def __call__(self, assumptions=None):
"""
Run 'command' and collect output.
INPUT:
- ``assumptions`` - ignored, accepted for compatibility with
other solvers (default: ``None``)
TESTS:
This class is not meant to be called directly::
sage: from sage.sat.solvers.dimacs import DIMACS
sage: fn = tmp_filename()
sage: solver = DIMACS(filename=fn)
sage: solver.add_clause( (1, -2 , 3) )
sage: solver()
Traceback (most recent call last):
...
ValueError: No SAT solver command selected.
"""
if assumptions is not None:
raise NotImplementedError("Assumptions are not supported for DIMACS based solvers.")
self.write()
output_filename = None
self._output = []
command = self._command.strip()
if not command:
raise ValueError("No SAT solver command selected.")
if "{output}" in command:
output_filename = tmp_filename()
command = command.format(input=self._headname, output=output_filename)
args = shlex.split(command)
try:
process = subprocess.Popen(args, stdout=subprocess.PIPE)
except OSError:
raise OSError("Could run '%s', perhaps you need to add your SAT solver to $PATH?"%(" ".join(args)))
try:
while process.poll() is None:
for line in iter(process.stdout.readline,''):
if get_verbose() or self._verbosity:
print line,
sys.stdout.flush()
self._output.append(line)
sleep(0.1)
if output_filename:
self._output.extend(open(output_filename).readlines())
except BaseException:
process.kill()
raise
def __call__(self, assumptions=None):
"""
Run 'command' and collect output.
INPUT:
- ``assumptions`` - ignored, accepted for compatibility with
other solvers (default: ``None``)
TESTS:
This class is not meant to be called directly::
sage: from sage.sat.solvers.dimacs import DIMACS
sage: fn = tmp_filename()
sage: solver = DIMACS(filename=fn)
sage: solver.add_clause( (1, -2 , 3) )
sage: solver()
Traceback (most recent call last):
...
ValueError: No SAT solver command selected.
"""
if assumptions is not None:
raise NotImplementedError("Assumptions are not supported for DIMACS based solvers.")
self.write()
output_filename = None
self._output = []
command = self._command.strip()
if not command:
raise ValueError("No SAT solver command selected.")
if "{output}" in command:
output_filename = tmp_filename()
command = command.format(input=self._headname, output=output_filename)
args = shlex.split(command)
try:
process = subprocess.Popen(args, stdout=subprocess.PIPE)
except OSError:
raise OSError("Could run '%s', perhaps you need to add your SAT solver to $PATH?"%(" ".join(args)))
try:
while process.poll() is None:
for line in iter(process.stdout.readline,''):
if get_verbose() or self._verbosity:
print(line)
sys.stdout.flush()
self._output.append(line)
sleep(0.1)
if output_filename:
self._output.extend(open(output_filename).readlines())
except BaseException:
process.kill()
raise
def series(self, n=2, quadratic_twist=+1, prec=5):
r"""
Returns the `n`-th approximation to the `p`-adic L-series as a
power series in `T` (corresponding to `\gamma-1` with
`\gamma=1+p` as a generator of `1+p\ZZ_p`). Each coefficient
is a `p`-adic number whose precision is provably correct.
Here the normalization of the `p`-adic L-series is chosen such
that `L_p(J,1) = (1-1/\alpha)^2 L(J,1)/\Omega_J` where
`\alpha` is the unit root
INPUT:
- ``n`` - (default: 2) a positive integer
- ``quadratic_twist`` - (default: +1) a fundamental
discriminant of a quadratic field, coprime to the
conductor of the curve
- ``prec`` - (default: 5) maximal number of terms of the
series to compute; to compute as many as possible just
give a very large number for ``prec``; the result will
still be correct.
ALIAS: power_series is identical to series.
