Showing papers on "Divide-and-conquer eigenvalue algorithm published in 1998"
TL;DR: In this article, the general correlation function for the eigenvalues of p complex Hermitian matrices coupled in a chain is given as a single determinant, using a slight generalization of a theorem of Dyson.
Abstract: The general correlation function for the eigenvalues of p complex Hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.
252 citations
Abstract: The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh–Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated. This procedure can be applied to the Arnoldi method, to preconditioning methods, and to other methods for nonsymmetric eigenvalue problems that use the Rayleigh–Ritz procedure. The subject of estimating the boundary of the entire spectrum is briefly discussed, and the importance of preconditioning for interior eigenvalue problems is mentioned. © 1998 John Wiley & Sons, Ltd.
95 citations
TL;DR: In this article, the eigenvalue problem for 2×2 and 3×3 octonionic Hermitian matrices was studied and general solutions for real and non-real eigenvalues were given.
Abstract: We discuss the eigenvalue problem for 2×2 and 3×3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with non-real eigenvalues.
74 citations
TL;DR: It is found that even though the Runge?Kutta method is of higher order, this extra accuracy can be lost because of the additional dependence of its numerical error on the eigenvalue, and it is shown that no such limitations exist for the piecewise-constant algorithm.
50 citations
TL;DR: In this paper, a procedure and related theories are developed to find loci of optimal support positions for a structure to maximize its fundamental eigenvalue by increasing the support stiffness, which is the upper bound of the fundamental value achieved by adding supports.
46 citations
TL;DR: In this article, the authors introduced general discrete linear Hamiltonian eigenvalue problems and characterised the eigenvalues, among them the new notion of strict controllability of a discrete system, that imply isolatedness and lower boundedness of the Eigenvalues.
Abstract: This paper introduces general discrete linear Hamiltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict controllability of a discrete system, that imply isolatedness and lower boundedness of the eigenvalues. Due to the quite general assumptions, discrete Sturm-Liouville eigenvalue problems of higher order are included in the presented theory.
43 citations
TL;DR: In this paper, a perturbation analysis is presented to determine approximate eigenvalue loci and stability conclusions in the vicinity of critical speeds and zero speed for general continuous gyroscopic systems.
Abstract: In order to provide analytical eigenvalue estimates for general continuous gyroscopic systems, this paper presents a perturbation analysis to determine approximate eigenvalue loci and stability conclusions in the vicinity of critical speeds and zero speed. The perturbation analysis relies on a formulation of the general continuous gyroscopic system eigenvalue problem in terms of matrix differential operators and vector eigenfunctions. The eigenvalue λ appears only as λ 2 in the formulation, and the smoothness of λ 2 at the critical speeds and zero speed is the essential feature. First-order eigenvalue perturbations are determined at the critical speeds and zero speed. The derived eigenvalue perturbations are simple expressions in terms of the original mass, gyroscopic, and stillness operators and the critical-speed/zero-speed eigenfunctions. Prediction of whether an eigenvalue passes to or from a region of divergence instability at the critical speed is determined by the sign of the eigenvalue perturbation. Additionally, eigenvalue perturbation at the critical speeds and zero speed yields approximations for the eigenvalue loci over a range of speeds. The results are limited to systems having one independent eigenfunction associated with each critical speed and each stationary system eigenvalue. Examples are presented for an axially moving tensioned beam, an axially moving string on an elastic foundation, and a rotating rigid body. The eigenvalue perturbations agree identically with exact solutions for the moving string and rotating rigid body.
42 citations
Journal Article•
TL;DR: This paper studies a number of Newton methods and uses them to define new secondary linear systems of equations for the Davidson eigenvalue method to avoid some common pitfalls of the existing ones.
Abstract: This paper studies a number of Newton methods and use them to define new secondary linear systems of equations for the Davidson eigenvalue method. The new secondary equations avoid some common pitfalls of the existing ones such as the correction equation and the Jacobi-Davidson preconditioning. We will also demonstrate that the new schemes can be used efficiently in test problems.
37 citations
TL;DR: Pfaffian expressions for the smallest eigenvalue distributions of Laguerre orthogonal and symplectic ensembles of random matrices have been derived in this paper, where they are evaluated in the limit of large matrix dimension.
