def test_primitive_element():
assert primitive_element([sqrt(2)]) == (PurePoly(x**2 - 2), [1], [[1, 0]])
assert (primitive_element([sqrt(2), sqrt(3)]) ==
(PurePoly(x**4 - 10*x**2 + 1), [1, 1], [[QQ(+1, 2), 0, -QQ(9, 2), 0],
[QQ(-1, 2), 0, QQ(11, 2), 0]]))
A = QQ.algebraic_field(sqrt(2))
assert (primitive_element([sqrt(2), sqrt(3)], domain=A) ==
(PurePoly(x**2 - 2*sqrt(2)*x - 1, x, domain=A), [1, 1],
[[A.unit], [1, -A.unit]]))
assert (primitive_element([sqrt(2), sqrt(2 + sqrt(2))], domain=A) ==
(PurePoly(x**2 - 2*sqrt(2)*x - sqrt(2), x, domain=A), [1, 1],
[[A.unit], [1, -A.unit]]))
A = QQ.algebraic_field(sqrt(2) + sqrt(3))
assert (primitive_element([sqrt(2), sqrt(3)], domain=A) ==
(PurePoly(x - sqrt(2) - sqrt(3), x, domain=A), [1, 1],
[[A([QQ(1, 2), 0, -QQ(9, 2), 0])], [A([-QQ(1, 2), 0, QQ(11, 2), 0])]]))
pytest.raises(ValueError, lambda: primitive_element([]))
# issue sympy/sympy#13849
assert (primitive_element([sqrt(2), sqrt(2) + sqrt(5)]) ==
(PurePoly(x**4 - 76*x**2 + 4), [1, 2], [[QQ(1, 12), 0, QQ(-37, 6), 0],
[QQ(-1, 24), 0, QQ(43, 12), 0]]))
# issue sympy/sympy#14117
assert (primitive_element([I*sqrt(2*sqrt(2) + 3), I*sqrt(-2*sqrt(2) + 3), I]) ==
(PurePoly(x**4 + 54*x**2 + 81), [1, 2, 4], [[QQ(1, 3), 0], [QQ(1, 27), 0, 2, 0],
[QQ(-1, 54), 0, QQ(-5, 6), 0]]))
def test_primitive_element():
assert primitive_element([sqrt(2)]) == (PurePoly(x**2 - 2), [1], [[1, 0]])
assert (primitive_element([sqrt(2), sqrt(3)]) ==
(PurePoly(x**4 - 10*x**2 + 1), [1, 1], [[QQ(+1, 2), 0, -QQ(9, 2), 0],
[QQ(-1, 2), 0, QQ(11, 2), 0]]))
A = QQ.algebraic_field(sqrt(2))
assert (primitive_element([sqrt(2), sqrt(3)], domain=A) ==
(PurePoly(x**2 - 2*sqrt(2)*x - 1, x, domain=A), [1, 1],
[[A.unit], [1, -A.unit]]))
assert (primitive_element([sqrt(2), sqrt(2 + sqrt(2))], domain=A) ==
(PurePoly(x**2 - 2*sqrt(2)*x - sqrt(2), x, domain=A), [1, 1],
[[A.unit], [1, -A.unit]]))
A = QQ.algebraic_field(sqrt(2) + sqrt(3))
assert (primitive_element([sqrt(2), sqrt(3)], domain=A) ==
(PurePoly(x - sqrt(2) - sqrt(3), x, domain=A), [1, 1],
[[A([QQ(1, 2), 0, -QQ(9, 2), 0])], [A([-QQ(1, 2), 0, QQ(11, 2), 0])]]))
pytest.raises(ValueError, lambda: primitive_element([]))
# issue sympy/sympy#13849
assert (primitive_element([sqrt(2), sqrt(2) + sqrt(5)]) ==
(PurePoly(x**4 - 76*x**2 + 4), [1, 2], [[QQ(1, 12), 0, QQ(-37, 6), 0],
[QQ(-1, 24), 0, QQ(43, 12), 0]]))
# issue sympy/sympy#14117
assert (primitive_element([I*sqrt(2*sqrt(2) + 3), I*sqrt(-2*sqrt(2) + 3), I]) ==
(PurePoly(x**4 + 54*x**2 + 81), [1, 2, 4], [[QQ(1, 3), 0], [QQ(1, 27), 0, 2, 0],
[QQ(-1, 54), 0, QQ(-5, 6), 0]]))
def test_dmp_ext_factor():
R, x, y = ring("x,y", QQ.algebraic_field(sqrt(2)))
assert R.dmp_ext_factor(0) == (R.domain(0), [])
f = x + 1
assert R.dmp_ext_factor(f) == (R.domain(1), [(f, 1)])
g = 2 * x + 2
assert R.dmp_ext_factor(g) == (R.domain(2), [(f, 1)])
f = x**2 - 2 * y**2
assert R.dmp_ext_factor(f) == (R.domain(1), [(x - sqrt(2) * y, 1),
(x + sqrt(2) * y, 1)])
f = 2 * x**2 - 4 * y**2
assert R.dmp_ext_factor(f) == (R.domain(2), [(x - sqrt(2) * y, 1),
(x + sqrt(2) * y, 1)])
# issue sympy/sympy#5786
R, x, y, z, t = ring("x, y, z, t", QQ.algebraic_field(I))
f = -I * t * x - t * y + x * z - I * y * z
assert (R.dmp_ext_factor(f) == (R.domain(1), [(z - I * t, 1),
(x - I * y, 1)]))
R, x = ring("x", QQ.algebraic_field(I))
f = x**2 + 1
assert R.dmp_ext_factor(f) == (R.domain(1), [(x - I, 1), (x + I, 1)])
def test_to_number_field():
A = QQ.algebraic_field(sqrt(2))
assert A.convert(sqrt(2)) == A([1, 0])
B = QQ.algebraic_field(sqrt(2), sqrt(3))
assert B.convert(sqrt(2) + sqrt(3)) == B([1, 0])
K = QQ.algebraic_field(sqrt(2) + sqrt(3))
a = K([Rational(1, 2), Integer(0), -Rational(9, 2), Integer(0)])
assert B.from_expr(sqrt(2)) == a
pytest.raises(CoercionFailed,
lambda: QQ.algebraic_field(sqrt(3)).convert(sqrt(2)))
p = x**6 - 6 * x**4 - 6 * x**3 + 12 * x**2 - 36 * x + 1
r0, r1 = p.as_poly(x).all_roots()[:2]
A = QQ.algebraic_field(r0)
a = A([
Rational(-96, 755),
Rational(-54, 755),
Rational(128, 151),
Rational(936, 755),
Rational(-1003, 755),
Rational(2184, 755)
])
assert A.from_expr(r1) == a
def test_AlgebraicElement():
K = QQ.algebraic_field(sqrt(2))
a = K.unit
sT(a, "AlgebraicField(%s, Pow(Integer(2), Rational(1, 2)))([Integer(1), Integer(0)])" % repr(QQ))
K = QQ.algebraic_field(root(-2, 3))
a = K.unit
sT(a, "AlgebraicField(%s, Pow(Integer(-2), Rational(1, 3)))([Integer(1), Integer(0)])" % repr(QQ))
def test_PolyElement_sqf_norm():
R, x = ring("x", QQ.algebraic_field(sqrt(3)))
X = R.to_ground().x
assert (x**2 - 2).sqf_norm() == (1, x**2 - 2*sqrt(3)*x + 1, X**4 - 10*X**2 + 1)
R, x = ring("x", QQ.algebraic_field(sqrt(2)))
X = R.to_ground().x
assert (x**2 - 3).sqf_norm() == (1, x**2 - 2*sqrt(2)*x - 1, X**4 - 10*X**2 + 1)
def test_PolyElement_sqf_norm():
R, x = ring("x", QQ.algebraic_field(sqrt(3)))
X = R.to_ground().x
assert (x**2 - 2).sqf_norm() == (1, x**2 - 2*sqrt(3)*x + 1, X**4 - 10*X**2 + 1)
R, x = ring("x", QQ.algebraic_field(sqrt(2)))
X = R.to_ground().x
assert (x**2 - 3).sqf_norm() == (1, x**2 - 2*sqrt(2)*x - 1, X**4 - 10*X**2 + 1)
def test_modgcd_algebraic_field():
A = QQ.algebraic_field(sqrt(2))
R, x = ring("x", A)
one = A.one
f, g = 2 * x, R(2)
assert func_field_modgcd(f, g) == (one, f, g)
f, g = 2 * x, R(sqrt(2))
assert func_field_modgcd(f, g) == (one, f, g)
f, g = 2 * x + 2, 6 * x**2 - 6
assert func_field_modgcd(f, g) == (x + 1, R(2), 6 * x - 6)
R, x, y = ring("x, y", A)
f, g = x + sqrt(2) * y, x + y
assert func_field_modgcd(f, g) == (one, f, g)
f, g = x * y + sqrt(2) * y**2, R(sqrt(2)) * y
assert func_field_modgcd(f, g) == (y, x + sqrt(2) * y, R(sqrt(2)))
f, g = x**2 + 2 * sqrt(2) * x * y + 2 * y**2, x + sqrt(2) * y
assert func_field_modgcd(f, g) == (g, g, one)
A = QQ.algebraic_field(sqrt(2), sqrt(3))
R, x, y, z = ring("x, y, z", A)
h = x**2 * y**7 + sqrt(6) / 21 * z
f, g = h * (27 * y**3 + 1), h * (y + x)
assert func_field_modgcd(f, g) == (h, 27 * y**3 + 1, y + x)
h = x**13 * y**3 + 1 / 2 * x**10 + 1 / sqrt(2)
f, g = h * (x + 1), h * sqrt(2) / sqrt(3)
assert func_field_modgcd(f, g) == (h, x + 1, R(sqrt(2) / sqrt(3)))
h = x**4 * y**9 + sqrt(6) / 22 * z
f, g = h * (21 * y**3 + 1), h * (y + x)
assert func_field_modgcd(f, g) == (x**4 * y**9 + sqrt(6) / 22 * z,
21 * y**3 + 1, x + y)
h = x**4 * y**3 + sqrt(6) / 22 * z
f, g = h * (11 * y**3 + 1), h * (y + x)
assert func_field_modgcd(f, g) == (x**4 * y**3 + sqrt(6) / 22 * z,
11 * y**3 + 1, x + y)
A = QQ.algebraic_field(sqrt(2)**(-1) * sqrt(3))
R, x = ring("x", A)
f, g = x + 1, x - 1
assert func_field_modgcd(f, g) == (A.one, f, g)
def test_AlgebraicElement():
K = QQ.algebraic_field(sqrt(2))
a = K.unit
sT(
a,
"AlgebraicField(%s, Pow(Integer(2), Rational(1, 2)))([Integer(1), Integer(0)])"
% repr(QQ))
K = QQ.algebraic_field(root(-2, 3))
a = K.unit
sT(
a,
"AlgebraicField(%s, Pow(Integer(-2), Rational(1, 3)))([Integer(1), Integer(0)])"
% repr(QQ))
def test_modgcd_algebraic_field():
A = QQ.algebraic_field(sqrt(2))
R, x = ring("x", A)
one = A.one
f, g = 2*x, R(2)
assert func_field_modgcd(f, g) == (one, f, g)
f, g = 2*x, R(sqrt(2))
assert func_field_modgcd(f, g) == (one, f, g)
f, g = 2*x + 2, 6*x**2 - 6
assert func_field_modgcd(f, g) == (x + 1, R(2), 6*x - 6)
R, x, y = ring("x, y", A)
f, g = x + sqrt(2)*y, x + y
assert func_field_modgcd(f, g) == (one, f, g)
f, g = x*y + sqrt(2)*y**2, R(sqrt(2))*y
assert func_field_modgcd(f, g) == (y, x + sqrt(2)*y, R(sqrt(2)))
f, g = x**2 + 2*sqrt(2)*x*y + 2*y**2, x + sqrt(2)*y
assert func_field_modgcd(f, g) == (g, g, one)
A = QQ.algebraic_field(sqrt(2), sqrt(3))
R, x, y, z = ring("x, y, z", A)
h = x**2*y**7 + sqrt(6)/21*z
f, g = h*(27*y**3 + 1), h*(y + x)
assert func_field_modgcd(f, g) == (h, 27*y**3+1, y+x)
h = x**13*y**3 + 1/2*x**10 + 1/sqrt(2)
f, g = h*(x + 1), h*sqrt(2)/sqrt(3)
assert func_field_modgcd(f, g) == (h, x + 1, R(sqrt(2)/sqrt(3)))
h = x**4*y**9 + sqrt(6)/22*z
f, g = h*(21*y**3 + 1), h*(y + x)
assert func_field_modgcd(f, g) == (x**4*y**9 + sqrt(6)/22*z, 21*y**3 + 1, x + y)
h = x**4*y**3 + sqrt(6)/22*z
f, g = h*(11*y**3 + 1), h*(y + x)
assert func_field_modgcd(f, g) == (x**4*y**3 + sqrt(6)/22*z, 11*y**3 + 1, x + y)
A = QQ.algebraic_field(sqrt(2)**(-1)*sqrt(3))
R, x = ring("x", A)
f, g = x + 1, x - 1
assert func_field_modgcd(f, g) == (A.one, f, g)
def test_dup_count_complex_roots_9():
R, x = ring("x", QQ.algebraic_field(sqrt(2)))
f = -x**3 + sqrt(2)*x - 1
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
R, x = ring("x", QQ.algebraic_field(sqrt(2)).algebraic_field(I))
f = -x**3 + I*x**2 + sqrt(2)*x - 1
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
def test_dup_count_complex_roots_9():
R, x = ring("x", QQ.algebraic_field(sqrt(2)))
f = -x**3 + sqrt(2) * x - 1
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
R, x = ring("x", QQ.algebraic_field(sqrt(2)).algebraic_field(I))
f = -x**3 + I * x**2 + sqrt(2) * x - 1
assert R.dup_count_complex_roots(f, a, b) == 2
assert R.dup_count_complex_roots(f, c, d) == 1
def test_dup_ext_factor():
R, x = ring("x", QQ.algebraic_field(I))
assert R.dup_ext_factor(0) == (R.domain(0), [])
f = x + 1
assert R.dup_ext_factor(f) == (R.domain(1), [(f, 1)])
g = 2 * x + 2
assert R.dup_ext_factor(g) == (R.domain(2), [(f, 1)])
f = 7 * x**4 + 1
g = x**4 + QQ(1, 7)
assert R.dup_ext_factor(f) == (R.domain(7), [(g, 1)])
f = x**4 + 1
assert R.dup_ext_factor(f) == (R.domain(1), [(x**2 - I, 1), (x**2 + I, 1)])
f = 4 * x**2 + 9
assert R.dup_ext_factor(f) == (R.domain(4), [(x - 3 * I / 2, 1),
(x + 3 * I / 2, 1)])
f = 4 * x**4 + 8 * x**3 + 77 * x**2 + 18 * x + 153
assert R.dup_ext_factor(f) == (4, [(x - 3 * I / 2, 1), (x + 1 + 4 * I, 1),
(x + 1 - 4 * I, 1), (x + 3 * I / 2, 1)])
R, x = ring("x", QQ.algebraic_field(sqrt(2)))
f = x**4 + 1
assert R.dup_ext_factor(f) == (R.domain(1), [(x**2 - sqrt(2) * x + 1, 1),
