Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2005"
Book•
02 Sep 2005
TL;DR: Inverse Eigenvalue problems have been studied in this paper, where the objective is to find the least square inverse eigenvalue problem with the minimum number of vertices in the least squares.
Abstract: Preface 1. Introduction 2. Applications 3. Parameterized Inverse Eigenvalue Problems 4. Structured Inverse Eigenvalue Problems 5. Partially Described Inverse Eigenvalue Problems 6. Least Squares Inverse Eigenvalue Problems 7. Spectrally Constrained Approximation 8. Structured Low Rank Approximation 9. Group Orbitally Constrained Approximation Index
285 citations
TL;DR: In this paper, it was shown that the principal eigenvalue of linear elliptic equations with high first-order coefficients is bounded as the amplitude of the coefficients of the first order derivatives goes to infinity if and only if the associated dynamical system has a first integral.
Abstract: This paper is concerned with the asymptotic behaviour of the principal eigenvalue of some linear elliptic equations in the limit of high first-order coefficients. Roughly speaking, one of the main results says that the principal eigenvalue, with Dirichlet boundary conditions, is bounded as the amplitude of the coefficients of the first-order derivatives goes to infinity if and only if the associated dynamical system has a first integral, and the limiting eigenvalue is then determined through the minimization of the Dirichlet functional over all first integrals. A parabolic version of these results, as well as other results for more general equations, are given. Some of the main consequences concern the influence of high advection or drift on the speed of propagation of pulsating travelling fronts.
152 citations
Book•
01 Jan 2005
TL;DR: The QR Algorithm, the QZ Al algorithm, and the Krylov-Schur Algorithm are algorithms used for solving Structured Eigenvalue Problems.
Abstract: The QR Algorithm.- The QZ Algorithm.- The Krylov-Schur Algorithm.- Structured Eigenvalue Problems.- Background in Control Theory Structured Eigenvalue Problems.- Software.
141 citations
TL;DR: This iteration, applied to generalized companion matrices, provides new O(n2) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations.
Abstract: We introduce a class ** of n×n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of **, we show that if A ∈ ** then A′=RQ ∈ **, where A=QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A ∈ ** with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n2) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.
109 citations
TL;DR: In this paper, the structural vibration problem involving uncertain material or geometric parameters, specified as bounds on these parameters, is transformed into a generalized interval eigenvalue problem in interval mathematics, and tighter bounds on the eigenvalues may be obtained by using the formulation of the structural dynamic problem.
78 citations
01 Jan 2005
TL;DR: This work will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures.
Abstract: Skew-Hamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computational methods. We will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures. Applications of Hamiltonian and skew-Hamiltonian eigenproblems are briefly described.
60 citations
TL;DR: In this article, the photonic band structure of periodic materials such as photonic crystals is calculated using the solution of a Hermitian nonlinear eigenvalue problem, where bordered matrices are used to compute critical points in singular systems.
57 citations
TL;DR: In this article, the problem of infinite eigenvalue assignment by output-feedbacks is considered and necessary and sufficient conditions for the existence of a solution to the problem are established, and a procedure for computation of the outputfeedback gain matrix is given and illustrated by a numerical example.
54 citations
TL;DR: A new numerical method for computing selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem does not require good initial approximations and is able to tackle large problems that are too expensive for methods that compute all eigen values.
Abstract: We present a new numerical method for computing selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. The method does not require good initial approximations and is able to tackle large problems that are too expensive for methods that compute all eigenvalues. The new method uses a two-sided approach and is a generalization of the Jacobi--Davidson type method for right definite two-parameter eigenvalue problems [M. E. Hochstenbach and B. Plestenjak, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 392--410]. Here we consider the much wider class of nonsingular problems. In each step we first compute Petrov triples of a small projected two-parameter eigenvalue problem and then expand the left and right search spaces using approximate solutions to appropriate correction equations. Using a selection technique, it is possible to compute more than one eigenpair. Some numerical examples are presented.
50 citations
TL;DR: The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept and that the standard algorithms for solving them are instances of a generic $GR$ algorithm applied to a related cyclic matrix.
Abstract: Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix $A$ are wanted, but $A$ is not given explicitly. Instead it is presented as a product of several factors: $A = A_{k}A_{k-1}\cdots A_{1}$. Usually more accurate results are obtained by working with the factors rather than forming $A$ explicitly. For example, if we want eigenvalues/vectors of $B^{T}B$, it is better to work directly with $B$ and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic $GR$ algorithm applied to a related cyclic matrix.
