Provably monotone approximations

Abstract

In the preceding note (Dunham, 1987) were given forms of approximation that were provably monotone for suitable arithmetic, but left open the forms(1) c + x * r (x),where r could be a rational (the Kuki form on middle p. 9) or the next to last stage in Horner's method (top p. 10). If r is positive and non-decreasing on a positive interval I, (1) is obviously monotone on I for monotone arithmetic. This leaves two cases, (a)r positive and (slowly) decreasing on I, and (b) r negative on I. We consider them separately.