Fixed Points and The Fixed Point Algorithm. Fixed Points A fixed point for a function f(x) is a value x 0 in the domain of the function such that f(x 0 ) = x 0 . We say the function f(x) fixes the value x 0 . Geometry
Fixed PointsandThe Fixed Point Algorithm
A fixed point for a function f(x) is a value x0 in the domain of the function such that f(x0) = x0. We say the function f(x) fixes the value x0.
Geometrically the fixed point occurs where the graph of y=f(x) crosses the graph of y=x. A function may have none, one or many fixed points.
In terms of algebra the fixed point(s) is(are) the solutions to the equation f(x)=x.
In the example to the right we see the fixed point for the function f(x) is 2. If you compute f(2) you get 2 (i.e. f(2)=2).
Complicated Fixed Points
Finding the fixed point for some functions results in a very complicated or impossible equation to solve that would find and exact value for the fixed point. For example if we consider the function f(x)=cos(x) it is apparent from the graph that (or you could prove using the Intermediate Value Theorem) this functions has a fixed point.
It has been proven there is no algebraic combination of number to express the solution to the equation cos(x)=x. This is why we need to rely on Numerical Method to estimate solutions.
The Fixed Point Algorithm
The Fixed Point Algorithm (FPA) is an algorithm that generates a recursively defined sequence that will find the fixed point for a function under the correct conditions. One of the big advantages of the algorithm is that it is no very difficult to implement.