Small-Sample Statistical Condition Estimates for General Matrix Functions

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Abstract
A new condition estimation procedure for general matrix functions is presented that accurately gauges sensitivity by measuring the effect of random perturbations at the point of evaluation. In this procedure 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 broad range of problems in which the function maps between spaces of different dimensions. This is in sharp contrast to the more common power method condition estimation procedure that is limited, in the usual case where the Frechet derivative is known only implicitly, to maps between spaces of equal dimension. 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 componentwise condition estimates.

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