def test_Domain_get_ring():
assert ZZ.has_assoc_Ring is True
assert QQ.has_assoc_Ring is True
assert ZZ[x].has_assoc_Ring is True
assert QQ[x].has_assoc_Ring is True
assert ZZ[x, y].has_assoc_Ring is True
assert QQ[x, y].has_assoc_Ring is True
assert ZZ.frac_field(x).has_assoc_Ring is True
assert QQ.frac_field(x).has_assoc_Ring is True
assert ZZ.frac_field(x, y).has_assoc_Ring is True
assert QQ.frac_field(x, y).has_assoc_Ring is True
assert EX.has_assoc_Ring is False
assert RR.has_assoc_Ring is False
assert ALG.has_assoc_Ring is False
assert ZZ.get_ring() == ZZ
assert QQ.get_ring() == ZZ
assert ZZ[x].get_ring() == ZZ[x]
assert QQ[x].get_ring() == QQ[x]
assert ZZ[x, y].get_ring() == ZZ[x, y]
assert QQ[x, y].get_ring() == QQ[x, y]
assert ZZ.frac_field(x).get_ring() == ZZ[x]
assert QQ.frac_field(x).get_ring() == QQ[x]
assert ZZ.frac_field(x, y).get_ring() == ZZ[x, y]
assert QQ.frac_field(x, y).get_ring() == QQ[x, y]
assert EX.get_ring() == EX
raises(DomainError, lambda: RR.get_ring())
raises(DomainError, lambda: ALG.get_ring())
def test_Domain_get_ring():
assert ZZ.has_assoc_Ring is True
assert QQ.has_assoc_Ring is True
assert ZZ[x].has_assoc_Ring is True
assert QQ[x].has_assoc_Ring is True
assert ZZ[x, y].has_assoc_Ring is True
assert QQ[x, y].has_assoc_Ring is True
assert ZZ.frac_field(x).has_assoc_Ring is True
assert QQ.frac_field(x).has_assoc_Ring is True
assert ZZ.frac_field(x, y).has_assoc_Ring is True
assert QQ.frac_field(x, y).has_assoc_Ring is True
assert EX.has_assoc_Ring is False
assert RR.has_assoc_Ring is False
assert ALG.has_assoc_Ring is False
assert ZZ.get_ring() == ZZ
assert QQ.get_ring() == ZZ
assert ZZ[x].get_ring() == ZZ[x]
assert QQ[x].get_ring() == QQ[x]
assert ZZ[x, y].get_ring() == ZZ[x, y]
assert QQ[x, y].get_ring() == QQ[x, y]
assert ZZ.frac_field(x).get_ring() == ZZ[x]
assert QQ.frac_field(x).get_ring() == QQ[x]
assert ZZ.frac_field(x, y).get_ring() == ZZ[x, y]
assert QQ.frac_field(x, y).get_ring() == QQ[x, y]
assert EX.get_ring() == EX
assert RR.get_ring() == RR
# XXX: This should also be like RR
raises(DomainError, lambda: ALG.get_ring())
def test_Domain_get_exact():
assert EX.get_exact() == EX
assert ZZ.get_exact() == ZZ
assert QQ.get_exact() == QQ
assert RR.get_exact() == QQ
assert ALG.get_exact() == ALG
assert ZZ[x].get_exact() == ZZ[x]
assert QQ[x].get_exact() == QQ[x]
assert ZZ[x,y].get_exact() == ZZ[x,y]
assert QQ[x,y].get_exact() == QQ[x,y]
assert ZZ.frac_field(x).get_exact() == ZZ.frac_field(x)
assert QQ.frac_field(x).get_exact() == QQ.frac_field(x)
assert ZZ.frac_field(x,y).get_exact() == ZZ.frac_field(x,y)
assert QQ.frac_field(x,y).get_exact() == QQ.frac_field(x,y)
def test_Domain_get_field():
assert EX.has_assoc_Field == True
assert ZZ.has_assoc_Field == True
assert QQ.has_assoc_Field == True
assert RR.has_assoc_Field == False
assert ALG.has_assoc_Field == True
assert ZZ[x].has_assoc_Field == True
assert QQ[x].has_assoc_Field == True
assert ZZ[x,y].has_assoc_Field == True
assert QQ[x,y].has_assoc_Field == True
assert EX.get_field() == EX
assert ZZ.get_field() == QQ
assert QQ.get_field() == QQ
raises(DomainError, "RR.get_field()")
assert ALG.get_field() == ALG
assert ZZ[x].get_field() == ZZ.frac_field(x)
assert QQ[x].get_field() == QQ.frac_field(x)
assert ZZ[x,y].get_field() == ZZ.frac_field(x,y)
assert QQ[x,y].get_field() == QQ.frac_field(x,y)
def test_Domain_get_field():
assert EX.has_assoc_Field is True
assert ZZ.has_assoc_Field is True
assert QQ.has_assoc_Field is True
assert RR.has_assoc_Field is True
assert ALG.has_assoc_Field is True
assert ZZ[x].has_assoc_Field is True
assert QQ[x].has_assoc_Field is True
assert ZZ[x, y].has_assoc_Field is True
assert QQ[x, y].has_assoc_Field is True
assert EX.get_field() == EX
assert ZZ.get_field() == QQ
assert QQ.get_field() == QQ
assert RR.get_field() == RR
assert ALG.get_field() == ALG
assert ZZ[x].get_field() == ZZ.frac_field(x)
assert QQ[x].get_field() == QQ.frac_field(x)
assert ZZ[x, y].get_field() == ZZ.frac_field(x, y)
assert QQ[x, y].get_field() == QQ.frac_field(x, y)
def test_Domain__unify():
assert ZZ.unify(ZZ) == ZZ
assert QQ.unify(QQ) == QQ
assert ZZ.unify(QQ) == QQ
assert QQ.unify(ZZ) == QQ
assert EX.unify(EX) == EX
assert ZZ.unify(EX) == EX
assert QQ.unify(EX) == EX
assert EX.unify(ZZ) == EX
assert EX.unify(QQ) == EX
assert ZZ.poly_ring('x').unify(EX) == EX
assert ZZ.frac_field('x').unify(EX) == EX
assert EX.unify(ZZ.poly_ring('x')) == EX
assert EX.unify(ZZ.frac_field('x')) == EX
assert ZZ.poly_ring('x', 'y').unify(EX) == EX
assert ZZ.frac_field('x', 'y').unify(EX) == EX
assert EX.unify(ZZ.poly_ring('x', 'y')) == EX
assert EX.unify(ZZ.frac_field('x', 'y')) == EX
assert QQ.poly_ring('x').unify(EX) == EX
assert QQ.frac_field('x').unify(EX) == EX
assert EX.unify(QQ.poly_ring('x')) == EX
assert EX.unify(QQ.frac_field('x')) == EX
assert QQ.poly_ring('x', 'y').unify(EX) == EX
assert QQ.frac_field('x', 'y').unify(EX) == EX
assert EX.unify(QQ.poly_ring('x', 'y')) == EX
assert EX.unify(QQ.frac_field('x', 'y')) == EX
assert ZZ.poly_ring('x').unify(ZZ) == ZZ.poly_ring('x')
assert ZZ.poly_ring('x').unify(QQ) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(ZZ) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(QQ) == QQ.poly_ring('x')
assert ZZ.unify(ZZ.poly_ring('x')) == ZZ.poly_ring('x')
assert QQ.unify(ZZ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.poly_ring('x', 'y').unify(ZZ) == ZZ.poly_ring('x', 'y')
assert ZZ.poly_ring('x', 'y').unify(QQ) == QQ.poly_ring('x', 'y')
assert QQ.poly_ring('x', 'y').unify(ZZ) == QQ.poly_ring('x', 'y')
assert QQ.poly_ring('x', 'y').unify(QQ) == QQ.poly_ring('x', 'y')
assert ZZ.unify(ZZ.poly_ring('x', 'y')) == ZZ.poly_ring('x', 'y')
assert QQ.unify(ZZ.poly_ring('x', 'y')) == QQ.poly_ring('x', 'y')
assert ZZ.unify(QQ.poly_ring('x', 'y')) == QQ.poly_ring('x', 'y')
assert QQ.unify(QQ.poly_ring('x', 'y')) == QQ.poly_ring('x', 'y')
assert ZZ.frac_field('x').unify(ZZ) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(QQ) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(ZZ) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ) == QQ.frac_field('x')
assert ZZ.unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert QQ.unify(ZZ.frac_field('x')) == EX # QQ.frac_field('x')
assert ZZ.unify(QQ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x', 'y').unify(ZZ) == ZZ.frac_field('x', 'y')
assert ZZ.frac_field('x', 'y').unify(QQ) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x', 'y').unify(ZZ) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x', 'y').unify(QQ) == QQ.frac_field('x', 'y')
assert ZZ.unify(ZZ.frac_field('x', 'y')) == ZZ.frac_field('x', 'y')
assert QQ.unify(ZZ.frac_field('x', 'y')) == EX # QQ.frac_field('x','y')
assert ZZ.unify(QQ.frac_field('x', 'y')) == EX # QQ.frac_field('x','y')
assert QQ.unify(QQ.frac_field('x', 'y')) == QQ.frac_field('x', 'y')
assert ZZ.poly_ring('x').unify(ZZ.poly_ring('x')) == ZZ.poly_ring('x')
assert ZZ.poly_ring('x').unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(ZZ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.poly_ring('x', 'y').unify(ZZ.poly_ring('x')) == ZZ.poly_ring(
'x', 'y')
assert ZZ.poly_ring('x', 'y').unify(QQ.poly_ring('x')) == QQ.poly_ring(
'x', 'y')
assert QQ.poly_ring('x', 'y').unify(ZZ.poly_ring('x')) == QQ.poly_ring(
'x', 'y')
assert QQ.poly_ring('x', 'y').unify(QQ.poly_ring('x')) == QQ.poly_ring(
'x', 'y')
assert ZZ.poly_ring('x').unify(ZZ.poly_ring('x', 'y')) == ZZ.poly_ring(
'x', 'y')
assert ZZ.poly_ring('x').unify(QQ.poly_ring('x', 'y')) == QQ.poly_ring(
'x', 'y')
assert QQ.poly_ring('x').unify(ZZ.poly_ring('x', 'y')) == QQ.poly_ring(
'x', 'y')
assert QQ.poly_ring('x').unify(QQ.poly_ring('x', 'y')) == QQ.poly_ring(
'x', 'y')
assert ZZ.poly_ring('x', 'y').unify(ZZ.poly_ring('x',
'z')) == ZZ.poly_ring(
'x', 'y', 'z')
assert ZZ.poly_ring('x', 'y').unify(QQ.poly_ring('x',
'z')) == QQ.poly_ring(
'x', 'y', 'z')
assert QQ.poly_ring('x', 'y').unify(ZZ.poly_ring('x',
'z')) == QQ.poly_ring(
'x', 'y', 'z')
assert QQ.poly_ring('x', 'y').unify(QQ.poly_ring('x',
'z')) == QQ.poly_ring(
'x', 'y', 'z')
assert ZZ.frac_field('x').unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert QQ.frac_field('x').unify(ZZ.frac_field('x')) == QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x', 'y').unify(ZZ.frac_field('x')) == ZZ.frac_field(
'x', 'y')
assert ZZ.frac_field('x', 'y').unify(QQ.frac_field('x')) == QQ.frac_field(
'x', 'y')
assert QQ.frac_field('x', 'y').unify(ZZ.frac_field('x')) == QQ.frac_field(
'x', 'y')
assert QQ.frac_field('x', 'y').unify(QQ.frac_field('x')) == QQ.frac_field(
'x', 'y')
assert ZZ.frac_field('x').unify(ZZ.frac_field('x', 'y')) == ZZ.frac_field(
'x', 'y')
