def test_DMP_exclude():
f = [[[[[[[[[[[[[[[[[[[[[[[[[[1]], [[]]]]]]]]]]]]]]]]]]]]]]]]]]
J = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 24, 25]
assert DMP(f, ZZ).exclude() == (J, DMP([1, 0], ZZ))
assert DMP([[1], [1, 0]], ZZ).exclude() == ([], DMP([[1], [1, 0]], ZZ))
def test___hash__():
# issue sympy/sympy#5571
assert DMP([[1, 2], [3]], ZZ) == DMP([[int(1), int(2)], [int(3)]], ZZ)
assert hash(DMP([[1, 2], [3]], ZZ)) == hash(DMP([[int(1), int(2)], [int(3)]], ZZ))
assert DMF(
([[1, 2], [3]], [[1]]), ZZ) == DMF(([[int(1), int(2)], [int(3)]], [[int(1)]]), ZZ)
assert hash(DMF(([[1, 2], [3]], [[1]]), ZZ)) == hash(DMF(([[int(1),
int(2)], [int(3)]], [[int(1)]]), ZZ))
assert ANP([1, 1], [1, 0, 1], ZZ) == ANP([int(1), int(1)], [int(1), int(0), int(1)], ZZ)
assert hash(
ANP([1, 1], [1, 0, 1], ZZ)) == hash(ANP([int(1), int(1)], [int(1), int(0), int(1)], ZZ))
def test_DMP___eq__():
assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) == \
DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ)
assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) == \
DMP([[QQ(1), QQ(2)], [QQ(3)]], QQ)
assert DMP([[QQ(1), QQ(2)], [QQ(3)]], QQ) == \
DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ)
assert DMP([[[ZZ(1)]]], ZZ) != DMP([[ZZ(1)]], ZZ)
assert DMP([[ZZ(1)]], ZZ) != DMP([[[ZZ(1)]]], ZZ)
def test_pickling_polys_polyclasses():
for c in (DMP, DMP([[ZZ(1)], [ZZ(2)], [ZZ(3)]], ZZ)):
check(c)
for c in (DMF, DMF(([ZZ(1), ZZ(2)], [ZZ(1), ZZ(3)]), ZZ)):
check(c)
for c in (ANP, ANP([QQ(1), QQ(2)], [QQ(1), QQ(2), QQ(3)], QQ)):
check(c)
def test_DMP_to_dict():
f = DMP([[3], [], [2], [], [8]], ZZ)
assert f.to_dict() == \
{(4, 0): 3, (2, 0): 2, (0, 0): 8}
assert f.to_diofant_dict() == \
{(4, 0): ZZ.to_expr(3), (2, 0): ZZ.to_expr(2), (0, 0):
ZZ.to_expr(8)}
def test_DMP___init__():
f = DMP([[0], [], [0, 1, 2], [3]], ZZ)
assert f.rep == [[1, 2], [3]]
assert f.domain == ZZ
assert f.lev == 1
f = DMP([[1, 2], [3]], ZZ, 1)
assert f.rep == [[1, 2], [3]]
assert f.domain == ZZ
assert f.lev == 1
f = DMP({(1, 1): 1, (0, 0): 2}, ZZ, 1)
assert f.rep == [[1, 0], [2]]
assert f.domain == ZZ
assert f.lev == 1
def test_DMP___init__():
f = DMP([[0], [], [0, 1, 2], [3]], ZZ)
assert f.rep == [[1, 2], [3]]
assert f.domain == ZZ
assert f.lev == 1
f = DMP([[1, 2], [3]], ZZ, 1)
assert f.rep == [[1, 2], [3]]
assert f.domain == ZZ
assert f.lev == 1
f = DMP({(1, 1): 1, (0, 0): 2}, ZZ, 1)
assert f.rep == [[1, 0], [2]]
assert f.domain == ZZ
assert f.lev == 1
assert f == DMP.from_monoms_coeffs(f.monoms(), f.coeffs(), f.lev, f.domain)
def test_to_algebraic_integer():
a = AlgebraicNumber(sqrt(3), gen=x).to_algebraic_integer()
assert a.minpoly == x**2 - 3
assert a.root == sqrt(3)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(2 * sqrt(3), gen=x).to_algebraic_integer()
assert a.minpoly == x**2 - 12
assert a.root == 2 * sqrt(3)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(3) / 2, gen=x).to_algebraic_integer()
assert a.minpoly == x**2 - 12
