Error Analysis for Polynomial Evaluation

Abstract

Abstract.A floating-point error analysis is given for the evaluation of a real polynomial at a real argument by Horner's scheme. A computable error bound is derived. It is observed that when a polynomial has coefficients of constant sign or of strictly alternating sign, one cannot expect better accuracy by reformulating the problem in terms of Chebyshev polynomials.Given that the real polynomial P{x) = 22 A*' is to be evaluated at a real argument a under conditions of normalized floating-point arithmetic, we wish to study the bounds for accumulated round-off effects. Two algorithms for evaluation will be compared: (i) the standard Homer scheme and (ii) Clenshaw's algorithm [1] applied to the Chebyshev form.First, we note that for practical purposes there is no loss of generality in assuming that |a| Si 1. For instance, on a binary machine one could define H{x) = P{2kx), and, in the absence of overflow/underflow, the coefficients of H are exactly determined in terms of those of P. The problem of evaluating P(a) can be replaced by that of evaluating H{x) at x = 2~ka, where |2~*a| Si 1. The two computations give rise to identically the same sequence of significands; only the exponents may differ. Bauer [2] made an analogous observation with respect to the scaling of linear equation problems. In order to have a fair basis for comparison with Clenshaw's algorithm, we shall assume the problem has been normalized so that |a| Si 1.According to the Horner scheme, we have (1) P{a) = q0, where qn = p", and q, = pr + aqr+x, r = n -1, n -2, ..., 0.Computationally, since the result of each arithmetical operation in (1) is subject to a relative error in the range ±e, we shall generate a sequence {<?*} given by q* = Pn, q* = pr + aq*+] + 8r, r = n -1, n -2, ..., 0, where 8r denotes the difference between the floating-point and true evaluation of pr + otq*+ x.