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
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 , which has two fixed points, -1 and +1, and the orbit of any other value x 0 is the cycle x , , x , , x , and so on. |
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