assert a.root == 2 * sqrt(3)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(3) / 2, [Rational(7, 19), 3],
gen=x).to_algebraic_integer()
assert a.minpoly == x**2 - 12
assert a.root == 2 * sqrt(3)
assert a.rep == DMP([QQ(7, 19), QQ(3)], QQ)
def legendre_poly(n, x=None, **args):
"""Generates Legendre polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Legendre polynomial of degree %s" % n)
poly = DMP(dup_legendre(int(n), QQ), QQ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def cyclotomic_poly(n, x=None, **args):
"""Generates cyclotomic polynomial of order `n` in `x`. """
if n <= 0:
raise ValueError("can't generate cyclotomic polynomial of order %s" %
n)
poly = DMP(dup_zz_cyclotomic_poly(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def chebyshevt_poly(n, x=None, **args):
"""Generates Chebyshev polynomial of the first kind of degree `n` in `x`. """
if n < 0:
raise ValueError(
"can't generate 1st kind Chebyshev polynomial of degree %s" % n)
poly = DMP(dup_chebyshevt(int(n), ZZ), ZZ)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def jacobi_poly(n, a, b, x=None, **args):
"""Generates Jacobi polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Jacobi polynomial of degree %s" % n)
K, v = construct_domain([a, b], field=True)
poly = DMP(dup_jacobi(int(n), v[0], v[1], K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def gegenbauer_poly(n, a, x=None, **args):
"""Generates Gegenbauer polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Gegenbauer polynomial of degree %s" %
n)
K, a = construct_domain(a, field=True)
poly = DMP(dup_gegenbauer(int(n), a, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def spherical_bessel_fn(n, x=None, **args):
"""
Coefficients for the spherical Bessel functions.
Those are only needed in the jn() function.
The coefficients are calculated from:
fn(0, z) = 1/z
fn(1, z) = 1/z**2
fn(n-1, z) + fn(n+1, z) == (2*n+1)/z * fn(n, z)
Examples
========
>>> from diofant import Symbol
>>> z = Symbol("z")
>>> spherical_bessel_fn(1, z)
z**(-2)
>>> spherical_bessel_fn(2, z)
-1/z + 3/z**3
>>> spherical_bessel_fn(3, z)
-6/z**2 + 15/z**4
>>> spherical_bessel_fn(4, z)
1/z - 45/z**3 + 105/z**5
"""
if n < 0:
dup = dup_spherical_bessel_fn_minus(-int(n), ZZ)
else:
dup = dup_spherical_bessel_fn(int(n), ZZ)
poly = DMP(dup, ZZ)
if x is not None:
poly = Poly.new(poly, 1 / x)
else:
poly = PurePoly.new(poly, 1 / Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def laguerre_poly(n, x=None, alpha=None, **args):
"""Generates Laguerre polynomial of degree `n` in `x`. """
if n < 0:
raise ValueError("can't generate Laguerre polynomial of degree %s" % n)
if alpha is not None:
K, alpha = construct_domain(alpha,
field=True) # XXX: ground_field=True
else:
K, alpha = QQ, QQ(0)
poly = DMP(dup_laguerre(int(n), alpha, K), K)
if x is not None:
poly = Poly.new(poly, x)
else:
poly = PurePoly.new(poly, Dummy('x'))
if not args.get('polys', False):
return poly.as_expr()
else:
return poly
def test___hash__():
# issue sympy/sympy#5571
assert DMP([[1, 2], [3]], ZZ) == DMP([[int(1), int(2)], [int(3)]], ZZ)
