On Jun 3, 2008, at 12:22 PM, Henryk Trappmann wrote: > On 31 Mai, 15:59, "William Stein" <[EMAIL PROTECTED]> wrote: >> At a bare minimum there is never a canonical (automatic) >> coercion from elements of R to elements of S unless that coercion >> is defined (as a homomorphism) on all of R. > > I dont want to be heretical by why is it so important that coercion is > totally defined? > > Considering all the current discussion, I get the impression that > allowing partial coercion would solve much of the arising problems > nearly without effort. > > SR can be easily partially coerced to any number system. Just try to > evaluate the SR term in that number system. Either it succeeds for > example > log(2) or sqrt(2) directly translates to RR by log(RR(2)) and > sqrt(RR(2)) or it fails, for example RR(x) or RR(1)/RR(0). > > While sqrt(RR(-2)) is somehow mixed, depending on the sign of the > argument sqrt returns either something of a complex or a real field. > Which is quite desirable. So that means if one would try to coerce > from SR to RR it can happen that the result is a CC, what however > would perfectly fit, i.e. would perfectly be what one expects when one > writes: sqrt(-2)*1.0. > > I hope everyone agrees that the above two (but at least the above > first) paragraph(s) is what one would expect/desire. > > Then for coercion would only matter the exactness of a structure and > the subset relation. The general rule would be: If there one structure > is less exact then coerce to this structure. If we have two structures > with equal exactness coerce to the substructure if possible, in doubt > coerce to the superstructure. > > This would also solve the performance and representation issue. As > long as no coercing takes places just compute in whatever Ring you > want. But if it comes to coercion/evaluation/decision it may take > somewhat longer (what also would be what I expect).
This is a good question, and has been extensively discussed before, but I will try and justify it a bit here. In Sage there are two kinds of conversions from one ring to another: casting and coercion. If I have an element a in S, and I want to view it as an element in R, I can write R(a). This should work whenever it's possible to make sense of a in R. Coercion, on the other hand, is implicit and thus should be fairly conservative. To reduce ambiguity, it should also go exactly one direction (unless, perhaps, there is a canonical isomorphism). It makes more sense for there to be a coercion RR -> SR than the other way around, so that when one writes "sqrt(x) * 2.4" it tries to give a result in SR rather than an error due to failure to give a result in RR. I'm not sure how allowing partial coercions would help the situation. The problems are (1) given a and b in different rings, how to quickly find a' and b' so that a' + b' makes sense and dispatch to that operation and (2) whether or not sqrt(2) should start out as a SR object, or something more specific the same way integers and rationals do, and then coerced down to SR if need be. - Robert --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
