Stochastic Conditioning of Matrix Functions

We investigate the sensitivity of matrix functions to random noise in their input. We propose the notion of a stochastic condition number, which determines, to first order, the sensitivity of a matrix function to random noise. We derive an upper bound on the stochastic condition number that can be estimated efficiently by using “small-sample" estimation techniques. The bound can be used to estimate the median, or any other quantile, of the error in a function's output when its input is subjected to random noise. We give numerical experiments illustrating the effectiveness of our stochastic error estimate.