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Horner scheme
From Wikipedia, the free encyclopedia
In numerical analysis, the Horner scheme or Horner algorithm, named after William George Horner, is an algorithm for the efficient evaluation of polynomials in monomial form. Horner's method describes a manual process by which one may approximate the roots of a polynomial equation. The Horner scheme can also be viewed as a fast algorithm for dividing a polynomial by a linear polynomial with Ruffini's rule.
Contents |
Given the polynomial
where 
are real numbers, we wish to evaluate the polynomial at a specific value of 
, say 
.
To accomplish this, we define a new sequence of constants as follows:
Then 
is the value of 
.
To see why this works, note that the polynomial can be written in the form
Thus, by iteratively substituting the bi into the expression,
Evaluate 
for 
. By repeatedly factoring out x, f1 may be rewritten as 
for . We use a synthetic diagram to organize these calculations and make the process faster.
x0 | x3 x2 x1 x0
3 | 2 -6 2 -1
| 6 0 6
|----------------------
2 0 2 5
The entries in the third row are the sum of those in the first two. Each entry in the second row is the product of the x-value (3 in this example) with the third-row entry immediately to the left. The entries in the first row are the coefficients of the polynomial to be evaluated. The answer is 5.
As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial that is the quotient of f1/(x-3). The remainder is 5. This makes Horner's method useful for polynomial long division.
Divide 
by x − 2:
2 | 1 -6 11 -6
| 2 -8 6
|----------------------
1 -4 3 0
The quotient is x2 − 4x + 3.
Let 
and 
. Divide 
by 
using Horner's scheme.
2 | 4 -6 0 3 | -5
---------------------------|------
1 | 2 -2 -1 | 1
| |
|----------------------|-------
2 -2 -1 1 | -4
The third row is the sum of the first two rows, divided by 2. Each entry in the second row is the product of 1 with the third-row entry to the left. The answer is
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 coefficients are the digits of the base-x representation of a given number — and can also be used if x is a matrix, in which case the gain in computational efficiency is even greater.
Evaluation using the monomial form of a degree-n polynomial requires at most n additions and (n2 + n)/2 multiplications, if powers are calculated by repeated multiplication and each monomial is evaluated individually. (This can be reduced to n additions and 2n + 1 multiplications by evaluating the powers of x iteratively.) If numerical data are represented in terms of digits (or bits), then the naive algorithm also entails storing approximately 2n times the number of bits of x (the evaluated polynomial has approximate magnitude xn, and one must also store xn itself). By contrast, Horner's scheme requires only n additions and n multiplications, and its storage requirements are only n times the number of bits of x. Alternatively, Horner's scheme can be computed with n fused multiply-adds.
It has been shown that the Horner scheme is optimal, in the sense that any algorithm to evaluate an arbitrary polynomial must use at least as many operations. That the number of additions required is minimal was shown by Alexander Ostrowski in 1954; that the number of multiplications is minimal by Victor Pan, in 1966. When x is a matrix, the Horner scheme is not optimal.
This assumes that the polynomial is evaluated in monomial form and no preconditioning of the representation is allowed, which makes sense if the polynomial is evaluated only once. However, if preconditioning is allowed and the polynomial is to be evaluated many times, then faster algorithms are possible. They involve a transformation of the representation of the polynomial. In general, a degree-n polynomial can be evaluated using only 
multiplications and n additions (see Knuth: The Art of Computer Programming, Vol.2).
Even though the algorithm is named after William George Horner, who described it in 1819, the method was already known to Isaac Newton in 1669, and even earlier to the Chinese mathematician Ch'in Chiu-Shao in the 13th century.
- Clenshaw algorithm to evaluate polynomials in Chebyshev form
- De Casteljau's algorithm to evaluate polynomials in Bézier form
- De Boor's algorithm to evaluate splines in B-spline form
- William George Horner. A new method of solving numerical equations of all orders, by continuous approximation. In Philosophical Transactions of the Royal Society of London, pp. 308-335, July 1819.
- Spiegel, Murray R. (1956). Schaum's Outline of Theory and Problems of College Algebra. McGraw-Hill Book Company.
- Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Pages 486–488 in section 4.6.4.
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Problem 2-3 (pg.39) and page 823 of section 30.1: Representation of polynomials.