def from_factored_str(cls, numerator, denomenator, factor_numerator):
if factor_numerator:
unpacked_numerator = cls.no_powers(factor_list(numerator))
else:
unpacked_numerator = [str(numerator.expand())]
unpacked_denomenator = cls.no_powers(factor_list(denomenator))
return cls(unpacked_numerator, unpacked_denomenator)
def apply_list(self, expr, vars, evaluation):
"FactorTermsList[expr_, vars_List]"
if expr == Integer(0):
return Expression("List", Integer(1), Integer(0))
elif isinstance(expr, Number):
return Expression("List", expr, Integer(1))
for x in vars.leaves:
if not (isinstance(x, Atom)):
return evaluation.message("CoefficientList", "ivar", x)
sympy_expr = expr.to_sympy()
if sympy_expr is None:
return Expression("List", Integer(1), expr)
sympy_expr = sympy.together(sympy_expr)
sympy_vars = [
x.to_sympy() for x in vars.leaves
if isinstance(x, Symbol) and sympy_expr.is_polynomial(x.to_sympy())
]
result = []
numer, denom = sympy_expr.as_numer_denom()
try:
if denom == 1:
# Get numerical part
num_coeff, num_polys = sympy.factor_list(sympy.Poly(numer))
result.append(num_coeff)
# Get factors are independent of sub list of variables
if (sympy_vars and isinstance(expr, Expression) and any(
x.free_symbols.issubset(sympy_expr.free_symbols)
for x in sympy_vars)):
for i in reversed(range(len(sympy_vars))):
numer = sympy.factor(numer) / sympy.factor(num_coeff)
num_coeff, num_polys = sympy.factor_list(
sympy.Poly(numer),
*[x for x in sympy_vars[:(i + 1)]])
result.append(sympy.expand(num_coeff))
# Last factor
numer = sympy.factor(numer) / sympy.factor(num_coeff)
result.append(sympy.expand(numer))
else:
num_coeff, num_polys = sympy.factor_list(sympy.Poly(numer))
den_coeff, den_polys = sympy.factor_list(sympy.Poly(denom))
result = [
num_coeff / den_coeff,
sympy.expand(
sympy.factor(numer) / num_coeff /
(sympy.factor(denom) / den_coeff)),
]
except sympy.PolynomialError: # MMA does not raise error for non poly
result.append(sympy.expand(numer))
# evaluation.message(self.get_name(), 'poly', expr)
return Expression("List", *[from_sympy(i) for i in result])
def get_lc(self):
numerator_factors = factor_list(self.get_n_str())
n_no_lc = self.no_powers(numerator_factors, include_lc=False)
n_lc = int(numerator_factors[0])
denomenator_factors = factor_list(self.get_d_str())
d_no_lc = self.no_powers(denomenator_factors, include_lc=False)
d_lc = int(denomenator_factors[0])
return n_lc, n_no_lc, d_lc, d_no_lc
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
terms = factor_list(eq)[1]
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
okay = True
while len(s) and okay:
solution = s.pop()
okay = False
for term in terms:
subeq = term[0]
if simplify(_mexpand(Subs(subeq, var, solution).doit())) == 0:
okay = True
break
return okay
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
terms = factor_list(eq)[1]
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
okay = True
while len(s) and okay:
solution = s.pop()
okay = False
for term in terms:
subeq = term[0]
if simplify(_mexpand(Subs(subeq, var, solution).doit())) == 0:
okay = True
break
return okay
def reduce_squarefree(self):
eqs_sqf = []
factored_eqs_sqf = {}
flag = False
for eq in self.eqs:
# If factored before, just copy the factorization
eq_expr = eq.expr
if eq_expr in self.factored_eqs:
eqs_sqf.append(eq)
factored_eqs_sqf[eq_expr] = self.factored_eqs[eq_expr]
continue
factors = sp.factor_list(eq, extension = True)
factors = [ factor[0] for factor in factors[1] ]
eq_sqf = to_poly(1, self.X_vars)
for factor in factors:
eq_sqf *= factor
eq_sqf = to_poly(eq_sqf / eq_sqf.LC(order = 'grevlex'), self.X_vars)
eqs_sqf.append(eq_sqf)
factored_eqs_sqf[eq_sqf.expr] = factors
if not flag and eq_sqf.LM() != eq.LM():
flag = True
self.eqs = eqs_sqf
self.factored_eqs = factored_eqs_sqf
return flag
def apply_list(self, expr, vars, evaluation):
'FactorTermsList[expr_, vars_List]'
if expr == Integer(0):
return Expression('List', Integer(1), Integer(0))
elif isinstance(expr, Number):
return Expression('List', expr, Integer(1))
