论文标题
Crouzeix-Palencia结果的双变量扩展，并应用于矩阵函数的Fréchet衍生物
A bivariate extension of the Crouzeix-Palencia result with an application to Fréchet derivatives of matrix functions
论文作者
论文摘要
Crouzeix和Palencia的结果指出，矩阵函数$ f（a）$的频谱规范由$ k = 1+ \ sqrt {2} $ timple $ w（a）$上的$ f $的最大值，$ a $ a $的数值范围。这项工作的目的是指出，该结果扩展到了双变量矩阵函数的一定概念。 $ f \ {a，b \} $的频谱规范由$ k^2 $限制$ w（a）\ times w（b）$的最大$ f $。作为一种特殊情况，$ f（a）$的fréchet导数的光谱规范由$ f^\ prime $ w（a）$的最大$ k^2 $限制。讨论了对某些Krylov子空间方法的收敛分析的应用，并讨论了两个以上变量的功能扩展。
A result by Crouzeix and Palencia states that the spectral norm of a matrix function $f(A)$ is bounded by $K = 1+\sqrt{2}$ times the maximum of $f$ on $W(A)$, the numerical range of $A$. The purpose of this work is to point out that this result extends to a certain notion of bivariate matrix functions; the spectral norm of $f\{A,B\}$ is bounded by $K^2$ times the maximum of $f$ on $W(A)\times W(B)$. As a special case, it follows that the spectral norm of the Fréchet derivative of $f(A)$ is bounded by $K^2$ times the maximum of $f^\prime$ on $W(A)$. An application to the convergence analysis of certain Krylov subspace methods and the extension to functions in more than two variables are discussed.
