Looking at the winding of f°0.5(x) at the second fixpoint and in contrast at the straight line at first fixpoint gives an idea for an explanation for another problem. I was always wondering, why the alternating series of consecutive iterates gives matching results for the serial summation and the matrix-computed summation in one direction and not in the other.
It seems, that this agrees insofar, that, if the trajectory converges straightly to the fixpoint the results match; and if the trajectory spirals the results don't match.
This seems to be more generalizable: it does not only occur with spiralling but also with oscillating trajectories as we observe this with the f(x)-function itself when iterated towards the second fixpoint.
I get, for the alternating sums, beginning at x=1, iterates towards 2'nd versus 1'st fixpoint
serial summation
0.709801988103 towards 2'nd fixpoint: \( \sum_{h=0}^{\infty} (-1)^h * f^{\circ h}(1.0) \)
0.419756033790 towards 1'st fixpoint: \( \sum_{h=0}^{\infty} (-1)^h * f^{\circ -h}(1.0) \)
Matrix-method:
0.580243966210 towards 2'nd fixpoint // incorrect, doesn't match serial summation
0.419756033790 towards 1'st fixpoint // matches serial summation
where systematically with the matrix-method the two results sum up to the parameter x in f°h(x). If we could find the functional equation for the relation between the series of the negative and of the postive heights we could give reliable values for the "sums" for very badly diverging series by that...
Gottfried
It seems, that this agrees insofar, that, if the trajectory converges straightly to the fixpoint the results match; and if the trajectory spirals the results don't match.
This seems to be more generalizable: it does not only occur with spiralling but also with oscillating trajectories as we observe this with the f(x)-function itself when iterated towards the second fixpoint.
I get, for the alternating sums, beginning at x=1, iterates towards 2'nd versus 1'st fixpoint
serial summation
0.709801988103 towards 2'nd fixpoint: \( \sum_{h=0}^{\infty} (-1)^h * f^{\circ h}(1.0) \)
0.419756033790 towards 1'st fixpoint: \( \sum_{h=0}^{\infty} (-1)^h * f^{\circ -h}(1.0) \)
Matrix-method:
0.580243966210 towards 2'nd fixpoint // incorrect, doesn't match serial summation
0.419756033790 towards 1'st fixpoint // matches serial summation
where systematically with the matrix-method the two results sum up to the parameter x in f°h(x). If we could find the functional equation for the relation between the series of the negative and of the postive heights we could give reliable values for the "sums" for very badly diverging series by that...
Gottfried
Gottfried Helms, Kassel