On the Computation of Particular Eigenvectors of Hamiltonian Matrix Pencils

We discuss a structure-preserving algorithm for the accurate solution of generalized eigenvalue problems for skew-Hamiltonian/Hamiltonian matrix pencils λN − ℋ. By embedding the matrix pencil λ𝒩 − ℋ into a skew-Hamiltonian/Hamiltonian matrix pencil of double size it is possible to avoid the problem of non-existence of a structured Schur form. For these embedded matrix pencils we can compute a particular condensed form to accurately compute the simple, finite, purely imaginary eigenvalues of λ𝒩 − ℋ. In this paper we describe a new method to compute also the corresponding eigenvectors by using the information contained in the condensed form of the embedded matrix pencils and associated transformation matrices. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Newton-based method for the calculation of the distance to instability

In this paper, a new fast algorithm for the computation of the distance of a stable matrix to the unstable matrices is provided. The method uses Newton’s method to find a two-dimensional Jordan block corresponding to a pure imaginary eigenvalue in a certain two-parameter Hamiltonian eigenvalue problem introduced by Byers [R. Byers, A bisection method for measuring the distance of a stable matrix to the unstable matrices, SIAM J. Sci. Statist. Comput. 9 (1988) 875-881]. This local method is augmented by a test step, previously used by other authors, to produce a global method. Numerical results are presented for several examples and comparison is made with the methods of Boyd and Balakrishnan [S. Boyd, V. Balakrishnan, A regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its L_∞-norm, Systems Control Lett. 15 (1990) 1-7] and He and Watson [C. He, G.A. Watson, An algorithm for computing the distance to instability, SIAM J. Matrix Anal. Appl. 20 (1999) 101-116].     

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．     

On the quadratic two-parameter eigenvalue problem and its linearization

We introduce the quadratic two-parameter eigenvalue problem and linearize it as a singular two-parameter eigenvalue problem. This, together with an example from model updating, shows the need for numerical methods for singular two-parameter eigenvalue problems and for a better understanding of such problems.There are various numerical methods for two-parameter eigenvalue problems, but only few for nonsingular ones. We present a method that can be applied to singular two-parameter eigenvalue problems including the linearization of the quadratic two-parameter eigenvalue problem. It is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils.     

Stability of equilibrium solutions of Hamiltonian systems under the presence of a single resonance in the non-diagonalizable case

The problem of knowing the stability of one equilibrium solution of an analytic autonomous Hamiltonian system in a neighborhood of the equilibrium point in the case where all eigenvalues are pure imaginary and the matrix of the linearized system is non-diagonalizable is considered. We give information about the stability of the equilibrium solution of Hamiltonian systems with two degrees of freedom in the critical case. We make a partial generalization of the results to Hamiltonian systems with n degrees of freedom, in particular, this generalization includes those in [1].      

Local finitely smooth equivalence of real autonomous systems with two pure imaginary eigenvalues

The paper deals with real autonomous systems of ordinary differential equations in a neighborhood of a nondegenerate singular point such that the matrix of the linearized system has two pure imaginary eigenvalues, all other eigenvalues lying outside the imaginary axis. It is proved that, for such systems having a focus on the center manifold, the problem of finitely smooth equivalence is solved in terms of the finite segments of the Taylor series of their right-hand sides.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

Note on structured indefinite perturbations to Hermitian matrices

In a recent paper, Overton and Van Dooren have considered structured indefinite perturbations to a given Hermitian matrix. We extend their results to skew-Hermitian, Hamiltonian and skew-Hamiltonian matrices. As an application, we give a formula for computation of the smallest perturbation with a special structure, which makes a given Hamiltonian matrix own a purely imaginary eigenvalue.     

Finitely smooth local equivalence of autonomous systems with one zero root

In this paper, in a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has one zero eigenvalue, while the other eigenvalues lie outside the imaginary axis. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.     

Sign patterns that require real, nonreal or pure imaginary eigenvalues

We characterize the n-by-n sign pattern matrices that require all real, all nonreal, and all pure imaginary eigenvalues. Characterization of sign patterns that allow a real eigenvalue and those that allow a nonreal eigenvalue then follow. Some related specialized results and a characterization of sign patterns that allow a positive real eigenvalue are included.     