48 citations
TL;DR: In this paper, a stochastic meshless method for probabilistic analysis of linear-elastic structures with spatially varying random material properties was presented, where the homogeneous random field representing material properties were discretized by a set of orthonormal eigenfunctions and uncorrelated random variables.
Abstract: This paper presents a stochastic meshless method for probabilistic analysis of linear-elastic structures with spatially varying random material properties. Using Karhunen-Loeve (K-L) expansion, the homogeneous random field representing material properties was discretized by a set of orthonormal eigenfunctions and uncorrelated random variables. Two numerical methods were developed for solving the integral eigenvalue problem associated with K-L expansion. In the first method, the eigenfunctions were approximated as linear sums of wavelets and the integral eigenvalue problem was converted to a finite-dimensional matrix eigenvalue problem that can be easily solved. In the second method, a Galerkin-based approach in conjunction with meshless discretization was developed in which the integral eigenvalue problem was also converted to a matrix eigenvalue problem. The second method is more general than the first, and can solve problems involving a multi-dimensional random field with arbitrary covariance functions....
TL;DR: It is believed that elongation induced crystallization occurring during the adiabatic expansion process has resulted in an increase in crystallization rate, eventually leading to a faster growth rate of spherulites and a increase in the nucleation density.
Abstract: Several Jacobi–Davidson type methods are proposed for computing interior eigenpairs of large-scale cubic eigenvalue problems. To successively compute the eigenpairs, a novel explicit non-equivalence deflation method with low-rank updates is developed and analysed. Various techniques such as locking, search direction transformation, restarting, and preconditioning are incorporated into the methods to improve stability and efficiency. A semiconductor quantum dot model is given as an example to illustrate the cubic nature of the eigenvalue system resulting from the finite difference approximation. Numerical results of this model are given to demonstrate the convergence and effectiveness of the methods. Comparison results are also provided to indicate advantages and disadvantages among the various methods. Copyright © 2004 John Wiley & Sons, Ltd.
TL;DR: Eigensolvers for computing a few of the smallest eigenvalues of a generalized eigenvalue problem resulting from the finite element discretization of the time independent Maxwell equation are investigated.
TL;DR: This method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigen value approximation of nonsymmetric problems).
Abstract: A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.
TL;DR: It is proved that the unitary factor appearing in the QR factorization of a suitably dened rational Krylov matrix transforms a Hermitian matrix having pairwise distinct eigenvalues into a diagonal-plus- semiseparable form with prescribed diagonal term.
Abstract: SUMMARY We prove that the unitary factor appearing in the QR factorization of a suitably dened rational Krylov matrix transforms a Hermitian matrix having pairwise distinct eigenvalues into a diagonal-plus- semiseparable form with prescribed diagonal term. This transformation is essentially uniquely dened by itsrst column. Furthermore, we prove that the set of Hermitian diagonal-plus-semiseparable matrices is invariant under QR iteration. These and other results are shown to be the rational counterpart of known facts involving structured matrices related to polynomial computations. Copyright ? 2005 John Wiley & Sons, Ltd.
TL;DR: An approach to solving the quadratic eigenvalue problem directly without linearizing it is introduced and perturbation subspaces for block eigenvector matrices are used to reduce the modified problem to a sequence of problems of smaller dimension.
Abstract: Quadratic eigenvalue problems involving large matrices arise frequently in areas such as the vibration analysis of structures, micro-electro-mechanical systems (MEMS) simulation, and the solution of quadratically constrained least squares problems. The typical approach is to solve the quadratic eigenvalue problem using a mathematically equivalent linearized formulation, resulting in a doubled dimension and, in many cases, a lack of backward stability.
This paper introduces an approach to solving the quadratic eigenvalue problem directly without linearizing it. Perturbation subspaces for block eigenvector matrices are used to reduce the modified problem to a sequence of problems of smaller dimension. These perturbation subspaces are shown to be contained in certain generalized Krylov subspaces of the n-dimensional space, where n is the undoubled dimension of the matrices in the quadratic problem. The method converges at least as fast as the corresponding Taylor series, and the convergence can be accelerated further by applying a block generalization of the quadratically convergent Rayleigh quotient iteration. Numerical examples are presented to illustrate the applicability of the method.
