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))
raises(ExactQuotientFailed, lambda: m/Monomial((5, 2, 0)))
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))
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 == 1) is False
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))
raises(ExactQuotientFailed, lambda: m / Monomial((5, 2, 0)))
mm = Monomial((1, 2, 3))
raises(ValueError, lambda: mm.as_expr())
assert str(mm) == 'Monomial((1, 2, 3))'
assert str(m) == 'x**3*y**4*z**1'
raises(NotImplementedError, lambda: m * 1)
raises(NotImplementedError, lambda: m / 1)
raises(ValueError, lambda: m**-1)
raises(TypeError, lambda: m.gcd(3))
raises(TypeError, lambda: m.lcm(3))
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))
raises(ExactQuotientFailed, lambda: m / Monomial((5, 2, 0)))
def itermonomials_degree_list_AGA(variables, max_degrees, min_degrees):
"""
Returns a list of all monomials such that
min_degrees[i] <= degree_list(monom)[i] <= max_degrees[i]
for all monom in the list and all i.
If max_degrees = [d_1, ..., d_n] and min_degrees = [e_1, ..., e_n],
the number of monomials in the list is:
(d_1 - e_1 + 1) * ... * (d_n - e_n + 1)
"""
# input lists of same length?
if len(variables) != len(max_degrees) or len(variables) != len(
min_degrees):
raise ValueError('all three input lists should be of same length')
# length of each row to be formed
columns = len(variables)
# divider of row j while filling a column k.
div_by = [1]
# min_degrees = [0, ..., 0]?
if sum(min_degrees) == 0:
# compute the number of rows (monomials)
rows = 1
rows = Mul(*[rows * (max_degrees[i] + 1) for i in range(columns)])
# initialize table m having 'rows' rows, where
# each row is filled with 'columns' zeros.
m = [[0 for i in range(columns)] for i in range(rows)]
# copy list max_degrees
temp = max_degrees[:]
# update table m
while sum(temp) != 0:
# pick k-th column to fill
k = temp.index(max(temp))
temp[k] = 0
# fill in k-th column
for j in range(rows):
m[j][k] = floor(j / Mul(*div_by)) % (max_degrees[k] + 1)
# update div_by for next column
div_by.append(max_degrees[k] + 1)
# min_degrees != [0, ..., 0]
else:
# here diff_degrees in the role of max_degrees
diff_degrees = [
max_degrees[i] - min_degrees[i] for i in range(columns)
]
if any(x < 0 for x in diff_degrees):
raise ValueError(
'min_degrees[i] should be <= max_degrees[i] for all i.')
# compute the number of rows (monomials)
rows = 1
rows = Mul(*[rows * (diff_degrees[i] + 1) for i in range(columns)])
# initialize table m having 'rows' rows, where
# each row is filled with 'columns' zeros.
m = [[0 for i in range(columns)] for i in range(rows)]
# copy the list diff_degrees
temp = diff_degrees[:]
# update table m
while sum(temp) != 0:
# pick k-th column to fill
k = temp.index(max(temp))
temp[k] = 0
# fill in k-th column
for j in range(rows):
m[j][k] = floor(j / Mul(*div_by)) % (diff_degrees[k] + 1)
# update div_by for next column
div_by.append(diff_degrees[k] + 1)
# add min_degrees to each row
m = [[m[j][k] + min_degrees[k] for k in range(columns)]
for j in range(rows)]
return [Monomial(m[j]).as_expr(*variables) for j in range(rows)]
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 == 1) is False
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))
raises(ExactQuotientFailed, lambda: m/Monomial((5, 2, 0)))
mm = Monomial((1, 2, 3))
raises(ValueError, lambda: mm.as_expr())
assert str(mm) == 'Monomial((1, 2, 3))'
assert str(m) == 'x**3*y**4*z**1'
raises(NotImplementedError, lambda: m*1)
raises(NotImplementedError, lambda: m/1)
raises(ValueError, lambda: m**-1)
raises(TypeError, lambda: m.gcd(3))
raises(TypeError, lambda: m.lcm(3))