#11725: Generate random elements of the algebraic field
----------------------------+-----------------------------------------------
Reporter: rbeezer | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-4.7.2
Component: algebra | Keywords: random, QQbar
Work_issues: | Upstream: N/A
Reviewer: Simon Spicer | Author: Rob Beezer
Merged: | Dependencies:
----------------------------+-----------------------------------------------
Changes (by spice):
* keywords: => random, QQbar
* reviewer: => Simon Spicer
Old description:
> Random elements of QQbar default to the method implemented for the
> integers. This patch creates algebraic numbers via roots of random
> polynomials with integer coefficients.
>
> Implementation notes:
> 1. Degree 2 is the default so that some complex numbers are generated.
> 1. Might create constant polynomials, so a monic term (of random
> degree) may be added.
> 1. Could be some efficiencies to be gained by not creating all the
> roots and choosing just one, but this should be a good start.
> 1. Real goal was random vectors over `QQbar`. A doctest demonstrates
> how control of the integer coefficients can be passed from
> `random_vector()` to the `QQbar.random_element()` to
> `ZZ.random_element()`.
>
> '''Apply''':
> 1. [attachment:trac_11725-random-algebraic-numbers.patch]
New description:
Random elements of QQbar default to the method implemented for the
integers. This patch creates algebraic numbers via roots of random
polynomials with integer coefficients.
Implementation notes:
1. Degree 2 is the default so that some complex numbers are generated.
1. Might create constant polynomials, so a monic term (of random
degree) may be added.
1. Could be some efficiencies to be gained by not creating all the
roots and choosing just one, but this should be a good start.
1. Real goal was random vectors over `QQbar`. A doctest demonstrates
how control of the integer coefficients can be passed from
`random_vector()` to the `QQbar.random_element()` to
`ZZ.random_element()`.
'''Apply''':
1. [attachment:trac_11725-random-algebraic-numbers2.patch]
--
Comment:
Seems mostly fine to me. It would be easy to get bogged down as to what
the default behaviour of {{{QQbar.random_element()}}} should be, but as
implemented here this seems a reasonable first step. I did a test and
found that with default parameters {{{QQbar.random_element()}}} returns
zero about 15% of the time, nonzero real numbers about 60% of the time,
and nonzero imaginary numbers about 7% of the time:
{{{
sage: L = []
sage: for j in range(5000):
....: L.append(QQbar.random_element())
....:
sage: RDF(L.count(0))/len(L)
0.133
sage: C = [r.imag() == 0 and r != 0 for r in L]
sage: RDF(C.count(True))/len(C)
0.6194
sage: D = [r.real() == 0 and r != 0 for r in L]
sage: RDF(D.count(True))/len(D)
0.066
}}}
The doctests all work and illustrate the new behaviour well. I fixed a few
spelling errors, one grammatical error and added extra info in one other
place:
{{{
- ``poly_degree`` - default: 2 - degree of the random
polynomial over the integers that the algebraic number
is a root of.
}}}
becomes
{{{
- ``poly_degree`` - default: 2 - degree of the random
polynomial over the integers of which the returned
algebraic number is a root.
}}}
And
{{{
For example if we do not include zero as a possible
coefficient, there will never be a zero constant term, and
thus never a zero root. ::
}}}
becomes
{{{
For example, current default behaviour of this method
returns zero about 15% of the time; if we do not include zero as a
possible coefficient, there will never be a zero constant term, and
thus never a zero root. ::
}}}
I've uploaded the updated patch. These are trivial changes, but I suppose
this should be re-reviewed before it can be given a positive review.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11725#comment:2>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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