16.1 Fixed-Point Iteration and Orbits

   

 
Java Number Cruncher: The Java Programmer's Guide to Numerical Computing
By Ronald  Mak

Table of Contents
Chapter  16.   Fractals

16.1 Fixed-Point Iteration and Orbits

In Chapter 5, we saw how we can iterate a function by applying it to itself repeatedly. If there is a value that remains unchanged by the iteration, then that value is a fixed point, and we are performing fixed-point iteration. The function f ( x ) = x 2 has a fixed point at 0, since

graphics/16equ01.gif


and another fixed point at 1.

If we perform fixed-point iteration on a function with a starting value x , then the values that the iteration generates is the orbit of x . For f ( x ) = x 2 , the orbit of 0.5 is 0.5, 0.25, 0.0625, 0.00390625, . . . and the orbit of 3 is 3, 9, 81, 6561, . . . .

0 is an attracting fixed point for f ( x ) = x 2 , since the orbit of any x in the region -1 < x < 1 converges to 0. This region is the fixed point's basin of attraction. On the other hand, 1 is a repelling fixed point for the function. If we choose an x value slightly less than 1, its orbit converges to 0, and any x value greater than 1 has an orbit that " escapes " to infinity.

Some orbits neither converge nor escape but are periodic?athe orbit values cycle. A simple example is the function graphics/16inl01.gif , which has two fixed points, -1 and +1, and the orbit of any other value x 0 is the cycle x , graphics/onebyx.gif , x , graphics/onebyx.gif , x , and so on.


   
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Java Number Cruncher. The Java Programmer's Guide to Numerical Computing
Java Number Cruncher: The Java Programmers Guide to Numerical Computing
ISBN: 0130460419
EAN: 2147483647
Year: 2001
Pages: 141
Authors: Ronald Mak

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