Wow, thanks Vincent. This is awesome! Could you shortly explain what the
crucial difference is here?
On this occasion, may I ask you a slightly off-topic question? Is there an
effective way to compute the polarization of homogeneous polynomials in
Sage?
Best
Michael
Am Dienstag, 21. April 2020 12:08:56 UTC+2 schrieb vdelecroix:
>
> Hi Michael,
>
> Indeed, Sym.e().from_polynomial(P(seq_taylor)) is taking ages!
> Using the function from the attached you can compute todd(12)
> is computed in 18ms :-)
>
> sage: todd(12)
> e[] + 1/2*e[1] + 1/12*e[1, 1] - 1/720*e[1, 1, 1, 1] + 1/30240*e[1, 1, 1,
> 1, 1, 1] + 1/12*e[2] + 1/24*e[2, 1] + 1/180*e[2, 1, 1] - 1/1440*e[2, 1,
> 1, 1] - 1/5040*e[2, 1, 1, 1, 1] + 1/240*e[2, 2] + 1/480*e[2, 2, 1] +
> 11/60480*e[2, 2, 1, 1] + 1/6048*e[2, 2, 2] + 1/720*e[3, 1] + 1/1440*e[3,
> 1, 1] + 1/12096*e[3, 1, 1, 1] + 11/60480*e[3, 2, 1] - 1/60480*e[3, 3] -
> 1/720*e[4] - 1/1440*e[4, 1] - 1/12096*e[4, 1, 1] - 1/6720*e[4, 2] -
> 1/30240*e[5, 1] + 1/30240*e[6]
> sage: %time t = todd(12)
> CPU times: user 18.5 ms, sys: 112 µs, total: 18.6 ms
> Wall time: 18.1 ms
>
> Best
> Vincent
>
>
>
> Le 21/04/2020 à 11:20, Michael Jung a écrit :
> > Hi Travis,
> > yes, that is exactly what I want to do. I want to compute multiplicative
> > sequences from arbitrary formal power series, which has use in geometry.
> > However, in that paper, they compare two different approaches. And even
> the
> > general (slower) approach (elimination) seems to be faster than Sage's
> > algorithm. Let me show you what I have tried so far.
> >
> > sage: n = 12
> > sage: f = x / (1-exp(-x))
> > sage: Sym = SymmetricFunctions(QQ)
> > sage: P = PolynomialRing(QQ, 'x', n)
> > sage: seq = prod(f.subs({f.default_variable(): var}) for var in
> P.gens())
> > sage: inp = [(var, 0) for var in P.gens()]
> > sage: seq_taylor = seq.taylor(*inp, n // 2)
> > sage: Sym.e().from_polynomial(P(seq_taylor))
> >
> > This code is to compute Todd numbers up the the 6th degree and takes
> hours
> > of computation. Did I do use symmetric functions wrong here?
> >
> > Regards
> > Michael
> >
> > Am Dienstag, 21. April 2020 02:44:19 UTC+2 schrieb Travis Scrimshaw:
> >>
> >> Hi Michael,
> >> Perhaps I am misunderstanding what you are trying to do. I was
> thinking
> >> you were starting with a polynomial in, say, R.<x,y,z> that you know is
> >> symmetric and you want its expression in expressed in terms of
> elementary
> >> symmetric functions. Whereas in that paper, they seem to be computing
> very
> >> specific symmetric polynomials by applying the constraints of the
> geometry.
> >> However, there are some specialized change of bases that could be done,
> >> such as p -> e using the contents of section 2, that could be faster
> >> (although in that case, Sage seems to be pretty fast). Or have I
> >> misunderstood something?
> >>
> >> Thanks,
> >> Travis
> >>
> >>
> >> On Tuesday, April 21, 2020 at 10:00:35 AM UTC+10, Michael Jung wrote:
> >>>
> >>> Thanks for your reply Travis.
> >>>
> >>> Here are different computations compared, which take milliseonds up to
> >>> seconds: https://orbilu.uni.lu/bitstream/10993/21949/2/ChernLib.pdf
> >>>
> >>> In contrast, doing a similar computation with the same polynomials
> using
> >>> `SymmetricFunctions` takes hours.
> >>>
> >>> However, I think this is a special case they consider, which makes the
> >>> computation probably faster.
> >>>
> >>> Kind regards
> >>> Michael
> >>>
> >>> Am Dienstag, 21. April 2020 01:52:43 UTC+2 schrieb Travis Scrimshaw:
> >>>>
> >>>> Hi Michael,
> >>>> The process is to take a polynomial, convert it to the monomial
> sym
> >>>> func basis, then to the elementary basis (which is outsourced to
> >>>> symmetrica). Do you have some references for these efficient
> algorithms to
> >>>> convert a polynomial directly to the elementary basis?
> >>>>
> >>>> Best,
> >>>> Travis
> >>>>
> >>>>
> >>>> On Tuesday, April 21, 2020 at 3:18:37 AM UTC+10, Michael Jung wrote:
> >>>>>
> >>>>> Dear Sage developers,
> >>>>>
> >>>>> currently, I am working on an alternative algorithm to compute
> >>>>> characteristic forms. I hope to gain a speed-up here. For this
> reason, I
> >>>>> need to express symmetric polynomials in terms of elementary
> symmetric
> >>>>> functions. At the moment, I am playing around with
> `SymmetricFunctions`.
> >>>>> However, the implemented algorithm seems to be very slow. I know
> that there
> >>>>> are quite efficient algorithms out there, but are they accessible in
> Sage?
> >>>>>
> >>>>> Thanks for your help and best wishes
> >>>>> Michael
> >>>>>
> >>>>
> >
>
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