The computation of off-diagonal blocks of matrix functions $f(T)$, where $T$ is block triangular, poses a challenging problem in scientific computing. We present a novel algorithm that exploits the structure of block triangular matrices, generalizing the algorithm of Al-Mohy and Higham for computing the Fr\'echet derivative of the matrix exponential. This work has significant applications in fields such as exponential integrators for solving systems of first-order differential equations, Hamiltonian linear systems in control theory, and option pricing in finance. Our approach introduces a linear operator that maps off-diagonal blocks of $T$ into their counterparts in $f(T)$. By studying the algebraic properties of the operator, we establish a comprehensive computational framework, paving the way to extend existing Fr\'echet derivative algorithms of matrix functions to more general settings. For the matrix exponential, in particular, the algorithm employs the scaling and squaring method with diagonal Pad\'e approximants to $\exp(x)$, with parameters chosen based on a rigorous backward error analysis, which notably does not depend on the norm of the off-diagonal blocks. The numerical experiment demonstrates that our algorithm surpasses existing algorithms in terms of accuracy and efficiency, making it highly valuable for a wide range of applications.

翻译：块三角矩阵函数 $f(T)$ 的非对角块计算是科学计算中的一个具有挑战性的问题。我们提出了一种新颖的算法，该算法利用块三角矩阵的结构，推广了 Al-Mohy 和 Higham 用于计算矩阵指数 Fréchet 导数的算法。这项工作在诸多领域具有重要应用，例如求解一阶微分方程组的指数积分器、控制理论中的哈密顿线性系统以及金融领域的期权定价。我们的方法引入了一个线性算子，该算子将 $T$ 的非对角块映射到 $f(T)$ 中的对应块。通过研究该算子的代数性质，我们建立了一个全面的计算框架，为将现有的矩阵函数 Fréchet 导数算法推广到更一般的场景铺平了道路。特别地，对于矩阵指数，该算法采用了基于对角 Padé 逼近 $\exp(x)$ 的缩放与平方方法，其参数选择基于严格的后向误差分析，且显著地不依赖于非对角块的范数。数值实验表明，我们的算法在精度和效率方面均超越了现有算法，使其在广泛的应用中具有极高的价值。