Horner's rule

最新推荐文章于 2025-06-05 20:59:06 发布

最新推荐文章于 2025-06-05 20:59:06 发布 · 103 阅读

介绍了一种高效计算多项式的算法——霍纳法则。通过重新组织和嵌套多项式表达式，该法则显著减少了计算所需的浮点运算次数，对于高次多项式的求值尤为重要。

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Horner's Rule for Polynomials

A general polynomial of degree

can be written as

$\begin{displaymath} P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n = \sum_{i=0}^n a_i x^i \end{displaymath}$

(1)

If we use the Newton-Raphson method for finding roots of the polynomial we need to evaluate both

and its derivative $P^\prime(x)$ for any

It is often important to write efficient algorithms to complete a project in a timely manner. So let us try to design the algorithm for evaluating a polynomial so it takes the fewest flops (floating point operations, counting both additions and multiplications). For concreteness, consider the polynomial

$\begin{displaymath} 7x^3 + 5x^2 - 4x + 2. \end{displaymath}$

The most direct evaluation computes each monomial

one by one. It takes

multiplications for each monomial and

additions, resulting in

flops for a polynomial of degree

. That is, the example polynomial takes three flops for the first term, two for the second, one for the third, and three to add them together, for a total of nine. If we reuse

from monomial to monomial we can reduce the effort. In the above example, working backwards, we can save

from the second term and get

for the first in one multiplication by

. This strategy reduces the work to

flops overall or eight flops for the example polynomial. For short polynomials, the difference is trivial, but for high degree polynomials it is huge. A still more economical approach regroups and nests the terms as follows:

$\begin{displaymath} 2 - 4x + 5x^2 + 7x^3 = 2 + x[-4 + x(5 + 7x)]. \end{displaymath}$

(Check the identity by multiplying it out.) This procedure can be generalized to an arbitrary polynomial. Computation starts with the innermost parentheses using the coefficients of the highest degree monomials and works outward, each time multiplying the previous result by

and adding the coefficient of the monomial of the next lower degree. Now it takes

flops or six for the above example. This is the efficient Horner's algorithm.

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