def __init__(self, verbosity=0, confl_limit=None, threads=None):
r"""
Constuct a new CryptoMiniSat instance.
See the documentation class for the description of inputs.
EXAMPLES::
sage: from sage.sat.solvers.cryptominisat import CryptoMiniSat
sage: solver = CryptoMiniSat(threads=1) # optional - cryptominisat
"""
if threads is None:
from sage.parallel.ncpus import ncpus
threads = ncpus()
if confl_limit is None:
from sys import maxint
confl_limit = maxint
try:
from pycryptosat import Solver
except ImportError:
from sage.misc.package import PackageNotFoundError
raise PackageNotFoundError("cryptominisat")
self._solver = Solver(verbose=int(verbosity), confl_limit=int(confl_limit), threads=int(threads))
self._nvars = 0
self._clauses = []
def _start(self):
try:
Expect._start(self)
except RuntimeError:
from sage.misc.package import PackageNotFoundError
raise PackageNotFoundError("kash")
# Turn off the annoying timer.
self.eval('Time(false);')
def _start(self):
try:
Expect._start(self)
except RuntimeError:
# TODO: replace this error with something more accurate.
from sage.misc.package import PackageNotFoundError
raise PackageNotFoundError("kash")
# Turn off the annoying timer.
self.eval('Time(false);')
def RandomLinearCodeGuava(n, k, F):
r"""
The method used is to first construct a `k \times n` matrix of the block
form `(I,A)`, where `I` is a `k \times k` identity matrix and `A` is a
`k \times (n-k)` matrix constructed using random elements of `F`. Then
the columns are permuted using a randomly selected element of the symmetric
group `S_n`.
INPUT:
- ``n,k`` -- integers with `n>k>1`.
OUTPUT:
Returns a "random" linear code with length `n`, dimension `k` over field `F`.
EXAMPLES::
sage: C = codes.RandomLinearCodeGuava(30,15,GF(2)); C # optional - gap_packages (Guava package)
[30, 15] linear code over GF(2)
sage: C = codes.RandomLinearCodeGuava(10,5,GF(4,'a')); C # optional - gap_packages (Guava package)
[10, 5] linear code over GF(4)
AUTHOR: David Joyner (11-2005)
"""
current_randstate().set_seed_gap()
q = F.order()
if not is_package_installed('gap_packages'):
raise PackageNotFoundError('gap_packages')
gap.load_package("guava")
gap.eval("C:=RandomLinearCode(" + str(n) + "," + str(k) + ", GF(" +
str(q) + "))")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[
gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i, j)), F)
for j in range(1, n + 1)
] for i in range(1, k + 1)]
MS = MatrixSpace(F, k, n)
return LinearCode(MS(G))
def QuasiQuadraticResidueCode(p):
r"""
A (binary) quasi-quadratic residue code (or QQR code).
Follows the definition of Proposition 2.2 in [BM]. The code has a generator
matrix in the block form `G=(Q,N)`. Here `Q` is a `p \times p` circulant
matrix whose top row is `(0,x_1,...,x_{p-1})`, where `x_i=1` if and only if
`i` is a quadratic residue `\mod p`, and `N` is a `p \times p` circulant
matrix whose top row is `(0,y_1,...,y_{p-1})`, where `x_i+y_i=1` for all
`i`.
INPUT:
- ``p`` -- a prime `>2`.
OUTPUT:
Returns a QQR code of length `2p`.
EXAMPLES::
sage: C = codes.QuasiQuadraticResidueCode(11); C # optional - gap_packages (Guava package)
[22, 11] linear code over GF(2)
These are self-orthogonal in general and self-dual when $p \\equiv 3 \\pmod 4$.
AUTHOR: David Joyner (11-2005)
"""
if not is_package_installed('gap_packages'):
raise PackageNotFoundError('gap_packages')
F = GF(2)
gap.load_package("guava")
gap.eval("C:=QQRCode(" + str(p) + ")")
gap.eval("G:=GeneratorMat(C)")
k = int(gap.eval("Length(G)"))
n = int(gap.eval("Length(G[1])"))
G = [[
gfq_gap_to_sage(gap.eval("G[%s][%s]" % (i, j)), F)
for j in range(1, n + 1)
] for i in range(1, k + 1)]
MS = MatrixSpace(F, k, n)
return LinearCode(MS(G))
def best_linear_code_in_guava(n, k, F):
r"""
Returns the linear code of length ``n``, dimension ``k`` over field ``F``
with the maximal minimum distance which is known to the GAP package GUAVA.
The function uses the tables described in ``bounds_on_minimum_distance_in_guava`` to
construct this code. This requires the optional GAP package GUAVA.
INPUT:
- ``n`` -- the length of the code to look up
- ``k`` -- the dimension of the code to look up
- ``F`` -- the base field of the code to look up
OUTPUT:
- A :class:`LinearCode` which is a best linear code of the given parameters known to GUAVA.
