On Thursday, December 11, 2014 8:16:09 AM UTC-8, Joachim Durchholz wrote: > > Am 11.12.2014 um 00:40 schrieb Richard Fateman: > > 1994 paper by Adam Dingle and Richard Fateman > > Branch Cuts in Computer Algebra, (ISSAC '94 proceedings. also search > > online). > > That paper assumes that everything can be refactored to logarithms plus > arithmetic. > Does that assumption hold? I could imagine that expressions containing > irreducible integrals might not be normalizable in that fashion. > > > When you say things about sqrt(), does it generalize to cuberoot? If it > > does not, you are in trouble, or will be down the road. > > > > What is the principal value of (1)^(1/6) ? >
Well, there are 6 possible values. You could pick (same as sqrt rule?) the positive one, namely 1. That doesn't strike me as a good choice, since powers of 1 do not generate the other roots. > > Not sure what you mean with that. > > > Several systems have RootOf( ...) expressions. e.g. RootOf(s^2=x, > s). > > There are 2 values for s. What systems? > > Maple > > Macsyma > > Axiom/Fricas > > Mathematica > > probably others. > > Not sure how that relates to the point you're making. > If instead of sqrt(x^2) --> abs(x) or some such thing, you could consider an expression h= RootOf(y=x^2,x). Then the only "simplifications" that would be allowed would be ones that are valid for ANY choice of root. for example, squaring h gives you y. > > > You think you can use sqrt because the quadratic formula says > > -b+-sqrt(b^2....) > > etc. > > So you think you know what sqrt means. > > Not sure what assumptions you assume. > No assumptions whatsoever. There are 2 roots to a quadratic. Well known fundamental theorem of algebra, or some such. > > > But in that formula you can switch the values +- and the formula > > is still valid. > > Not sure what you mean with that - switching the signs means I still get > the same set of expressions. > That's why the formula works. switching the sign on sqrt just exchanges the roots. > > > How many other formulas do you expect to fiddle with where that is true? > > > > Certainly not this one: sqrt(y^2) = abs(y). > > That could still be handled by doing case distinctions. > If you plot abs(y) you get a V-shaped curve, with a singularity at 0. neither square root of y^2 has such a plot. > > This is an interesting case though, since it tells us that we need a > quantor: Are we interested in "there exists a branch where the > expression holds" or in "the expression holds in all branches"? > I think we need the former when determining all solutions to an > equation, and the latter when verifying an assumption. > I'm not familiar with the term quantor, but it makes sense to me to have 2 separate questions here. Maxima's radcan() does something like the first. It produces a particular interpretation of radicals that is intended to be canonical (and often is) > > > So if you go off and do the wrong thing, it is probably prudent to > > understand that you are doing the wrong thing. > > Actually the "wrong thing" could be the right thing in specific > circumstances (experimental physics is full of this kind of stuff), perhaps that is why some people advise against learning math from a physicist. (I was a physics major as an undergraduate though) > but > I agree that you need to be aware of the kind of shortcut you're taking > and where the boundaries are where the approach would break. > > I'm not sure what kind of "wrong thing" you are referring to in this > particular exchange though. > You paper takes a geometric approach (defining cuts as lines across the > complex plane), and I can see that that might be more appropriate than > blindly enumerating all branches, which might run into performance > issues, and would need quite a bit of algorithmic work to deal with > I think the paper has had little impact on how people write systems. It has had a slight impact on how people write papers. -- You received this message because you are subscribed to the Google Groups "sympy" group. To unsubscribe from this group and stop receiving emails from it, send an email to sympy+unsubscr...@googlegroups.com. To post to this group, send email to sympy@googlegroups.com. Visit this group at http://groups.google.com/group/sympy. To view this discussion on the web visit https://groups.google.com/d/msgid/sympy/b3c1429c-2741-4c3b-93d7-9f4eefa70ede%40googlegroups.com. For more options, visit https://groups.google.com/d/optout.
