Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

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