EXAMPLES::
sage: codes.databases.best_linear_code_in_guava(10,5,GF(2)) # long time; optional - gap_packages (Guava package)
[10, 5] linear code over GF(2)
sage: gap.eval("C:=BestKnownLinearCode(10,5,GF(2))") # long time; optional - gap_packages (Guava package)
'a linear [10,5,4]2..4 shortened code'
This means that the best possible binary linear code of length 10 and
dimension 5 is a code with minimum distance 4 and covering radius s somewhere
between 2 and 4. Use ``bounds_on_minimum_distance_in_guava(10,5,GF(2))``
for further details.
"""
if not is_package_installed('gap_packages'):
raise PackageNotFoundError('gap_packages')
gap.load_package("guava")
q = F.order()
C = gap("BestKnownLinearCode(%s,%s,GF(%s))" % (n, k, q))
from .linear_code import LinearCode
return LinearCode(C.GeneratorMat()._matrix_(F))
def get(self, S, var='a'):
"""
Return all fields in the database ramified exactly at
the primes in S.
INPUT:
- ``S`` - list or set of primes, or a single prime
- ``var`` - the name used for the generator of the number fields (default 'a').
EXAMPLES::
sage: J = JonesDatabase() # optional - database_jones_numfield
sage: J.get(163, var='z') # optional - database_jones_numfield
[Number Field in z with defining polynomial x^2 + 163,
Number Field in z with defining polynomial x^3 - x^2 - 54*x + 169,
Number Field in z with defining polynomial x^4 - x^3 - 7*x^2 + 2*x + 9]
sage: J.get([3, 4]) # optional - database_jones_numfield
Traceback (most recent call last):
...
ValueError: S must be a list of primes
"""
if self.root is None:
if os.path.exists(JONESDATA + "/jones.sobj"):
self.root = load(JONESDATA + "/jones.sobj")
else:
raise PackageNotFoundError("database_jones_numfield")
try:
S = list(S)
except TypeError:
S = [S]
if not all([p.is_prime() for p in S]):
raise ValueError("S must be a list of primes")
S.sort()
s = tuple(S)
if s not in self.root:
return []
return [NumberField(f, var, check=False) for f in self.root[s]]
def __init__(self, verbosity=0, prop_limit=0):
r"""
Constuct a new PicoSAT instance.
See the documentation class for the description of inputs.
EXAMPLES::
sage: from sage.sat.solvers.picosat import PicoSAT
sage: solver = PicoSAT() # optional - pycosat
"""
self._verbosity = int(verbosity)
if prop_limit is None:
self._prop_limit = 0
else:
self._prop_limit = int(prop_limit)
try:
import pycosat
except ImportError:
from sage.misc.package import PackageNotFoundError
raise PackageNotFoundError("pycosat")
self._solve = pycosat.solve
self._nvars = 0
self._clauses = []
def lovasz_theta(graph):
r"""
Return the value of Lovász theta-function of graph
For a graph `G` this function is denoted by `\theta(G)`, and it can be
computed in polynomial time. Mathematically, its most important property is the following:
.. MATH::
\alpha(G)\leq\theta(G)\leq\chi(\overline{G})
with `\alpha(G)` and `\chi(\overline{G})` being, respectively, the maximum
size of an :meth:`independent set <sage.graphs.graph.Graph.independent_set>`
set of `G` and the :meth:`chromatic number
<sage.graphs.graph.Graph.chromatic_number>` of the :meth:`complement
<sage.graphs.generic_graph.GenericGraph.complement>` `\overline{G}` of `G`.
For more information, see the :wikipedia:`Lovász_number`.
.. NOTE::
- Implemented for undirected graphs only. Use to_undirected to convert a
digraph to an undirected graph.
- This function requires the optional package ``csdp``, which you can
install with with ``sage -i csdp``.
EXAMPLES::
sage: C=graphs.PetersenGraph()
sage: C.lovasz_theta() # optional csdp
4.0
sage: graphs.CycleGraph(5).lovasz_theta() # optional csdp
2.236068
TEST::
sage: g = Graph()
sage: g.lovasz_theta() # indirect doctest
0
"""
n = graph.order()
if n == 0:
return 0
from networkx import write_edgelist
from sage.misc.temporary_file import tmp_filename
import os, subprocess
from sage.env import SAGE_LOCAL
from sage.misc.package import is_package_installed, PackageNotFoundError
if not is_package_installed('csdp'):
raise PackageNotFoundError("csdp")
g = graph.relabel(inplace=False, perm=range(1, n + 1)).networkx_graph()
tf_name = tmp_filename()
tf = open(tf_name, 'wb')
tf.write(str(n) + '\n' + str(g.number_of_edges()) + '\n')
write_edgelist(g, tf, data=False)
tf.close()
lines = subprocess.check_output(
[os.path.join(SAGE_LOCAL, 'bin', 'theta'), tf_name])
return float(lines.split()[-1])
def bounds_on_minimum_distance_in_guava(n, k, F):
r"""
Computes a lower and upper bound on the greatest minimum distance of a
`[n,k]` linear code over the field ``F``.
