Crouzeix-Palestinian结果的二元推广及其对矩阵函数Fréchet导数的应用
A bivariate extension of the Crouzeix-Palencia result with an application to Fréchet derivatives of matrix functions
摘要：
Crouzeix和Palestinian的一个结果表明，矩阵函数$f(A)$的谱范数由$K = 1+\sqrt{2}$乘以$W(A)$上的$f$的最大值（$A$的数值域）来界定。这项工作的目的是指出，这一结果扩展到一定的概念，二元矩阵函数的谱范数的$f\{A,B\}$是有界的$K^2$倍的最大值的$f$上$W(A)\times W(B)$。作为一个特殊情况，它遵循$f(A)$的Fréchet导数的谱范数由$K^2$乘以$W(A)$上的$f^\prime$的最大值来界定。应用某些Krylov子空间方法的收敛性分析和扩展到功能在两个以上的变量进行了讨论。
英文摘要：
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.
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