用随机奇异值分解算法求解矩阵恢复问题

本文研究了大型低秩矩阵恢复问题.利用随机奇异值分解（RSVD）算法,对稀疏矩阵做奇异值分解.该算法与Lanczos方法相比,在误差精度一致的同时运算时间大大降低,且该算法对相对低秩矩阵也有效.     

On the pseudonormal form of real autonomous systems with two pure imaginary eigenvalues

The paper deals with real autonomous systems of ordinary differential equations in a neighborhood of a nondegenerate singular point such that the matrix of the linearized system has two pure imaginary eigenvalues, all other eigenvalues lying outside the imaginary axis. The reducibility of such systems to pseudonormal form is studied. The notion of resonance is refined, and the notions of removable and irremovable resonances are introduced.     

A Quadratic Eigenproblem in the Analysis of a Time Delay System

In this work we solve a quadratic eigenvalue problem occurring in a method to compute the set of delays of a linear time delay system (TDS) such that the system has an imaginary eigenvalue. The computationally dominating part of the method is to find all eigenvalues z of modulus one of the quadratic eigenvalue problem where φ ₁, …, φ _{m –1} ∈ ℝ are free parameters and u a vectorization of a Hermitian rank one matrix. Because of its origin in the vectorization of a Lyapunov type matrix equation, the quadratic eigenvalue problem is, even for moderate size problems, of very large size. We show one way to treat this problem by exploiting the Lyapunov type structure of the quadratic eigenvalue problem when constructing an iterative solver. More precisely, we show that the shift-invert operation for the companion form of the quadratic eigenvalue problem can be efficiently computed by solving a Sylvester equation. The usefulness of this exploitation is demonstrated with an example. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sign patterns that require real,nonreal or pure imaginary eigenvalues

We characterize the n-by-n sign pattern matrices that require all real, all nonreal, and all pure imaginary eigenvalues. Characterization of sign patterns that allow a real eigenvalue and those that allow a nonreal eigenvalue then follow. Some related specialized results and a characterization of sign patterns that allow a positive real eigenvalue are included.     

Detecting Codimension Two Bifurcations with a Pure Imaginary Pair and a Simple Zero Eigenvalue

An extended system for codimension two bifurcation with a pure imaginary pair and a simple zero eigenvalue is proposed. Its regularity is proved. An efficient algorithm for solving the extended system is constructed. Finally, some results on the axial dispersion problem in a tubular non-adiabatic reactor is given.     

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.     

The calculation of the distance to a nearby defective matrix

The distance of a matrix to a nearby defective matrix is an important classical problem in numerical linear algebra, as it determines how sensitive or ill‐conditioned an eigenvalue decomposition of a matrix is. The concept has been discussed throughout the history of numerical linear algebra, and the problem of computing the nearest defective matrix first appeared in Wilkinsons famous book on the algebraic eigenvalue problem. In this paper, a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam and Bora introduced in (2005) and reduces to finding when a parameter‐dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the implicit determinant method introduced by Spence and Poulton in (2005). Numerical results for several examples illustrate the performance of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

The stability of periodic Hamiltonian systems with multiple fourth-order resonance

The problem of the stability of the periodic motion of a non-linear periodic Hamiltonian system is considered in the case of pure imaginary characteristic exponents which also satisfy several fourth-order resonance conditions. Conditions for stability and instability are formulated based on terms of the third order inclusive. Some conclusions generalize results obtained previously [1].     

Locating a nearest matrix with an eigenvalue of prespecified algebraic multiplicity

The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization problems. In particular the numerical technique is applicable to Malyshev’s formula to compute the Wilkinson distance as well as to retrieve a nearest matrix with a multiple eigenvalue.     

广义块Toeplitz特征值问题的基于sine变换的预处理子

在求块Toeplitz矩阵束（Amn，Bmn）特征值的Lanczos过程中，通过对移位块Toepltz矩阵Amn-ρBmn进行基于sine变换的块预处理，从而改进了位移块Toeplitz矩阵的谱分布，加速了Lanczos过程的收敛速度．该块预处理方法能通过快速算法有效快速执行．本文证明了预处理后Lanczos过程收敛迅速，并通过实验证明该算法求解大规模矩阵问题尤其有效．