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Fixed Points and The Fixed Point Algorithm Fixed Points y y=x 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. y 2 x Geometry x 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). 2 2 x x 2 x 2 x2 2 x x2 0 x2 x 2 0 ( x 2)( x 1) 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. 2 1.5 1 0.5 1 2 3 4 5 6 -0.5 -1 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