def staircase(n):
"""
Compute all monomials with degree less than ``n`` that are
not divisible by any element of ``leading_monomials``.
"""
if n == 0:
return [1]
S = []
for mi in combinations_with_replacement(range(len(opt.gens)), n):
m = [0] * len(opt.gens)
for i in mi:
m[i] += 1
if all([monomial_div(m, lmg) is None
for lmg in leading_monomials]):
S.append(m)
return [Monomial(s).as_expr(*opt.gens) for s in S] + staircase(n - 1)
def test_Monomial():
m = Monomial((3, 4, 1), (x, y, z))
l = Monomial((3, 4, 1))
n = Monomial((1, 2, 0), (x, y, z))
assert m.as_expr() == x**3*y**4*z
assert n.as_expr() == x**1*y**2
assert m.as_expr(a, b, c) == a**3*b**4*c
assert n.as_expr(a, b, c) == a**1*b**2
pytest.raises(ValueError, lambda: l.as_expr())
assert m.exponents == (3, 4, 1)
assert m.gens == (x, y, z)
assert n.exponents == (1, 2, 0)
assert n.gens == (x, y, z)
assert m == (3, 4, 1)
assert n != (3, 4, 1)
assert m != (1, 2, 0)
assert n == (1, 2, 0)
assert n != object()
assert m != n
assert hash(m) != hash(n)
assert m[0] == m[-3] == 3
assert m[1] == m[-2] == 4
assert m[2] == m[-1] == 1
assert n[0] == n[-3] == 1
assert n[1] == n[-2] == 2
assert n[2] == n[-1] == 0
assert m[:2] == (3, 4)
assert n[:2] == (1, 2)
assert m*n == Monomial((4, 6, 1))
assert m/n == Monomial((2, 2, 1))
assert m*(1, 2, 0) == Monomial((4, 6, 1))
assert m/(1, 2, 0) == Monomial((2, 2, 1))
pytest.raises(TypeError, lambda: m*object())
pytest.raises(TypeError, lambda: m/object())
assert m.gcd(n) == Monomial((1, 2, 0))
assert m.lcm(n) == Monomial((3, 4, 1))
assert m.gcd((1, 2, 0)) == Monomial((1, 2, 0))
assert m.lcm((1, 2, 0)) == Monomial((3, 4, 1))
pytest.raises(TypeError, lambda: m.gcd(object()))
pytest.raises(TypeError, lambda: m.lcm(object()))
assert m**0 == Monomial((0, 0, 0))
assert m**1 == m
assert m**2 == Monomial((6, 8, 2))
assert m**3 == Monomial((9, 12, 3))
pytest.raises(ValueError, lambda: m**-3)
pytest.raises(ExactQuotientFailed, lambda: m/Monomial((5, 2, 0)))
assert str(m) == "x**3*y**4*z**1"
assert str(l) == "Monomial((3, 4, 1))"
def test_monomial_ops():
m1 = Monomial((3, 4, 1))
assert m1 * (1, 2, 0) == (4, 6, 1)
assert m1 / (1, 2, 0) == (2, 2, 1)
assert m1**2 == (6, 8, 2)
assert m1.gcd((1, 2, 0)) == (1, 2, 0)
m2 = Monomial((1, 4, 1))
assert m2.gcd((3, 2, 0)) == (1, 2, 0)
m3 = Monomial((3, 4, 5))
assert functools.reduce(Monomial.gcd,
(m3, (0, 5, 1), (6, 3, 9))) == (0, 3, 1)
assert m1.lcm((1, 2, 0)) == (3, 4, 1)
m4 = Monomial((1, 4, 1))
assert m4.lcm((3, 2, 0)) == (3, 4, 1)
m5 = Monomial((1, 2, 3))
assert m5.divides((4, 5, 6)) is True
assert m5.divides((0, 5, 6)) is False
m6 = Monomial((1, 2))
assert m6.divides((3, 4)) is True
assert m6.divides((0, 2)) is False
pytest.raises(TypeError, lambda: m6.divides(2 * x))
def test_pickling_polys_monomials():
for c in (Monomial, Monomial((1, 2, 3), (x, y, z))):
check(c)
def test_Monomial():
m = Monomial((3, 4, 1), (x, y, z))
n = Monomial((1, 2, 0), (x, y, z))
assert m.as_expr() == x**3 * y**4 * z
assert n.as_expr() == x**1 * y**2
assert m.as_expr(a, b, c) == a**3 * b**4 * c
assert n.as_expr(a, b, c) == a**1 * b**2
assert m.exponents == (3, 4, 1)
assert m.gens == (x, y, z)
assert n.exponents == (1, 2, 0)
assert n.gens == (x, y, z)
assert m == (3, 4, 1)
assert n != (3, 4, 1)
assert m != (1, 2, 0)
assert n == (1, 2, 0)
assert m[0] == m[-3] == 3
assert m[1] == m[-2] == 4
assert m[2] == m[-1] == 1
assert n[0] == n[-3] == 1
assert n[1] == n[-2] == 2
assert n[2] == n[-1] == 0
assert m[:2] == (3, 4)
assert n[:2] == (1, 2)
assert m * n == Monomial((4, 6, 1))
assert m / n == Monomial((2, 2, 1))
assert m * (1, 2, 0) == Monomial((4, 6, 1))
assert m / (1, 2, 0) == Monomial((2, 2, 1))
assert m.gcd(n) == Monomial((1, 2, 0))
assert m.lcm(n) == Monomial((3, 4, 1))
assert m.gcd((1, 2, 0)) == Monomial((1, 2, 0))
assert m.lcm((1, 2, 0)) == Monomial((3, 4, 1))
assert m**0 == Monomial((0, 0, 0))
assert m**1 == m
assert m**2 == Monomial((6, 8, 2))
assert m**3 == Monomial((9, 12, 3))
pytest.raises(ExactQuotientFailed, lambda: m / Monomial((5, 2, 0)))