16.[A] Verify that whatever value is given to the complex number c, z = 1
is a solution of F_{c}(z) = 0, where F_{c}(z) =
^{3} + (c-1)*z - c

17.[A] For the function F_{c}(z) = ^{3} + (c-1)*z - c

(a) Verify that F'_{c}(z) = ^{2} + c - 1

(b) Show z_{n+1} =
_{n}^{3} + c)/(3z_{n}^{2} + (c-1))

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

x_{n+1} =
_{1}(n)*denom_{1}(n) + num_{2}(n)*denom_{2}(n))/
(denom_{1}(n)^{2} + denom_{2}(n)^{2})

and

y_{n+1} =
_{2}(n)*denom_{1}(n) - num_{1}(n)*denom_{2}(n))/
(denom_{1}(n)^{2} + denom_{2}(n)^{2})

where

num_{1}(n) = _{n}^{3} -
6*x_{n}*y_{n}^{2} + a

num_{2}(n) = _{n}^{3} +
6*x_{n}^{2}*y_{n} + b

denom_{1}(n) = _{n}^{2} - 3*y_{n}^{2} + a -1

and denom_{2}(n) = _{n}*y_{n} + b

18.[N] Using the iteration rules presented in exercise 16 (c), take a = 0.31,
b = 1.65, x_{0} = 0, and y_{0} = 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 x_{1}, y_{1}, ..., x_{14}, y_{14}.
Deduce for this A + B*i the Newton method converges to a 2-cycle, not to a fixed point.
Answer

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