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}
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___add__():
F, x, y = field("x,y", QQ)
f, g = 1 / x, 1 / y
assert f + g == g + f == (x + y) / (x * y)
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___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 == x * Rational(3, 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_evaluate():
F, x,y,z = field("x,y,z", ZZ)
Fyz = field("y,z", ZZ)[0]
f = (x**2 + 3*y)/z
assert f.evaluate(x, 0) == 3*Fyz.y/Fyz.z
raises(ZeroDivisionError, lambda: f.evaluate(z, 0))
def test_FracElement_evaluate():
F, x, y, z = field("x,y,z", ZZ)
Fyz = field("y,z", ZZ)[0]
f = (x**2 + 3 * y) / z
assert f.evaluate(x, 0) == 3 * Fyz.y / Fyz.z
raises(ZeroDivisionError, lambda: f.evaluate(z, 0))
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}
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)
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___div__():
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
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
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)
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
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)
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
def test_FracElement___div__():
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
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
def test_FracElement___call__():
F, x,y,z = field("x,y,z", ZZ)
f = (x**2 + 3*y)/z
r = f(1, 1, 1)
assert r == 4 and not isinstance(r, FracElement)
raises(ZeroDivisionError, lambda: f(1, 1, 0))
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 = QQ(1,2)*x**3 + x**2 + QQ(1,2)*x
g = QQ(1,3)*x**2 + QQ(1,3)*x
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_PolyElement___mul__():
Rt, t = ring("t", ZZ)
Ruv, u,v = ring("u,v", ZZ)
Rxyz, x,y,z = ring("x,y,z", Ruv.to_domain())
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}
raises(TypeError, lambda: t*x + z)
raises(TypeError, lambda: x*t + z)
raises(TypeError, lambda: t*u + z)
raises(TypeError, lambda: u*t + z)
Fuv, u,v = field("u,v", ZZ)
Rxyz, x,y,z = ring("x,y,z", Fuv.to_domain())
assert dict(u*x) == dict(x*u) == {(1, 0, 0): u}
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 + 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}
raises(TypeError, lambda: t + x)
raises(TypeError, lambda: x + t)
raises(TypeError, lambda: t + u)
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___sub__():
Rt, t = ring("t", ZZ)
Ruv, u,v = ring("u,v", ZZ)
Rxyz, x,y,z = ring("x,y,z", Ruv)
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}
raises(TypeError, lambda: t - x)
raises(TypeError, lambda: x - t)
raises(TypeError, lambda: t - u)
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_FracElement___call__():
F, x, y, z = field("x,y,z", ZZ)
f = (x**2 + 3 * y) / z
r = f(1, 1, 1)
assert r == 4 and not isinstance(r, FracElement)
raises(ZeroDivisionError, lambda: f(1, 1, 0))
def test_FracElement___neg__():
F, x, y = field("x,y", QQ)
f = (7 * x - 9) / y
g = (-7 * x + 9) / y
assert -f == g
assert -g == f
def test_FracElement___neg__():
F, x,y = field("x,y", QQ)
f = (7*x - 9)/y
g = (-7*x + 9)/y
assert -f == g
assert -g == f
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_FractionField_from_PolynomialRing():
R, x,y = ring("x,y", QQ)
F, X,Y = field("x,y", ZZ)
f = 3*x**2 + 5*y**2
g = x**2/3 + y**2/5
assert F.to_domain().from_PolynomialRing(f, R.to_domain()) == 3*X**2 + 5*Y**2
assert F.to_domain().from_PolynomialRing(g, R.to_domain()) == (5*X**2 + 3*Y**2)/15
