Pseudospectra and delay differential equations

In this paper, we present a new method for computing the pseudospectra of delay differential equations (DDEs) with fixed finite delay. This provides information on the sensitivity of eigenvalues under arbitrary perturbations of a given size, and hence insight into how stability may change under variation of parameters. We also investigate how differently weighted perturbations applied to the individual matrices of the delayed eigenvalue problem affect the pseudospectra. Furthermore, we compute pseudospectra of the infinitesimal generator of the DDE, from which a lower bound on the maximum transient growth can be inferred. To illustrate our method, we consider a DDE modelling a semiconductor laser subject to external feedback.     

Approximated structured pseudospectra

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank‐one or projected rank‐one perturbations of the given matrix is proposed. The choice of rank‐one or projected rank‐one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank‐one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra.     

A note on structured pseudospectra

In this note, we study the notion of structured pseudospectra. We prove that for Toeplitz, circulant, Hankel and symmetric structures, the structured pseudospectrum equals the unstructured pseudospectrum. We show that this is false for Hermitian and skew-Hermitian structures. We generalize the result to pseudospectra of matrix polynomials. Indeed, we prove that the structured pseudospectrum equals the unstructured pseudospectrum for matrix polynomials with Toeplitz, circulant, Hankel and symmetric structures. We conclude by giving a formula for structured pseudospectra of real matrix polynomials. The particular type of perturbations used for these pseudospectra arise in control theory.     

基于伪谱理论分析电力系统的暂态增长

使用伪谱理论进行电力系统的稳定分析,通过计算电力系统状态矩阵的伪谱得到其特征值的敏感性,并利用伪谱横坐标来度量系统的鲁棒性.矩阵指数和暂态仿真给出了敏感特征值与系统暂态增长的关系.     

A fast algorithm for computing the pseudospectra of Toeplitz matrices

The concept of pseudospectra was introduced by Trefethen during the 1990s and became a popular tool to explain the behavior of non-normal matrices. In this paper, we propose a fast algorithm for computing the pseudospectra of Toeplitz matrices by using fast Toeplitz QR factorization. Numerical experiments are given to illustrate the efficiency of the new algorithm.     

Pseudospectra of rectangular matrices

Pseudospectra of rectangular matrices vary continuously withthe matrix entries, a feature that eigenvalues of these matricesdo not have. Some properties of eigenvalues and pseudospectraof rectangular matrices are explored, and an efficient algorithmfor the computation of pseudospectra is proposed. Applicationsare given in (square) eigenvalue computation (Lanczos iteration),square pseudospectra approximation (Arnoldi iteration), controltheory (nearest uncontrollable system) and game theory.     

A matrix lower bound

Four essentially different interpretations of a lower bound for linear operators are shown to be equivalent for matrices (involving inequalities, convex sets, minimax problems, and quotient spaces). Properties stated by von Neumann in a restricted case are satisfied by the lower bound. Applications are made to rank reduction, s-numbers, condition numbers, and pseudospectra. In particular, the matrix lower bound is the distance to the nearest matrix with strictly contained row or column spaces, and it occurs in a condition number formula for any consistent system of linear equations, including those that are underdetermined.     

A note on the essential pseudospectra and application

In this research article, we work with the notion of essential pseudospectra of closed, densely defined linear operators in the Banach space. We start by giving the definition and we investigate the characterization, the stability and some properties of these essential pseudospectra.     

Computing the field of values and pseudospectra using the Lanczos method with continuation

The field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes. To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm. We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices. The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type.This work was supported by Engineering and Physical Sciences Research Council grants GR/H/52139 and GR/H/94528.     

Spectral conditioning and pseudospectral growth

Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii’s analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra. J. V. Burke’s research was supported in part by National Science Foundation Grant DMS-0505712. A. S. Lewis’s research was supported in part by National Science Foundation Grant DMS-0504032. M. L. Overton’s research was supported in part by National Science Foundation Grant DMS-0412049.     

Measures of noncompactness and essential pseudospectra on Banach space

In this research article, we work with the notion of the measures of noncompactness in order to establish some results concerning the essential pseudospectra of closed, densely defined linear operators in the Banach space. We start by giving a refinement of the definition of the essential pseudospectra by means of the measure of noncompactness, and we give sufficient conditions on the perturbed operator to have its invariance. Copyright © 2013 John Wiley & Sons, Ltd.     