EXAMPLES::
sage: J = J0(188)[0]
sage: p = 7
sage: L = J.padic_lseries(p)
sage: L.is_ordinary()
True
sage: f = L.series(2)
sage: f[0]
O(7^20)
sage: f[1].norm()
3 + 4*7 + 3*7^2 + 6*7^3 + 5*7^4 + 5*7^5 + 6*7^6 + 4*7^7 + 5*7^8 + 7^10 + 5*7^11 + 4*7^13 + 4*7^14 + 5*7^15 + 2*7^16 + 5*7^17 + 7^18 + 7^19 + O(7^20)
"""
n = ZZ(n)
if n < 1:
raise ValueError("n (={0}) must be a positive integer".format(n))
if not self.is_ordinary():
raise ValueError("p (={0}) must be an ordinary prime".format(
self._p))
# check if the conditions on quadratic_twist are satisfied
D = ZZ(quadratic_twist)
if D != 1:
if D % 4 == 0:
d = D // 4
if not d.is_squarefree() or d % 4 == 1:
raise ValueError(
"quadratic_twist (={0}) must be a fundamental discriminant of a quadratic field"
.format(D))
else:
if not D.is_squarefree() or D % 4 != 1:
raise ValueError(
"quadratic_twist (={0}) must be a fundamental discriminant of a quadratic field"
.format(D))
if gcd(D, self._p) != 1:
raise ValueError(
"quadratic twist (={0}) must be coprime to p (={1}) ".
format(D, self._p))
if gcd(D, self._E.conductor()) != 1:
for ell in prime_divisors(D):
if valuation(self._E.conductor(), ell) > valuation(D, ell):
raise ValueError(
"can not twist a curve of conductor (={0}) by the quadratic twist (={1})."
.format(self._E.conductor(), D))
p = self._p
if p == 2 and self._normalize:
print('Warning : For p=2 the normalization might not be correct !')
#verbose("computing L-series for p=%s, n=%s, and prec=%s"%(p,n,prec))
# bounds = self._prec_bounds(n,prec)
# padic_prec = max(bounds[1:]) + 5
padic_prec = 10
# verbose("using p-adic precision of %s"%padic_prec)
res_series_prec = min(p**(n - 1), prec)
verbose("using series precision of %s" % res_series_prec)
ans = self._get_series_from_cache(n, res_series_prec, D)
if not ans is None:
verbose("found series in cache")
return ans
K = QQ
gamma = K(1 + p)
R = PowerSeriesRing(K, 'T', res_series_prec)
T = R(R.gen(), res_series_prec)
#L = R(0)
one_plus_T_factor = R(1)
gamma_power = K(1)
teich = self.teichmuller(padic_prec)
p_power = p**(n - 1)
# F = Qp(p,padic_prec)
verbose("Now iterating over %s summands" % ((p - 1) * p_power))
verbose_level = get_verbose()
count_verb = 0
alphas = self.alpha()
#print len(alphas)
Lprod = []
self._emb = 0
if len(alphas) == 2:
split = True
else:
split = False
for alpha in alphas:
L = R(0)
self._emb = self._emb + 1
for j in range(p_power):
s = K(0)
if verbose_level >= 2 and j / p_power * 100 > count_verb + 3:
verbose("%.2f percent done" % (float(j) / p_power * 100))
count_verb += 3
for a in range(1, p):
if split:
b = (teich[a]) % ZZ(p**n)
b = b * gamma_power
else:
b = teich[a] * gamma_power
s += self.measure(b, n, padic_prec, D, alpha)
L += s * one_plus_T_factor
one_plus_T_factor *= 1 + T
gamma_power *= gamma
Lprod = Lprod + [L]
if len(Lprod) == 1:
return Lprod[0]
else:
return Lprod[0] * Lprod[1]
def series(self, n=2, quadratic_twist=+1, prec=5):
r"""
Returns the `n`-th approximation to the `p`-adic L-series as a
power series in `T` (corresponding to `\gamma-1` with
`\gamma=1+p` as a generator of `1+p\ZZ_p`). Each coefficient
is a `p`-adic number whose precision is provably correct.
Here the normalization of the `p`-adic L-series is chosen such
that `L_p(J,1) = (1-1/\alpha)^2 L(J,1)/\Omega_J` where
`\alpha` is the unit root
INPUT:
- ``n`` - (default: 2) a positive integer
- ``quadratic_twist`` - (default: +1) a fundamental
discriminant of a quadratic field, coprime to the
conductor of the curve
- ``prec`` - (default: 5) maximal number of terms of the
series to compute; to compute as many as possible just
give a very large number for ``prec``; the result will
still be correct.
