Nonsymmetric Eigenvalue Problem Research Articles

A continuation method for a weakly elliptic two-parameter eigenvalue problem

We show that the continuation method can be used to solve a weakly elliptic two-parameter eigenvalue problem. We generalize the continuation method for a nonsymmetric eigenvalue problem Ax = λx by T. Y. Li, Z. Zeng and L. Cong (1992 SIAM J. Numer. Anal. 29, 229-248) to two-parameter problems.

Condition Number Bounds for Problems with Integer Coefficients

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 polynomials. It is assumed that the input has integer coefficients and is not on the degeneracy locus of the respective problem (i.e., the condition number is finite). Our bounds are stated in terms of the dimension and of the bit-size of the input. In the same setting, bounds are given for the speed of convergence of the following iterative algorithms: QR iteration without shift for the symmetric eigenvalue problem and Graeffe iteration for univariate polynomials.

311 有理摂動法を用いた大規模連成解析手法の開発

In this paper, we introduce a improved perturbation analysis of eigenvalue for structural-acoustic coupled system. Two years ago, we gave the derivation of perturbation series from a standard symmetric eigenvalue, this time we give it from a non-symmetric eigenvalue problem. The coefficients of the perturbation series will be more compact because it is unnecessary to calculate the part from non-symmetric problem into symmetric one. We confirm the adequacy through some numerical examples which are calculated by NASTRAN software.

108 有限摂動法を用いた室内騒音低減の最適化

In this paper, we introduce a improved perturbation analysis of eigenvalue for structural-acoustic coupled system. Two years ago, we gave the derivation of perturbation series from a standard symmetric eigenvalue, this time we give it from a non-symmetric eigenvalue problem. The coefficients of the perturbation series will be more compact because it is unnecessary to calculate the part from non-symmetric problem into symmetric one. We confirm the adequacy through some numerical examples.

The Homotopy Continuation Algorithm for the Real Nonsymmetric Eigenproblem: Further Development and Implementation

New development and implementation of a homotopy continuation algorithm for the nonsymmetric eigenvalue problem are presented, along with thorough investigation on the advantages and limitations of our code. Numerical results on substantial varieties of matrices arising in applications indicate excellent efficiency and stability of our algorithm and a drastic improvement over previous implementations.

Harmonic projection methods for large non-symmetric eigenvalue problems

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.

A reformulated Arnoldi algorithm for non-classically damped eigenvalue problems

In applying Arnoldi method to non-symmetric eigenvalue problems for damped structures, a structure of the projected upper Hessenberg matrix is obtained in this paper. By exploiting the structure of the upper Hessenberg matrix and taking advantages of the block properties of system matrices, the Arnoldi reduction algorithm is reformulated for less computation and higher accuracy. In conjunction with the reformulated Arnoldi algorithm, real Schur decomposition instead of Jordan decomposition is adopted aiming at non-complex arithmetic, non-discriminative processing of defective and non-defective systems and numeric stability. A concise reduction algorithm for eigenproblems for undamped gyroscopic systems is obtained by directly degenerating from the reformulated Arnoldi algorithm. For safely solving engineering problems without omitting eigenvalues, a restart reduction procedure is proposed in terms of the reformulated reduction algorithm with deflation developed in this paper. Numerical examples once solved with algorithms originated from Lanczos methods were re-solved. In addition, the non-symmetric eigenvalue problem for a shear wall by BEM modeling and a damped gyroscopic system with eigenvalues of high multiplicity were also used to demonstrate the efficacy of the presented methods. © 1997 John Wiley & Sons, Ltd.

Reflection coefficients for weak anisotropic media

SUMMARY The interaction of plane elastic waves with a plane boundary between two anisotropic elastic half-spaces is investigated. The anisotropy dealt with in this study is of a general type. Explicit expressions for energy-related reflection and transmission coefficients are derived. They represent an approximation which is valid for a small deviation of the elastic parameters from isotropy. Classical perturbation theory is applied on a 6x 6 non-symmetric real eigenvalue problem to calculate first-order corrections for the polarization and stress of the plane waves. The explicit solution of the isotropic problem is used as a reference case. Degenerate perturbation theory is used to consider the splitting of the isotropic S-wave into two anisotropic qS-waves. The boundary conditions for two half-spaces in welded contact lead to a 6x6 system of linear equations. A correction to the isotropic solution is calculated by linearization. The resultant coefficients are functions of horizontal slowness, Lam6 parameters and densities of the reference media, and of the perturbation of the elasticity tensors from isotropy.