26 citations
TL;DR: Several previously known backward stable QR algorithms are tied together in a common framework of which each is a special case, and their connection to an explicit expression for the feedback vector is exposed.
Abstract: A close look is taken at the single-input eigenvalue assignment methods. Several previously known backward stable QR algorithms are tied together in a common framework of which each is a special case, and their connection to an explicit expression for the feedback vector is exposed. A simple new algorithm is presented and its backward stability is established by round-off error analysis. The differences between this new algorithm and the other QR algorithms are discussed. Also, the round-off error analysis of a simple recursive algorithm for the problem [B. N. Datta, IEEE Trans. Automat. Control, AC-32 (1987), pp. 414--417] is presented. The analysis shows that the latter is reliable, and the reliability can be determined during the execution of the algorithm rather cheaply. Finally, some numerical experiments comparing some of the methods are reported.
TL;DR: In this article, conditions for existence of eigenvalues with positive decaying eigenfunctions were discussed for the eigenvalue problem with respect to the weight function q changing sign.
Abstract: For the eigenvalue problem—λΔu = q(x)u in IRd, with the weight function q changing sign, conditions are discussed for existence of eigenvalues with positive decaying eigenfunctions.
TL;DR: A method for solving the joint eigenpair problem of a family of matrix pencils is presented, which gives all solutions of the matrix inverse eigenvalue problem.
Abstract: This paper discusses the matrix inverse eigenvalue problem: given matrices , find scalars such that has the prescribed eigenvalues . It is proven that this problem is equivalent to the joint eigenpair problem of a family of related matrix pencils. Furthermore, the conditions under which this problem possesses a solution are given. A method for solving the joint eigenpair problem of a family of matrix pencils is then presented, which, in turn, gives all solutions of the problem. Some results in this paper may be of individual interest.
TL;DR: For an Hermitian matrix the QR transform is diagonally similar to two steps of the LR transforms, and for non-Hermitian matrices the QR Transform may be written in rational form.
Abstract: For an Hermitian matrix the QR transform is diagonally similar to two steps of the LR transforms. Even for non-Hermitian matrices the QR transform may be written in rational form.
TL;DR: An analogue of the Schwarz alternating method is considered and a minimal eigenvalue and its corresponding eigenvector of the generalized symmetric eigen value problem are found and it is shown that a discretization of the multiplicative variation of the Schwartz method is equivalent to the block coordinate relaxation method.
Abstract: In this paper an analogue of the Schwarz alternating method is considered and a minimal eigenvalue and its corresponding eigenvector of the generalized symmetric eigenvalue problem are found. The technique suggested is based on decomposition of the original domain into overlapping subdomains and on consideration of local eigenvalue problems in subdomains. Both multiplicative and additive variations of the method are constructed and studied. It is shown that a discretization of the multiplicative variation of the Schwarz method is equivalent to the block coordinate relaxation method. An additive variation of the method is suitable for realization on parallel architecture. In this paper we propose an analogue of the Schwarz alternating method to evaluate the principal eigenvalue and its corresponding eigenfunction of a second-order symmetric elliptic operator in a bounded domain in R. The technique suggested is based on decomposition of the original domain into overlapping subdomains and on consideration of local eigenvalue problems in subspaces connected with these subdomains. Domain decomposition methods (DD) are powerful techniques for solving boundary value problems. Currently DD algorithms have become increasingly popular because they take full advantage of modern parallel computing technology. Although there are many papers on domain decomposition for the linear applications (see, e.g., [1,5,9,13]), there are relatively few results concerning the application of DD methods to eigenvalue problems. One of the approaches is presented in [11], where an approximation to the principal eigenpair is computed by solving a sequence of linear problems in the subdomains. Several other domain decomposition methods, which also use linearization, were proposed in [10,15]. These works are based on a nonoverlapping partitioning of the computational domain and on the use of some iterative techniques for the Schur complement of the block corresponding to the interface variables. Another way to apply the domain decomposition idea to an eigenvalue problem is a divide-and-conquer method proposed in [4]. The authors introduced a parallel algorithm for computing all the eigenpairs of the symmetric and positive definite matrix, first, by parallelizing the Householder transformation and then providing the multilevel parallel method of solving an eigenvalue problem for a three-diagonal symmetric matrix. A different approach was presented by the author in the earlier work [12], where only a multiplicative algorithm and its discretization were described with some limiting assumptions. In that algorithm an approximation to the principal eigenpair is computed by solving sequentially a series of minimization problems for the Rayleigh quotient in the subdomains. On the algebraic level, this method is equivalent to the one developed in * Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455 This research was supported by the National Science Foundation through the Institute for Mathematics and its Applications.
