def test_FracElement___mul__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f*g == g*f == 1/(x*y)
assert x*F.ring.gens[0] == F.ring.gens[0]*x == x**2
F, x, y = field("x,y", ZZ)
assert x*3 == 3*x
assert x*QQ(3, 7) == QQ(3, 7)*x == 3*x/7
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)
assert dict(f.numer) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1}
assert dict(f.denom) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)
assert dict(f.numer) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1}
assert dict(f.denom) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1}
def test_FracElement_eval():
F, x, y, z = field("x,y,z", ZZ)
Fyz = field("y,z", ZZ)[0]
f = (x**2 + 3*y)/z
assert f.eval(x, 0) == 3*Fyz.y/Fyz.z
pytest.raises(ZeroDivisionError, lambda: f.eval(z, 0))
def test_FracElement___add__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f + g == g + f == (x + y)/(x*y)
z = symbols('z')
pytest.raises(TypeError, lambda: x + z)
assert x + F.ring.gens[0] == F.ring.gens[0] + x == 2*x
F, x, y = field("x,y", ZZ)
assert x + 3 == 3 + x
assert x + QQ(3, 7) == QQ(3, 7) + x == (7*x + 3)/7
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = (u*v + x)/(y + u*v)
assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = (u*v + x)/(y + u*v)
assert dict(f.numer) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
def test_FracElement_diff():
F, x, y, z = field("x,y,z", ZZ)
assert ((x**2 + y)/(z + 1)).diff(x) == 2*x/(z + 1)
F, x, y = field('x,y', QQ.algebraic_field(I))
assert ((x - y)/x).diff(x) == y/x**2
def test_PolyElement___sub__():
Rt, t = ring("t", ZZ)
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert u - x == -x + u
assert (x + u) - 2*u == x - u
assert dict(x - 3*y) == {(1, 0, 0): 1, (0, 1, 0): -3}
assert dict(-u + x) == dict(x - u) == {(1, 0, 0): 1, (0, 0, 0): -u}
assert dict(-u + x*y) == dict(x*y - u) == {(1, 1, 0): 1, (0, 0, 0): -u}
assert dict(-u + x*y + z) == dict(x*y + z - u) == {(1, 1, 0): 1, (0, 0, 1): 1, (0, 0, 0): -u}
assert dict(-u*x + x) == dict(x - u*x) == {(1, 0, 0): -u + 1}
assert dict(-u*x + x*y) == dict(x*y - u*x) == {(1, 1, 0): 1, (1, 0, 0): -u}
assert dict(-u*x + x*y + z) == dict(x*y + z - u*x) == {(1, 1, 0): 1, (0, 0, 1): 1, (1, 0, 0): -u}
pytest.raises(TypeError, lambda: t - x)
pytest.raises(TypeError, lambda: x - t)
pytest.raises(TypeError, lambda: t - u)
pytest.raises(TypeError, lambda: u - t)
Fuv, u, v = field("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Fuv)
assert dict(-u + x) == dict(x - u) == {(1, 0, 0): 1, (0, 0, 0): -u}
Rxyz, x, y, z = ring("x,y,z", EX)
assert dict(-EX(pi) + x*y*z) == dict(x*y*z - EX(pi)) == {(1, 1, 1): EX(1), (0, 0, 0): -EX(pi)}
def test_PolyElement_cancel():
R, x, y = ring("x,y", ZZ)
f = 2*x**3 + 4*x**2 + 2*x
g = 3*x**2 + 3*x
F = 2*x + 2
G = 3
assert f.cancel(g) == (F, G)
assert (-f).cancel(g) == (-F, G)
assert f.cancel(-g) == (-F, G)
R, x, y = ring("x,y", QQ)
f = x**3/2 + x**2 + x/2
g = x**2/3 + x/3
F = 3*x + 3
G = 2
assert f.cancel(g) == (F, G)
assert (-f).cancel(g) == (-F, G)
assert f.cancel(-g) == (-F, G)
Fx, x = field("x", ZZ)
Rt, t = ring("t", Fx)
f = (-x**2 - 4)/4*t
g = t**2 + (x**2 + 2)/2
assert f.cancel(g) == ((-x**2 - 4)*t, 4*t**2 + 2*x**2 + 4)
def test_FracElement_copy():
F, x, y, z = field("x,y,z", ZZ)
f = x*y/3*z
g = f.copy()
assert f == g
g.numer[(1, 1, 1)] = 7
assert f != g
def test_FracElement__pos_neg__():
F, x, y = field("x,y", QQ)
f = (7*x - 9)/y
g = (-7*x + 9)/y
assert +f == f
