Wednesday, 20 August 2008

computer science - Closed form of a nonlinear recurrence sequence.

As has already been explained, there is no hope in general of finding explicit solutions to nonlinear recurrences. However, for your example, it is possible to find limntoinftyfn(X) for all real X.



The function g(x)=(x2+x)/2 has two fixed points: x=0 (atractor) and x=1 (repulsor). Its respective stable sets are (2,1) and 2,1; (infty,2)cup(1,+infty) is the stable set of +infty. Thus,



lim_{ntoinfty}f_n(X)=left{matrix{0, & Xin(-2,1)cr 1, & Xin{-2,1}cr +infty, & Xin(-infty,-2)cup(1,+infty)}right.

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