On 25 November 2014 at 14:51, Ondřej Čertík <ondrej.cer...@gmail.com> wrote: > On Tue, Nov 25, 2014 at 11:30 AM, Bill Page <bill.p...@newsynthesis.org> > wrote: ... >>>> Try it this way: >>>> >>>> a*b = exp(?1) >>>> a = exp(?2) >>>> b = exp(?3) >>>> >>>> I think 'normalize' is saying that there is a solution that makes >>>> >>>> ?1 - ?2 - ?3 = 0. >>> >>> Ok, but why wouldn't normalize return 2*pi*i instead? Or 4*pi*i? >> >> These are equivalent in the sense of having the same number of >> algebraically independent transcendental kernels, i.e. none. > > I don't understand that. Is the result of normalize() multivalued?
No. > Or how else could 0 be equivalent to 2*pi*i or 4*pi*i? It is not equality it is an equivalence relation i.e. "modulo constants". To dig deeper on this I think would need to consult the source code and someone who is much more of an expert in this subject: Waldek Hebisch. >>> In other words, how exactly are the operations on the multivalued >>> sets log(x) defined? >> >> FriCAS does not perform operations on multivalued sets to determine > the above. > > Ok. Though my question stands, how are the operations defined in your > approach? > Does it help if a say the operations are defined "symbolically"? Maybe we need to define exactly what operations we are talking about. > ... > Essentially the [derivative] formula with theta is equivalent to just > returning a tuple of the two Wirtinger derivatives. So what holds for > one approach holds for the other one. > Yes, so we agree that in general more than one derivative operator is necessary. > ... > My current best solution is to define a function `diff(x, theta=0)`, > where the theta argument is 0 by default, but you can pass any > angle into it, or a symbol theta if you want. That way you won't get > the theta factors by default, but if in doubt, you can always get them. > It seems that you prefer an "infinite" number of derivative operators while I still think it is best to define only two. > Let me know if you have a better proposal. > After continued thinking about this and my current experiments in FriCAS I am still of the opinion that the best option is to implement just the Wirtinger derivative (only one since the other can be obtained by 'conjugate'). This has the affect of making the derivative of non-analytic functions subtly different than what you call the conventional "real derivative" (e.g. factor of 1/2 in derivative of 'abs'). I have decided that I would prefer to explain this difference to a less experienced user, rather than to get into a discussion of theta and directional derivatives. Bill. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
