8 HILLEL FURSTENBERG Figure 14. The Mandelbrot set Suppose that we need to solve the equation f(x) = 0 and know some initial approximation x0 to the root. Linearizing f at x0, we get the equation f(x0) + f (x0)(x x0) = 0, whose solution is x1 = x0 f(x0) f (x0) . If the function f is nice enough, and x0 is close enough to the root we are seeking, x1 is closer to the root than x0: However, that is not always the case: The classical Newton method uses this iteration process to approximate the root. Note that, when the function f has several roots, it is by no means trivial to determine which of them the Newton method will converge to starting with some fixed x0 (if it converges at all) and the set of initial values leading to a particular root is referred to as the basin of attraction of that root. This becomes an even more complicated question when we apply the Newton method to an analytic mapping on the complex plane. The basin of attraction of the root z = 1 of the function f(z) = z3 1 is depicted in Figure 17. The boundary of this basin is very far from being smooth. It, too, has fractal structure. For more information regarding fractals in the complex plane and their relation to dynamics, the reader is referred to [B] and [Mi].
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