def test_FracField_index():
a = symbols("a")
F, x, y, z = field('x y z', QQ)
assert F.index(x) == 0
assert F.index(y) == 1
raises(ValueError, lambda: F.index(1))
raises(ValueError, lambda: F.index(a))
pass
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_FractionField_from_PolynomialRing():
R, x,y = ring("x,y", QQ)
F, X,Y = field("x,y", ZZ)
f = 3*x**2 + 5*y**2
g = x**2/3 + y**2/5
assert F.to_domain().from_PolynomialRing(f, R.to_domain()) == 3*X**2 + 5*Y**2
assert F.to_domain().from_PolynomialRing(g, R.to_domain()) == (5*X**2 + 3*Y**2)/15
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
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
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),
]
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
raises(ZeroDivisionError, lambda: (x - x)**-3)
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
raises(ZeroDivisionError, lambda: (x - x)**-3)
def test_PolynomialRing_from_FractionField():
F, x,y = field("x,y", ZZ)
R, X,Y = ring("x,y", ZZ)
f = (x**2 + y**2)/(x + 1)
g = (x**2 + y**2)/4
h = x**2 + y**2
assert R.to_domain().from_FractionField(f, F.to_domain()) is None
assert R.to_domain().from_FractionField(g, F.to_domain()) == X**2/4 + Y**2/4
assert R.to_domain().from_FractionField(h, F.to_domain()) == X**2 + Y**2
F, x,y = field("x,y", QQ)
R, X,Y = ring("x,y", QQ)
f = (x**2 + y**2)/(x + 1)
g = (x**2 + y**2)/4
h = x**2 + y**2
assert R.to_domain().from_FractionField(f, F.to_domain()) is None
assert R.to_domain().from_FractionField(g, F.to_domain()) == X**2/4 + Y**2/4
assert R.to_domain().from_FractionField(h, F.to_domain()) == X**2 + Y**2
def test_PolynomialRing_from_FractionField():
F, x,y = field("x,y", ZZ)
R, X,Y = ring("x,y", ZZ)
f = (x**2 + y**2)/(x + 1)
g = (x**2 + y**2)/4
h = x**2 + y**2
assert R.to_domain().from_FractionField(f, F.to_domain()) is None
assert R.to_domain().from_FractionField(g, F.to_domain()) == X**2/4 + Y**2/4
assert R.to_domain().from_FractionField(h, F.to_domain()) == X**2 + Y**2
F, x,y = field("x,y", QQ)
R, X,Y = ring("x,y", QQ)
f = (x**2 + y**2)/(x + 1)
g = (x**2 + y**2)/4
h = x**2 + y**2
assert R.to_domain().from_FractionField(f, F.to_domain()) is None
assert R.to_domain().from_FractionField(g, F.to_domain()) == X**2/4 + Y**2/4
assert R.to_domain().from_FractionField(h, F.to_domain()) == X**2 + Y**2
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_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_FracElement___div__():
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 == x * Rational(7, 3)
raises(ZeroDivisionError, lambda: x / 0)
raises(ZeroDivisionError, lambda: 1 / (x - x))
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}
def test_FracElement___div__():
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
raises(ZeroDivisionError, lambda: x/0)
raises(ZeroDivisionError, lambda: 1/(x - x))
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}
def test_solve_lin_sys_6x6_2():
ground, d,r,e,g,i,j,l,o,m,p,q = field("d,r,e,g,i,j,l,o,m,p,q", ZZ)
domain, c,f,h,k,n,b = ring("c,f,h,k,n,b", ground)
eqs = [b + r/d - c/d, c*(1/d + 1/e + 1/g) - f/g - r/d, f*(1/g + 1/i + 1/j) - c/g - h/i, h*(1/i + 1/l + 1/m) - f/i - k/m, k*(1/m + 1/o + 1/p) - h/m - n/p, n*(1/p + 1/q) - k/p]
sol = {
b: -((l*q*e*o + l*q*g*o + i*m*q*e + i*l*q*e + i*l*p*e + i*j*o*q + j*e*o*q + g*j*o*q + i*e*o*q + g*i*o*q + e*l*o*p + e*l*m*p + e*l*m*o + e*i*o*p + e*i*m*p + e*i*m*o + e*i*l*o + j*e*o*p + j*e*m*q + j*e*m*p + j*e*m*o + j*l*m*q + j*l*m*p + j*l*m*o + i*j*m*p + i*j*m*o + i*j*l*q + i*j*l*o + i*j*m*q + j*l*o*p + j*e*l*o + g*j*o*p + g*j*m*q + g*j*m*p + i*j*l*p + i*j*o*p + j*e*l*q + j*e*l*p + j*l*o*q + g*j*m*o + g*j*l*q + g*j*l*p + g*j*l*o + g*l*o*p + g*l*m*p + g*l*m*o + g*i*m*o + g*i*o*p + g*i*m*q + g*i*m*p + g*i*l*q + g*i*l*p + g*i*l*o + l*m*q*e + l*m*q*g)*r)/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
