STATISTICAL CONDITION ESTIMATION FOR LINEAR-SYSTEMS

The standard approach to measuring the condition of a linear system co mpresses all sensitivity information into one number. Thus a loss of i nformation can occur in situations in which the standard condition num ber with respect to inversion does not accurately re ect the actual se nsitivity of a solution or particular entries of a solution. It is sho wn that a new method for estimating the sensitivity of linear systems addresses these difficulties. The new procedure measures the effects o n the solution of small random changes in the input data and, by prope rly scaling the results, obtains reliable condition estimates for each entry of the computed solution. Moreover, this approach, which is ref erred to as small-sample statistical condition estimation, is no more costly than the standard 1-norm or power method 2-norm condition estim ates, and it has the advantage of considerable flexibility. For exampl e, it easily accommodates restrictions on, or structure associated wit h, allowable perturbations. The method also has a rigorous statistical theory available for the probability of accuracy of the condition est imates. However, it gives no estimate of an approximate null vector fo r nearly singular systems. The theory of this approach is discussed al ong with several illustrative examples.