Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2009"
TL;DR: A new numerical algorithm for solving the symmetric eigenvalue problem is presented, which takes its inspiration from the contour integration and density matrix representation in quantum mechanics.
Abstract: A fast and stable numerical algorithm for solving the symmetric eigenvalue problem is presented. The technique deviates fundamentally from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms) or other Davidson-Jacobi techniques and takes its inspiration from the contour integration and density-matrix representation in quantum mechanics. It will be shown that this algorithm---named FEAST---exhibits high efficiency, robustness, accuracy, and scalability on parallel architectures. Examples from electronic structure calculations of carbon nanotubes are presented, and numerical performances and capabilities are discussed.
379 citations
TL;DR: A contour integral method is proposed to solve nonlinear eigenvalue problems numerically by reducing the original problem to a linear eigen value problem that has identical eigenvalues in the domain.
Abstract: A contour integral method is proposed to solve nonlinear eigenvalue problems numerically. The target equation is F (λ)x = 0, where the matrix F (λ) is an analytic matrix function of λ. The method can extract only the eigenvalues λ in a domain defined by the integral path, by reducing the original problem to a linear eigenvalue problem that has identical eigenvalues in the domain. Theoretical aspects of the method are discussed, and we illustrate how to apply of the method with some numerical examples.
187 citations
TL;DR: This paper derives an expression for the limiting eigenvalue ratio distribution, which turns out to be much more accurate than the previous approximations also in the non-asymptotical region, and is applied to calculate the decision sensing threshold as a function of a target probability of false alarm.
Abstract: Recent advances in random matrix theory have spurred the adoption of eigenvalue-based detection techniques for cooperative spectrum sensing in cognitive radio. These techniques use the ratio between the largest and the smallest eigenvalues of the received signal covariance matrix to infer the presence or absence of the primary signal. The results derived so far are based on asymptotical assumptions, due to the difficulties in characterizing the exact eigenvalues ratio distribution. By exploiting a recent result on the limiting distribution of the smallest eigenvalue in complex Wishart matrices, in this paper we derive an expression for the limiting eigenvalue ratio distribution, which turns out to be much more accurate than the previous approximations also in the non-asymptotical region. This result is then applied to calculate the decision sensing threshold as a function of a target probability of false alarm. Numerical simulations show that the proposed detection rule provides a substantial improvement compared to the other eigenvalue-based algorithms.
180 citations
TL;DR: The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon, and to demonstrate the use of this concept in the development of numerical methods, a novel block Newton method is developed.
Abstract: We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability.
116 citations
TL;DR: By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version and the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence.
Abstract: In the spirit of the Hamiltonian QR algorithm and other bidirectional chasing algorithms, a structure-preserving variant of the implicit QR algorithm for palindromic eigenvalue problems is proposed. This new palindromic QR algorithm is strongly backward stable and requires less operations than the standard QZ algorithm, but is restricted to matrix classes where a preliminary reduction to structured Hessenberg form can be performed. By an extension of the implicit Q theorem, the palindromic QR algorithm is shown to be equivalent to a previously developed explicit version. Also, the classical convergence theory for the QR algorithm can be extended to prove local quadratic convergence. We briefly demonstrate how even eigenvalue problems can be addressed by similar techniques.
90 citations
84 citations
TL;DR: Eigenvalue bounds for perturbations of Hermitian matrices are presented and the change in eigenvalues are expressed in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms.
Abstract: We present eigenvalue bounds for perturbations of Hermitian matrices and express the change in eigenvalues in terms of a projection of the perturbation onto a particular eigenspace, rather than in terms of the full perturbation. The perturbations we consider are Hermitian of rank one, and Hermitian or non-Hermitian with norm smaller than the spectral gap of a specific eigenvalue. Applications include principal component analysis under a spiked covariance model, and pseudo-arclength continuation methods for the solution of nonlinear systems.
76 citations
TL;DR: New optimization based algorithms for solving partial quadratic eigenvalue assignment problems with attractive features, coupled with minimal computational requirements, such as solutions of small diagonal Sylvester equations make the proposed algorithms ideally suited for application to large real-life structures.
