Applications of Iteration

Date: 05/29/97 at 10:01:09
From: Erin Nichols
Subject: iterated functions

Hi,

I'm currently doing a research project in my precalc class with the 
topic "applications of iteration."  I've spent two days with my group 
trying to figure out what iteration is (it seems easy, so I guess we 
don't get it?) and what we can say about it.  I'm also wondering what 
it has to do with fractals (and what fractals are, not that I couldn't 
look that up as well).  Whatever you can tell me (especially web sites 
that deal with this topic) would be appreciated.

Erin

Date: 05/29/97 at 11:48:07
From: Doctor Sarah
Subject: Re: iterated functions

Hi Erin,

To "iterate" means to repeat.  An iteration is a repetition of an 
operation. In math, iterations often involve taking the output of a 
function and plugging it back in - repeating a process using each 
output as the input for the following iteration.

Here's an example of a very simple iteration using the function 
5 + x/2:  

Pick a number, say 0, and put it in for x. What do you get out? 
5 + 0/2 = 5. 

Now *iterate*: put the output of the function (5) back in for x. What 
do you get out this time? 5 + 5/2 = 7.5 

Now put that output (7.5) in for x.
   5 + 7.5/2 = ...   

Etc., etc.  

You can see that by continuing these iterations (repetitions), you 
will get the infinite sequence: 0, 5, 7.5, 8.75, ..

You can research iteration and fractals together. A great place to 
start is Cynthia Lanius' web unit on fractals:

  http://math.rice.edu:80/~lanius/frac/   

For explanations of iteration, look at her Jurassic Park fractal page:

  http://math.rice.edu/~lanius/frac/jurra.html   

and Properties of Fractions - formation by iteration:

  http://math.rice.edu/~lanius/fractals/iter.html    

Good luck with your project!


-Doctor Sarah,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 06/04/97 at 19:24:21
From: Erin Nichols
Subject: Re: iterated functions

Dr. Math wrote:

Your definition is, essentially, what our limited research has yielded 
(although in a far simplified and abbreviated manner).  This is why we 
are curious about why we would have been assigned this topic if there 
isn't something more complicated about it. Where does it get 
complicated?

Thanks,
Erin

Date: 06/04/97 at 20:21:25
From: Doctor Tom
Subject: Re: iterated functions

Hi Erin,

Well, the idea is simple, but it can be VERY difficult to figure out
the results of iteration.  If you iterate forever, does it converge?
Diverge to infinity?  If it converges, how many possible points of 
convergence are there (given different starting points)?  Are there 
fixed points (where iteration leaves them fixed)?  If so, are those 
fixed points stable?  In other words, if you choose a starting point 
"nearby", does it converge back to the fixed point?

If you look at complex numbers z and iterate with a function as simple 
as z -> z^2 - c, and begin iteration at zero, for which constants does 
it converge?  This generates the Mandelbrot fractal which you've no 
doubt seen illustrations of.

In biology, if you look at allele frequencies under selection, 
crossing over, et cetera, each generation is an iteration.  If you can 
understand iteration completely, you can figure out how evolution will 
go.

Many math problems are solved by successive approximation.  This is 
just an interation technique.  There are hundreds of them with amazing 
properties.

So like much of mathematics, the idea is simple, but the ramifications
can be unimaginably complex.

Good luck with your project.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/