A comparison of solution accuracy resulting from factoring and inverting ill-conditioned matrices

In previous related work a study was described that investigated the behavior of ill-conditioned matrices having the goal of numerically characterizing their information content. One numerical result from that study was that the solution accuracy (SA) is related to the coefficient accuracy (CA) and condition number (CN). This conclusion was based on using, as one measure of the SA, a comparison of [ZI[Y] with [I], where [Z] is a matrix under study, [Y] is its computed inverse and [I] is the identity matrix. The author's particular reason for examining this result is the implication they may have when using residuals in determining the convergence of an iterative solution. It seems reasonable, if the residual error is smaller than the actual error in an iterated sequence of solution estimates, to conclude that relying on the residuals as an indicator of solution accuracy could be misleading.