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13.2 Poor Man's Newton

To apply Newton's method as just described, it is necessary to differentiate the function f at each successive guess. This is not difficult, but requires a small amount of effort.

We can instead apply what is essentially the same method, using an approximation to f ' ( x i ) of the form f ( x i + d ) − f ( x i ) d for some d . We then have to decide what to use for d , but with a spreadsheet we can pick a small value to begin with and let it slowly go to zero as we iterate. If we wanted to be fancier, we could use a symmetric approximation to the derivative f ( x i + d ) − f ( x i − d ) 2 d .

How do we do all this?

In box e2 we can put our initial value of d , say 10^(-3); then we can put "= e2*9/10" in e3 and copy it down, so that d will slowly decline to 0. (Why slowly like this? If you go too fast round-off error might ruin you before you find your solution.) The iteration here is

x j + 1 = x j − d f ( x j ) f ( x j + d ) − f ( x j )

We can therefore put our guess in f2, put "= e2+f2"in g2, set h2 to "= f(f2)", copy it into i2 and then set f3 to "= f2-e2*h2/(i2-h2)", and copy g2, h2, i2 and e3 and f3 down, and we are done.

Exercise 13.7 Do this for each of the functions in the exercises of the section 13.1. Do you find a difference between the results here and with the regular Newton's method? If so, what?

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