On Sep 23, 2009, at 6:11 PM, rjf wrote: > If you want to look at possible definitions of solve that have been > refined more recently than Maxima's solve, you can look at > Mathematica's > Solve, NSolve, RSolve, Reduce, and maybe some others like Eliminate. > > Maxima's solve dates to 1971, but there is also linsolve, algsys, > realroots, and some other programs that you can probably find in the > documentation ... roots, newton... and there may also be numerical > rootfinders in the Fortran library. The decision as to whether > to_poly_solve does the job for you (apparently not, at least at the > moment) or not, is only one aspect. > > I am pleased that you consider, at least as an option, augmenting > Maxima by loading in programs into Maxima written in the Maxima > language (to_poly_solve). Maybe someone would define what you want > Sage's "solve" to do in all cases, and feed the disambiguation > information to a short Maxima program that then calls solve, > to_poly_solve, roots, etc etc as > necessary. You might even find that your short program would be > useful to people directly using Maxima. > > There is, in my experience, a gap between people who think that all > roots of a polynomial can be found [by complex rootfinders] and those > people who want exact algebraic expressions. Or those who want > rational isolating intervals for each real root, computed by Sturm > sequences. > > And then there are the people who need to represent infinite sets like > the roots of sin(x)=0. > > The gap between these groups is kind of cognitive. Like "what do you > mean you can't find the roots of a quintic because it is unsolvable?? > It has 5 roots!"
Yep, we support various root numerical root finders, linear solvers, grobner bases, and symbolic solvers (via Maxima--you're right, we should look into exposing and using the various package here). The non-symbolic ones are typically orders of magnitude faster, if you're in that special case. This is one area where the idea of a domain like Sage has is useful. sage: x = ZZ['x'].gen() sage: f = x^6 - 2*x^5 - x^2 + x + 2 sage: f.roots(QQ) [(2, 1)] sage: f.roots(QQbar) # these are exact [(1.167303978261419?, 1), (2, 1), (-0.7648844336005847? - 0.3524715460317263?*I, 1), (-0.7648844336005847? + 0.3524715460317263? *I, 1), (0.1812324444698754? - 1.083954101317711?*I, 1), (0.1812324444698754? + 1.083954101317711?*I, 1)] sage: f.roots(RR) [(1.16730397826142, 1), (2.00000000000000, 1)] sage: f.roots(RealField(500)) [(1.16730397826141868425604589985484218072056037152548903914008244927565 190342952705318068520504972867289535916899524104793645129596750871791336 957872256, 1), (2.000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000 00000000, 1)] sage: f.roots(CDF) [(1.16730397826, 1), (2.0, 1), (-0.764884433601 - 0.352471546032*I, 1), (-0.764884433601 + 0.352471546032*I, 1), (0.18123244447 - 1.08395410132*I, 1), (0.18123244447 + 1.08395410132*I, 1)] sage: f.roots(GF(17)) [(8, 1), (6, 1), (2, 1)] sage: sage: f.roots(Qp(17, 5)) [(8 + 17 + 2*17^2 + 17^3 + 11*17^4 + O(17^5), 1), (6 + 17 + 8*17^2 + 14*17^3 + 7*17^4 + O(17^5), 1), (2 + O(17^5), 1)] Alternatively, sage: x = RR['x'].gen() sage: f = x^6 - 2*x^5 - x^2 + x + 2 sage: parent(f) Univariate Polynomial Ring in x over Real Field with 53 bits of precision sage: f.roots() [(1.16730397826142, 1), (2.00000000000000, 1)] etc. - Robert --~--~---------~--~----~------------~-------~--~----~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
