“Prediction” means here mathematicaly as follows, with an well-known example of function F(x)=x^{2}+a.

x_{1}= x_{0}^{2}+ a x_{2}= x_{1}^{2}+ a .... x_{n}= x_{n-1}^{2}+ a

What will be the value of x_{n} under a very large value of n?

The situation will be different the two parameters: the initial value, i.e. x_{0} and a.

The most simple prediction can be done when

F(x)=x

Then the value stays x_{0} forever.

Beautiful! It would sound like snobbish, if a person as me use such a word, but it is beautiful.

Anyway such x_{0} for the present example can be obtained as the solution of following equation.

x_{0}^{2}- x_{0}+ a = 0

Features of this solution are effected by the value of a.

Graphically this x_{0} can be obtained as intersections of palaboric curve y = x^{2} + a and y = x. Dependent on a, it might be a single solution, or nothing.

Advertisements