Flat Iterators

Experimentation suggests that it is the flatnessof the Babylonian iterator

at its fixed points that makes it so remarkably effective in approximating the square root of k. Indeed, in order for any iterative process

to be effective in approximating a fixed point, it is necessary the slope m of F(x) satisfy |m| < 1 at that point. (If m > 1, then the iterative process moves away from the fixed point.) On this basis, iterators for which m = 0 at their fixed points are especially prized.

To demonstrate our assertion regarding the flatness of the Babylonian iterator we shall show that for k > 0,

While we could turn to calculus in this regard, let us instead recall a particular case of the Arithmetic-Geometric Mean Inequality. For arbitrary positive numbers a and b:

Setting a = x and b = k/x, we obtain ab = k and

In order to adapt these techniques to the solution of quadratic equations, we shall seek an iterator G(x) whose fixed points are the (real) solutions of ax² + bx + c = 0 and has slope zero at those fixed points.

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