Structured level-2 condition numbers of matrix functions

Abstract

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 present a novel upper bound on the structured level-2 condition number, focusing on perturbation matrices within 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.