(x**2 + sqrt(2) * x + 1, 1)])
f = x**2 + 2 * sqrt(2) * x + 2
assert R.dup_ext_factor(f) == (R.domain(1), [(x + sqrt(2), 2)])
assert R.dup_ext_factor(f**3) == (R.domain(1), [(x + sqrt(2), 6)])
f *= 2
assert R.dup_ext_factor(f) == (R.domain(2), [(x + sqrt(2), 2)])
assert R.dup_ext_factor(f**3) == (R.domain(8), [(x + sqrt(2), 6)])
def test_dup_count_real_roots():
R, x = ring("x", ZZ)
assert R.dup_count_real_roots(0) == 0
assert R.dup_count_real_roots(7) == 0
f = x - 1
assert R.dup_count_real_roots(f) == 1
assert R.dup_count_real_roots(f, inf=1) == 1
assert R.dup_count_real_roots(f, sup=0) == 0
assert R.dup_count_real_roots(f, sup=1) == 1
assert R.dup_count_real_roots(f, inf=0, sup=1) == 1
assert R.dup_count_real_roots(f, inf=0, sup=2) == 1
assert R.dup_count_real_roots(f, inf=1, sup=2) == 1
f = x**2 - 2
assert R.dup_count_real_roots(f) == 2
assert R.dup_count_real_roots(f, sup=0) == 1
assert R.dup_count_real_roots(f, inf=-1, sup=1) == 0
R, x = ring("x", QQ.algebraic_field(I))
f = x**3 + I * x + 2
assert R.dup_count_real_roots(f) == 0
f *= (x - 1) * (x + 1)
assert R.dup_count_real_roots(f) == 2
def test_dup_count_real_roots():
R, x = ring("x", ZZ)
assert R.dup_count_real_roots(0) == 0
assert R.dup_count_real_roots(7) == 0
f = x - 1
assert R.dup_count_real_roots(f) == 1
assert R.dup_count_real_roots(f, inf=1) == 1
assert R.dup_count_real_roots(f, sup=0) == 0
assert R.dup_count_real_roots(f, sup=1) == 1
assert R.dup_count_real_roots(f, inf=0, sup=1) == 1
assert R.dup_count_real_roots(f, inf=0, sup=2) == 1
assert R.dup_count_real_roots(f, inf=1, sup=2) == 1
f = x**2 - 2
assert R.dup_count_real_roots(f) == 2
assert R.dup_count_real_roots(f, sup=0) == 1
assert R.dup_count_real_roots(f, inf=-1, sup=1) == 0
R, x = ring("x", QQ.algebraic_field(I))
f = x**3 + I*x + 2
assert R.dup_count_real_roots(f) == 0
f *= x**2 - 1
assert R.dup_count_real_roots(f) == 2
def test_minpoly_domain():
F = QQ.algebraic_field(sqrt(2))
assert minimal_polynomial(sqrt(2), domain=F) == PurePoly(x - sqrt(2), x, domain=F)
assert minimal_polynomial(sqrt(8), domain=F)(x) == x - 2*sqrt(2)
assert minimal_polynomial(sqrt(Rational(3, 2)), domain=F)(x) == 2*x**2 - 3
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(y, domain=QQ))
# issue sympy/sympy#14494
F = QQ.algebraic_field(I)
assert minimal_polynomial(I, domain=F)(x) == x - I
F = QQ.algebraic_field(sqrt(3)*I)
assert minimal_polynomial(exp(I*pi/3), domain=F)(x) == 2*x - sqrt(3)*I - 1
def test_minpoly_domain():
F = QQ.algebraic_field(sqrt(2))
assert minimal_polynomial(sqrt(2), domain=F) == PurePoly(x - sqrt(2), x, domain=F)
assert minimal_polynomial(sqrt(8), domain=F)(x) == x - 2*sqrt(2)
assert minimal_polynomial(sqrt(Rational(3, 2)), domain=F)(x) == 2*x**2 - 3
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(y, domain=QQ))
# issue sympy/sympy#14494
F = QQ.algebraic_field(I)
assert minimal_polynomial(I, domain=F)(x) == x - I
F = QQ.algebraic_field(sqrt(3)*I)
assert minimal_polynomial(exp(I*pi/3), domain=F)(x) == 2*x - sqrt(3)*I - 1
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_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_dmp_lift():
A = QQ.algebraic_field(I)
f = [A(1), A(0), A(0), A(I), A(17*I)]
assert dmp_lift(f, 0, A) == [1, 0, 0, 0, 0, 0, 2, 0, 578, 0, 0, 0,
1, 0, -578, 0, 83521]
pytest.raises(DomainError, lambda: dmp_lift([EX(1), EX(2)], 0, EX))
def test_solve_lin_sys_2x2_2():
domain = QQ.algebraic_field(sqrt(2))
coeff_ring, A0, A1 = ring('A:2', domain)
R, x = ring('x', coeff_ring)
expr = (A1 - sqrt(2))*R.one
sol = solve_lin_sys(expr.coeffs(), coeff_ring)
assert all(isinstance(s, coeff_ring.dtype) for s in sol.values())
assert sol == {A1: coeff_ring.convert(sqrt(2))}
def test_Gaussian_postprocess():
opt = {'gaussian': True}
Gaussian.postprocess(opt)
assert opt == {
'gaussian': True,
'extension': {I},
'domain': QQ.algebraic_field(I),
}
def test_Gaussian_postprocess():
opt = {'gaussian': True}
Gaussian.postprocess(opt)
assert opt == {
'gaussian': True,
'extension': {I},
'domain': QQ.algebraic_field(I),
}
def test_dup_real_imag():
R, x, y = ring("x y", ZZ)
assert R.dup_real_imag(R.zero) == (0, 0)
assert R.dup_real_imag(R.one) == (1, 0)
assert R.dup_real_imag(x + 1) == (x + 1, y)
assert R.dup_real_imag(x + 2) == (x + 2, y)
assert R.dup_real_imag(x**2 + 2 * x + 3) == (x**2 - y**2 + 2 * x + 3,
2 * x * y + 2 * y)
f = x**3 + x**2 + x + 1
assert R.dup_real_imag(f) == (x**3 + x**2 - 3 * x * y**2 + x - y**2 + 1,
3 * x**2 * y + 2 * x * y - y**3 + y)
R, x, y = ring("x y", EX)
pytest.raises(DomainError, lambda: R.dup_real_imag(x + 1))
R = QQ.algebraic_field(I).poly_ring("x", "y")
x, y = R.to_ground().gens
f = R.x**4 + I * R.x**3 - R.x + 1
r = x**4 - 6 * x**2 * y**2 - 3 * x**2 * y - x + y**4 + y**3 + 1
i = 4 * x**3 * y + x**3 - 4 * x * y**3 - 3 * x * y**2 - y
assert R.dup_real_imag(f) == (r, i)
K = QQ.algebraic_field(sqrt(2))
R = K.poly_ring("x", "y")
x, y = R.gens
f = R.x**2 + sqrt(2) * R.x - 1
assert R.dup_real_imag(f) == (x**2 - y**2 + sqrt(2) * x - 1,
2 * x * y + sqrt(2) * y)
K = K.algebraic_field(I)
R = K.poly_ring("x", "y")
x, y = R.to_ground().gens
f = R.x**2 + 2 * sqrt(2) * I * R.x - 1 + I
assert R.dup_real_imag(f) == (x**2 - y**2 - 2 * sqrt(2) * y - 1,
2 * x * y + 2 * sqrt(2) * x + 1)
def test_to_number_field():
A = QQ.algebraic_field(sqrt(2))
assert A.convert(sqrt(2)) == A([1, 0])
B = QQ.algebraic_field(sqrt(2), sqrt(3))
assert B.convert(sqrt(2) + sqrt(3)) == B([1, 0])
K = QQ.algebraic_field(sqrt(2) + sqrt(3))
a = K([Rational(1, 2), Integer(0), -Rational(9, 2), Integer(0)])
assert B.from_expr(sqrt(2)) == a
pytest.raises(CoercionFailed, lambda: QQ.algebraic_field(sqrt(3)).convert(sqrt(2)))
p = x**6 - 6*x**4 - 6*x**3 + 12*x**2 - 36*x + 1
r0, r1 = p.as_poly(x).all_roots()[:2]
A = QQ.algebraic_field(r0)
a = A([Rational(-96, 755), Rational(-54, 755), Rational(128, 151), Rational(936, 755),
Rational(-1003, 755), Rational(2184, 755)])
assert A.from_expr(r1) == a
def test_dup_real_imag():
R, x, y = ring("x y", ZZ)
assert R.dup_real_imag(R.zero) == (0, 0)
assert R.dup_real_imag(R.one) == (1, 0)
assert R.dup_real_imag(x + 1) == (x + 1, y)
assert R.dup_real_imag(x + 2) == (x + 2, y)
assert R.dup_real_imag(x**2 + 2*x + 3) == (x**2 - y**2 + 2*x + 3,
2*x*y + 2*y)
f = x**3 + x**2 + x + 1
assert R.dup_real_imag(f) == (x**3 + x**2 - 3*x*y**2 + x - y**2 + 1,
3*x**2*y + 2*x*y - y**3 + y)
R, x, y = ring("x y", EX)
pytest.raises(DomainError, lambda: R.dup_real_imag(x + 1))
R = QQ.algebraic_field(I).poly_ring("x", "y")
x, y = R.to_ground().gens
f = R.x**4 + I*R.x**3 - R.x + 1
r = x**4 - 6*x**2*y**2 - 3*x**2*y - x + y**4 + y**3 + 1
i = 4*x**3*y + x**3 - 4*x*y**3 - 3*x*y**2 - y
assert R.dup_real_imag(f) == (r, i)
K = QQ.algebraic_field(sqrt(2))
R = K.poly_ring("x", "y")
x, y = R.gens
f = R.x**2 + sqrt(2)*R.x - 1
assert R.dup_real_imag(f) == (x**2 - y**2 + sqrt(2)*x - 1, 2*x*y + sqrt(2)*y)
K = K.algebraic_field(I)
R = K.poly_ring("x", "y")
x, y = R.to_ground().gens
f = R.x**2 + 2*sqrt(2)*I*R.x - 1 + I
assert R.dup_real_imag(f) == (x**2 - y**2 - 2*sqrt(2)*y - 1,
2*x*y + 2*sqrt(2)*x + 1)
def test_sympyissue_5786():
R, x, y, z, t = ring("x, y, z, t", QQ.algebraic_field(I))
f = (z - I*t)*(x - I*y)
assert (R.dmp_ext_factor(f) == (R.domain(1), [(z - I*t, 1), (x - I*y, 1)]))
f = (z - I*t)**2*(x - I*y)
assert (R.dmp_ext_factor(f) == (R.domain(1), [(z - I*t, 2), (x - I*y, 1)]))
f = (z - I*t)*(x - I*y)**3
assert (R.dmp_ext_factor(f) == (R.domain(1), [(z - I*t, 1), (x - I*y, 3)]))
def test_sympyissue_5786():
R, x, y, z, t = ring("x, y, z, t", QQ.algebraic_field(I))
f = (z - I * t) * (x - I * y)
assert f.factor_list() == (1, [(z - I * t, 1), (x - I * y, 1)])
f = (z - I * t)**2 * (x - I * y)
assert f.factor_list() == (1, [(z - I * t, 2), (x - I * y, 1)])
f = (z - I * t) * (x - I * y)**3
assert f.factor_list() == (1, [(z - I * t, 1), (x - I * y, 3)])
def test_modgcd_algebraic_field_random():
A = QQ.algebraic_field(sqrt(2), sqrt(3))
R, x, y, z = ring("x, y, z", A)
h = x**2 * y**3 + 1111 * sqrt(6) / 12 * z
a, b = 11 * y**3 + 2, (y + x - 1) * h
assert func_field_modgcd(h * a, h * b) == (h, a, b)
a, b = 12 * y + 2 * x - 1, (y + x - 1) * h
assert func_field_modgcd(h * a, h * b) == (h, a, b)
def test_conversion():
L = QQ.poly_ring(x, y, order="ilex")
G = QQ.poly_ring(x, y)
assert L.convert(x) == L.convert(G.convert(x), G)
assert G.convert(x) == G.convert(L.convert(x), L)
pytest.raises(CoercionFailed, lambda: G.convert(L.convert(1 / (1 + x)), L))
R = ALG.poly_ring(x, y)
assert R.convert(ALG(1), ALG) == R(1)
pytest.raises(CoercionFailed,
lambda: R.convert(ALG(1), QQ.algebraic_field(sqrt(2))))
def test_Domain_unify_algebraic():
sqrt5 = QQ.algebraic_field(sqrt(5))
sqrt7 = QQ.algebraic_field(sqrt(7))
sqrt57 = QQ.algebraic_field(sqrt(5), sqrt(7))
assert sqrt5.unify(sqrt7) == sqrt57
assert sqrt5.unify(sqrt5.poly_ring(x, y)) == sqrt5.poly_ring(x, y)
assert sqrt5.poly_ring(x, y).unify(sqrt5) == sqrt5.poly_ring(x, y)
assert sqrt5.unify(sqrt5.frac_field(x, y)) == sqrt5.frac_field(x, y)
assert sqrt5.frac_field(x, y).unify(sqrt5) == sqrt5.frac_field(x, y)
assert sqrt5.unify(sqrt7.poly_ring(x, y)) == sqrt57.poly_ring(x, y)
assert sqrt5.poly_ring(x, y).unify(sqrt7) == sqrt57.poly_ring(x, y)
assert sqrt5.unify(sqrt7.frac_field(x, y)) == sqrt57.frac_field(x, y)
assert sqrt5.frac_field(x, y).unify(sqrt7) == sqrt57.frac_field(x, y)
sqrt2 = QQ.algebraic_field(sqrt(2))
r = RootOf(x**7 - x + 1, 0)
rootof = QQ.algebraic_field(r)
ans = QQ.algebraic_field(r + sqrt(2))
assert sqrt2.unify(rootof) == rootof.unify(sqrt2) == ans
# here domain created from tuple, not Expr
p = Poly(x**3 - sqrt(2) * x - 1, x)
sqrt2 = p.domain
assert sqrt2.unify(rootof) == rootof.unify(sqrt2) == ans
def test_Domain_unify_algebraic():
sqrt5 = QQ.algebraic_field(sqrt(5))