71 citations
TL;DR: In this article, the authors present a method to compute the delays of delay-differential equations (DDEs) such that the DDE has a purely imaginary eigenvalue.
60 citations
TL;DR: In this article, a general analytic approach for nonlinear eigenvalue problems is described and two physical problems are used as examples to show the validity of this approach for eigen value problems with either periodic or non-periodic eigenfunctions.
Abstract: A general analytic approach for nonlinear eigenvalue problems is described. Two physical problems are used as examples to show the validity of this approach for eigenvalue problems with either periodic or non-periodic eigenfunctions. Unlike perturbation techniques, this approach is independent of any small physical parameters. Besides, different from all other analytic techniques, it provides a simple way to ensure the convergence of series of eigenvalues and eigenfunctions so that one can always get accurate enough approximations. Finally, unlike all other analytic techniques, this approach provides great freedom to choose an auxiliary linear operator so as to approximate the eigenfunction more effectively by means of better base functions. This approach provides us a new way to investigate eigenvalue problems with strong nonlinearity.
57 citations
TL;DR: In this article, computable a posteriori error bounds are derived based on employing the generalization of the standard dual-weighted-residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems.
Abstract: In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard dual-weighted-residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem and the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.
TL;DR: This work considers the Stokes eigenvalue problem and derives both upper and lower a‐posteriori error bounds for the eigenvalues.
Abstract: We consider the Stokes eigenvalue problem. For the eigenvalues we derive both upper and lower a-posteriori error bounds. The estimates are verified by numerical computations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009
TL;DR: It is shown that the simple finite regular eigenvalues of a two-parameter eigen value problem and the associated system of generalized eigenvalue problems agree.
Abstract: In the 1960s, Atkinson introduced an abstract algebraic setting for multiparameter eigenvalue problems. He showed that a nonsingular multiparameter eigenvalue problem is equivalent to the associated system of generalized eigenvalue problems. Many theoretical results and numerical methods for nonsingular multiparameter eigenvalue problems are based on this relation. In this paper, the above relation to singular two-parameter eigenvalue problems is extended, and it is shown that the simple finite regular eigenvalues of a two-parameter eigenvalue problem and the associated system of generalized eigenvalue problems agree. This enables one to solve a singular two-parameter eigenvalue problem by computing the common regular eigenvalues of the associated system of two singular generalized eigenvalue problems.
14 Jun 2009
TL;DR: This paper shows that under a mild condition, a class of generalized eigenvalue problems in machine learning can be formulated as a least squares problem, and reports experimental results that confirm the established equivalence relationship.
Abstract: Many machine learning algorithms can be formulated as a generalized eigenvalue problem. One major limitation of such formulation is that the generalized eigenvalue problem is computationally expensive to solve especially for large-scale problems. In this paper, we show that under a mild condition, a class of generalized eigenvalue problems in machine learning can be formulated as a least squares problem. This class of problems include classical techniques such as Canonical Correlation Analysis (CCA), Partial Least Squares (PLS), and Linear Discriminant Analysis (LDA), as well as Hypergraph Spectral Learning (HSL). As a result, various regularization techniques can be readily incorporated into the formulation to improve model sparsity and generalization ability. In addition, the least squares formulation leads to efficient and scalable implementations based on the iterative conjugate gradient type algorithms. We report experimental results that confirm the established equivalence relationship. Results also demonstrate the efficiency and effectiveness of the equivalent least squares formulations on large-scale problems.
TL;DR: The nonlinear elliptic eigenvalue problem is studied in this paper, where the key ingredient is a special constrained minimization method, which is the same as in this paper.
Abstract: The nonlinear elliptic eigenvalue problem
, where
and
are studied. The key ingredient is a special constrained minimization method.
TL;DR: The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach but here an alternative approach is proposed which is in some sense equivalent to the above.
