Condition Estimation for Matrix Functions via the Schur Decomposition

We show how to cheaply estimate the Frechet derivative and the condition number for a general class of matrix functions (the class includes the matrix sign function and functions that can be expressed as power series) via the Schur decomposition. In the case of the matrix sign function we also give a method to compute the Frechet derivative exactly. We also show that often this general method, based on the Schur decomposition, when applied the matrix sign function and the matrix exponential, enables one to compute the function and estimate its condition number more cheaply than the various special techniques that exploit special properties of these two functions.