This function requires the optional GAP package GUAVA.
The function returns a GAP record with the two bounds and an explanation for
each bound. The function Display can be used to show the explanations.
The values for the lower and upper bound are obtained from a table
constructed by Cen Tjhai for GUAVA, derived from the table of
Brouwer. See http://www.codetables.de/ for the most recent data.
These tables contain lower and upper bounds for `q=2` (when ``n <= 257``),
`q=3` (when ``n <= 243``), `q=4` (``n <= 256``). (Current as of
11 May 2006.) For codes over other fields and for larger word lengths,
trivial bounds are used.
INPUT:
- ``n`` -- the length of the code to look up
- ``k`` -- the dimension of the code to look up
- ``F`` -- the base field of the code to look up
OUTPUT:
- A GAP record object. See below for an example.
EXAMPLES::
sage: gap_rec = codes.databases.bounds_on_minimum_distance_in_guava(10,5,GF(2)) # optional - gap_packages (Guava package)
sage: print(gap_rec) # optional - gap_packages (Guava package)
rec(
construction :=
[ <Operation "ShortenedCode">,
[
[ <Operation "UUVCode">,
[
[ <Operation "DualCode">,
[ [ <Operation "RepetitionCode">, [ 8, 2 ] ] ] ],
[ <Operation "UUVCode">,
[
[ <Operation "DualCode">,
[ [ <Operation "RepetitionCode">, [ 4, 2 ] ] ] ]
, [ <Operation "RepetitionCode">, [ 4, 2 ] ] ] ]
] ], [ 1, 2, 3, 4, 5, 6 ] ] ],
k := 5,
lowerBound := 4,
lowerBoundExplanation := ...
n := 10,
q := 2,
references := rec(
),
upperBound := 4,
upperBoundExplanation := ... )
"""
if not is_package_installed('gap_packages'):
raise PackageNotFoundError('gap_packages')
gap.load_package("guava")
q = F.order()
gap.eval("data := BoundsMinimumDistance(%s,%s,GF(%s))" % (n, k, q))
Ldata = gap.eval("Display(data)")
return Ldata
def ehrhart_polynomial(self,
verbose=False,
dual=None,
irrational_primal=None,
irrational_all_primal=None,
maxdet=None,
no_decomposition=None,
compute_vertex_cones=None,
smith_form=None,
dualization=None,
triangulation=None,
triangulation_max_height=None,
**kwds):
r"""
Return the Ehrhart polynomial of this polyhedron.
Let `P` be a lattice polytope in `\RR^d` and define `L(P,t) = \# (tP
\cap \ZZ^d)`. Then E. Ehrhart proved in 1962 that `L` coincides with a
rational polynomial of degree `d` for integer `t`. `L` is called the
*Ehrhart polynomial* of `P`. For more information see the
:wikipedia:`Ehrhart_polynomial`.
INPUT:
- ``verbose`` - (boolean, default to ``False``) if ``True``, print the
whole output of the LattE command.
The following options are passed to the LattE command, for details you
should consult `the LattE documentation
<https://www.math.ucdavis.edu/~latte/software/packages/latte_current/>`__:
- ``dual`` - (boolean) triangulate and signed-decompose in the dual
space
- ``irrational_primal`` - (boolean) triangulate in the dual space,
signed-decompose in the primal space using irrationalization.
- ``irrational_all_primal`` - (boolean) Triangulate and signed-decompose
in the primal space using irrationalization.
- ``maxdet`` -- (integer) decompose down to an index (determinant) of
``maxdet`` instead of index 1 (unimodular cones).
- ``no_decomposition`` -- (boolean) do not signed-decompose simplicial cones.
- ``compute_vertex_cones`` -- (string) either 'cdd' or 'lrs' or '4ti2'
- ``smith_form`` -- (string) either 'ilio' or 'lidia'
- ``dualization`` -- (string) either 'cdd' or '4ti2'
- ``triangulation`` - (string) 'cddlib', '4ti2' or 'topcom'
- ``triangulation_max_height`` - (integer) use a uniform distribution of
height from 1 to this number
.. NOTE::
Any additional argument is forwarded to LattE's executable
``count``. All occurrences of '_' will be replaced with a '-'.
ALGORITHM:
This method calls the program ``count`` from LattE integrale, a program
for lattice point enumeration (see
https://www.math.ucdavis.edu/~latte/).
