3 ms·
How to do (c)? I tried to model the reals as lazily computed sequences of intervals with rational bounds where every next interval is subset of previous one, an
by chromoblob 3y ago
How to do (c)? I tried to model the reals as lazily computed sequences of intervals with rational bounds where every next interval is subset of previous one, and couldn't write a higher-order differentiator function because on every step for a given input interval there is output interval with non-zero (in most cases) size and the differentiated function may contain quick enough oscillations that the derivative is unbounded, and thus the derivative will stay unbounded no matter what you do. What do you mean by rate of convergence?
- orlp 3y agohttps://en.wikipedia.org/wiki/Modulus_of_convergence https://en.wikipedia.org/wiki/Modulus_of_convergence
- chromoblob 3y ago(I assume you refer to the sequence of reals that is used to compute the limit in the definition of derivative) But is there a way to somehow construct this modulus automatically for some functions? And if there isn't, then isn't requiring manual input of the modulus like just requiring manual input of the derivative itself? Can you typically find the modulus manually more easily than the derivative?
- orlp 3y ago> But is there a way to somehow construct this modulus automatically for some functions? In a sense. You would end up with https://en.wikipedia.org/wiki/Automatic_differentiation https://en.wikipedia.org/wiki/Automatic_differentiation. The amount of functions you can construct in this manner is massive, along with their derivatives, but it is not an operator that given an arbitrary computable function gives you its derivative.
- chromoblob 3y agoThank you.