The Behavior of Differential Correction in Difficult Situations

Abstract

A paper published in 1980 points out that if the differential correction algorithm for generalized uniform rational approximation is applied when the best approximation is degenerate, and the kth approximation produced by the algorithm is near optimal, then a small error in the linear programming subroutine could cause the $(k + 1)$st approximation to be poor. In the present paper we point out that this in itself is not a difficulty in practice, since most computer implementations of the algorithm would return the (near-optimal) kth approximation to the user. Furthermore, examining more closely the behavior of the algorithm under the assumption of small inaccuracies in its linear programming subroutine, we show that the algorithm is stable in the sense that an approximation with error norm arbitrarily close to optimal will be produced if the linear programming error is sufficiently small. Both theoretical and numerical considerations indicate that the algorithm is usually reliable even if the best approxi...