assert ZZ.frac_field('x').unify(QQ.frac_field('x', 'y')) == QQ.frac_field(
'x', 'y')
assert QQ.frac_field('x').unify(ZZ.frac_field('x', 'y')) == QQ.frac_field(
'x', 'y')
assert QQ.frac_field('x').unify(QQ.frac_field('x', 'y')) == QQ.frac_field(
'x', 'y')
assert ZZ.frac_field('x', 'y').unify(ZZ.frac_field('x',
'z')) == ZZ.frac_field(
'x', 'y', 'z')
assert ZZ.frac_field('x', 'y').unify(QQ.frac_field('x',
'z')) == QQ.frac_field(
'x', 'y', 'z')
assert QQ.frac_field('x', 'y').unify(ZZ.frac_field('x',
'z')) == QQ.frac_field(
'x', 'y', 'z')
assert QQ.frac_field('x', 'y').unify(QQ.frac_field('x',
'z')) == QQ.frac_field(
'x', 'y', 'z')
assert ZZ.poly_ring('x').unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert ZZ.poly_ring('x').unify(
QQ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.poly_ring('x').unify(
ZZ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.poly_ring('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.poly_ring('x', 'y').unify(ZZ.frac_field('x')) == ZZ.frac_field(
'x', 'y')
assert ZZ.poly_ring('x', 'y').unify(
QQ.frac_field('x')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x', 'y').unify(
ZZ.frac_field('x')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x', 'y').unify(QQ.frac_field('x')) == QQ.frac_field(
'x', 'y')
assert ZZ.poly_ring('x').unify(ZZ.frac_field('x', 'y')) == ZZ.frac_field(
'x', 'y')
assert ZZ.poly_ring('x').unify(QQ.frac_field(
'x', 'y')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x').unify(ZZ.frac_field(
'x', 'y')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x').unify(QQ.frac_field('x', 'y')) == QQ.frac_field(
'x', 'y')
assert ZZ.poly_ring('x', 'y').unify(ZZ.frac_field('x',
'z')) == ZZ.frac_field(
'x', 'y', 'z')
assert ZZ.poly_ring('x', 'y').unify(QQ.frac_field(
'x', 'z')) == EX # QQ.frac_field('x','y','z')
assert QQ.poly_ring('x', 'y').unify(ZZ.frac_field(
'x', 'z')) == EX # QQ.frac_field('x','y','z')
assert QQ.poly_ring('x', 'y').unify(QQ.frac_field('x',
'z')) == QQ.frac_field(
'x', 'y', 'z')
assert ZZ.frac_field('x').unify(ZZ.poly_ring('x')) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(
QQ.poly_ring('x')) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(
ZZ.poly_ring('x')) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ.poly_ring('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x', 'y').unify(ZZ.poly_ring('x')) == ZZ.frac_field(
'x', 'y')
assert ZZ.frac_field('x', 'y').unify(
QQ.poly_ring('x')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x', 'y').unify(
ZZ.poly_ring('x')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x', 'y').unify(QQ.poly_ring('x')) == QQ.frac_field(
'x', 'y')
assert ZZ.frac_field('x').unify(ZZ.poly_ring('x', 'y')) == ZZ.frac_field(
'x', 'y')
assert ZZ.frac_field('x').unify(QQ.poly_ring(
'x', 'y')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(ZZ.poly_ring(
'x', 'y')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(QQ.poly_ring('x', 'y')) == QQ.frac_field(
'x', 'y')
assert ZZ.frac_field('x', 'y').unify(ZZ.poly_ring('x',
'z')) == ZZ.frac_field(
'x', 'y', 'z')
assert ZZ.frac_field('x', 'y').unify(QQ.poly_ring(
'x', 'z')) == EX # QQ.frac_field('x','y','z')
assert QQ.frac_field('x', 'y').unify(ZZ.poly_ring(
'x', 'z')) == EX # QQ.frac_field('x','y','z')
assert QQ.frac_field('x', 'y').unify(QQ.poly_ring('x',
'z')) == QQ.frac_field(
'x', 'y', 'z')
alg = QQ.algebraic_field(sqrt(5))
assert alg.unify(alg['x', 'y']) == alg['x', 'y']
assert alg['x', 'y'].unify(alg) == alg['x', 'y']
assert alg.unify(alg.frac_field('x', 'y')) == alg.frac_field('x', 'y')
assert alg.frac_field('x', 'y').unify(alg) == alg.frac_field('x', 'y')
ext = QQ.algebraic_field(sqrt(7))
raises(NotImplementedError, "alg.unify(ext)")
raises(UnificationFailed,
"ZZ.poly_ring('x','y').unify(ZZ, gens=('y', 'z'))")
raises(UnificationFailed,
"ZZ.unify(ZZ.poly_ring('x','y'), gens=('y', 'z'))")
def test_dmp_factor_list():
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, y = ring("x,y", ZZ)
assert R.dmp_factor_list_include(0) == [(0, 1)]
assert R.dmp_factor_list_include(7) == [(7, 1)]
R, X = xring("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(QQ(1, 2) * x**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(QQ(1, 2) * x**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)])
assert R.dmp_factor_list_include(f) == \
[(4*y, 1),
(x, 1),
(x + y, 1)]
R, x, y = ring("x,y", QQ)
f = QQ(1, 2) * x**2 * y + QQ(1, 2) * x * y**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
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, 1),
(x, 1),
(x + t, 1)])
Rt, t = ring("t", QQ)
R, x, y = ring("x,y", Rt)
f = QQ(1, 2) * t * x**2 + QQ(1, 2) * t**2 * x
assert R.dmp_factor_list(f) == \
(QQ(1, 2), [(t, 1),
(x, 1),
(x + t, 1)])
R, x, y = ring("x,y", FF(2))
raises(NotImplementedError, lambda: R.dmp_factor_list(x**2 + y**2))
R, x, y = ring("x,y", EX)
raises(DomainError, lambda: R.dmp_factor_list(EX(sin(1))))
def test_dup_factor_list():
R, x = ring("x", ZZ)
assert R.dup_factor_list(0) == (0, [])
assert R.dup_factor_list(7) == (7, [])
R, x = ring("x", QQ)
assert R.dup_factor_list(0) == (0, [])
assert R.dup_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, x = ring("x", ZZ['t'])
assert R.dup_factor_list(0) == (0, [])
assert R.dup_factor_list(7) == (7, [])
R, x = ring("x", QQ['t'])
assert R.dup_factor_list(0) == (0, [])
assert R.dup_factor_list(QQ(1, 7)) == (QQ(1, 7), [])
R, x = ring("x", ZZ)
assert R.dup_factor_list_include(0) == [(0, 1)]
assert R.dup_factor_list_include(7) == [(7, 1)]
assert R.dup_factor_list(x**2 + 2 * x + 1) == (1, [(x + 1, 2)])
assert R.dup_factor_list_include(x**2 + 2 * x + 1) == [(x + 1, 2)]
R, x = ring("x", QQ)
assert R.dup_factor_list(QQ(1, 2) * x**2 + x + QQ(1, 2)) == (QQ(1, 2),
[(x + 1, 2)])
R, x = ring("x", FF(2))
assert R.dup_factor_list(x**2 + 1) == (1, [(x + 1, 2)])
R, x = ring("x", RR)
assert R.dup_factor_list(1.0 * x**2 + 2.0 * x + 1.0) == (1.0, [
(1.0 * x + 1.0, 2)
])
assert R.dup_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.dup_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
Rt, t = ring("t", ZZ)
R, x = ring("x", Rt)
f = 4 * t * x**2 + 4 * t**2 * x
assert R.dup_factor_list(f) == \
(4, [(t, 1),
(x, 1),
(x + t, 1)])
Rt, t = ring("t", QQ)
R, x = ring("x", Rt)
f = QQ(1, 2) * t * x**2 + QQ(1, 2) * t**2 * x
assert R.dup_factor_list(f) == \
(QQ(1, 2), [(t, 1),
(x, 1),
(x + t, 1)])
R, x = ring("x", QQ.algebraic_field(I))
def anp(element):
return ANP(element, [QQ(1), QQ(0), QQ(1)], QQ)
f = anp([QQ(1, 1)]) * x**4 + anp([QQ(2, 1)]) * x**2
assert R.dup_factor_list(f) == \
(anp([QQ(1, 1)]), [(anp([QQ(1, 1)])*x, 2),
(anp([QQ(1, 1)])*x**2 + anp([])*x + anp([QQ(2, 1)]), 1)])
R, x = ring("x", EX)
raises(DomainError, lambda: R.dup_factor_list(EX(sin(1))))
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([S(1), S(2), S(3)]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
assert construct_domain([S(1), S(2), S(3)],
field=True) == (QQ, [QQ(1), QQ(2),
QQ(3)])
assert construct_domain([S(1) / 2, S(2)]) == (QQ, [QQ(1, 2), QQ(2)])
assert construct_domain([3.14, 1,
S(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([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, S(1)/2, sqrt(2)], extension=True) == \
(alg, [alg.convert(7), alg.convert(S(1)/2), alg.convert(sqrt(2))])
alg = QQ.algebraic_field(sqrt(2) + sqrt(3))
assert construct_domain([7, sqrt(2), sqrt(3)], extension=True) == \
(alg, [alg.convert(7), alg.convert(sqrt(2)), alg.convert(sqrt(3))])
dom = ZZ[x]
assert construct_domain([2*x, 3]) == \
(dom, [dom.convert(2*x), dom.convert(3)])
dom = ZZ[x, y]
assert construct_domain([2*x, 3*y]) == \
(dom, [dom.convert(2*x), dom.convert(3*y)])
dom = QQ[x]
assert construct_domain([x/2, 3]) == \
(dom, [dom.convert(x/2), dom.convert(3)])
dom = QQ[x, y]
assert construct_domain([x/2, 3*y]) == \
(dom, [dom.convert(x/2), dom.convert(3*y)])
dom = RR[x]
assert construct_domain([x/2, 3.5]) == \
(dom, [dom.convert(x/2), dom.convert(3.5)])
dom = RR[x, y]
assert construct_domain([x/2, 3.5*y]) == \
(dom, [dom.convert(x/2), dom.convert(3.5*y)])
dom = ZZ.frac_field(x)
assert construct_domain([2/x, 3]) == \
(dom, [dom.convert(2/x), dom.convert(3)])
dom = ZZ.frac_field(x, y)
assert construct_domain([2/x, 3*y]) == \
(dom, [dom.convert(2/x), dom.convert(3*y)])
dom = RR.frac_field(x)
assert construct_domain([2/x, 3.5]) == \
(dom, [dom.convert(2/x), dom.convert(3.5)])
dom = RR.frac_field(x, y)
assert construct_domain([2/x, 3.5*y]) == \
(dom, [dom.convert(2/x), dom.convert(3.5*y)])
assert construct_domain(2) == (ZZ, ZZ(2))
assert construct_domain(S(2) / 3) == (QQ, QQ(2, 3))
assert construct_domain({}) == (ZZ, {})
def test_PolyElement___mul__():
Rt, t = ring("t", ZZ)
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert dict(u * x) == dict(x * u) == {(1, 0, 0): u}
