Timeline for How to find the best convergence rate of a dynamical system $x_{n+1} = g(x_n),\ n\ge 0,\ x_0\in \mathbb{R}$?
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| Jun 28, 2019 at 8:22 | comment | added | Greg Martin | @PietroMajer: see this answer and thank you for your comments here! | |
| Apr 23, 2018 at 15:17 | comment | added | Samrat Mukhopadhyay | @PietroMajer, thanks for the comments, they were really helpful. | |
| Apr 23, 2018 at 8:31 | comment | added | Pietro Majer | 3)Assuming $|g'(0)|<1$ completely changes your question. Then g is locally a contraction (there is an invariant nbd of 0 where it is a contraction). In this case the first non zero derivative of g at 0 gives the convergence rate. | |
| Apr 23, 2018 at 8:27 | comment | added | Pietro Majer | 2) If $g'(0)=1$ and $g''(o)<0$ then locally the behavior is the same as for $x-x^2$. | |
| Apr 23, 2018 at 8:25 | comment | added | Pietro Majer | @SamratMukhopadhyay: 1) consider the sequence $y_n:=1/x_n$, satisfying $y_{n+1}=y_n+{1\over 1-1/y_n}=y_n+o(1)$, so that $y_n=n(1+o(1))$, then boot-strap. | |
| Apr 23, 2018 at 8:05 | comment | added | Samrat Mukhopadhyay | @PietroMajer, One other thing, for this particular map, the fixed point $0$ has the property $g'(0)=1$, which is why this is not a contraction, locally in a neighborhood of $0$. However, my situation is simpler. I want to analyze the rate of convergences for mappings which has $|g'(0)|<1$, but might not be a contraction globally. For example the $x^2$ map. Though it is locally a contraction at $0$, the convergence rate is actually quadratic rather than linear. I want to understand this kind of situation for general functions. | |
| Apr 23, 2018 at 7:44 | comment | added | Samrat Mukhopadhyay | @PietroMajer, also can you kindly tell me how you obtained the expansion for $x_n$? And can that method be applied to any ``sufficiently smooth function'' to get an idea of the convergence rate? | |
| Apr 23, 2018 at 7:29 | comment | added | Samrat Mukhopadhyay | @PietroMajer, for $g(x) = x - x^2$, I see that the convergence is sublinear. So I guess what I am asking should be applicable for superlinearly converging sequences. | |
| Apr 20, 2018 at 8:51 | comment | added | RaphaelB4 | A Taylor expansion around $0$ of $g$ should give you an answer for most of the practical cases. | |
| Apr 20, 2018 at 8:27 | comment | added | Pietro Majer | But note e.g. that for $g(x)=x-x^2$, that has $g'(0)=1$, starting with $0<x_0\le1$ produces $x_n=1/n +o(1/n)$ (more precisely $x_n=1/n-(\log n)/n^2+o(1/n^2)$ I think) | |
| Apr 20, 2018 at 7:47 | history | edited | Samrat Mukhopadhyay | CC BY-SA 3.0 |
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| Apr 20, 2018 at 7:35 | history | asked | Samrat Mukhopadhyay | CC BY-SA 3.0 |