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/
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