Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1984"
TL;DR: A new class of algorithms which is based on rational functions of the matrix is described, and there are also new algorithms which correspond to rational functions with several poles.
343 citations
Book•
01 Jan 1984
140 citations
TL;DR: In this article, a generalized Toeplitz matrix whose eigenvalue distribution can be found using a theorem of Kac, Murdock, and Szego is presented.
131 citations
TL;DR: In this paper, the entire dispersive spectra of a cylinder with cylindrical anisotropy is determined from three different algebraic eigenvalue problems deducible from the same finite element formulation.
95 citations
01 Dec 1984
TL;DR: Numerical algorithms are described for solving the eigenvalue assignment (EVA) problem for multi-input systems by means of state feedback by way of an algorithm based on the well-known implicitly shifted QR algorithm.
Abstract: Numerical algorithms are described for solving the eigenvalue assignment (EVA) problem for multi-input systems by means of state feedback. The multi-input EVA problem is first reduced to one or more single-input EVA problems where the single-input systems are in "upper Hessenberg form." An algorithm based on the well-known implicitly shifted QR algorithm is then described for solving the single-input EVA problem. Some variations on the use of these algorithms are also proposed to handle certain special cases. Numerical properties of the algorithms are discussed and examples are given to illustrate their numerical performance.
95 citations
TL;DR: In this article, a method for solving a form of the inverse Sturm-Liouville problem is presented, which is based on modifying the given differential eigenvalues so that the tridiagonal matrix eigenvalue problem recovered from the first N eigen values can be identified with the finite difference approximation of the required differential Eigen value problem.
Abstract: In this paper we present a method for solving a form of the inverse Sturm–Liouville problem. The basis of the method is to modify the given differential eigenvalues so that the $N \times N$ tridiagonal matrix eigenvalue problem recovered from the first N eigenvalues can (after a suitable transformation) be identified with the finite difference approximation of the required differential eigenvalue problem. Numerical results are presented to illustrate the effectiveness of this method.
60 citations
TL;DR: It is shown that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs and these curves are solutions of a certain ordinary differential equation with different initial values.
47 citations
TL;DR: In this article, a new method is presented for the solution of the matrix eigenvalue problem Ax=λBx, where A and B are real symmetric square matrices and B is positive semidefinite.
46 citations
01 Jan 1984
44 citations
TL;DR: In this article, the eigenvalue problem for a particular class of "arrow" matrices Z=(γB†ACB), where A is a Hermitian N×N matrix, C a real diagonal M×M matrix, B an arbitrary complex N×M mat, and γ a real number, is investigated by means of a partitioning technique.
Abstract: The eigenvalue problem for a particular class of ‘‘arrow’’ matrices Z=(γB†ACB), where A is a Hermitian N×N matrix, C a real diagonal M×M matrix, B an arbitrary complex N×M matrix, and γ a real number, is investigated by means of a partitioning technique. Both Hermitian (γ=1) and non‐Hermitian (γ≠1) arrow matrices Z are studied. The one‐dimensional case (dimension N of A equal to 1) is briefly reviewed and a detailed treatment of the multidimensional case (N>1) is presented. For Hermitian arrow matrices, the analysis leads to a new algorithm for computing the eigenvalues and eigenvectors of Z which is particularly efficient if M≫N.
24 citations
TL;DR: In this article, the convergence of the power method for the algebraic eigenvalue problem is accelerated by an aggregation/disaggregation process, where a lower dimensional aggregated eigen value problem is created from time to time during the course of power iteration and the solution of this problem is used to produce acceleration.
Abstract: Convergence of the power method for the algebraic eigenvalue problem is accelerated by an aggregation/disaggregation process. A lower dimensional aggregated eigenvalue problem is created from time to time during the course of the power iteration and the solution of this problem is used to produce acceleration. Analysis and numerical experiments show the effectiveness of the method, the more so, the more poorly conditioned is the original eigenvalue problem.
TL;DR: In this paper, the squared sine-Gordon eigenvalue problem in laboratory coordinates has been analyzed in the form of a standard eigen value problem, in which an independent operator operating on an eigenfunction generates the eigenvalues.
Abstract: An analysis of the squared sine–Gordon eigenvalue problem in laboratory coordinates is presented. It is shown that unlike the unsquared laboratory coordinate eigenvalue problem, the squared laboratory coordinate eigenvalue problem may be cast into the form of a standard eigenvalue problem, wherein an eigenvalue independent operator operating on an eigenfunction generates the eigenvalue. With this form, it becomes rather elementary to obtain the squared‐eigenfunction expansion of the sine–Gordon potentials as well as to demonstrate closure.
TL;DR: In this paper, the authors studied the asymptotic behavior of the Toda lattice when acting on real normal matrices, and they showed that the solution flow eventually converges to a diagonal block form where for a real eigenvalue the associated block is of size $1 \times 1$ with that eigen value as its element and for complex-conjugate pairs of eigenvalues, for real part as its diagonal elements and the (negative) imaginary part as off-diagonal elements.
Abstract: The asymptotic behavior of the Toda lattice, when acting on real normal matrices, is studied. It is shown that the solution flow eventually converges to a diagonal block form where for a real eigenvalue the associated block is of size $1 \times 1$ with that eigenvalue as its element and for complex-conjugate pairs of eigenvalues the associated block is of size $2 \times 2$ with the real part as its diagonal elements and the (negative) imaginary part as its off-diagonal elements. This result generalizes the well-known asymptotic behavior of Jacobi matrices and is consistent with that from the $QR$-algorithm.
