## Fixed Point Iteration

A fixed point of a function is a value such that If is a continuous function such that converges to as then is a stable fixed point and Thus, provided is sufficiently close to we can use the simple iteration to find stable fixed points.

The problem of finding the root of an equation can be easily converted to a fixed point iteration by choosing, e.g., It is important to note that there exists an infinite number of such functions but the precise

form of the we choose will prove crucial for the convergence of the fixed point iteration.

Example: Let us look at the zeros (or roots) of so a zero of becomes a fixed point of If we choose and start at the fixed point iteration converges. The fixed point of is If we write  0 1 2 3 4 5 1 1.5 1.371 1.4297 1.4077 1.4169 0.4142 0.0858 0.0392 0.0155 0.0065 0.0027

However, for with the procedure diverges to infinity. 1 2 3 4 5 1 -0.5 -3.13 8.52 114.5 0.41 1.91 4.54 7.11 113.08

Clearly, we need to establish the criteria for a stable fixed point. We consider the problem graphically. The fixed point of is at the intersection of the lines and The different types of convergence are shown below. Consider an iterate close to say. Then Thus and so if then is further away from than so there is no convergence. 