assert hash(DMP([[1, 2], [3]],
ZZ)) == hash(DMP([[int(1), int(2)], [int(3)]], ZZ))
def test_AlgebraicNumber():
minpoly, root = x**2 - 2, sqrt(2)
a = AlgebraicNumber(root, gen=x)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [Integer(1), Integer(0)]
assert a.native_coeffs() == [QQ(1), QQ(0)]
a = AlgebraicNumber(root, gen=x, alias='y')
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
a = AlgebraicNumber(root, gen=x, alias=Symbol('y'))
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [Rational(8, 3)]).rep == DMP([QQ(8, 3)],
QQ)
assert AlgebraicNumber(sqrt(2), [7, 3]).rep == DMP([QQ(7), QQ(3)], QQ)
assert AlgebraicNumber(sqrt(2),
[Rational(7, 9), Rational(3, 2)]).rep == DMP(
[QQ(7, 9), QQ(3, 2)], QQ)
assert AlgebraicNumber(sqrt(2), [1, 2, 3]).rep == DMP([QQ(2), QQ(5)], QQ)
a = AlgebraicNumber(AlgebraicNumber(root, gen=x), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [Integer(1), Integer(2)]
assert a.native_coeffs() == [QQ(1), QQ(2)]
a = AlgebraicNumber((minpoly, root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
a = AlgebraicNumber((Poly(minpoly), root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert AlgebraicNumber(sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
assert AlgebraicNumber(-sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(2))
assert a == b
c = AlgebraicNumber(sqrt(2), gen=x)
d = AlgebraicNumber(sqrt(2), gen=x)
assert a == b
assert a == c
a = AlgebraicNumber(sqrt(2), [1, 2])
b = AlgebraicNumber(sqrt(2), [1, 3])
assert a != b and a != sqrt(2) + 3
assert (a == x) is False and (a != x) is True
a = AlgebraicNumber(sqrt(2), [1, 0])
b = AlgebraicNumber(sqrt(2), [1, 0], alias=y)
assert a.as_poly(x) == Poly(x)
assert b.as_poly() == Poly(y)
assert a.as_expr() == sqrt(2)
assert a.as_expr(x) == x
assert b.as_expr() == sqrt(2)
assert b.as_expr(x) == x
a = AlgebraicNumber(sqrt(2), [2, 3])
b = AlgebraicNumber(sqrt(2), [2, 3], alias=y)
p = a.as_poly()
assert p == Poly(2 * p.gen + 3)
assert a.as_poly(x) == Poly(2 * x + 3)
assert b.as_poly() == Poly(2 * y + 3)
assert a.as_expr() == 2 * sqrt(2) + 3
assert a.as_expr(x) == 2 * x + 3
assert b.as_expr() == 2 * sqrt(2) + 3
assert b.as_expr(x) == 2 * x + 3
a = AlgebraicNumber(sqrt(2))
b = to_number_field(sqrt(2))
assert a.args == b.args == (sqrt(2), Tuple(1, 0))
b = AlgebraicNumber(sqrt(2), alias='alpha')
assert b.args == (sqrt(2), Tuple(1, 0), Symbol('alpha'))
a = AlgebraicNumber(sqrt(2), [1, 2, 3])
assert a.args == (sqrt(2), Tuple(2, 5))
pytest.raises(ValueError,
lambda: AlgebraicNumber(RootOf(x**3 + y * x + 1, x, 0)))
a = AlgebraicNumber(RootOf(x**3 + 2 * x - 1, x, 1), alias='alpha')
assert a.free_symbols == set()
# integer powers:
assert a**0 == 1
assert a**2 == AlgebraicNumber(a, (1, 0, 0), alias='alpha')
assert a**5 == AlgebraicNumber(a, (1, 0, 0, 0, 0, 0), alias='alpha')
assert a**110 == AlgebraicNumber(a, ([1] + [0] * 110), alias='alpha')
assert (a**pi).is_Pow
b = AlgebraicNumber(sqrt(2), (1, 0), alias='theta')
c = b + 1
assert c**2 == 2 * b + 3
assert c**5 == 29 * b + 41
