Showing papers on "Divide-and-conquer eigenvalue algorithm published in 2000"
TL;DR: In this article, the authors developed normwise backward errors and condition numbers for the polynomial eigenvalue problem, and showed that solving the QEP by applying the QZ algorithm to a corresponding generalized eigen value problem can be backward unstable.
227 citations
TL;DR: In this paper, the eigenvalues of the solutions to a class of continuous and discrete-time Lyapunov equations with symmetric coefficient matrices and right-hand side matrices of low rank were studied.
185 citations
TL;DR: In this paper, the authors studied the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternion vector spaces, and they gave a necessary and sufficient condition for the diagonalization of their representations.
Abstract: We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n -dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.
118 citations
TL;DR: A nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n×n symmetric matrices is presented and some strict complementarity and non-degeneracy assumptions are needed.
Abstract: In this paper we present a nonsmooth algorithm to minimize the maximum eigenvalue of matrices belonging to an affine subspace of n×n symmetric matrices. We show how a simple bundle method, the approximate eigenvalue method can be used to globalize the second-order method developed by M.L. Overton in the eighties and recently revisited in the framework of the ?-Lagrangian theory. With no additional assumption, the resulting algorithm generates a minimizing sequence. A geometrical and constructive proof is given. To prove that quadratic convergence is achieved asymptotically, some strict complementarity and non-degeneracy assumptions are needed. We also introduce new variants of bundle methods for semidefinite programming.
73 citations
TL;DR: In this article, it was shown that the upper and lower bounds of the first eigenvalue can be further bounded by one or two constants depending on the coefficients of the corresponding operators only.
Abstract: It is proved that the general formulas, obtained recently for the lower bound of the first eigenvalue, can be further bounded by one or two constants depending on the coefficients of the corresponding operators only. Moreover, the ratio of the upper and lower bounds is no more than four.
59 citations
TL;DR: In this article, the semi-classical limit of the Zakharov-Shabat eigenvalue problem for the focusing of NLS with some specific initial data is studied.
48 citations
TL;DR: An eigenvalue-based characterization of positive realness of transfer functions of a single-input single output time-invariant linear system is derived and an efficient computational procedure to determine if a given transfer function is positive real is proposed.
Abstract: An eigenvalue-based characterization of positive realness of transfer functions of a single-input single output time-invariant linear system is derived. Based on this characterization, we propose an efficient computational procedure to determine if a given transfer function is positive real. The input for this eigenvalue-based test is any given, not necessarily minimal, state-space representation of the linear system. The test only involves standard matrix computations, such as computing eigenvalues of a matrix or a matrix pencil. Results of numerical experiments are reported.
47 citations
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TL;DR: In this article, the eigenvalue problem on nite intervals of length n with periodic boundary conditions is studied and the limit eigen value distribution is supported by curves in the complex plane.
Abstract: Random Schrodinger operators with imaginary vector potentials are studied in di- mension one. These operators are non-Hermitian and their spectra lie in the complex plane. We consider the eigenvalue problem on nite intervals of length n with periodic boundary conditions and describe the limit eigenvalue distribution when n !1 . We prove that this limit distribu- tion is supported by curves in the complex plane. We also obtain equations for these curves and for the corresponding eigenvalue density in terms of the Lyapunov exponent and the integrated density of states of a \reference" symmetric eigenvalue problem. In contrast to these results, the spectrum of the limit operator in l 2 (Z) is a two dimensional set which is not approximated by the spectra of the nite-interval operators.
39 citations
TL;DR: The generalized integral transform technique (GITT) is employed in obtaining formal solutions for eigenvalue problems of the Sturm-Liouville type, described by multidimensional partial differential models within irregularly shaped domains.
Abstract: The generalized integral transform technique (GITT) is employed in obtaining formal solutions for eigenvalue problems of the Sturm-Liouville type, described by multidimensional partial differential models within irregularly shaped domains. The successive elimination of independent variables in the appropriate order through integral transformations produces the associated algebraic eigenvalue problem, which is then readily solved by algorithms from scientific subroutine libraries. A representative example of the known exact solution is presented, of particular interest to the field of heat and mass diffusion, for validation purposes.
37 citations
TL;DR: The family of GR algorithms is discussed, with emphasis on the QR algorithm, with historical remarks, an outline of what GR algorithms are and why they work, and descriptions of the latest, highly parallelizable, versions of the QR algorithms.
