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/ -~----------~----~----~----~------~----~------~--~---