assert c**-2 == 3 - 2 * b
assert c**-11 == 5741 * b - 8119
# arithmetics
assert a**3 == -2 * a + 1 == a * (-2) + 1 == 1 + (-2) * a == 1 - 2 * a
assert a**5 == a**2 + 4 * a - 2
assert a**4 == -2 * a**2 + a == a - 2 * a**2
assert a**110 == (-2489094528619081871 * a**2 + 3737645722703173544 * a -
1182958048412500088)
assert a + a == 2 * a
assert 2 * a - a == a
assert Integer(1) - a == (-a) + 1
assert (a + pi).is_Add
assert (pi + a).is_Add
assert (a - pi).is_Add
assert (pi - a).is_Add
assert (a * pi).is_Mul
assert (pi * a).is_Mul
def test_DMP_properties():
assert DMP([[]], ZZ).is_zero is True
assert DMP([[1]], ZZ).is_zero is False
assert DMP([[1]], ZZ).is_one is True
assert DMP([[2]], ZZ).is_one is False
assert DMP([[1]], ZZ).is_ground is True
assert DMP([[1], [2], [1]], ZZ).is_ground is False
assert DMP([[1], [2, 0], [1, 0]], ZZ).is_sqf is True
assert DMP([[1], [2, 0], [1, 0, 0]], ZZ).is_sqf is False
assert DMP([[1, 2], [3]], ZZ).is_monic is True
assert DMP([[2, 2], [3]], ZZ).is_monic is False
assert DMP([[1, 2], [3]], ZZ).is_primitive is True
assert DMP([[2, 4], [6]], ZZ).is_primitive is False
def test_DMP_arithmetics():
f = DMP([[2], [2, 0]], ZZ)
assert f.mul_ground(2) == DMP([[4], [4, 0]], ZZ)
assert f.quo_ground(2) == DMP([[1], [1, 0]], ZZ)
pytest.raises(ExactQuotientFailed, lambda: f.exquo_ground(3))
f = DMP([[-5]], ZZ)
g = DMP([[5]], ZZ)
assert f.abs() == g
assert abs(f) == g
assert g.neg() == f
assert -g == f
h = DMP([[]], ZZ)
assert f.add(g) == h
assert f + g == h
assert g + f == h
assert f + 5 == h
assert 5 + f == h
h = DMP([[-10]], ZZ)
assert f.sub(g) == h
assert f - g == h
assert g - f == -h
assert f - 5 == h
assert 5 - f == -h
h = DMP([[-25]], ZZ)
assert f.mul(g) == h
assert f * g == h
assert g * f == h
assert f * 5 == h
assert 5 * f == h
h = DMP([[25]], ZZ)
assert f.sqr() == h
assert f.pow(2) == h
assert f**2 == h
pytest.raises(TypeError, lambda: f.pow('x'))
f = DMP([[1], [], [1, 0, 0]], ZZ)
g = DMP([[2], [-2, 0]], ZZ)
q = DMP([[2], [2, 0]], ZZ)
r = DMP([[8, 0, 0]], ZZ)
assert f.pdiv(g) == (q, r)
assert f.pquo(g) == q
assert f.prem(g) == r
pytest.raises(ExactQuotientFailed, lambda: f.pexquo(g))
f = DMP([[1], [], [1, 0, 0]], ZZ)
g = DMP([[1], [-1, 0]], ZZ)
q = DMP([[1], [1, 0]], ZZ)
r = DMP([[2, 0, 0]], ZZ)
assert f.div(g) == (q, r)
assert f.quo(g) == q
assert f.rem(g) == r
assert divmod(f, g) == (q, r)
assert f // g == q
assert f // 2 == DMP([[0], [], [0, 0, 0]], QQ)
assert f % g == r
pytest.raises(ExactQuotientFailed, lambda: f.exquo(g))
f = DMP([[-5]], ZZ)
g = DMP([[5]], QQ)
h = DMP([[]], QQ)
assert f + g == g + f == h
def test_pickling_polys_polyclasses():
for c in (DMP, DMP([[ZZ(1)], [ZZ(2)], [ZZ(3)]], ZZ)):
check(c)
def test_DMP___bool__():
assert bool(DMP([[]], ZZ)) is False
assert bool(DMP([[1]], ZZ)) is True
def test_DMP___eq__():
f = DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ)
assert f == f
assert f.eq(f)
assert DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ) == \
DMP([[QQ(1), QQ(2)], [QQ(3)]], QQ)
