CONDITION ESTIMATION FOR MATRIX FUNCTIONS VIA THE SCHUR DECOMPOSITION

We show how to cheaply estimate the Frechet derivative and the conditi on 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 fu nction we also give a method to compute the Frechet derivative exactly . We also show that often this general method, based on the Schur deco mposition, when applied the matrix sign function and the matrix expone ntial, enables one to compute the function and estimate its condition number more cheaply than the various special techniques that exploit s pecial properties of these two functions.