Abstract: The solution of eigenvalue problems for partial differential operators by using boundary integral equation methods usually involves some Newton potentials which may be resolved by using a multiple reciprocity approach. Here we propose an alternative approach which is in some sense equivalent to the above. Instead of a linear eigenvalue problem for the partial differential operator we consider a nonlinear eigenvalue problem for an associated boundary integral operator. This nonlinear eigenvalue problem can be solved by using some appropriate iterative scheme, here we will consider a Newton scheme. We will discuss the convergence and the boundary element discretization of this algorithm, and give some numerical results.
TL;DR: In this article, two numerical techniques for solving cone-constrained eigenvalue problem are discussed, namely, the Power Iteration Method and the Scaling and Projection Algorithm.
Abstract: Given a convex cone K and matrices A and B, one wishes to find a scalar λ and a nonzero vector x satisfying the complementarity system K ∋ x ⊥(Ax-λ Bx) ∈ K+. This problem arises in mechanics and in other areas of applied mathematics. Two numerical techniques for solving such kind of cone-constrained eigenvalue problem are discussed, namely, the Power Iteration Method and the Scaling and Projection Algorithm.
TL;DR: Experimental results show that the proposed tracking algorithm gives a similar performance in tracking to the direct computation of singular value decomposition (SVD), while the computation order of the proposed algorithm is lower than the SVD.
Abstract: This letter develops a power method-based algorithm for tracking generalized eigenvectors when stochastic signals having unknown correlation matrices are observed. The proposed approach is based on the fact that the generalized eigenvalue problem is reduced to a standard eigenvalue problem, for which the power method can be easily applied by changing the metric of the vector space. The difficulty of applying the power method to this problem lies in the computational load of the inverse of the square root of a matrix at every update. The proposed algorithm can avoid the computation of eigenvalue decomposition to obtain the inverse of the matrix square root. Experimental results show that the proposed tracking algorithm gives a similar performance in tracking to the direct computation of singular value decomposition (SVD), while the computation order of the proposed algorithm is lower than the SVD.
01 Jan 2009
TL;DR: In this article, it was shown that the multiplicity of an eigenvalue of a periodic Jacobi matrix is at most 2, and that the complexity of the complex analogue of a Jacobi matrices can be constructed using the Lanzcos algorithm.
Abstract: In 1979, Ferguson characterized the periodic Jacobi matrices with given eigenvalues and showed how to use the Lanzcos Algorithm to construct each such matrix. This article provides general characterizations and constructions for the complex analogue of periodic Jacobi matrices. As a consequence of the main procedure, we prove that the multiplicity of an eigenvalue of a periodic Jacobi matrix is at most 2.
TL;DR: The smallest eigenvalue of the non-smooth operator under consideration is shown to be the same for all bounded, sufficiently smooth, domains in two space dimensions.
Abstract: Abstract In this article, we address the numerical solution of a non-smooth eigenvalue problem, which has implications in plasticity theory and image processing. The smallest eigenvalue of the non-smooth operator under consideration is shown to be the same for all bounded, sufficiently smooth, domains in two space dimensions. Piecewise linear finite elements are used for the discretization of eigenfunctions and eigenvalues. An augmented Lagrangian method is proposed for the computation of the minima of the associated non-convex optimization problem. The convergence of finite element approximations of generalized eigenpairs is investigated. Numerical solutions are presented for the first eigenvalue and eigenfunction. For non-simply connected domains, the augmented Lagrangian method also captures larger eigenvalues as local minima. Bifurcation between the first and second eigenvalues is investigated numerically.
TL;DR: In this article, the authors analyzed the stream function-vorticity-pressure method for the Stokes eigenvalue problem and obtained full order convergence rate of the eigen value approximations for two nonconforming finite elements.
Abstract: In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q
1
rot
and EQ
1
rot
. 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.
TL;DR: In this paper, the authors consider graphs with three distinct eigenvalues and characterize those with the largest eigenvalue less than 8, and give an upper bound on the number of vertices of graphs with a given number of distinct Eigenvalues in terms of the largest Eigenvalue.
