The optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm

We obtain the optimal perturbation bounds of the Moore-Penrose inverse under the Frobenius norm by using Singular Value Decomposition, which improved the results in the earlier paper [P.-Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13 (1973) 217-232]. In addition, a perturbation bound of the Moore-Penrose inverse under the Frobenius norm in the case of the multiplicative perturbation model is also given.     

Some New Perturbation Bounds for Subunitary Polar Factors

In this paper, we present some new perturbation bounds for subunitary polar factors in a special unitarily invariant norm called a Q-norm. Some recent results in the Frobenius norm and the spectral norm are extended to the Q-norm on one hand. On the other hand we also present some relative perturbation bounds for subunitary polar factors.     

关于极分解和广义极分解的一些新结果

本文研究极分解和广义极分解.孙和陈提出的Frobenius范数下的逼近定理被推广至任何酉不变范数情形.得到了次酉极因子的一个新的表达式.通过新的表达式,我们得到了次酉极因子在任何酉不变范数下的扰动界.最后,讨论了数值计算方法.     

求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法

该文建立了求矩阵方程AXB+CXD=F的中心对称最小二乘解的迭代算法.使用该算法不仅可以判断该矩阵方程的中心对称解的存在性,而且无论中心对称解是否存在,都能够在有限步迭代计算之后得到中心对称最小二乘解.选取特殊的初始矩阵时,可求得极小范数中心对称最小二乘解.同时,也能给出指定矩阵的最佳逼近中心对称矩阵.     

Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.     

An optimal perturbation bound

Cai and Zhang establish separate perturbation bounds for distances with spectral and Frobenius norms (Cai T, Zhang A. Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics. The Annals of Statistics. 2018; Vol. 46, No. 1: 60?89). We extend their theorem to each unitarily invariant norm. It turns out that our estimation is optimal as well.     

LSQR iterative method for generalized coupled Sylvester matrix equations

An iterative method is proposed to solve generalized coupled Sylvester matrix equations, based on a matrix form of the least-squares QR-factorization (LSQR) algorithm. By this iterative method on the selection of special initial matrices, we can obtain the minimum Frobenius norm solutions or the minimum Frobenius norm least-squares solutions over some constrained matrices, such as symmetric, generalized bisymmetric and (R, S)-symmetric matrices. Meanwhile, the optimal approximate solutions to the given matrices can be derived by solving the corresponding new generalized coupled Sylvester matrix equations. Finally, numerical examples are given to illustrate the effectiveness of the present method.     

矩阵广义逆新的扰动上界

利用矩阵的奇异值分解方法,研究了矩阵广义逆的扰动上界,得到了在F-范数下矩阵广义逆的扰动上界定理,所得定理推广并彻底改进了近期的相关结果.相应的数值算例验证了定理的有效性.     

Residual bounds of approximate solutions of the algebraic Riccati equation

Summary. Let approximate the unique Hermitian positive semi-definite solution to the algebraic Riccati equation (ARE) where , is stabilizable, and is detectable. Let be the residual of the ARE with respect to , and define the linear operator by By applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution , we obtained the following result: If is stable, and if for any unitarily invariant norm , then Received April 28, 1995 / Revised version received August 30, 1995     

Condition number related to the outer inverse of a complex matrix

In this paper we obtain the formula for computing the condition number of a complex matrix, which is related to the outer generalized inverse of a given matrix. We use the Schur decomposition of a matrix. We characterize the spectral norm and the Frobenius norm of the relative condition number of the generalized inverse, and the level-2 condition number of the generalized inverse. The sensitivity for the generalized Drazin-inverse solution of linear systems is presented. We also present the structured perturbation of the generalized inverse.     

An optimal bound for the spectral variation of two matrices

We establish a bound for the spectral variation of two complex n × n matrices A,B in terms of ∥A∥, ∥B∥, and ∥A ? B∥. Here ∥ ∥ denotes the spectral norm. It is always better than a bound previously given by Bhatia and Friedland, and it is optimal. We describe the set of pairs A,B for which the bound is attained.     