EXAMPLES::
sage: P = Polyhedron(vertices=[(0,0,0),(3,3,3),(-3,2,1),(1,-1,-2)])
sage: p = P.ehrhart_polynomial() # optional - latte_int
sage: p # optional - latte_int
7/2*t^3 + 2*t^2 - 1/2*t + 1
sage: p(1) # optional - latte_int
6
sage: len(P.integral_points())
6
sage: p(2) # optional - latte_int
36
sage: len((2*P).integral_points())
36
The unit hypercubes::
sage: from itertools import product
sage: def hypercube(d):
....: return Polyhedron(vertices=list(product([0,1],repeat=d)))
sage: hypercube(3).ehrhart_polynomial() # optional - latte_int
t^3 + 3*t^2 + 3*t + 1
sage: hypercube(4).ehrhart_polynomial() # optional - latte_int
t^4 + 4*t^3 + 6*t^2 + 4*t + 1
sage: hypercube(5).ehrhart_polynomial() # optional - latte_int
t^5 + 5*t^4 + 10*t^3 + 10*t^2 + 5*t + 1
sage: hypercube(6).ehrhart_polynomial() # optional - latte_int
t^6 + 6*t^5 + 15*t^4 + 20*t^3 + 15*t^2 + 6*t + 1
An empty polyhedron::
sage: P = Polyhedron(ambient_dim=3, vertices=[])
sage: P.ehrhart_polynomial() # optional - latte_int
0
sage: parent(_) # optional - latte_int
Univariate Polynomial Ring in t over Rational Field
TESTS:
Test options::
sage: P = Polyhedron(ieqs=[[1,-1,1,0], [-1,2,-1,0], [1,1,-2,0]], eqns=[[-1,2,-1,-3]], base_ring=ZZ)
sage: p = P.ehrhart_polynomial(maxdet=5, verbose=True) # optional - latte_int
This is LattE integrale ...
...
Invocation: count --ehrhart-polynomial '--redundancy-check=none' '--maxdet=5' --cdd ...
...
sage: p # optional - latte_int
1/2*t^2 + 3/2*t + 1
sage: p = P.ehrhart_polynomial(dual=True, verbose=True) # optional - latte_int
This is LattE integrale ...
...
Invocation: count --ehrhart-polynomial '--redundancy-check=none' --dual --cdd ...
...
sage: p # optional - latte_int
1/2*t^2 + 3/2*t + 1
sage: p = P.ehrhart_polynomial(irrational_primal=True, verbose=True) # optional - latte_int
This is LattE integrale ...
...
Invocation: count --ehrhart-polynomial '--redundancy-check=none' --irrational-primal --cdd ...
...
sage: p # optional - latte_int
1/2*t^2 + 3/2*t + 1
sage: p = P.ehrhart_polynomial(irrational_all_primal=True, verbose=True) # optional - latte_int
This is LattE integrale ...
...
Invocation: count --ehrhart-polynomial '--redundancy-check=none' --irrational-all-primal --cdd ...
sage: p # optional - latte_int
1/2*t^2 + 3/2*t + 1
Test bad options::
sage: P.ehrhart_polynomial(bim_bam_boum=19) # optional - latte_int
Traceback (most recent call last):
...
RuntimeError: LattE integrale failed with exit code 1 to execute...
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
R = PolynomialRing(QQ, 't')
if self.is_empty():
return R.zero()
from sage.misc.misc import SAGE_TMP
from subprocess import Popen, PIPE
ine = self.cdd_Hrepresentation()
args = ['count', '--ehrhart-polynomial']
if 'redundancy_check' not in kwds:
args.append('--redundancy-check=none')
# note: the options below are explicitely written in the function
# declaration in order to keep tab completion (see #18211).
kwds.update({
'dual': dual,
'irrational_primal': irrational_primal,
'irrational_all_primal': irrational_all_primal,
'maxdet': maxdet,
'no_decomposition': no_decomposition,
'compute_vertex_cones': compute_vertex_cones,
'smith_form': smith_form,
'dualization': dualization,
'triangulation': triangulation,
'triangulation_max_height': triangulation_max_height
})
for key, value in kwds.items():
if value is None or value is False:
continue
key = key.replace('_', '-')
if value is True:
args.append('--{}'.format(key))
else:
args.append('--{}={}'.format(key, value))
args += ['--cdd', '/dev/stdin']
try:
# The cwd argument is needed because latte
# always produces diagnostic output files.
latte_proc = Popen(args,
stdin=PIPE,
stdout=PIPE,
stderr=(None if verbose else PIPE),
cwd=str(SAGE_TMP))
except OSError:
from sage.misc.package import PackageNotFoundError
raise PackageNotFoundError('latte_int')
ans, err = latte_proc.communicate(ine)
ret_code = latte_proc.poll()
if ret_code:
if err is None:
err = ", see error message above"
else:
err = ":\n" + err
raise RuntimeError(
"LattE integrale failed with exit code {} to execute {}".
format(ret_code, ' '.join(args)) + err.strip())
p = ans.splitlines()[-2]
return R(p)
def integrate(arg, polynomial=None, algorithm='triangulate', raw_output=False, verbose=False, **kwds):
r"""
Call to the function integrate from LattE integrale.
INPUT:
- ``arg`` -- a cdd or LattE description string.
- ``polynomial`` -- multivariate polynomial or valid LattE polynomial description string.
If given, the valuation parameter of LattE is set to integrate, and is set to volume otherwise.
- ``algorithm`` -- (default: 'triangulate') the integration method. Use 'triangulate' for
polytope triangulation or 'cone-decompose' for tangent cone decomposition method.