sqrt7 = QQ.algebraic_field(sqrt(7))
sqrt57 = QQ.algebraic_field(sqrt(5), sqrt(7))
assert sqrt5.unify(sqrt7) == sqrt57
assert sqrt5.unify(sqrt5.poly_ring(x, y)) == sqrt5.poly_ring(x, y)
assert sqrt5.poly_ring(x, y).unify(sqrt5) == sqrt5.poly_ring(x, y)
assert sqrt5.unify(sqrt5.frac_field(x, y)) == sqrt5.frac_field(x, y)
assert sqrt5.frac_field(x, y).unify(sqrt5) == sqrt5.frac_field(x, y)
assert sqrt5.unify(sqrt7.poly_ring(x, y)) == sqrt57.poly_ring(x, y)
assert sqrt5.poly_ring(x, y).unify(sqrt7) == sqrt57.poly_ring(x, y)
assert sqrt5.unify(sqrt7.frac_field(x, y)) == sqrt57.frac_field(x, y)
assert sqrt5.frac_field(x, y).unify(sqrt7) == sqrt57.frac_field(x, y)
sqrt2 = QQ.algebraic_field(sqrt(2))
r = RootOf(x**7 - x + 1, 0)
rootof = QQ.algebraic_field(r)
ans = QQ.algebraic_field(r + sqrt(2))
assert sqrt2.unify(rootof) == rootof.unify(sqrt2) == ans
# here domain created from tuple, not Expr
p = Poly(x**3 - sqrt(2)*x - 1, x)
sqrt2 = p.domain
assert sqrt2.unify(rootof) == rootof.unify(sqrt2) == ans
def test_Extension_postprocess():
opt = {'extension': {sqrt(2)}}
Extension.postprocess(opt)
assert opt == {
'extension': {sqrt(2)},
'domain': QQ.algebraic_field(sqrt(2)),
}
opt = {'extension': True}
Extension.postprocess(opt)
assert opt == {'extension': True}
def test_Extension_postprocess():
opt = {'extension': {sqrt(2)}}
Extension.postprocess(opt)
assert opt == {
'extension': {sqrt(2)},
'domain': QQ.algebraic_field(sqrt(2)),
}
opt = {'extension': True}
Extension.postprocess(opt)
assert opt == {'extension': True}
def test_ComplexInterval():
R, x = ring("x", QQ.algebraic_field(I))
f = x**3 + x + I
_, r1, r2 = R.dup_isolate_complex_roots_sqf(f, blackbox=True)
assert r1.is_disjoint(r2) is True
assert r1.is_disjoint(r2, check_re_refinement=True) is False
for i in range(4):
r1, r2 = r1.refine(), r2.refine()
assert r1.is_disjoint(r2, check_re_refinement=True) is True
def test_dup_real_imag():
assert dup_real_imag([], ZZ) == ([[]], [[]])
assert dup_real_imag([ZZ(1)], ZZ) == ([[1]], [[]])
assert dup_real_imag([ZZ(1), ZZ(1)], ZZ) == ([[1], [1]], [[1, 0]])
assert dup_real_imag([ZZ(1), ZZ(2)], ZZ) == ([[1], [2]], [[1, 0]])
assert dup_real_imag([ZZ(1), ZZ(2), ZZ(3)], ZZ) == ([[1], [2], [-1, 0, 3]],
[[2, 0], [2, 0]])
f = dmp_normal([1, 1, 1, 1], 0, ZZ)
assert dup_real_imag(f, ZZ) == ([[1], [1], [-3, 0, 1], [-1, 0, 1]],
[[3, 0], [2, 0], [-1, 0, 1, 0]])
f = dmp_normal([1, 1], 0, EX)
pytest.raises(DomainError, lambda: dup_real_imag(f, EX))
A = QQ.algebraic_field(I)
f = [A(1), A(I), A(0), A(-1), A(1)]
assert dup_real_imag(f, A) == ([[1], [], [-6, -3, 0], [-1],
[1, 1, 0, 0, 1]],
[[4, 1], [], [-4, -3, 0, 0], [-1, 0]])
A = QQ.algebraic_field(sqrt(2))
f = [A(1), A(sqrt(2)), A(-1)]
assert dup_real_imag(f, A) == ([[1], [A.unit], [-1, 0, -1]],
[[2, 0], [A.unit, 0]])
A2 = A.algebraic_field(I)
f = [A2(1), A2(2*sqrt(2)*I), A2(I - 1)]
assert dup_real_imag(f, A2) == ([[1], [], [-1, -2*A.unit, -1]],
[[2, 2*A.unit], [1]])
def test_Domain_convert():
assert QQ.convert(10e-52) == QQ(
1684996666696915,
1684996666696914987166688442938726917102321526408785780068975640576)
R, x = ring("x", ZZ)
assert ZZ.convert(x - x) == 0
assert ZZ.convert(x - x, R) == 0
F3 = FF(3)
assert F3.convert(Float(2.0)) == F3.dtype(2)
assert F3.convert(PythonRational(2, 1)) == F3.dtype(2)
pytest.raises(CoercionFailed, lambda: F3.convert(PythonRational(1, 2)))
assert F3.convert(2.0) == F3.dtype(2)
pytest.raises(CoercionFailed, lambda: F3.convert(2.1))
assert RR.convert(CC(1)) == RR(1)
pytest.raises(CoercionFailed, lambda: RR.convert(CC(1, 2)))
assert QQ.convert(ALG(1), ALG) == QQ(1)
pytest.raises(CoercionFailed, lambda: QQ.convert(ALG([1, 1]), ALG))
assert ZZ.convert(ALG(1), ALG) == ZZ(1)
pytest.raises(CoercionFailed, lambda: ZZ.convert(ALG([1, 1]), ALG))
assert EX.convert(ALG([1, 1]), ALG) == sqrt(2) + sqrt(3) + 1
ALG2 = QQ.algebraic_field(sqrt(2))
a2 = ALG2.convert(sqrt(2))
a = ALG.convert(a2, ALG2)
assert a.rep.to_dense() == [QQ(1, 2), 0, -QQ(9, 2), 0]
assert RR.convert(a) == RR(1.4142135623730951)
assert CC.convert(a) == CC(1.4142135623730951)
assert ZZ_python.convert(3.0) == ZZ_python.dtype(3)
pytest.raises(CoercionFailed, lambda: ZZ_python.convert(3.2))
assert CC.convert(QQ_python(1, 2)) == CC(0.5)
CC01 = ComplexField(tol=0.1)
assert CC.convert(CC01(0.3)) == CC(0.3)
assert RR.convert(complex(2 + 0j)) == RR(2)
pytest.raises(CoercionFailed, lambda: RR.convert(complex(2 + 3j)))
assert ALG.convert(EX(sqrt(2)), EX) == ALG.from_expr(sqrt(2))
pytest.raises(CoercionFailed, lambda: ALG.convert(EX(sqrt(5)), EX))
pytest.raises(CoercionFailed, lambda: ALG2.convert(ALG.unit))
def test_dup_count_complex_roots_exclude():
R, x = ring("x", ZZ)
f = (x**2 + 1) * (x**2 - 1) * x
a, b = (-QQ(1), QQ(0)), (QQ(1), QQ(1))
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['S']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['N']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['S', 'N']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['E']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['W']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['E', 'W']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['N', 'S', 'E',
'W']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['SW']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['SE']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S']) == 1
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S',
'N']) == 0
a, b = (QQ(0), QQ(0)), (QQ(1), QQ(1))
assert R.dup_count_complex_roots(f, a, b, exclude=True) == 1
R, x = ring("x", QQ.algebraic_field(I))
f = x**4 + I * x**3 - x + 1
assert R.dup_count_complex_roots(f, inf=(QQ(0), QQ(0)),
sup=(QQ(1), QQ(1))) == 1
r = R.dup_isolate_complex_roots_sqf(f)
assert r == [((QQ(-201, 100), QQ(-201, 100)), (0, 0)),
((QQ(-201, 100), 0), (0, QQ(201, 100))),
((0, QQ(-201, 100)), (QQ(201, 100), 0)),
((0, 0), (QQ(201, 100), QQ(201, 100)))]
assert all(R.dup_count_complex_roots(f, inf=i, sup=s) == 1 for i, s in r)
def test_dup_count_complex_roots_exclude():
R, x = ring("x", ZZ)
f = (x**2 + 1)*(x**2 - 1)*x
a, b = (-QQ(1), QQ(0)), (QQ(1), QQ(1))
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['S']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['N']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['S', 'N']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['E']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['W']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['E', 'W']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['N', 'S', 'E', 'W']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['SW']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['SE']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S']) == 1
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S', 'N']) == 0
a, b = (QQ(0), QQ(0)), (QQ(1), QQ(1))
assert R.dup_count_complex_roots(f, a, b, exclude=True) == 1
R, x = ring("x", QQ.algebraic_field(I))
f = x**4 + I*x**3 - x + 1
assert R.dup_count_complex_roots(f, inf=(QQ(0), QQ(0)), sup=(QQ(1), QQ(1))) == 1
r = R.dup_isolate_complex_roots_sqf(f)
assert r == [((QQ(-201, 100), QQ(-201, 100)), (0, 0)),
((QQ(-201, 100), 0), (0, QQ(201, 100))),
((0, QQ(-201, 100)), (QQ(201, 100), 0)),
((0, 0), (QQ(201, 100), QQ(201, 100)))]
assert all(R.dup_count_complex_roots(f, inf=i, sup=s) == 1
for i, s in r)
def test_Domain_convert():
assert QQ.convert(10e-52) == QQ(1684996666696915, 1684996666696914987166688442938726917102321526408785780068975640576)
R, x = ring("x", ZZ)
assert ZZ.convert(x - x) == 0
assert ZZ.convert(x - x, R) == 0
F3 = FF(3)
assert F3.convert(Float(2.0)) == F3.dtype(2)
assert F3.convert(PythonRational(2, 1)) == F3.dtype(2)
pytest.raises(CoercionFailed, lambda: F3.convert(PythonRational(1, 2)))
assert F3.convert(2.0) == F3.dtype(2)
pytest.raises(CoercionFailed, lambda: F3.convert(2.1))
assert RR.convert(CC(1)) == RR(1)
pytest.raises(CoercionFailed, lambda: RR.convert(CC(1, 2)))
assert QQ.convert(ALG.new(1), ALG) == QQ(1)
pytest.raises(CoercionFailed, lambda: QQ.convert(ALG.new([1, 1]), ALG))
assert ZZ.convert(ALG.new(1), ALG) == ZZ(1)
pytest.raises(CoercionFailed, lambda: ZZ.convert(ALG.new([1, 1]), ALG))
assert EX.convert(ALG.new([1, 1]), ALG) == sqrt(2) + sqrt(3) + 1
ALG2 = QQ.algebraic_field(sqrt(2))
a2 = ALG2.convert(sqrt(2))
a = ALG.convert(a2, ALG2)
assert a.rep.to_dense() == [QQ(1, 2), 0, -QQ(9, 2), 0]
assert ZZ_python.convert(3.0) == ZZ_python.dtype(3)
pytest.raises(CoercionFailed, lambda: ZZ_python.convert(3.2))
assert CC.convert(QQ_python(1, 2)) == CC(0.5)
CC01 = ComplexField(tol=0.1)
assert CC.convert(CC01(0.3)) == CC(0.3)
assert RR.convert(complex(2 + 0j)) == RR(2)
pytest.raises(CoercionFailed, lambda: RR.convert(complex(2 + 3j)))
assert ALG.convert(EX(sqrt(2)), EX) == ALG.from_expr(sqrt(2))
pytest.raises(CoercionFailed, lambda: ALG.convert(EX(sqrt(5)), EX))
pytest.raises(CoercionFailed, lambda: ALG2.convert(ALG.unit))
def test_to_ZZ_ANP_poly():
A = QQ.algebraic_field(sqrt(2))
R, x = ring("x", A)
f = x * (sqrt(2) + 1)
T, x_, z_ = ring("x_, z_", ZZ)
f_ = x_ * z_ + x_
assert _to_ZZ_poly(f, T) == f_
assert _to_ANP_poly(f_, R) == f
R, x, t, s = ring("x, t, s", A)
f = x * t**2 + x * s + sqrt(2)
D, t_, s_ = ring("t_, s_", ZZ)
T, x_, z_ = ring("x_, z_", D)
f_ = (t_**2 + s_) * x_ + z_
assert _to_ZZ_poly(f, T) == f_
assert _to_ANP_poly(f_, R) == f
def test_ComplexInterval():
R, x = ring("x", QQ.algebraic_field(I))
f = x**3 + x + I
_, r1, r2 = R.dup_isolate_complex_roots_sqf(f, blackbox=True)
assert r1.is_disjoint(r2) is True
assert r1.is_disjoint(r2, check_re_refinement=True) is False
for i in range(4):
r1, r2 = r1.refine(), r2.refine()
assert r1.is_disjoint(r2, check_re_refinement=True) is True
(u1, v1), (s1, t1) = r1.as_tuple()