TL;DR: This paper proposes a concise functional neural network expressed as a differential equation and designs steps to do this work, which can compute the smallest eigenvalue and the largest eigen value whether the matrix is non-definite, positive definite or negative definite.
08 Jul 2005
TL;DR: In this article, the Ackermann-like formulae for both SISO and MIMO linear time-varying systems are proposed for eigenvalue assignment.
Abstract: Eigenvalue assignment techniques for linear time-varying systems are presented as a way of achieving feedback stabilisation. For this, novel eigenvalue concepts, which are the time-varying counterparts of conventional (time-invariant) eigenvalue ideas, are introduced. Ackermann-like formulae for both SISO and MIMO linear time-varying systems are proposed. It is believed that these techniques are the generalised versions of the Ackermann formulae for linear time-invariant systems. The advantages of the proposed Ackermann-like formulae are that they neither require the transformation of the original system into a phase-variable form nor the computation of the eigenvalues of the original system, and they can allow the closed-loop system to have desired eigenvalue trajectories rather than fixed eigenvalue locations. Two examples are given to demonstrate the capabilities of the proposed techniques.
Dissertation•
01 Aug 2005
TL;DR: An implementation of the HZ algorithm that allows stability in most cases and comparable results with other classical algorithms for ill conditioned problems is proposed and an explicit expression for the backward error of an approximate eigenpair of a matrix polynomial written in homogeneous form is derived.
Abstract: In this thesis, we consider polynomial eigenvalue problems. We extend results on
eigenvalue and eigenvector condition numbers of matrix polynomials to condition
numbers with perturbations measured with a weighted Frobenius norm. We derive
an explicit expression for the backward error of an approximate eigenpair of a
matrix polynomial written in homogeneous form. We consider structured eigenvalue
condition numbers for which perturbations have a certain structure such
as symmetry, Hermitian or sparsity. We also obtain explicit and/or computable
expressions for the structured backward error of an eigenpair.
We present a robust implementation of the HZ (or HR) algorithm for symmetric
generalized eigenvalue problems. This algorithm has the advantage of preserving
pseudosymmetric tridiagonal forms. It has been criticized for its numerical
instability. We propose an implementation of the HZ algorithm that allows stability
in most cases and comparable results with other classical algorithms for ill
conditioned problems. The HZ algorithm is based on the HR factorization, an
extension of the QR factorization in which the H factor is hyperbolic. This yields
us to the sensitivity analysis of hyperbolic factorizations.
TL;DR: In this paper, the authors consider the evolution of a system of finite faults by considering the non-linear eigenvalue problems associated to static and dynamic solutions on unbounded domains.
Abstract: We analyse the evolution of a system of finite faults by considering the non-linear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the first eigenvalue (Rayleigh quotient). We point out its physical significance through a stability analysis and we give an efficient numerical algorithm able to compute it together with the corresponding eigenfunction.
We consider the anti-plane shearing on a system of finite faults under a slip-dependent friction in a linear elastic domain, not necessarily bounded. The static problem is formulated in terms of local minima of the energy functional. We introduce the non-linear (static) eigenvalue problem and we prove the existence of a first eigenvalue/eigenfunction characterizing the isolated local minima. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) non-linear eigenvalue problem. We prove the existence of a first dynamic eigenvalue and we analyse its behaviour with respect to the friction parameter. We deduce a mixed finite element discretization of the non-linear spectral problem and we give a numerical algorithm to approach the first eigenvalue/eigenfunction. Finally we give some numerical results which include convergence tests, on a single fault and a two-faults system, and a comparison between the non-linear spectral results and the time evolution results. Copyright © 2004 John Wiley & Sons, Ltd.
TL;DR: A novel method for solving the unitary Hessenberg eigenvalue problem by exploiting the structure of the problem to yield a quadratic time using a linear memory space and relying on the earlier adaptation of the QR algorithm to solve the dpss eigen value problem in a fast and robust way.
01 Jan 2005
TL;DR: In this article, the authors provide an overview of the recent development of eigenvalue computation in the context of two SciDAC applications and discuss the value of alternative approaches that are more amenable to the use of preconditioners, and report the progress on using the multi-level algebraic substructuring techniques to speed up eigen value calculation.