- ``raw_output`` -- if ``True`` then return directly the output string from LattE.
- ``verbose`` -- if ``True`` then return directly verbose output from LattE.
- For all other options of the integrate program, consult the LattE manual.
OUTPUT:
Either a string (if ``raw_output`` if set to ``True``) or a rational.
EXAMPLES::
sage: from sage.interfaces.latte import integrate
sage: P = 2 * polytopes.cube()
sage: x, y, z = polygen(QQ, 'x, y, z')
Integrating over a polynomial over a polytope in either the H or V representation::
sage: integrate(P.cdd_Hrepresentation(), x^2*y^2*z^2, cdd=True) # optional - latte_int
4096/27
sage: integrate(P.cdd_Vrepresentation(), x^2*y^2*z^2, cdd=True) # optional - latte_int
4096/27
Computing the volume of a polytope in either the H or V representation::
sage: integrate(P.cdd_Hrepresentation(), cdd=True) # optional - latte_int
64
sage: integrate(P.cdd_Vrepresentation(), cdd=True) # optional - latte_int
64
Polynomials given as a string in LattE description are also accepted::
sage: integrate(P.cdd_Hrepresentation(), '[[1,[2,2,2]]]', cdd=True) # optional - latte_int
4096/27
TESTS::
Testing raw output::
sage: from sage.interfaces.latte import integrate
sage: P = polytopes.cuboctahedron()
sage: cddin = P.cdd_Vrepresentation()
sage: x, y, z = polygen(QQ, 'x, y, z')
sage: f = 3*x^2*y^4*z^6 + 7*y^3*z^5
sage: integrate(cddin, f, cdd=True, raw_output=True) # optional - latte_int
'629/47775'
Testing the ``verbose`` option to integrate over a polytope::
sage: ans = integrate(cddin, f, cdd=True, verbose=True, raw_output=True) # optional - latte_int
This is LattE integrale ...
...
Invocation: integrate --valuation=integrate --triangulate --redundancy-check=none --cdd --monomials=... /dev/stdin
...
Testing triangulate algorithm::
sage: from sage.interfaces.latte import integrate
sage: P = polytopes.cuboctahedron()
sage: cddin = P.cdd_Vrepresentation()
sage: integrate(cddin, algorithm='triangulate', cdd=True) # optional - latte_int
20/3
Testing convex decomposition algorithm::
sage: from sage.interfaces.latte import integrate
sage: P = polytopes.cuboctahedron()
sage: cddin = P.cdd_Vrepresentation()
sage: integrate(cddin, algorithm='cone-decompose', cdd=True) # optional - latte_int
20/3
Testing raw output::
sage: from sage.interfaces.latte import integrate
sage: P = polytopes.cuboctahedron()
sage: cddin = P.cdd_Vrepresentation()
sage: integrate(cddin, cdd=True, raw_output=True) # optional - latte_int
'20/3'
Testing polynomial given as a string in LattE description::
sage: from sage.interfaces.latte import integrate
sage: P = polytopes.cuboctahedron()
sage: integrate(P.cdd_Hrepresentation(), '[[3,[2,4,6]],[7,[0, 3, 5]]]', cdd=True) # optional - latte_int
629/47775
Testing the ``verbose`` option to compute the volume of a polytope::
sage: from sage.interfaces.latte import integrate
sage: P = polytopes.cuboctahedron()
sage: cddin = P.cdd_Vrepresentation()
sage: ans = integrate(cddin, cdd=True, raw_output=True, verbose=True) # optional - latte_int
This is LattE integrale ...
...
Invocation: integrate --valuation=volume --triangulate --redundancy-check=none --cdd /dev/stdin
...
"""
from subprocess import Popen, PIPE
from sage.misc.misc import SAGE_TMP
from sage.rings.integer import Integer
args = ['integrate']
got_polynomial = True if polynomial is not None else False
if got_polynomial:
args.append('--valuation=integrate')
else:
args.append('--valuation=volume')
if algorithm=='triangulate':
args.append('--triangulate')
elif algorithm=='cone-decompose':
args.append('--cone-decompose')
if 'redundancy_check' not in kwds:
args.append('--redundancy-check=none')
for key,value in kwds.items():
if value is None or value is False:
continue
key = key.replace('_','-')
if value is True:
args.append('--{}'.format(key))
else:
args.append('--{}={}'.format(key, value))
if got_polynomial:
if not isinstance(polynomial, six.string_types):
# transform polynomial to LattE description
monomials_list = to_latte_polynomial(polynomial)
else:
monomials_list = str(polynomial)
from sage.misc.temporary_file import tmp_filename
filename_polynomial = tmp_filename()
with open(filename_polynomial, 'w') as f:
f.write(monomials_list)
args += ['--monomials=' + filename_polynomial]
args += ['/dev/stdin']
try:
# The cwd argument is needed because latte
# always produces diagnostic output files.
latte_proc = Popen(args,
stdin=PIPE, stdout=PIPE,
stderr=(None if verbose else PIPE),
cwd=str(SAGE_TMP))
except OSError:
from sage.misc.package import PackageNotFoundError
raise PackageNotFoundError('latte_int')
ans, err = latte_proc.communicate(arg)
ret_code = latte_proc.poll()
if ret_code:
if err is None:
err = ", see error message above"
else:
err = ":\n" + err
raise RuntimeError("LattE integrale program failed (exit code {})".format(ret_code) + err.strip())
ans = ans.splitlines()
ans = ans[-5].split()
assert(ans[0]=='Answer:')
ans = ans[1]
if raw_output:
return ans
else:
from sage.rings.rational import Rational
return Rational(ans)
def count(arg, ehrhart_polynomial=False, multivariate_generating_function=False, raw_output=False, verbose=False, **kwds):
r"""
Call to the program count from LattE integrale
INPUT:
- ``arg`` -- a cdd or LattE description string
- ``ehrhart_polynomial``, ``multivariate_generating_function`` -- to
compute Ehrhart polynomial or multivariate generating function instead of
just counting points
- ``raw_output`` -- if ``True`` then return directly the output string from LattE
- For all other options of the count program, consult the LattE manual
OUTPUT:
Either a string (if ``raw_output`` if set to ``True``) or an integer (when
counting points), or a polynomial (if ``ehrhart_polynomial`` is set to
``True``) or a multivariate THING (if ``multivariate_generating_function``
is set to ``True``)
EXAMPLES::
sage: from sage.interfaces.latte import count
sage: P = 2 * polytopes.cube()
Counting integer points from either the H or V representation::
sage: count(P.cdd_Hrepresentation(), cdd=True) # optional - latte_int
125
sage: count(P.cdd_Vrepresentation(), cdd=True) # optional - latte_int
125
Ehrhart polynomial::
sage: count(P.cdd_Hrepresentation(), cdd=True, ehrhart_polynomial=True) # optional - latte_int
64*t^3 + 48*t^2 + 12*t + 1
Multivariate generating function currently only work with ``raw_output=True``::
sage: opts = {'cdd': True,
....: 'multivariate_generating_function': True,
....: 'raw_output': True}
sage: cddin = P.cdd_Hrepresentation()
sage: print(count(cddin, **opts)) # optional - latte_int
x[0]^2*x[1]^(-2)*x[2]^(-2)/((1-x[1])*(1-x[2])*(1-x[0]^(-1)))
+ x[0]^(-2)*x[1]^(-2)*x[2]^(-2)/((1-x[1])*(1-x[2])*(1-x[0]))
+ x[0]^2*x[1]^(-2)*x[2]^2/((1-x[1])*(1-x[0]^(-1))*(1-x[2]^(-1)))
+ x[0]^(-2)*x[1]^(-2)*x[2]^2/((1-x[1])*(1-x[0])*(1-x[2]^(-1)))
+ x[0]^2*x[1]^2*x[2]^(-2)/((1-x[2])*(1-x[0]^(-1))*(1-x[1]^(-1)))
+ x[0]^(-2)*x[1]^2*x[2]^(-2)/((1-x[2])*(1-x[0])*(1-x[1]^(-1)))
+ x[0]^2*x[1]^2*x[2]^2/((1-x[0]^(-1))*(1-x[1]^(-1))*(1-x[2]^(-1)))
+ x[0]^(-2)*x[1]^2*x[2]^2/((1-x[0])*(1-x[1]^(-1))*(1-x[2]^(-1)))
TESTS:
Testing raw output::
sage: from sage.interfaces.latte import count
sage: P = polytopes.cuboctahedron()
sage: cddin = P.cdd_Vrepresentation()
sage: count(cddin, cdd=True, raw_output=True) # optional - latte_int
'19'
sage: count(cddin, cdd=True, raw_output=True, ehrhart_polynomial=True) # optional - latte_int
' + 1 * t^0 + 10/3 * t^1 + 8 * t^2 + 20/3 * t^3'
sage: count(cddin, cdd=True, raw_output=True, multivariate_generating_function=True) # optional - latte_int
'x[0]^(-1)*x[1]^(-1)/((1-x[0]*x[2])*(1-x[0]^(-1)*x[1])*...x[0]^(-1)*x[2]^(-1)))\n'
Testing the ``verbose`` option::
sage: n = count(cddin, cdd=True, verbose=True, raw_output=True) # optional - latte_int
This is LattE integrale ...
...
Invocation: count '--redundancy-check=none' --cdd /dev/stdin
...
Total Unimodular Cones: ...
Maximum number of simplicial cones in memory at once: ...