assert +g == g
assert -f == g
assert -g == f
def test_FracElement_subs():
F, x, y, z = field("x,y,z", ZZ)
f = (x**2 + 3*y)/z
assert f.subs({(x, 0)}) == 3*y/z
assert f.subs({x: 0}) == 3*y/z
pytest.raises(ValueError, lambda: f.subs(object()))
pytest.raises(ZeroDivisionError, lambda: f.subs({z: 0}))
assert f.subs([(x, 0), (y, 1)]) == 3/z
def test_FracElement___call__():
F, x, y, z = field("x,y,z", ZZ)
f = (x**2 + 3*y)/z
pytest.raises(ValueError, lambda: f(1, 1, 1, 1))
r = f(1, 1, 1)
assert r == 4 and not isinstance(r, FracElement)
pytest.raises(ZeroDivisionError, lambda: f(1, 1, 0))
Fz = ZZ.frac_field("z")
assert f(1, 1) == 4/Fz.z
def test_FracElement___pow__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f**3 == 1/x**3
assert g**3 == 1/y**3
assert (f*g)**3 == 1/(x**3*y**3)
assert (f*g)**-3 == (x*y)**3
pytest.raises(ZeroDivisionError, lambda: (x - x)**-3)
def test_FracElement___sub__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f - g == (-x + y)/(x*y)
assert x - F.ring.gens[0] == F.ring.gens[0] - x == 0
F, x, y = field("x,y", ZZ)
assert x - 3 == -(3 - x)
assert x - QQ(3, 7) == -(QQ(3, 7) - x) == (7*x - 3)/7
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = (u*v - x)/(y - u*v)
assert dict(f.numer) == {(1, 0, 0, 0): -1, (0, 0, 0, 0): u*v}
assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): -u*v}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = (u*v - x)/(y - u*v)
assert dict(f.numer) == {(1, 0, 0, 0): -1, (0, 0, 0, 0): u*v}
assert dict(f.denom) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): -u*v}
Fuv, u, v = field("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Fuv)
f = u - x
assert dict(f) == {(0, 0, 0): u, (1, 0, 0): -Fuv.one}
def test_FracElement_from_expr():
x, y, z = symbols("x,y,z")
F, X, Y, Z = field((x, y, z), ZZ)
f = F.convert(1)
assert f == 1 and isinstance(f, F.dtype)
f = F.convert(Rational(3, 7))
assert f == F(3)/7 and isinstance(f, F.dtype)
f = F.convert(x)
assert f == X and isinstance(f, F.dtype)
f = F.convert(Rational(3, 7)*x)
assert f == 3*X/7 and isinstance(f, F.dtype)
f = F.convert(1/x)
assert f == 1/X and isinstance(f, F.dtype)
f = F.convert(x*y*z)
assert f == X*Y*Z and isinstance(f, F.dtype)
f = F.convert(x*y/z)
assert f == X*Y/Z and isinstance(f, F.dtype)
f = F.convert(x*y*z + x*y + x)
assert f == X*Y*Z + X*Y + X and isinstance(f, F.dtype)
f = F.convert((x*y*z + x*y + x)/(x*y + 7))
assert f == (X*Y*Z + X*Y + X)/(X*Y + 7) and isinstance(f, F.dtype)
f = F.convert(x**3*y*z + x**2*y**7 + 1)
assert f == X**3*Y*Z + X**2*Y**7 + 1 and isinstance(f, F.dtype)
pytest.raises(CoercionFailed, lambda: F.convert(2**x))
pytest.raises(CoercionFailed, lambda: F.convert(7*x + sqrt(2)))
F, X, Y = field((2**x, y), ZZ)
f = F.convert(2**(2*x) + 1)
assert f == X**2 + 1
def test_FracElement__lt_le_gt_ge__():
F, x, y = field("x,y", ZZ)
assert F(1) < 1/x < 1/x**2 < 1/x**3
assert F(1) <= 1/x <= 1/x**2 <= 1/x**3
assert -7/x < 1/x < 3/x < y/x < 1/x**2
assert -7/x <= 1/x <= 3/x <= y/x <= 1/x**2
assert 1/x**3 > 1/x**2 > 1/x > F(1)
assert 1/x**3 >= 1/x**2 >= 1/x >= F(1)
assert 1/x**2 > y/x > 3/x > 1/x > -7/x
assert 1/x**2 >= y/x >= 3/x >= 1/x >= -7/x
def test_PolyElement_sturm():
F, t = field("t", ZZ)
_, x = ring("x", F)
f = 1024/(15625*t**8)*x**5 - 4096/(625*t**8)*x**4 + 32/(15625*t**4)*x**3 - 128/(625*t**4)*x**2 + F(1)/62500*x - F(1)/625
assert f.sturm() == [
x**3 - 100*x**2 + t**4/64*x - 25*t**4/16,
3*x**2 - 200*x + t**4/64,
(-t**4/96 + F(20000)/9)*x + 25*t**4/18,
