Well that answered my next question, which is whether this method could be used for Qbar.
Carl, what language is your code in. I would be interested in taking a look. Bill. On 23 Sep, 20:15, "William Stein" <[EMAIL PROTECTED]> wrote: > On 9/23/07, cwitty <[EMAIL PROTECTED]> wrote: > > > Here's one (heavily biased) example: > > > sage: sum([sqrt(AA(i)) for i in range(1, 1000)]) > > [21065.833110879048 .. 21065.833110879056] > > > I'm pretty sure that doing computations with this number algebraically > > requires dealing with polynomials of degree at least 2^168 (there are > > 168 primes less than 1000), which is obviously impossible. > > It is also possible to do very fast arithmetic in QQbar building on AA > as illustrated below: > > sage: R.<x> = AA[] > sage: Q.<i> = R.quotient(x^2 + 1) > sage: w = i * AA(3).sqrt(); w > [1.7320508075688771 .. 1.7320508075688775]*i > sage: omega = (1 + w)/2; omega > [0.86602540378443859 .. 0.86602540378443871]*i + 1/2 > sage: omega^3 > [-1.0000000000000003 .. -0.99999999999999988] > sage: omega + AA(5).sqrt() > [0.86602540378443859 .. 0.86602540378443871]*i + [2.7360679774997893 > .. 2.7360679774997899] > sage: z1 = omega = (1 + w)/2; omega > [0.86602540378443859 .. 0.86602540378443871]*i + 1/2 > sage: z1 = omega = (1 + w)/2 + sqrt(AA(5)) > sage: z1 > [0.86602540378443859 .. 0.86602540378443871]*i + [2.7360679774997893 > .. 2.7360679774997899] > sage: z1^10 > [2882.8577427505738 .. 2882.8577427505789]*i + [-37786.733390210728 .. > -37786.733390210720] > sage: (z1^10)[0].minpoly() > x^2 + 37851*x + 9713701/4 > sage: (z1^10)[1].minpoly() > x^4 - 17754903/2*x^2 + 75340716089409/16 > > This could probably be useful for some applications. > > 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://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