for x in vars.leaves:
if not(isinstance(x, Atom)):
return evaluation.message('CoefficientList', 'ivar', x)
sympy_expr = expr.to_sympy()
if sympy_expr is None:
return Expression('List', Integer(1), expr)
sympy_expr = sympy.together(sympy_expr)
sympy_vars = [x.to_sympy() for x in vars.leaves if isinstance(x, Symbol) and sympy_expr.is_polynomial(x.to_sympy())]
result = []
numer, denom = sympy_expr.as_numer_denom()
try:
from sympy import factor, factor_list, Poly
if denom == 1:
# Get numerical part
num_coeff, num_polys = factor_list(Poly(numer))
result.append(num_coeff)
# Get factors are independent of sub list of variables
if (sympy_vars and isinstance(expr, Expression)
and any(x.free_symbols.issubset(sympy_expr.free_symbols) for x in sympy_vars)):
for i in reversed(range(len(sympy_vars))):
numer = factor(numer) / factor(num_coeff)
num_coeff, num_polys = factor_list(Poly(numer), *[x for x in sympy_vars[:(i+1)]])
result.append(sympy.expand(num_coeff))
# Last factor
numer = factor(numer) / factor(num_coeff)
result.append(sympy.expand(numer))
else:
num_coeff, num_polys = factor_list(Poly(numer))
den_coeff, den_polys = factor_list(Poly(denom))
result = [num_coeff / den_coeff, sympy.expand(factor(numer)/num_coeff / (factor(denom)/den_coeff))]
except sympy.PolynomialError: # MMA does not raise error for non poly
result.append(sympy.expand(numer))
# evaluation.message(self.get_name(), 'poly', expr)
return Expression('List', *[from_sympy(i) for i in result])
def factorlist():
try:
print('factor list: {}'.format(request.json))
ps = parse_expr(request.json['expression'], locals())
return jsonify({'in': latex(ps), 'out': latex(factor_list(ps))})
except Exception as e:
print(e)
return str(e), 400
def geratriz(basedir, gx, gy):
count = 0
limit = 1000000
while count < limit:
p = pij(gx, gy) + pij(gx, gy) - pij(gx, gy)
if len(factor_list(p)[1]) == 1:
dirname = f"{basedir}/gx={gx}/gy={gy}"
filename = f"{dirname}/{hash(p)}"
pathlib.Path(dirname).mkdir(parents=True, exist_ok=True)
np.save(filename, np.array([p]))
count = count + 1
def get_max_even_divisor(polynomial):
"""
:param polynomial: sympy polynomial
:return: leading coefficient of poly, max polynomial divisor of poly that's even power, remainder of poly / max_divisor
"""
_factors = factor_list(polynomial)
_coeff_leading = _factors[0]
_factors_non_constant = _factors[1]
_factors_max_even_divisor = [(_p, 2 * (n // 2)) for (_p, n) in _factors_non_constant]
_factors_remainder = [(_p, n - 2 * (n // 2)) for (_p, n) in _factors_non_constant]
_max_even_divisor = np.prod([_p.as_expr() ** n for (_p, n) in _factors_max_even_divisor])
_remainder = np.prod([_p.as_expr() ** n for (_p, n) in _factors_remainder])
return _coeff_leading, _max_even_divisor, _remainder
def eqn_to_mul_add(eqn):
atoms = {s: 0 for s in eqn.atoms() if s.is_symbol}
if not atoms:
return eqn
coeff, factors = sp.factor_list(eqn)
if len(factors) > 1:
for factor, power in factors:
coeff *= eqn_to_mul_add(factor)
return coeff
elif not factors:
return coeff
this_factor, power = factors[0]
_, terms = this_factor.as_coeff_add()
for term in terms:
for atom in term.atoms():
if atom in atoms:
atoms[atom] += 1
root, _ = max(atoms.items(), key=lambda x: x[1])
b = this_factor.coeff(root, 0)
a = 0
for i in range(10):
a += this_factor.coeff(root, i + 1) * (root**i)
a = eqn_to_mul_add(a)
b = eqn_to_mul_add(b)
result = coeff
for i in range(power):
result *= (a * root + b)
return result
def divides(a,b):
x = b/a
print(re(x))
real = factor_list(re(x)*Number(1))[0]
imaginary = factor_list(im(x)*Number(1))[0]
return real.is_Integer and imaginary.is_Integer
def sympy_factor(self, p):
sympy_p = self.sympy_from_upolynomial(p)
if (p.ring().modulus() is None):
return sympy.factor_list(sympy_p)
else:
return sympy.factor_list(sympy_p, modulus=p.ring().modulus())
def compute_ifactors(poly):
for (ipoly, _) in sp.factor_list(poly)[1]:
yield ipoly
def integral_basis(f,x,y):
"""
Compute the integral basis {b1, ..., bg} of the algebraic function
field C[x,y] / (f).
