A statistical approach to condition estimation

A new approach is presented for estimating the conditioning of general matrix functions by measuring the effect of random perturbations at the point of evaluation. This method is efficient in the sense that the number of extra function evaluations used to evaluate the condition estimate determines the order of the estimate. That is, the probability that the estimate is off by a given factor is inversely proportional to the factor raised to the order of the method. The “transpose-free” nature of this new method allows it to be applied to a much broader range of problems than the commonly used power method of condition estimation. A group of examples illustrates the flexibility of the new estimation procedure in handling a variety of problems and types of sensitivity estimates, such as mixed and component wise condition estimates. Short MATLAB routines are included to demonstrate the ease with which the new condition method can be implemented in a general setting