Rounding error propagation in polynomial evaluation schemes

Abstract

The error propagation characteristics of the polynomial evaluation schemes of Horner, Clenshaw and Reinsch are analysed, and Newbery's transformation of the Horner method is extended to Clenshaw's algorithm. In each case computable error bounds are derived in terms of the coefficients of both the power series and Chebyshev series form of the polynomial, and are found to be of particular relevance to polynomials having equisign or strictly alternating sign coefficients or which are similar to Chebyshev polynomials. The analysis confirms the earlier hypotheses and empirical results of Newbery, but affords greater insight into how the error propagation varies with the argument of the polynomial.