assert dict(2 * u * x + z) == dict(x * 2 * u + z) == {
(1, 0, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(u * 2 * x + z) == dict(2 * x * u + z) == {
(1, 0, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(2 * u * x + z) == dict(x * 2 * u + z) == {
(1, 0, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(u * x * 2 + z) == dict(x * u * 2 + z) == {
(1, 0, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(2 * u * x * y + z) == dict(x * y * 2 * u + z) == {
(1, 1, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(u * 2 * x * y + z) == dict(2 * x * y * u + z) == {
(1, 1, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(2 * u * x * y + z) == dict(x * y * 2 * u + z) == {
(1, 1, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(u * x * y * 2 + z) == dict(x * y * u * 2 + z) == {
(1, 1, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(2 * u * y * x + z) == dict(y * x * 2 * u + z) == {
(1, 1, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(u * 2 * y * x + z) == dict(2 * y * x * u + z) == {
(1, 1, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(2 * u * y * x + z) == dict(y * x * 2 * u + z) == {
(1, 1, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(u * y * x * 2 + z) == dict(y * x * u * 2 + z) == {
(1, 1, 0): 2 * u,
(0, 0, 1): 1
}
assert dict(3 * u * (x + y) + z) == dict((x + y) * 3 * u + z) == {
(1, 0, 0): 3 * u,
(0, 1, 0): 3 * u,
(0, 0, 1): 1
}
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)
assert dict(u * x) == dict(x * u) == {(1, 0, 0): u}
Rxyz, x, y, z = ring("x,y,z", EX)
assert dict(EX(pi) * x * y * z) == dict(x * y * z * EX(pi)) == {
(1, 1, 1): EX(pi)
}
def test_Domain__unify():
assert ZZ.unify(ZZ) == ZZ
assert QQ.unify(QQ) == QQ
assert ZZ.unify(QQ) == QQ
assert QQ.unify(ZZ) == QQ
assert EX.unify(EX) == EX
assert ZZ.unify(EX) == EX
assert QQ.unify(EX) == EX
assert EX.unify(ZZ) == EX
assert EX.unify(QQ) == EX
assert ZZ.poly_ring('x').unify(EX) == EX
assert ZZ.frac_field('x').unify(EX) == EX
assert EX.unify(ZZ.poly_ring('x')) == EX
assert EX.unify(ZZ.frac_field('x')) == EX
assert ZZ.poly_ring('x','y').unify(EX) == EX
assert ZZ.frac_field('x','y').unify(EX) == EX
assert EX.unify(ZZ.poly_ring('x','y')) == EX
assert EX.unify(ZZ.frac_field('x','y')) == EX
assert QQ.poly_ring('x').unify(EX) == EX
assert QQ.frac_field('x').unify(EX) == EX
assert EX.unify(QQ.poly_ring('x')) == EX
assert EX.unify(QQ.frac_field('x')) == EX
assert QQ.poly_ring('x','y').unify(EX) == EX
assert QQ.frac_field('x','y').unify(EX) == EX
assert EX.unify(QQ.poly_ring('x','y')) == EX
assert EX.unify(QQ.frac_field('x','y')) == EX
assert ZZ.poly_ring('x').unify(ZZ) == ZZ.poly_ring('x')
assert ZZ.poly_ring('x').unify(QQ) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(ZZ) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(QQ) == QQ.poly_ring('x')
assert ZZ.unify(ZZ.poly_ring('x')) == ZZ.poly_ring('x')
assert QQ.unify(ZZ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.poly_ring('x','y').unify(ZZ) == ZZ.poly_ring('x','y')
assert ZZ.poly_ring('x','y').unify(QQ) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x','y').unify(ZZ) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x','y').unify(QQ) == QQ.poly_ring('x','y')
assert ZZ.unify(ZZ.poly_ring('x','y')) == ZZ.poly_ring('x','y')
assert QQ.unify(ZZ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert ZZ.unify(QQ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert QQ.unify(QQ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert ZZ.frac_field('x').unify(ZZ) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(QQ) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(ZZ) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ) == QQ.frac_field('x')
assert ZZ.unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert QQ.unify(ZZ.frac_field('x')) == EX # QQ.frac_field('x')
assert ZZ.unify(QQ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x','y').unify(ZZ) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(QQ) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(ZZ) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(QQ) == QQ.frac_field('x','y')
assert ZZ.unify(ZZ.frac_field('x','y')) == ZZ.frac_field('x','y')
assert QQ.unify(ZZ.frac_field('x','y')) == EX # QQ.frac_field('x','y')
assert ZZ.unify(QQ.frac_field('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.unify(QQ.frac_field('x','y')) == QQ.frac_field('x','y')
assert ZZ.poly_ring('x').unify(ZZ.poly_ring('x')) == ZZ.poly_ring('x')
assert ZZ.poly_ring('x').unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(ZZ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.poly_ring('x','y').unify(ZZ.poly_ring('x')) == ZZ.poly_ring('x','y')
assert ZZ.poly_ring('x','y').unify(QQ.poly_ring('x')) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x','y').unify(ZZ.poly_ring('x')) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x','y').unify(QQ.poly_ring('x')) == QQ.poly_ring('x','y')
assert ZZ.poly_ring('x').unify(ZZ.poly_ring('x','y')) == ZZ.poly_ring('x','y')
assert ZZ.poly_ring('x').unify(QQ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x').unify(ZZ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x').unify(QQ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert ZZ.poly_ring('x','y').unify(ZZ.poly_ring('x','z')) == ZZ.poly_ring('x','y','z')
assert ZZ.poly_ring('x','y').unify(QQ.poly_ring('x','z')) == QQ.poly_ring('x','y','z')
assert QQ.poly_ring('x','y').unify(ZZ.poly_ring('x','z')) == QQ.poly_ring('x','y','z')
assert QQ.poly_ring('x','y').unify(QQ.poly_ring('x','z')) == QQ.poly_ring('x','y','z')
assert ZZ.frac_field('x').unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert QQ.frac_field('x').unify(ZZ.frac_field('x')) == QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x','y').unify(ZZ.frac_field('x')) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(QQ.frac_field('x')) == QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(ZZ.frac_field('x')) == QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(QQ.frac_field('x')) == QQ.frac_field('x','y')
assert ZZ.frac_field('x').unify(ZZ.frac_field('x','y')) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x').unify(QQ.frac_field('x','y')) == QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(ZZ.frac_field('x','y')) == QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(QQ.frac_field('x','y')) == QQ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(ZZ.frac_field('x','z')) == ZZ.frac_field('x','y','z')
assert ZZ.frac_field('x','y').unify(QQ.frac_field('x','z')) == QQ.frac_field('x','y','z')
assert QQ.frac_field('x','y').unify(ZZ.frac_field('x','z')) == QQ.frac_field('x','y','z')
assert QQ.frac_field('x','y').unify(QQ.frac_field('x','z')) == QQ.frac_field('x','y','z')
assert ZZ.poly_ring('x').unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert ZZ.poly_ring('x').unify(QQ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.poly_ring('x').unify(ZZ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.poly_ring('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.poly_ring('x','y').unify(ZZ.frac_field('x')) == ZZ.frac_field('x','y')
assert ZZ.poly_ring('x','y').unify(QQ.frac_field('x')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x','y').unify(ZZ.frac_field('x')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x','y').unify(QQ.frac_field('x')) == QQ.frac_field('x','y')
assert ZZ.poly_ring('x').unify(ZZ.frac_field('x','y')) == ZZ.frac_field('x','y')
assert ZZ.poly_ring('x').unify(QQ.frac_field('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x').unify(ZZ.frac_field('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x').unify(QQ.frac_field('x','y')) == QQ.frac_field('x','y')
assert ZZ.poly_ring('x','y').unify(ZZ.frac_field('x','z')) == ZZ.frac_field('x','y','z')
assert ZZ.poly_ring('x','y').unify(QQ.frac_field('x','z')) == EX # QQ.frac_field('x','y','z')
assert QQ.poly_ring('x','y').unify(ZZ.frac_field('x','z')) == EX # QQ.frac_field('x','y','z')
assert QQ.poly_ring('x','y').unify(QQ.frac_field('x','z')) == QQ.frac_field('x','y','z')
assert ZZ.frac_field('x').unify(ZZ.poly_ring('x')) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(QQ.poly_ring('x')) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(ZZ.poly_ring('x')) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ.poly_ring('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x','y').unify(ZZ.poly_ring('x')) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(QQ.poly_ring('x')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(ZZ.poly_ring('x')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(QQ.poly_ring('x')) == QQ.frac_field('x','y')