(u2, v2), (s2, t2) = r1.refine(vertical=True).as_tuple()
assert v1 == v2 and t1 == t2
assert u1 <= u2 < s2 < s1
def test_to_ZZ_ANP_poly():
A = QQ.algebraic_field(sqrt(2))
R, x = ring("x", A)
f = x*(sqrt(2) + 1)
T, x_, z_ = ring("x_, z_", ZZ)
f_ = x_*z_ + x_
assert _to_ZZ_poly(f, T) == f_
assert _to_ANP_poly(f_, R) == f
R, x, t, s = ring("x, t, s", A)
f = x*t**2 + x*s + sqrt(2)
D, t_, s_ = ring("t_, s_", ZZ)
T, x_, z_ = ring("x_, z_", D)
f_ = (t_**2 + s_)*x_ + z_
assert _to_ZZ_poly(f, T) == f_
assert _to_ANP_poly(f_, R) == f
def test_PolyElement_gcd():
R, x, y = ring("x,y", QQ)
f = x**2/2 + x + QQ(1, 2)
g = x/2 + QQ(1, 2)
assert f.gcd(g) == x + 1
with using(use_heu_gcd=False):
assert f.gcd(g) == x + 1
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
assert f.gcd(g) == x + 1
R, x, y = ring("x,y", QQ.algebraic_field(sqrt(2)))
f, g = (x + sqrt(2)*y)**2, x + sqrt(2)*y
assert f.gcd(g) == g
with using(gcd_aa_method='modgcd'):
assert f.gcd(g) == g
def test_dup_isolate_real_roots_sqf():
R, x = ring("x", ZZ)
assert R.dup_isolate_real_roots_sqf(0) == []
assert R.dup_isolate_real_roots_sqf(5) == []
f = x * (x + 1)
assert R.dup_isolate_real_roots_sqf(f) == [(-1, -1), (0, 0)]
assert R.dup_isolate_real_roots_sqf(f, inf=+1) == []
assert R.dup_isolate_real_roots_sqf(f, sup=-1) == [(-1, -1)]
assert R.dup_isolate_real_roots_sqf(f, sup=-2) == []
f = x * (x - 1)
assert R.dup_isolate_real_roots_sqf(f) == [(0, 0), (1, 1)]
assert R.dup_isolate_real_roots_sqf(x**4 + x + 1) == []
i = [(-2, -1), (1, 2)]
f = x**2 - 2
assert R.dup_isolate_real_roots_sqf(+f) == i
assert R.dup_isolate_real_roots_sqf(-f) == i
for r in range(2, 7):
for s in (1, 10, -1, -10):
f = prod(x - s * _ for _ in range(1, r))
ans = sorted((s * _, s * _) for _ in range(1, r))
assert R.dup_isolate_real_roots_sqf(f) == ans
assert R.dup_isolate_real_roots_sqf(x**2 - 5) == [(-3, -2), (2, 3)]
assert R.dup_isolate_real_roots_sqf(x**3 - 5) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(x**4 - 5) == [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(x**5 - 5) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(x**6 - 5) == [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(x**7 - 5) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(x**8 - 5) == [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(x**9 - 5) == [(1, 2)]
for roots in subsets(range(1, 4)):
f = prod(x - r for r in roots)
ans = sorted((_, _) for _ in roots)
assert R.dup_isolate_real_roots_sqf(f) == ans
assert R.dup_isolate_real_roots_sqf((x - 3)*(x - 2)*(x - 1)*(x + 1)*(x + 2)*(x + 3)*(2*x + 1)) == \
[(-3, -3), (-2, -2), (-1, -1), (-1, 0), (1, 1), (2, 2), (3, 3)]
assert R.dup_isolate_real_roots_sqf((x - 3)*(x - 2)*(x - 1)*(x + 1)*(x + 2)*(x + 3)*(2*x - 1)*(2*x + 1)) == \
[(-3, -3), (-2, -2), (-1, -1), (-1, 0), (0, 1), (1, 1), (2, 2), (3, 3)]
f = 9 * x**2 - 2
assert R.dup_isolate_real_roots_sqf(f) == \
[(-1, 0), (0, 1)]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10)) == \
[(QQ(-1, 2), QQ(-3, 7)), (QQ(3, 7), QQ(1, 2))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100)) == \
[(QQ(-9, 19), QQ(-8, 17)), (QQ(8, 17), QQ(9, 19))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000)) == \
[(QQ(-33, 70), QQ(-8, 17)), (QQ(8, 17), QQ(33, 70))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10000)) == \
[(QQ(-33, 70), QQ(-107, 227)), (QQ(107, 227), QQ(33, 70))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000)) == \
[(QQ(-305, 647), QQ(-272, 577)), (QQ(272, 577), QQ(305, 647))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000000)) == \
[(QQ(-1121, 2378), QQ(-272, 577)), (QQ(272, 577), QQ(1121, 2378))]
f = (x - 2) * (x - 1) * (2 * x - 1) * (10002 * x - 1) * (10003 * x - 1)
assert R.dup_isolate_real_roots_sqf(f) == \
[(QQ(15, 150046), QQ(47, 470110)), (QQ(47, 470110), QQ(17, 170018)),
(QQ(1, 2), QQ(1, 2)), (1, 1), (2, 2)]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000000000)) == \
[(QQ(1, 10003), QQ(1, 10003)), (QQ(1, 10002), QQ(1, 10002)),
(QQ(1, 2), QQ(1, 2)), (1, 1), (2, 2)]
a, b, c, d = 10000090000001, 2000100003, 10000300007, 10000005000008
f = (x - d) * (x + a) * (b * x + 1) * (c * x - 1)
assert R.dup_isolate_real_roots_sqf(f) == \
[(-13194139533313, -8796093022209), (-1, 0), (0, 1),
(8796093022209, 13194139533313)]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000000000)) == \
[(-a, -a), (QQ(-7, 13958643719), QQ(-1, 2013265921)),
(QQ(3, 30064771075), QQ(1, 9663676417)),
(QQ(1328823874562668133568119, 132882321015),
QQ(37336367728494399224248237, 3733634906029))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100000000000000000000000000000)) == \
[(-a, -a), (-QQ(1, b), -QQ(1, b)), (QQ(1, c), QQ(1, c)), (d, d)]
f = -2 * (x - 2) * (x + 2) * (5 * x**2 - 4 * x - 20)
assert R.dup_isolate_real_roots_sqf(f) == \
[(-2, -2), (-2, -1), (2, 2), (2, 3)]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 100)) == \
[(-2, -2), (-QQ(23, 14), -QQ(18, 11)), (2, 2), (QQ(39, 16), QQ(22, 9))]
f = x - 1
assert R.dup_isolate_real_roots_sqf(f, inf=2) == []
assert R.dup_isolate_real_roots_sqf(f, sup=0) == []
assert R.dup_isolate_real_roots_sqf(f) == [(1, 1)]
assert R.dup_isolate_real_roots_sqf(f, inf=1) == [(1, 1)]
assert R.dup_isolate_real_roots_sqf(f, sup=1) == [(1, 1)]
assert R.dup_isolate_real_roots_sqf(f, inf=1, sup=1) == [(1, 1)]
f = x**2 - 2
assert R.dup_isolate_real_roots_sqf(f, inf=QQ(7, 4)) == []
assert R.dup_isolate_real_roots_sqf(f, inf=QQ(7, 5)) == [(QQ(7,
5), QQ(3, 2))]
assert R.dup_isolate_real_roots_sqf(f, sup=QQ(7, 5)) == [(-2, -1)]
assert R.dup_isolate_real_roots_sqf(f, sup=QQ(7, 4)) == [(-2, -1),
(1, QQ(3, 2))]
assert R.dup_isolate_real_roots_sqf(f, sup=-QQ(7, 4)) == []
assert R.dup_isolate_real_roots_sqf(f, sup=-QQ(7, 5)) == [(-QQ(3, 2),
-QQ(7, 5))]
assert R.dup_isolate_real_roots_sqf(f, inf=-QQ(7, 5)) == [(1, 2)]
assert R.dup_isolate_real_roots_sqf(f, inf=-QQ(7, 4)) == [(-QQ(3, 2), -1),
(1, 2)]
i = [(-2, -1), (1, 2)]
assert R.dup_isolate_real_roots_sqf(f, inf=-2) == i
assert R.dup_isolate_real_roots_sqf(f, sup=+2) == i
assert R.dup_isolate_real_roots_sqf(f, inf=-2, sup=2) == i
assert R.dup_isolate_real_roots_sqf(f, inf=+1) == [i[1]]
assert R.dup_isolate_real_roots_sqf(f, sup=-1) == [i[0]]
R, x = ring("x", QQ)
f = (6 * x - 85) * (1028 * x + 1) / 3855
assert R.dup_isolate_real_roots_sqf(f) == [(-1, 0), (14, 15)]
assert [
_.as_tuple() for _ in R.dup_isolate_real_roots_sqf(f, blackbox=True)
] == [(-1, 0), (14, 15)]
R, x = ring("x", EX)
pytest.raises(DomainError, lambda: R.dup_isolate_real_roots_sqf(x + 3))
R, x = ring("x", QQ.algebraic_field(I))
f = (x - 1) * (x**3 + I * x - 2)
assert R.dup_isolate_real_roots_sqf(f) == [(1, 1)]
assert R.dup_isolate_real_roots_sqf(f, sup=0) == []
f = (x**2 - 2) * (x**3 - x + I)
assert R.dup_isolate_real_roots_sqf(f) == [(QQ(-3, 2), QQ(-4, 3)),
(QQ(4, 3), QQ(3, 2))]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 10),
inf=0) == [(QQ(7, 5), QQ(10, 7))]
assert R.dup_isolate_real_roots_sqf(x) == [(0, 0)]
assert R.dup_isolate_real_roots_sqf(x - 1) == [(1, 1)]
assert R.dup_isolate_real_roots_sqf(x - I) == []
R, x = ring("x", QQ.algebraic_field(sqrt(2)))
f = (-x**3 + sqrt(2) * x - 1) * (x**2 + 1)
assert R.dup_isolate_real_roots_sqf(f) == [(-2, -1)]
assert R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000)) == [(QQ(-132, 91),
QQ(-29, 20))]
f = (x - sqrt(2)) * (x + 2 * sqrt(2)) * (x - 7 + sqrt(2)) * (
x + 3 * sqrt(2)) * (x - 1) * (x + 1 - sqrt(2))
assert R.dup_isolate_real_roots_sqf(f) == [(-5, -4), (-3, -2), (0, 1),
(1, 1), (1, 2), (5, 6)]
R, x = ring("x", QQ.algebraic_field(sqrt(2), sqrt(3)))
f = (x - sqrt(2)) * (x - sqrt(3)) * (x - 2 * sqrt(6)) * (x - sqrt(6)) * (
x**2 + 2)
assert R.dup_isolate_real_roots_sqf(f) == [(1, QQ(3, 2)), (QQ(3, 2), 2),
(2, 3), (4, 5)]
assert (R.dup_isolate_real_roots_sqf(f, eps=QQ(1, 1000)) == [
(QQ(41, 29), QQ(58, 41)), (QQ(71, 41), QQ(97, 56)),
(QQ(218, 89), QQ(49, 20)), (QQ(436, 89), QQ(485, 99))
])
def test_Domain_preprocess():
assert Domain.preprocess(ZZ) == ZZ
assert Domain.preprocess(QQ) == QQ
assert Domain.preprocess(EX) == EX
assert Domain.preprocess(FF(2)) == FF(2)
assert Domain.preprocess(ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert Domain.preprocess('Z') == ZZ
assert Domain.preprocess('Q') == QQ
assert Domain.preprocess('ZZ') == ZZ
assert Domain.preprocess('QQ') == QQ
assert Domain.preprocess('EX') == EX
assert Domain.preprocess('FF(23)') == FF(23)
assert Domain.preprocess('GF(23)') == GF(23)
pytest.raises(OptionError, lambda: Domain.preprocess('Z[]'))
assert Domain.preprocess('Z[x]') == ZZ.poly_ring(x)
assert Domain.preprocess('Q[x]') == QQ.poly_ring(x)
assert Domain.preprocess('ZZ[x]') == ZZ.poly_ring(x)
assert Domain.preprocess('QQ[x]') == QQ.poly_ring(x)
assert Domain.preprocess('Z[x,y]') == ZZ.poly_ring(x, y)
assert Domain.preprocess('Q[x,y]') == QQ.poly_ring(x, y)
assert Domain.preprocess('ZZ[x,y]') == ZZ.poly_ring(x, y)
assert Domain.preprocess('QQ[x,y]') == QQ.poly_ring(x, y)
pytest.raises(OptionError, lambda: Domain.preprocess('Z()'))
assert Domain.preprocess('Z(x)') == ZZ.frac_field(x)
assert Domain.preprocess('Q(x)') == QQ.frac_field(x)
assert Domain.preprocess('ZZ(x)') == ZZ.frac_field(x)
assert Domain.preprocess('QQ(x)') == QQ.frac_field(x)
assert Domain.preprocess('Z(x,y)') == ZZ.frac_field(x, y)
assert Domain.preprocess('Q(x,y)') == QQ.frac_field(x, y)
assert Domain.preprocess('ZZ(x,y)') == ZZ.frac_field(x, y)
assert Domain.preprocess('QQ(x,y)') == QQ.frac_field(x, y)
assert Domain.preprocess('Q<I>') == QQ.algebraic_field(I)
assert Domain.preprocess('QQ<I>') == QQ.algebraic_field(I)
assert Domain.preprocess('Q<sqrt(2), I>') == QQ.algebraic_field(sqrt(2), I)
assert Domain.preprocess(
'QQ<sqrt(2), I>') == QQ.algebraic_field(sqrt(2), I)
pytest.raises(OptionError, lambda: Domain.preprocess('abc'))
