def test_RootOf___eval_Eq__():
f = Function('f')
r = RootOf(x**3 + x + 3, 2)
r1 = RootOf(x**3 + x + 3, 1)
assert Eq(r, r1) is false
assert Eq(r, r) is true
assert Eq(r, x) is false
assert Eq(r, 0) is false
assert Eq(r, oo) is false
assert Eq(r, I) is false
assert Eq(r, f(0)) is false
assert Eq(r, f(0)) is false
sol = solve(r.expr, x)
for s in sol:
if s[x].is_real:
assert Eq(r, s[x]) is false
r = RootOf(r.expr, 0)
for s in sol:
if s[x].is_real:
assert Eq(r, s[x]) is true
eq = x**3 + x + 1
assert ([Eq(RootOf(eq, i), j[x])
for i in range(3) for j in solve(eq)] ==
[False, False, True, False, True, False, True, False, False])
assert Eq(RootOf(eq, 0), 1 + I) is false
def test_unrad_fail():
# this only works if we check real_root(eq.subs(x, Rational(1, 3)))
# but checksol doesn't work like that
assert solve(root(x**3 - 3 * x**2, 3) + 1 - x) == [Rational(1, 3)]
assert solve(root(x + 1, 3) + root(x**2 - 2, 5) + 1) == [
-1, -1 + RootOf(x**5 + x**4 + 5 * x**3 + 8 * x**2 + 10 * x + 5, 0)**3
]
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs({x: x - 2})
assert solve(f, x) == [{x: 2}]
assert solve(f - 1, x) == [{x: 1}, {x: 3}]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [{x: 2}]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [{x: 2}, {x: 5}]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [{x: 2}, {x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (-x + 2, x - 2 <= 0),
(x - 2, x - 2 > 0))
assert solve(g, x) == [{x: 5}]
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs({x: x - 2})
assert solve(f, x) == [{x: 2}]
assert solve(f - 1, x) == [{x: 1}, {x: 3}]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [{x: 2}]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [{x: 2}, {x: 5}]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [{x: 2}, {x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [{x: 5}]
g = Piecewise(((x - 5)**5, x >= 2),
(-x + 2, x - 2 <= 0), (x - 2, x - 2 > 0))
assert solve(g, x) == [{x: 5}]
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs(x, x - 2)
assert solve(f, x) == [2]
assert solve(f - 1, x) == [1, 3]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [2]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (-x + 2, x - 2 <= 0),
(x - 2, x - 2 > 0))
assert solve(g, x) == [5]
def test_idiff():
# the use of idiff in ellipse also provides coverage
circ = x**2 + y**2 - 4
ans = -3 * x * (x**2 + y**2) / y**5
assert ans == idiff(circ, y, x, 3).simplify()
assert ans == idiff(circ, [y], x, 3).simplify()
assert idiff(circ, y, x, 3).simplify() == ans
explicit = 12 * x / sqrt(-x**2 + 4)**5
assert ans.subs(solve(circ, y)[0]).equals(explicit)
assert True in (sol[y].diff(x, 3).equals(explicit)
for sol in solve(circ, y))
assert idiff(x + t + y, [y, t], x) == -Derivative(t, x) - 1
def test_nfloat():
x = Symbol('x')
eq = x**Rational(4, 3) + 4*cbrt(x)/3
assert _aresame(nfloat(eq), x**Rational(4, 3) + (4.0/3)*cbrt(x))
assert _aresame(nfloat(eq, exponent=True), x**(4.0/3) + (4.0/3)*x**(1.0/3))
eq = x**Rational(4, 3) + 4*x**(x/3)/3
assert _aresame(nfloat(eq), x**Rational(4, 3) + (4.0/3)*x**(x/3))
big = 12345678901234567890
# specify precision to match value used in nfloat
Float_big = Float(big, 15)
assert _aresame(nfloat(big), Float_big)
assert _aresame(nfloat(big*x), Float_big*x)
assert _aresame(nfloat(x**big, exponent=True), x**Float_big)
assert nfloat({x: sqrt(2)}) == {x: nfloat(sqrt(2))}
assert nfloat({sqrt(2): x}) == {sqrt(2): x}
assert nfloat(cos(x + sqrt(2))) == cos(x + nfloat(sqrt(2)))
# issue sympy/sympy#6342
lamda = Symbol('lamda')
f = x*lamda + lamda**3*(x/2 + Rational(1, 2)) + lamda**2 + Rational(1, 4)
assert not any(a[lamda].free_symbols
for a in solve(f.subs({x: -0.139})))
# issue sympy/sympy#6632
assert nfloat(-100000*sqrt(2500000001) + 5000000001) == \
9.99999999800000e-11
# issue sympy/sympy#7122
eq = cos(3*x**4 + y)*RootOf(x**5 + 3*x**3 + 1, 0)
assert str(nfloat(eq, exponent=False, n=1)) == '-0.7*cos(3.0*x**4 + y)'
def idiff(eq, y, x, n=1):
"""Return ``dy/dx`` assuming that ``eq == 0``.
Parameters
==========
y : the dependent variable or a list of dependent variables (with y first)
x : the variable that the derivative is being taken with respect to
n : the order of the derivative (default is 1)
Examples
========
>>> from diofant.abc import x, y, a
>>> from diofant.geometry.util import idiff
>>> circ = x**2 + y**2 - 4
>>> idiff(circ, y, x)
-x/y
>>> idiff(circ, y, x, 2).simplify()
-(x**2 + y**2)/y**3
Here, ``a`` is assumed to be independent of ``x``:
>>> idiff(x + a + y, y, x)
-1
Now the x-dependence of ``a`` is made explicit by listing ``a`` after
``y`` in a list.
>>> idiff(x + a + y, [y, a], x)
-Derivative(a, x) - 1
See Also
========
diofant.core.function.Derivative: represents unevaluated derivatives
diofant.core.function.diff: explicitly differentiates wrt symbols
"""
if is_sequence(y):
dep = set(y)
y = y[0]
elif isinstance(y, (Dummy, Symbol)):
dep = {y}
else:
raise ValueError("expecting x-dependent symbol(s) but got: %s" % y)
f = {s: Function(s.name)(x) for s in eq.free_symbols if s != x and s in dep}
dydx = Function(y.name)(x).diff(x)
eq = eq.subs(f)
derivs = {}
for i in range(n):
yp = solve(eq.diff(x), dydx)[0].subs(derivs)
if i == n - 1:
return yp.subs([(v, k) for k, v in f.items()])
derivs[dydx] = yp
eq = dydx - yp
dydx = dydx.diff(x)
def ratint_ratpart(f, g, x):
"""
Horowitz-Ostrogradsky algorithm.
Given a field K and polynomials f and g in K[x], such that f and g
are coprime and deg(f) < deg(g), returns fractions A and B in K(x),
such that f/g = A' + B and B has square-free denominator.
Examples
========
>>> from diofant.integrals.rationaltools import ratint_ratpart
>>> from diofant.abc import x, y
>>> from diofant import Poly
>>> ratint_ratpart(Poly(1, x, domain='ZZ'),
... Poly(x + 1, x, domain='ZZ'), x)
(0, 1/(x + 1))
>>> ratint_ratpart(Poly(1, x, domain='EX'),
... Poly(x**2 + y**2, x, domain='EX'), x)
(0, 1/(x**2 + y**2))
>>> ratint_ratpart(Poly(36, x, domain='ZZ'),
... Poly(x**5 - 2*x**4 - 2*x**3 + 4*x**2 + x - 2, x, domain='ZZ'), x)
((12*x + 6)/(x**2 - 1), 12/(x**2 - x - 2))
See Also
========
diofant.integrals.rationaltools.ratint
diofant.integrals.rationaltools.ratint_logpart
"""
from diofant import solve
f = Poly(f, x)
g = Poly(g, x)
u, v, _ = g.cofactors(g.diff())
n = u.degree()
m = v.degree()
A_coeffs = [Dummy('a' + str(n - i)) for i in range(0, n)]
B_coeffs = [Dummy('b' + str(m - i)) for i in range(0, m)]
C_coeffs = A_coeffs + B_coeffs
A = Poly(A_coeffs, x, domain=ZZ[C_coeffs])
B = Poly(B_coeffs, x, domain=ZZ[C_coeffs])
H = f - A.diff() * v + A * (u.diff() * v).quo(u) - B * u
result = solve(H.coeffs(), C_coeffs)
A = A.as_expr().subs(result)
B = B.as_expr().subs(result)
rat_part = cancel(A / u.as_expr(), x)
log_part = cancel(B / v.as_expr(), x)
return rat_part, log_part
def given(expr, condition=None, **kwargs):
""" Conditional Random Expression
From a random expression and a condition on that expression creates a new
probability space from the condition and returns the same expression on that
conditional probability space.
