Boundary value method for inverse Sturm-Liouville problems

In this paper we present a method to recover symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. The linear multistep method coupled with suitable boundary conditions known as boundary value method (BVM) is the main tool to approximate the eigenvalues in each iteration step of the used Newton method. The BVM was extended to work for Neumann-Neumann boundary conditions. Moreover, a suitable approximation for the asymptotic correction of the eigenvalues is given. Numerical results demonstrate that the method is able to give good results for both symmetric and non-symmetric potentials.     

Domains of divergence of the USSOR method applied onp-cyclic matrices

Summary Using a recently derived classical type general functional equation, relating the eigenvalues of a weakly cyclic Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix, we obtain bounds for the convergence of the USSOR method, when applied to systems with ap-cyclic coefficient matrix.     

A generalisation of systematic relaxation methods for consistently ordered matrices

Summary A systematic relaxation method is analysed for consistently ordered matrices as defined by Broyden (1964). The method is a generalisation of successive over-relaxation (S.O.R.). A relation is derived between the eigenvalues of the iteration matrix of the method and the eigenvalues of the Jacobi iteration matrix. Forp-cyclic matrices, the method corresponds to using a special type of diagonal matrix instead of a single relaxation factor. For certain choices of this diagonal matrix, the method has a better asymptotic rate of convergence than S.O.R. and requires less calculations and computer store.     

A reflection on the implicitly restarted Arnoldi method for computing eigenvalues near a vertical line

In this article, we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted Arnoldi method. The extension to computing eigenvalues closest to a vertical line is straightforward, by incorporating a shift. Without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis.In a recent publication, Meerbergen and Spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to Hopf bifurcations, which is of interest for the stability of dynamical systems. The novel method is based on inverse iteration (inverse power method) applied on a Lyapunov-like eigenvalue problem. To reduce the computational overhead significantly a projection was added.This method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane. We will prove in this paper that the combination of inverse iteration with the projection step is equivalent to Sorensen’s implicitly restarted Arnoldi method utilizing well-chosen shifts.     

Generalized consistent orderings and the Accelerated Overrelaxation method

In this paper, the behavior of the block Accelerated Overrelaxation (AOR) iterative method, when applied to linear systems with a generalized consistently ordered coefficient matrix, is investigated. An equation, relating the eigenvalues of the block Jacobi iteration matrix to the eigenvalues of its associated block AOR iteration matrix, as well as sufficient conditions for the convergence of the block AOR method, are obtained.     

Optimal parameters in the HSS‐like methods for saddle‐point problems

For the Hermitian and skew‐Hermitian splitting iteration method and its accelerated variant for solving the large sparse saddle‐point problems, we compute their quasi‐optimal iteration parameters and the corresponding quasi‐optimal convergence factors for the more practical but more difficult case that the (1, 1)‐block of the saddle‐point matrix is not algebraically equivalent to the identity matrix. In addition, the algebraic behaviors and the clustering properties of the eigenvalues of the preconditioned matrices with respect to these two iterations are investigated in detail, and the formulas for computing good iteration parameters are given under certain principle for optimizing the distribution of the eigenvalues. Copyright © 2008 John Wiley & Sons, Ltd.     

Preconditioned AHSS iteration method for singular saddle point problems

In this paper, for solving the singular saddle point problems, we present a new preconditioned accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration method. The semi-convergence of this method and the eigenvalue distribution of the preconditioned iteration matrix are studied. In addition, we prove that all eigenvalues of the iteration matrix are clustered for any positive iteration parameters α and β. Numerical experiments illustrate the theoretical results and examine the numerical effectiveness of the AHSS iteration method served either as a preconditioner or as a solver.     

The extrapolated first order method for solving systems with complex eigenvalues

An extrapolated form of the basic first order stationary iterative method for solving linear systems when the associated iteration matrix possesses complex eigenvalues, is investigated. Sufficient (and necessary) conditions are given such that convergence is assured. An analytic determination of good (and sometimes optimum) values of the involved real parameter is presented in terms of certain bounds on the eigenvalues of the iteration matrix. The usefulness of the developed theory is shown through a simple application to the conventional Jacobi method.     

A hybrid iterative method for symmetric positive definite linear systems

A hybrid iterative scheme that combines the Conjugate Gradient (CG) method with Richardson iteration is presented. This scheme is designed for the solution of linear systems of equations with a large sparse symmetric positive definite matrix. The purpose of the CG iterations is to improve an available approximate solution, as well as to determine an interval that contains all, or at least most, of the eigenvalues of the matrix. This interval is used to compute iteration parameters for Richardson iteration. The attraction of the hybrid scheme is that most of the iterations are carried out by the Richardson method, the simplicity of which makes efficient implementation on modern computers possible. Moreover, the hybrid scheme yields, at no additional computational cost, accurate estimates of the extreme eigenvalues of the matrix. Knowledge of these eigenvalues is essential in some applications.Research supported in part by NSF grant DMS-9409422.Research supported in part by NSF grant DMS-9205531.     

