The Horner scheme
is an algorithm
for the efficient evaluation of polynomial
functions, and for dividing polynomials by linear polynomials.
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
) = b0
+ ... + bn-1T n-1
- 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 numeral 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.
There is another way to describe the Horner scheme. Given the ai coefficients and the number x, first rewrite p with x factored out:
- p(x) = a0 + a1x + a2x2 + ... + an-1 xn-1 + an xn
- = a0 + x(a1 + x(a2 + ... + x(an-1 + x(an)) ... ))
then evaluate this expression in the obvious way, starting from the innermost parentheses and working out. The value of the expression in the innermost parentheses is bn-1
. The value of the expression in the second-to-innermost parentheses is bn-2
, and so on until the value of the contents of the outermost parentheses is b0