ALIAS: power_series is identical to series.
EXAMPLES:
sage: J = J0(188)[0]
sage: p = 7
sage: L = J.padic_lseries(p)
sage: L.is_ordinary()
True
sage: f = L.series(2)
sage: f[0]
O(7^20)
sage: f[1].norm()
3 + 4*7 + 3*7^2 + 6*7^3 + 5*7^4 + 5*7^5 + 6*7^6 + 4*7^7 + 5*7^8 + 7^10 + 5*7^11 + 4*7^13 + 4*7^14 + 5*7^15 + 2*7^16 + 5*7^17 + 7^18 + 7^19 + O(7^20)
"""
n = ZZ(n)
if n < 1:
raise ValueError, "n (=%s) must be a positive integer"%n
if not self.is_ordinary():
raise ValueError, "p (=%s) must be an ordinary prime"%p
# check if the conditions on quadratic_twist are satisfied
D = ZZ(quadratic_twist)
if D != 1:
if D % 4 == 0:
d = D//4
if not d.is_squarefree() or d % 4 == 1:
raise ValueError, "quadratic_twist (=%s) must be a fundamental discriminant of a quadratic field"%D
else:
if not D.is_squarefree() or D % 4 != 1:
raise ValueError, "quadratic_twist (=%s) must be a fundamental discriminant of a quadratic field"%D
if gcd(D,self._p) != 1:
raise ValueError, "quadratic twist (=%s) must be coprime to p (=%s) "%(D,self._p)
if gcd(D,self._E.conductor())!= 1:
for ell in prime_divisors(D):
if valuation(self._E.conductor(),ell) > valuation(D,ell) :
raise ValueError, "can not twist a curve of conductor (=%s) by the quadratic twist (=%s)."%(self._E.conductor(),D)
p = self._p
if p == 2 and self._normalize :
print 'Warning : For p=2 the normalization might not be correct !'
#verbose("computing L-series for p=%s, n=%s, and prec=%s"%(p,n,prec))
# bounds = self._prec_bounds(n,prec)
# padic_prec = max(bounds[1:]) + 5
padic_prec = 10
# verbose("using p-adic precision of %s"%padic_prec)
res_series_prec = min(p**(n-1), prec)
verbose("using series precision of %s"%res_series_prec)
ans = self._get_series_from_cache(n, res_series_prec,D)
if not ans is None:
verbose("found series in cache")
return ans
K = QQ
gamma = K(1 + p)
R = PowerSeriesRing(K,'T',res_series_prec)
T = R(R.gen(),res_series_prec )
#L = R(0)
one_plus_T_factor = R(1)
gamma_power = K(1)
teich = self.teichmuller(padic_prec)
p_power = p**(n-1)
# F = Qp(p,padic_prec)
verbose("Now iterating over %s summands"%((p-1)*p_power))
verbose_level = get_verbose()
count_verb = 0
alphas = self.alpha()
#print len(alphas)
Lprod = []
self._emb = 0
if len(alphas) == 2:
split = True
else:
split = False
for alpha in alphas:
L = R(0)
self._emb = self._emb + 1
for j in range(p_power):
s = K(0)
if verbose_level >= 2 and j/p_power*100 > count_verb + 3:
verbose("%.2f percent done"%(float(j)/p_power*100))
count_verb += 3
for a in range(1,p):
if split:
# b = ((F.teichmuller(a)).lift() % ZZ(p**n))
b = (teich[a]) % ZZ(p**n)
b = b*gamma_power
else:
b = teich[a] * gamma_power
s += self.measure(b, n, padic_prec,D,alpha)
L += s * one_plus_T_factor
one_plus_T_factor *= 1+T
gamma_power *= gamma
Lprod = Lprod + [L]
if len(Lprod)==1:
return Lprod[0]
else:
return Lprod[0]*Lprod[1]