assert DMP([[QQ(1), QQ(2)], [QQ(3)]], QQ) == \
DMP([[ZZ(1), ZZ(2)], [ZZ(3)]], ZZ)
assert DMP([[[ZZ(1)]]], ZZ) != DMP([[ZZ(1)]], ZZ)
assert DMP([[[ZZ(1)]]], ZZ).ne(DMP([[ZZ(1)]], ZZ))
assert DMP([[ZZ(1)]], ZZ) != DMP([[[ZZ(1)]]], ZZ)
def test_AlgebraicNumber():
minpoly, root = x**2 - 2, sqrt(2)
a = AlgebraicNumber(root, gen=x)
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [Integer(1), Integer(0)]
assert a.native_coeffs() == [QQ(1), QQ(0)]
a = AlgebraicNumber(root, gen=x, alias='y')
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
a = AlgebraicNumber(root, gen=x, alias=Symbol('y'))
assert a.rep == DMP([QQ(1), QQ(0)], QQ)
assert a.root == root
assert a.alias == Symbol('y')
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is True
assert AlgebraicNumber(sqrt(2), []).rep == DMP([], QQ)
assert AlgebraicNumber(sqrt(2), [8]).rep == DMP([QQ(8)], QQ)
assert AlgebraicNumber(sqrt(2), [Rational(8, 3)]).rep == DMP([QQ(8, 3)],
QQ)
assert AlgebraicNumber(sqrt(2), [7, 3]).rep == DMP([QQ(7), QQ(3)], QQ)
assert AlgebraicNumber(sqrt(2),
[Rational(7, 9), Rational(3, 2)]).rep == DMP(
[QQ(7, 9), QQ(3, 2)], QQ)
assert AlgebraicNumber(sqrt(2), [1, 2, 3]).rep == DMP([QQ(2), QQ(5)], QQ)
a = AlgebraicNumber(AlgebraicNumber(root, gen=x), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert a.coeffs() == [Integer(1), Integer(2)]
assert a.native_coeffs() == [QQ(1), QQ(2)]
a = AlgebraicNumber((minpoly, root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
a = AlgebraicNumber((Poly(minpoly), root), [1, 2])
assert a.rep == DMP([QQ(1), QQ(2)], QQ)
assert a.root == root
assert a.alias is None
assert a.minpoly == minpoly
assert a.is_number
assert a.is_aliased is False
assert AlgebraicNumber(sqrt(3)).rep == DMP([QQ(1), QQ(0)], QQ)
assert AlgebraicNumber(-sqrt(3)).rep == DMP([-QQ(1), QQ(0)], QQ)
a = AlgebraicNumber(sqrt(2))
b = AlgebraicNumber(sqrt(2))
assert a == b
c = AlgebraicNumber(sqrt(2), gen=x)
d = AlgebraicNumber(sqrt(2), gen=x)
assert a == b
assert a == c
a = AlgebraicNumber(sqrt(2), [1, 2])
b = AlgebraicNumber(sqrt(2), [1, 3])
assert a != b and a != sqrt(2) + 3
assert (a == x) is False and (a != x) is True
a = AlgebraicNumber(sqrt(2), [1, 0])
b = AlgebraicNumber(sqrt(2), [1, 0], alias=y)
assert a.as_poly(x) == Poly(x)
assert b.as_poly() == Poly(y)
assert a.as_expr() == sqrt(2)
assert a.as_expr(x) == x
assert b.as_expr() == sqrt(2)
assert b.as_expr(x) == x
a = AlgebraicNumber(sqrt(2), [2, 3])
b = AlgebraicNumber(sqrt(2), [2, 3], alias=y)
p = a.as_poly()
assert p == Poly(2 * p.gen + 3)
assert a.as_poly(x) == Poly(2 * x + 3)
assert b.as_poly() == Poly(2 * y + 3)
assert a.as_expr() == 2 * sqrt(2) + 3
assert a.as_expr(x) == 2 * x + 3
assert b.as_expr() == 2 * sqrt(2) + 3
assert b.as_expr(x) == 2 * x + 3
a = AlgebraicNumber(sqrt(2))
b = to_number_field(sqrt(2))
assert a.args == b.args == (sqrt(2), Tuple())
b = AlgebraicNumber(sqrt(2), alias='alpha')
assert b.args == (sqrt(2), Tuple(), Symbol('alpha'))
a = AlgebraicNumber(sqrt(2), [1, 2, 3])
assert a.args == (sqrt(2), Tuple(1, 2, 3))
