Am 2016-09-20 um 20:22 schrieb Marc Mezzarobba: > Clemens Heuberger wrote: >> x = polygen(QQ) >> equation = -960000000*x^7 + 416000000*x^6 - 66400000*x^5 + 5600000*x^4 >> - 280000*x^3 + 8400*x^2 - 140*x + 1 roots = equation.roots(QQbar) >> a_root = roots[-1][0] >> abs_root = abs(a_root) > [...] >> Is this expected behaviour? > > Well, QQbar has a number of well-known but not yet fixed efficiency > problems... > >> I am intersted in the smallest root(s) in >> absolute value only, any suggestions for achieving that in less time? > > You could perhaps compute a polynomial whose roots include the z·conj(z) > for all roots z of equation (e.g., with a resultant), factor that > polynomial, and sort its root numerically while increasing the > precision until you can tell which of the factors correspond to > dominant roots. Or something like that :-/

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Thank you for your comments. The problem is the square root: sage: x = polygen(QQ) sage: polynomial = x^5 - 1/3*x^4 + 1/30*x^3 - 1/600*x^2 + 1/24000*x - 1/2400000 sage: %time root = polynomial.roots(QQbar)[1][0] CPU times: user 20 ms, sys: 0 ns, total: 20 ms Wall time: 19.7 ms sage: %time norm_root = root.norm() CPU times: user 0 ns, sys: 0 ns, total: 0 ns Wall time: 74.1 µs sage: %time norm_root.minpoly() CPU times: user 136 ms, sys: 4 ms, total: 140 ms Wall time: 137 ms x^10 - 1/30*x^9 + 37/72000*x^8 - 97/21600000*x^7 + 613/25920000000*x^6 - 2009/25920000000000*x^5 + 19/115200000000000*x^4 - 97/414720000000000000*x^3 + 1/4608000000000000000*x^2 - 1/8294400000000000000000*x + 1/33177600000000000000000000 sage: %time abs_root = sqrt(norm_root) CPU times: user 8 ms, sys: 0 ns, total: 8 ms Wall time: 9.44 ms sage: %time abs_root.minpoly() CPU times: user 2min 17s, sys: 31.3 s, total: 2min 48s Wall time: 2min 48s x^20 - 1/30*x^18 + 37/72000*x^16 - 97/21600000*x^14 + 613/25920000000*x^12 - 2009/25920000000000*x^10 + 19/115200000000000*x^8 - 97/414720000000000000*x^6 + 1/4608000000000000000*x^4 - 1/8294400000000000000000*x^2 + 1/33177600000000000000000000 Especially for taking the square root, it would be rather easy to get a minimal polynomial by hand: sage: %time AA.polynomial_root(norm_root.minpoly().subs(x=x^2), ....: sqrt(RIF(norm_root))).minpoly() CPU times: user 124 ms, sys: 0 ns, total: 124 ms Wall time: 124 ms x^20 - 1/30*x^18 + 37/72000*x^16 - 97/21600000*x^14 + 613/25920000000*x^12 - 2009/25920000000000*x^10 + 19/115200000000000*x^8 - 97/414720000000000000*x^6 + 1/4608000000000000000*x^4 - 1/8294400000000000000000*x^2 + 1/33177600000000000000000000 For my particular problem (finding the root of minimal absolute value), the solution is actually much simpler: comparing the norms is certainly sufficient and works instanteneouly. Regards, Clemens -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.