assert ZZ.frac_field('x').unify(ZZ.poly_ring('x','y')) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x').unify(QQ.poly_ring('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(ZZ.poly_ring('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(QQ.poly_ring('x','y')) == QQ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(ZZ.poly_ring('x','z')) == ZZ.frac_field('x','y','z')
assert ZZ.frac_field('x','y').unify(QQ.poly_ring('x','z')) == EX # QQ.frac_field('x','y','z')
assert QQ.frac_field('x','y').unify(ZZ.poly_ring('x','z')) == EX # QQ.frac_field('x','y','z')
assert QQ.frac_field('x','y').unify(QQ.poly_ring('x','z')) == QQ.frac_field('x','y','z')
raises(UnificationFailed, "ZZ.poly_ring('x','y').unify(ZZ, gens=('y', 'z'))")
raises(UnificationFailed, "ZZ.unify(ZZ.poly_ring('x','y'), gens=('y', 'z'))")
def test_dmp_factor_list():
assert dmp_factor_list([[]], 1, ZZ) == (ZZ(0), [])
assert dmp_factor_list([[]], 1, QQ) == (QQ(0), [])
assert dmp_factor_list([[]], 1, ZZ['y']) == (DMP([], ZZ), [])
assert dmp_factor_list([[]], 1, QQ['y']) == (DMP([], QQ), [])
assert dmp_factor_list_include([[]], 1, ZZ) == [([[]], 1)]
assert dmp_factor_list([[ZZ(7)]], 1, ZZ) == (ZZ(7), [])
assert dmp_factor_list([[QQ(1, 7)]], 1, QQ) == (QQ(1, 7), [])
assert dmp_factor_list([[DMP([ZZ(7)], ZZ)]], 1, ZZ['y']) == (DMP([ZZ(7)],
ZZ), [])
assert dmp_factor_list([[DMP([QQ(1, 7)], QQ)]], 1,
QQ['y']) == (DMP([QQ(1, 7)], QQ), [])
assert dmp_factor_list_include([[ZZ(7)]], 1, ZZ) == [([[ZZ(7)]], 1)]
f, g = [ZZ(1), ZZ(2), ZZ(1)], [ZZ(1), ZZ(1)]
assert dmp_factor_list(dmp_nest(f, 200, ZZ), 200, ZZ) == \
(ZZ(1), [(dmp_nest(g, 200, ZZ), 2)])
assert dmp_factor_list(dmp_raise(f, 200, 0, ZZ), 200, ZZ) == \
(ZZ(1), [(dmp_raise(g, 200, 0, ZZ), 2)])
assert dmp_factor_list([ZZ(1),ZZ(2),ZZ(1)], 0, ZZ) == \
(ZZ(1), [([ZZ(1), ZZ(1)], 2)])
assert dmp_factor_list([QQ(1,2),QQ(1),QQ(1,2)], 0, QQ) == \
(QQ(1,2), [([QQ(1),QQ(1)], 2)])
assert dmp_factor_list([[ZZ(1)],[ZZ(2)],[ZZ(1)]], 1, ZZ) == \
(ZZ(1), [([[ZZ(1)], [ZZ(1)]], 2)])
assert dmp_factor_list([[QQ(1,2)],[QQ(1)],[QQ(1,2)]], 1, QQ) == \
(QQ(1,2), [([[QQ(1)],[QQ(1)]], 2)])
f = [[ZZ(4), ZZ(0)], [ZZ(4), ZZ(0), ZZ(0)], []]
assert dmp_factor_list(f, 1, ZZ) == \
(ZZ(4), [([[ZZ(1),ZZ(0)]], 1),
([[ZZ(1)],[]], 1),
([[ZZ(1)],[ZZ(1),ZZ(0)]], 1)])
assert dmp_factor_list_include(f, 1, ZZ) == \
[([[ZZ(4),ZZ(0)]], 1),
([[ZZ(1)],[]], 1),
([[ZZ(1)],[ZZ(1),ZZ(0)]], 1)]
f = [[QQ(1, 2), QQ(0)], [QQ(1, 2), QQ(0), QQ(0)], []]
assert dmp_factor_list(f, 1, QQ) == \
(QQ(1,2), [([[QQ(1),QQ(0)]], 1),
([[QQ(1)],[]], 1),
([[QQ(1)],[QQ(1),QQ(0)]], 1)])
f = [[RR(2.0)], [], [-RR(8.0), RR(0.0), RR(0.0)]]
assert dmp_factor_list(f, 1, RR) == \
(RR(2.0), [([[RR(1.0)],[-RR(2.0),RR(0.0)]], 1),
([[RR(1.0)],[ RR(2.0),RR(0.0)]], 1)])
f = [[DMP([ZZ(4), ZZ(0)], ZZ)], [DMP([ZZ(4), ZZ(0), ZZ(0)], ZZ)],
[DMP([], ZZ)]]
assert dmp_factor_list(f, 1, ZZ['y']) == \
(DMP([ZZ(4)],ZZ), [([[DMP([ZZ(1),ZZ(0)],ZZ)]], 1),
([[DMP([ZZ(1)],ZZ)],[]], 1),
([[DMP([ZZ(1)],ZZ)],[DMP([ZZ(1),ZZ(0)],ZZ)]], 1)])
f = [[DMP([QQ(1, 2), QQ(0)], ZZ)],
[DMP([QQ(1, 2), QQ(0), QQ(0)], ZZ)], [DMP([], ZZ)]]
assert dmp_factor_list(f, 1, QQ['y']) == \
(DMP([QQ(1,2)],QQ), [([[DMP([QQ(1),QQ(0)],QQ)]], 1),
([[DMP([QQ(1)],QQ)],[]], 1),
([[DMP([QQ(1)],QQ)],[DMP([QQ(1),QQ(0)],QQ)]], 1)])
K = FF(2)
raises(DomainError,
lambda: dmp_factor_list([[K(1)], [], [K(1), K(0), K(0)]], 1, K))
raises(DomainError, lambda: dmp_factor_list([[EX(sin(1))]], 1, EX))
def test_sin():
R, x, y = ring('x, y', QQ)
assert rs_sin(x, x, 9) == \
x - x**3/6 + x**5/120 - x**7/5040
assert rs_sin(x*y + x**2*y**3, x, 9) == x**8*y**11/12 - \
x**8*y**9/720 + x**7*y**9/12 - x**7*y**7/5040 - x**6*y**9/6 + \
x**6*y**7/24 - x**5*y**7/2 + x**5*y**5/120 - x**4*y**5/2 - \
x**3*y**3/6 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
assert rs_sin(x + a, x, 5) == sin(a)*x**4/24 - cos(a)*x**3/6 - \
sin(a)*x**2/2 + cos(a)*x + sin(a)
assert rs_sin(x + x**2*y + a, x, 5) == -sin(a)*x**4*y**2/2 - \
cos(a)*x**4*y/2 + sin(a)*x**4/24 - sin(a)*x**3*y - cos(a)*x**3/6 + \
cos(a)*x**2*y - sin(a)*x**2/2 + cos(a)*x + sin(a)
R, x, y = ring('x, y', EX)
assert rs_sin(x + a, x, 5) == EX(sin(a)/24)*x**4 - EX(cos(a)/6)*x**3 - \
EX(sin(a)/2)*x**2 + EX(cos(a))*x + EX(sin(a))
assert rs_sin(x + x**2*y + a, x, 5) == -EX(sin(a)/2)*x**4*y**2 - \
EX(cos(a)/2)*x**4*y + EX(sin(a)/24)*x**4 - EX(sin(a))*x**3*y - \
EX(cos(a)/6)*x**3 + EX(cos(a))*x**2*y - EX(sin(a)/2)*x**2 + \
EX(cos(a))*x + EX(sin(a))
def test_tan():
R, x, y = ring('x, y', QQ)
assert rs_tan(x, x, 9) == \
x + x**3/3 + 2*x**5/15 + 17*x**7/315
assert rs_tan(x*y + x**2*y**3, x, 9) == 4*x**8*y**11/3 + 17*x**8*y**9/45 + \
4*x**7*y**9/3 + 17*x**7*y**7/315 + x**6*y**9/3 + 2*x**6*y**7/3 + \
x**5*y**7 + 2*x**5*y**5/15 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[tan(a), a])
assert rs_tan(x + a, x, 5) == (tan(a)**5 + 5*tan(a)**3/3 + \
2*tan(a)/3)*x**4 + (tan(a)**4 + 4*tan(a)**2/3 + 1/3)*x**3 + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
assert rs_tan(x + x**2*y + a, x, 4) == (2*tan(a)**3 + 2*tan(a))*x**3*y + \
(tan(a)**4 + 4/3*tan(a)**2 + 1/3)*x**3 + (tan(a)**2 + 1)*x**2*y + \
(tan(a)**3 + tan(a))*x**2 + (tan(a)**2 + 1)*x + tan(a)
R, x, y = ring('x, y', EX)
assert rs_tan(x + a, x, 5) == EX(tan(a)**5 + 5*tan(a)**3/3 + \
2*tan(a)/3)*x**4 + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**3 + tan(a))*x**2 + EX(tan(a)**2 + 1)*x + EX(tan(a))
assert rs_tan(x + x**2*y + a, x, 4) == EX(2*tan(a)**3 + \
2*tan(a))*x**3*y + EX(tan(a)**4 + 4*tan(a)**2/3 + EX(1)/3)*x**3 + \
EX(tan(a)**2 + 1)*x**2*y + EX(tan(a)**3 + tan(a))*x**2 + \
EX(tan(a)**2 + 1)*x + EX(tan(a))
p = x + x**2 + 5
assert rs_atan(p, x, 10).compose(x, 10) == EX(atan(5) + 67701870330562640/ \
668083460499)
def test_atan():
R, x, y = ring('x, y', QQ)
assert rs_atan(x, x, 9) == -x**7/7 + x**5/5 - x**3/3 + x
assert rs_atan(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 - x**8*y**9 + \
2*x**7*y**9 - x**7*y**7/7 - x**6*y**9/3 + x**6*y**7 - x**5*y**7 + \
x**5*y**5/5 - x**4*y**5 - x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_atan(x + a, x, 5) == -EX((a**3 - a)/(a**8 + 4*a**6 + 6*a**4 + \
4*a**2 + 1))*x**4 + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + \
9*a**2 + 3))*x**3 - EX(a/(a**4 + 2*a**2 + 1))*x**2 + \
EX(1/(a**2 + 1))*x + EX(atan(a))
assert rs_atan(x + x**2*y + a, x, 4) == -EX(2*a/(a**4 + 2*a**2 + 1)) \
*x**3*y + EX((3*a**2 - 1)/(3*a**6 + 9*a**4 + 9*a**2 + 3))*x**3 + \
EX(1/(a**2 + 1))*x**2*y - EX(a/(a**4 + 2*a**2 + 1))*x**2 + EX(1/(a**2 \
+ 1))*x + EX(atan(a))
def test_nth_root():
R, x, y = ring('x, y', QQ)
r1 = rs_nth_root(1 + x**2*y, 4, x, 10)
assert rs_nth_root(1 + x**2*y, 4, x, 10) == -77*x**8*y**4/2048 + \
7*x**6*y**3/128 - 3*x**4*y**2/32 + x**2*y/4 + 1
assert rs_nth_root(1 + x*y + x**2*y**3, 3, x, 5) == -x**4*y**6/9 + \
5*x**4*y**5/27 - 10*x**4*y**4/243 - 2*x**3*y**4/9 + 5*x**3*y**3/81 + \
x**2*y**3/3 - x**2*y**2/9 + x*y/3 + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_nth_root(x + a, 3, x, 4) == EX(5/(81*a**QQ(8, 3)))*x**3 - \
EX(1/(9*a**QQ(5, 3)))*x**2 + EX(1/(3*a**QQ(2, 3)))*x + EX(a**QQ(1, 3))
assert rs_nth_root(x**QQ(2, 3) + x**2*y + 5, 2, x, 3) == -EX(sqrt(5)/100)*\
x**QQ(8, 3)*y - EX(sqrt(5)/16000)*x**QQ(8, 3) + EX(sqrt(5)/10)*x**2*y + \
EX(sqrt(5)/2000)*x**2 - EX(sqrt(5)/200)*x**QQ(4, 3) + \
EX(sqrt(5)/10)*x**QQ(2, 3) + EX(sqrt(5))
def test_exp():
R, x = ring('x', QQ)
p = x + x**4
for h in [10, 30]:
q = rs_series_inversion(1 + p, x, h) - 1
p1 = rs_exp(q, x, h)
q1 = rs_log(p1, x, h)
assert q1 == q
p1 = rs_exp(p, x, 30)
assert p1.coeff(x**29) == QQ(74274246775059676726972369, 353670479749588078181744640000)
prec = 21
p = rs_log(1 + x, x, prec)
p1 = rs_exp(p, x, prec)
assert p1 == x + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[exp(a), a])
assert rs_exp(x + a, x, 5) == exp(a)*x**4/24 + exp(a)*x**3/6 + \
exp(a)*x**2/2 + exp(a)*x + exp(a)
assert rs_exp(x + x**2*y + a, x, 5) == exp(a)*x**4*y**2/2 + \
exp(a)*x**4*y/2 + exp(a)*x**4/24 + exp(a)*x**3*y + \
exp(a)*x**3/6 + exp(a)*x**2*y + exp(a)*x**2/2 + exp(a)*x + exp(a)
R, x, y = ring('x, y', EX)
assert rs_exp(x + a, x, 5) == EX(exp(a)/24)*x**4 + EX(exp(a)/6)*x**3 + \
EX(exp(a)/2)*x**2 + EX(exp(a))*x + EX(exp(a))
assert rs_exp(x + x**2*y + a, x, 5) == EX(exp(a)/2)*x**4*y**2 + \
EX(exp(a)/2)*x**4*y + EX(exp(a)/24)*x**4 + EX(exp(a))*x**3*y + \
EX(exp(a)/6)*x**3 + EX(exp(a))*x**2*y + EX(exp(a)/2)*x**2 + \
EX(exp(a))*x + EX(exp(a))
def test_log():
R, x = ring('x', QQ)
p = 1 + x
p1 = rs_log(p, x, 4)
assert p1 == x - x**2/2 + x**3/3
p = 1 + x +2*x**2/3
p1 = rs_log(p, x, 9)
assert p1 == -17*x**8/648 + 13*x**7/189 - 11*x**6/162 - x**5/45 + \
7*x**4/36 - x**3/3 + x**2/6 + x
p2 = rs_series_inversion(p, x, 9)
p3 = rs_log(p2, x, 9)
assert p3 == -p1
R, x, y = ring('x, y', QQ)
p = 1 + x + 2*y*x**2