assert Domain.preprocess('RR') == RR
assert Domain.preprocess('RR_5') == RealField(prec=5)
assert Domain.preprocess('CC') == CC
assert Domain.preprocess('CC_5') == ComplexField(prec=5)
pytest.raises(OptionError, lambda: Domain.preprocess(()))
from diofant.abc import x, y, z
from diofant.domains import CC, EX, FF, GF, QQ, RR, ZZ, QQ_python, ZZ_python
from diofant.domains.algebraicfield import AlgebraicField
from diofant.domains.complexfield import ComplexField
from diofant.domains.domainelement import DomainElement
from diofant.domains.groundtypes import PythonRational
from diofant.domains.realfield import RealField
from diofant.polys import RootOf, field, ring, roots
from diofant.polys.polyerrors import (CoercionFailed, DomainError,
GeneratorsError, GeneratorsNeeded,
NotInvertible, UnificationFailed)
__all__ = ()
ALG = QQ.algebraic_field(sqrt(2), sqrt(3))
def unify(K0, K1):
return K0.unify(K1)
def test_Domain_interface():
pytest.raises(TypeError, lambda: DomainElement().parent)
assert RR(1).parent is RR
assert CC(1).parent is CC
assert RR.has_default_precision
assert CC.has_default_precision
"""Tests for the PolynomialRing classes. """
import pytest
from diofant import sqrt
from diofant.abc import x, y
from diofant.domains import QQ, ZZ
from diofant.polys.orderings import build_product_order
from diofant.polys.polyerrors import CoercionFailed, GeneratorsNeeded
__all__ = ()
ALG = QQ.algebraic_field(sqrt(2), sqrt(3))
def test_build_order():
R = QQ.poly_ring(x,
y,
order=build_product_order((("lex", x), ("ilex", y)),
(x, y)))
assert R.order((1, 5)) == ((1, ), (-5, ))
def test_globalring():
Qxy = QQ.frac_field(x, y)
R = QQ.poly_ring(x, y)
X = R.convert(x)
Y = R.convert(y)
assert x in R
assert 1 / x not in R
def test_sympyissue_14294():
A = QQ.algebraic_field(I)
a = A.convert(I)
assert A.convert(a) == a
def test_dup_isolate_complex_roots_sqf():
R, x = ring("x", ZZ)
f = x**2 - 2 * x + 3
assert R.dup_isolate_complex_roots_sqf(f) == \
[((0, -6), (6, 0)), ((0, 0), (6, 6))]
assert [r.as_tuple() for r in R.dup_isolate_complex_roots_sqf(f, blackbox=True)] == \
[((0, -6), (6, 0)), ((0, 0), (6, 6))]
assert R.dup_isolate_complex_roots_sqf(f, inf=1,
sup=3) == [((1, -3), (3, 0)),
((1, 0), (3, 3))]
assert R.dup_isolate_complex_roots_sqf(f, inf=(1, 0),
sup=3) == [((1, 0), (3, 3))]
assert R.dup_isolate_complex_roots_sqf(f, inf=(1, QQ(-1, 2)),
sup=3) == [((1, 0), (3, 3))]
assert R.dup_isolate_complex_roots_sqf(f, inf=(1, -3),
sup=(3, -1)) == [((1, -3), (3, -1))]
assert R.dup_isolate_complex_roots_sqf(f, inf=0, sup=QQ(1, 6)) == []
assert R.dup_isolate_complex_roots_sqf(R.zero) == []
pytest.raises(ValueError,
lambda: R.dup_isolate_complex_roots_sqf(f, inf=1, sup=1))
assert R.dup_isolate_complex_roots_sqf(f, eps=QQ(1, 10)) == \
[((QQ(15, 16), -QQ(3, 2)), (QQ(33, 32), -QQ(45, 32))),
((QQ(15, 16), QQ(45, 32)), (QQ(33, 32), QQ(3, 2)))]
assert R.dup_isolate_complex_roots_sqf(f, eps=QQ(1, 100)) == \
[((QQ(255, 256), -QQ(363, 256)), (QQ(513, 512), -QQ(723, 512))),
((QQ(255, 256), QQ(723, 512)), (QQ(513, 512), QQ(363, 256)))]
f = 7 * x**4 - 19 * x**3 + 20 * x**2 + 17 * x + 20
assert R.dup_isolate_complex_roots_sqf(f) == \
[((-QQ(40, 7), -QQ(40, 7)), (0, 0)), ((-QQ(40, 7), 0), (0, QQ(40, 7))),
((0, -QQ(40, 7)), (QQ(40, 7), 0)), ((0, 0), (QQ(40, 7), QQ(40, 7)))]
R, x = ring("x", QQ)
f = x**2 / 2 - 3 * x / 7 + 1
assert R.dup_isolate_complex_roots_sqf(f) == [((0, -4), (4, 0)),
((0, 0), (4, 4))]
R, x = ring("x", EX)
pytest.raises(
DomainError,
lambda: R.dup_isolate_complex_roots_sqf(x, inf=(-1, 0), sup=(1, 1)))
R, x = ring("x", QQ.algebraic_field(I))
f = x**4 + I * x**3 - x + 1
assert R.dup_isolate_complex_roots_sqf(f, inf=(0, 0), sup=(1, 1)) == [
((0, 0), (1, QQ(1, 2)))
]
assert R.dup_isolate_complex_roots_sqf(f,
inf=(0, 0),
sup=(1, 1),
eps=QQ(1, 100)) == [
((QQ(79, 128), QQ(19, 64)),
(QQ(5, 8), QQ(39, 128)))
]
assert R.dup_isolate_complex_roots_sqf(f, inf=(0, -1), sup=(1, 1)) == [
((0, -1), (1, QQ(-1, 2))), ((0, 0), (1, QQ(1, 2)))
]
assert R.dup_isolate_complex_roots_sqf(f,
inf=(0, -1),
sup=(1, 1),
eps=QQ(1, 100)) == [
((QQ(79, 128), QQ(19, 64)),
(QQ(5, 8), QQ(39, 128))),
((QQ(45, 64), QQ(-91, 128)),
(QQ(91, 128), QQ(-45, 64)))
]
f *= (x - 1)
assert R.dup_isolate_complex_roots_sqf(f) == [
((QQ(-401, 100), QQ(-401, 100)), (0, 0)),
((QQ(-401, 100), 0), (0, QQ(401, 100))),
((0, QQ(-401, 100)), (QQ(401, 100), 0)),
((0, 0), (QQ(401, 100), QQ(401, 100)))
]
f = x**7 + I * x**4 - (2 + I) * x**3 - 3 * x + 5
assert R.dup_isolate_complex_roots_sqf(f) == [
((QQ(-1001, 100), 0), (0, QQ(1001, 100))),
((QQ(-1001, 400), QQ(-1001, 800)), (QQ(-1001, 800), 0)),
((QQ(-1001, 800), QQ(-1001, 800)), (0, 0)),
((0, QQ(-1001, 400)), (QQ(1001, 400), QQ(-1001, 800))),
((0, QQ(-1001, 800)), (QQ(1001, 400), 0)),
((0, 0), (QQ(1001, 400), QQ(1001, 800))),
((0, QQ(1001, 800)), (QQ(1001, 400), QQ(1001, 400)))
]
R, x = ring("x", QQ.algebraic_field(sqrt(2)))
f = -x**3 + sqrt(2) * x - 1
assert R.dup_isolate_complex_roots_sqf(f) == [
((0, QQ(-283, 100)), (QQ(283, 100), 0)),
((0, 0), (QQ(283, 100), QQ(283, 100)))
]
R, x = ring("x", QQ.algebraic_field(sqrt(2)).algebraic_field(I))
f = -x**3 + I * x**2 + sqrt(2) * x - 1
assert R.dup_isolate_complex_roots_sqf(f) == [
((QQ(-283, 100), 0), (0, QQ(283, 100))),
((0, QQ(-283, 100)), (QQ(283, 100), 0)),
((0, 0), (QQ(283, 100), QQ(283, 100)))
]
R, x = ring("x", EX)
pytest.raises(DomainError, lambda: R.dup_isolate_complex_roots_sqf(x))
def test_dmp_gcd():
R, x = ring("x", ZZ)
f, g = 0, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (0, 0, 0)
f, g = 2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 0)
f, g = -2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 0)
f, g = 0, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 0, -1)
f, g = 0, 2*x + 4
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 0, 1)
f, g = 2*x + 4, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2*x + 4, 1, 0)
f, g = 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, 1)
f, g = -2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, 1)
f, g = 2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, -1)
f, g = -2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, -1, -1)
f, g = x**2 + 2*x + 1, 1
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1)
f, g = x**2 + 2*x + 1, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
f, g = 2*x**2 + 4*x + 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1)
f, g = 2, 2*x**2 + 4*x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1)
f, g = 2*x**2 + 4*x + 2, x + 1
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1)
f, g = x + 1, 2*x**2 + 4*x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2)
f, g = x - 31, x
assert R.dmp_zz_heu_gcd(f, g) == R.dup_rr_prs_gcd(f, g) == (1, f, g)
f = x**4 + 8*x**3 + 21*x**2 + 22*x + 8
g = x**3 + 6*x**2 + 11*x + 6
h = x**2 + 3*x + 2
cff = x**2 + 5*x + 4
cfg = x + 3
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
f = x**4 - 4
g = x**4 + 4*x**2 + 4
h = x**2 + 2
cff = x**2 - 2
cfg = x**2 + 2
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
h = 1
cff = f
cfg = g
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dup_rr_prs_gcd(f, g) == (h, cff, cfg)
R, x = ring("x", QQ)
f = x**8 + x**6 - 3*x**4 - 3*x**3 + 8*x**2 + 2*x - 5
g = 3*x**6 + 5*x**4 - 4*x**2 - 9*x + 21
h = 1
cff = f
cfg = g
assert R.dmp_qq_heu_gcd(f, g) == (h, cff, cfg)
assert R.dup_ff_prs_gcd(f, g) == (h, cff, cfg)
assert R.dup_ff_prs_gcd(R.zero, R.zero) == ([], [], [])
R, x = ring("x", ZZ)
f = - 352518131239247345597970242177235495263669787845475025293906825864749649589178600387510272*x**49 \
+ 46818041807522713962450042363465092040687472354933295397472942006618953623327997952*x**42 \
+ 378182690892293941192071663536490788434899030680411695933646320291525827756032*x**35 \
+ 112806468807371824947796775491032386836656074179286744191026149539708928*x**28 \
- 12278371209708240950316872681744825481125965781519138077173235712*x**21 \
+ 289127344604779611146960547954288113529690984687482920704*x**14 \
+ 19007977035740498977629742919480623972236450681*x**7 \
+ 311973482284542371301330321821976049
g = 365431878023781158602430064717380211405897160759702125019136*x**21 \
+ 197599133478719444145775798221171663643171734081650688*x**14 \
- 9504116979659010018253915765478924103928886144*x**7 \
- 311973482284542371301330321821976049
assert R.dmp_zz_heu_gcd(f, R.dmp_diff_in(f, 1, 0))[0] == g
assert R.dup_rr_prs_gcd(f, R.dmp_diff_in(f, 1, 0))[0] == g
R, x = ring("x", QQ)
f = x**2/2 + x + QQ(1, 2)
g = x/2 + QQ(1, 2)
h = x + 1
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dup_ff_prs_gcd(f, g) == (h, g, QQ(1, 2))
R, x = ring("x", ZZ)
f = 1317378933230047068160*x + 2945748836994210856960
g = 120352542776360960*x + 269116466014453760
h = 120352542776360960*x + 269116466014453760
cff = 10946
cfg = 1
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
with using(heu_gcd_max=0):
pytest.raises(HeuristicGCDFailed, lambda: R.dmp_zz_heu_gcd(f, g))
R, x = ring("x", CC)
f, g = (x**2 - 1, x**3 - 3*x + 2)
assert R.dmp_inner_gcd(f, g) == (1, f, g)
R, x, y = ring("x,y", CC)
f, g = (x**2 - y, x**3 - y*x + 2)
assert R.dmp_inner_gcd(f, g) == (1, f, g)
R, x, y = ring("x,y", ZZ)
f, g = 0, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (0, 0, 0)
f, g = 2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 0)
f, g = -2, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 0)
f, g = 0, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 0, -1)
f, g = 0, 2*x + 4
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 0, 1)
f, g = 2*x + 4, 0
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2*x + 4, 1, 0)
f, g = 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, 1)
f, g = -2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, 1)
