def mu(f, a, n, x, y):
binom = sympy.binomial_coefficients_list(n)
binom = map(numpy.double, binom)
normf = numpy.sqrt(
sum(numpy.abs(a[k](x))**2 * 1 / binom[k] for k in xrange(n + 1)))
Delta = numpy.sqrt(numpy.double(n)) * (1 + numpy.abs(y))
normy = 1 + numpy.abs(y)
return normf * Delta / (2 * normy)
def test_binomial_coefficients_list():
assert binomial_coefficients_list(0) == [1]
assert binomial_coefficients_list(1) == [1, 1]
assert binomial_coefficients_list(2) == [1, 2, 1]
assert binomial_coefficients_list(3) == [1, 3, 3, 1]
assert binomial_coefficients_list(4) == [1, 4, 6, 4, 1]
assert binomial_coefficients_list(5) == [1, 5, 10, 10, 5, 1]
assert binomial_coefficients_list(6) == [1, 6, 15, 20, 15, 6, 1]
def test_binomial_coefficients_list():
assert binomial_coefficients_list(0) == [1]
assert binomial_coefficients_list(1) == [1, 1]
assert binomial_coefficients_list(2) == [1, 2, 1]
assert binomial_coefficients_list(3) == [1, 3, 3, 1]
assert binomial_coefficients_list(4) == [1, 4, 6, 4, 1]
assert binomial_coefficients_list(5) == [1, 5, 10, 10, 5, 1]
assert binomial_coefficients_list(6) == [1, 6, 15, 20, 15, 6, 1]
def _compute_num_chambers(self):
"""Computes the number of chambers defined by the hyperplanes
corresponding to the normal vectors."""
d = self.rank
n = self.num_normal_vectors
raw_cfs = sp.binomial_coefficients_list(n)
cfs = np.array([(-1)**i * raw_cfs[i] for i in range(n + 1)])
powers = np.array([max(entry, 0)
for entry in [d - k for k in range(n + 1)]])
ys = np.array([-1] * len(powers))
return (-1)**d * sum(cfs * (ys**powers))
def mu(f,a,n,x,y):
binom = sympy.binomial_coefficients_list(n)
binom = map(numpy.double, binom)
normf = numpy.sqrt(sum(numpy.abs(a[k](x))**2 * 1/binom[k]
for k in xrange(n+1)
)
)
Delta = numpy.sqrt(numpy.double(n)) * (1 + numpy.abs(y))
normy = 1 + numpy.abs(y)
return normf * Delta / (2 * normy)
def _new_polynomial(F,X,Y,tau,l):
"""
Computes the next iterate of the newton-puiseux algorithms. Given
the Puiseux data `\tau = (q,\mu,m,\beta)` ``_new_polynomial``
returns .. math:
\tilde{F} = F(\mu X^q,X^m(\beta+Y))/X \in \mathbb{L}(\mu,\beta)(X)[Y]
In this algorithm, the choice of parameters will always result in
.. math:
\tilde{F} \in \mathbb{L}(\mu,\beta)[X,Y]
Algorithm:
Calling sympy.Poly() with new generators (i.e. [x,y]) takes a long
time. Hence, the technique below.
"""
q,mu,m,beta,eta = tau
d = {}
# for each monomial of the form
#
# c * x**a * y**b
#
# compute the appropriate new terms after applying the
# transformation
#
# x |--> mu * x**q
# y |--> eta * x**m * (beta+y)
#
for (a,b),c in F.as_dict().iteritems():
binom = sympy.binomial_coefficients_list(b)
new_a = int(q*a + m*b)
for i in xrange(b+1):
# the coefficient of the x***(qa+mb) * y**i term
new_c = c * (mu**a) * (eta**b) * (binom[i]) * (beta**(b-i))
try:
d[(new_a,i)] += new_c
except KeyError:
d[(new_a,i)] = new_c
# now perform the polynomial division by x**l. In the case when
# the curve is singular there will be a cancellation resulting in
# a term of the form (0,0):0 . Creating a new polynomial
# containing such a term will result in the zero polynomial.
new_d = dict([((a-l,b),c) for (a,b),c in d.iteritems()])
Fnew = sympy.Poly.from_dict(new_d, gens=[X,Y], domain=sympy.EX)
return Fnew
def test_binomial_coefficients():
for n in range(15):
c = binomial_coefficients(n)
l = [c[k] for k in sorted(c)]
assert l == binomial_coefficients_list(n)
def test_binomial_coefficients():
for n in range(15):
c = binomial_coefficients(n)
l = [c[k] for k in sorted(c)]
assert l == binomial_coefficients_list(n)