Pseudospectra Analysis, Nonlinear Eigenvalue Problems, and Studying Linear Systems with Delays

Time delays occur naturally in many physical and social systems. Computer simulations of such systems require that models of these systems be stable, and perhaps even passive, if several such systems are to be joined together in the simulation. We present a visual procedure for studying the stability and passivity of such systems. This procedure uses ideas from pseudospectra analysis. It is applicable to systems of linear, delay-differential-algebraic equations. There are no a priori restrictions on the types or sizes of the delays. No approximations to the original system are made. All approximations are confined to the grid used in the visualization procedure, and the procedure parallelizes readily. We apply this procedure to the study of the stability and passivity of proposed models for simulations of the behavior of currents and voltages in packaged VLSI interconnects (wires and planes) in computers. Simulations are required to verify that internal electromagnetic fields do not significantly delay or distort circuit signals.This revised version was published online in October 2005 with corrections to the Cover Date.     

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of “good” nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied. This work was partially funded by the Natural Sciences and Engineering Research Council of Canada, and by the MITACS Network of Centres of Excellence.     

Pseudospectra localizations and their applications

Usefulness of spectral localizations in analysis of various matrix properties, such as stability of dynamical systems, has led us to derive a pseudospectra localization technique using the ideas that come from diagonally dominant matrices. In such way, many theoretical and practical applications of pseudospectra (robust stability, transient behavior, nonnormal dynamics, etc.) can be linked with specific relationships between matrix entries. This allows one to understand certain phenomena that occur in practice better, as we show for the realistic model of soil energetic food web. The novelty of the presented results, therefore, lies not only in new mathematical formulations but also in the conceptual sense because it links stability with empirical data and their uncertainty limitations. Copyright © 2015 John Wiley & Sons, Ltd.     

Structured pseudospectra for nonlinear eigenvalue problems

In this paper we introduce structured pseudospectra for nonlinear eigenvalue problems and derive computable formulae. The results are applied to the sensitivity analysis of the eigenvalues of a second-order system arising from structural dynamics and of a time-delay system arising from laser physics. In the former case, a comparison is made with the results obtained in the framework of random eigenvalue problems.     

A distance bound for pseudospectra of matrix polynomials

In this note, we obtain a lower bound for the distance between the pseudospectrum of a matrix polynomial and a given point that lies out of it, generalizing a known result on pseudospectra of matrices.     

Pseudospectra, critical points and multiple eigenvalues of matrix polynomials

We develop a general framework for perturbation analysis of matrix polynomials. More specifically, we show that the normed linear space L_m(C^n×n) of n-by-n matrix polynomials of degree at most m provides a natural framework for perturbation analysis of matrix polynomials in L_m(C^n×n). We present a family of natural norms on the space L_m(C^n×n) and show that the norms on the spaces C^m+1 and C^n×n play a crucial role in the perturbation analysis of matrix polynomials. We define pseudospectra of matrix polynomials in the general framework of the normed space L_m(C^n×n) and show that the pseudospectra of matrix polynomials well known in the literature follow as special cases. We analyze various properties of pseudospectra in the unified framework of the normed space L_m(C^n×n). We analyze critical points of backward errors of approximate eigenvalues of matrix polynomials and show that each critical point is a multiple eigenvalue of an appropriately perturbed polynomial. We show that common boundary points of components of pseudospectra of matrix polynomials are critical points. As a consequence, we show that a solution of Wilkinson’s problem for matrix polynomials can be read off from the pseudospectra of matrix polynomials.     

Spectra,pseudospectra, and localization for random bidiagonal matrices

There has been much recent interest, initiated by work of the physicists Hatano and Nelson, in the eigenvalues of certain random, non‐Hermitian, periodic tridiagonal matrices and their bidiagonal limits. These eigenvalues cluster along a “bubble with wings” in the complex plane, and the corresponding eigenvectors are localized in the wings, delocalized in the bubble. Here, in addition to eigenvalues, pseudospectra are analyzed, making it possible to treat the nonperiodic analogues of these random matrix problems. Inside the bubble, the resolvent norm grows exponentially with the dimension. Outside, it grows subexponentially in a bounded region that is the spectrum of the infinite‐dimensional operator. Localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase, respectively, with distance from the diagonal. This article presents theorems that characterize the spectra, pseudospectra, and numerical range for the four cases of finite bidiagonal matrices, infinite bidiagonal matrices (“stochastic Toeplitz operators”), finite periodic matrices, and doubly infinite bidiagonal matrices (“stochastic Laurent operators”). © 2001 John Wiley & Sons, Inc.     

Pseudospectra do not determine norm behavior, even for matrices with only simple eigenvalues

We present an example of a pair of 4×4 matrices having identical pseudospectra but whose squares have different norms. The novelty of the example lies in the fact that the matrices in question have only simple eigenvalues.