On the conditioning of the matrix-matrix exponentiation
Polytechnic Institute of Coimbra/ISEC（科英布拉理工学院/ISEC）；Institute of Systems and Robotics, University of Coimbra（科英布拉大学系统机器人研究所）；Department of Mathematics, Robat Karim Branch, Islamic Azad University（伊斯兰阿扎德大学罗巴特卡里姆分校数学系）
If ${A}$ has no eigenvalues on the closed negative real axis, and $B$ is arbitrary square complex, the matrix-matrix exponentiation is defined as $A^B:=e^{\log({A}){B}}$. This function arises, for instance, in Von Newmann's quantum-mechanical entropy, which in turn finds applications in other areas of science and engineering. Since in general $A$ and $B$ do not commute, this bivariate matrix function may not be a primary matrix function as commonly defined, which raises many challenging issues. In this paper, we revisit this function and derive new related results. Particular emphasis is given to its Fréchet derivative and conditioning. We present a general result on the Fréchet derivative of bivariate matrix functions with applications not only to the matrix-matrix exponentiation but also to other functions, such as the second order Fréchet derivatives and some iteration functions arising in matrix iterative methods. The numerical computation of the Fréchet derivative is discussed and an algorithm for computing the relative condition number of $A^B$ is proposed. Some numerical experiments are included.
Conditioning and backward error of block-symmetric block-tridiagonal linearizations of matrix polynomials
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Universidade de Lisboa（里斯本大学）；Faculdade de Economia do Porto（波尔图经济与管理学院）；Boston University（波士顿大学）
当已知具有n列的矩阵A被基矩阵B_1，.的线性组合很好地近似时，B_p，我们可以将A应用于随机向量并求解线性系以恢复此线性组合。同样的方法也可以用来恢复A^-1的近似值。一个基本的问题是这个线性系是否可逆和良态。本文证明了如果B_j的Gram矩阵是充分良态的，并且每个B_j都有高的数值秩，那么n {proportional} p log^2 n将保证线性系以高概率良态.我们的主要应用是探测具有光滑伪微分符号的线性算子，例如地震成像中的波动方程Hessian。我们数值证明，矩阵探测也可以产生良好的预条件，在可变介质中的反椭圆算子。
Computing the Conditioning of the Components of a Linear Least Squares Solution
CERFACS（法国计算与实验流体力学研究中心）；University of Tennessee（田纳西大学）；Oak Ridge National Laboratory（橡树岭国家实验室）；University of Manchester（曼彻斯特大学）；Centre National d’Etudes Spatiales（法国国家空间研究中心）；University of Colorado at Denver and Health Sciences Center（科罗拉多大学丹佛分校健康科学中心）