c: (r*e*(l*q*g*o + i*j*o*q + g*j*o*q + g*i*o*q + j*l*m*q + j*l*m*p + j*l*m*o + i*j*m*p + i*j*m*o + i*j*l*q + i*j*l*o + i*j*m*q + j*l*o*p + g*j*o*p + g*j*m*q + g*j*m*p + i*j*l*p + i*j*o*p + j*l*o*q + g*j*m*o + g*j*l*q + g*j*l*p + g*j*l*o + g*l*o*p + g*l*m*p + g*l*m*o + g*i*m*o + g*i*o*p + g*i*m*q + g*i*m*p + g*i*l*q + g*i*l*p + g*i*l*o + l*m*q*g))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
f: (r*e*j*(l*q*o + l*o*p + l*m*q + l*m*p + l*m*o + i*o*q + i*o*p + i*m*q + i*m*p + i*m*o + i*l*q + i*l*p + i*l*o))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
h: (j*e*r*l*(o*q + o*p + m*q + m*p + m*o))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
k: (j*e*r*o*l*(q + p))/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
n: (j*e*r*o*q*l)/(l*q*d*e*o + l*q*d*g*o + l*q*e*g*o + i*j*d*o*q + i*j*e*o*q + j*d*e*o*q + g*j*d*o*q + g*j*e*o*q + g*i*e*o*q + i*d*e*o*q + g*i*d*o*q + g*i*d*o*p + g*i*d*m*q + g*i*d*m*p + g*i*d*m*o + g*i*d*l*q + g*i*d*l*p + g*i*d*l*o + g*e*l*m*p + g*e*l*o*p + g*j*e*l*q + g*e*l*m*o + g*j*e*m*p + g*j*e*m*o + d*e*l*m*p + d*e*l*m*o + i*d*e*m*p + g*j*e*l*p + g*j*e*l*o + d*e*l*o*p + i*j*d*l*o + i*j*e*o*p + i*j*e*m*q + i*j*d*m*q + i*j*d*m*p + i*j*d*m*o + i*j*d*l*q + i*j*d*l*p + i*j*e*m*p + i*j*e*m*o + i*j*e*l*q + i*j*e*l*p + i*j*e*l*o + i*d*e*m*q + i*d*e*m*o + i*d*e*l*q + i*d*e*l*p + j*d*l*o*p + j*d*e*l*o + g*j*d*o*p + g*j*d*m*q + g*j*d*m*p + g*j*d*m*o + g*j*d*l*q + g*j*d*l*p + g*j*d*l*o + g*j*e*o*p + g*j*e*m*q + g*d*l*o*p + g*d*l*m*p + g*d*l*m*o + j*d*e*m*p + i*d*e*o*p + j*e*o*q*l + j*e*o*p*l + j*e*m*q*l + j*d*e*o*p + j*d*e*m*q + i*j*d*o*p + g*i*e*o*p + j*d*e*m*o + j*d*e*l*q + j*d*e*l*p + j*e*m*p*l + j*e*m*o*l + g*i*e*m*q + g*i*e*m*p + g*i*e*m*o + g*i*e*l*q + g*i*e*l*p + g*i*e*l*o + j*d*l*o*q + j*d*l*m*q + j*d*l*m*p + j*d*l*m*o + i*d*e*l*o + l*m*q*d*e + l*m*q*d*g + l*m*q*e*g),
}
assert solve_lin_sys(eqs, domain) == sol
def test_solve_lin_sys_6x6_1():
ground, d,r,e,g,i,j,l,o,m,p,q = field("d,r,e,g,i,j,l,o,m,p,q", ZZ)
domain, c,f,h,k,n,b = ring("c,f,h,k,n,b", ground)
eqs = [b + q/d - c/d, c*(1/d + 1/e + 1/g) - f/g - q/d, f*(1/g + 1/i + 1/j) - c/g - h/i, h*(1/i + 1/l + 1/m) - f/i - k/m, k*(1/m + 1/o + 1/p) - h/m - n/p, n/p - k/p]
sol = {
b: (e*i*l*q + e*i*m*q + e*i*o*q + e*j*l*q + e*j*m*q + e*j*o*q + e*l*m*q + e*l*o*q + g*i*l*q + g*i*m*q + g*i*o*q + g*j*l*q + g*j*m*q + g*j*o*q + g*l*m*q + g*l*o*q + i*j*l*q + i*j*m*q + i*j*o*q + j*l*m*q + j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o),
c: (-e*g*i*l*q - e*g*i*m*q - e*g*i*o*q - e*g*j*l*q - e*g*j*m*q - e*g*j*o*q - e*g*l*m*q - e*g*l*o*q - e*i*j*l*q - e*i*j*m*q - e*i*j*o*q - e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o),
f: (-e*i*j*l*q - e*i*j*m*q - e*i*j*o*q - e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o),
h: (-e*j*l*m*q - e*j*l*o*q)/(-d*e*i*l - d*e*i*m - d*e*i*o - d*e*j*l - d*e*j*m - d*e*j*o - d*e*l*m - d*e*l*o - d*g*i*l - d*g*i*m - d*g*i*o - d*g*j*l - d*g*j*m - d*g*j*o - d*g*l*m - d*g*l*o - d*i*j*l - d*i*j*m - d*i*j*o - d*j*l*m - d*j*l*o - e*g*i*l - e*g*i*m - e*g*i*o - e*g*j*l - e*g*j*m - e*g*j*o - e*g*l*m - e*g*l*o - e*i*j*l - e*i*j*m - e*i*j*o - e*j*l*m - e*j*l*o),
k: e*j*l*o*q/(d*e*i*l + d*e*i*m + d*e*i*o + d*e*j*l + d*e*j*m + d*e*j*o + d*e*l*m + d*e*l*o + d*g*i*l + d*g*i*m + d*g*i*o + d*g*j*l + d*g*j*m + d*g*j*o + d*g*l*m + d*g*l*o + d*i*j*l + d*i*j*m + d*i*j*o + d*j*l*m + d*j*l*o + e*g*i*l + e*g*i*m + e*g*i*o + e*g*j*l + e*g*j*m + e*g*j*o + e*g*l*m + e*g*l*o + e*i*j*l + e*i*j*m + e*i*j*o + e*j*l*m + e*j*l*o),
