Iteration exercises: f(x)=1/(1+x)

[Gottfried]

Hi friends -

just for my own exercise I've looked at the iteration of f(x) = 1/(1+x).
I've posed my discussion in a sci.math-thread with some interesting answers and have put things together in an article
iteration 1/(1+x)

I asked in sci.math, whether concurring methods for continuous iteration are possible; but with that simple function it seems, that all formulae including the matrix-diagonalization-method (and regular iteration with fixpoint-shift) lead (exactly) to the same results.
I'd still like to see, whether there are other iteration-approaches for this function to improve my understanding of the more general whereabouts an possible limitations of the regular iteration.

Gottfried

[bo198214]

just for my own exercise I've looked at the iteration of f(x) = 1/(1+x).

The interesting thing about fractional linear function is indeed that it does not depend on the fixed point. Particularly interesting does this become with non-real fixed points, like for example:



This function has two non-real fixed points: i and -i.
The regular iterations at both fixed points coincide (which is only the case with linear fractions, and we know for example that this is not the case for ).
Particularly it is real.

There is a different way to compute the regular iteration.
These linear fractions have the interesting property that they can be represented by its 2x2 matrices, in this case:

composition of two linear fractions corresponds to multiplication of their matrices.
And hence we can use here also matrix powers to obtain the regular iteration.

Without making the calculations too explicit, I give the result here:


To optically verify the iteration, I give the graphs for .
   

If you wonder where this factor comes from, its the angle of the eigenvalue(s) of the matrix, which are in this case and .

[tommy1729]

regular iterations at n fixed points correspond when the function can be represented by an n x n matrix with real or positive real entries.

for finite n of course.

if i recall correctly ...

[bo198214]

regular iterations at n fixed points correspond when the function can be represented by an n x n matrix with real or positive real entries.
...
if i recall correctly ...

Recall from where? I dont know about other finite matrix representations than those 2x2 of linear fractions (but was wondering whether it exists). Matrix representation in the sense that . For example which kind of function is representable with 3x3 matrices?

[tommy1729]

i dont know if it helps , but there is a relation to number representations.

extensions or variations on complex numbers can be given by square matrices.

then finding their parts ( e.g. real part ) leads to functions.

those functions satisfy differential equations and they link the matrix to functions ...

well at least for some 2n x 2n matrices. not sure about 3x3.

that might sound complicated ...

tommy1729