A note on the normwise perturbation theory for the regular generalized eigenproblem

In this paper, we present a normwise perturbation theory for the regular generalized eigenproblem Ax = λBx, when λ is a semi-simple and finite eigenvalue, which departs from the classical analysis with the chordal norm [9]. A backward error and a condition number are derived for a choice of flexible measure to represent independent perturbations in the matrices A and B. The concept of optimal backward error associated with an eigenvalue only is also discussed. © 1998 John Wiley & Sons, Ltd.     

Towards backward perturbation bounds for approximate dual Krylov subspaces

Given a matrix A,n by n, and two subspaces K and L of dimension m, we consider how to determine a backward perturbation E whose norm is as small as possible, such that k and L are Krylov subspaces of A+E and its adjoint, respectively. We first focus on determining a perturbation matrix for a given pair of biorthonormal bases, and then take into account how to choose an appropriate biorthonormal pair and express the Krylov residuals as a perturbation of the matrix A. Specifically, the perturbation matrix is globally optimal when A is Hermitian and K=L. The results show that the norm of the perturbation matrix can be assessed by using the norms of the Krylov residuals and those of the biorthonormal bases. Numerical experiments illustrate the efficiency of our strategy.     

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.     

On the stability of large matrices

The distance rstab(A) of a stable matrix A to the set of unstable matrices and the norm of the exponential of matrices constitute two important topics in stability theory. We treat in this note the case of large matrices. The method proposed partitions the matrix into two blocks: a small block in which the stability is studied and a large block whose field of values is located in the complex plane. Using the information on the blocks and some results on perturbation theory, we give sufficient conditions for the stability of the original matrix, a lower bound of rstab(A) and an upper bound on the norm of the exponential of A. We illustrate these theoretical bounds on a practical test problem.     

A new bound for the distance to singularity of a regular matrix pencil

For regular matrix pencils 𝒜(s) = sE − A the distance to the nearest singular pencil in the Frobenius norm of the coefficients is called the distance to singularity. We derive a new lower bound for this distance by using the spectral theory of tridiagonal Toeplitz matrices. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

矩阵方程AXA^T＋BYB^T=C的双对称最小二乘解及其最佳逼近

基于共轭梯度法的思想,通过特殊的变形,建立了一类求矩阵方程AXA^T＋BYB^T=C的双对称最小二乘解的迭代算法．对任意的初始双对称矩阵．在没有舍人误差的情况下,经过有限步迭代得到它的双对称最小二乘解;在选取特殊的初始双对称矩阵时,能得到它的的极小范数双对称最小二乘解．另外,给定任意矩阵,利用此方法可得到它的最佳逼近双对称解,数值例子表明,这种方法是有效的．     

THE STRUCTURE SENSITIVITY ANALYSIS FOR TRIANGULAR SYSTEMS

Triangular systems play a fundamental role in matrix computations. It has become commonplace that triangular systems are solved to be more accurate even if they are ill-conditioned. In this paper, we define structured condition number and give structured (forward) perturbation bound. In addition, we derive the representation of optimal structured backward perturbation bound.     

矩阵Frobenius范数不等式

1 引言与引理 矩阵范数与矩阵奇异值问题是数值代数的重要课题,并在矩阵扰动分析,数值计算等分支中起着重要作用.国内外学者对此已作了大量研究.     

A minimum norm approach for low-rank approximations of a matrix

The problems of calculating a dominant eigenvector or a dominant pair of singular vectors, arise in several large scale matrix computations. In this paper we propose a minimum norm approach for solving these problems. Given a matrix, A, the new method computes a rank-one matrix that is nearest to A, regarding the Frobenius matrix norm. This formulation paves the way for effective minimization techniques. The methods proposed in this paper illustrate the usefulness of this idea. The basic iteration is similar to that of the power method, but the rate of convergence is considerably faster. Numerical experiments are included.