Algorithm Implementation/Mathematics/Polynomial evaluation

Horner's method Horner's method allows the efficient computation of the value of p(x0) for any polynomial p(x) = a0 + a1x + ... + anxn and value x0. Algorithm Rather than the naive method of computing each term individually, in Horner's method you rewrite the polynomial as p(x) = a0 + a1x + ... + anxn = a0 + x(a1 + x(a2 + ... x(an))...)), and then use a recursive method to compute its value for a specific x0. The result is an algorithm requiring n multiplies, rather than the 2n multiplies needed by the best variant of the naive approach (and much more if each xi is computed separately). In both the pseudocode and each implementation below, the polynomial p(x) is represented as an array of it's coefficients. Pseudocode input: (a0, ..., an) input: x0 output: p(x0) accum := 0 for i = n, n-1, n-2, ..., 2, 1, 0 accum := x0(accum + ai) return accum C float horner(float a[], int deg, float x0) { float accum = 0.0; int i; for (i=deg; i>=0; i--) { // equivalently using a fused multiply-add: // accum = fmaf(x0, accum, a[i]); accum = x0*accum + a[i]; } return accum; } Fortran F77 real function horner(p,deg,x0) real p(*), x0 integer deg horner = 0.0 do 10 i=deg,0,-1 horner = x0*horner + p(i+1) 10 continue return end Python from typing import List def horner(a: List[float], x0: float) -> float: accum = 0.0 for i in reversed(a): accum = x0 * accum + i return accum Variants The Compensated Horner Scheme is a variant of Horner's Algorithm that compensates for floating-point inaccuracies. It takes 1.5x the time of normal Horner to calculate for 2x the accuracy.S. Graillat and Nicolas Louvet (2005). Compensated Horner Scheme Loop unrolling can be used with Horner's Scheme to evalulate multiple multiplications at once. This is called a kth-order Horner.Timothée Ewart, Francesco Cremonesi, Felix Schürmann, and Fabien Delalondre. 2020. Polynomial Evaluation on Superscalar Architecture, Applied to the Elementary Function ex. ACM Trans. Math. Softw. 46, 3, Article 28 (September 2020), 22 pages. https://doi.org/10.1145/3408893 There are also non-Horner schemes for evalulating polynomials: the naive method, the Estrin method, and factorization. References