- What is the best way on SAGE to get all the roots? - Can't I just ask SAGE to find the maximum root and tell me its value if it sees that all the roots are real?
On Monday, May 11, 2015 at 1:27:04 AM UTC-5, Kristoffer Ryhl-Johansen wrote: > > If you figure out that all roots are real, I'm pretty sure the best you > can do about checking if k is larger than all roots, is to take the > smallest of the bounds on roots here > <https://en.wikipedia.org/wiki/Properties_of_polynomial_roots#Bounds_on_.28complex.29_polynomial_roots>, > > and checking if k is larger than that. > > Note that the previous checks the absolute value, so if you don't care > about negative roots, take a look at this paper > <http://www.inf.uth.gr/wp-content/uploads/formidable/phd_thesis_vigklas.pdf> > or this paper > <http://www.jucs.org/jucs_15_3/linear_and_quadratic_complexity>. The > things mentioned in those papers should be implemented in sage, but I don't > know where. > > man. 11. maj 2015 kl. 07.43 skrev Phoenix <[email protected] > <javascript:>>: > >> >> Here is an example of a real-rooted polynomial generated by my code: >> >> 1296*x^24 - 20736*x^22 + 129600*x^20 - 393984*x^18 + 584496*x^16 - >> 362880*x^14 + 62208*x^12 >> >> >> This I know to be real-rooted because I tested this in Mathematica. >> >> >> In general my code will generate polynomials of degree ~200-300 >> >> Then is there a way to test real-rootedness inside SAGE itseld? >> >> And given some other number (typically as a squareroot of some other >> integer) can I test if the largest root is below that threshold? >> >> >> >> >> On Sunday, May 10, 2015 at 3:09:48 PM UTC-5, vdelecroix wrote: >>> >>> On 10/05/15 21:51, Dima Pasechnik wrote: >>> > On Sunday, 10 May 2015 19:11:27 UTC+1, vdelecroix wrote: >>> >> On 10/05/15 19:43, Dima Pasechnik wrote: >>> >>> A random polynomial for sure won't have the properties your >>> polynomial >>> >>> probably has (e.g. all roots real). >>> >> >>> >> Nope, but the characteristic polynomial of a "random" matrix in >>> SL(d,R) >>> >> would ;-) >>> >> >>> > >>> > really? I'd say a random symmetric matrix in M_d(R), yes, surely, but >>> not >>> > in SL_d(R)... >>> >>> Depends on your definition of random. If you pick any non-degenerate >>> generating set and look at a random product of length > d^2 then yes. >>> >>> Though, I do not know for the definition of random as "uniform on all >>> matrices in SL_d(R) with coefficients less than N". >>> >>> Vincent >>> >> -- >> You received this message because you are subscribed to the Google Groups >> "sage-support" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected] <javascript:>. >> To post to this group, send email to [email protected] >> <javascript:>. >> Visit this group at http://groups.google.com/group/sage-support. >> For more options, visit https://groups.google.com/d/optout. >> > -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
