## The Babylonian Method, Part III

OK. Why does it work? Better still, why does it work so well? Let's begin by checking the claim made at the outset: If is an estimate for , then the average of and / is an even better estimate for .

First consider

And now compare distances

What this means, in terms of our algorithm, is that a new estimate and its companion estimate / are only half as far apart as and its companion estimate /. In other words, using induction, we would have

Since the right-hand side of this inequality tends to zero, we find that and / converge to the same limit, say . This common limit is necessarily because it satisfies

Finally, now that we know the limit, we can speak of the rate of convergence. For this we (essentially) repeat our first calculation:

where is a constant.

Now in our case, if we set = , then we have (at least for ). Hence,

This is what is meant by "quadratic convergence". It's a good thing to know: A quadratically convergent sequence zooms toward its limit! For example, if < 1, then we would have

Thus, if (or any other ) is less than 1 unit away from , then tends to faster than tends to 0, where < 1. (In other words, pretty darn fast!)

Et voila! We now know that the Babylonian method for computing converges, and converges rapidly.

You can call it quits, of course, but you might also be interested in a related algorithm.

Neal Carothers - carother@bgnet.bgsu.edu