Perturbation analysis of Horner's method for nice cases

Under favourable conditions, evaluation of polynomials by Homer's rule has an error not exceeding a few units in the last place.It has been observed that for moderate sized x and coefficients decreasing rapidly in magnitude that floating point evaluation of polynomials by Homer's method gives results accurate to around one or two units in the last place. A hand calculated example together with an informal justification is given by Fike (1968, pp. 52--53). In this note the observation is justified by a Wilkinson-type backward error analysis. It would be helpful for the reader to be familiar with the approach of Wilkinson (1963, Chap. 1), in which fl (expr) denotes the effect of evaluation of expression expr in floating point.We will assume that floating point is done using guard digits (Johnston, 1982, p. 11: Sterbenz, 1974) in which case we have by analysis similar to those of Wilkinson (1963, Chap. 1).