Horner's Method - Description of The Algorithm

Description of The Algorithm

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:

\begin{align}
b_n & := a_n \\
b_{n-1} & := a_{n-1} + b_n x_0 \\
& {}\ \ \vdots \\
b_0 & := a_0 + b_1 x_0.
\end{align}

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 into the expression,


\begin{align}
p(x_0) & = a_0 + x_0(a_1 + x_0(a_2 + \cdots + x_0(a_{n-1} + b_n x_0)\cdots)) \\
& = a_0 + x_0(a_1 + x_0(a_2 + \cdots + x_0(b_{n-1})\cdots)) \\
& {} \ \ \vdots \\
& = a_0 + x_0(b_1) \\
& = b_0.
\end{align}

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