The tensor t-function, a formalism that generalizes the well-known concept of matrix functions to third-order tensors, is introduced in [K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numer. Linear Algebra Appl. 27 (3), e2288]. In this work, we investigate properties of the Fr\'echet derivative of the tensor t-function and derive algorithms for its efficient numerical computation. Applications in condition number estimation and nuclear norm minimization are explored. Numerical experiments implemented by the \texttt{t-Frechet} toolbox hosted at \url{https://gitlab.com/katlund/t-frechet} illustrate properties of the t-function Fr\'echet derivative, as well as the efficiency and accuracy of the proposed algorithms.

翻译：张量t-函数是一种将矩阵函数的概念推广到三阶张量的形式化方法，该概念由[K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numer. Linear Algebra Appl. 27 (3), e2288]提出。本文研究了张量t-函数的Fréchet导数的性质，并推导了其高效数值计算算法。同时探讨了在条件数估计和核范数最小化中的应用。基于托管在\url{https://gitlab.com/katlund/t-frechet}的\texttt{t-Frechet}工具箱实现的数值实验，验证了t-函数Fréchet导数的特性，以及所提算法的效率与准确性。