component of each iteration is an eigenvalue-eigenvector decomposition, while for the other algorithm, it is a Schur matrix decomposition. Convergence properties of the algorithms are investigated and numerical results are also presented. While the paper deals with two specific types of inverse eigenvalue problems, the ideas presented here should be applicable to many other inverse eigenvalue problems, including those involving nonsymmetric matrices.
TL;DR: In this article, an algorithm to compare the zero-structured individual condition numbers of a set of simple eigenvalues with the traditional ones is presented, and numerical tests highlight how the algorithm provides interesting information about eigenvalue sensitivity when the perturbations in the matrix have an arbitrarily assigned zero-structure.
TL;DR: A treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew‐symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices.
Abstract: Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices.
TL;DR: In this article, an eigenvalue problem that involves uncertain-but-non-random parameters is discussed, and a new method is developed to evaluate the reliable upper and lower bounds on frequencies of structures for these problems.
Abstract: In this paper, an eigenvalue problem that involves uncertain-but-non-random parameters is discussed. A new method is developed to evaluate the reliable upper and lower bounds on frequencies of structures for these problems. In this method the matrix in the deviation amplitude interval is considered to be a perturbation around the nominal value of the interval matrix, and the upper and lower bounds to the maximum and minimum eigenvalues of this perturbation matrix are computed, respectively. Then based on the matrix perturbation theory, the eigenvalue bounds of the original interval eigenvalue problem can be obtained. Finally, two numerical examples are provided and the results show that the proposed method is reliable and efficient. Copyright © 2006 John Wiley & Sons, Ltd.
TL;DR: In this article, the authors studied the asymptotic behavior of the eigenvalues of a family of nonlinear monotone elliptic operators of the form Ae = −div(ae (x, ∇u)), which are sub-differentials of even, positively homogeneous convex functionals.
Abstract: In this article we study the asymptotic behaviour of the eigenvalues of a family of nonlinear monotone elliptic operators of the form Ae = −div(ae (x, ∇u)), which are sub-differentials of even, positively homogeneous convex functionals, under the assumption that the operators G-converge to an operator Ahom = −div(ahom(x, ∇u)). We show that any limit point λ of a sequence of eigenvalues λe is an eigenvalue of the limit operator Ahom, where λe is an eigenvalue corresponding to the operator Ae. We also show the convergence of the sequence of first eigenvalues to the corresponding first eigenvalue of the homogenized operator.
TL;DR: In this article, the authors presented a modification of the domaindecomposition method of Descloux and Tolley for planar eigenvalue problems, which is based on the generalized singular value decomposition.
Abstract: In this article we present a modification of the domain
decomposition method of Descloux and Tolley for planar eigenvalue problems. Instead of formulating a generalized eigenvalue problem our method is based on the generalized singular value decomposition. This approach is robust and at the same time highly accurate. Furthermore, we give an improved convergence analysis based on results from complex approximation theory. Several examples show the effectiveness of our method.
17 Jun 2006
TL;DR: This paper shows that four non-iterative algorithms for simultaneous low rank approximations of matrices (SLRAM) are equivalent to each other because they are reduced to the eigenvalue problems of row-row and column-column covariance matrices of given matrices.
Abstract: Recently four non-iterative algorithms for simultaneous low rank approximations of matrices (SLRAM) have been presented by several researchers. In this paper, we show that those algorithms are equivalent to each other because they are reduced to the eigenvalue problems of row-row and column-column covariance matrices of given matrices. Also, we show a relationship between the non-iterative algorithms and another algorithm which is claimed to be an analytical algorithm for the SLRAM. Experimental results show that the analytical algorithm does not necessarily give the optimal solution of the SLRAM.
TL;DR: In this article, the application of the quadratic eigenvalue problem in electrical power systems is reviewed and the spectrum and pseudospectrum of an electrical power system are defined.
Abstract: The application of the quadratic eigenvalue problem in electrical power systems is reviewed. The spectrum and pseudospectrum of an electrical power system are defined.
TL;DR: Some bounds on the largest eigenvalue are presented, which generalize those given in the linear case, and are compared to the classical bounds in the particular case of the trees.
TL;DR: In this article, a Robin type boundary condition with a sign-changing coefficient is treated, where the existence of a principal eigenvalue is discussed by the use of a variational approach.