(-9*t**12 - 11520000*t**8 - 3686400000000*t**4)/(576*t**8 - 245760000*t**4 + 26214400000000),
]
R, x, y = ring("x,y", ZZ)
pytest.raises(MultivariatePolynomialError, lambda: (x + y).sturm())
def test_PolyElement___mul__():
Rt, t = ring("t", ZZ)
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert dict(u*x) == dict(x*u) == {(1, 0, 0): u}
assert dict(2*u*x + z) == dict(x*2*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1}
assert dict(u*2*x + z) == dict(2*x*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1}
assert dict(2*u*x + z) == dict(x*2*u + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1}
assert dict(u*x*2 + z) == dict(x*u*2 + z) == {(1, 0, 0): 2*u, (0, 0, 1): 1}
assert dict(2*u*x*y + z) == dict(x*y*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
assert dict(u*2*x*y + z) == dict(2*x*y*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
assert dict(2*u*x*y + z) == dict(x*y*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
assert dict(u*x*y*2 + z) == dict(x*y*u*2 + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
assert dict(2*u*y*x + z) == dict(y*x*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
assert dict(u*2*y*x + z) == dict(2*y*x*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
assert dict(2*u*y*x + z) == dict(y*x*2*u + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
assert dict(u*y*x*2 + z) == dict(y*x*u*2 + z) == {(1, 1, 0): 2*u, (0, 0, 1): 1}
assert dict(3*u*(x + y) + z) == dict((x + y)*3*u + z) == {(1, 0, 0): 3*u, (0, 1, 0): 3*u, (0, 0, 1): 1}
pytest.raises(TypeError, lambda: t*x + z)
pytest.raises(TypeError, lambda: x*t + z)
pytest.raises(TypeError, lambda: t*u + z)
pytest.raises(TypeError, lambda: u*t + z)
Fuv, u, v = field("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Fuv)
assert dict(u*x) == dict(x*u) == {(1, 0, 0): u}
Rxyz, x, y, z = ring("x,y,z", EX)
assert dict(EX(pi)*x*y*z) == dict(x*y*z*EX(pi)) == {(1, 1, 1): EX(pi)}
assert (x + 2*y).mul_ground(0) == Rxyz.zero
R, x, y = ring("x,y", ZZ)
p = x + y**2
p1 = p._imul_num(3)
assert p == p1 and p1 == 3*x + 3*y**2
p2 = p._imul_num(0)
assert p == p2 and p2 == R.zero
def test_PolyElement___add__():
Rt, t = ring("t", ZZ)
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert dict(+x) == dict(x)
assert dict(x + 3*y) == {(1, 0, 0): 1, (0, 1, 0): 3}
assert dict(u + x) == dict(x + u) == {(1, 0, 0): 1, (0, 0, 0): u}
assert dict(u + x*y) == dict(x*y + u) == {(1, 1, 0): 1, (0, 0, 0): u}
assert dict(u + x*y + z) == dict(x*y + z + u) == {(1, 1, 0): 1, (0, 0, 1): 1, (0, 0, 0): u}
assert dict(u*x + x) == dict(x + u*x) == {(1, 0, 0): u + 1}
assert dict(u*x + x*y) == dict(x*y + u*x) == {(1, 1, 0): 1, (1, 0, 0): u}
assert dict(u*x + x*y + z) == dict(x*y + z + u*x) == {(1, 1, 0): 1, (0, 0, 1): 1, (1, 0, 0): u}
pytest.raises(TypeError, lambda: t + x)
pytest.raises(TypeError, lambda: x + t)
pytest.raises(TypeError, lambda: t + u)
pytest.raises(TypeError, lambda: u + t)
Fuv, u, v = field("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Fuv)
assert u + (x - u) == x
assert dict(u + x) == dict(x + u) == {(1, 0, 0): 1, (0, 0, 0): u}
Rxyz, x, y, z = ring("x,y,z", EX)
assert dict(EX(pi) + x*y*z) == dict(x*y*z + EX(pi)) == {(1, 1, 1): EX(1), (0, 0, 0): EX(pi)}
R, x, y = ring('x, y', ZZ)
p = x**4 + 2*y
m = (1, 2)
p1 = p._iadd_monom((m, 5))
assert p == p1 and p1 == x**4 + 5*x*y**2 + 2*y
p2 = p._iadd_monom(((0, 1), 2))
assert p == p2 and p2 == x**4 + 5*x*y**2 + 4*y
p3 = p._iadd_monom(((0, 1), -4))
