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Iteration is the process of
repeating a task over and over again. In reference to functions, the
process of iterating a function refers to obtaining a value from a
function, using that value to produce a subsequent value, using that
subsequent value to produce yet another value, and so on.
Example of function
iteration:
f (x) = x2
+ 2 ; starting value of
x = 0.
f (0) = 2
f (2) = 6
f (6) = 38
f (38) = 1446
and so on.....
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Example of iteration
with a complex number:
f (x) = 2x + (4 - 2i) ;
starting value of x = 0.
f (0) = 4 - 2i
f (4 - 2i) = 2(4 - 2i) + (4 - 2i) = 12 - 6i
f (12 - 6i) = 2(12 - 6i) + (4 - 2i) = 28 - 14i
f (28 - 14i) = 2(28 - 14i) + (4 - 2i) = 60 - 30i
and so on ...... |
This process of iteration can serve as a good exercise for students to
practice their basic skills for manipulating complex numbers. Have
students consider iterations on the following functions (or create your
own):
f (x) = x2 + (1 - 3i) ;
starting value of x = 0; four iterations
f (x) = 3x - (2 - i) ; starting value of x = 0; three iterations
f (x) = 2x2 + (1 +
i); starting
value of x = 0; three iterations
Graphically speaking, a function of the
form f (x) = x2 + c
(where c
is a complex
number) is a special function under iteration. If you plot the
results of the iterations (starting at
x
= 0) in the complex plane, you obtain what is called the
critical orbits of c. If
these critical orbits repeat (where the same point in the complex plane
repeats), the complex number is in the
Mandelbrot set. If the critical orbits simply move
further and further away from the origin, the complex numbers are not in
the Mandelbrot set.
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Examine the following functions to see where
the orbits fall in the Mandelbrot set:
f
(x) =
x2 + (-1.1 + 0.2i)
f
(x) =
x2 + (-0.05 + 0.75i)
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Use the graphing calculator
to see if critical orbits repeat and lie in the Mandelbrot set:
(instructions for the
TI-83+/84+)
f (x) =
x2
+ (- 0.6 + 0.2i)
The screens below show the iterations of the function and
the start of a repeating nature of the orbits.
Put calculator in a+bi mode.
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Further Investigations:
Examine the critical orbits of any complex
number and the critical orbits of its complex conjugate. A
discussion of the symmetric nature of the Mandelbrot set should
follow this examination.. |
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