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 = [max(degree_list(poly)[i] for poly in self.polynomials)
for i in range(self.n)]
def decompose(expr, separate=False):
"""Decomposes an input polynomial into homogeneous ones of
smaller or equal degree.
Returns a dictionary with keys as the degree of the smaller
constituting polynomials. Values are the constituting polynomials.
Parameters
==========
expr : Polynomial(SymPy expression)
Optional Parameters :
separate : If True then simply return a list of the constituent monomials
If not then break up the polynomial into constituent homogeneous
polynomials.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.integrals.intpoly import decompose
>>> decompose(x**2 + x*y + x + y + x**3*y**2 + y**5)
{1: x + y, 2: x**2 + x*y, 5: x**3*y**2 + y**5}
>>> decompose(x**2 + x*y + x + y + x**3*y**2 + y**5, True)
{x, x**2, y, y**5, x*y, x**3*y**2}
"""
expr = S(expr)
poly_dict = {}
if isinstance(expr, Expr) and not expr.is_number:
if expr.is_Symbol:
poly_dict[1] = expr
elif expr.is_Add:
symbols = expr.atoms(Symbol)
degrees = [(sum(degree_list(monom, *symbols)), monom)
for monom in expr.args]
if separate:
return {monom[1] for monom in degrees}
else:
for monom in degrees:
degree, term = monom
if poly_dict.get(degree):
poly_dict[degree] += term
else:
poly_dict[degree] = term
elif expr.is_Pow:
_, degree = expr.args
poly_dict[degree] = expr
else: # Now expr can only be of `Mul` type
degree = 0
for term in expr.args:
term_type = len(term.args)
if term_type == 0 and term.is_Symbol:
degree += 1
elif term_type == 2:
degree += term.args[1]
poly_dict[degree] = expr
else:
poly_dict[0] = expr
if separate:
return set(poly_dict.values())
return poly_dict
def decompose(expr, separate=False):
"""Decomposes an input polynomial into homogeneous ones of
smaller or equal degree.
Returns a dictionary with keys as the degree of the smaller
constituting polynomials. Values are the constituting polynomials.
Parameters
==========
expr : Polynomial(SymPy expression)
Optional Parameters :
---------------------
separate : If True then simply return a list of the constituent monomials
If not then break up the polynomial into constituent homogeneous
polynomials.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.integrals.intpoly import decompose
>>> decompose(x**2 + x*y + x + y + x**3*y**2 + y**5)
{1: x + y, 2: x**2 + x*y, 5: x**3*y**2 + y**5}
>>> decompose(x**2 + x*y + x + y + x**3*y**2 + y**5, True)
{x, x**2, y, y**5, x*y, x**3*y**2}
"""
expr = S(expr)
poly_dict = {}
if isinstance(expr, Expr) and not expr.is_number:
if expr.is_Symbol:
poly_dict[1] = expr
elif expr.is_Add:
symbols = expr.atoms(Symbol)
degrees = [(sum(degree_list(monom, *symbols)), monom)
for monom in expr.args]
if separate:
return {monom[1] for monom in degrees}
else:
for monom in degrees:
degree, term = monom
if poly_dict.get(degree):
poly_dict[degree] += term
else:
poly_dict[degree] = term
elif expr.is_Pow:
_, degree = expr.args
poly_dict[degree] = expr
else: # Now expr can only be of `Mul` type
degree = 0
for term in expr.args:
term_type = len(term.args)
if term_type == 0 and term.is_Symbol:
degree += 1
elif term_type == 2:
degree += term.args[1]
poly_dict[degree] = expr
else:
poly_dict[0] = expr
if separate:
return set(poly_dict.values())
return poly_dict
def get_max_degrees(self, polynomial):
r"""
Returns a list of the maximum degree of each variable appearing
in the coefficients of the Dixon polynomial. The coefficients are
viewed as polys in $x_1, x_2, \dots, x_n$.
"""
deg_lists = [degree_list(Poly(poly, self.variables))
for poly in polynomial.coeffs()]
max_degrees = [max(degs) for degs in zip(*deg_lists)]
return max_degrees
