Horner's method

In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation. It is named after William George Horner, although it is much older, attributed by Horner to Joseph-Louis Lagrange, and was discovered hundreds of years earlier by Chinese and Persian mathematicians. After the introduction of computers, this algorithm became fundamental for computing efficiently with polynomials.

The algorithm is based on Horner's rule, in which a polynomial is written in nested form: $${\displaystyle {\begin{aligned}&a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n}\\={}&a_{0}+x{\bigg (}a_{1}+x{\Big (}a_{2}+x{\big (}a_{3}+\cdots +x(a_{n-1}+x\,a_{n})\cdots {\big )}{\Big )}{\bigg )}.\end{aligned}}}$$

This allows the evaluation of a polynomial of degree n with only ${\displaystyle n}$ multiplications and ${\displaystyle n}$ additions. This is optimal, since there are polynomials of degree n that cannot be evaluated with fewer arithmetic operations.

Horner's method and Horner–Ruffini method also refers to a method for approximating the roots of polynomials, described by Horner in 1819. It is a variant of the Newton–Raphson method made more efficient for hand calculation by application of Horner's rule. It was widely used until computers came into general use around 1970.