Examples
========
>>> from diofant.stats import given, density, Die
>>> X = Die('X', 6)
>>> Y = given(X, X > 3)
>>> density(Y).dict
{4: 1/3, 5: 1/3, 6: 1/3}
Following convention, if the condition is a random symbol then that symbol
is considered fixed.
>>> from diofant.stats import Normal
>>> from diofant import pprint
>>> from diofant.abc import z
>>> X = Normal('X', 0, 1)
>>> Y = Normal('Y', 0, 1)
>>> pprint(density(X + Y, Y)(z), use_unicode=False)
2
-(-Y + z)
-----------
___ 2
\/ 2 *E
------------------
____
2*\/ pi
"""
if not random_symbols(condition) or pspace_independent(expr, condition):
return expr
if isinstance(condition, RandomSymbol):
condition = Eq(condition, condition.symbol)
condsymbols = random_symbols(condition)
if (isinstance(condition, Equality) and len(condsymbols) == 1 and
not isinstance(pspace(expr).domain, ConditionalDomain)):
rv = tuple(condsymbols)[0]
results = solve(condition, rv)
return sum(expr.subs(rv, res) for res in results)
# Get full probability space of both the expression and the condition
fullspace = pspace(Tuple(expr, condition))
# Build new space given the condition
space = fullspace.conditional_space(condition, **kwargs)
# Dictionary to swap out RandomSymbols in expr with new RandomSymbols
# That point to the new conditional space
swapdict = rs_swap(fullspace.values, space.values)
# Swap random variables in the expression
expr = expr.xreplace(swapdict)
return expr
def test_diofantissue_453():
x = Symbol('x', real=True)
assert isolve(abs((x - 1) / (x - 5)) <= Rational(1, 3),
x) == And(Integer(-1) <= x, x <= 2)
assert solve(abs((x - 1) / (x - 5)) - Rational(1, 3), x) == [{
x: -1
}, {
x: 2
}]
def test_sympyissue_3557():
f_1 = x * a + y * b + z * c - 1
f_2 = x * d + y * e + z * f - 1
f_3 = x * g + y * h + z * i - 1
solutions = solve([f_1, f_2, f_3], x, y, z, simplify=False)
assert simplify(solutions[0][y]) == \
(a*i + c*d + f*g - a*f - c*g - d*i) / \
(a*e*i + b*f*g + c*d*h - a*f*h - b*d*i - c*e*g)
def test_RootOf___eval_Eq__():
f = Function('f')
r = RootOf(x**3 + x + 3, 2)
r1 = RootOf(x**3 + x + 3, 1)
assert Eq(r, r1) is S.false
assert Eq(r, r) is S.true
assert Eq(r, x) is S.false
assert Eq(r, 0) is S.false
assert Eq(r, S.Infinity) is S.false
assert Eq(r, I) is S.false
assert Eq(r, f(0)) is S.false
assert Eq(r, f(0)) is S.false
sol = solve(r.expr)
for s in sol:
if s.is_real:
assert Eq(r, s) is S.false
r = RootOf(r.expr, 0)
for s in sol:
if s.is_real:
assert Eq(r, s) is S.true
eq = (x**3 + x + 1)
assert [Eq(RootOf(eq, i), j) for i in range(3) for j in solve(eq)] == \
[False, False, True, False, True, False, True, False, False]
assert Eq(RootOf(eq, 0), 1 + S.ImaginaryUnit) is S.false
def _inverse_cdf_expression(self):
""" Inverse of the CDF
Used by sample
"""
x, z = symbols('x, z', extended_real=True, positive=True, cls=Dummy)
# Invert CDF
try:
inverse_cdf = solve(self.cdf(x) - z, x)
except NotImplementedError:
inverse_cdf = None
if not inverse_cdf or len(inverse_cdf) != 1:
raise NotImplementedError("Could not invert CDF")
return Lambda(z, inverse_cdf[0])
def test_RootOf___eval_Eq__():
f = Function('f')
r = RootOf(x**3 + x + 3, 2)
r1 = RootOf(x**3 + x + 3, 1)
assert Eq(r, r1) is false
assert Eq(r, r) is true
assert Eq(r, x) is false
assert Eq(r, 0) is false
assert Eq(r, oo) is false
assert Eq(r, I) is false
assert Eq(r, f(0)) is false
assert Eq(r, f(0)) is false
sol = solve(r.expr, x)
for s in sol:
if s[x].is_real:
assert Eq(r, s[x]) is false
r = RootOf(r.expr, 0)
for s in sol:
if s[x].is_real:
assert Eq(r, s[x]) is true
eq = x**3 + x + 1
assert ([Eq(RootOf(eq, i), j[x]) for i in range(3) for j in solve(eq)
] == [False, False, True, False, True, False, True, False, False])
assert Eq(RootOf(eq, 0), 1 + I) is false
def main():
print("Initial metric:")
pprint(gdd)
print("-" * 40)
print("Christoffel symbols:")
pprint_Gamma_udd(0, 1, 0)
pprint_Gamma_udd(0, 0, 1)
print()
pprint_Gamma_udd(1, 0, 0)
pprint_Gamma_udd(1, 1, 1)
pprint_Gamma_udd(1, 2, 2)
pprint_Gamma_udd(1, 3, 3)
print()
pprint_Gamma_udd(2, 2, 1)
pprint_Gamma_udd(2, 1, 2)
pprint_Gamma_udd(2, 3, 3)
print()
pprint_Gamma_udd(3, 2, 3)
pprint_Gamma_udd(3, 3, 2)
pprint_Gamma_udd(3, 1, 3)
pprint_Gamma_udd(3, 3, 1)
print("-" * 40)
print("Ricci tensor:")
pprint_Rmn_dd(0, 0)
e = Rmn.dd(1, 1)
pprint_Rmn_dd(1, 1)
pprint_Rmn_dd(2, 2)
pprint_Rmn_dd(3, 3)
print("-" * 40)
print("Solve Einstein's equations:")
e = e.subs(nu(r), -lam(r)).doit()
l = dsolve(e, lam(r))
pprint(l)
lamsol = solve(l, lam(r))[0]
metric = gdd.subs(lam(r), lamsol).subs(nu(r), -lamsol) # .combine()
print("metric:")
pprint(metric)
def scale(self, **kwargs):
# TODO: ensure provided kwargs are keys in saufs
# pass the value of the scaling constant with key its name
self.scaling_constants = kwargs
## TODO: Assuming order, should find a way to pair each ki to its value by name
## may not be necessary since we do not care about the value of each UF
scaling_constants = np.array(list(self.scaling_constants.values()))
if sum(scaling_constants) == 1:
self.k = 0
k = diofant.symbols("k")
rhs = reduce(mul, k * scaling_constants + 1, 1)
lhs = k + 1
sols = diofant.solve(lhs - rhs)
# as opposed to complex, filter real solutions
real_solutions = filter(
lambda sol: -1 <= sol.get(k) <= 1,
filter(lambda sol: sol.get(k).is_Float, sols),
)
not_null = [sol.get(k) for sol in real_solutions if sol.get(k) != 0]
if len(not_null) == 0:
self.k = 0
else:
self.k = not_null[0]
return self.k
def test_unrad2():
assert solve(root(x**3 - 3 * x**2, 3) + 1 - x) == []
assert solve(root(x + 1, 3) + root(x**2 - 2, 5) + 1) == []
def test_unrad_slow():
# this has roots with multiplicity > 1; there should be no
# repeats in roots obtained, however
eq = (sqrt(1 + sqrt(1 - 4 * x**2)) - x *
((1 + sqrt(1 + 2 * sqrt(1 - 4 * x**2)))))
assert solve(eq) == [{x: Rational(1, 2)}]
def test_unrad1_fail():
assert solve(sqrt(x + root(x, 3)) + root(x - y, 5), y) != []
def test_unrad1():
pytest.raises(NotImplementedError,
lambda: unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
pytest.raises(NotImplementedError,
lambda: unrad(sqrt(x) + cbrt(x + 1) + 2 * sqrt(y)))
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue sympy/sympy#5203)
def check(rv, ans):
assert bool(rv[1]) == bool(ans[1])
if ans[1]:
return s_check(rv, ans)
e = rv[0].expand()
a = ans[0].expand()
return e in [a, -a] and rv[1] == ans[1]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = list(zip(d, [s] * len(d)))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(),
[rv[1][0].subs(reps), rv[1][1].subs(reps)])
ans = (ans[0].subs(reps).expand(),
