An e cient algorithm for evaluating polynomials in the P olya basis

A new O(n) algorithm is given for evaluating univariate polynomials of degree n in the P olya basis. Since the Lagrange, Bernstein, and monomial bases are all special instances of the P olya basis, this technique leads to e cient evaluation algorithms for these special bases. For the monomial basis, this algorithm is shown to be equivalent to Horner's rule. 1 P olya basis functions Let nk(t) be the restriction of the standard B-spline basis functions [dB72] of degree n over the nondecreasing knot sequence t1; t2; :::; t2n to the interval tn < t < tn+1. The P olya basis functions are the unique polynomial functions dk(t) that satisfy Marsden's identity [Mar70]. (x t) = n X k=0 dk(t)nk(x): The P olya basis functions of degree n can be written explicitly as dk(t) = n Y