n: e*j*l*o*q/(d*e*i*l + d*e*i*m + d*e*i*o + d*e*j*l + d*e*j*m + d*e*j*o + d*e*l*m + d*e*l*o + d*g*i*l + d*g*i*m + d*g*i*o + d*g*j*l + d*g*j*m + d*g*j*o + d*g*l*m + d*g*l*o + d*i*j*l + d*i*j*m + d*i*j*o + d*j*l*m + d*j*l*o + e*g*i*l + e*g*i*m + e*g*i*o + e*g*j*l + e*g*j*m + e*g*j*o + e*g*l*m + e*g*l*o + e*i*j*l + e*i*j*m + e*i*j*o + e*j*l*m + e*j*l*o),
}
assert solve_lin_sys(eqs, domain) == sol
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
raises(ValueError, lambda: f.as_expr(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
raises(ValueError, lambda: f.as_expr(X))
def test_FracElement_from_expr():
x, y, z = symbols("x,y,z")
F, X, Y, Z = field((x, y, z), ZZ)
f = F.from_expr(1)
assert f == 1 and isinstance(f, F.dtype)
f = F.from_expr(Rational(3, 7))
assert f == F(3)/7 and isinstance(f, F.dtype)
f = F.from_expr(x)
assert f == X and isinstance(f, F.dtype)
f = F.from_expr(Rational(3,7)*x)
assert f == X*Rational(3, 7) and isinstance(f, F.dtype)
f = F.from_expr(1/x)
assert f == 1/X and isinstance(f, F.dtype)
f = F.from_expr(x*y*z)
assert f == X*Y*Z and isinstance(f, F.dtype)
f = F.from_expr(x*y/z)
assert f == X*Y/Z and isinstance(f, F.dtype)
f = F.from_expr(x*y*z + x*y + x)
assert f == X*Y*Z + X*Y + X and isinstance(f, F.dtype)
f = F.from_expr((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.from_expr(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)
raises(ValueError, lambda: F.from_expr(2**x))
raises(ValueError, lambda: F.from_expr(7*x + sqrt(2)))
assert isinstance(ZZ[2**x].get_field().convert(2**(-x)),
FracElement)
assert isinstance(ZZ[x**2].get_field().convert(x**(-6)),
FracElement)
assert isinstance(ZZ[exp(Rational(1, 3))].get_field().convert(E),
FracElement)
def test_FracElement_from_expr():
x, y, z = symbols("x,y,z")
F, X, Y, Z = field((x, y, z), ZZ)
f = F.from_expr(1)
assert f == 1 and isinstance(f, F.dtype)
f = F.from_expr(Rational(3, 7))
assert f == F(3)/7 and isinstance(f, F.dtype)
f = F.from_expr(x)
assert f == X and isinstance(f, F.dtype)
f = F.from_expr(Rational(3,7)*x)
assert f == 3*X/7 and isinstance(f, F.dtype)
f = F.from_expr(1/x)
assert f == 1/X and isinstance(f, F.dtype)
f = F.from_expr(x*y*z)
assert f == X*Y*Z and isinstance(f, F.dtype)
f = F.from_expr(x*y/z)
assert f == X*Y/Z and isinstance(f, F.dtype)
f = F.from_expr(x*y*z + x*y + x)
assert f == X*Y*Z + X*Y + X and isinstance(f, F.dtype)
f = F.from_expr((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.from_expr(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)
raises(ValueError, lambda: F.from_expr(2**x))
raises(ValueError, lambda: F.from_expr(7*x + sqrt(2)))
assert isinstance(ZZ[2**x].get_field().convert(2**(-x)),
FracElement)
assert isinstance(ZZ[x**2].get_field().convert(x**(-6)),
FracElement)
assert isinstance(ZZ[exp(S(1)/3)].get_field().convert(E),
FracElement)
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_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_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
raises(ZeroDivisionError, lambda: f.subs(z, 0))
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_FractionField__init():
F, = field("", ZZ)
assert ZZ.frac_field() == F.to_domain()
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
raises(ZeroDivisionError, lambda: f.subs(z, 0))
def test_FracElement___hash__():
F, x, y, z = field("x,y,z", QQ)
assert hash(x * y / z)
def test_FractionField__init():
F, = field("", ZZ)
assert ZZ.frac_field() == F.to_domain()