一类二次矩阵方程的条件数和后向误差

主要讨论一类二次矩阵方程X^2-EX-F=0的条件数和后向误差,其中E是一个对角矩阵,F是一个M矩阵.这类二次矩阵方程来源于Markov链的噪声Wiener-Hopf问题.实际问题中人们感兴趣的是它的M矩阵的解.应用Rice创立的基于Frobenius范数下的条件数理论,导出此类二次矩阵方程的M矩阵解的条件数的显式表达式.同时,也给出近似解的后向误差的定义以及一个可计算的表达式.最后,通过数值例子验证理论结果是有效的.     

Condition numbers and backward perturbation bound for linear matrix equations

This paper deals with the normwise perturbation theory for linear (Hermitian) matrix equations. The definition of condition number for the linear (Hermitian) matrix equations is presented. The lower and upper bounds for the condition number are derived. The estimation for the optimal backward perturbation bound for the Hermitian matrix equations is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

A unified framework for the numerical solution of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: N.karcanias{at}city.ac.uk^**** Email: ixp{at}le.ac.uk Algebraic quadratic equations are special cases of a singlegeneralized algebraic quadratic matrix equation (GQME). Thispaper focuses on the numerical solution of the GQME using probability-1homotopy methods. A synoptic review of these methods and theirapplication to algebraic matrix equations is provided as background.A large variety of analysis and design problems in systems andcontrol are reported as special cases of the presented frameworkand some of them are illustrated via numerical examples fromthe literature.     

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h²). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.     

Backward errors for eigenvalues and eigenvectors of Hermitian,skew-Hermitian,H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.     

Reflexive periodic solutions of general periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The objective of this paper is to provide four new iterative methods based on the conjugate gradient normal equation error (CGNE), conjugate gradient normal equation residual (CGNR), and least‐squares QR factorization (LSQR) algorithms to find the reflexive periodic solutions (X₁,Y₁,X₂,Y₂,…,X_σ,Y_σ) of the general periodic matrix equations for i = 1,2,…,σ. The iterative methods are guaranteed to converge in a finite number of steps in the absence of round‐off errors. Finally, some numerical results are performed to illustrate the efficiency and feasibility of new methods.     

Stability analysis of a general toeplitz systems solver

We show that a fast algorithm for theQR factorization of a Toeplitz or Hankel matrixA is weakly stable in the sense thatR ^T R is close toA ^T A. Thus, when the algorithm is used to solve the semi-normal equationsR ^TRx=A^T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear systemAx=b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min ||Ax-b||₂.     

A class of one-parameter alternating direction implicit methods for two-dimensional wave equations with discrete and distributed time-variable delays

For solving the initial-boundary value problem of two-dimensional wave equations with discrete and distributed time-variable delays, in the present paper, we first construct a class of basic one-parameter methods. In order to raise the computational efficiency of this class methods, we remold the methods as one-parameter alternating direction implicit (ADI) methods. Under the suitable conditions, the remolded methods are proved to be stable and convergent of second order in both of time and space. With several numerical experiments, the computational effectiveness and theoretical exactness of the methods are confirmed. Moreover, it is illustrated that the proposed one-parameter ADI method has the better advantage in computational efficiency than the basic one-parameter methods.     

Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix

^** Email: mengi{at}cs.nyu.edu^*** Email: overton{at}cs.nyu.edu Two useful measures of the robust stability of the discrete-timedynamical system x_k+1 = Ax_k are the -pseudospectral radius andthe numerical radius of A. The -pseudospectral radius of A isthe largest of the moduli of the points in the -pseudospectrumof A, while the numerical radius is the largest of the moduliof the points in the field of values. We present globally convergentalgorithms for computing the -pseudospectral radius and thenumerical radius. For the former algorithm, we discuss conditionsunder which it is quadratically convergent and provide a detailedaccuracy analysis giving conditions under which the algorithmis backward stable. The algorithms are inspired by methods ofByers, Boyd–Balakrishnan, He–Watson and Burke–Lewis–Overtonfor related problems and depend on computing eigenvalues ofsymplectic pencils and Hamiltonian matrices.     

The Equality Constrained Indefinite Least Squares Problem: Theory and Algorithms

We present theory and algorithms for the equality constrained indefinite least squares problem, which requires minimization of an indefinite quadratic form subject to a linear equality constraint. A generalized hyperbolic QR factorization is introduced and used in the derivation of perturbation bounds and to construct a numerical method. An alternative method is obtained by employing a generalized QR factorization in combination with a Cholesky factorization. Rounding error analysis is given to show that both methods have satisfactory numerical stability properties and numerical experiments are given for illustration. This work builds on recent work on the unconstrained indefinite least squares problem by Chandrasekaran, Gu, and Sayed and by the present authors.     

Analysis of Error Growth Via Stability Regions in Numerical Initial Value Problems

This paper concerns the stability analysis of numerical methods for solving time dependent ordinary and partial differential equations. In the literature stability estimates for such methods were derived, under a condition which can be viewed as a transplantation of the Kreiss resolvent condition (from the unit disk to the stability region S of the numerical method). These estimates tell us that errors in the numerical time stepping process cannot grow faster than linearly with min{s,n}. Here n denotes the number of time steps, and s stands for the order of the (spatial discretization) matrices involved.In this paper we address the natural question of whether the above stability estimates can be improved so as to imply an error growth at a slower rate than min{s,n} (when n, s). Our results concerning this question are as follows: (a) for all (practical) Runge–Kutta and other one-step formulas, we show that the estimates from the literature are sharp in that error growth at the rate min{s,n} can actually occur, (b) for linear multistep formulas we find that, rather surprisingly, some of the stability estimates can substantially be improved and extended, whereas others are sharp.The results proved in this paper are also relevant to (suitably scaled spatial discretization) matrices whose -pseudo-eigenvalues lie at a distance not exceeding K from the stability region S of the time stepping method, for all >0 and fixed constant K.