accessing some pari types

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jocaps
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accessing some pari types

Message par jocaps » mar. janv. 23, 2018 12:57 pm

Hi,

I want to know how to access some member variable of a pari type that comes out of giac/giacpy. Here is an example:

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from giacpy import giac
giac("printpow(1)")
giac("pari()")
nfinit = giac("nfinit")
galoisinit = giac("galoisinit")

f = giac("t^3-3*t+1") #splitting field has primitive element which is a root of f
G=galoisinit(nfinit(f))
G.group
G.group gives me an error in giac. This is internally defined in Pari as the 6-th element of G (G is a sort of a list).
What I wanted to do is the following:

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permtopol=giac("galoispermtopol")
permtopol(G,G.group)
#even permtopol(G,G[5]) does not work, result is "undef"
In gp this outputs as [t,t^2-2,-t^2-t+4]

If interested, these are all the roots of t^3-3*t+1 expressed as an algebraic expression with respect to one specific root of the polynomial.

I could of course, straight away run a gp script.. but I wanted to take advantage of python and giacpy here as well (I am doing a lot of algebra here which giacpy has better features than pari)

Jose

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Re: accessing some pari types

Message par frederic han » mar. janv. 23, 2018 7:17 pm

Why not using rootof?

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>>> from giacpy import giac,rootof
>>> t=giac('t')
>>> f=t**3-3*t+1
>>> f.factor(rootof(f))
(t+rootof([[-1,0],[1,0,-3,1]]))*(t+rootof([[-1,0,2],[1,0,-3,1]]))*(t+rootof([[1,1,-2],[1,0,-3,1]]))
>>> f.factors(rootof(f))
[t+rootof([[-1,0],[1,0,-3,1]]),1,t+rootof([[-1,0,2],[1,0,-3,1]]),1,t+rootof([[1,1,-2],[1,0,-3,1]]),1]
NB: indeed your pari compution has also a problem in giac/xcas so it won't work in giacpy.

jocaps
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Re: accessing some pari types

Message par jocaps » mer. janv. 24, 2018 5:32 am

Hi Frederic,

Thank you for your answer. I do not understand the output of rootof, I do not see [t,t^2-2,-t^2-t+4] in the output of your sample code. Could you explain a bit?

More generally, if f is a polynomial over Q (of arbitrary degree) such that the splitting field of f is Q(a) for some root "a" of f, then I would like to have all the roots of f expressed as a polynomial of "a".

Jose

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Re: accessing some pari types

Message par parisse » mer. janv. 24, 2018 7:34 am

Well, there is a confusion in the meaning of t, your t is alpha=rootof([[1,0],[1,0,-3,1]]), rootof(P,M) where P and M are list polynomials means P(alpha) where M(alpha)=0 (M irreducible).
The last command of Frederic is the list of factors of t^3-3t+1, i.e. t+(-alpha), t+(-alpha^2+2), t+(alpha^2+alpha-2). Inside giac, you can get a more readable answer (without rootof) by running

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rootof(t^3-3t+1):=alpha;
factor(t^3-3t+1,alpha);

frederic han
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Re: accessing some pari types

Message par frederic han » mer. janv. 24, 2018 9:12 am

On giacpy you can do this.

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In [1]: from giacpy import giac,rootof
// Giac share root-directory:/usr/share/giac/
// Giac share root-directory:/usr/share/giac/
Help file /usr/share/giac/doc/fr/aide_cas not found
Added 0 synonyms

In [2]: t=giac('t')

In [3]: f=t**3-3*t+1

In [4]: j=rootof(f)

In [5]: (j**4).normal()
Out[5]: rootof([[3,-1,0],[1,0,-3,1]])

In [6]: f.factor(j)
Out[6]: (t+rootof([[-1,0],[1,0,-3,1]]))*(t+rootof([[-1,0,2],[1,0,-3,1]]))*(t+rootof([[1,1,-2],[1,0,-3,1]]))

In [7]: giac(str(j)+":=j")  # this trick let the rootof be printed as j. But the variable j inside giac will be assigned to this rootof so don't forget to purge it.
Out[7]: j

In [8]: f.factor(j)
Out[8]: (t+(-j))*(t+(-j**2+2))*(t+(j**2+j-2))

In [9]: j.purge()
Out[9]: rootof([1,0],poly1[1,0,-3,1])

In [10]: f.factor(j)
Out[10]: (t+rootof([[-1,0],[1,0,-3,1]]))*(t+rootof([[-1,0,2],[1,0,-3,1]]))*(t+rootof([[1,1,-2],[1,0,-3,1]]))
NB on windows the giac.dll provided by giacpy is built without NTL (because I think that my NTL built had optimisations for my processor and it gives crashes with lower processors) (so?) the following example seems too long on windows.