TL;DR: In this paper, an efficient solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of nonproportionally damped structural systems, which is obtained by applying the modified Newton-Raphson technique and the orthonormal condition of eigenvectors to the linear eigen problem through matrix augmentation of the quadratic eigen value problem.
Abstract: An efficient solution method is presented to solve the eigenvalue problem arising in the dynamic analysis of nonproportionally damped structural systems. The proposed method is obtained by applying the modified Newton-Raphson technique and the orthonormal condition of the eigenvectors to the linear eigenproblem through matrix augmentation of the quadratic eigenvalue problem. In the iteration methods, such as the vector inverse iteration and subspace iteration methods, singularity may occur during the factorizing process when the shift value is close to an eigenvalue of the system. However, even though the shift value is an eigenvalue of the system, the proposed method provides nonsingularity, if the desired eigenvalue is not multiple, and this is analytically proved. Because the modified Newton-Raphson technique is adapted to the proposed method, initial values are needed. The initial values of the proposed method can be obtained by the intermediate results of iteration methods or results of approximate m...
01 Sep 1998
TL;DR: Two new solvers for large sparse symmetric matrix eigenvalue problems are presented: the implicitly restarted Lanczos algorithm and the Jacobi-Davidson algorithm.
Abstract: We present experiments with two new solvers for large sparse symmetric matrix eigenvalue problems: (1) the implicitly restarted Lanczos algorithm and (2) the Jacobi-Davidson algorithm The eigenvalue problems originate from in the computation of a few of the lowest frequencies of standing electromagnetic waves in cavities that have been discretized by the finite element method The experiments have been conducted on up to 12 processors of an HP Exemplar X-Class multiprocessor computer
TL;DR: In this paper, a general global-outer-inner iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem is presented.
TL;DR: In this paper, the N-coupled symmetric rotor problem is solved exactly by using an infinite-dimensional algebra and a formalism for solving the corresponding Hamiltonian eigenvalue problem is also proposed.
Abstract: The N-coupled symmetric rotor problem is solved exactly by using an infinite-dimensional algebra. A formalism for solving the corresponding Hamiltonian eigenvalue problem is also proposed. The system of equations that solves a special Hamiltonian eigenvalue problem is shown to yield coupling coefficients of the corresponding Lie algebra.
07 Sep 1998
TL;DR: The problem of evaluating the dominant eigenvalue of real matrices using Monte Carlo numerical methods is considered and the results are compared to those obtained using classical methods.
Abstract: The problem of evaluating the dominant eigenvalue of real matrices using Monte Carlo numerical methods is considered.
21 Jun 1998
TL;DR: In this article, a unified approach to the design of different parallel block-Jacobi methods for solving the Symmetric Eigenvalue Problem is presented, where the problem can be solved designing a logical algorithm by considering the matrices divided into square blocks, and considering each block as a process.
Abstract: In this paper we present a unified approach to the design of different parallel block-Jacobi methods for solving the Symmetric Eigenvalue Problem. The problem can be solved designing a logical algorithm by considering the matrices divided into square blocks, and considering each block as a process. Finally, the processes of the logical algorithm are mapped on the processors to obtain an algorithm for a particular system. Algorithms designed in this way for ring, square mesh and triangular mesh topologies are theoretically compared.
21 Jun 1998
TL;DR: Numerical algorithms based on the well-known Gauss-type quadrature rules and Lanczos process are reviewed for computing quadratic forms and these algorithms are well suited for large sparse problems.