A fast iterative algorithm for eigenvalue determination

The task of establishing analytically the natural frequencies of vibration of a partially embedded beam gives rise to a particular difficulty because, frequently, the modes are clustered in close proximity to each other. Due to this and the fact that the exact formulated solution is in the form of a nonsymmetrical eigenvalue problem, neither of the established techniques (using either crude graphical methods or the infallible algorithmic methods which are based on a stiffness-matrix approach) can be utilized directly in the solution of this problem. Consequently an alternative technique is proposed which makes use of the concept of sign counting of the main diagonal of the upper triangular form of the solution matrix introduced in the earlier methods. An efficient algorithm is developed which allows near-infallible detection of eigenvalues by using a dynamic increment in the search for diagonal element sign pattern changes. This paper outlines the main features of this procedure and provides a typical example to illustrate its operation.

On restarting the Arnoldi method for large nonsymmetric eigenvalue problems

The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues are desired. We analyze several approaches to restarting and show why Sorensen’s implicit QR approach is generally far superior to the others. Ritz vectors are combined in precisely the right way for an effective new starting vector. Also, a new method for restarting Arnoldi is presented. It is mathematically equivalent to the Sorensen approach but has additional uses.

A parallel algorithm for the reduction of a nonsymmetric matrix to block upper-Hessenberg form

In this paper, we present an algorithm for the reduction to block upper-Hessenberg form which can be used to solve the nonsymmetric eigenvalue problem on message-passing multicomputers. On such multicomputers, a nonsymmetric matrix can be distributed across processing nodes logically configured into a two-dimensional mesh using the block-cyclic data distribution. Based on the matrix partitioning and mapping, the algorithm employs both Householder reflectors and Givens rotations within each reduction step. We analyze the arithmetic and communication complexities and describe the implementation details of the algorithm on message-passing multicomputers. We discuss two different implementations — synchronous and asynchronous — and present performance results on the Intel iPSC/860 and DELTA. We conclude with an evaluation of the algorithm's communication cost, and suggest areas for further improvement.

Application of the Implicitly Updated Arnoldi Method with a Complex Shift-and-Invert Strategy in MHD

The implicitly updated Arnoldi method introduced by Sorensen with an internal QR-iteration is a very useful eigenvalue solver for nonsymmetric eigenvalue problems. To make this method rigorous in finding internal eigenvalues, a complex shift-and-invert strategy is used. Therefore a complex variant of the method has been constructed and the method has been compared with a Lanczos method, as implemented by Cullum et al., for a practical problem in magnetohydrodynamics.

Progress in the numerical solution of the nonsymmetric eigenvalue problem

AbstractWith the growing demands from disciplinary and interdisciplinary fields of science and engineering for the numerical solution of the nonsymmetric eigenvalue problem, competitive new techniques have been developed for solving the problem. In this paper we examine the state of the art of the algorithmic techniques and the software scene for the problem. Some current developments are also outlined.

Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays

A symmetrizer of a nonsymmetric matrix A is the symmetric matrix X that satisfies the equation XA = A(t)X, where t indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.

Towards a Divide and Conquer Algorithm for the Real Nonsymmetric Eigenvalue Problem

Theory is developed that couldd be used towards developing a divide and conquer algorithm for the nonsymmetric eigenvalue problem. The shortcomings of this theory and its application to the Hessenberg and nonsymmetric tridiagonal problems are discussed. The conclusion is made that the method may not be as promising as the divide and conquer methods for symmetric problems.