Abstract: Large-scale eigenvalue problems arise in a number of DOE applications. This paper provides an overview of the recent development of eigenvalue computation in the context of two SciDAC applications. We emphasize the importance of Krylov subspace methods, and point out its limitations. We discuss the value of alternative approaches that are more amenable to the use of preconditioners, and report the progress on using the multi-level algebraic sub-structuring techniques to speed up eigenvalue calculation. In addition to methods for linear eigenvalue problems, we also examine new approaches to solving two types of non-linear eigenvalue problems arising from SciDAC applications.
TL;DR: In this paper, a non-matrix-analytic procedure to compute the performance measures of the discrete-time BMAP/G/1 queueing system when the order of parameter matrices is very low is presented.
TL;DR: In this article, an eigenvalue problem in RN with respect to the Laplacian was studied and the existence of certain open intervals of eigenvalues was guaranteed for which the eigen value problem has two nonzero, radially symmetric solutions.
Abstract: We study an eigenvalue problem in RN which involves the p -Laplacian (p > N ≥ 2) and the nonlinear term has a global (p – 1)-sublinear growth. The existence of certain open intervals of eigenvalues is guaranteed for which the eigenvalue problem has two nonzero, radially symmetric solutions. Some stability properties of solutions with respect to the eigenvalues are also obtained. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
01 Jan 2005
TL;DR: In this paper, the spectrum for a Neumann eigenvalue problem involving the p-Laplacian operator with weight in a bounded domain is studied, and the authors show that the spectrum of the spectrum is bounded.
Abstract: This paper is devoted to study the spectrum for a Neumann eigen- value problem involving the p-Laplacian operator with weight in a bounded domain.
Posted Content•
TL;DR: In this article, the adjoint sector of the c=1 matrix model was considered and the exact wave function and a scattering phase were obtained, which matched the string theory calculation of the non-local eigenvalue problem.
Abstract: We solve the non-local eigenvalue problem that arose from consideration of the adjoint sector of the c=1 matrix model in hep-th/0503112. We obtain the exact wavefunction and a scattering phase that matches the string theory calculation.
TL;DR: In this paper, sufficient conditions for existence of error bounds for systems expressed in terms of eigenvalue functions or positive semidefiniteness are given.
Abstract: In this paper we give sufficient conditions for existence of error bounds for systems expressed in terms of eigenvalue functions (such as in eigenvalue optimization) or positive semidefiniteness (such as in semidefinite programming).
TL;DR: In this paper, it was shown that the spectrum of a given quadratic eigenvalue problem possesses a Hamiltonian structure, which can be exploited for an efficient computation of the eigenvalues.
Abstract: SFB393/04-09 October 2004 Abstract When the eigenvalues of a given eigenvalue problems are symmetric with respect to the real and the imaginary axes, we speak about a Hamiltonian eigenvalue symmetry or a Hamiltonian structure of the spectrum. This property can be exploited for an efficient computation of the eigenvalues. For some elliptic boundary value problems it is known that the derived eigenvalue problems have this Hamiltonian symmetry. Without having a specific application in mind, we trace the question, under which assumptions the spectrum of a given quadratic eigenvalue problem possesses the Hamiltonian structure.
TL;DR: Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric ToePLitz matrices.
Abstract: Some inverse eigenvalue problems for matrices with Toeplitz-related structure are considered in this paper In particular, the solutions of the inverse eigenvalue problems for Toeplitz-plus-Hankel matrices and for Toeplitz matrices having all double eigenvalues are characterized, respectively, in close form Being centrosymmetric itself, the Toeplitz-plus-Hankel solution can be used as an initial value in a continuation method to solve the more difficult inverse eigenvalue problem for symmetric Toeplitz matrices Numerical testing results show a clear advantage of such an application
TL;DR: Based on the Davidson method for solving generalized eigenvalue problems, a new method for synchro calculation of eigenpairs and their partial derivatives of generalized Eigen value problems is presented in this paper, where the equation systems that are solved for eigenvector partial derivatives can be greatly reduced from the original matrix sizes.
Abstract: Based on Davidson method for solving generalized eigenvalue problems, a new method for synchro calculation of eigenpairs and their partial derivatives of generalized eigenvalue problems is presented. Eigenpairs and their partial derivatives are computed simultaneously. The equation systems that are solved for eigenvector partial derivatives can be greatly reduced from the original matrix sizes, thus the efficiency of computing eigenvector partial derivatives is improved. Numerical results show that the proposed method is efficient, especially for the large-scale symmetric generalized eigenvalue problems. Copyright © 2005 John Wiley & Sons, Ltd.