def __init__(self, polynomials, variables):
"""
A class that takes two lists, a list of polynomials and list of
variables. Returns the Dixon matrix of the multivariate system.
Parameters
----------
polynomials : list of polynomials
A list of m n-degree polynomials
variables: list
A list of all n variables
"""
self.polynomials = polynomials
self.variables = variables
self.n = len(self.variables)
self.m = len(self.polynomials)
a = IndexedBase("alpha")
# A list of n alpha variables (the replacing variables)
self.dummy_variables = [a[i] for i in range(self.n)]
# A list of the d_max of each variable.
self.max_degrees = [
total_degree(poly, *self.variables) for poly in self.polynomials
]
def __init__(self, polynomials, variables):
"""
A class that takes two lists, a list of polynomials and list of
variables. Returns the Dixon matrix of the multivariate system.
Parameters
----------
polynomials : list of polynomials
A list of m n-degree polynomials
variables: list
A list of all n variables
"""
self.polynomials = polynomials
self.variables = variables
self.n = len(self.variables)
self.m = len(self.polynomials)
a = IndexedBase("alpha")
# A list of n alpha variables (the replacing variables)
self.dummy_variables = [a[i] for i in range(self.n)]
# A list of the d_max of each variable.
self.max_degrees = [total_degree(poly, *self.variables) for poly
in self.polynomials]
def get_reduced_nonreduced(self):
r"""
Returns
-------
reduced: list
A list of the reduced monomials
non_reduced: list
A list of the monomials that are not reduced
Definition.
---------
A polynomial is said to be reduced in x_i, if its degree (the
maximum degree of its monomials) in x_i is less than d_i. A
polynomial that is reduced in all variables but one is said
simply to be reduced.
"""
divisible = []
for m in self.monomial_set:
temp = []
for i, v in enumerate(self.variables):
temp.append(bool(total_degree(m, v) >= self.degrees[i]))
divisible.append(temp)
reduced = [i for i, r in enumerate(divisible) if sum(r) < self.n - 1]
non_reduced = [
i for i, r in enumerate(divisible) if sum(r) >= self.n - 1
]
return reduced, non_reduced
def get_reduced_nonreduced(self):
r"""
Returns
-------
reduced: list
A list of the reduced monomials
non_reduced: list
A list of the monomials that are not reduced
Definition.
---------
A polynomial is said to be reduced in x_i, if its degree (the
maximum degree of its monomials) in x_i is less than d_i. A
polynomial that is reduced in all variables but one is said
simply to be reduced.
"""
divisible = []
for m in self.monomial_set:
temp = []
for i, v in enumerate(self.variables):
temp.append(bool(total_degree(m, v) >= self.degrees[i]))
divisible.append(temp)
reduced = [i for i, r in enumerate(divisible)
if sum(r) < self.n - 1]
non_reduced = [i for i, r in enumerate(divisible)
if sum(r) >= self.n -1]
return reduced, non_reduced
def enum_ipolys(xs, max_degree, max_summands, max_coeff, max_ipolys):
"""
Enumerates _some_ irreducible polynomials upto degree `max_degree`.
Args:
- xs (list of sympy symbols): the variables composing the monomials
- max_degree (int): the maximum degree of the monomials
- max_summands (int): the maximum number of summands in a polynomial
- max_coeff (int): the maximum coefficient magnitude of a monomial
- max_ipolys (int): the maximum number of irreducible polynomials to generate
Returns:
- stats (dict): statistics of the generation process
- ipolys (dict): maps (degree, def_nneg) to a set of irreducible polynomials
"""
stats = collections.defaultdict(int)
ipolys = collections.defaultdict(set)
for ums in util.flatten([itertools.combinations(enum_umonomials(xs, max_degree), n_summands) for n_summands in range(1, max_summands+1)]):
d_poly = max([sp.total_degree(um) for um in ums])
for cs in itertools.product(*([list(range(-max_coeff, 0) ) + list(range(1, max_coeff + 1))] * len(ums))):
if all([c < 0 for c in cs]) or functools.reduce(math.gcd, cs) != 1: continue
poly = sum([c * um for c, um in zip(cs, ums)])
for ipoly in util.compute_ifactors(poly):
d_ipoly = sp.total_degree(ipoly)
def_nneg = not util.contains_negation(ipoly)
if ipoly in ipolys[(d_ipoly, def_nneg)]:
stats['n_dups_%d' % d_ipoly] += 1
else:
stats['n_new_%d' % d_ipoly] += 1
ipolys[(d_ipoly, def_nneg)].add(ipoly)
stats['n_ipolys'] += 1
if stats['n_ipolys'] > 0 and stats['n_ipolys'] % 1000 == 0: print(dict(stats))
if max_ipolys is not None and stats['n_ipolys'] >= max_ipolys: return stats, ipolys
return stats, ipolys
def __init__(self, polynomials, variables):
"""
Parameters
----------
variables: list
A list of all n variables
polynomials : list of sympy polynomials
A list of m n-degree polynomials
"""
self.polynomials = polynomials
self.variables = variables
self.n = len(variables)
# A list of the d_max of each variable.
self.degrees = [total_degree(poly, *self.variables) for poly
in self.polynomials]
self.degree_m = self._get_degree_m()
self.monomials_size = self.get_size()
def __init__(self, polynomials, variables):
"""
Parameters
----------
variables: list
A list of all n variables
polynomials : list of sympy polynomials
A list of m n-degree polynomials
"""
self.polynomials = polynomials
self.variables = variables
self.n = len(variables)
# A list of the d_max of each variable.
self.degrees = [
total_degree(poly, *self.variables) for poly in self.polynomials
]
self.degree_m = self._get_degree_m()
self.monomials_size = self.get_size()
def reduce_direct_solve(self):
# Look for equations which are linear in some variable
for eq in self.eqs:
for x in [ x for x in eq.free_symbols.difference(self.S_vars) if eq.degree(x) == 1 ]:
# See if we can directly solve for x
r = eq.subs(x, 0)
q = (eq - r) / x
if sp.total_degree(q) != 0:
continue
# Solve for x and remove x from X_vars
x_value = -r / q
self.X_vars = self.X_vars.copy()
self.X_vars.remove(x)
self.eqs = [ to_poly(f.subs(x, x_value), self.X_vars) for f in self.eqs if f != eq ]
self.op_eqs = [ to_poly(f.subs(x, x_value), self.X_vars) for f in self.op_eqs ]
return True
else:
return False