Shifting Strategies for the Parallel $QR$ Algorithm

The use of high-order generalized Rayleigh quotient shifting strategies is advocated as a means of improving the parallel performance of the double-shift $QR$ algorithm for nonsymmetric eigenvalue problems.

Magnetohydrodynamic spectroscopy: Large scale computation of the spectrum of waves in plasmas

A systematic study of the spectrum of magnetohydrodynamic (MHD) waves in thermonuclear and astrophysical plasmas is presently undertaken at the FOM Institute for Plasma Physics. This study is based on the fact that the macroscopic waves and instabilities in both thermonuclear confinement machines and in astrophysical magnetic flux loops are described by the MHD equations. The linearised interaction between plasma and magnetic field leads to large-scale non-symmetric eigenvalue problems for free oscillations and to temporal evolution problems for forced (externally driven) oscillations. These problems have been cast in the form of a set of two modular computer programs, called CASTOR for tokamaks and POLLUX for coronal flux tubes, exploiting identical solvers for the resulting very large systems. CASTOR is extensively used at JET (the Joint European Torus experiment at Culham) for the interpretation of measured MHD spectra, whereas POLLUX is presently used for the study of Alfvén wave heating of the solar corona and will be used in the near future for the interpretation of observed x-ray emissions from the SOHO satellite which will be launched by ESA and NASA in 1995. For the interpretation of actual plasma dynamics in tokamaks and solar coronal loops it is crucial to understand the non-linear phase of the evolution of MHD phenomena as well. This adds two complications: non-linear mode coupling of hundreds of small-scale modes and enormous discrepancies between the different dynamical time scales. We have developed numerical codes that simulate the non-linear temporal evolution of plasma columns under external excitation. These codes have been modelled after the linear ones, with a finite difference discretisation in the radial direction and a spectral discretisation in the two other directions. A relatively simple and flexible semi-implicit predictor-corrector scheme has been used for the time stepping. This allows time steps up to a factor 1000 larger than the largest time steps allowed for explicit methods. Because of the many modes involved, memory requirements and CPU time requirements soon become prohibitive. Hence, the new physics in the non-linear plasma dynamics regime requires parallel computing.

Error Analysis of the Lanczos Algorithm for the Nonsymmetric Eigenvalue Problem

This paper presents an error analysis of the Lanczos algorithm in finite-precision arithmetic for solving the standard nonsymmetric eigenvalue problem, if no breakdown occurs. An analog of Paige’s theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. The theory developed illustrates that in the nonsymmetric Lanczos scheme, if Ritz values are well conditioned, then the loss of biorthogonality among the computed Lanczos vectors implies the convergence of a group of Ritz triplets in terms of small residuals. Numerical experimental results confirm this observation.

Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem

This paper presents an error analysis of the Lanczos algorithm in finite-precision arithmetic for solving the standard nonsymmetric eigenvalue problem, if no breakdown occurs. An analog of Paige’s theory on the relationship between the loss of orthogonality among the Lanczos vectors and the convergence of Ritz values in the symmetric Lanczos algorithm is discussed. The theory developed illustrates that in the nonsymmetric Lanczos scheme, if Ritz values are well conditioned, then the loss of biorthogonality among the computed Lanczos vectors implies the convergence of a group of Ritz triplets in terms of small residuals. Numerical experimental results confirm this observation.

Subspace Iteration for Nonsymmetric Eigenvalue Problems Applied to the λ-Eigenvalue Problem

Often in reactor dynamics, higher eigenfunctions of the multigroup diffusion equation must be determined. An algorithm to calculate higher eigenfunctions (modes) of the [lambda]-eigenvalue problem corresponding to the steady-state two-group neutron diffusion equation is presented. The method is based on a special type of subspace iteration for large sparse nonsymmetric eigenvalue problems. Having been tested using an International Atomic Energy Agency benchmark problem and also applied to a VVER-1000 pressurized water reactor assembly, the algorithm was found to work very effectively and reliably. In its application, the algorithm presented is not restricted to the [lambda]-eigenvalue problem only but is also generally applicable to large sparse nonsymmetric eigenvalue problems even with multiple and complex eigenvalues.