1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

4.2 • JULIA AND MANDELBROT SETS 133

quadratic polynomials and indkated his intention to publish a subsequent paper
for cubic polynomials. Unfortunately, Cayley died before producing this paper.
As you will see, the extension of Newton's method to the complex domain and
the more general question of iteration are quite complicated.

• EXAMPLE 4.7 Trace the next five iterates of Newton's method for an initial
  guess of zo = ~ + ~i as a solution to the equation f(z) = 0, where J(z) = z^2 +1.

Solut ion For any guess z for a solution-;-Newton's method gives as the next

guess the number z-/J(J) = ·~~^1. Table 4.1 gives the required iterates, rounded
to five decimal places.
F igure 4.2 shows the relative positions of these points on the z plane. Note
that the points z 4 and z 5 are so close together that they appear to coincide and
that the value for zs agrees to five decimal places with the actual solution z = i.

k Zk f(zk)

0 0.25000 + 0.25000i 1.00000 + 0.12500i

1 - 0.87500 + l.12500i 0.50000 - l.96875i

2 -0.22212 + 0.83942i 0.34470 - 0 .37290i

3 0.03624 + 0.97638i 0.04799 + 0.07077i

4 -0.00086 + 0.99958i 0.00084- 0.00172i

(^5) 0.00000 + l.OOOOOi 0.00000 + O.OOOOOi
Table 4.1 The iterates of zo = i + ii for Newton's method applied to f ( z) = z^2 + 1.
y

1. 2

Z2 • 0.8

0.6

0.4

0.2 ·z.i

--0.75 --0.5 --0.25 0.25 0.5 0.75

--0.2

Figure 4.2 The iterates of zo = ~+ii for Newton's method applied to f (z) = z^2 +1.