Ruffini-Horner Method/Examples/Arbitrary Example 1

Examples of Use of the Ruffini-Horner method

$\map f x = x^2 - x - 1 = 0$

We have that:

$\ds \map f 1$

$=$

$\ds -1$

$\ds \map f 2$

$=$

$\ds 1$

so we observe there is a root between $x = 1$ and $x = 2$.

Then:

$\ds \map {f_1} x$

$=$

$\ds \map f {x - 1}$

$\ds $

$=$

$\ds \paren {x - 1}^2 - \paren {x - 1} - 1$

$\ds $

$=$

$\ds x^2 - 3 x + 1$

We then identify a root between $x = 0.6$ and $x = 0.7$.

This leads to calculating:

$\map {f_2} x = \map {f_1} {x - 0.6}$

Hence and so, until the required accuracy is achieved.