Iterating Linear Functions
Sequences of iterates of nonlinear functions can behave in complicated ways as can be seen from a variety of Demonstrations (see Related Links).[more]
The iterates of linear functions, however, are easily understood. If the absolute value of the slope is greater than one, the absolute values of the iterates grow without bound. If the absolute value of the slope is less than one, the iterates converge to the solution of the equation.
The top plot shows the function being iterated (purple), the line (blue), and the web diagram of the iteration. The bottom plot is simply a plot of the points where is the term in the sequence of iterates.[less]
Given a function and an initial value , the sequence of iterates of is the sequence defined recursively by . If , then . If is smaller than one in absolute value, then clearly , which is the solution to the equation . If >1, then the sequence of iterates diverges to infinity or minus infinity depending on the sign of ; that is, depending on which side of the fixed point the sequence of iterates starts. If , then the sequence of iterates alternates between values to the left and right of the fixed point whose distance from the fixed point grows without bound.