[ans[1][0].subs(reps), ans[1][1].subs(reps)])
return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \
str(rv[1]) == str(ans[1])
assert check(unrad(sqrt(x)), (x, []))
assert check(unrad(sqrt(x) + 1), (x - 1, []))
assert check(unrad(sqrt(x) + root(x, 3) + 2),
(s**3 + s**2 + 2, [s, s**6 - x]))
assert check(unrad(sqrt(x) * root(x, 3) + 2), (x**5 - 64, []))
assert check(unrad(sqrt(x) + cbrt(x + 1)), (x**3 - (x + 1)**2, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2 * x)),
(-2 * sqrt(2) * x - 2 * x + 1, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + 2), (16 * x - 9, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)),
(5 * x**2 - 4 * x, []))
assert check(unrad(a * sqrt(x) + b * sqrt(x) + c * sqrt(y) + d * sqrt(y)),
((a * sqrt(x) + b * sqrt(x))**2 -
(c * sqrt(y) + d * sqrt(y))**2, []))
assert check(unrad(sqrt(x) + sqrt(1 - x)), (2 * x - 1, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) - 3), (x**2 - x + 16, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(5 * x**2 - 2 * x + 1, []))
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [
(25 * x**4 + 376 * x**3 + 1256 * x**2 - 2272 * x + 784, []),
(25 * x**8 - 476 * x**6 + 2534 * x**4 - 1468 * x**2 + 169, [])
]
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \
(41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487
assert check(unrad(sqrt(x) + sqrt(x + 1)), (Integer(1), []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq), (16 * x**2 - 9 * x, []))
assert {s[x] for s in solve(eq, check=False)} == {0, Rational(9, 16)}
assert solve(eq) == []
# but this one really does have those solutions
assert ({s[x]
for s in solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))
} == {0, Rational(9, 16)})
assert check(unrad(sqrt(x) + root(x + 1, 3) + 2 * sqrt(y), y),
(2 * sqrt(x) * cbrt(x + 1) + x - 4 * y +
(x + 1)**Rational(2, 3), []))
assert check(unrad(sqrt(x / (1 - x)) + cbrt(x + 1)),
(x**5 - x**4 - x**3 + 2 * x**2 + x - 1, []))
assert check(unrad(sqrt(x / (1 - x)) + 2 * sqrt(y), y),
(4 * x * y + x - 4 * y, []))
assert check(unrad(sqrt(x) * sqrt(1 - x) + 2, x), (x**2 - x + 4, []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [{x: 3}]
assert solve(Eq(x + sqrt(x - 4), 4)) == [{x: 4}]
assert solve(Eq(1, x + sqrt(2 * x - 3))) == []
assert {s[x] for s in solve(Eq(sqrt(5 * x + 6) - 2, x))} == {-1, 2}
assert {s[x]
for s in solve(Eq(sqrt(2 * x - 1) - sqrt(x - 4), 2))} == {5, 13}
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [{x: -6}]
# http://www.purplemath.com/modules/solverad.htm
assert solve(cbrt(2 * x - 5) - 3) == [{x: 16}]
assert {s[x]
for s in solve(x + 1 - root(x**4 + 4 * x**3 - x, 4))
} == {-Rational(1, 2), -Rational(1, 3)}
assert {s[x] for s in solve(sqrt(2 * x**2 - 7) - (3 - x))} == {-8, 2}
assert solve(sqrt(2 * x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [{x: 0}]
assert solve(sqrt(x + 4) + sqrt(2 * x - 1) - 3 * sqrt(x - 1)) == [{x: 5}]
assert solve(sqrt(x) * sqrt(x - 7) - 12) == [{x: 16}]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [{x: 4}]
assert solve(sqrt(9 * x**2 + 4) - (3 * x + 2)) == [{x: 0}]
assert solve(sqrt(x) - 2 - 5) == [{x: 49}]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [{x: 10}]
assert solve(sqrt(x - 2) - 5) == [{x: 27}]
assert solve(sqrt(17 * x - sqrt(x**2 - 5)) - 7) == [{x: 3}]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expression in unrad and do use _mexpand
z = sqrt(2 * x + 1) / sqrt(x) - sqrt(2 + 1 / x)
p = posify(z)[0]
assert solve(p) == []
assert solve(z) == []
assert solve(z + 6 * I) == [{x: -Rational(1, 11)}]
assert solve(p + 6 * I) == []
# issue sympy/sympy#8622
assert unrad(
(root(x + 1, 5) - root(x, 3))) == (x**5 - x**3 - 3 * x**2 - 3 * x - 1,
[])
# issue sympy/sympy#8679
assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x),
(s**3 + s**2 + s + sqrt(y), [s, s**3 - x]))
# for coverage
assert check(unrad(sqrt(x) + root(x, 3) + y),
(s**3 + s**2 + y, [s, s**6 - x]))
assert solve(sqrt(x) + root(x, 3) - 2) == [{x: 1}]
pytest.raises(NotImplementedError,
lambda: solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2))
# fails through a different code path
pytest.raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x) / x))
# unrad some
e = root(x + 1, 3) + root(x, 3)
assert unrad(e) == (2 * x + 1, [])
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5)
assert check(unrad(eq), (15625 * x**4 + 173000 * x**3 + 355600 * x**2 -
817920 * x + 331776, []))
assert check(unrad(root(x, 4) + root(x, 4)**3 - 1),
(s**3 + s - 1, [s, s**4 - x]))
assert check(unrad(root(x, 2) + root(x, 2)**3 - 1),
(x**3 + 2 * x**2 + x - 1, []))
assert unrad(x**0.5) is None
assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3),
(s**3 + s + t, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y),
(s**3 + s + x, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x),
(s**5 + s**3 + s - y, [s, s**5 - x - y]))
assert check(
unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)),
(s**5 + 5 * root(2, 5) * s**4 + s**3 + 10 * 2**Rational(2, 5) * s**3 +
10 * 2**Rational(3, 5) * s**2 + 5 * 2**Rational(4, 5) * s + 4,
[s, s**3 - x + 1]))
pytest.raises(
NotImplementedError, lambda: unrad(
(root(x, 2) + root(x, 3) + root(x, 4)).subs({x: x**5 - x + 1})))
# the simplify flag should be reset to False for unrad results;
# if it's not then this next test will take a long time
assert solve(root(x, 3) + root(x, 5) - 2) == [{x: 1}]
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6 * sqrt(5) / 5)
assert check(unrad(eq),
((5 * x - 4) *
(3125 * x**3 + 37100 * x**2 + 100800 * x - 82944), []))
ans = [{
x: Rational(4, 5)
}, {
x:
Rational(-1484, 375) + 172564 /
(140625 *
cbrt(114 * sqrt(12657) / 78125 + Rational(12459439, 52734375))) +
4 * cbrt(114 * sqrt(12657) / 78125 + Rational(12459439, 52734375))
}]
assert solve(eq) == ans
# duplicate radical handling
assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2),
(s**3 - s**2 - 3 * s - 5, [s, s**3 - x - 1]))
# cov post-processing
e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2
assert check(unrad(e),
(s**5 - 10 * s**4 + 39 * s**3 - 80 * s**2 + 80 * s - 30,
[s, s**3 - x**2 - 1]))
e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2
assert check(unrad(e), (s**6 - 2 * s**5 - 7 * s**4 - 3 * s**3 + 26 * s**2 +
40 * s + 25, [s, s**3 - x - 1]))
assert check(unrad(e, _reverse=True),
(s**6 - 14 * s**5 + 73 * s**4 - 187 * s**3 + 276 * s**2 -
228 * s + 89, [s, s**2 - x - sqrt(x + 1)]))
# this one needs r0, r1 reversal to work
assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2),
(s**12 - 2 * s**8 - 8 * s**7 - 8 * s**6 + s**4 + 8 * s**3 +
23 * s**2 + 32 * s + 17, [s, s**6 - x]))