assert p == p3 and p3 == x**4 + 5*x*y**2
assert x._iadd_monom((m, 5)) == 5*x*y**2 + x
def test_FracElement_as_expr():
F, x, y, z = field("x,y,z", ZZ)
f = (3*x**2*y - x*y*z)/(7*z**3 + 1)
X, Y, Z = F.symbols
g = (3*X**2*Y - X*Y*Z)/(7*Z**3 + 1)
assert f != g
assert f.as_expr() == g
X, Y, Z = symbols("x,y,z")
g = (3*X**2*Y - X*Y*Z)/(7*Z**3 + 1)
assert f != g
assert f.as_expr(X, Y, Z) == g
pytest.raises(ValueError, lambda: f.as_expr(X))
def test_FracElement___truediv__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f/g == y/x
assert x/F.ring.gens[0] == F.ring.gens[0]/x == 1
F, x, y = field("x,y", ZZ)
assert x*3 == 3*x
assert x/QQ(3, 7) == (QQ(3, 7)/x)**-1 == 7*x/3
pytest.raises(ZeroDivisionError, lambda: x/0)
pytest.raises(ZeroDivisionError, lambda: 1/(x - x))
pytest.raises(ZeroDivisionError, lambda: x/(x - x))
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = (u*v)/(x*y)
assert dict(f.numer) == {(0, 0, 0, 0): u*v}
assert dict(f.denom) == {(1, 1, 0, 0): 1}
g = (x*y)/(u*v)
assert dict(g.numer) == {(1, 1, 0, 0): 1}
assert dict(g.denom) == {(0, 0, 0, 0): u*v}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = (u*v)/(x*y)
assert dict(f.numer) == {(0, 0, 0, 0): u*v}
assert dict(f.denom) == {(1, 1, 0, 0): 1}
g = (x*y)/(u*v)
assert dict(g.numer) == {(1, 1, 0, 0): 1}
assert dict(g.denom) == {(0, 0, 0, 0): u*v}
Fuv, u, v = field("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Fuv)
pytest.raises(TypeError, lambda: u/x)
def test_FracElement_diff():
F, x, y, z = field("x,y,z", ZZ)
assert ((x**2 + y) / (z + 1)).diff(x) == 2 * x / (z + 1)
def test_FracElement___hash__():
F, x, y, z = field("x,y,z", QQ)
assert hash(x * y / z)
def test_FracField___eq__():
assert field("x,y,z", QQ)[0] == field("x,y,z", QQ)[0]
assert field("x,y,z", QQ)[0] is field("x,y,z", QQ)[0]
assert field("x,y,z", QQ)[0] != field("x,y,z", ZZ)[0]
assert field("x,y,z", QQ)[0] is not field("x,y,z", ZZ)[0]
assert field("x,y,z", ZZ)[0] != field("x,y,z", QQ)[0]
assert field("x,y,z", ZZ)[0] is not field("x,y,z", QQ)[0]
assert field("x,y,z", QQ)[0] != field("x,y", QQ)[0]
assert field("x,y,z", QQ)[0] is not field("x,y", QQ)[0]
assert field("x,y", QQ)[0] != field("x,y,z", QQ)[0]
assert field("x,y", QQ)[0] is not field("x,y,z", QQ)[0]
def test_FractionField___eq__():
assert field("x,y,z", QQ)[0] == field("x,y,z", QQ)[0]
assert field("x,y,z", QQ)[0] is field("x,y,z", QQ)[0]
assert field("x,y,z", QQ)[0] != field("x,y,z", ZZ)[0]
assert field("x,y,z", QQ)[0] is not field("x,y,z", ZZ)[0]
assert field("x,y,z", ZZ)[0] != field("x,y,z", QQ)[0]
assert field("x,y,z", ZZ)[0] is not field("x,y,z", QQ)[0]
assert field("x,y,z", QQ)[0] != field("x,y", QQ)[0]
assert field("x,y,z", QQ)[0] is not field("x,y", QQ)[0]
assert field("x,y", QQ)[0] != field("x,y,z", QQ)[0]
assert field("x,y", QQ)[0] is not field("x,y,z", QQ)[0]
def test_FracElement___hash__():
F, x, y, z = field("x,y,z", QQ)
assert hash(x*y/z)
def test_FracElement_to_poly():
F, x, y = field("x y", ZZ)
pytest.raises(ValueError, lambda: (x / y).to_poly())
def test_FracElement_to_poly():
F, x, y = field("x y", ZZ)
pytest.raises(ValueError, lambda: (x/y).to_poly())
def test_FracField___hash__():
F, x, y, z = field("x,y,z", QQ)
assert hash(F)
def test_FracElement_subs():
F, x, y, z = field("x,y,z", ZZ)
f = (x**2 + 3 * y) / z
assert f.subs(x, 0) == 3 * y / z
pytest.raises(ZeroDivisionError, lambda: f.subs(z, 0))
def test_FractionField___hash__():
F, x, y, z = field("x,y,z", QQ)
assert hash(F)