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# example from afactor.xws (in xcas examples)
u=rootof([1,0,-1176,0,345744,-537824])
giac(str(u)+":=u")
X,Y=giac('X,Y')
F2=Y**20+X**20+5*X**16*Y**4+Y**5*X**5+9*X**8*Y**4-6*X**14*Y**2-18*X**10*Y**6-18*X**6*Y**10+10*X**12*Y**8+10*X**8*Y**12+5*Y**16*X**4+9*Y**8*X**4-6*Y**14*X**2
F2.factor()
print("factorisation of F2 over QQ[u]:")
from time import time
t=time()
print(F2.factor(u))  # 5.5s on linux64bit, but too long on win32 because giacpy is built without NTL?
print(time()-t)

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Re: accessing some pari types

Message par frederic han » mer. janv. 24, 2018 9:16 am

NB: rootof is not any root of the polynomial but the greatest for lex ordering. So it makes sense to take approximations:
with j from the above example

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In [11]: j.approx()
Out[11]: 1.53208888624

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Re: accessing some pari types

Message par frederic han » mer. janv. 24, 2018 9:29 am

Indeed with giacpy-0.5.3-cp27-cp27m-win32.whl that was built with NTL I am able to compute this factorisation on a core i7 but only 6times slower that with linux 64bits.

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>>> from time import time
>>> t=time(); F2.factor(u); time()-t
(Y**4+u/14*Y*X+X**4)*(Y**16+(-u)/14*Y**13*X+4*Y**12*X**4+(u**2-1176)/196*Y**10*X**2+(-3*u)/14*Y**9*X**5+6*Y**8*X**8+(-u**3+1176*u)/2744*Y**7*X**3+(u**2-1176)/98*Y**6*X**6+(-3*u)/14*Y**5*X**9+4*Y**4*X**12+(u**4-1176*u**2+345744)/38416*Y**4*X**4+(-u**3+1176*u)/2744*Y**3*X**7+(u**2-1176)/196*Y**2*X**10+(-u)/14*Y*X**13+X**16)
36.61199998855591

jocaps
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Re: accessing some pari types

Message par jocaps » mer. janv. 24, 2018 12:46 pm

parisse a écrit :Well, there is a confusion in the meaning of t, your t is alpha=rootof([[1,0],[1,0,-3,1]]), rootof(P,M) where P and M are list polynomials means P(alpha) where M(alpha)=0 (M irreducible).
The last command of Frederic is the list of factors of t^3-3t+1, i.e. t+(-alpha), t+(-alpha^2+2), t+(alpha^2+alpha-2). Inside giac, you can get a more readable answer (without rootof) by running

Code : Tout sélectionner

rootof(t^3-3t+1):=alpha;
factor(t^3-3t+1,alpha);
Ah, I follow now. Thanks Bernard. I think this explains it to me.
frederic han a écrit :NB: rootof is not any root of the polynomial but the greatest for lex ordering. So it makes sense to take approximations:
with j from the above example

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In [11]: j.approx()
Out[11]: 1.53208888624
This I do not quite understand. What is meant by root of the polynomial with the greatest lex ordering (how do you lex order the roots? do you mean you take the complex number as tuple of two reals and lex order them?)? Sorry for the ignorance, maybe there is a terminology that is commonly used that I do not really know.

Jose

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Re: accessing some pari types

Message par parisse » mer. janv. 24, 2018 12:57 pm

It's not lex ordering. The root is choosen with the largest real part. In case of equality, the largest module, and for conjugates, positive imaginary part.

jocaps
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Re: accessing some pari types

Message par jocaps » mer. janv. 24, 2018 3:35 pm

Thanks. Regarding the ntl issue and windows I can remind everyone about the link of the discussion (crash in factor if giac is linked with ntl in windows):
http://xcas.e.ujf-grenoble.fr/XCAS/view ... f=3&t=1901

What if I am able to compile giac with mpir, will the lack of ntl still make factor slower? I am able to compile giac with mpir in windows (and visual studio).

Jose

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Re: accessing some pari types

Message par parisse » mer. janv. 24, 2018 4:41 pm

giac factor without NTL and PARI is slower for some polynomials, those who have many factors in Z/pZ and only a few in Z. If giac is compiled without NTL but with PARI, it should not be too slow.

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Re: accessing some pari types

Message par frederic han » ven. févr. 02, 2018 11:08 am

I have rebuilt NTL with:
'NATIVE' => 'off'
'TUNE' => 'generic'
in DoConfig

now I can't reproduce the crash on a core i3. Is it OK for you?
so with this package
http://webusers.imj-prg.fr/~frederic.ha ... -win32.whl
you should be able to run the above:

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F2.factor(u)

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