p1 = rs_log(p, x, 6)
assert p1 == (4*x**5*y**2 - 2*x**5*y - 2*x**4*y**2 + x**5/5 + 2*x**4*y -
x**4/4 - 2*x**3*y + x**3/3 + 2*x**2*y - x**2/2 + x)
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_log(x + a, x, 5) == -EX(1/(4*a**4))*x**4 + EX(1/(3*a**3))*x**3 \
- EX(1/(2*a**2))*x**2 + EX(1/a)*x + EX(log(a))
assert rs_log(x + x**2*y + a, x, 4) == -EX(a**(-2))*x**3*y + \
EX(1/(3*a**3))*x**3 + EX(1/a)*x**2*y - EX(1/(2*a**2))*x**2 + \
EX(1/a)*x + EX(log(a))
p = x + x**2 + 3
assert rs_log(p, x, 10).compose(x, 5) == EX(log(3) + 19281291595/9920232)
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([S.One, S(2), S(3)]) == (ZZ, [ZZ(1), ZZ(2), ZZ(3)])
assert construct_domain([S.One, S(2), S(3)],
field=True) == (QQ, [QQ(1), QQ(2),
QQ(3)])
assert construct_domain([S.Half, S(2)]) == (QQ, [QQ(1, 2), QQ(2)])
result = construct_domain([3.14, 1, S.Half])
assert isinstance(result[0], RealField)
assert result[1] == [RR(3.14), RR(1.0), RR(0.5)]
result = construct_domain([3.14, I, S.Half])
assert isinstance(result[0], ComplexField)
assert result[1] == [CC(3.14), CC(1.0j), CC(0.5)]
assert construct_domain([1, I]) == (ZZ_I, [ZZ_I(1, 0), ZZ_I(0, 1)])
assert construct_domain([1,
I / 2]) == (QQ_I, [QQ_I(1, 0),
QQ_I(0, S.Half)])
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([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, S.Half, sqrt(2)], extension=True) == \
(alg, [alg.convert(7), alg.convert(S.Half), alg.convert(sqrt(2))])
alg = QQ.algebraic_field(sqrt(2) + sqrt(3))
assert construct_domain([7, sqrt(2), sqrt(3)], extension=True) == \
(alg, [alg.convert(7), alg.convert(sqrt(2)), alg.convert(sqrt(3))])
dom = ZZ[x]
assert construct_domain([2*x, 3]) == \
(dom, [dom.convert(2*x), dom.convert(3)])
dom = ZZ[x, y]
assert construct_domain([2*x, 3*y]) == \
(dom, [dom.convert(2*x), dom.convert(3*y)])
dom = QQ[x]
assert construct_domain([x/2, 3]) == \
(dom, [dom.convert(x/2), dom.convert(3)])
dom = QQ[x, y]
assert construct_domain([x/2, 3*y]) == \
(dom, [dom.convert(x/2), dom.convert(3*y)])
dom = ZZ_I[x]
assert construct_domain([2*x, I]) == \
(dom, [dom.convert(2*x), dom.convert(I)])
dom = ZZ_I[x, y]
assert construct_domain([2*x, I*y]) == \
(dom, [dom.convert(2*x), dom.convert(I*y)])
dom = QQ_I[x]
assert construct_domain([x/2, I]) == \
(dom, [dom.convert(x/2), dom.convert(I)])
dom = QQ_I[x, y]
assert construct_domain([x/2, I*y]) == \
(dom, [dom.convert(x/2), dom.convert(I*y)])
dom = RR[x]
assert construct_domain([x/2, 3.5]) == \
(dom, [dom.convert(x/2), dom.convert(3.5)])
dom = RR[x, y]
assert construct_domain([x/2, 3.5*y]) == \
(dom, [dom.convert(x/2), dom.convert(3.5*y)])
dom = CC[x]
assert construct_domain([I*x/2, 3.5]) == \
(dom, [dom.convert(I*x/2), dom.convert(3.5)])
dom = CC[x, y]
assert construct_domain([I*x/2, 3.5*y]) == \
(dom, [dom.convert(I*x/2), dom.convert(3.5*y)])
dom = CC[x]
assert construct_domain([x/2, I*3.5]) == \
(dom, [dom.convert(x/2), dom.convert(I*3.5)])
dom = CC[x, y]
assert construct_domain([x/2, I*3.5*y]) == \
(dom, [dom.convert(x/2), dom.convert(I*3.5*y)])
dom = ZZ.frac_field(x)
assert construct_domain([2/x, 3]) == \
(dom, [dom.convert(2/x), dom.convert(3)])
dom = ZZ.frac_field(x, y)
assert construct_domain([2/x, 3*y]) == \
(dom, [dom.convert(2/x), dom.convert(3*y)])
dom = RR.frac_field(x)
assert construct_domain([2/x, 3.5]) == \
(dom, [dom.convert(2/x), dom.convert(3.5)])
dom = RR.frac_field(x, y)
assert construct_domain([2/x, 3.5*y]) == \
(dom, [dom.convert(2/x), dom.convert(3.5*y)])
dom = RealField(prec=336)[x]
assert construct_domain([pi.evalf(100)*x]) == \
(dom, [dom.convert(pi.evalf(100)*x)])
assert construct_domain(2) == (ZZ, ZZ(2))
assert construct_domain(S(2) / 3) == (QQ, QQ(2, 3))
assert construct_domain(Rational(2, 3)) == (QQ, QQ(2, 3))
assert construct_domain({}) == (ZZ, {})
def test_Domain__unify():
assert ZZ.unify(ZZ) == ZZ
assert QQ.unify(QQ) == QQ
assert ZZ.unify(QQ) == QQ
assert QQ.unify(ZZ) == QQ
assert EX.unify(EX) == EX
assert ZZ.unify(EX) == EX
assert QQ.unify(EX) == EX
assert EX.unify(ZZ) == EX
assert EX.unify(QQ) == EX
assert ZZ.poly_ring(x).unify(EX) == EX
assert ZZ.frac_field(x).unify(EX) == EX
assert EX.unify(ZZ.poly_ring(x)) == EX
assert EX.unify(ZZ.frac_field(x)) == EX
assert ZZ.poly_ring(x, y).unify(EX) == EX
assert ZZ.frac_field(x, y).unify(EX) == EX
assert EX.unify(ZZ.poly_ring(x, y)) == EX
assert EX.unify(ZZ.frac_field(x, y)) == EX
assert QQ.poly_ring(x).unify(EX) == EX
assert QQ.frac_field(x).unify(EX) == EX
assert EX.unify(QQ.poly_ring(x)) == EX
assert EX.unify(QQ.frac_field(x)) == EX
assert QQ.poly_ring(x, y).unify(EX) == EX
assert QQ.frac_field(x, y).unify(EX) == EX
assert EX.unify(QQ.poly_ring(x, y)) == EX
assert EX.unify(QQ.frac_field(x, y)) == EX
assert ZZ.poly_ring(x).unify(ZZ) == ZZ.poly_ring(x)
assert ZZ.poly_ring(x).unify(QQ) == QQ.poly_ring(x)
assert QQ.poly_ring(x).unify(ZZ) == QQ.poly_ring(x)
assert QQ.poly_ring(x).unify(QQ) == QQ.poly_ring(x)
assert ZZ.unify(ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert QQ.unify(ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert ZZ.unify(QQ.poly_ring(x)) == QQ.poly_ring(x)
assert QQ.unify(QQ.poly_ring(x)) == QQ.poly_ring(x)
assert ZZ.poly_ring(x, y).unify(ZZ) == ZZ.poly_ring(x, y)
assert ZZ.poly_ring(x, y).unify(QQ) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x, y).unify(ZZ) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x, y).unify(QQ) == QQ.poly_ring(x, y)
assert ZZ.unify(ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert QQ.unify(ZZ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert ZZ.unify(QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert QQ.unify(QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert ZZ.frac_field(x).unify(ZZ) == ZZ.frac_field(x)
assert ZZ.frac_field(x).unify(QQ) == EX # QQ.frac_field(x)
assert QQ.frac_field(x).unify(ZZ) == EX # QQ.frac_field(x)
assert QQ.frac_field(x).unify(QQ) == QQ.frac_field(x)
assert ZZ.unify(ZZ.frac_field(x)) == ZZ.frac_field(x)
assert QQ.unify(ZZ.frac_field(x)) == EX # QQ.frac_field(x)
assert ZZ.unify(QQ.frac_field(x)) == EX # QQ.frac_field(x)
assert QQ.unify(QQ.frac_field(x)) == QQ.frac_field(x)
assert ZZ.frac_field(x, y).unify(ZZ) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x, y).unify(QQ) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x, y).unify(ZZ) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x, y).unify(QQ) == QQ.frac_field(x, y)
assert ZZ.unify(ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert QQ.unify(ZZ.frac_field(x, y)) == EX # QQ.frac_field(x,y)
assert ZZ.unify(QQ.frac_field(x, y)) == EX # QQ.frac_field(x,y)
assert QQ.unify(QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert ZZ.poly_ring(x).unify(ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert ZZ.poly_ring(x).unify(QQ.poly_ring(x)) == QQ.poly_ring(x)
assert QQ.poly_ring(x).unify(ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert QQ.poly_ring(x).unify(QQ.poly_ring(x)) == QQ.poly_ring(x)
assert ZZ.poly_ring(x, y).unify(ZZ.poly_ring(x)) == ZZ.poly_ring(x, y)
assert ZZ.poly_ring(x, y).unify(QQ.poly_ring(x)) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x, y).unify(ZZ.poly_ring(x)) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x, y).unify(QQ.poly_ring(x)) == QQ.poly_ring(x, y)
assert ZZ.poly_ring(x).unify(ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert ZZ.poly_ring(x).unify(QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x).unify(ZZ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x).unify(QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert ZZ.poly_ring(x, y).unify(ZZ.poly_ring(x, z)) == ZZ.poly_ring(x, y, z)
assert ZZ.poly_ring(x, y).unify(QQ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert QQ.poly_ring(x, y).unify(ZZ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert QQ.poly_ring(x, y).unify(QQ.poly_ring(x, z)) == QQ.poly_ring(x, y, z)
assert ZZ.frac_field(x).unify(ZZ.frac_field(x)) == ZZ.frac_field(x)
assert ZZ.frac_field(x).unify(QQ.frac_field(x)) == QQ.frac_field(x)
assert QQ.frac_field(x).unify(ZZ.frac_field(x)) == QQ.frac_field(x)
assert QQ.frac_field(x).unify(QQ.frac_field(x)) == QQ.frac_field(x)
assert ZZ.frac_field(x, y).unify(ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x, y).unify(QQ.frac_field(x)) == QQ.frac_field(x, y)
assert QQ.frac_field(x, y).unify(ZZ.frac_field(x)) == QQ.frac_field(x, y)
assert QQ.frac_field(x, y).unify(QQ.frac_field(x)) == QQ.frac_field(x, y)
assert ZZ.frac_field(x).unify(ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x).unify(QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert QQ.frac_field(x).unify(ZZ.frac_field(x, y)) == QQ.frac_field(x, y)