<BLANKLINE>
**** The number of lattice points is: ****
Total time: ... sec
Trivial input for which LattE's preprocessor does all the work::
sage: P = Polyhedron(vertices=[[0,0,0]])
sage: cddin = P.cdd_Hrepresentation()
sage: count(cddin, cdd=True, raw_output=False) # optional - latte_int
1
"""
from subprocess import Popen, PIPE
from sage.misc.misc import SAGE_TMP
from sage.rings.integer import Integer
args = ['count']
if ehrhart_polynomial and multivariate_generating_function:
raise ValueError
if ehrhart_polynomial:
args.append('--ehrhart-polynomial')
elif multivariate_generating_function:
args.append('--multivariate-generating-function')
if 'redundancy_check' not in kwds:
args.append('--redundancy-check=none')
for key,value in kwds.items():
if value is None or value is False:
continue
key = key.replace('_','-')
if value is True:
args.append('--{}'.format(key))
else:
args.append('--{}={}'.format(key, value))
if multivariate_generating_function:
from sage.misc.temporary_file import tmp_filename
filename = tmp_filename()
with open(filename, 'w') as f:
f.write(arg)
args += [filename]
else:
args += ['/dev/stdin']
try:
# The cwd argument is needed because latte
# always produces diagnostic output files.
latte_proc = Popen(args,
stdin=PIPE, stdout=PIPE,
stderr=(None if verbose else PIPE),
cwd=str(SAGE_TMP))
except OSError:
from sage.misc.package import PackageNotFoundError
raise PackageNotFoundError('latte_int')
ans, err = latte_proc.communicate(arg)
ret_code = latte_proc.poll()
if ret_code:
if err is None:
err = ", see error message above"
else:
err = ":\n" + err
raise RuntimeError("LattE integrale program failed (exit code {})".format(ret_code) + err.strip())
if ehrhart_polynomial:
ans = ans.splitlines()[-2]
if raw_output:
return ans
else:
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.rational_field import QQ
R = PolynomialRing(QQ, 't')
return R(ans)
elif multivariate_generating_function:
with open(filename + '.rat') as f:
ans = f.read()
if raw_output:
return ans
else:
raise NotImplementedError("there is no Sage object to handle multivariate series from LattE, use raw_output=True")
else:
if ans: # Sometimes (when LattE's preproc does the work), no output appears on stdout.
ans = ans.splitlines()[-1]
if not ans:
# opening a file is slow (30e-6s), so we read the file
# numOfLatticePoints only in case of a IndexError above
with open(SAGE_TMP+'/numOfLatticePoints', 'r') as f:
ans = f.read()
if raw_output:
return ans
else:
from sage.rings.integer import Integer
return Integer(ans)
def structure_description(G, latex=False):
r"""
Return a string that tries to describe the structure of ``G``.
This methods wraps GAP's ``StructureDescription`` method.
Requires the *optional* ``database_gap`` package.
For full details, including the form of the returned string and the
algorithm to build it, see `GAP's documentation
<http://www.gap-system.org/Manuals/doc/ref/chap39.html>`_.
INPUT:
- ``latex`` -- a boolean (default: ``False``). If ``True`` return a
LaTeX formatted string.
OUTPUT:
- string
.. WARNING::
From GAP's documentation: The string returned by
``StructureDescription`` is **not** an isomorphism invariant:
non-isomorphic groups can have the same string value, and two
isomorphic groups in different representations can produce different
strings.
EXAMPLES::
sage: G = CyclicPermutationGroup(6)
sage: G.structure_description() # optional - database_gap
'C6'
sage: G.structure_description(latex=True) # optional - database_gap
'C_{6}'
sage: G2 = G.direct_product(G, maps=False)
sage: LatexExpr(G2.structure_description(latex=True)) # optional - database_gap
C_{6} \times C_{6}
This method is mainly intended for small groups or groups with few
normal subgroups. Even then there are some surprises::
sage: D3 = DihedralGroup(3)
sage: D3.structure_description() # optional - database_gap
'S3'
We use the Sage notation for the degree of dihedral groups::
sage: D4 = DihedralGroup(4)
sage: D4.structure_description() # optional - database_gap
'D4'
Works for finitely presented groups (:trac:`17573`)::
sage: F.<x, y> = FreeGroup()
sage: G=F / [x^2*y^-1, x^3*y^2, x*y*x^-1*y^-1]
sage: G.structure_description() # optional - database_gap
'C7'
And matrix groups (:trac:`17573`)::
sage: groups.matrix.GL(4,2).structure_description() # optional - database_gap
'A8'
"""
import re
from sage.misc.package import is_package_installed, PackageNotFoundError
def correct_dihedral_degree(match):
return "%sD%d" % (match.group(1), int(match.group(2)) / 2)
try:
description = str(G._gap_().StructureDescription())
except RuntimeError:
if not is_package_installed('database_gap'):
raise PackageNotFoundError("database_gap")
raise
description = re.sub(r"(\A|\W)D(\d+)", correct_dihedral_degree,
description)
if not latex:
return description
description = description.replace("x", r"\times").replace(":", r"\rtimes")
description = re.sub(r"([A-Za-z]+)([0-9]+)", r"\g<1>_{\g<2>}", description)
description = re.sub(r"O([+-])", r"O^{\g<1>}", description)
return description
def CossidentePenttilaGraph(q):
r"""
Cossidente-Penttila `((q^3+1)(q+1)/2,(q^2+1)(q-1)/2,(q-3)/2,(q-1)^2/2)`-strongly regular graph
For each odd prime power `q`, one can partition the points of the `O_6^-(q)`-generalized
quadrange `GQ(q,q^2)` into two parts, so that on any of them the induced subgraph of
the point graph of the GQ has parameters as above [CP05]_.
