1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

134 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES

The complex version of Newton's method also appears to work quite well.
Recall, however, that with functions defined on the reals, not every initial guess
produces a sequence that converges to a solution. Example 4.8 shows that the
same is true in the complex case.

• EXAMPLE 4.8 Show that Newton's method fails for the function f (z) =
  z^2 + 1 if the initial guess is a real number.

Solution From Example 4.7 we know that, for any guess z as a solution of

z^2 + 1 = 0, the next guess at a solution is N (z) = z - /JtJi = z~~^1. We let z 0 be

any real number and {zk} be the sequence of iterations produced by the initial

seed zo. If for any k, Zk = 0, the procedure terminates, as zk+ 1 will be undefined.

If all the terms of the sequence {zk} are defined, an easy induction argument

shows that all the terms of t he sequence are real. The solutions of z^2 + 1 = 0

a.re ± i, so the sequence { zk} cannot possibly converge to either solution. In the

exercises we ask you to explore in detail what happens when z 0 is in the upper

or lower half-plane.

The case for cubic polynomials is more complicated than that for quadratics.

Fortunately, we can get an idea of what's going on by doing some experimentation

with computer graphics. We begin with the cubic polynomial f (z) = z^8 + 1.

(Recall that the roots of this polynomial are at -1,! + v;i, and ~ - 4i.) We

associate a color with each root (blue, red, and green, respectively). We form a

rectangular region R, which contains the three roots off (z), and partition this

region into equal rectangles R.;. We then choose a point z;; at the center of each

rectangle and for each of these points we apply the following algorithm.

1. With N (z) = z - /!()), compute N (z.;;). Continue computing successive

iterates of this initial point either until we are within a certain preassigned

tolerance (say, e) of one of the roots off (z) = 0, or until the number of

iterations has exceeded a preassigned maximum.

2. If Step 1 leaves us within e of one of the roots of f (z), we color the entire
   rectangle R;; with the color associated with that root. Otherwise, we assume
   that the initial point z;; does not converge to any root, and we color the entire
   rectangle yellow.

Note that this algorithm doesn't prove anything. In Step 2, there is no a

priori reason to justify the assumption mentioned, nor is there any necessity for

an initial point z.;; to have its sequence of iterates converging to one of the roots

off (z) = 0 just because a particular iteration is within c of that root. Finally,

the fact that one point in a rectangle behaves in a certain way does not imply

that all the points in that rectangle behave in a like manner. Nevertheless,

we can use this algorithm as a basis for mathematical explorations. Indeed,

computer experiments such as the one described have contributed to a lot of

exciting mathematics during the past 30 years. T he color plates located on the