def test_DMP_functionality():
f = DMP([[1], [2, 0], [1, 0, 0]], ZZ)
g = DMP([[1], [1, 0]], ZZ)
h = DMP([[1]], ZZ)
assert f.degree() == 2
assert f.degree_list() == (2, 2)
assert f.total_degree() == 2
pytest.raises(TypeError, lambda: f.degree("spam"))
assert f.LC() == ZZ(1)
assert f.TC() == ZZ(0)
assert f.nth(1, 1) == ZZ(2)
pytest.raises(TypeError, lambda: f.nth(0, 'x'))
assert f.max_norm() == 2
assert f.l1_norm() == 4
u = DMP([[2], [2, 0]], ZZ)
assert f.diff(m=1, j=0) == u
assert f.diff(m=1, j=1) == u
pytest.raises(TypeError, lambda: f.diff(m='x', j=0))
pytest.raises(TypeError, lambda: f.diff(j="spam"))
u = DMP([1, 2, 1], ZZ)
v = DMP([1, 2, 1], ZZ)
assert f.eval(a=1, j=0) == u
assert f.eval(a=1, j=1) == v
pytest.raises(TypeError, lambda: f.eval(a=1, j="spam"))
assert f.eval(1).eval(1) == ZZ(4)
assert f.cofactors(g) == (g, g, h)
assert f.gcd(g) == g
assert f.lcm(g) == f
u = DMP([[QQ(45), QQ(30), QQ(5)]], QQ)
v = DMP([[QQ(1), QQ(2, 3), QQ(1, 9)]], QQ)
assert u.monic() == v
assert (4 * f).content() == ZZ(4)
assert (4 * f).primitive() == (ZZ(4), f)
f = DMP([[1], [2], [3], [4], [5], [6]], ZZ)
assert f.trunc(3) == DMP([[1], [-1], [], [1], [-1], []], ZZ)
f = DMP(f_4, ZZ)
assert f.sqf_part() == -f
assert f.sqf_list() == (ZZ(-1), [(-f, 1)])
f = DMP([[-1], [], [], [5]], ZZ)
g = DMP([[3, 1], [], []], ZZ)
h = DMP([[45, 30, 5]], ZZ)
r = DMP([675, 675, 225, 25], ZZ)
assert f.subresultants(g) == [f, g, h]
assert f.resultant(g) == r
f = DMP([1, 3, 9, -13], ZZ)
assert f.discriminant() == -11664
f = DMP([QQ(2), QQ(0)], QQ)
g = DMP([QQ(1), QQ(0), QQ(-16)], QQ)
s = DMP([QQ(1, 32), QQ(0)], QQ)
t = DMP([QQ(-1, 16)], QQ)
h = DMP([QQ(1)], QQ)
assert f.half_gcdex(g) == (s, h)
assert f.gcdex(g) == (s, t, h)
assert f.invert(g) == s
f = DMP([[1], [2], [3]], QQ)
pytest.raises(ValueError, lambda: f.half_gcdex(f))
pytest.raises(ValueError, lambda: f.gcdex(f))
pytest.raises(ValueError, lambda: f.invert(f))
f = DMP([1, 0, 20, 0, 150, 0, 500, 0, 625, -2, 0, -10, 9], ZZ)
g = DMP([1, 0, 0, -2, 9], ZZ)
h = DMP([1, 0, 5, 0], ZZ)
assert g.compose(h) == f
assert f.decompose() == [g, h]
f = DMP([[1], [2], [3]], QQ)
pytest.raises(ValueError, lambda: f.decompose())
pytest.raises(ValueError, lambda: f.sturm())
pytest.raises(PolynomialError, lambda: f.all_coeffs())
pytest.raises(PolynomialError, lambda: f.all_monoms())
pytest.raises(PolynomialError, lambda: f.all_terms())
pytest.raises(ValueError, lambda: f.revert(1))
pytest.raises(ValueError, lambda: f.shift(1))
pytest.raises(ValueError, lambda: f.gff_list())
pytest.raises(PolynomialError, lambda: f.intervals())
pytest.raises(PolynomialError, lambda: f.refine_root(1, 2))
assert f.integrate() == DMP([[QQ(1, 3)], [QQ(1, 1)], [QQ(3, 1)], []], QQ)
pytest.raises(TypeError, lambda: f.integrate(m="spam"))
pytest.raises(TypeError, lambda: f.integrate(j="spam"))
f = DMP([[-1], [], [], [5]], ZZ)
g = DMP([[3, 1], [], []], QQ)
r = DMP([675, 675, 225, 25], QQ)
assert f.resultant(g) == r
assert DMP([1, 2], QQ).resultant(DMP([3], ZZ)) == 3
assert f.is_cyclotomic is False