35 citations
TL;DR: Several domain decomposition methods to compute the smallest eigenvalue of linear self-adjoint partial differential operators, including a scheme which determines an appropriate boundary condition at the interface separating the two regions.
TL;DR: In this paper, an explicit-implicit scheme for the integration with respect to time of the equations of the finite-element method of non-stationary acoustoelectric elasticity is proposed and discussed.
TL;DR: In this paper, the authors consider the nonlinear right focal eigenvalue problem and determine the value of the parameter for which this problem has a positive solution, by extending recent works by allowing for a broader class of functions for a(t).
Abstract: We consider the nonlinear right focal eigenvalue problem value of the parameterλ are determined for which this problem has a positive solution. The methods used here extend recent works by allowing for a broader class of functions for a(t). Optimal eigenvalue interclas are given for some relevant examples.
TL;DR: In this paper, the authors extend previous work on the eigenvalue problem for Hermitian octonionic matrices by discussing the case where the eigvalues are not real, giving a complete treatment of the 2 × 2 case and summarizing some preliminary results for the 3 × 3 case.
Abstract: We extend previous work on the eigenvalue problem for Hermitian octonionic matrices by discussing the case where the eigenvalues are not real, giving a complete treatment of the 2 × 2 case, and summarizing some preliminary results for the 3 × 3 case.
TL;DR: Modifications to the LAPACK subroutine for reducing a symmetric banded matrix to tridiagonal form improve the performance for larger-bandwidth problems and reduce the number of operations when accumulating the transformations onto the identity matrix, by taking advantage of the structure of the initial matrix.
Abstract: In this paper we explain some of the changes that have been incorporated in the latest version of the LAPACK subroutine for reducing a symmetric banded matrix to tridiagonal form. These modifications improve the performance for larger-bandwidth problems and reduce the number of operations when accumulating the transformations onto the identity matrix, by taking advantage of the structure of the initial matrix. We show that similar modifications can be made to the LAPACK subroutines for reducing a symmetric positive definite generalized eigenvalue problem to a standard symmetric banded eigenvalue problem and for reducing a general banded matrix to bidiagonal form to facilitate the computation of the singular values of the matrix.
TL;DR: The proposed algorithm is computationally efficient because it not only retains the speed inherent in the Remez exchange algorithm but also simplifies the interpolation step.
Abstract: This paper presents an efficient method for designing complex infinite impulse response digital filters in the complex Chebyshev sense. The proposed method is based on the formulation of a generalized eigenvalue problem by using the Remez multiple exchange algorithm. Hence, the filter coefficients can be easily obtained by solving the eigenvalue problem to find the absolute minimum eigenvalue, and then the complex Chebyshev approximation is attained through a few iterations starting from a given initial guess. The proposed algorithm is computationally efficient because it not only retains the speed inherent in the Remez exchange algorithm but also simplifies the interpolation step. Some design examples are presented and compared with the conventional methods. It is shown that the results obtained by using the method proposed in this paper are better than those obtained by the conventional methods.
TL;DR: In this article, the sampling method was applied to compute the eigenvalues of a regular Sturm Liouville problem when a boundary condition contains the Eigenvalue parameter, and truncation errors were worked out and help to obtain eigenvalue enclosures.
Abstract: . We shall apply the sampling method to compute the eigenvalues of a regular Sturm Liouville problem when a boundary condition contains the eigenvalue parameter. Truncation errors will be worked out and help us obtain eigenvalue enclosures. Examples are provided to illustrate the theory.
TL;DR: The recursive inverse eigen value problem for matrices is studied, where for each leading principal submatrix an eigenvalue and associated left and right eigenvectors are assigned.
Abstract: The recursive inverse eigenvalue problem for matrices is studied, where for each leading principal submatrix an eigenvalue and associated left and right eigenvectors are assigned. Existence and uniqueness results as well as explicit formulas are proven, and applications to nonnegative matrices, Z-matrices, M-matrices, symmetric matrices, Stieltjes matrices, and inverse M-matrices are considered.
TL;DR: In this article, the eigenvalue problem on finite intervals of length $n$ with periodic boundary conditions was studied and the limit eigen value distribution when $n $ goes to infinity.