assert QQ.frac_field(x).unify(QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert ZZ.frac_field(x, y).unify(ZZ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert ZZ.frac_field(x, y).unify(QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert QQ.frac_field(x, y).unify(ZZ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert QQ.frac_field(x, y).unify(QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert ZZ.poly_ring(x).unify(ZZ.frac_field(x)) == ZZ.frac_field(x)
assert ZZ.poly_ring(x).unify(QQ.frac_field(x)) == EX # QQ.frac_field(x)
assert QQ.poly_ring(x).unify(ZZ.frac_field(x)) == EX # QQ.frac_field(x)
assert QQ.poly_ring(x).unify(QQ.frac_field(x)) == QQ.frac_field(x)
assert ZZ.poly_ring(x, y).unify(ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert ZZ.poly_ring(x, y).unify(QQ.frac_field(x)) == EX # QQ.frac_field(x,y)
assert QQ.poly_ring(x, y).unify(ZZ.frac_field(x)) == EX # QQ.frac_field(x,y)
assert QQ.poly_ring(x, y).unify(QQ.frac_field(x)) == QQ.frac_field(x, y)
assert ZZ.poly_ring(x).unify(ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert ZZ.poly_ring(x).unify(QQ.frac_field(x, y)) == EX # QQ.frac_field(x,y)
assert QQ.poly_ring(x).unify(ZZ.frac_field(x, y)) == EX # QQ.frac_field(x,y)
assert QQ.poly_ring(x).unify(QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert ZZ.poly_ring(x, y).unify(ZZ.frac_field(x, z)) == ZZ.frac_field(x, y, z)
assert ZZ.poly_ring(x, y).unify(QQ.frac_field(x, z)) == EX # QQ.frac_field(x,y,z)
assert QQ.poly_ring(x, y).unify(ZZ.frac_field(x, z)) == EX # QQ.frac_field(x,y,z)
assert QQ.poly_ring(x, y).unify(QQ.frac_field(x, z)) == QQ.frac_field(x, y, z)
assert ZZ.frac_field(x).unify(ZZ.poly_ring(x)) == ZZ.frac_field(x)
assert ZZ.frac_field(x).unify(QQ.poly_ring(x)) == EX # QQ.frac_field(x)
assert QQ.frac_field(x).unify(ZZ.poly_ring(x)) == EX # QQ.frac_field(x)
assert QQ.frac_field(x).unify(QQ.poly_ring(x)) == QQ.frac_field(x)
assert ZZ.frac_field(x, y).unify(ZZ.poly_ring(x)) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x, y).unify(QQ.poly_ring(x)) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x, y).unify(ZZ.poly_ring(x)) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x, y).unify(QQ.poly_ring(x)) == QQ.frac_field(x, y)
assert ZZ.frac_field(x).unify(ZZ.poly_ring(x, y)) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x).unify(QQ.poly_ring(x, y)) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x).unify(ZZ.poly_ring(x, y)) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x).unify(QQ.poly_ring(x, y)) == QQ.frac_field(x, y)
assert ZZ.frac_field(x, y).unify(ZZ.poly_ring(x, z)) == ZZ.frac_field(x, y, z)
assert ZZ.frac_field(x, y).unify(QQ.poly_ring(x, z)) == EX # QQ.frac_field(x,y,z)
assert QQ.frac_field(x, y).unify(ZZ.poly_ring(x, z)) == EX # QQ.frac_field(x,y,z)
assert QQ.frac_field(x, y).unify(QQ.poly_ring(x, z)) == QQ.frac_field(x, y, z)
alg = QQ.algebraic_field(sqrt(5))
assert alg.unify(alg[x, y]) == alg[x, y]
assert alg[x, y].unify(alg) == alg[x, y]
assert alg.unify(alg.frac_field(x, y)) == alg.frac_field(x, y)
assert alg.frac_field(x, y).unify(alg) == alg.frac_field(x, y)
ext = QQ.algebraic_field(sqrt(7))
raises(NotImplementedError, lambda: alg.unify(ext))
raises(UnificationFailed, lambda: ZZ.poly_ring(x, y).unify(ZZ, gens=(y, z)))
raises(UnificationFailed, lambda: ZZ.unify(ZZ.poly_ring(x, y), gens=(y, z)))
def test_dup_factor_list():
assert dup_factor_list([], ZZ) == (ZZ(0), [])
assert dup_factor_list([], QQ) == (QQ(0), [])
assert dup_factor_list([], ZZ['y']) == (DMP([], ZZ), [])
assert dup_factor_list([], QQ['y']) == (DMP([], QQ), [])
assert dup_factor_list_include([], ZZ) == [([], 1)]
assert dup_factor_list([ZZ(7)], ZZ) == (ZZ(7), [])
assert dup_factor_list([QQ(1, 7)], QQ) == (QQ(1, 7), [])
assert dup_factor_list([DMP([ZZ(7)], ZZ)], ZZ['y']) == (DMP([ZZ(7)],
ZZ), [])
assert dup_factor_list([DMP([QQ(1, 7)], QQ)], QQ['y']) == (DMP([QQ(1, 7)],
QQ), [])
assert dup_factor_list_include([ZZ(7)], ZZ) == [([ZZ(7)], 1)]
assert dup_factor_list([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
(ZZ(1), [([ZZ(1), ZZ(1)], 2)])
assert dup_factor_list([QQ(1,2),QQ(1),QQ(1,2)], QQ) == \
(QQ(1,2), [([QQ(1),QQ(1)], 2)])
assert dup_factor_list_include([ZZ(1),ZZ(2),ZZ(1)], ZZ) == \
[([ZZ(1), ZZ(1)], 2)]
K = FF(2)
assert dup_factor_list([K(1),K(0),K(1)], K) == \
(K(1), [([K(1), K(1)], 2)])
assert dup_factor_list([RR(1.0),RR(2.0),RR(1.0)], RR) == \
(RR(1.0), [([RR(1.0),RR(1.0)], 2)])
assert dup_factor_list([RR(2.0),RR(4.0),RR(2.0)], RR) == \
(RR(2.0), [([RR(1.0),RR(1.0)], 2)])
f = [DMP([ZZ(4), ZZ(0)], ZZ), DMP([ZZ(4), ZZ(0), ZZ(0)], ZZ), DMP([], ZZ)]
assert dup_factor_list(f, ZZ['y']) == \
(DMP([ZZ(4)],ZZ), [([DMP([ZZ(1),ZZ(0)],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([],ZZ)], 1),
([DMP([ZZ(1)],ZZ),DMP([ZZ(1),ZZ(0)],ZZ)], 1)])
f = [
DMP([QQ(1, 2), QQ(0)], ZZ),
DMP([QQ(1, 2), QQ(0), QQ(0)], ZZ),
DMP([], ZZ)
]
assert dup_factor_list(f, QQ['y']) == \
(DMP([QQ(1,2)],QQ), [([DMP([QQ(1),QQ(0)],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([],QQ)], 1),
([DMP([QQ(1)],QQ),DMP([QQ(1),QQ(0)],QQ)], 1)])
K = QQ.algebraic_field(I)
h = [QQ(1, 1), QQ(0, 1), QQ(1, 1)]
f = [
ANP([QQ(1, 1)], h, QQ),
ANP([], h, QQ),
ANP([QQ(2, 1)], h, QQ),
ANP([], h, QQ),
ANP([], h, QQ)
]
assert dup_factor_list(f, K) == \
(ANP([QQ(1,1)], h, QQ), [([ANP([QQ(1,1)], h, QQ), ANP([], h, QQ)], 2),
([ANP([QQ(1,1)], h, QQ), ANP([], h, QQ), ANP([QQ(2,1)], h, QQ)], 1)])
raises(DomainError, lambda: dup_factor_list([EX(sin(1))], EX))
def test_cos():
R, x, y = ring('x, y', QQ)
assert rs_cos(x, x, 9) == \
x**8/40320 - x**6/720 + x**4/24 - x**2/2 + 1
assert rs_cos(x*y + x**2*y**3, x, 9) == x**8*y**12/24 - \
x**8*y**10/48 + x**8*y**8/40320 + x**7*y**10/6 - \
x**7*y**8/120 + x**6*y**8/4 - x**6*y**6/720 + x**5*y**6/6 - \
x**4*y**6/2 + x**4*y**4/24 - x**3*y**4 - x**2*y**2/2 + 1
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', QQ[sin(a), cos(a), a])
assert rs_cos(x + a, x, 5) == cos(a)*x**4/24 + sin(a)*x**3/6 - \
cos(a)*x**2/2 - sin(a)*x + cos(a)
assert rs_cos(x + x**2*y + a, x, 5) == -cos(a)*x**4*y**2/2 + \
sin(a)*x**4*y/2 + cos(a)*x**4/24 - cos(a)*x**3*y + sin(a)*x**3/6 - \
sin(a)*x**2*y - cos(a)*x**2/2 - sin(a)*x + cos(a)
R, x, y = ring('x, y', EX)
assert rs_cos(x + a, x, 5) == EX(cos(a)/24)*x**4 + EX(sin(a)/6)*x**3 - \
EX(cos(a)/2)*x**2 - EX(sin(a))*x + EX(cos(a))
assert rs_cos(x + x**2*y + a, x, 5) == -EX(cos(a)/2)*x**4*y**2 + \
EX(sin(a)/2)*x**4*y + EX(cos(a)/24)*x**4 - EX(cos(a))*x**3*y + \
EX(sin(a)/6)*x**3 - EX(sin(a))*x**2*y - EX(cos(a)/2)*x**2 - \
EX(sin(a))*x + EX(cos(a))
def test_dup_clear_denoms():
assert dup_clear_denoms([], QQ, ZZ) == (ZZ(1), [])
assert dup_clear_denoms([QQ(1)], QQ, ZZ) == (ZZ(1), [QQ(1)])
assert dup_clear_denoms([QQ(7)], QQ, ZZ) == (ZZ(1), [QQ(7)])
assert dup_clear_denoms([QQ(7, 3)], QQ) == (ZZ(3), [QQ(7)])
assert dup_clear_denoms([QQ(7, 3)], QQ, ZZ) == (ZZ(3), [QQ(7)])
assert dup_clear_denoms([QQ(3), QQ(1), QQ(0)], QQ, ZZ) == (
ZZ(1),
[QQ(3), QQ(1), QQ(0)],
)
assert dup_clear_denoms([QQ(1), QQ(1, 2), QQ(0)], QQ, ZZ) == (
ZZ(2),
[QQ(2), QQ(1), QQ(0)],
)
assert dup_clear_denoms([QQ(3), QQ(1), QQ(0)], QQ, ZZ, convert=True) == (
ZZ(1),
[ZZ(3), ZZ(1), ZZ(0)],
)
assert dup_clear_denoms([QQ(1), QQ(1, 2), QQ(0)], QQ, ZZ,
convert=True) == (
ZZ(2),
[ZZ(2), ZZ(1), ZZ(0)],
)
assert dup_clear_denoms([EX(S(3) / 2), EX(S(9) / 4)],
EX) == (EX(4), [EX(6), EX(9)])
assert dup_clear_denoms([EX(7)], EX) == (EX(1), [EX(7)])
assert dup_clear_denoms([EX(sin(x) / x), EX(0)],
EX) == (EX(x), [EX(sin(x)), EX(0)])
def test_Domain__unify():
assert ZZ.unify(ZZ) == ZZ
assert QQ.unify(QQ) == QQ
assert ZZ.unify(QQ) == QQ
assert QQ.unify(ZZ) == QQ
assert EX.unify(EX) == EX
assert ZZ.unify(EX) == EX
assert QQ.unify(EX) == EX
assert EX.unify(ZZ) == EX
assert EX.unify(QQ) == EX
assert ZZ.poly_ring('x').unify(EX) == EX
assert ZZ.frac_field('x').unify(EX) == EX
assert EX.unify(ZZ.poly_ring('x')) == EX
assert EX.unify(ZZ.frac_field('x')) == EX
assert ZZ.poly_ring('x','y').unify(EX) == EX
assert ZZ.frac_field('x','y').unify(EX) == EX
assert EX.unify(ZZ.poly_ring('x','y')) == EX
assert EX.unify(ZZ.frac_field('x','y')) == EX
assert QQ.poly_ring('x').unify(EX) == EX
assert QQ.frac_field('x').unify(EX) == EX
assert EX.unify(QQ.poly_ring('x')) == EX
assert EX.unify(QQ.frac_field('x')) == EX
assert QQ.poly_ring('x','y').unify(EX) == EX
assert QQ.frac_field('x','y').unify(EX) == EX
assert EX.unify(QQ.poly_ring('x','y')) == EX
assert EX.unify(QQ.frac_field('x','y')) == EX
assert ZZ.poly_ring('x').unify(ZZ) == ZZ.poly_ring('x')
assert ZZ.poly_ring('x').unify(QQ) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(ZZ) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(QQ) == QQ.poly_ring('x')
assert ZZ.unify(ZZ.poly_ring('x')) == ZZ.poly_ring('x')
