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.     

The condition numbers for weighted Moore-Penrose inverse and weighted linear least squares problem

Condition numbers play an important role in numerical analysis. Classical normise condition numbers are used to measure the size of both input perturbations and output errors. In this paper, we study the weighted normwise relative condition numbers for the weighted Moore-Penrose inverse and the weighted linear least-squares (WLS) problems in the case of the full-column rank matrix. The bounds or formulas for the weighted condition numbers are presented. The obtained results can be viewed as extensions of the earlier works studied by others.     

在不平地面上把椅子放稳的充分必要条件

把椅子放在不平的地面上,通常只有三只脚着地,放不稳,然而只需稍挪动几次,就可以使四只脚同时着地、放稳。本文指出,当且仅当椅子的四脚共圆时,才能在一般不平的地面上放稳,并对此建立了数学模型,给出了理论上的证明。     

一类具有广义Lipschitz条件的非线性映象的迭代过程

本文研究了广义Ｌｉｐｓｃｈｉｔｚ强增生映象的Ｉｓｈｉｋａｗａ型迭代和Ｍａｎｎ型迭代过程的收敛性．所得结果统一和扩展了近期相关结果     

动力学参数数目与法方程条件数的关系

多相催化反应动力学研究的基本问题是动力学方程中的参数估计，它一直是动力学研究中的活跃领域·在参数估计中最常用的方法是最小二乘法．早在1947年［刘开始用线性最小二乘法处理动力学模型的数据．1958年在线性最小二乘法中应用置信区间问题［21．1959年MarguardM出非线性最小二乘法问．前人提出如何求非线性最小二乘法的初值［4］，及加权最小二乘法的重要性问，并对最小二乘法在动力学中的应用做了综述卜］．在多相催化动力学研究中，关干线性最小二乘法遇到根本性困难的问题前人未涉及．我们曾在动力学研究中发现线性最小二乘法的法…     

CONSTRUCTION OF A PRECONDITIONER FOR DOMAIN DECOMPOSITION METHODS WITH POLYNOMIAL LAGRANGIAN MULTIPLIERS

1. IntroductionIn recent years the non-overlapping domain decomposition methods (DDMs) with nonmatching grids have attracted particular attention of computional eXPerts and engineers (see[1]--[9]). This kind of DDM allows non--coincidence of nodal points at common edges (or commonfaces) of two neighbouring subdomains. Thus it can be applied to solving the problems ofchanging meshes (for example, the multi-body contact problems in solid mechanics an'd therelative motion problems in oil explo…     

Pullback attractors for the norm-to-weak continuous process and application to the nonautonomous reaction–diffusion equations

In this paper, we introduce the concept of norm-to-weak continuous process in a Banach space, and obtain the existence of pullback attractors for this kind of process. Then we give a new method for proving the existence of the pullback attractors. As an application, we obtain the existence of pullback attractors for nonautonomous reaction–diffusion equation in with exponential growth of the external force.     

Condition measures and properties of the central trajectory of a linear program

Given a data instanced=(A, b, c) of a linear program, we show that certain properties of solutions along the central trajectory of the linear program are inherently related to the condition number C(d) of the data instanced=(A, b, c), where C(d) is a scale-invariant reciprocal of a closely-related measure ρ(d) called the “distance to ill-posedness”. (The distance to ill-posedness essentially measures how close the data instanced=(A,b,c) is to being primal or dual infeasible.) We present lower and upper bounds on sizes of optimal solutions along the central trajectory, and on rates of change of solutions along the central trajectory, as either the barrier parameter μ or the datad=(A, b, c) of the linear program is changed. These bounds are all linear or polynomial functions of certain natural parameters associated with the linear program, namely the condition number C(d), the distance to ill-posedness ρ(d), the norm of the data ‖d‖, and the dimensionsm andn.     

Condition numbers of algebraic Riccati equations in the Frobenius norm

Consider the continuous-time algebraic Riccati equation (CARE) and the discrete-time algebraic Riccati equation (DARE) which arise in linear control and system theory. It is known that appropriate assumptions on the coefficient matrices guarantee the existence and uniqueness of Hermitian positive semidefinite stabilizing solutions. In this note, we apply the theory of condition developed by Rice to define condition numbers of the CARE and DARE in the Frobenius norm, and derive explicit expressions of the condition numbers in a uniform manner. Both the complex case and real case are considered, and connections to certain existing condition numbers of the CARE and DARE are discussed.     

A perturbation analysis of the intrinsic conditioning of an approximate null vector computed with a SVD

Let A be an m ×n real matrix with singular values σ₁ ? ··· ? σ_n?1 ? σ_n ? 0. In cases where σ_n ? 0, the corresponding right singular vector υ_n is a natural choice to use for an approximate null vector ofA. Using an elementary perturbation analysis, we show that κ = σ₁/(σ_n?1 ? σ_n) provides a quantitative measure of the intrinsic conditioning of the computation of υ_n from A.