On Sat, Sep 17, 2016 at 10:39 AM, Jonathan Bober wrote:
> Ok, so I figured out what is going on here, at least.
>
> When S = CuspForms(N, k), a call to S.hecke_polynomial() passes through a
> bunch of abstract layers and then eventually arrives at
> S._compute_hecke_matrix_prime(self, p, prec=None),
> in sage/modular/modform/space.py. To compute the hecke matrix, this
> function computes a basis of cusp forms, explicitly computes the action of
> the Hecke operator on this basis, and then does linear algebra to find the
> matrix of this operator on this basis. Which is all rather roundabout,
> since it ends up using modular symbols to compute hecke matrices to compute
> a basis. I understand why this might take 5 hours.
>
> Maybe there is some sense to this very generic functionality, if, e.g., I
> can use it to more efficiently compute the same thing for a small
> dimensional subspace.
>
You should implement a more optimized approach in the case of just
computing the Hecke polynomial. There's a million things in Sage (or any
math software) where the most optimal strategy isn't implemented, so the
system falls back to something generic that works.
-- William
>
> On Sat, Sep 17, 2016 at 3:48 PM, Jonathan Bober wrote:
>
>> What is going on with the timings in the following examples?
>>
>> sage: S = CuspForms(1728, 2)
>> sage: %time f = S.hecke_polynomial(2)
>> CPU times: user 17276.68 s, sys: 13.46 s, total: 17290.14 s
>> Wall time: 17293.59 s
>> sage: f
>> x^253 + x^251 - 2*x^249
>>
>> (Meanwhile, in a separate Sage session...)
>>
>> sage: %time M = ModularSymbols(1728, 2, 1)
>> CPU times: user 0.47 s, sys: 0.03 s, total: 0.50 s
>> Wall time: 0.54 s
>> sage: %time S = M.cuspidal_subspace()
>> CPU times: user 1.57 s, sys: 0.00 s, total: 1.57 s
>> Wall time: 1.54 s
>> sage: %time f = S.hecke_polynomial(2)
>> CPU times: user 19.09 s, sys: 0.00 s, total: 19.09 s
>> Wall time: 19.07 s
>> sage: f
>> x^253 + x^251 - 2*x^249
>>
>> That's almost 5 hours using CuspForms() and just over 20 seconds using
>> ModularSymbols() directly. To compute the same thing. Is this expected
>> behavior? Am I being stupid somehow?
>>
>> (Note: I did this on a "very old" (2 years) version of Sage. But as other
>> checks, I also tried the first method on a relatively up to date version on
>> my laptop and stopped after about 7 minutes when it was using 15% of my
>> ram. I also tried Magma, where the standard ModularForms(),
>> CuspidalSubspace(), HeckePolynomial() commands complete quickly, and I
>> tried cloud.sagemath.com, where I again killed the computation after a
>> few minutes. Maybe if I find an old enough version of Sage the computations
>> will be the same speed...)
>>
>> For a long time, and until recently, I would have just tried the first
>> method, not knowing what I was doing, and would have said to myself
>> something like "I guess Sage is probably pretty good at computations with
>> trivial character, so this is probably just something that takes a long
>> time." Now instead I would guess that for at least some functionality the
>> ModularForms()/CuspForms() classes are so slow that they are effectively
>> broken (or, at least, not much more than a toy) and have been for a long
>> time. This is a shame, because I shouldn't even need to know that modular
>> symbols are a thing that exist if I just want to do some simple
>> computations like this.
>>
>> I would expect ModularForms/CuspForms to mostly be a light wrapper around
>> ModularSymbols, but something else must be going on here. One explanation
>> would be that for some reason the first computation triggers some large
>> computation that would make all subsequent computations faster, but the
>> computation of hecke_polynomial(3) didn't seem like it was going to be
>> fast. (I didn't wait for it to finish.)
>>
>
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--
William (http://wstein.org)
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