3 ms·

Beautifully, iterating any continuous function will yield a fixed point of the function if it converges. https://en.wikipedia.org/wiki/Fixed-point_iteration ht

by ja3k 6y ago

Beautifully, iterating any continuous function will yield a fixed point of the function if it converges. https://en.wikipedia.org/wiki/Fixed-point_iteration https://en.wikipedia.org/wiki/Fixed-point_iteration

Sharlin 6y ago
And in case it neither converges to a single value nor diverges to infinity, a whole new worldof wondrous math emerges [1][2][3] [1] https://en.wikipedia.org/wiki/Logistic_map https://en.wikipedia.org/wiki/Logistic_map [2] https://en.wikipedia.org/wiki/Mandelbrot_set https://en.wikipedia.org/wiki/Mandelbrot_set [3] https://en.wikipedia.org/wiki/Attractor#Strange_attractor https://en.wikipedia.org/wiki/Attractor#Strange_attractor
Enginerrrd 6y ago
Interestingly, I've found this kinda sorta occasionally works on functionals of a differential equation too if you start with a good enough guess. Someone has probably made this concept and the necessary/sufficient conditions rigorous but its getting into territory that is a good bit too advanced for me to follow. As I recall from my fiddling, you are probably most likely to end up with a series solution of sorts, so it's a good idea to guess with polynomials or exponentials so you end up with component functions that form a basis for analytic functions.
- hansvm 6y ago
  It's not too bad to formalize :) The [https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikipedia https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikip... article on fixed point iteration) gives lip service to everything working fine in arbitrary metric spaces. It's easy to define a useful metric between functions (e.g. via integrating their difference, ignoring edge cases like occasional pointwise differences since a more careful treatment can give a presentation where those don't matter for the problem at hand), so the same kinds of theorems that work with cos(x) on the reals also work with differential operators on function spaces.

by ja3k 6y ago

Sharlin 6y ago

And in case it neither converges to a single value nor diverges to infinity, a whole new worldof wondrous math emerges [1][2][3] [1] https://en.wikipedia.org/wiki/Logistic_map https://en.wikipedia.org/wiki/Logistic_map [2] https://en.wikipedia.org/wiki/Mandelbrot_set https://en.wikipedia.org/wiki/Mandelbrot_set [3] https://en.wikipedia.org/wiki/Attractor#Strange_attractor https://en.wikipedia.org/wiki/Attractor#Strange_attractor

Enginerrrd 6y ago

Interestingly, I've found this kinda sorta occasionally works on functionals of a differential equation too if you start with a good enough guess. Someone has probably made this concept and the necessary/sufficient conditions rigorous but its getting into territory that is a good bit too advanced for me to follow. As I recall from my fiddling, you are probably most likely to end up with a series solution of sorts, so it's a good idea to guess with polynomials or exponentials so you end up with component functions that form a basis for analytic functions.

hansvm 6y ago
It's not too bad to formalize :) The [https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikipedia https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikip... article on fixed point iteration) gives lip service to everything working fine in arbitrary metric spaces. It's easy to define a useful metric between functions (e.g. via integrating their difference, ignoring edge cases like occasional pointwise differences since a more careful treatment can give a presentation where those don't matter for the problem at hand), so the same kinds of theorems that work with cos(x) on the reals also work with differential operators on function spaces.