论文标题
矩阵函数的高阶Fréchet衍生物的积分表示：正交档位和2级条件编号的新结果
Integral representations for higher-order Fréchet derivatives of matrix functions: Quadrature algorithms and new results on the level-2 condition number
论文作者
论文摘要
我们提出了分析矩阵函数的高阶Fréchet导数$ f（a）$的积分表示，该代码$ f（a）$统一了通用分析矩阵函数的一阶Fréchet衍生物的已知结果和$ a^{ - 1} $的$ a^{-1} $的高阶Fréchet衍生物的一阶Fréchet衍生物。我们重点介绍了此积分表示的两个应用程序：一方面，它允许找到一类大型函数$ f $ $ f $ $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ f $ f $ f（a）$的确切值（即条件编号的条件编号）的确切值。另一方面，它还允许使用数值正交方法近似高阶Fréchet衍生物。我们证明，在某些情况下 - 尤其是当衍生级顺序$ k $中等且衍生物中的方向项具有低级别结构时，所得算法可以优于从文献中较大的差距从文献中建立的方法。
We propose an integral representation for the higher-order Fréchet derivative of analytic matrix functions $f(A)$ which unifies known results for the first-order Fréchet derivative of general analytic matrix functions and for higher-order Fréchet derivatives of $A^{-1}$. We highlight two applications of this integral representation: On the one hand, it allows to find the exact value of the level-2 condition number (i.e., the condition number of the condition number) of $f(A)$ for a large class of functions $f$ when $A$ is Hermitian. On the other hand, it also allows to use numerical quadrature methods to approximate higher-order Fréchet derivatives. We demonstrate that in certain situations -- in particular when the derivative order $k$ is moderate and the direction terms in the derivative have low-rank structure -- the resulting algorithm can outperform established methods from the literature by a large margin.