"""
# If the curve is not monic then map y |-> y/lc(x) where lc(x)
# is the leading coefficient of f
T = sympy.Dummy('T')
d = sympy.degree(f,y)
lc = sympy.LC(f,y)
if x in lc:
f = sympy.ratsimp( f.subs(y,y/lc)*lc**(d-1) )
else:
f = f/lc
lc = 1
#
# Compute df
#
p = sympy.Poly(f,[x,y])
n = p.degree(y)
res = sympy.resultant(p,p.diff(y),y)
factors = sympy.factor_list(res)[1]
df = [k for k,deg in factors if (deg > 1) and (sympy.LC(k) == 1)]
#
# Compute series truncations at appropriate x points
#
alpha = []
r = []
for l in range(len(df)):
k = df[l]
alphak = sympy.roots(k).keys()[0]
rk = compute_series_truncations(f,x,y,alphak,T)
alpha.append(alphak)
r.append(rk)
#
# Main Loop
#
a = [sympy.Dummy('a%d'%k) for k in xrange(n)]
b = [1]
for d in range(1,n):
bd = y*b[-1]
for l in range(len(df)):
k = df[l]
alphak = alpha[l]
rk = r[l]
found_something = True
while found_something:
# construct system of equations consisting of the coefficients
# of negative powers of (x-alphak) in the substitutions
# A(r_{k,1}),...,A(r_{k,n})
A = (sum(ak*bk for ak,bk in zip(a,b)) + bd) / (x - alphak)
coeffs = []
for rki in rk:
# substitute and extract coefficients
A_rki = A.subs(y,rki)
coeffs.extend(_negative_power_coeffs(A_rki, x, alphak))
# solve the coefficient equations for a0,...,a_{d-1}
coeffs = [coeff.as_numer_denom()[0] for coeff in coeffs]
sols = sympy.solve_poly_system(coeffs, a[:d])
if sols is None or sols == []:
found_something = False
else:
sol = sols[0]
bdm1 = sum( sol[i]*bk for i,bk in zip(range(d),b) )
bd = (bdm1 + bd) / k
# bd found. Append to list of basis elements
b.append( bd )
# finally, convert back to singularized curve if necessary
for i in xrange(1,len(b)):
b[i] = b[i].subs(y,y*lc).ratsimp()
return b
f= sym.lambdify(x,sym1,'numpy') a=np.arange(1,10,0.01) f(a) # return the np.ndarray # it's fast Eq(f,2) or f-2 from sympy.utilities.autowrap import ufuncify g = ufuncify([x],f) ufuncify is better than lambdify for larger than 1 s; do to its compile overhead ############################################################ # 1.1 sym1.simplify() sym1.expand() sym1.factor() sym.factor_list(expr) sym1.collect(x) sym1.coeff(x,2) sym1.cancel() sym1.apart() # tajziye kasr ha # 1.2 sym1.trigsimp() sym1.rewrite(sin) # tan(x).rewrite(sin) , foctorial(x).rewrite(gamma) ###### # 2.1 sym1.series(x,x0,n) sym1.removeO() ########################################################### # 3.1 # 3.2 sym.solveset(f,x,domain=sym.S.Complexes) sym.solveset(f<0,domain=sym.S.Reals)
def sympy_factor(self, p):
sympy_p = self.sympy_from_upolynomial(p)
if (p.ring().modulus() is None):
return sympy.factor_list(sympy_p)
else:
return sympy.factor_list(sympy_p, modulus=p.ring().modulus())
