]> 12.3 Spreadsheet Implementation of this Procedure

## 12.3 Spreadsheet Implementation of this Procedure

How can you do such things?
First put $x$ and $j$ in fixed locations $X$ and $Y$ .
Then set up the following columns on the spreadsheet:

$f ( x j )$ :

In the first column enter the successive values of $f ( x j )$ starting with the first, known value.

For roots you can start with $x 0 = f ( x 0 ) = 1$ .

Compute later values $f ( x j )$ by using the linear approximation tangent line at $x j − 1$ evaluated at argument $x , f L x j − 1 ( x )$ or $f ( x j − 1 ) + f ' ( x j − 1 ) ( x − x j − 1 )$ . ( $x j − 1$ is the entry in the second column in the previous row.)

$x j$ :

In the second column apply the inverse function, $f − 1$ to the value in the first column.

Once you have entered your instructions for $f ( x 1 )$ and $x 1$ , you can copy these down a hundred rows, and you are done.

What happens if f is a root, $x 1 / m$ ?

In general we have

$f ( x j ) = f L x j − 1 ( x ) = f ( x j − 1 ) + ( x − x j − 1 ) f ' ( x j − 1 )$

For j-th root, $f ' ( x j ) = f ( x j ) / x j j$ so that this formula reduces to

$f ( x j ) = f ( x j − 1 ) ( 1 − ( 1 j − 1 + x / ( j − 1 ) x 0 ) )$

And that is all you need enter. The rest is copying down.