assert QQ.unify(ZZ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.poly_ring('x','y').unify(ZZ) == ZZ.poly_ring('x','y')
assert ZZ.poly_ring('x','y').unify(QQ) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x','y').unify(ZZ) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x','y').unify(QQ) == QQ.poly_ring('x','y')
assert ZZ.unify(ZZ.poly_ring('x','y')) == ZZ.poly_ring('x','y')
assert QQ.unify(ZZ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert ZZ.unify(QQ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert QQ.unify(QQ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert ZZ.frac_field('x').unify(ZZ) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(QQ) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(ZZ) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ) == QQ.frac_field('x')
assert ZZ.unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert QQ.unify(ZZ.frac_field('x')) == EX # QQ.frac_field('x')
assert ZZ.unify(QQ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x','y').unify(ZZ) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(QQ) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(ZZ) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(QQ) == QQ.frac_field('x','y')
assert ZZ.unify(ZZ.frac_field('x','y')) == ZZ.frac_field('x','y')
assert QQ.unify(ZZ.frac_field('x','y')) == EX # QQ.frac_field('x','y')
assert ZZ.unify(QQ.frac_field('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.unify(QQ.frac_field('x','y')) == QQ.frac_field('x','y')
assert ZZ.poly_ring('x').unify(ZZ.poly_ring('x')) == ZZ.poly_ring('x')
assert ZZ.poly_ring('x').unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(ZZ.poly_ring('x')) == QQ.poly_ring('x')
assert QQ.poly_ring('x').unify(QQ.poly_ring('x')) == QQ.poly_ring('x')
assert ZZ.poly_ring('x','y').unify(ZZ.poly_ring('x')) == ZZ.poly_ring('x','y')
assert ZZ.poly_ring('x','y').unify(QQ.poly_ring('x')) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x','y').unify(ZZ.poly_ring('x')) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x','y').unify(QQ.poly_ring('x')) == QQ.poly_ring('x','y')
assert ZZ.poly_ring('x').unify(ZZ.poly_ring('x','y')) == ZZ.poly_ring('x','y')
assert ZZ.poly_ring('x').unify(QQ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x').unify(ZZ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert QQ.poly_ring('x').unify(QQ.poly_ring('x','y')) == QQ.poly_ring('x','y')
assert ZZ.poly_ring('x','y').unify(ZZ.poly_ring('x','z')) == ZZ.poly_ring('x','y','z')
assert ZZ.poly_ring('x','y').unify(QQ.poly_ring('x','z')) == QQ.poly_ring('x','y','z')
assert QQ.poly_ring('x','y').unify(ZZ.poly_ring('x','z')) == QQ.poly_ring('x','y','z')
assert QQ.poly_ring('x','y').unify(QQ.poly_ring('x','z')) == QQ.poly_ring('x','y','z')
assert ZZ.frac_field('x').unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert QQ.frac_field('x').unify(ZZ.frac_field('x')) == QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x','y').unify(ZZ.frac_field('x')) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(QQ.frac_field('x')) == QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(ZZ.frac_field('x')) == QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(QQ.frac_field('x')) == QQ.frac_field('x','y')
assert ZZ.frac_field('x').unify(ZZ.frac_field('x','y')) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x').unify(QQ.frac_field('x','y')) == QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(ZZ.frac_field('x','y')) == QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(QQ.frac_field('x','y')) == QQ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(ZZ.frac_field('x','z')) == ZZ.frac_field('x','y','z')
assert ZZ.frac_field('x','y').unify(QQ.frac_field('x','z')) == QQ.frac_field('x','y','z')
assert QQ.frac_field('x','y').unify(ZZ.frac_field('x','z')) == QQ.frac_field('x','y','z')
assert QQ.frac_field('x','y').unify(QQ.frac_field('x','z')) == QQ.frac_field('x','y','z')
assert ZZ.poly_ring('x').unify(ZZ.frac_field('x')) == ZZ.frac_field('x')
assert ZZ.poly_ring('x').unify(QQ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.poly_ring('x').unify(ZZ.frac_field('x')) == EX # QQ.frac_field('x')
assert QQ.poly_ring('x').unify(QQ.frac_field('x')) == QQ.frac_field('x')
assert ZZ.poly_ring('x','y').unify(ZZ.frac_field('x')) == ZZ.frac_field('x','y')
assert ZZ.poly_ring('x','y').unify(QQ.frac_field('x')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x','y').unify(ZZ.frac_field('x')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x','y').unify(QQ.frac_field('x')) == QQ.frac_field('x','y')
assert ZZ.poly_ring('x').unify(ZZ.frac_field('x','y')) == ZZ.frac_field('x','y')
assert ZZ.poly_ring('x').unify(QQ.frac_field('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x').unify(ZZ.frac_field('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.poly_ring('x').unify(QQ.frac_field('x','y')) == QQ.frac_field('x','y')
assert ZZ.poly_ring('x','y').unify(ZZ.frac_field('x','z')) == ZZ.frac_field('x','y','z')
assert ZZ.poly_ring('x','y').unify(QQ.frac_field('x','z')) == EX # QQ.frac_field('x','y','z')
assert QQ.poly_ring('x','y').unify(ZZ.frac_field('x','z')) == EX # QQ.frac_field('x','y','z')
assert QQ.poly_ring('x','y').unify(QQ.frac_field('x','z')) == QQ.frac_field('x','y','z')
assert ZZ.frac_field('x').unify(ZZ.poly_ring('x')) == ZZ.frac_field('x')
assert ZZ.frac_field('x').unify(QQ.poly_ring('x')) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(ZZ.poly_ring('x')) == EX # QQ.frac_field('x')
assert QQ.frac_field('x').unify(QQ.poly_ring('x')) == QQ.frac_field('x')
assert ZZ.frac_field('x','y').unify(ZZ.poly_ring('x')) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(QQ.poly_ring('x')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(ZZ.poly_ring('x')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x','y').unify(QQ.poly_ring('x')) == QQ.frac_field('x','y')
assert ZZ.frac_field('x').unify(ZZ.poly_ring('x','y')) == ZZ.frac_field('x','y')
assert ZZ.frac_field('x').unify(QQ.poly_ring('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(ZZ.poly_ring('x','y')) == EX # QQ.frac_field('x','y')
assert QQ.frac_field('x').unify(QQ.poly_ring('x','y')) == QQ.frac_field('x','y')
assert ZZ.frac_field('x','y').unify(ZZ.poly_ring('x','z')) == ZZ.frac_field('x','y','z')
assert ZZ.frac_field('x','y').unify(QQ.poly_ring('x','z')) == EX # QQ.frac_field('x','y','z')
assert QQ.frac_field('x','y').unify(ZZ.poly_ring('x','z')) == EX # QQ.frac_field('x','y','z')
assert QQ.frac_field('x','y').unify(QQ.poly_ring('x','z')) == QQ.frac_field('x','y','z')
alg = QQ.algebraic_field(sqrt(5))
assert alg.unify(alg['x','y']) == alg['x','y']
assert alg['x','y'].unify(alg) == alg['x','y']
assert alg.unify(alg.frac_field('x','y')) == alg.frac_field('x','y')
assert alg.frac_field('x','y').unify(alg) == alg.frac_field('x','y')
ext = QQ.algebraic_field(sqrt(7))
raises(NotImplementedError, "alg.unify(ext)")
raises(UnificationFailed, "ZZ.poly_ring('x','y').unify(ZZ, gens=('y', 'z'))")
raises(UnificationFailed, "ZZ.unify(ZZ.poly_ring('x','y'), gens=('y', 'z'))")
def test_atanh():
R, x, y = ring('x, y', QQ)
assert rs_atanh(x, x, 9) == x**7/7 + x**5/5 + x**3/3 + x
assert rs_atanh(x*y + x**2*y**3, x, 9) == 2*x**8*y**11 + x**8*y**9 + \
2*x**7*y**9 + x**7*y**7/7 + x**6*y**9/3 + x**6*y**7 + x**5*y**7 + \
x**5*y**5/5 + x**4*y**5 + x**3*y**3/3 + x**2*y**3 + x*y
# Constant term in series
a = symbols('a')
R, x, y = ring('x, y', EX)
assert rs_atanh(x + a, x, 5) == EX((a**3 + a)/(a**8 - 4*a**6 + 6*a**4 - \
4*a**2 + 1))*x**4 - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + \
9*a**2 - 3))*x**3 + EX(a/(a**4 - 2*a**2 + 1))*x**2 - EX(1/(a**2 - \
1))*x + EX(atanh(a))
assert rs_atanh(x + x**2*y + a, x, 4) == EX(2*a/(a**4 - 2*a**2 + \
1))*x**3*y - EX((3*a**2 + 1)/(3*a**6 - 9*a**4 + 9*a**2 - 3))*x**3 - \
EX(1/(a**2 - 1))*x**2*y + EX(a/(a**4 - 2*a**2 + 1))*x**2 - \
EX(1/(a**2 - 1))*x + EX(atanh(a))
p = x + x**2 + 5
assert rs_atanh(p, x, 10).compose(x, 10) == EX(-733442653682135/5079158784 \
+ atanh(5))
def test_dmp_clear_denoms():
assert dmp_clear_denoms([[]], 1, QQ, ZZ) == (ZZ(1), [[]])
assert dmp_clear_denoms([[QQ(1)]], 1, QQ, ZZ) == (ZZ(1), [[QQ(1)]])
assert dmp_clear_denoms([[QQ(7)]], 1, QQ, ZZ) == (ZZ(1), [[QQ(7)]])
assert dmp_clear_denoms([[QQ(7, 3)]], 1, QQ) == (ZZ(3), [[QQ(7)]])
assert dmp_clear_denoms([[QQ(7, 3)]], 1, QQ, ZZ) == (ZZ(3), [[QQ(7)]])
assert dmp_clear_denoms([[QQ(3)], [QQ(1)], []], 1, QQ,
ZZ) == (ZZ(1), [[QQ(3)], [QQ(1)], []])
assert dmp_clear_denoms([[QQ(1)], [QQ(1, 2)], []], 1, QQ,
ZZ) == (ZZ(2), [[QQ(2)], [QQ(1)], []])
assert dmp_clear_denoms([QQ(3), QQ(1), QQ(0)], 0, QQ, ZZ,
convert=True) == (ZZ(1), [ZZ(3),
ZZ(1),
ZZ(0)])
