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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.

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