Abstract: We survey some unusual eigenvalue problems arising in different applications. We show that all these problems can be cast as problems of estimating quadratic forms. Numerical algorithms based on the well-known Gauss-type quadrature rules and Lanczos process are reviewed for computing these quadratic forms. These algorithms reference the matrix in question only through a matrix-vector product operation. Hence it is well suited for large sparse problems. Some selected numerical examples are presented to illustrate the efficiency of such an approach.
18 Aug 1998
TL;DR: In this paper, an efficient eigenvalue algorithm for large power systems is presented, which is based on the Arnoldi method and the IIA method with the complex shift Cayley transformation.
Abstract: In this paper an efficient eigenvalue algorithm for large power systems is presented. The proposed algorithm is based on the Arnoldi method and the IIA method with the complex shift Cayley transformation. Furthermore, a reduced admittance matrix preserving the sparsity is applied for solving a network equation which is the most time consuming part of eigenvalue calculation. Test calculations were performed to confirm the effectiveness of the proposed method. The results showed that the proposed algorithm could obtain more eigenvalues with high accuracy compared to the conventional methods.
21 Jun 1998
TL;DR: It is shown that the concomitant requirements for vectorisation and parallelisation lead both to novel algorithms and novel implementation techniques in the development and implementation of eigensolvers on distributed memory parallel arrays of vector processors.
Abstract: We consider the development and implementation of eigensolvers on distributed memory parallel arrays of vector processors and show that the concomitant requirements for vectorisation and parallelisation lead both to novel algorithms and novel implementation techniques. Performance results are given for several large-scale applications and some performance comparisons made with LAPACK and ScaLAPACK.
02 Nov 1998
TL;DR: In this paper, the generalized eigenvalue problem of the quadratic matrix beam is reduced to a linear problem and modified successive approximations method is used for calculation of the eigenvalues and eigenvectors.
Abstract: The case of the eigenvalue problem of the quadratic matrix beam is proposed. The method permits one to calculate all eigenpairs of the matrix beam. With this aim, the generalized eigenvalue problem is reduced to a linear problem. Then, the modified successive approximations method is used for calculation of the eigenvalues and eigenvectors. This permits the use of the information obtained from the iterations, to obtain the first eigenpair and all the following ones more quickly. Some numerical examples are considered.
01 Oct 1998
DOI•
01 Nov 1998
TL;DR: It is proved that the bound for the smallest eigen value of the projection of the given eigenvalue problem onto a Krylov space of T n of dimension is the size of the principal eigenvector of Tn.
Abstract: In a recent paper Melman derived upper bounds for the smallest eigen value of a real symmetric Toeplitz matrix in terms of the smallest roots of rational and polynomial approximations of the secular equation f the best of which being constructed by the Pad e approximation of f In this paper we prove that this bound is the smallest eigenvalue of the projection of the given eigenvalue problem onto a Krylov space of T n of dimension This interpretation of the bound suggests enhanced bounds of increasing ac curacy They can be substantially improved further by exploiting symmetry properties of the principal eigenvector of Tn
TL;DR: In this paper, constructive solutions to a general nonlinear nonhomogenious two-parameter eigenvalue problem of the form Ax + λBx + μCx = f are established.
Abstract: Based on the three well-known Banach, Mann, and Ishikawa iteration processes, constructive solutions to a general nonlinear nonhomogenious two-parameter eigenvalue problem of the form Ax + λBx + μCx = f are established. The allowed range of the complex parameters λ and μ is discussed. Error estimates are studied. The obtained results are natural extensions, to the two-parameter case, of the corresponding one-parameter problem discussed in the author's paper [1].
TL;DR: The first eigenvalue for the Dirichlet problem is often a very important characterization of a system in Physics, Elasticity, Biology, and so on as mentioned in this paper, and it serves for estimation of the domain of existence or uniqueness of solutions.
Abstract: The first eigenvalue for the Dirichlet problem is often a very important characterization of a system in Physics, Elasticity, Biology and so on. In the theory of boundary value problems it serves for estimation of the domain of existence or uniqueness of solutions. We consider the problem of existence of the extremal values for the first eigenvalue for a number of boundary value problems. For example, the boundary value problem of the form Lu=λV (x)u is studied, where L is an elliptic operator in a domain Ω ⊂ ℝn. The first eigenvalue is estimated from above or from below in terms of the integral with some real α.