assert dmp_clear_denoms([QQ(1), QQ(1, 2), QQ(0)], 0, QQ, ZZ,
convert=True) == (ZZ(2), [ZZ(2),
ZZ(1),
ZZ(0)])
assert dmp_clear_denoms([[QQ(3)], [QQ(1)], []], 1, QQ, ZZ,
convert=True) == (ZZ(1), [[QQ(3)], [QQ(1)], []])
assert dmp_clear_denoms([[QQ(1)], [QQ(1, 2)], []], 1, QQ, ZZ,
convert=True) == (ZZ(2), [[QQ(2)], [QQ(1)], []])
assert dmp_clear_denoms([[EX(S(3) / 2)], [EX(S(9) / 4)]], 1,
EX) == (EX(4), [[EX(6)], [EX(9)]])
assert dmp_clear_denoms([[EX(7)]], 1, EX) == (EX(1), [[EX(7)]])
assert dmp_clear_denoms([[EX(sin(x) / x), EX(0)]], 1,
EX) == (EX(x), [[EX(sin(x)), EX(0)]])
def test_Domain__unify():
assert ZZ.unify(ZZ) == ZZ
assert QQ.unify(QQ) == QQ
assert ZZ.unify(QQ) == QQ
assert QQ.unify(ZZ) == QQ
assert EX.unify(EX) == EX
assert ZZ.unify(EX) == EX
assert QQ.unify(EX) == EX
assert EX.unify(ZZ) == EX
assert EX.unify(QQ) == EX
assert ZZ.poly_ring(x).unify(EX) == EX
assert ZZ.frac_field(x).unify(EX) == EX
assert EX.unify(ZZ.poly_ring(x)) == EX
assert EX.unify(ZZ.frac_field(x)) == EX
assert ZZ.poly_ring(x, y).unify(EX) == EX
assert ZZ.frac_field(x, y).unify(EX) == EX
assert EX.unify(ZZ.poly_ring(x, y)) == EX
assert EX.unify(ZZ.frac_field(x, y)) == EX
assert QQ.poly_ring(x).unify(EX) == EX
assert QQ.frac_field(x).unify(EX) == EX
assert EX.unify(QQ.poly_ring(x)) == EX
assert EX.unify(QQ.frac_field(x)) == EX
assert QQ.poly_ring(x, y).unify(EX) == EX
assert QQ.frac_field(x, y).unify(EX) == EX
assert EX.unify(QQ.poly_ring(x, y)) == EX
assert EX.unify(QQ.frac_field(x, y)) == EX
assert ZZ.poly_ring(x).unify(ZZ) == ZZ.poly_ring(x)
assert ZZ.poly_ring(x).unify(QQ) == QQ.poly_ring(x)
assert QQ.poly_ring(x).unify(ZZ) == QQ.poly_ring(x)
assert QQ.poly_ring(x).unify(QQ) == QQ.poly_ring(x)
assert ZZ.unify(ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert QQ.unify(ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert ZZ.unify(QQ.poly_ring(x)) == QQ.poly_ring(x)
assert QQ.unify(QQ.poly_ring(x)) == QQ.poly_ring(x)
assert ZZ.poly_ring(x, y).unify(ZZ) == ZZ.poly_ring(x, y)
assert ZZ.poly_ring(x, y).unify(QQ) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x, y).unify(ZZ) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x, y).unify(QQ) == QQ.poly_ring(x, y)
assert ZZ.unify(ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert QQ.unify(ZZ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert ZZ.unify(QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert QQ.unify(QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert ZZ.frac_field(x).unify(ZZ) == ZZ.frac_field(x)
assert ZZ.frac_field(x).unify(QQ) == EX # QQ.frac_field(x)
assert QQ.frac_field(x).unify(ZZ) == EX # QQ.frac_field(x)
assert QQ.frac_field(x).unify(QQ) == QQ.frac_field(x)
assert ZZ.unify(ZZ.frac_field(x)) == ZZ.frac_field(x)
assert QQ.unify(ZZ.frac_field(x)) == EX # QQ.frac_field(x)
assert ZZ.unify(QQ.frac_field(x)) == EX # QQ.frac_field(x)
assert QQ.unify(QQ.frac_field(x)) == QQ.frac_field(x)
assert ZZ.frac_field(x, y).unify(ZZ) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x, y).unify(QQ) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x, y).unify(ZZ) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x, y).unify(QQ) == QQ.frac_field(x, y)
assert ZZ.unify(ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert QQ.unify(ZZ.frac_field(x, y)) == EX # QQ.frac_field(x,y)
assert ZZ.unify(QQ.frac_field(x, y)) == EX # QQ.frac_field(x,y)
assert QQ.unify(QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert ZZ.poly_ring(x).unify(ZZ.poly_ring(x)) == ZZ.poly_ring(x)
assert ZZ.poly_ring(x).unify(QQ.poly_ring(x)) == QQ.poly_ring(x)
assert QQ.poly_ring(x).unify(ZZ.poly_ring(x)) == QQ.poly_ring(x)
assert QQ.poly_ring(x).unify(QQ.poly_ring(x)) == QQ.poly_ring(x)
assert ZZ.poly_ring(x, y).unify(ZZ.poly_ring(x)) == ZZ.poly_ring(x, y)
assert ZZ.poly_ring(x, y).unify(QQ.poly_ring(x)) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x, y).unify(ZZ.poly_ring(x)) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x, y).unify(QQ.poly_ring(x)) == QQ.poly_ring(x, y)
assert ZZ.poly_ring(x).unify(ZZ.poly_ring(x, y)) == ZZ.poly_ring(x, y)
assert ZZ.poly_ring(x).unify(QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x).unify(ZZ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert QQ.poly_ring(x).unify(QQ.poly_ring(x, y)) == QQ.poly_ring(x, y)
assert ZZ.poly_ring(x, y).unify(ZZ.poly_ring(x,
z)) == ZZ.poly_ring(x, y, z)
assert ZZ.poly_ring(x, y).unify(QQ.poly_ring(x,
z)) == QQ.poly_ring(x, y, z)
assert QQ.poly_ring(x, y).unify(ZZ.poly_ring(x,
z)) == QQ.poly_ring(x, y, z)
assert QQ.poly_ring(x, y).unify(QQ.poly_ring(x,
z)) == QQ.poly_ring(x, y, z)
assert ZZ.frac_field(x).unify(ZZ.frac_field(x)) == ZZ.frac_field(x)
assert ZZ.frac_field(x).unify(QQ.frac_field(x)) == QQ.frac_field(x)
assert QQ.frac_field(x).unify(ZZ.frac_field(x)) == QQ.frac_field(x)
assert QQ.frac_field(x).unify(QQ.frac_field(x)) == QQ.frac_field(x)
assert ZZ.frac_field(x, y).unify(ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x, y).unify(QQ.frac_field(x)) == QQ.frac_field(x, y)
assert QQ.frac_field(x, y).unify(ZZ.frac_field(x)) == QQ.frac_field(x, y)
assert QQ.frac_field(x, y).unify(QQ.frac_field(x)) == QQ.frac_field(x, y)
assert ZZ.frac_field(x).unify(ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x).unify(QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert QQ.frac_field(x).unify(ZZ.frac_field(x, y)) == QQ.frac_field(x, y)
assert QQ.frac_field(x).unify(QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert ZZ.frac_field(x,
y).unify(ZZ.frac_field(x,
z)) == ZZ.frac_field(x, y, z)
assert ZZ.frac_field(x,
y).unify(QQ.frac_field(x,
z)) == QQ.frac_field(x, y, z)
assert QQ.frac_field(x,
y).unify(ZZ.frac_field(x,
z)) == QQ.frac_field(x, y, z)
assert QQ.frac_field(x,
y).unify(QQ.frac_field(x,
z)) == QQ.frac_field(x, y, z)
assert ZZ.poly_ring(x).unify(ZZ.frac_field(x)) == ZZ.frac_field(x)
assert ZZ.poly_ring(x).unify(QQ.frac_field(x)) == EX # QQ.frac_field(x)
assert QQ.poly_ring(x).unify(ZZ.frac_field(x)) == EX # QQ.frac_field(x)
assert QQ.poly_ring(x).unify(QQ.frac_field(x)) == QQ.frac_field(x)
assert ZZ.poly_ring(x, y).unify(ZZ.frac_field(x)) == ZZ.frac_field(x, y)
assert ZZ.poly_ring(x, y).unify(
QQ.frac_field(x)) == EX # QQ.frac_field(x,y)
assert QQ.poly_ring(x, y).unify(
ZZ.frac_field(x)) == EX # QQ.frac_field(x,y)
assert QQ.poly_ring(x, y).unify(QQ.frac_field(x)) == QQ.frac_field(x, y)
assert ZZ.poly_ring(x).unify(ZZ.frac_field(x, y)) == ZZ.frac_field(x, y)
assert ZZ.poly_ring(x).unify(QQ.frac_field(x,
y)) == EX # QQ.frac_field(x,y)
assert QQ.poly_ring(x).unify(ZZ.frac_field(x,
y)) == EX # QQ.frac_field(x,y)
assert QQ.poly_ring(x).unify(QQ.frac_field(x, y)) == QQ.frac_field(x, y)
assert ZZ.poly_ring(x,
y).unify(ZZ.frac_field(x,
z)) == ZZ.frac_field(x, y, z)
assert ZZ.poly_ring(x, y).unify(QQ.frac_field(
x, z)) == EX # QQ.frac_field(x,y,z)
assert QQ.poly_ring(x, y).unify(ZZ.frac_field(
x, z)) == EX # QQ.frac_field(x,y,z)
assert QQ.poly_ring(x,
y).unify(QQ.frac_field(x,
z)) == QQ.frac_field(x, y, z)
assert ZZ.frac_field(x).unify(ZZ.poly_ring(x)) == ZZ.frac_field(x)
assert ZZ.frac_field(x).unify(QQ.poly_ring(x)) == EX # QQ.frac_field(x)
assert QQ.frac_field(x).unify(ZZ.poly_ring(x)) == EX # QQ.frac_field(x)
assert QQ.frac_field(x).unify(QQ.poly_ring(x)) == QQ.frac_field(x)
assert ZZ.frac_field(x, y).unify(ZZ.poly_ring(x)) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x, y).unify(
QQ.poly_ring(x)) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x, y).unify(
ZZ.poly_ring(x)) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x, y).unify(QQ.poly_ring(x)) == QQ.frac_field(x, y)
assert ZZ.frac_field(x).unify(ZZ.poly_ring(x, y)) == ZZ.frac_field(x, y)
assert ZZ.frac_field(x).unify(QQ.poly_ring(x,
y)) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x).unify(ZZ.poly_ring(x,
y)) == EX # QQ.frac_field(x,y)
assert QQ.frac_field(x).unify(QQ.poly_ring(x, y)) == QQ.frac_field(x, y)
assert ZZ.frac_field(x,
y).unify(ZZ.poly_ring(x,
z)) == ZZ.frac_field(x, y, z)
assert ZZ.frac_field(x, y).unify(QQ.poly_ring(
x, z)) == EX # QQ.frac_field(x,y,z)
assert QQ.frac_field(x, y).unify(ZZ.poly_ring(
x, z)) == EX # QQ.frac_field(x,y,z)
assert QQ.frac_field(x,
y).unify(QQ.poly_ring(x,
z)) == QQ.frac_field(x, y, z)
alg = QQ.algebraic_field(sqrt(5))
assert alg.unify(alg[x, y]) == alg[x, y]
assert alg[x, y].unify(alg) == alg[x, y]
assert alg.unify(alg.frac_field(x, y)) == alg.frac_field(x, y)
assert alg.frac_field(x, y).unify(alg) == alg.frac_field(x, y)
ext = QQ.algebraic_field(sqrt(7))
raises(NotImplementedError, lambda: alg.unify(ext))
raises(UnificationFailed,
lambda: ZZ.poly_ring(x, y).unify(ZZ, gens=(y, z)))
raises(UnificationFailed,
lambda: ZZ.unify(ZZ.poly_ring(x, y), gens=(y, z)))
