Statistical condition estimation for linear systems

The standard approach to measuring the condition of a linear system compresses all sensitivity information into one number. Thus a loss of information can occur in situations in which the standard condition number with respect to inversion does not accurately reflect the actual sensitivity of a solution or particular entries of a solution. It is shown that a new method for estimating the sensitivity of linear systems addresses these difficulties. The new procedure measures the effects on the solution of small random changes in the input data and, by properly scaling the results, obtains reliable condition estimates for each entry of the computed solution. Moreover, this approach, which is referred to as small-sample statistical condition estimation, is no more costly than the standard 1-norm or power method 2-norm condition estimates, and it has the advantage of considerable flexibility. For example, it easily accommodates restrictions on, or structure associated with, allowable perturbations. The method also has a rigorous statistical theory available for the probability of accuracy of the condition estimates. However, it gives no estimate of an approximate null vector for nearly singular systems. The theory of this approach is discussed along with several illustrative examples.