On Mon, Jun 2, 2008 at 1:30 AM, Henryk Trappmann <[EMAIL PROTECTED]> wrote: > > On 1 Jun., 18:56, Carl Witty <[EMAIL PROTECTED]> wrote: >> Sage has such a decision procedure built in to its implementation of >> QQbar, the algebraic numbers. > > But I have to say they are quite hidden in the reference manual. > Thank you for this hint. I also found AA the algebraic reals if your > computations only involve reals. > >> sage: QQbar(sqrt(-2)*sqrt(-3) + sqrt(6)) == 0 >> True > > Using QQbar or AA also my previous example works: > QQbar(sqrt(5+2*sqrt(2)*sqrt(3))-sqrt(2)-sqrt(3)) == 0 > True > AA(sqrt(5+2*sqrt(2)*sqrt(3))-sqrt(2)-sqrt(3)) == 0 > True > > However is it a *bug* that this does not work in the symbolic Ring > with is_zero? > (sqrt(5+2*sqrt(2)*sqrt(3))-sqrt(2)-sqrt(3)).is_zero() > False
The definition of is_zero in the Sage symbolic ring is currently *precisely* the definition in Maxima. In fact, most of the functionality of the Sage symbolic ring is just a light wrapper around Maxima, though when we found bugs that Maxima couldn't fix (e.g., in floor/ceiling), we have worked around them. Work is under way to completely rewrite the symbolic ring. > And good to know that with QQbar or AA the rule of coercing to the > lowest precision works. > AA(sqrt(2))*1.0 > 1.41421356237309 > QQbar(sqrt(2)) * CC(1.0) > 1.41421356237309 > > But back to SymbolicRing and SymbolicConstant. > I have the following improvement > SUGGESTION: when creating sqrt(2) or other roots from integers, then > assign to them the parent AlgebraicReal or AlgebraicNumer accordingly > instead of the too general Symbolic Ring. > That's definitely planned. > Coercing (see above) and calculating with AA and QQbar works well, for > example: > sqrt(AA(sqrt(2))).parent() > Algebraic Real Field > sqrt(-AA(sqrt(2))).parent() > Algebraic Field > > For future development maybe one can introduce a RealSymbolicConstant > (and ComplexSymbolicConstant) which allow application of the common > real functions like +,-,*,/,^, exp, log, sin, cos, etc. However the > difficulty I see here is that we must be able to decide whether such > an expression is 0 if we want to reject expr1/expr2 or log(expr2) for > expr2 being 0. We also must be able to decide whether such an > expression is greater 0. When we want to determine the outcome > (RealSymbolicConstant or ComplexSymbolicConstant) of log(expr) or expr > ** (1/n). So maybe this isnt possible from a theoretical point of view > as those questions maybe undecidable (opposed to the algebraic case) > which however is not yet clear for me. Thanks for your comments. William --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
