The Evaluation of Matrix Inversion Programs

It is tedious and difficult to make error estimates in a calculation of significant size. The number of genuine general error estimates which have been made is very small indeed, even in the case of matrix calculations, which are in many respects the basic and simplest calculations. Our feeling however is that it is not economically possible to carry out rigorous error estimates for all computational problems. We must be content to carry out "experimental" calculations on "representative" problems for which exact results are known, to observe the errors and to extrapolate from these to predict the errors in less academic problems. Once we have found a process which is experimentally satisfactory, then, if we can find theoretical error estimates, we can compare these with the observed errors and if they are in reasonable agreement use the theoretical estimates for prediction. These general remarks being made, we shall confine our attention to the problem of matrix inversion. Most groups operating large scale computing installations have prepared their own programs for inverting matrices and solving systems of equations, and all have a considerable amount of experience with these problems. In order to pick out the better methods we present in Chapter II a set of matrices which we have reason to believe form a representative sample of those which occur in practice. We have decided, for various reasons, generally to interpret "better" in the sense of "least error", disregarding time and storage considerations. We present in Chapter IV the results of some computational experiments carried out on these matrices at our own organization, and some results communicated to us by colleagues elsewhere. We also give a preliminary discussion of these results. We are continuing our experiments in directions indicated by the available results, and we hope that other organizations will also do so.3 We expect to prepare a more penetrating report later; this will include comparisons of the direct and indirect methods-at present, we have restricted ourselves to direct methods.