Download presentation

Presentation is loading. Please wait.

Published byEve Hamlett Modified over 4 years ago

1
Fixed Points and The Fixed Point Algorithm

2
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 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 ). x y y=x 2

3
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.

4
The Fixed Point Algorithm (FPA) use a value x 0 (ideally chosen close to the fixed point you want to find) and a function f(x) and generates a recursively defined sequence given by: x 0 for n=0 and x n+1 =f(x n ) for n>0. The FPA will be able to estimate a fixed point if and only if the sequence x n 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

Similar presentations

OK

Rolle’s theorem and Mean Value Theorem ( Section 3.2) Alex Karassev.

Rolle’s theorem and Mean Value Theorem ( Section 3.2) Alex Karassev.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

Ppt on international accounting standards Ppt on rbi reforms in education Ppt on latest technology in electrical engineering What does appt only means that Glass fiber post ppt online Ppt on diode as rectifier tube Ppt on input and output devices of computer download Ppt on biogas in india Ppt on trade union act 1926 in india Ppt on solar system for class 8