TL;DR: In this article, the LRP 90 Reference CRPP-REPORT-1975-002 Record created on 2008-04-18, modified on 2017-05-12 was used.
TL;DR: A generalization of the Lanczos algorithm to make it applicable directly to the generalized matrix eigenvalue problem Ax = λBx, where A, B are real symmetric matrices of high order ( n × n ) and B is positive definite.
TL;DR: Numerical results for difference approximations of nonlinear elliptic eigenvalue problems in one and two space dimensions confirm the efficiency of the algorithm proposed here.
Abstract: This paper deals with an application of the multi-grid iteration for large sparse linear systems to the problem of computing stable branches of solutions of nonlinear eigenvalue problems which bifurcate from simple eigenvalues. The theoretical background of the algorithm considered here is the selective Picard iteration. We present numerical results for difference approximations of nonlinear elliptic eigenvalue problems in one and two space dimensions. They confirm the efficiency of the algorithm proposed here.
TL;DR: In this article, sufficient conditions for a local extremum for single and double eigenvalues are obtained for discrete and continuous systems, and the conditions obtained are constructive in nature and can be utilized in different eigenvalue optimization problems.
01 Mar 1984
TL;DR: Finite forms of the two common eigenvalue problems associated with the uncertainty principle are introduced and it is shown that feature extraction filters produced using these methods are effective in the processing of natural images.
Abstract: Finite forms of the two common eigenvalue problems associated with the uncertainty principle are introduced. The results are extended to two dimensions and it is shown that feature extraction filters produced using these methods are effective in the processing of natural images.
TL;DR: In this article, it was shown that one can calculate in a simple analytical way the asymptotic eigenvalue density of all these matrices by means of its moments without solving the corresponding eigen value problem.
Abstract: In several branches of physics the Hamiltonian of a many-body system can be reduced to a rational Jacobi matrix (i.e. a Jacobi matrix whose elements are rational functions of the suffix) by means of a method of the Lanczos type. Here it is shown that one can calculate in a simple analytical way the asymptotic eigenvalue density of all these matrices by means of its moments without solving the corresponding eigenvalue problem. The method is applied to a large class of quantum mechanical models of Hamiltonians.
TL;DR: The conformal transformation method overcomes the difficulties associated with the computation of the eigenvalues of problems involving curved boundaries and/or boundary singularities and produces accurate numerical approximations.
Abstract: A conformal transformation method is used for the numerical solution of the membrane eigenvalue problem. The method overcomes, in many cases, the difficulties associated with the computation of the eigenvalues of problems involving curved boundaries and/or boundary singularities and produces accurate numerical approximations. This is achieved by transforming the original problem into another computationally simpler one.
01 Mar 1984
TL;DR: An algorithm is presented that can obtain the QR iterates at step 1, 2, 4, 8, 16, etc for every sweep over the matrix and can be implemented on a highly regular array of computing elements with only neighborhood communication between processors.
Abstract: We present in this paper an algorithm that doubles-up on Francis's QR algorithm. By this we mean that we can obtain the QR iterates at step 1, 2, 4, 8, 16, etc for every sweep over the matrix. We also show that the algorithm can be implemented on a highly regular array of computing elements with only neighborhood communication between processors. Simulations are presented which suggest that algorithm is stable.
TL;DR: In this paper, a class of non-Hermitian eigenvalue equations, which are aimed to determine positions and widths of resonant (decaying) levels, is derived quite generally through a projection operator procedure whereby the effects of the continuum states responsible for the decay is represented in an effective manner.
Abstract: A class of non-Hermitian eigenvalue equations, which are aimed to determine positions and widths of resonant (decaying) levels, is derived quite generally through a projection operator procedure whereby the effects of the continuum states responsible for the decay is represented in an effective manner. The boundary conditions appropriate to these eigenvalue problems are discussed and the wave function renormalization is evaluated. A connection is drawn with the effective eigenvalue problems occurring in the many-body Green’s functions theory. Features like the energy dependence of the self-energy for the single-particle Green’s function and of the screened Coulomb interaction for the electron-hole Green’s function are accordingly interpreted in terms of elimination of continuum channels.
TL;DR: In this paper, the authors consider numerical solutions of holomorphic multiparameter eigenvalue problems and prove general convergence results for general continuoustime eigen value problems, and establish algebraic bifurcation equations for these methods to switch branches in bifurlcation points.
Abstract: We consider numerical solutions of holomorphic multiparameter eigenvalue problems and prove general convergence results. We give continuation methods for the computation of eigenvalue curves and manifolds of limit points. We establish algebraic bifurcation equations for these methods to switch branches in bifurcation points. We illustrate the results with some numerical examples in mechanics.
01 Jan 1984
TL;DR: In this paper, sufficient conditions are given for existence of a solution to the eigenvalues assignment problem for three-dimensional linear systems with separable closed-loop characteristic polynomials.
Abstract: Sufficient conditions are given for existence of a solution to the eigenvalues assignment problem for three-dimensional (3-D) linear systems with separable closed-loop characteristic polynomials Three methods for finding the feedback gain matrix are presented The method 3 is an extension for 3-D systems of the method presented in [3] for 2-D systems
TL;DR: Eigenvalue problems that depend on a finite number of parameters, if the eigenvalue is considered as a function of the parameters, then its gradient and level surfaces (or curves) can be evaluated.