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,

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

Read more about this topic: Horner's Method

Famous quotes containing the words description of the, description of and/or description:

“God damnit, why must all those journalists be such sticklers for detail? Why, they’d hold you to an accurate description of the first time you ever made love, expecting you to remember the color of the room and the shape of the windows.”
—Lyndon Baines Johnson (1908–1973)

“As they are not seen on their way down the streams, it is thought by fishermen that they never return, but waste away and die, clinging to rocks and stumps of trees for an indefinite period; a tragic feature in the scenery of the river bottoms worthy to be remembered with Shakespeare’s description of the sea-floor.”
—Henry David Thoreau (1817–1862)

“To give an accurate description of what has never occurred is not merely the proper occupation of the historian, but the inalienable privilege of any man of parts and culture.”
—Oscar Wilde (1854–1900)