f, g = 2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, -1)
f, g = -2, -2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, -1, -1)
f, g = x**2 + 2*x + 1, 1
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 1)
f, g = x**2 + 2*x + 1, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
with using(use_simplify_gcd=0):
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (1, x**2 + 2*x + 1, 2)
f, g = 2*x**2 + 4*x + 2, 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, x**2 + 2*x + 1, 1)
f, g = 2, 2*x**2 + 4*x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (2, 1, x**2 + 2*x + 1)
f, g = 2*x**2 + 4*x + 2, x + 1
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 2*x + 2, 1)
f, g = x + 1, 2*x**2 + 4*x + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (x + 1, 1, 2*x + 2)
with using(heu_gcd_max=0):
pytest.raises(HeuristicGCDFailed, lambda: R.dmp_zz_heu_gcd(f, g))
f = x**2 + 2*x*y + y**2
g = x**2 + x*y
assert R.dmp_rr_prs_gcd(f, g) == (x + y, x + y, x)
R, x, y, z, u = ring("x,y,z,u", ZZ)
f, g = u**2 + 2*u + 1, 2*u + 2
assert R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) == (u + 1, u + 1, 2)
f, g = z**2*u**2 + 2*z**2*u + z**2 + z*u + z, u**2 + 2*u + 1
h, cff, cfg = u + 1, z**2*u + z**2 + z, u + 1
assert R.dmp_zz_heu_gcd(f, g) == (h, cff, cfg)
assert R.dmp_rr_prs_gcd(f, g) == (h, cff, cfg)
assert R.dmp_zz_heu_gcd(g, f) == (h, cfg, cff)
assert R.dmp_rr_prs_gcd(g, f) == (h, cfg, cff)
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(4, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v, a, b = ring("x,y,z,u,v,a,b", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(6, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v, a, b, c, d = ring("x,y,z,u,v,a,b,c,d", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_1(8, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z = ring("x,y,z", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_2(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(2, ZZ))
H, cff, cfg = R.dmp_zz_heu_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
H, cff, cfg = R.dmp_rr_prs_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y, z, u, v = ring("x,y,z,u,v", ZZ)
f, g, h = map(R.from_dense, dmp_fateman_poly_F_3(4, ZZ))
H, cff, cfg = R.dmp_inner_gcd(f, g)
assert H == h and R.dmp_mul(H, cff) == f \
and R.dmp_mul(H, cfg) == g
R, x, y = ring("x,y", QQ)
f = x**2/2 + x + QQ(1, 2)
g = x/2 + QQ(1, 2)
h = x + 1
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1, 2))
with using(use_simplify_gcd=0):
assert R.dmp_qq_heu_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(f, g) == (h, g, QQ(1, 2))
assert R.dmp_ff_prs_gcd(R.zero, R.zero) == (0, 0, 0)
assert R.dmp_qq_heu_gcd(R.zero, R.zero) == (0, 0, 0)
assert R.dmp_ff_prs_gcd(R.zero, g) == (x + 1, R.zero, QQ(1, 2))
assert R.dmp_qq_heu_gcd(R.zero, g) == (x + 1, R.zero, QQ(1, 2))
R, x, y = ring("x,y", RR)
f = 2.1*x*y**2 - 2.2*x*y + 2.1*x
g = 1.0*x**3
assert R.dmp_ff_prs_gcd(f, g) == \
(1.0*x, 2.1*y**2 - 2.2*y + 2.1, 1.0*x**2)
R, x, y = ring("x,y", ZZ)
f = (-17434367009167300000000000000000000000000000000000000000000000000000000*x**4*y -
250501827896299135568887342575961783764139560000000000000000000000000000000000000000000*x**3*y -
2440935909299672540738135183426056447877858000000000000000000000000000000*x**3 -
1349729941723537919695626818065131519270095220127010623905326719279566297660000000000000000000000000000*x**2*y -
26304033868956978374552886858060487282904504027042515077682955951658838800000000000000000*x**2 -
3232215785736369696036755035364398565076440134133908303058376297547504030528179314849416971379040931276000000000000000*x*y -
94485916261760032526508027937078714464844205539023800247528621905831259414691631156161537919255129011800*x -
2902585888465621357542575571971656665554321652262249362701116665830760628936600958940851960635161420991047110815678789984677193092993*y -
113133324167442997472440652189550843502029192913459268196939183295294085146407870078840385860571627108778756267503630290)
g = (10000000000000000000000000000*x**2 + 71841388839807267676152024786000000000000000*x +
129029628760809605749020969023932901278290735413660734705971)
assert (R.dmp_zz_heu_gcd(f, g) == R.dmp_rr_prs_gcd(f, g) ==
(g,
-1743436700916730000000000000000000000000000*x**2*y -
12525091394814956778444367128798089188206978000000000000000*x*y -
244093590929967254073813518342605644787785800*x -
22495499028725631994927113634418779135935898997901327211111875586270479483*y -
876801128965234839118530545935732755107147297241756982389990, 1))
R, x = ring("x", ZZ)
f, g = x**2 - 1, x**2 - 3*x + 2
assert R.dmp_gcd(f, g) == x - 1
with using(use_heu_gcd=False, fallback_gcd_zz_method='modgcd'):
R.dmp_gcd(f, g) == x - 1
R, x = ring("x", QQ)
f, g = x**2/2 + x + QQ(1, 2), x/2 + QQ(1, 2)
assert R.dmp_gcd(f, g) == x + 1
with using(use_heu_gcd=False):
R.dmp_gcd(f, g) == x + 1
R, x, y = ring("x,y", QQ.algebraic_field(sqrt(2)))
f, g = (x + sqrt(2)*y)**2, x + sqrt(2)*y
assert R.dmp_gcd(f, g) == g
with using(gcd_aa_method='modgcd'):
assert R.dmp_gcd(f, g) == g
def test_AlgebraicElement():
A = QQ.algebraic_field(I)
rep = [QQ(1), QQ(1)]
mod = [QQ(1), QQ(0), QQ(1)]
f = A(rep)
assert f.rep.to_dense() == rep
assert f.mod.to_dense() == mod
assert f.domain.domain == QQ
f = A(1)
assert f.rep.to_dense() == [QQ(1)]
assert f.mod.to_dense() == mod
assert f.domain.domain == QQ
f = A([QQ(3, 2)])
assert f.rep.to_dense() == [QQ(3, 2)]
assert f.mod.to_dense() == mod
assert f.domain.domain == QQ
B = QQ.algebraic_field(I*sqrt(2))
a = A([QQ(1), QQ(1)])
b = B([QQ(1), QQ(1)])
assert (a == a) is True
assert (a != a) is False
assert (a == b) is False
assert (a != b) is True
b = A([QQ(1), QQ(2)])
assert (a == b) is False
assert (a != b) is True
assert A([1, 1]) == A([int(1), int(1)])
assert hash(A([1, 1])) == hash(A([int(1), int(1)]))
assert a.to_dict() == {(0,): QQ(1), (1,): QQ(1)}
assert bool(A([])) is False
assert bool(A([QQ(1)])) is True
a = A([QQ(1), -QQ(1), QQ(2)])
assert a.LC() == -1
assert a.rep.to_dense() == [-1, 1]
A = QQ.algebraic_field(root(2, 3))
assert A.unit > 0
assert A.unit >= 0
assert (A.unit < 0) is False
assert (A.unit <= 0) is False
pytest.raises(TypeError, lambda: A.unit > x)
pytest.raises(TypeError, lambda: QQ.algebraic_field(I).unit > 0)
assert abs(+A.unit) == A.unit
assert abs(-A.unit) == A.unit
a = A([QQ(2), QQ(-1), QQ(1)])
b = A([QQ(1), QQ(2)])
c = A([QQ(-2), QQ(1), QQ(-1)])
assert +a == a
assert -a == c
c = A([QQ(2), QQ(0), QQ(3)])
assert a + b == c
assert b + a == c
assert c + 1 == A([QQ(2), QQ(0), QQ(4)])
pytest.raises(TypeError, lambda: c + "x")
pytest.raises(TypeError, lambda: "x" + c)
c = A([QQ(2), QQ(-2), QQ(-1)])
assert a - b == c
c = A([QQ(-2), QQ(2), QQ(1)])
assert b - a == c
assert c - 1 == A([QQ(-2), QQ(2), QQ(0)])
pytest.raises(TypeError, lambda: c - "x")
pytest.raises(TypeError, lambda: "x" - c)
c = A([QQ(3), QQ(-1), QQ(6)])
assert a*b == c
assert b*a == c
assert c*2 == A([QQ(6), QQ(-2), QQ(12)])
pytest.raises(TypeError, lambda: c*"x")
pytest.raises(TypeError, lambda: "x"*c)
c = A([QQ(11, 10), -QQ(1, 5), -QQ(3, 5)])
assert c/2 == A([QQ(11, 20), -QQ(1, 10), -QQ(3, 10)])
pytest.raises(TypeError, lambda: c/"x")
pytest.raises(TypeError, lambda: "x"/c)
c = A([QQ(-1, 43), QQ(9, 43), QQ(5, 43)])
assert a**0 == A(1)
assert a**1 == a
assert a**-1 == c
pytest.raises(TypeError, lambda: a**QQ(1, 2))
assert a*a**(-1) == A(1)
A = QQ.algebraic_field(I)
a = A([QQ(1, 2), QQ(1), QQ(2)])
b = A([ZZ(1), ZZ(1), ZZ(2)])
c = A([QQ(3, 2), QQ(2), QQ(4)])
assert a + b == b + a == c
def test_dmp_ext_factor():
R, x = ring("x", QQ.algebraic_field(I))
assert R.dmp_ext_factor(0) == (R.domain(0), [])
f = x + 1
assert R.dmp_ext_factor(f) == (R.domain(1), [(f, 1)])
g = 2*x + 2
assert R.dmp_ext_factor(g) == (R.domain(2), [(f, 1)])
f = 7*x**4 + 1
g = x**4 + QQ(1, 7)
assert R.dmp_ext_factor(f) == (R.domain(7), [(g, 1)])
f = x**4 + 1
assert R.dmp_ext_factor(f) == (R.domain(1), [(x**2 - I, 1), (x**2 + I, 1)])
f = 4*x**2 + 9
assert R.dmp_ext_factor(f) == (R.domain(4), [(x - 3*I/2, 1), (x + 3*I/2, 1)])
f = 4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153
assert R.dmp_ext_factor(f) == (4, [(x - 3*I/2, 1), (x + 1 + 4*I, 1),
(x + 1 - 4*I, 1), (x + 3*I/2, 1)])
R, x = ring("x", QQ.algebraic_field(sqrt(2)))
f = x**4 + 1
assert R.dmp_ext_factor(f) == (R.domain(1), [(x**2 - sqrt(2)*x + 1, 1),
(x**2 + sqrt(2)*x + 1, 1)])
f = x**2 + 2*sqrt(2)*x + 2
assert R.dmp_ext_factor(f) == (R.domain(1), [(x + sqrt(2), 2)])
assert R.dmp_ext_factor(f**3) == (R.domain(1), [(x + sqrt(2), 6)])
f *= 2
assert R.dmp_ext_factor(f) == (R.domain(2), [(x + sqrt(2), 2)])
assert R.dmp_ext_factor(f**3) == (R.domain(8), [(x + sqrt(2), 6)])
R, x, y = ring("x,y", QQ.algebraic_field(sqrt(2)))
assert R.dmp_ext_factor(0) == (R.domain(0), [])
f = x + 1
assert R.dmp_ext_factor(f) == (R.domain(1), [(f, 1)])
g = 2*x + 2
assert R.dmp_ext_factor(g) == (R.domain(2), [(f, 1)])
f = x**2 - 2*y**2
assert R.dmp_ext_factor(f) == (R.domain(1), [(x - sqrt(2)*y, 1),
(x + sqrt(2)*y, 1)])
f = 2*x**2 - 4*y**2
assert R.dmp_ext_factor(f) == (R.domain(2), [(x - sqrt(2)*y, 1),
(x + sqrt(2)*y, 1)])
R, x = ring("x", QQ.algebraic_field(I))
f = x**2 + 1
assert R.dmp_ext_factor(f) == (R.domain(1), [(x - I, 1), (x + I, 1)])
def test_dmp_factor_list():
R, x = ring("x", ZZ)
assert R.dmp_factor_list(0) == (0, [])
assert R.dmp_factor_list(7) == (7, [])
R, x = ring("x", QQ)
assert R.dmp_factor_list(0) == (0, [])
assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, x = ring("x", ZZ.poly_ring('t'))
assert R.dmp_factor_list(0) == (0, [])
assert R.dmp_factor_list(7) == (7, [])
R, x = ring("x", QQ.poly_ring('t'))
assert R.dmp_factor_list(0) == (0, [])
assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, x = ring("x", ZZ)
assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
# issue sympy/sympy#8037
assert R.dmp_factor_list(6*x**2 - 5*x - 6) == (1, [(2*x - 3, 1), (3*x + 2, 1)])
R, x = ring("x", QQ)
assert R.dmp_factor_list(x**2/2 + x + QQ(1, 2)) == (QQ(1, 2), [(x + 1, 2)])
R, x = ring("x", FF(2))
assert R.dmp_factor_list(x**2 + 1) == (1, [(x + 1, 2)])
R, x = ring("x", RR)
assert R.dmp_factor_list(1.0*x**2 + 2.0*x + 1.0) == (1.0, [(1.0*x + 1.0, 2)])
assert R.dmp_factor_list(2.0*x**2 + 4.0*x + 2.0) == (2.0, [(1.0*x + 1.0, 2)])
f = 6.7225336055071*x**2 - 10.6463972754741*x - 0.33469524022264
coeff, factors = R.dmp_factor_list(f)
assert coeff == RR(1.0) and len(factors) == 1 and factors[0][0].almosteq(f, 1e-10) and factors[0][1] == 1
# issue diofant/diofant#238
f = 0.1*x**2 + 1.1*x + 1.0
assert R.dmp_factor_list(f) == (10.0, [(0.1*x + 0.1, 1), (0.1*x + 1.0, 1)])
f = 0.25 + 1.0*x + 1.0*x**2
assert R.dmp_factor_list(f) == (4.0, [(0.25 + 0.5*x, 2)])
Rt, t = ring("t", ZZ)
R, x = ring("x", Rt)
f = 4*t*x**2 + 4*t**2*x
assert R.dmp_factor_list(f) == (4*t, [(x, 1), (x + t, 1)])
Rt, t = ring("t", QQ)
R, x = ring("x", Rt)
f = t*x**2/2 + t**2*x/2
assert R.dmp_factor_list(f) == (t/2, [(x, 1), (x + t, 1)])
R, x = ring("x", QQ.algebraic_field(I))
f = x**4 + 2*x**2
assert R.dmp_factor_list(f) == (R.domain(1), [(x, 2), (x**2 + 2, 1)])
R, x = ring("x", EX)
pytest.raises(DomainError, lambda: R.dmp_factor_list(EX(sin(1))))
R, x, y = ring("x,y", ZZ)
assert R.dmp_factor_list(0) == (ZZ(0), [])
assert R.dmp_factor_list(7) == (7, [])
R, x, y = ring("x,y", QQ)
assert R.dmp_factor_list(0) == (QQ(0), [])
assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
Rt, t = ring("t", ZZ)
R, x, y = ring("x,y", Rt)
assert R.dmp_factor_list(0) == (0, [])
assert R.dmp_factor_list(7) == (ZZ(7), [])
Rt, t = ring("t", QQ)
R, x, y = ring("x,y", Rt)
assert R.dmp_factor_list(0) == (0, [])
assert R.dmp_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, *X = ring("x:200", ZZ)
f, g = X[0]**2 + 2*X[0] + 1, X[0] + 1
assert R.dmp_factor_list(f) == (1, [(g, 2)])
f, g = X[-1]**2 + 2*X[-1] + 1, X[-1] + 1
assert R.dmp_factor_list(f) == (1, [(g, 2)])
R, x = ring("x", ZZ)
assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
R, x = ring("x", QQ)
assert R.dmp_factor_list(x**2/2 + x + QQ(1, 2)) == (QQ(1, 2), [(x + 1, 2)])
R, x, y = ring("x,y", ZZ)
assert R.dmp_factor_list(x**2 + 2*x + 1) == (1, [(x + 1, 2)])
R, x, y = ring("x,y", QQ)
assert R.dmp_factor_list(x**2/2 + x + QQ(1, 2)) == (QQ(1, 2), [(x + 1, 2)])
R, x, y = ring("x,y", ZZ)
f = 4*x**2*y + 4*x*y**2
assert R.dmp_factor_list(f) == (4, [(y, 1), (x, 1), (x + y, 1)])
R, x, y = ring("x,y", QQ)
f = x**2*y/2 + x*y**2/2
assert R.dmp_factor_list(f) == (QQ(1, 2), [(y, 1), (x, 1), (x + y, 1)])
R, x, y = ring("x,y", RR)
f = 2.0*x**2 - 8.0*y**2
assert R.dmp_factor_list(f) == (RR(2.0), [(1.0*x - 2.0*y, 1), (1.0*x + 2.0*y, 1)])
f = 6.7225336055071*x**2*y**2 - 10.6463972754741*x*y - 0.33469524022264
coeff, factors = R.dmp_factor_list(f)
assert coeff == RR(1.0) and len(factors) == 1 and factors[0][0].almosteq(f, 1e-10) and factors[0][1] == 1
# issue diofant/diofant#238
R, x, y, z = ring("x,y,z", RR)
f = x*y + x*z + 0.1*y + 0.1*z
assert R.dmp_factor_list(f) == (10.0, [(0.1*y + 0.1*z, 1), (x + 0.1, 1)])
f = 0.25*x**2 + 1.0*x*y*z + 1.0*y**2*z**2
assert R.dmp_factor_list(f) == (4.0, [(0.25*x + 0.5*y*z, 2)])
Rt, t = ring("t", ZZ)
R, x, y = ring("x,y", Rt)
f = 4*t*x**2 + 4*t**2*x
assert R.dmp_factor_list(f) == (4*t, [(x, 1), (x + t, 1)])
Rt, t = ring("t", QQ)
R, x, y = ring("x,y", Rt)
f = t*x**2/2 + t**2*x/2
assert R.dmp_factor_list(f) == (t/2, [(x, 1), (x + t, 1)])
R, x, y = ring("x,y", FF(2))
pytest.raises(NotImplementedError, lambda: R.dmp_factor_list(x**2 + y**2))
R, x, y = ring("x,y", EX)
pytest.raises(DomainError, lambda: R.dmp_factor_list(EX(sin(1))))
def test_AlgebraicElement():
K = QQ.algebraic_field(sqrt(2))
assert str(K([1, 0])) == 'sqrt(2)'
def test_AlgebraicField():
sT(QQ.algebraic_field(sqrt(2)),
"AlgebraicField(%s, Pow(Integer(2), Rational(1, 2)))" % repr(QQ))
def test_field_isomorphism():
a = QQ.algebraic_field(sqrt(2))
b = QQ.algebraic_field(sqrt(3))
c = QQ.algebraic_field(sqrt(7))
d = QQ.algebraic_field(sqrt(2) + sqrt(3))
e = QQ.algebraic_field(sqrt(2) + sqrt(3) + sqrt(7))
assert field_isomorphism(a, a) == [1, 0]
assert field_isomorphism(a, b) is None
assert field_isomorphism(a, c) is None
assert field_isomorphism(a, d) == [QQ(1, 2), 0, -QQ(9, 2), 0]
assert field_isomorphism(a, e) == [QQ(1, 80), 0, -QQ(1, 2), 0, QQ(59, 20), 0]
assert field_isomorphism(b, a) is None
assert field_isomorphism(b, b) == [1, 0]
assert field_isomorphism(b, c) is None
assert field_isomorphism(b, d) == [-QQ(1, 2), 0, QQ(11, 2), 0]
assert field_isomorphism(b, e) == [-QQ(3, 640), 0, QQ(67, 320), 0, -QQ(297, 160), 0, QQ(313, 80), 0]
assert field_isomorphism(c, a) is None
assert field_isomorphism(c, b) is None
assert field_isomorphism(c, c) == [1, 0]
assert field_isomorphism(c, d) is None
assert field_isomorphism(c, e) == [QQ(3, 640), 0, -QQ(71, 320), 0, QQ(377, 160), 0, -QQ(469, 80), 0]
assert field_isomorphism(d, a) is None
assert field_isomorphism(d, b) is None
assert field_isomorphism(d, c) is None
assert field_isomorphism(d, d) == [1, 0]
assert field_isomorphism(d, e) == [-QQ(3, 640), 0, QQ(71, 320), 0, -QQ(377, 160), 0, QQ(549, 80), 0]
assert field_isomorphism(e, a) is None
assert field_isomorphism(e, b) is None
assert field_isomorphism(e, c) is None
assert field_isomorphism(e, d) is None
assert field_isomorphism(e, e) == [1, 0]
f = QQ.algebraic_field(3*sqrt(2) + 8*sqrt(7) - 5)
assert field_isomorphism(f, e) == [QQ(3, 80), 0, -QQ(139, 80), 0, QQ(347, 20), 0, -QQ(761, 20), -5]
assert field_isomorphism(QQ.algebraic_field(+I*sqrt(3)), QQ.algebraic_field(I*sqrt(3)/2)) == [QQ(+1, 2), 0]
assert field_isomorphism(QQ.algebraic_field(-I*sqrt(3)), QQ.algebraic_field(I*sqrt(3)/2)) == [QQ(-1, 2), 0]
assert field_isomorphism(QQ.algebraic_field(+I*sqrt(3)), QQ.algebraic_field(-I*sqrt(3)/2)) == [QQ(-1, 2), 0]
assert field_isomorphism(QQ.algebraic_field(-I*sqrt(3)), QQ.algebraic_field(-I*sqrt(3)/2)) == [QQ(+1, 2), 0]
assert field_isomorphism(QQ.algebraic_field(+2*I*sqrt(3)/7), QQ.algebraic_field(5*I*sqrt(3)/3)) == [QQ(+14, 5), 0]
assert field_isomorphism(QQ.algebraic_field(-2*I*sqrt(3)/7), QQ.algebraic_field(5*I*sqrt(3)/3)) == [QQ(-14, 5), 0]
assert field_isomorphism(QQ.algebraic_field(+2*I*sqrt(3)/7), QQ.algebraic_field(-5*I*sqrt(3)/3)) == [QQ(-14, 5), 0]
assert field_isomorphism(QQ.algebraic_field(-2*I*sqrt(3)/7), QQ.algebraic_field(-5*I*sqrt(3)/3)) == [QQ(+14, 5), 0]
assert field_isomorphism(QQ.algebraic_field(+2*I*sqrt(3)/7 + 27), QQ.algebraic_field(5*I*sqrt(3)/3)) == [QQ(+14, 5), 1323]
assert field_isomorphism(QQ.algebraic_field(-2*I*sqrt(3)/7 + 27), QQ.algebraic_field(5*I*sqrt(3)/3)) == [QQ(-14, 5), 1323]
assert field_isomorphism(QQ.algebraic_field(+2*I*sqrt(3)/7 + 27), QQ.algebraic_field(-5*I*sqrt(3)/3)) == [QQ(-14, 5), 1323]
assert field_isomorphism(QQ.algebraic_field(-2*I*sqrt(3)/7 + 27), QQ.algebraic_field(-5*I*sqrt(3)/3)) == [QQ(+14, 5), 1323]
p = QQ.algebraic_field(+sqrt(2) + sqrt(3))
q = QQ.algebraic_field(-sqrt(2) + sqrt(3))
r = QQ.algebraic_field(+sqrt(2) - sqrt(3))
s = QQ.algebraic_field(-sqrt(2) - sqrt(3))
c = QQ.algebraic_field(cbrt(2))
pos_coeffs = [+QQ(1, 2), 0, -QQ(9, 2), 0]
neg_coeffs = [-QQ(1, 2), 0, +QQ(9, 2), 0]
a = QQ.algebraic_field(sqrt(2))
assert is_isomorphism_possible(a, c) is False
assert field_isomorphism(a, c) is None
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
a = QQ.algebraic_field(-sqrt(2))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
pos_coeffs = [+QQ(1, 2), 0, -QQ(11, 2), 0]
neg_coeffs = [-QQ(1, 2), 0, +QQ(11, 2), 0]
a = QQ.algebraic_field(sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = QQ.algebraic_field(-sqrt(3))
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_coeffs
assert field_isomorphism(a, q, fast=True) == pos_coeffs
assert field_isomorphism(a, r, fast=True) == neg_coeffs
assert field_isomorphism(a, s, fast=True) == neg_coeffs
assert field_isomorphism(a, p, fast=False) == pos_coeffs
assert field_isomorphism(a, q, fast=False) == pos_coeffs
assert field_isomorphism(a, r, fast=False) == neg_coeffs
assert field_isomorphism(a, s, fast=False) == neg_coeffs
pos_coeffs = [+QQ(3, 2), 0, -QQ(33, 2), -8]
neg_coeffs = [-QQ(3, 2), 0, +QQ(33, 2), -8]
a = QQ.algebraic_field(3*sqrt(3) - 8)
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == neg_coeffs
assert field_isomorphism(a, q, fast=True) == neg_coeffs
assert field_isomorphism(a, r, fast=True) == pos_coeffs
assert field_isomorphism(a, s, fast=True) == pos_coeffs
assert field_isomorphism(a, p, fast=False) == neg_coeffs
assert field_isomorphism(a, q, fast=False) == neg_coeffs
assert field_isomorphism(a, r, fast=False) == pos_coeffs
assert field_isomorphism(a, s, fast=False) == pos_coeffs
a = QQ.algebraic_field(3*sqrt(2) + 2*sqrt(3) + 1)
pos_1_coeffs = [+QQ(1, 2), 0, -QQ(5, 2), 1]
neg_1_coeffs = [-QQ(1, 2), 0, +QQ(5, 2), 1]
pos_5_coeffs = [+QQ(5, 2), 0, -QQ(49, 2), 1]
neg_5_coeffs = [-QQ(5, 2), 0, +QQ(49, 2), 1]
assert is_isomorphism_possible(a, p) is True
assert is_isomorphism_possible(a, q) is True
assert is_isomorphism_possible(a, r) is True
assert is_isomorphism_possible(a, s) is True
assert field_isomorphism(a, p, fast=True) == pos_1_coeffs
assert field_isomorphism(a, q, fast=True) == neg_5_coeffs
assert field_isomorphism(a, r, fast=True) == pos_5_coeffs
assert field_isomorphism(a, s, fast=True) == neg_1_coeffs
assert field_isomorphism(a, p, fast=False) == pos_1_coeffs
assert field_isomorphism(a, q, fast=False) == neg_5_coeffs
assert field_isomorphism(a, r, fast=False) == pos_5_coeffs
assert field_isomorphism(a, s, fast=False) == neg_1_coeffs