Increasing efficiency of inverse iteration

Inverse iteration is simple but not very efficient method for computing few eigenvalues with minimal absolute values and corresponding eigenvectors of a symmetric matrix. The idea is to increase its efficiency by technique similar to multigrid methods used for solving linear systems. This approach is not new, but until now multigrid was mostly used for solving linear system which appear in Rayleigh quotient iteration, inverse iteration and related iterative methods. Instead of choosing appropriate coordinates (grids), our algorithm performs inverse iteration on a sequence of subspaces with decreasing dimensions (multispace). Block Lanczos method is used for the selection of a smaller subspace. This will produce a banded matrix, which makes inverse iteration even faster in the smaller dimensions.      

On algebraic multi-level methods for non-symmetric systems - Comparison results

We establish theoretical comparison results for algebraic multi-level methods applied to non-singular non-symmetric M-matrices. We consider two types of multi-level approximate block factorizations or AMG methods, the AMLI and the MAMLI method. We compare the spectral radii of the iteration matrices of these methods. This comparison shows, that the spectral radius of the MAMLI method is less than or equal to the spectral radius of the AMLI method. Moreover, we establish how the quality of the approximations in the block factorization effects the spectral radii of the iteration matrices. We prove comparisons results for different approximations of the fine grid block as well as for the used Schur complement. We also establish a theoretical comparison between the AMG methods and the classical block Jacobi and block Gauss-Seidel methods.     

A note on the eigenvalues of a special class of matrices

In the analysis of stability of a variant of the Crank-Nicolson (C-N) method for the heat equation on a staggered grid a class of non-symmetric matrices appear that have an interesting property: their eigenvalues are all real and lie within the unit circle. In this note we shall show how this class of matrices is derived from the C-N method and prove that their eigenvalues are inside [−1,1] for all values of m (the order of the matrix) and all values of a positive parameter σ, the stability parameter. As the order of the matrix is general, and the parameter σ lies on the positive real line this class of matrices turns out to be quite general and could be of interest as a test set for eigenvalue solvers, especially as examples of very large matrices.     

A PROCESS FOR SOLVING A FEW EXTREME EIGENPAIRS OF LARGE SPARSE POSITIVE DEFINITE GENERALIZED EIGENVALUE PROBLEM

1. IntroductionWe are concerned in this work with finding a few extreme eigenvalues and theircorresponding eigenvectors of a generalized large scale eigenvalue problem in which thematrices are sparse and symmetric positive definite.Although finding a few extreme eigenpairs is of interest both in theory and practice,there are only few usable and efficient methods up to now. Reinsch and Baner ([12]),suggested a oR algorithm with Newton shift for the standard eigenproblem which included an ingen…     

A mixed method of subspace iteration for Dirichlet eigenvalue problems

A full multigrid scheme was used in computing some eigenvalues of the Laplace eigenvalue problem with the Dirichlet boundary condition. We get a system of algebraic equations with an aid of finite difference method and apply subspace iteration method to the system to compute first some eigenvalues. The result shows that this is very effective in calculating some eigenvalues of this problem.     

A Note on the Calculation of Step-Lengths in Interior-Point Methods for Semidefinite Programming

In each iteration of an interior-point method for semidefinite programming, the maximum step-length that can be taken by the iterate while maintaining the positive semidefiniteness constraint needs to be estimated. In this note, we show how the maximum step-length can be estimated via the Lanczos iteration, a standard iterative method for estimating the extremal eigenvalues of a matrix. We also give a posteriori error bounds for the estimate. Numerical results on the performance of the proposed method against two commonly used methods for calculating step-lengths (backtracking via Cholesky factorizations and exact eigenvalues computations) are included.     

On the analysis of the unsymmetric successive overrelaxation method when applied top-cyclic matrices

Summary A Determinantal Invariance, associated with consistently ordered weakly cyclic matrices, is given. The DI is then used to obtain a new functional equation which relates the eigenvalues of a particular block Jacobi iteration matrix to the eigenvalues of its associated Unsymmetric Successive Overrelaxation (USSOR) iteration matrix. This functional equation as well as the theory of nonnegative matrices and regular splittings are used to obtain convergence and divergence regions of the USSOR method.     

大型不正定陀螺系统本征值问题

陀螺动力系统可以导入哈密顿辛几何体系,在哈密顿陀螺系统的辛子空间迭代法的基础上提出了一种能够有效计算大型不正定哈密顿函数的陀螺系统本征值问题的算法．利用陀螺矩阵既为哈密顿矩阵而本征值又是纯虚数或零的特点,将对应哈密顿函数为负的本征值分离开来,构造出对应哈密顿函数全为正的本征值问题,利用陀螺系统的辛子空间迭代法计算出正定哈密顿矩阵的本征值,从而解决了大型不正定陀螺系统的本征值问题,算例证明,本征解收敛得很快．     

A Simultaneous Iteration Method for the Unsymmetric Eigenvalue Problem

This paper describes a method of obtaining all or a dominantsubset of the eigenvalues and corresponding left and right-handeigenvectors of unsymmetric matrices by simultaneous iteration.The method differs from Bauer's biiteration in that re-orientationof the trial vectors is achieved at each iteration by performingan "interaction analysis".     

The Quasi-Laguerre iteration

The quasi-Laguerre iteration has been successfully established, by the same authors, in the spirit of Laguerre's iteration for solving the eigenvalues of symmetric tridiagonal matrices. The improvement in efficiency over Laguerre's iteration is drastic. This paper supplements the theoretical background of this new iteration, including the proofs of the convergence properties.

     

Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.