Given a number x and a polynomial p(T) = a0 + a1T + ... + anT n, the Horner scheme computes the number
- p(x) = a0 + a1x + a2x2 + ... + an xn
as well as a polynomial q(T) = b0 + b1T + ... + bn-1T n-1 such that
- p(T) = (T - x) · q(T) + p(x).
The algorithm works as follows:
- set i := n - 1
- set bi := an
- if i < 0, stop; the result p(x) is in b-1.
- set i := i - 1
- set bi := bi+1 * x + ai+1
- Go to step 3.
This is the method of choice for evaluating polynomials; it is faster and more numerically stable than the "normal" method, which involves computing the powers of x and multiplying them with the coefficients. The Horner scheme is often used to convert between different positional number systems (in which case x is the base of the number system, and the ai are the digits) and can also be used if x is a matrix, in which case the gain is even larger.