a = QQ.algebraic_field(sqrt(2))
b = QQ.algebraic_field(sqrt(3))
c = QQ.algebraic_field(sqrt(7))
assert is_isomorphism_possible(a, b) is True
assert is_isomorphism_possible(b, a) is True
assert is_isomorphism_possible(c, p) is False
assert field_isomorphism(a, b, fast=True) is None
assert field_isomorphism(b, a, fast=True) is None
assert field_isomorphism(a, b, fast=False) is None
assert field_isomorphism(b, a, fast=False) is None
pytest.raises(ValueError, lambda: field_isomorphism(1, 2))
def test_dmp_sqf():
R, x = ring("x", ZZ)
assert R(0).sqf_part() == 0
assert R(0).is_squarefree is True
assert R(7).sqf_part() == 1
assert R(7).is_squarefree is True
assert (2*x + 2).sqf_part() == x + 1
assert (2*x + 2).is_squarefree is True
assert (x**3 + x + 1).sqf_part() == x**3 + x + 1
assert (x**3 + x + 1).is_squarefree is True
assert (-x**3 + x + 1).sqf_part() == x**3 - x - 1
assert (-x**3 + x + 1).is_squarefree is True
assert (2*x**3 + 3*x**2).sqf_part() == 2*x**2 + 3*x
assert (2*x**3 + 3*x**2).is_squarefree is False
assert (-2*x**3 + 3*x**2).sqf_part() == 2*x**2 - 3*x
assert (-2*x**3 + 3*x**2).is_squarefree is False
assert (x**3 - 3*x - 2).sqf_part() == x**2 - x - 2
assert (x**3 - 3*x - 2).is_squarefree is False
assert R(0).sqf_list() == (0, [])
assert R(1).sqf_list() == (1, [])
assert x.sqf_list() == (1, [(x, 1)])
assert (2*x**2).sqf_list() == (2, [(x, 2)])
assert (3*x**3).sqf_list() == (3, [(x, 3)])
assert (-x**5 + x**4 + x - 1).sqf_list() == (-1, [(x**3 + x**2 + x + 1, 1),
(x - 1, 2)])
assert (x**8 + 6*x**6 + 12*x**4 + 8*x**2).sqf_list() == (1, [(x, 2),
(x**2 + 2, 3)])
assert (2*x**2 + 4*x + 2).sqf_list() == (2, [(x + 1, 2)])
R, x = ring("x", QQ)
assert (2*x**2 + 4*x + 2).sqf_list() == (2, [(x + 1, 2)])
R, x = ring("x", FF(2))
assert (x**2 + 1).sqf_list() == (1, [(x + 1, 2)])
R, x = ring("x", FF(3))
assert (x**10 + 2*x**7 + 2*x**4 + x).sqf_list() == (1, [(x, 1), (x + 1, 3),
(x + 2, 6)])
R1, x = ring("x", ZZ)
R2, y = ring("y", FF(3))
f = x**3 + 1
g = y**3 + 1
assert f.sqf_part() == f
assert g.sqf_part() == y + 1
assert f.is_squarefree is True
assert g.is_squarefree is False
R, x = ring("x", FF(5))
f = x**8 + x**7 + 3*x**6 + x**4 + 2*x**2 + 2*x + 1
assert f.sqf_part() == x**2 + 4*x + 3
f = 3*x**2 + 2*x + 4
assert f.is_squarefree is True
f = 2*x**6 + 4*x**5 + 4*x**4 + 2*x**3 + 2*x**2 + x + 4
assert f.is_squarefree is False
R, x = ring("x", FF(11))
f = x**3 + 5*x**2 + 8*x + 4
assert f.sqf_part() == x**2 + 3*x + 2
assert R(0).is_squarefree is True
assert R(0).sqf_list() == (0, [])
assert R(1).is_squarefree is True
assert R(1).sqf_list() == (1, [])
assert (x + 1).is_squarefree is True
assert (x + 1).sqf_list() == (1, [(x + 1, 1)])
f = x**11 + 1
assert f.is_squarefree is False
assert f.sqf_list() == (1, [(x + 1, 11)])
f = x**3 + 5*x**2 + 8*x + 4
assert f.is_squarefree is False
assert f.sqf_list() == (1, [(x + 1, 1), (x + 2, 2)])
R, x = ring('x', FF(3))
f = x**10 + 2*x**7 + 2*x**4 + x
assert f.is_squarefree is False
assert f.sqf_list() == (1, [(x, 1), (x + 1, 3), (x + 2, 6)])
R, x = ring('x', FF(53))
f = x**6 + 2*x**5 + 5*x**4 + 26*x**3 + 41*x**2 + 39*x + 38
assert f.is_squarefree is True
R, x = ring('x', FF(102953))
f = x**15 + x + 1
assert f.is_squarefree is True
R, x, y = ring("x,y", ZZ)
A = x**4 - 3*x**2 + 6
D = x**6 - 5*x**4 + 5*x**2 + 4
f, g = D, R.dmp_sub(A, R.dmp_mul(R.dmp_diff_in(D, 1, 0), y))
res = R.dmp_resultant(f, g)
h = (4*y**2 + 1).drop(x)
assert res.sqf_list() == (45796, [(h, 3)])
pytest.raises(DomainError, lambda: (x**2 - 1).sqf_norm())
K = QQ.algebraic_field(sqrt(3))
R, x = ring("x", K)
_, X = ring("x", QQ)
assert (x**2 - 2).sqf_norm() == (1, x**2 + K([QQ(-2), QQ(0)])*x + 1,
X**4 - 10*X**2 + 1)
R, x, y = ring("x,y", ZZ)
assert R(0).sqf_part() == 0
assert R(0).is_squarefree is True
assert R(7).sqf_part() == 1
assert R(7).is_squarefree is True
assert R(3).sqf_list() == (3, [])
R, x, y, z = ring("x,y,z", ZZ)
assert f_0.is_squarefree is True
assert (f_0**2).is_squarefree is False
assert f_1.is_squarefree is True
assert (f_1**2).is_squarefree is False
assert f_2.is_squarefree is True
assert (f_2**2).is_squarefree is False
assert f_3.is_squarefree is True
assert (f_3**2).is_squarefree is False
assert f_5.is_squarefree is False
assert (f_5**2).is_squarefree is False
assert f_4.is_squarefree is True
assert f_4.sqf_part() == -f_4
assert f_5.sqf_part() == x + y - z
R, x, y, z, t = ring("x,y,z,t", ZZ)
assert f_6.is_squarefree is True
assert f_6.sqf_part() == f_6
R, x = ring("x", ZZ)
f = -x**5 + x**4 + x - 1
assert f.sqf_list() == (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)])
f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16
assert f.sqf_list() == (2, [(x + 1, 2), (x + 2, 3)])
R, x, y = ring("x,y", ZZ)
f = -x**5 + x**4 + x - 1
assert f.sqf_list() == (-1, [(x**3 + x**2 + x + 1, 1), (x - 1, 2)])
pytest.raises(DomainError, lambda: (x**2 + y**2).sqf_norm())
R, x, y = ring("x,y", FF(2))
pytest.raises(NotImplementedError, lambda: (y**2 + 1).sqf_list())
pytest.raises(NotImplementedError, lambda: (x**3 + 2*x**2*y + x*y**2).sqf_part())
R, x, y = ring("x,y", QQ.algebraic_field(I))
assert (x**2 + 2*I*x - 1).sqf_list() == (1, [(x + I, 2)])
def test_construct_domain():
assert construct_domain([1, 2, 3]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
assert construct_domain([1, 2, 3],
field=True) == (QQ, [QQ(1), QQ(2),
QQ(3)])
assert construct_domain([Integer(1), Integer(2),
Integer(3)]) == (ZZ, [ZZ(1), ZZ(2),
ZZ(3)])
assert construct_domain(
[Integer(1), Integer(2), Integer(3)],
field=True) == (QQ, [QQ(1), QQ(2), QQ(3)])
assert construct_domain([Rational(1, 2),
Integer(2)]) == (QQ, [QQ(1, 2), QQ(2)])
assert construct_domain([3.14, 1, Rational(1, 2)
]) == (RR, [RR(3.14), RR(1.0),
RR(0.5)])
assert construct_domain([3.14, sqrt(2)],
extension=None) == (EX, [EX(3.14),
EX(sqrt(2))])
assert construct_domain([3.14, sqrt(2)],
extension=True) == (EX, [EX(3.14),
EX(sqrt(2))])
assert construct_domain([sqrt(2), 3.14],
extension=True) == (EX, [EX(sqrt(2)),
EX(3.14)])
assert construct_domain([1, sqrt(2)],
extension=None) == (EX, [EX(1), EX(sqrt(2))])
assert construct_domain([x, sqrt(x)]) == (EX, [EX(x), EX(sqrt(x))])
assert construct_domain([x, sqrt(x), sqrt(y)
]) == (EX, [EX(x),
EX(sqrt(x)),
EX(sqrt(y))])
alg = QQ.algebraic_field(sqrt(2))
assert (construct_domain(
[7, Rational(1, 2), sqrt(2)],
extension=True) == (alg, [alg(7),
alg(Rational(1, 2)),
alg(sqrt(2))]))
alg = QQ.algebraic_field(sqrt(2) + sqrt(3))
assert (construct_domain([7, sqrt(2), sqrt(3)], extension=True) == (alg, [
alg(7), alg(sqrt(2)), alg(sqrt(3))
]))
dom = ZZ.poly_ring(x)
assert construct_domain([2 * x, 3]) == (dom, [dom(2 * x), dom(3)])
dom = ZZ.poly_ring(x, y)
assert construct_domain([2 * x, 3 * y]) == (dom, [dom(2 * x), dom(3 * y)])
dom = QQ.poly_ring(x)
assert construct_domain([x / 2, 3]) == (dom, [dom(x / 2), dom(3)])
dom = QQ.poly_ring(x, y)
assert construct_domain([x / 2, 3 * y]) == (dom, [dom(x / 2), dom(3 * y)])
dom = RR.poly_ring(x)
assert construct_domain([x / 2, 3.5]) == (dom, [dom(x / 2), dom(3.5)])
dom = RR.poly_ring(x, y)
assert construct_domain([x / 2,
3.5 * y]) == (dom, [dom(x / 2),
dom(3.5 * y)])
dom = ZZ.frac_field(x)
assert construct_domain([2 / x, 3]) == (dom, [dom(2 / x), dom(3)])
dom = ZZ.frac_field(x, y)
assert construct_domain([2 / x, 3 * y]) == (dom, [dom(2 / x), dom(3 * y)])
dom = RR.frac_field(x)
assert construct_domain([2 / x, 3.5]) == (dom, [dom(2 / x), dom(3.5)])
dom = RR.frac_field(x, y)
assert construct_domain([2 / x,
3.5 * y]) == (dom, [dom(2 / x),
dom(3.5 * y)])
assert construct_domain(2) == (ZZ, ZZ(2))
assert construct_domain(Rational(2, 3)) == (QQ, QQ(2, 3))
assert construct_domain({}) == (ZZ, {})
def test_Domain__algebraic_field():
alg = ZZ.algebraic_field(sqrt(3))
assert alg.minpoly == Poly(x**2 - 3)
assert alg.domain == QQ
assert alg.from_expr(sqrt(3)).denominator == 1
assert alg.from_expr(2*sqrt(3)).denominator == 1
assert alg.from_expr(sqrt(3)/2).denominator == 2
assert alg([QQ(7, 38), QQ(3, 2)]).denominator == 38
alg = QQ.algebraic_field(sqrt(2))
assert alg.minpoly == Poly(x**2 - 2)
assert alg.domain == QQ
alg = QQ.algebraic_field(sqrt(2), sqrt(3))
assert alg.minpoly == Poly(x**4 - 10*x**2 + 1)
assert alg.domain == QQ
assert alg.is_nonpositive(alg([-1, 1])) is True
assert alg.is_nonnegative(alg([2, -1])) is True
assert alg(1).numerator == alg(1)
assert alg.from_expr(sqrt(3)/2).numerator == alg.from_expr(2*sqrt(3))
assert alg.from_expr(sqrt(3)/2).denominator == 4
pytest.raises(DomainError, lambda: AlgebraicField(ZZ, sqrt(2)))
assert alg.characteristic == 0
assert alg.is_RealAlgebraicField is True
assert int(alg(2)) == 2
assert int(alg.from_expr(Rational(3, 2))) == 1
pytest.raises(TypeError, lambda: int(alg([1, 1])))
alg = QQ.algebraic_field(I)
assert alg.algebraic_field(I) == alg
assert alg.is_RealAlgebraicField is False
alg = QQ.algebraic_field(sqrt(2)).algebraic_field(sqrt(3))
assert alg.minpoly == Poly(x**2 - 3, x, domain=QQ.algebraic_field(sqrt(2)))
# issue sympy/sympy#14476
assert QQ.algebraic_field(Rational(1, 7)) is QQ
alg = QQ.algebraic_field(sqrt(2)).algebraic_field(I)
assert alg.from_expr(2*sqrt(2) + I/3) == alg([alg.domain(1)/3,
alg.domain(2*sqrt(2))])
alg2 = QQ.algebraic_field(sqrt(2))
assert alg2.from_expr(sqrt(2)) == alg2.convert(alg.from_expr(sqrt(2)))
eq = -x**3 + 2*x**2 + 3*x - 2
rs = roots(eq, multiple=True)
alg = QQ.algebraic_field(rs[0])
assert alg.is_RealAlgebraicField
alg1 = QQ.algebraic_field(I)
alg2 = QQ.algebraic_field(sqrt(2)).algebraic_field(I)
assert alg1 != alg2
alg3 = QQ.algebraic_field(RootOf(4*x**7 + x - 1, 0))
assert alg3.is_RealAlgebraicField
assert 2.772 > alg3.unit > 2.771
alg4 = QQ.algebraic_field(sqrt(2) + I)
assert alg4.convert(alg2.unit) == alg4.from_expr(I)