# is this needed?
# assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == (
# x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5 - cosh(x)**5, [])
pytest.raises(
NotImplementedError,
lambda: unrad(sqrt(cosh(x) / x) + root(x + 1, 3) * sqrt(x) - 1))
assert unrad((x + y)**(2 * y / 3) + cbrt(x + y) + 1) is None
assert check(unrad((x + y)**(2 * y / 3) + cbrt(x + y) + 1, x),
(s**(2 * y) + s + 1, [s, s**3 - x - y]))
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests that the use of
# composite
assert len(solve(sqrt(y) * x + x**3 - 1, x)) == 3
assert len(
solve(-512 * y**3 + 1344 * cbrt(x + 2) * y**2 - 1176 *
(x + 2)**Rational(2, 3) * y - 169 * x + 686,
y,
_unrad=False)) == 3
# watch out for when the cov doesn't involve the symbol of interest
eq = -x + (7 * y / 8 - cbrt(27 * x / 2 + 27 * sqrt(x**2) / 2) / 3)**3 - 1
assert solve(eq, y) == [{
y:
RootOf(
-768 * x + 343 * y**3 -
588 * cbrt(4) * y**2 * cbrt(x + sqrt(x**2)) +
672 * cbrt(2) * y * cbrt(x + sqrt(x**2))**2 - 256 * sqrt(x**2) -
512, y, 0)
}, {
y:
RootOf(
-768 * x + 343 * y**3 -
588 * cbrt(4) * y**2 * cbrt(x + sqrt(x**2)) +
672 * cbrt(2) * y * cbrt(x + sqrt(x**2))**2 - 256 * sqrt(x**2) -
512, y, 1)
}, {
y:
RootOf(
-768 * x + 343 * y**3 -
588 * cbrt(4) * y**2 * cbrt(x + sqrt(x**2)) +
672 * cbrt(2) * y * cbrt(x + sqrt(x**2))**2 - 256 * sqrt(x**2) -
512, y, 2)
}]
eq = root(x + 1, 3) - (root(x, 3) + root(x, 5))
assert check(unrad(eq), (3 * s**13 + 3 * s**11 + s**9 - 1, [s, s**15 - x]))
assert check(unrad(eq - 2),
(3 * s**13 + 3 * s**11 + 6 * s**10 + s**9 + 12 * s**8 +
6 * s**6 + 12 * s**5 + 12 * s**3 + 7, [s, s**15 - x]))
assert check(
unrad(root(x, 3) - root(x + 1, 4) / 2 + root(x + 2, 3)),
(4096 * s**13 + 960 * s**12 + 48 * s**11 - s**10 - 1728 * s**4,
[s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389
assert check(
unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3) / 2),
(343 * s**13 + 2904 * s**12 + 1344 * s**11 + 512 * s**10 -
1323 * s**9 - 3024 * s**8 - 1728 * s**7 + 1701 * s**5 + 216 * s**4 -
729 * s, [s, s**4 - x - 1])) # orig expr has one real root: -0.048
assert check(
unrad(root(x, 3) / 2 - root(x + 1, 4) + root(x + 2, 3)),
(729 * s**13 - 216 * s**12 + 1728 * s**11 - 512 * s**10 + 1701 * s**9 -
3024 * s**8 + 1344 * s**7 + 1323 * s**5 - 2904 * s**4 + 343 * s,
[s, s**4 - x - 1])) # orig expr has 2 real roots: -0.91, -0.15
# orig expr has 1 real root: 19.53
assert check(unrad(root(x, 3) / 2 - root(x + 1, 4) + root(x + 2, 3) - 2),
(729 * s**13 + 1242 * s**12 + 18496 * s**10 + 129701 * s**9 +
388602 * s**8 + 453312 * s**7 - 612864 * s**6 -
3337173 * s**5 - 6332418 * s**4 - 7134912 * s**3 -
5064768 * s**2 - 2111913 * s - 398034, [s, s**4 - x - 1]))
ans = solve(sqrt(x) + sqrt(x + 1) - sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0][x])[:4] == '0.73'
# the fence optimization problem
# https://github.com/sympy/sympy/issues/4793#issuecomment-36994519
eq = F - (2 * x + 2 * y + sqrt(x**2 + y**2))
ans = 2 * F / 7 - sqrt(2) * F / 14
X = solve(eq, x, check=False)
for xi in reversed(X): # reverse since currently, ans is the 2nd one
Y = solve((x * y).subs(xi).diff(y), y, simplify=False, check=False)
if any((a[y] - ans).expand().is_zero for a in Y):
break
else:
assert None # no answer was found
assert (solve(sqrt(x + 1) + root(x, 3) - 2) == [{
x: (-11 / (9 * cbrt(Rational(47, 54) + sqrt(93) / 6)) +
Rational(1, 3) + cbrt(Rational(47, 54) + sqrt(93) / 6))**3
}])
assert (solve(sqrt(sqrt(x + 1)) + cbrt(x) - 2) == [{
x:
(-sqrt(-2 * cbrt(Rational(-1, 16) + sqrt(6913) / 16) +
6 / cbrt(Rational(-1, 16) + sqrt(6913) / 16) + Rational(17, 2) +
121 / (4 * sqrt(-6 / cbrt(Rational(-1, 16) + sqrt(6913) / 16) +
2 * cbrt(Rational(-1, 16) + sqrt(6913) / 16) +
Rational(17, 4)))) / 2 +
sqrt(-6 / cbrt(Rational(-1, 16) + sqrt(6913) / 16) + 2 *
cbrt(Rational(-1, 16) + sqrt(6913) / 16) + Rational(17, 4)) / 2 +
Rational(9, 4))**3
}])
assert (solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) == [{
x: (-cbrt(Rational(81, 2) + 3 * sqrt(741) / 2) / 3 +
(Rational(81, 2) + 3 * sqrt(741) / 2)**Rational(-1, 3) + 2)**2
}])
eq = (-x + (Rational(1, 2) - sqrt(3) * I / 2) *
cbrt(3 * x**3 / 2 - x * (3 * x**2 - 34) / 2 + sqrt(
(-3 * x**3 + x *
(3 * x**2 - 34) + 90)**2 / 4 - Rational(39304, 27)) - 45) + 34 /
(3 * (Rational(1, 2) - sqrt(3) * I / 2) *
cbrt(3 * x**3 / 2 - x * (3 * x**2 - 34) / 2 + sqrt(
(-3 * x**3 + x *
(3 * x**2 - 34) + 90)**2 / 4 - Rational(39304, 27)) - 45)))
assert check(unrad(eq),
(s**6 - sqrt(3) * s**6 * I + 102 * cbrt(12) * s**4 +
102 * 2**Rational(2, 3) * 3**Rational(5, 6) * s**4 * I +
1620 * s**3 - 1620 * sqrt(3) * s**3 * I -
13872 * cbrt(18) * s**2 + 471648 - 471648 * sqrt(3) * I, [
s, s**3 - 306 * x -
sqrt(3) * sqrt(31212 * x**2 - 165240 * x + 61484) + 810
]))
assert solve(eq, x, check=False) != [] # not other code errors
# %%
mauf = access_to_facilities * access_to_pros * access_to_medicine
# %%
mauf.scale(ncenters_nbeds_ndocs_nparam=0.7, npharma=0.2)
# %%
kwargs = {"ncenters_nbeds_ndocs_nparam": 0.7, "npharma": 0.5}
# %%
mauf.saufs.keys()
# %%
X = diofant.symbols("X")
diofant.solve(X + 1 - (0.8 * X + 1) * (0.6 * X + 1) * (0.7 * X + 1))
# %% [markdown]
# ## Figures
# %% [markdown]
# #### Number of doctors
# %%
fig = data.set_index("governorate")["Number of doctors"].apply(
u_ndocs.normalized_model
).sort_values().iplot(
kind="bar",
title=
"Ranking of governorates based on the utility of number of doctors per 1000 inhabitants",
asFigure=True)
def test_piecewise_solve2():
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solve(f, x) == [2, Interval(0, oo, True, True)]
def test_sympyissue_11553():
eqs = (x + y + 1, x + GoldenRatio)
assert solve(eqs, x, y) == [{x: -GoldenRatio, y: -1 + GoldenRatio}]
def test_piecewise_solve2():
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solve(f, x) != [{x: 2}]
def test_piecewise_solve2():
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solve(f, x) != [{x: 2}]
def ratsimpmodprime(expr, G, *gens, **args):
"""
Simplifies a rational expression ``expr`` modulo the prime ideal
generated by ``G``. ``G`` should be a Groebner basis of the
ideal.