07 Jun 2009
TL;DR: This work reduces the computational complexity of the Arnoldi iteration from O(k2N) to O(N), thus paving the way for full-wave extraction of very large-scale on-chip interconnects, the k of which is hundreds of thousands.
Abstract: In general, the optimal computational complexity of Arnoldi iteration is O(k2N) for solving a generalized eigenvalue problem, with k being the number of dominant eigenvalues and N the matrix size. In this work, we reduce the computational complexity of the Arnoldi iteration from O(k2N) to O(N), thus paving the way for full-wave extraction of very large-scale on-chip interconnects, the k of which is hundreds of thousands. Numerical and experimental results have demonstrated the accuracy and efficiency of the proposed fast eigenvalue solver.
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TL;DR: In this article, the eigenvalue problem Lu = u for an elliptic partial differential operator L over with zero values for either Dirichlet or Neumann boundary conditions is considered.
Abstract: Let be an open, simply connected, and bounded region in R d , d � 2, and assume its boundary @ is smooth. Consider solving the eigenvalue problem Lu = �u for an elliptic partial differential operator L over with zero values for either Dirichlet or Neumann boundary conditions. We propose, analyze, and illustrate a ‘spectral method’ for solving numerically such an eigenvalue problem. This is an extension of the methods presented earlier in [5], [6].
TL;DR: In this article, the first eigenvalue and corresponding eigenfunctions were studied for a fully nonlinear equation involving Hessian operators, and a Faber-Krahn inequality and a Payne-Rayner type inequality were proved for the p-laplacian operator and the Monge-Ampere operator.
Abstract: In this paper we consider the eigenvalue problem for a fully nonlinear equation involving Hessian operators. In particular we study some properties of the first eigenvalue and of corresponding eigenfunctions. Using suitable symmetrization arguments, we prove a Faber–Krahn inequality for the first eigenvalue and a Payne–Rayner type inequality for eigenfunctions, which are well known for the p-laplacian operator and the Monge–Ampere operator.
06 Dec 2009
TL;DR: This work presents a new method based on bootstrapping to reduce the bias in the sample eigenvalues: the eigenvalue estimates are updated in several iterations, where in each iteration synthetic data is generated to determine how to update the population eigen value estimates.
Abstract: Eigenvalue analysis is an important aspect in many data modeling methods. Unfortunately, the eigenvalues of the sample covariance matrix (sample eigenvalues) are biased estimates of the eigenvalues of the covariance matrix of the data generating process (population eigenvalues). We present a new method based on bootstrapping to reduce the bias in the sample eigenvalues: the eigenvalue estimates are updated in several iterations, where in each iteration synthetic data is generated to determine how to update the population eigenvalue estimates. Comparison of the bootstrap eigenvalue correction with a state of the art correction method by Karoui shows that depending on the type of population eigenvalue distribution, sometimes the Karoui method performs better and sometimes our bootstrap method.
TL;DR: A new method for accelerating the convergence of the implicitly restarted Arnoldi (IRA) algorithm for the solution of large sparse nonsymmetric eigenvalue problems is proposed and a technique for dynamically switching the Krylov subspace dimension at successive cycles is developed.
TL;DR: The approximate Jacobi method is proposed, where for the special case of a 3×3 symmetric matrix, an algebraic-based method is introduced and the proposed methods are compared with various other approaches reported in the literature.
Abstract: Computation of eigenvalues is essential in many applications in the fields of science and engineering. When the application of interest requires the computation of eigenvalues of high throughput or real-time performance, a hardware implementation of an eigenvalue computation block is often employed. The problem of eigenvalue computation of real symmetric matrices is focused upon. For the general case of a symmetric matrix eigenvalue problem, the approximate Jacobi method is proposed, where for the special case of a 3×3 symmetric matrix, an algebraic-based method is introduced. The proposed methods are compared with various other approaches reported in the literature. Results obtained by mapping the above architectures on a field programmble gate array device illustrate the advantages of the proposed methods over the existing ones.