Directly follwing the construction in [CP05]_ is not efficient,
as one then needs to construct the dual `GQ(q^2,q)`. Thus we
describe here a more efficient approach that we came up with, following a suggestion by
T.Penttila. Namely, this partition is invariant
under the subgroup `H=\Omega_3(q^2)<O_6^-(q)`. We build the appropriate `H`, which
leaves the form `B(X,Y,Z)=XY+Z^2` invariant, and
pick up two orbits of `H` on the `F_q`-points. One them is `B`-isotropic, and we
take the representative `(1:0:0)`. The other one corresponds to the points of
`PG(2,q^2)` that have all the lines on them either missing the conic specified by `B`, or
intersecting the conic in two points. We take `(1:1:e)` as the representative. It suffices
to pick `e` so that `e^2+1` is not a square in `F_{q^2}`. Indeed,
The conic can be viewed as the union of `\{(0:1:0)\}` and `\{(1:-t^2:t) | t \in F_{q^2}\}`.
The coefficients of a generic line on `(1:1:e)` are `[1:-1-eb:b]`, for `-1\neq eb`.
Thus, to make sure the intersection with the conic is always even, we need that the
discriminant of `1+(1+eb)t^2+tb=0` never vanishes, and this is if and only if
`e^2+1` is not a square. Further, we need to adjust `B`, by multiplying it by appropriately
chosen `\nu`, so that `(1:1:e)` becomes isotropic under the relative trace norm
`\nu B(X,Y,Z)+(\nu B(X,Y,Z))^q`. The latter is used then to define the graph.
INPUT:
- ``q`` -- an odd prime power.
EXAMPLES:
For `q=3` one gets Sims-Gewirtz graph. ::
sage: G=graphs.CossidentePenttilaGraph(3) # optional - gap_packages (grape)
sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape)
(56, 10, 0, 2)
For `q>3` one gets new graphs. ::
sage: G=graphs.CossidentePenttilaGraph(5) # optional - gap_packages (grape)
sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape)
(378, 52, 1, 8)
TESTS::
sage: G=graphs.CossidentePenttilaGraph(7) # optional - gap_packages (grape) # long time
sage: G.is_strongly_regular(parameters=True) # optional - gap_packages (grape) # long time
(1376, 150, 2, 18)
sage: graphs.CossidentePenttilaGraph(2)
Traceback (most recent call last):
...
ValueError: q(=2) must be an odd prime power
REFERENCES:
.. [CP05] A.Cossidente and T.Penttila
Hemisystems on the Hermitian surface
Journal of London Math. Soc. 72(2005), 731-741
"""
p, k = is_prime_power(q,get_data=True)
if k==0 or p==2:
raise ValueError('q(={}) must be an odd prime power'.format(q))
from sage.libs.gap.libgap import libgap
from sage.misc.package import is_package_installed, PackageNotFoundError
if not is_package_installed('gap_packages'):
raise PackageNotFoundError('gap_packages')
adj_list=libgap.function_factory("""function(q)
local z, e, so, G, nu, G1, G0, B, T, s, O1, O2, x;
LoadPackage("grape");
G0:=SO(3,q^2);
so:=GeneratorsOfGroup(G0);
G1:=Group(Comm(so[1],so[2]),Comm(so[1],so[3]),Comm(so[2],so[3]));
B:=InvariantBilinearForm(G0).matrix;
z:=Z(q^2); e:=z; sqo:=(q^2-1)/2;
if IsInt(sqo/Order(e^2+z^0)) then
e:=z^First([2..q^2-2], x-> not IsInt(sqo/Order(z^(2*x)+z^0)));
fi;
nu:=z^First([0..q^2-2], x->z^x*(e^2+z^0)+(z^x*(e^2+z^0))^q=0*z);
T:=function(x)
local r;
r:=nu*x*B*x;
return r+r^q;
end;
s:=Group([Z(q)*IdentityMat(3,GF(q))]);
O1:=Orbit(G1, Set(Orbit(s,z^0*[1,0,0])), OnSets);
O2:=Orbit(G1, Set(Orbit(s,z^0*[1,1,e])), OnSets);
G:=Graph(G1,Concatenation(O1,O2),OnSets,
function(x,y) return x<>y and 0*z=T(x[1]+y[1]); end);
return List([1..OrderGraph(G)],x->Adjacency(G,x));
end;""")
adj = adj_list(q) # for each vertex, we get the list of vertices it is adjacent to
G = Graph(((i,int(j-1))
for i,ni in enumerate(adj) for j in ni),
format='list_of_edges', multiedges=False)
G.name('CossidentePenttila('+str(q)+')')
return G