>>> from diofant.simplify.ratsimp import ratsimpmodprime
>>> from diofant.abc import x, y
>>> eq = (x + y**5 + y)/(x - y)
>>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex')
(x**2 + x*y + x + y)/(x**2 - x*y)
If ``polynomial`` is False, the algorithm computes a rational
simplification which minimizes the sum of the total degrees of
the numerator and the denominator.
If ``polynomial`` is True, this function just brings numerator and
denominator into a canonical form. This is much faster, but has
potentially worse results.
References
==========
M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial
Ideal,
http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984
(specifically, the second algorithm)
"""
from diofant import solve
quick = args.pop('quick', True)
polynomial = args.pop('polynomial', False)
debug('ratsimpmodprime', expr)
# usual preparation of polynomials:
num, denom = cancel(expr).as_numer_denom()
try:
polys, opt = parallel_poly_from_expr([num, denom] + G, *gens, **args)
except PolificationFailed:
return expr
domain = opt.domain
if domain.has_assoc_Field:
opt.domain = domain.get_field()
else:
raise DomainError("can't compute rational simplification over %s" %
domain)
# compute only once
leading_monomials = [g.LM(opt.order) for g in polys[2:]]
tested = set()
def staircase(n):
"""
Compute all monomials with degree less than ``n`` that are
not divisible by any element of ``leading_monomials``.
"""
if n == 0:
return [1]
S = []
for mi in combinations_with_replacement(range(len(opt.gens)), n):
m = [0] * len(opt.gens)
for i in mi:
m[i] += 1
if all([monomial_div(m, lmg) is None
for lmg in leading_monomials]):
S.append(m)
return [Monomial(s).as_expr(*opt.gens) for s in S] + staircase(n - 1)
def _ratsimpmodprime(a, b, allsol, N=0, D=0):
"""
Computes a rational simplification of ``a/b`` which minimizes
the sum of the total degrees of the numerator and the denominator.
The algorithm proceeds by looking at ``a * d - b * c`` modulo
the ideal generated by ``G`` for some ``c`` and ``d`` with degree
less than ``a`` and ``b`` respectively.
The coefficients of ``c`` and ``d`` are indeterminates and thus
the coefficients of the normalform of ``a * d - b * c`` are
linear polynomials in these indeterminates.
If these linear polynomials, considered as system of
equations, have a nontrivial solution, then `\frac{a}{b}
\equiv \frac{c}{d}` modulo the ideal generated by ``G``. So,
by construction, the degree of ``c`` and ``d`` is less than
the degree of ``a`` and ``b``, so a simpler representation
has been found.
After a simpler representation has been found, the algorithm
tries to reduce the degree of the numerator and denominator
and returns the result afterwards.
As an extension, if quick=False, we look at all possible degrees such
that the total degree is less than *or equal to* the best current
solution. We retain a list of all solutions of minimal degree, and try
to find the best one at the end.
"""
c, d = a, b
steps = 0
maxdeg = a.total_degree() + b.total_degree()
if quick:
bound = maxdeg - 1
else:
bound = maxdeg
while N + D <= bound:
if (N, D) in tested:
break
tested.add((N, D))
M1 = staircase(N)
M2 = staircase(D)
debug('%s / %s: %s, %s' % (N, D, M1, M2))
Cs = symbols("c:%d" % len(M1), cls=Dummy)
Ds = symbols("d:%d" % len(M2), cls=Dummy)
ng = Cs + Ds
c_hat = Poly(sum(Cs[i] * M1[i] for i in range(len(M1))),
opt.gens + ng)
d_hat = Poly(sum(Ds[i] * M2[i] for i in range(len(M2))),
opt.gens + ng)
r = reduced(a * d_hat - b * c_hat,
G,
opt.gens + ng,
order=opt.order,
polys=True)[1]
S = Poly(r, gens=opt.gens).coeffs()
sol = solve(S, Cs + Ds, particular=True, quick=True)
if sol and not all([s == 0 for s in sol.values()]):
c = c_hat.subs(sol)
d = d_hat.subs(sol)
# The "free" variables occuring before as parameters
# might still be in the substituted c, d, so set them
# to the value chosen before:
c = c.subs(dict(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))
d = d.subs(dict(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))
c = Poly(c, opt.gens)
d = Poly(d, opt.gens)
if d == 0:
raise ValueError('Ideal not prime?')
