When are pivotal interchanges not necessary?

Abstract

Solving a linear system Ax = b by Gaussian Elimination usually entails pivotal inter-changes designed to inhibit that explosive growth of intermediate results which would otherwise, through roundoff, vitiate the calculation. But these interchanges, motivated by numerical desiderata, frequently conflict with combinatorial desiderata like "Sparsity". We shall show that two special cases in which interchanges are well known not to be needed for stability, namely, when A is positive definite or diagonally dominant, are examples of a more frequent situation; A's field of values lies in a half-plane not containing zero. This situation, which is associated with certain electric networks and some boundary value problems, allows at least in principle for an estimate of the number of extra guard digits that need be carried to prevent explosive growth from blighting results obtained without interchanges.