Algorithm Implementation/Mathematics/Polynomial evaluation

Horner's method allows the efficient computation of the value of p(x₀) for any polynomial p(x) = a₀ + a₁x + ... + a_nxⁿ 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 + ... + a_nxⁿ = a₀ + x(a₁ + x(a₂ + ... x(a_n))...)), 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, ..., a_n)
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.

Loop unrolling can be used with Horner's Scheme to evalulate multiple multiplications at once. This is called a kth-order Horner.

There are also non-Horner schemes for evalulating polynomials: the naive method, the Estrin method, and factorization.