allsol.append((c_hat, d_hat, S, Cs + Ds))
if N + D != maxdeg:
allsol = [allsol[-1]]
break
steps += 1
N += 1
D += 1
if steps > 0:
c, d, allsol = _ratsimpmodprime(c, d, allsol, N, D - steps)
c, d, allsol = _ratsimpmodprime(c, d, allsol, N - steps, D)
return c, d, allsol
# preprocessing. this improves performance a bit when deg(num)
# and deg(denom) are large:
num = reduced(num, G, opt.gens, order=opt.order)[1]
denom = reduced(denom, G, opt.gens, order=opt.order)[1]
if polynomial:
return (num / denom).cancel()
c, d, allsol = _ratsimpmodprime(Poly(num, opt.gens), Poly(denom, opt.gens),
[])
if not quick and allsol:
debug('Looking for best minimal solution. Got: %s' % len(allsol))
newsol = []
for c_hat, d_hat, S, ng in allsol:
sol = solve(S, ng, particular=True, quick=False)
newsol.append((c_hat.subs(sol), d_hat.subs(sol)))
c, d = min(newsol, key=lambda x: len(x[0].terms()) + len(x[1].terms()))
if not domain.has_Field:
cn, c = c.clear_denoms(convert=True)
dn, d = d.clear_denoms(convert=True)
cf, c, d = cancel((c, d), opt.gens, order=opt.order) # canonicalize signs
r = cf * Rational(cn, dn)
return (c * r.q) / (d * r.p)
def test_unrad_slow():
# this has roots with multiplicity > 1; there should be no
# repeats in roots obtained, however
eq = (sqrt(1 + sqrt(1 - 4*x**2)) - x*((1 + sqrt(1 + 2*sqrt(1 - 4*x**2)))))
assert solve(eq) == [{x: Rational(1, 2)}]
def test_unrad1():
pytest.raises(NotImplementedError, lambda:
unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
pytest.raises(NotImplementedError, lambda:
unrad(sqrt(x) + cbrt(x + 1) + 2*sqrt(y)))
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue sympy/sympy#5203)
def check(rv, ans):
assert bool(rv[1]) == bool(ans[1])
if ans[1]:
return s_check(rv, ans)
e = rv[0].expand()
a = ans[0].expand()
return e in [a, -a] and rv[1] == ans[1]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = list(zip(d, [s]*len(d)))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [rv[1][0].subs(reps), rv[1][1].subs(reps)])
ans = (ans[0].subs(reps).expand(), [ans[1][0].subs(reps), ans[1][1].subs(reps)])
return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \
str(rv[1]) == str(ans[1])
assert check(unrad(sqrt(x)),
(x, []))
assert check(unrad(sqrt(x) + 1),
(x - 1, []))
assert check(unrad(sqrt(x) + root(x, 3) + 2),
(s**3 + s**2 + 2, [s, s**6 - x]))
assert check(unrad(sqrt(x)*root(x, 3) + 2),
(x**5 - 64, []))
assert check(unrad(sqrt(x) + cbrt(x + 1)),
(x**3 - (x + 1)**2, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2*x)),
(-2*sqrt(2)*x - 2*x + 1, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + 2),
(16*x - 9, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)),
(5*x**2 - 4*x, []))
assert check(unrad(a*sqrt(x) + b*sqrt(x) + c*sqrt(y) + d*sqrt(y)),
((a*sqrt(x) + b*sqrt(x))**2 - (c*sqrt(y) + d*sqrt(y))**2, []))
assert check(unrad(sqrt(x) + sqrt(1 - x)),
(2*x - 1, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) - 3),
(x**2 - x + 16, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(5*x**2 - 2*x + 1, []))
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [
(25*x**4 + 376*x**3 + 1256*x**2 - 2272*x + 784, []),
(25*x**8 - 476*x**6 + 2534*x**4 - 1468*x**2 + 169, [])]
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \
(41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487
assert check(unrad(sqrt(x) + sqrt(x + 1)), (Integer(1), []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq),
(16*x**2 - 9*x, []))
assert {s[x] for s in solve(eq, check=False)} == {0, Rational(9, 16)}
assert solve(eq) == []
# but this one really does have those solutions
assert ({s[x] for s in solve(sqrt(x) - sqrt(x + 1) +
sqrt(1 - sqrt(x)))} ==
{0, Rational(9, 16)})
assert check(unrad(sqrt(x) + root(x + 1, 3) + 2*sqrt(y), y),
(2*sqrt(x)*cbrt(x + 1) + x - 4*y +
(x + 1)**Rational(2, 3), []))
assert check(unrad(sqrt(x/(1 - x)) + cbrt(x + 1)),
(x**5 - x**4 - x**3 + 2*x**2 + x - 1, []))
assert check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y),
(4*x*y + x - 4*y, []))
assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x),
(x**2 - x + 4, []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [{x: 3}]
assert solve(Eq(x + sqrt(x - 4), 4)) == [{x: 4}]
assert solve(Eq(1, x + sqrt(2*x - 3))) == []
assert {s[x] for s in solve(Eq(sqrt(5*x + 6) - 2, x))} == {-1, 2}
assert {s[x] for s in solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2))} == {5, 13}
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [{x: -6}]
# http://www.purplemath.com/modules/solverad.htm
assert solve(cbrt(2*x - 5) - 3) == [{x: 16}]
assert {s[x] for s in solve(x + 1 - root(x**4 + 4*x**3 - x, 4))} == {-Rational(1, 2),
-Rational(1, 3)}
assert {s[x] for s in solve(sqrt(2*x**2 - 7) - (3 - x))} == {-8, 2}
assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [{x: 0}]
assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [{x: 5}]
assert solve(sqrt(x)*sqrt(x - 7) - 12) == [{x: 16}]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [{x: 4}]
assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [{x: 0}]
assert solve(sqrt(x) - 2 - 5) == [{x: 49}]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [{x: 10}]
assert solve(sqrt(x - 2) - 5) == [{x: 27}]
assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [{x: 3}]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expression in unrad and do use _mexpand
z = sqrt(2*x + 1)/sqrt(x) - sqrt(2 + 1/x)
p = posify(z)[0]
assert solve(p) == []
assert solve(z) == []
assert solve(z + 6*I) == [{x: -Rational(1, 11)}]
assert solve(p + 6*I) == []
# issue sympy/sympy#8622
assert unrad((root(x + 1, 5) - root(x, 3))) == (
x**5 - x**3 - 3*x**2 - 3*x - 1, [])
# issue sympy/sympy#8679
assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x),
(s**3 + s**2 + s + sqrt(y), [s, s**3 - x]))
# for coverage
assert check(unrad(sqrt(x) + root(x, 3) + y),
(s**3 + s**2 + y, [s, s**6 - x]))
assert solve(sqrt(x) + root(x, 3) - 2) == [{x: 1}]
pytest.raises(NotImplementedError, lambda:
solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2))
# fails through a different code path
pytest.raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x)/x))
# unrad some
e = root(x + 1, 3) + root(x, 3)
assert unrad(e) == (2*x + 1, [])
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
(15625*x**4 + 173000*x**3 + 355600*x**2 - 817920*x + 331776, []))
assert check(unrad(root(x, 4) + root(x, 4)**3 - 1),
(s**3 + s - 1, [s, s**4 - x]))
assert check(unrad(root(x, 2) + root(x, 2)**3 - 1),
(x**3 + 2*x**2 + x - 1, []))
assert unrad(x**0.5) is None
assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3),
(s**3 + s + t, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y),
(s**3 + s + x, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x),
(s**5 + s**3 + s - y, [s, s**5 - x - y]))
assert check(unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)),
(s**5 + 5*root(2, 5)*s**4 + s**3 + 10*2**Rational(2, 5)*s**3 +
10*2**Rational(3, 5)*s**2 + 5*2**Rational(4, 5)*s + 4, [s, s**3 - x + 1]))
pytest.raises(NotImplementedError,
lambda: unrad((root(x, 2) + root(x, 3) +
root(x, 4)).subs({x: x**5 - x + 1})))
# the simplify flag should be reset to False for unrad results;
# if it's not then this next test will take a long time
assert solve(root(x, 3) + root(x, 5) - 2) == [{x: 1}]
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
((5*x - 4)*(3125*x**3 + 37100*x**2 + 100800*x - 82944), []))
ans = [{x: Rational(4, 5)},
{x: Rational(-1484, 375) + 172564/(140625*cbrt(114*sqrt(12657)/78125 +
Rational(12459439, 52734375))) +
4*cbrt(114*sqrt(12657)/78125 +
Rational(12459439, 52734375))}]
assert solve(eq) == ans
# duplicate radical handling
assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2),
(s**3 - s**2 - 3*s - 5, [s, s**3 - x - 1]))
# cov post-processing
e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2
assert check(unrad(e),
(s**5 - 10*s**4 + 39*s**3 - 80*s**2 + 80*s - 30,
[s, s**3 - x**2 - 1]))
e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2
assert check(unrad(e),
(s**6 - 2*s**5 - 7*s**4 - 3*s**3 + 26*s**2 + 40*s + 25,
[s, s**3 - x - 1]))
assert check(unrad(e, _reverse=True),
(s**6 - 14*s**5 + 73*s**4 - 187*s**3 + 276*s**2 - 228*s + 89,
[s, s**2 - x - sqrt(x + 1)]))
# this one needs r0, r1 reversal to work
assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2),
(s**12 - 2*s**8 - 8*s**7 - 8*s**6 + s**4 + 8*s**3 + 23*s**2 +
32*s + 17, [s, s**6 - x]))