Abstract: Random Schrodinger operators with imaginary vector potentials are studied in dimension one. These operators are non-Hermitian and their spectra lie in the complex plane. We consider the eigenvalue problem on finite intervals of length $n$ with periodic boundary conditions and describe the limit eigenvalue distribution when $n$ goes to infinity. We prove that this limit distribution is supported by curves in the complex plane. We also obtain equations for these curves and for the corresponding eigenvalue density in terms of the Lyapunov exponent and the integrated density of states of a "reference" symmetric eigenvalue problem. In contrast to these results, the spectrum of the limit operator in $\ell^2(Z)$ is a two dimensional set which is not approximated by the spectra of the finite-interval operators.
TL;DR: In this article, the controllability properties of matrix eigenvalue algorithms arising in numerical linear algebra were studied and a complete characterization of reachable sets and their closures was given via cyclic invariant subspaces.
01 Jan 2000
TL;DR: Test cases illustrating this problem will be presented, and reparations to the standard software proposed, are proposed.
Abstract: Elementary plane rotations are one of the building blocks of numerical linear algebra and are employed in reducing matrices to condensed form for eigenvalue computations and during the QR algorithm. Unfortunately, their implementation in standard packages such as EISPACK, the BLAS and LAPACK lack the continuity of their mathematical formulation, which makes results from software that use them sensitive to perturbations. Test cases illustrating this problem will be presented, and reparations to the standard software proposed.
TL;DR: This paper shows an effective way of resolving eigenvalue problems of full three-dimensional elasticity for thin elastic structures by invoking a special preconditioning technique associated with the effective dimensional reduction algorithm (EDRA).
Abstract: We present a methodology for the efficient numerical solution of eigenvalue problems of full three-dimensional elasticity for thin elastic structures, such as shells, plates and rods of arbitrary geometry, discretized by the finite element method. Such problems are solved by iterative methods, which, however, are known to suffer from slow convergence or even convergence failure, when the thickness is small. In this paper we show an effective way of resolving this difficulty by invoking a special preconditioning technique associated with the effective dimensional reduction algorithm (EDRA). As an example, we present an algorithm for computing the minimal eigenvalue of a thin elastic plate and we show both theoretically and numerically that it is robust with respect to both the thickness and discretization parameters, i.e. the convergence does not deteriorate with diminishing thickness or mesh refinement. This robustness is sine qua non for the efficient computation of large-scale eigenvalue problems for thin elastic structures.
TL;DR: An explicit a priori bound for the condition number associated to each of the following problems is given: general linear equation solving, least squares, nonsymmetric eigenvalue problems, solving univariate polynomials, and solving systems of multivariate poynomials.
TL;DR: In this article, the second order conjugate eigenvalue problem on a time scale y ΔΔ(t) was considered, where t is the time complexity of the problem.
Abstract: We consider the nonlinear second order conjugate eigenvalue problem on a time scale: y ΔΔ(t)
TL;DR: Several lower bounds of the minimal eigenvalue of a class of Hermitian positive-definite matrices are presented, which improve the previous bounds given by Dembo (1988) and Ma and Zarowski (1995).
Abstract: In this correspondence, we present several lower bounds of the minimal eigenvalue of a class of Hermitian positive-definite matrices, which improve the previous bounds given by Dembo (1988) and Ma and Zarowski (1995).
Journal Article•
TL;DR: In this article, the existence of a principal eigenvalue of minimum modulus with an associated positive eigenfunction was established for the Tricomi problem in normal domains, where the authors used prior results of the authors on generalized solvability in weighted Sobolev spaces and associated maximum/minimum principles coupled with known results of Krein-Rutman type.
Abstract: The existence of a principal eigenvalue is established for the Tricomi problem in normal domains; that is, the existence of a positive eigenvalue of minimum modulus with an associated positive eigenfunction. The argument here uses prior results of the authors on the generalized solvability in weighted Sobolev spaces and associated maximum/minimum principles [17] coupled with known results of Krein-Rutman type.
TL;DR: In this paper, the authors studied the criterion for a new eigenvalue to appear in the linear spectral problem associated with the intermediate long-wave equation and computed the asymptotic value of the new value in the limit of a small potential using a Fourier decomposition method.
Abstract: We study the criterion for a new eigenvalue to appear in the linear spectral problem associated with the intermediate long-wave equation. We compute the asymptotic value of the new eigenvalue in the limit of a small potential using a Fourier decomposition method. We compare the results with those for the Schrodinger operator with a radially symmetrical potential.