Matrix functions play an increasingly important role in many areas of scientific computing and engineering disciplines. In such real-world applications, algorithms working in floating-point arithmetic are used for computing matrix functions and additionally, input data might be unreliable, e.g., due to measurement errors. Therefore, it is crucial to understand the sensitivity of matrix functions to perturbations, which is measured by condition numbers. However, the condition number itself might not be computed exactly as well due to round-off and errors in the input. The sensitivity of the condition number is measured by the so-called level-2 condition number. For the usual (level-1) condition number, it is well-known that structured condition numbers (i.e., where only perturbations are taken into account that preserve the structure of the input matrix) might be much smaller than unstructured ones, which, e.g., suggests that structure-preserving algorithms for matrix functions might yield much more accurate results than general-purpose algorithms. In this work, we examine structured level-2 condition numbers in the particular case of restricting the perturbation matrix to an automorphism group, a Lie or Jordan algebra or the space of quasi-triangular matrices. In numerical experiments, we then compare the unstructured level-2 condition number with the structured one for some specific matrix functions such as the matrix logarithm, matrix square root, and matrix exponential.

翻译：矩阵函数在科学计算与工程学科的众多领域中发挥着日益重要的作用。在这些实际应用中，基于浮点运算的算法被用于计算矩阵函数，此外，输入数据可能由于测量误差等原因而不可靠。因此，理解矩阵函数对扰动的敏感性至关重要，这通过条件数来衡量。然而，由于舍入误差和输入误差，条件数本身也可能无法精确计算。条件数的敏感性由所谓的二级条件数来衡量。对于通常的（一级）条件数，众所周知，结构化条件数（即仅考虑保持输入矩阵结构的扰动）可能远小于非结构化条件数，这表明保持结构的矩阵函数算法可能比通用算法产生更精确的结果。本文中，我们特别考虑扰动矩阵被限制为自同构群、李代数或若尔当代数、或拟三角矩阵空间的情况，研究结构化二级条件数。在数值实验中，我们针对某些特定矩阵函数（如矩阵对数、矩阵平方根和矩阵指数），比较了非结构化二级条件数与结构化二级条件数。