# is this needed?
# assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == (
# x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5 - cosh(x)**5, [])
pytest.raises(NotImplementedError, lambda:
unrad(sqrt(cosh(x)/x) + root(x + 1, 3)*sqrt(x) - 1))
assert unrad((x+y)**(2*y/3) + cbrt(x+y) + 1) is None
assert check(unrad((x+y)**(2*y/3) + cbrt(x+y) + 1, x),
(s**(2*y) + s + 1, [s, s**3 - x - y]))
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests that the use of
# composite
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*cbrt(x + 2)*y**2 -
1176*(x + 2)**Rational(2, 3)*y -
169*x + 686, y, _unrad=False)) == 3
# watch out for when the cov doesn't involve the symbol of interest
eq = -x + (7*y/8 - cbrt(27*x/2 + 27*sqrt(x**2)/2)/3)**3 - 1
assert solve(eq, y) == [
{y: RootOf(-2304*x + 1029*y**3 - 1764*cbrt(4)*y**2*cbrt(x + sqrt(x**2)) +
2016*cbrt(2)*y*(x + sqrt(x**2))**Rational(2, 3) -
768*sqrt(x**2) - 1536, y, 0, evaluate=False)},
{y: RootOf(-2304*x + 1029*y**3 - 1764*cbrt(4)*y**2*cbrt(x + sqrt(x**2)) +
2016*cbrt(2)*y*(x + sqrt(x**2))**Rational(2, 3) -
768*sqrt(x**2) - 1536, y, 1, evaluate=False)},
{y: RootOf(-2304*x + 1029*y**3 - 1764*cbrt(4)*y**2*cbrt(x + sqrt(x**2)) +
2016*cbrt(2)*y*(x + sqrt(x**2))**Rational(2, 3) -
768*sqrt(x**2) - 1536, y, 2, evaluate=False)}]
eq = root(x + 1, 3) - (root(x, 3) + root(x, 5))
assert check(unrad(eq),
(3*s**13 + 3*s**11 + s**9 - 1, [s, s**15 - x]))
assert check(unrad(eq - 2),
(3*s**13 + 3*s**11 + 6*s**10 + s**9 + 12*s**8 + 6*s**6 + 12*s**5 +
12*s**3 + 7, [s, s**15 - x]))
assert check(unrad(root(x, 3) - root(x + 1, 4)/2 + root(x + 2, 3)),
(4096*s**13 + 960*s**12 + 48*s**11 - s**10 - 1728*s**4,
[s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389
assert check(unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3)/2),
(343*s**13 + 2904*s**12 + 1344*s**11 + 512*s**10 - 1323*s**9 -
3024*s**8 - 1728*s**7 + 1701*s**5 + 216*s**4 - 729*s, [s, s**4 - x -
1])) # orig expr has one real root: -0.048
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3)),
(729*s**13 - 216*s**12 + 1728*s**11 - 512*s**10 + 1701*s**9 -
3024*s**8 + 1344*s**7 + 1323*s**5 - 2904*s**4 + 343*s, [s, s**4 - x -
1])) # orig expr has 2 real roots: -0.91, -0.15
# orig expr has 1 real root: 19.53
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3) - 2),
(729*s**13 + 1242*s**12 + 18496*s**10 + 129701*s**9 + 388602*s**8 +
453312*s**7 - 612864*s**6 - 3337173*s**5 - 6332418*s**4 - 7134912*s**3
- 5064768*s**2 - 2111913*s - 398034, [s, s**4 - x - 1]))
ans = solve(sqrt(x) + sqrt(x + 1) -
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0][x])[:4] == '0.73'
# the fence optimization problem
# https://github.com/sympy/sympy/issues/4793#issuecomment-36994519
eq = F - (2*x + 2*y + sqrt(x**2 + y**2))
ans = 2*F/7 - sqrt(2)*F/14
X = solve(eq, x, check=False)
for xi in reversed(X): # reverse since currently, ans is the 2nd one
Y = solve((x*y).subs(xi).diff(y), y,
simplify=False, check=False)
if any((a[y] - ans).expand().is_zero for a in Y):
break
else:
assert None # no answer was found
assert (solve(sqrt(x + 1) + root(x, 3) - 2) ==
[{x: (-11/(9*cbrt(Rational(47, 54) + sqrt(93)/6)) +
Rational(1, 3) + cbrt(Rational(47, 54) +
sqrt(93)/6))**3}])
assert (solve(sqrt(sqrt(x + 1)) + cbrt(x) - 2) ==
[{x: (-sqrt(-2*cbrt(Rational(-1, 16) + sqrt(6913)/16) +
6/cbrt(Rational(-1, 16) + sqrt(6913)/16) +
Rational(17, 2) +
121/(4*sqrt(-6/cbrt(Rational(-1, 16) +
sqrt(6913)/16) +
2*cbrt(Rational(-1, 16) +
sqrt(6913)/16) +
Rational(17, 4))))/2 +
sqrt(-6/cbrt(Rational(-1, 16) + sqrt(6913)/16) +
2*cbrt(Rational(-1, 16) + sqrt(6913)/16) +
Rational(17, 4))/2 + Rational(9, 4))**3}])
assert (solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) ==
[{x: (-cbrt(Rational(81, 2) + 3*sqrt(741)/2)/3 +
(Rational(81, 2) + 3*sqrt(741)/2)**Rational(-1, 3) + 2)**2}])
eq = (-x + (Rational(1, 2) - sqrt(3)*I/2)*cbrt(3*x**3/2 - x*(3*x**2 - 34)/2 +
sqrt((-3*x**3 + x*(3*x**2 - 34) + 90)**2/4 - Rational(39304, 27)) -
45) + 34/(3*(Rational(1, 2) - sqrt(3)*I/2)*cbrt(3*x**3/2 -
x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + x*(3*x**2 - 34) + 90)**2/4 -
Rational(39304, 27)) - 45)))
assert check(unrad(eq),
(s**6 - sqrt(3)*s**6*I + 102*cbrt(12)*s**4 +
102*2**Rational(2, 3)*3**Rational(5, 6)*s**4*I + 1620*s**3 - 1620*sqrt(3)*s**3*I -
13872*cbrt(18)*s**2 + 471648 - 471648*sqrt(3)*I, [s, s**3 - 306*x
- sqrt(3)*sqrt(31212*x**2 - 165240*x + 61484) + 810]))
assert solve(eq, x, check=False) != [] # not other code errors
def test_piecewise_solve2():
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
C0 = Symbol('C0', negative=True)
assert solve(f, x) == [{x: 2}, {x: C0}]
def test_sympyissue_11553():
eqs = (x + y + 1, x + GoldenRatio)
assert solve(eqs, x, y) == [{x: -GoldenRatio, y: -1 + GoldenRatio}]
def test_RootOf_evalf():
real = RootOf(x**3 + x + 3, 0).evalf(20)
assert real.epsilon_eq(Float("-1.2134116627622296341"))
re, im = RootOf(x**3 + x + 3, 1).evalf(20).as_real_imag()
assert re.epsilon_eq(+Float("0.60670583138111481707"))
assert im.epsilon_eq(-Float("1.45061224918844152650"))
re, im = RootOf(x**3 + x + 3, 2).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("0.60670583138111481707"))
assert im.epsilon_eq(Float("1.45061224918844152650"))
p = legendre_poly(4, x, polys=True)
roots = [str(r.evalf(17)) for r in p.real_roots()]
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
re = RootOf(x**5 - 5 * x + 12, 0).evalf(20)
assert re.epsilon_eq(Float("-1.84208596619025438271"))
re, im = RootOf(x**5 - 5 * x + 12, 1).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("-1.709561043370328882010"))
re, im = RootOf(x**5 - 5 * x + 12, 2).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("+1.709561043370328882010"))
re, im = RootOf(x**5 - 5 * x + 12, 3).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("-0.719798681483861386681"))
re, im = RootOf(x**5 - 5 * x + 12, 4).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("+0.719798681483861386681"))
# issue sympy/sympy#6393
assert str(RootOf(x**5 + 2 * x**4 + x**3 - 68719476736,
0).evalf(3)) == '147.'
