Re: [sage-support] Re: Polynomial representation over GF(2^n)
On Wednesday 13 Jun 2012, Oleksandr Kazymyrov wrote:
> And one more.
>
> The following is quite strange:
> sage: K=GF(2^8,'a',modulus=ZZ['x']("x^8 + x^7 + x^6 + x^4 + x^3 + x^2 +
> 1")) sage: b=K.multiplicative_generator()
> sage: c=K(0)
> sage: c.log(b)
> 0
> sage: c=K(1)
> sage: c.log(b)
> 0
>
> How to determine when c is 0 and when 1? Perhaps log should return
> -1 for finite fields when c=0?
Yep, this looks like a bug. Can you open a ticket and perhaps provide a patch?
> Kind regards,
> Oleksandr
>
> On Wednesday, June 13, 2012 3:02:46 PM UTC+2, Oleksandr Kazymyrov wrote:
> > Hi all,
> >
> > Continuing the
> > question > /cvtz4Zh-WYQ> ...
> >
> > Input data:
> > sage: K=GF(2^8,'a',modulus=ZZ['x']("x^8 + x^7 + x^6 + x^4 + x^3 + x^2 +
> > 1"))
> > sage: K.multiplicative_generator()
> > a^4 + a^3 + a
> > sage: P=PolynomialRing(K,'x')
> > sage: pol=P.random_element(5)
> > sage: pol
> > (a^7 + a^5 + a^3 + 1)*x^5 + (a^5 + a^4 + a^3 + a^2 + a + 1)*x^4 + (a^7 +
> > a^3 + a^2 + a)*x^3 + (a^5 + a^2)*x^2 + (a^6 + a^5 + a^4 + 1)*x + a^3
> >
> > 1. How to represent *pol *using multiplicative generator? The output must
> > be b^(i1)*x^5 + b^(i2)*x^4 + b^(i3)*x^3 + b^(i4)*x^2 + b^(i5)*x + a^3,
> > where b = K.multiplicative_generator(), i1-i5 logarithmic representation
> > of (a^7 + a^5 + a^3 + 1), (a^5 + a^4 + a^3 + a^2 + a + 1)...
> >
> > I have a correct function:
> > sage: b=K.multiplicative_generator()
> > sage: P(["({0})^{1}".format(b,i.log(b)) for i in pol.coeffs()]) == pol
> > True
> > sage: ["({0})^{1}".format(b,i.log(b)) for i in pol.coeffs()]
> > ['(a^4 + a^3 + a)^27', '(a^4 + a^3 + a)^234', '(a^4 + a^3 + a)^195',
> > '(a^4 + a^3 + a)^134', '(a^4 + a^3 + a)^79', '(a^4 + a^3 + a)^101']
> >
> > If I have one program then the above method suits me. But if two, i.e.
> > one program represents polynomial in the form b^(i1)*x^5 ... and returns
> > b and another program tries to restore it, then I worry about the phrase
> > The docs for multiplicative_generator() say: "return a generator of the
> > multiplicative group", then add "Warning: This generator might change
> > from one version of Sage to another."
> >
> > 2. So the question is how to uniquely reconstruct the original *pol*, if
> > I have *modulus*, *multiplicative generator *and* representation of pol
> > as *b^(i1)*x^5...?
> >
> > Regards,
> > Oleksandr
Cheers,
Martin
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[sage-support] Numerically integrating over a region defined by linear inequalities
Hi I want to numerically evaluate the integral of a function f(x,y) over a region defined by linear inequalities, for example 1/8<=y<=x<=1/3 x+y<=1/3. I can do this with a repeated call to numerical_integral because I can re-write the constraints as y<=min(x,1/3-x). However, this solution isn't very satisfying in general. If I have more constraints and/or more variables then working out the limits on the repeated integrals by hand is going to be rather unpleasant. Is there any way I can get sage to compute the integral directly from the inequalities. For example, if I use a Polyhedron object to store the region of interest is there any way of getting functions from it which describe the limits of integration? Ideally, if P is a Polyhedron it would be nice to have something like P.integral(f) which computes the integral of f over P. Many thanks Alastair Irving -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
[sage-support] Re: Numerically integrating over a region defined by linear inequalities
On 2012-06-16, Alastair Irving wrote: > Hi > > I want to numerically evaluate the integral of a function f(x,y) over a > region defined by linear inequalities, for example > 1/8<=y<=x<=1/3 > x+y<=1/3. > I can do this with a repeated call to numerical_integral because I can > re-write the constraints as y<=min(x,1/3-x). However, this solution > isn't very satisfying in general. If I have more constraints and/or > more variables then working out the limits on the repeated integrals by > hand is going to be rather unpleasant. > > Is there any way I can get sage to compute the integral directly from > the inequalities. For example, if I use a Polyhedron object to store > the region of interest is there any way of getting functions from it > which describe the limits of integration? Ideally, if P is a Polyhedron > it would be nice to have something like P.integral(f) which computes the > integral of f over P. there are plans to create a Sage interface to LattE Integrale, a package that can in particular solve these kinds of tasks. (see http://www.math.ucdavis.edu/~latte/) For the time being you can roll your own, at least for compact P: compute a triangulation of P and for each simplex of triangulation use an explicit formula. HTH, Dmitrii > > Many thanks > > Alastair Irving > -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
[sage-support] Error message when use vectors in Sage 5 with Ubuntu 12.04...
I installed Ubuntu 12.04 about 2 weeks ago along with Sage 5. I just now tried using vectors for the first time and got this error... sage: vector( [-1,2] ) --- AttributeErrorTraceback (most recent call last) /home/cs/Ws/Lone_Star/1401/Exams/ in () /usr/local/sage-5.0-linux-32bit-ubuntu_12.04_lts-i686-Linux/local/lib/python2.7/site-packages/sage/modules/free_module_element.so in sage.modules.free_module_element.vector (sage/modules/free_module_element.c:3789)() /usr/local/sage-5.0-linux-32bit-ubuntu_12.04_lts-i686-Linux/local/lib/python2.7/site-packages/numpy/__init__.py in () 134 return loader(*packages, **options) 135 --> 136 import add_newdocs 137 __all__ = ['add_newdocs'] 138 /usr/local/sage-5.0-linux-32bit-ubuntu_12.04_lts-i686-Linux/local/lib/python2.7/site-packages/numpy/add_newdocs.py in () 7 # core/fromnumeric.py, core/defmatrix.py up-to-date. 8 > 9 from numpy.lib import add_newdoc 10 11 ### /usr/local/sage-5.0-linux-32bit-ubuntu_12.04_lts-i686-Linux/local/lib/python2.7/site-packages/numpy/lib/__init__.py in () 11 12 import scimath as emath ---> 13 from polynomial import * 14 #import convertcode 15 from utils import * /usr/local/sage-5.0-linux-32bit-ubuntu_12.04_lts-i686-Linux/local/lib/python2.7/site-packages/numpy/lib/polynomial.py in () 9 import re 10 import warnings ---> 11 import numpy.core.numeric as NX 12 13 from numpy.core import isscalar, abs, finfo, atleast_1d, hstack AttributeError: 'module' object has no attribute 'core' sage: -- To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URL: http://www.sagemath.org
