# Exercises for Chaos Under Control

## Chapter 8: Another Surprise in Newton's Method

16.[A] Verify that whatever value is given to the complex number c, z = 1 is a solution of Fc(z) = 0, where Fc(z) = z3 + (c-1)*z - c.

17.[A] For the function Fc(z) = z3 + (c-1)*z - c,

(a) Verify that F'c(z) = 3z2 + c - 1.

(b) Show zn+1 = (2*zn3 + c)/(3zn2 + (c-1)).

(c) Writing z = x + y*i and c = a + b*i, show that zn+1 in (b) has the following real and for the imaginary parts:

xn+1 = (num1(n)*denom1(n) + num2(n)*denom2(n))/ (denom1(n)2 + denom2(n)2)

and

yn+1 = (num2(n)*denom1(n) - num1(n)*denom2(n))/ (denom1(n)2 + denom2(n)2)

where

num1(n) = 2*xn3 - 6*xn*yn2 + a,

num2(n) = -2*yn3 + 6*xn2*yn + b,

denom1(n) = 3*xn2 - 3*yn2 + a -1,

and denom2(n) = 6*xn*yn + b.

18.[N] Using the iteration rules presented in exercise 16 (c), take a = 0.31, b = 1.65, x0 = 0, and y0 = 0. This point c = a + b*i lies in the main cardioid of the pseudo-Mandelbrot set in Figure 8.12. Using a calculator, compute x1, y1, ..., x14, y14. Deduce for this A + B*i the Newton method converges to a 2-cycle, not to a fixed point. Answer