使用复杂的步骤方法,使自地组群矩阵功能的Fréchet衍生物接近于Fréchet衍生物 (On using the complex step method for the approximation of Fréchet derivatives of matrix functions in automorphism groups)

We show, that the complex step approximation $\mathrm{Im}(f(A+ihE))/h$ to the Fr\'echet derivative of matrix functions $f:\mathbb{R}^{m,n}\rightarrow\mathbb{R}^{m,n}$ is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the matrix sign using Newtons method, we extend the research to the family of Pad\'e iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure.

翻译：我们显示, 复杂的步骤近似 $\ mathrm{ Im} (f( A+iHE) / h$ 到矩阵函数的 Fr\ echet 衍生物$f:\mathbb{R ⁇ m,n ⁇ rightror\mathbb{R ⁇ m,n}$f:\\ m,n ⁇ rightrr\mathb{R ⁇ m,n} 适用于使用迭接机制的矩阵符号、平根和极地绘图。虽然此属性已经被发现用于使用牛顿方法的矩阵符号, 我们将研究扩展至 Pad\ e 迭代函数的家族, 从而允许我们引入迭接机制来查找函数和衍生物值, 同时大致保护自体结构。