eq = (531441 * x**11 + 3857868 * x**10 + 13730229 * x**9 +
32597882 * x**8 + 55077472 * x**7 + 60452000 * x**6 +
32172064 * x**5 - 4383808 * x**4 - 11942912 * x**3 - 1506304 * x**2 +
1453312 * x + 512)
a, b = RootOf(eq, 1).evalf(2).as_real_imag()
c, d = RootOf(eq, 2).evalf(2).as_real_imag()
assert a == c
assert b < d
assert b == -d
# issue sympy/sympy#6451
r = RootOf(legendre_poly(64, x), 7)
assert r.evalf(2) == r.evalf(100).evalf(2)
# issue sympy/sympy#8617
ans = [w[x].evalf(2) for w in solve(x**3 - x - 4)]
assert RootOf(exp(x)**3 - exp(x) - 4, 0).evalf(2) in ans
# issue sympy/sympy#9019
r0 = RootOf(x**2 + 1, 0, radicals=False)
r1 = RootOf(x**2 + 1, 1, radicals=False)
assert r0.evalf(4, chop=True) == -1.0 * I
assert r1.evalf(4, chop=True) == +1.0 * I
# make sure verification is used in case a max/min traps the "root"
assert str(RootOf(4 * x**5 + 16 * x**3 + 12 * x**2 + 7,
0).evalf(3)) == '-0.976'
assert isinstance(RootOf(x**3 + y * x + 1, x, 0).evalf(2), RootOf)
assert RootOf(x**3 + I * x + 2,
0).evalf(7) == (Float('-1.260785326', dps=7) +
I * Float('0.2684419416', dps=7))
r = RootOf(x**2 - 4456178 * x + 60372201703370, 0, radicals=False)
assert r.evalf(2) == Float('2.2282e+6',
dps=2) - I * Float('7.4465e+6', dps=2)
def test_unrad1_fail():
assert solve(sqrt(x + root(x, 3)) + root(x - y, 5), y) != []
def test_RootOf_evalf():
real = RootOf(x**3 + x + 3, 0).evalf(20)
assert real.epsilon_eq(Float("-1.2134116627622296341"))
re, im = RootOf(x**3 + x + 3, 1).evalf(20).as_real_imag()
assert re.epsilon_eq( Float("0.60670583138111481707"))
assert im.epsilon_eq(-Float("1.45061224918844152650"))
re, im = RootOf(x**3 + x + 3, 2).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("0.60670583138111481707"))
assert im.epsilon_eq(Float("1.45061224918844152650"))
p = legendre_poly(4, x, polys=True)
roots = [str(r.evalf(17)) for r in p.real_roots()]
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
re = RootOf(x**5 - 5*x + 12, 0).evalf(20)
assert re.epsilon_eq(Float("-1.84208596619025438271"))
re, im = RootOf(x**5 - 5*x + 12, 1).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("-1.709561043370328882010"))
re, im = RootOf(x**5 - 5*x + 12, 2).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("+1.709561043370328882010"))
re, im = RootOf(x**5 - 5*x + 12, 3).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("-0.719798681483861386681"))
re, im = RootOf(x**5 - 5*x + 12, 4).evalf(20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("+0.719798681483861386681"))
# issue sympy/sympy#6393
assert str(RootOf(x**5 + 2*x**4 + x**3 - 68719476736, 0).evalf(3)) == '147.'
eq = (531441*x**11 + 3857868*x**10 + 13730229*x**9 + 32597882*x**8 +
55077472*x**7 + 60452000*x**6 + 32172064*x**5 - 4383808*x**4 -
11942912*x**3 - 1506304*x**2 + 1453312*x + 512)
a, b = RootOf(eq, 1).evalf(2).as_real_imag()
c, d = RootOf(eq, 2).evalf(2).as_real_imag()
assert a == c
assert b < d
assert b == -d
# issue sympy/sympy#6451
r = RootOf(legendre_poly(64, x), 7)
assert r.evalf(2) == r.evalf(100).evalf(2)
# issue sympy/sympy#8617
ans = [w[x].evalf(2) for w in solve(x**3 - x - 4)]
assert RootOf(exp(x)**3 - exp(x) - 4, 0).evalf(2) in ans
# issue sympy/sympy#9019
r0 = RootOf(x**2 + 1, 0, radicals=False)
r1 = RootOf(x**2 + 1, 1, radicals=False)
assert r0.evalf(4, chop=True) == -1.0*I
assert r1.evalf(4, chop=True) == +1.0*I
# make sure verification is used in case a max/min traps the "root"
assert str(RootOf(4*x**5 + 16*x**3 + 12*x**2 + 7, 0).evalf(3)) == '-0.976'
assert isinstance(RootOf(x**3 + y*x + 1, x, 0).evalf(2), RootOf)
assert RootOf(x**3 + I*x + 2, 0).evalf(7) == (Float('-1.260785326', dps=7) +
I*Float('0.2684419416', dps=7))
r = RootOf(x**2 - 4456178*x + 60372201703370, 0, radicals=False)
assert r.evalf(2) == Float('2.2282e+6', dps=2) - I*Float('7.4465e+6', dps=2)
def test_unrad2():
assert solve(root(x**3 - 3*x**2, 3) + 1 - x) == []
assert solve(root(x + 1, 3) + root(x**2 - 2, 5) + 1) == []
def _ratsimpmodprime(a, b, allsol, N=0, D=0):
"""
Computes a rational simplification of ``a/b`` which minimizes
the sum of the total degrees of the numerator and the denominator.
The algorithm proceeds by looking at ``a * d - b * c`` modulo
the ideal generated by ``G`` for some ``c`` and ``d`` with degree
less than ``a`` and ``b`` respectively.
The coefficients of ``c`` and ``d`` are indeterminates and thus
the coefficients of the normalform of ``a * d - b * c`` are
linear polynomials in these indeterminates.
If these linear polynomials, considered as system of
equations, have a nontrivial solution, then `\frac{a}{b}
\equiv \frac{c}{d}` modulo the ideal generated by ``G``. So,
by construction, the degree of ``c`` and ``d`` is less than
the degree of ``a`` and ``b``, so a simpler representation
has been found.
After a simpler representation has been found, the algorithm
tries to reduce the degree of the numerator and denominator
and returns the result afterwards.
As an extension, if quick=False, we look at all possible degrees such
that the total degree is less than *or equal to* the best current
solution. We retain a list of all solutions of minimal degree, and try
to find the best one at the end.
"""
c, d = a, b
steps = 0
maxdeg = a.total_degree() + b.total_degree()
if quick:
bound = maxdeg - 1
else:
bound = maxdeg
while N + D <= bound:
if (N, D) in tested:
break
tested.add((N, D))
M1 = staircase(N)
M2 = staircase(D)
debug('%s / %s: %s, %s' % (N, D, M1, M2))
Cs = symbols("c:%d" % len(M1), cls=Dummy)
Ds = symbols("d:%d" % len(M2), cls=Dummy)
ng = Cs + Ds
c_hat = Poly(sum(Cs[i] * M1[i] for i in range(len(M1))),
opt.gens + ng)
d_hat = Poly(sum(Ds[i] * M2[i] for i in range(len(M2))),
opt.gens + ng)
r = reduced(a * d_hat - b * c_hat,
G,
opt.gens + ng,
order=opt.order,
polys=True)[1]
S = Poly(r, gens=opt.gens).coeffs()
sol = solve(S, Cs + Ds, particular=True, quick=True)
if sol and not all([s == 0 for s in sol.values()]):
c = c_hat.subs(sol)
d = d_hat.subs(sol)
# The "free" variables occuring before as parameters
# might still be in the substituted c, d, so set them
# to the value chosen before:
c = c.subs(dict(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))
d = d.subs(dict(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))
c = Poly(c, opt.gens)
d = Poly(d, opt.gens)
if d == 0:
raise ValueError('Ideal not prime?')
allsol.append((c_hat, d_hat, S, Cs + Ds))
if N + D != maxdeg:
allsol = [allsol[-1]]
break
steps += 1
N += 1
D += 1
if steps > 0:
c, d, allsol = _ratsimpmodprime(c, d, allsol, N, D - steps)
c, d, allsol = _ratsimpmodprime(c, d, allsol, N - steps, D)
return c, d, allsol