Fixed points and the fixed point algorithm
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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

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Fixed points and the fixed point algorithm

Fixed PointsandThe Fixed Point Algorithm


Fixed Points

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.

Geometry

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.

Algebra

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).

y

y=x

x

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.


  • The Fixed Point Algorithm (FPA) use a value x0 (ideally chosen close to the fixed point you want to find) and a function f(x) and generates a recursively defined sequence given by:

  • x0 for n=0 and xn+1=f(xn) for n>0.

  • The FPA will be able to estimate a fixed point if and only if the sequence xn converges.

  • There are several conditions that will that would imply convergence.

    • f(x) is increasing and bounded

    • f(x) satisfies a Lipshitz condition

    • f(x) is decreasing and contractive

    • others


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