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 # optional - latte_int
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 # optional - latte_int
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
"""
# Check that LattE is present
Latte().require()
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']
# 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))
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:
return Integer(ans)
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 # optional - latte_int
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 # optional - latte_int
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 # optional - latte_int
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 # optional - latte_int
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 # optional - latte_int
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 # optional - latte_int
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 # optional - latte_int
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
...
"""
# Check that LattE is present
Latte().require()
from sage.rings.rational import Rational
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']
# 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))
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:
return Rational(ans)