Abstract: A Robin type boundary condition with a sign-changing coefficient is treated. First, the associated linear elliptic eigenvalue problem is studied, where the existence of a principal eigenvalue is discussed by the use of a variational approach. Second, the associated semilinear elliptic boundary value problem of logistic type is studied and the one parameter-dependent structure of positive solutions is investigated, where results obtained are due to the construction of suitable super- and subsolutions by using the principal positive eigenfunctions of the linear eigenvalue problem.
06 Mar 2006
TL;DR: The paper illustrates that the proposed architecture is more efficient than previous architectures reported in the literature and shows that the pipelined architecture based on the algebraic method has a significant advantage in terms of area.
Abstract: Eigenvalue computation is essential in many fields of science and engineering. For high performance and real-time applications, this may need to be done in hardware. This paper focuses on the exploration of hardware architectures which compute eigenvalues of symmetric matrices. We propose to use the approximate Jacobi method for general case symmetric matrix eigenvalue problem. The paper illustrates that the proposed architecture is more efficient than previous architectures reported in the literature. Moreover, for the special case of 3times3 symmetric matrices, we propose to use an algebraic method. It is shown that the pipelined architecture based on the algebraic method has a significant advantage in terms of area
TL;DR: This contribution discusses how to handle the nonlinear eigen value problem and how to determine derivatives of the eigenvalue problem.
Abstract: Maxwell's equation for modeling the guided waves in a circularly symmetric fiber leads to a family of partial differential equation-eigenvalue systems. Incorporating the boundary condition into a discretized system leads to an eigenvalue problem which is nonlinear in only one element. In fiber design one would like to determine the index profile which is involved in Maxwell's equation so that certain optical properties, which sometimes involve derivatives of the eigenvalues, are satisfied. This contribution discusses how to handle the nonlinear eigenvalue problem and how to determine derivatives of the eigenvalue problem.
TL;DR: New algorithms of the Trefftz method are presented, which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions and gives a better performance in numerical testing, compared with the other popular methods of rootfinding.
Abstract: For Laplace's eigenvalue problems, this paper presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use an iteration process to yield approximate eigenvalues and eigenfunctions. The new iterative method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.
09 Jul 2006
TL;DR: The Schoof-Elkies-Atkin algorithm is the best known algorithm for counting the number of points of an elliptic curve defined over a finite field of large characteristic.
Abstract: The Schoof-Elkies-Atkin algorithm is the best known algorithm for counting the number of points of an elliptic curve defined over a finite field of large characteristic. Several practical and asymptotical improvements for the phase called eigenvalue computation are proposed.
Dissertation•
01 May 2006
TL;DR: In this article, a Rice University thesis/dissertation was published as a paper entitled "Rice University thesis and dissertation: A.handle.net/1911/17630".
Abstract: This work was also published as a Rice University thesis/dissertation: http://hdl.handle.net/1911/17630
TL;DR: Numerical results show that the presented methods are more efficient than solving a linearized generalized eigenvalue problem.
Abstract: We consider numerical methods for the computation of the eigenvalues of the tridiagonal hyperbolic quadratic eigenvalue problem. The eigenvalues are computed as zeros of the characteristic polynomial using the bisection, Laguerre’s method, and the Ehrlich-Aberth method. Initial approximations are provided by a divide-and-conquer approach using rank two modifications, and we show that these initial approximations interlace with the exact eigenvalues. The above methods need a stable and efficient evaluation of the quadratic eigenvalue problem’s characteristic polynomial and its derivatives. We discuss how to obtain these values using three-term recurrences, the QR factorization, and the LU factorization. Numerical results show that the presented methods are more efficient than solving a linearized generalized eigenvalue problem.
TL;DR: In this article, the authors considered the problem of preserving the basis property of a self-adjoint differential operator under a weak perturbation of the original selfadjoint operator with discrete spectrum.
Abstract: It is well known that the system of eigenfunctions of a formally self-adjoint differential operator with arbitrary self-adjoint boundary conditions providing a point spectrum is an orthonormal basis in the space L2. The problem as to whether the basis property is preserved under a weak, in a sense, perturbation of the original self-adjoint operator with discrete spectrum was considered in numerous papers (e.g., see [1, 2]). For a nonself-adjoint operator whose system of root functions is a Riesz basis in L2, a similar problem was investigated in [3]. In the present paper, on the interval [0, 1], we consider the problem