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ai-content-maker/.venv/Lib/site-packages/sympy/polys/numberfields/tests/test_modules.py

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from sympy.abc import x, zeta
from sympy.polys import Poly, cyclotomic_poly
from sympy.polys.domains import FF, QQ, ZZ
from sympy.polys.matrices import DomainMatrix, DM
from sympy.polys.numberfields.exceptions import (
ClosureFailure, MissingUnityError, StructureError
)
from sympy.polys.numberfields.modules import (
Module, ModuleElement, ModuleEndomorphism, PowerBasis, PowerBasisElement,
find_min_poly, is_sq_maxrank_HNF, make_mod_elt, to_col,
)
from sympy.polys.numberfields.utilities import is_int
from sympy.polys.polyerrors import UnificationFailed
from sympy.testing.pytest import raises
def test_to_col():
c = [1, 2, 3, 4]
m = to_col(c)
assert m.domain.is_ZZ
assert m.shape == (4, 1)
assert m.flat() == c
def test_Module_NotImplemented():
M = Module()
raises(NotImplementedError, lambda: M.n)
raises(NotImplementedError, lambda: M.mult_tab())
raises(NotImplementedError, lambda: M.represent(None))
raises(NotImplementedError, lambda: M.starts_with_unity())
raises(NotImplementedError, lambda: M.element_from_rational(QQ(2, 3)))
def test_Module_ancestors():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
D = B.submodule_from_matrix(5 * DomainMatrix.eye(4, ZZ))
assert C.ancestors(include_self=True) == [A, B, C]
assert D.ancestors(include_self=True) == [A, B, D]
assert C.power_basis_ancestor() == A
assert C.nearest_common_ancestor(D) == B
M = Module()
assert M.power_basis_ancestor() is None
def test_Module_compat_col():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
col = to_col([1, 2, 3, 4])
row = col.transpose()
assert A.is_compat_col(col) is True
assert A.is_compat_col(row) is False
assert A.is_compat_col(1) is False
assert A.is_compat_col(DomainMatrix.eye(3, ZZ)[:, 0]) is False
assert A.is_compat_col(DomainMatrix.eye(4, QQ)[:, 0]) is False
assert A.is_compat_col(DomainMatrix.eye(4, ZZ)[:, 0]) is True
def test_Module_call():
T = Poly(cyclotomic_poly(5, x))
B = PowerBasis(T)
assert B(0).col.flat() == [1, 0, 0, 0]
assert B(1).col.flat() == [0, 1, 0, 0]
col = DomainMatrix.eye(4, ZZ)[:, 2]
assert B(col).col == col
raises(ValueError, lambda: B(-1))
def test_Module_starts_with_unity():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
assert A.starts_with_unity() is True
assert B.starts_with_unity() is False
def test_Module_basis_elements():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
basis = B.basis_elements()
bp = B.basis_element_pullbacks()
for i, (e, p) in enumerate(zip(basis, bp)):
c = [0] * 4
assert e.module == B
assert p.module == A
c[i] = 1
assert e == B(to_col(c))
c[i] = 2
assert p == A(to_col(c))
def test_Module_zero():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
assert A.zero().col.flat() == [0, 0, 0, 0]
assert A.zero().module == A
assert B.zero().col.flat() == [0, 0, 0, 0]
assert B.zero().module == B
def test_Module_one():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
assert A.one().col.flat() == [1, 0, 0, 0]
assert A.one().module == A
assert B.one().col.flat() == [1, 0, 0, 0]
assert B.one().module == A
def test_Module_element_from_rational():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
rA = A.element_from_rational(QQ(22, 7))
rB = B.element_from_rational(QQ(22, 7))
assert rA.coeffs == [22, 0, 0, 0]
assert rA.denom == 7
assert rA.module == A
assert rB.coeffs == [22, 0, 0, 0]
assert rB.denom == 7
assert rB.module == A
def test_Module_submodule_from_gens():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
gens = [2*A(0), 2*A(1), 6*A(0), 6*A(1)]
B = A.submodule_from_gens(gens)
# Because the 3rd and 4th generators do not add anything new, we expect
# the cols of the matrix of B to just reproduce the first two gens:
M = gens[0].column().hstack(gens[1].column())
assert B.matrix == M
# At least one generator must be provided:
raises(ValueError, lambda: A.submodule_from_gens([]))
# All generators must belong to A:
raises(ValueError, lambda: A.submodule_from_gens([3*A(0), B(0)]))
def test_Module_submodule_from_matrix():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
e = B(to_col([1, 2, 3, 4]))
f = e.to_parent()
assert f.col.flat() == [2, 4, 6, 8]
# Matrix must be over ZZ:
raises(ValueError, lambda: A.submodule_from_matrix(DomainMatrix.eye(4, QQ)))
# Number of rows of matrix must equal number of generators of module A:
raises(ValueError, lambda: A.submodule_from_matrix(2 * DomainMatrix.eye(5, ZZ)))
def test_Module_whole_submodule():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.whole_submodule()
e = B(to_col([1, 2, 3, 4]))
f = e.to_parent()
assert f.col.flat() == [1, 2, 3, 4]
e0, e1, e2, e3 = B(0), B(1), B(2), B(3)
assert e2 * e3 == e0
assert e3 ** 2 == e1
def test_PowerBasis_repr():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
assert repr(A) == 'PowerBasis(x**4 + x**3 + x**2 + x + 1)'
def test_PowerBasis_eq():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = PowerBasis(T)
assert A == B
def test_PowerBasis_mult_tab():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
M = A.mult_tab()
exp = {0: {0: [1, 0, 0, 0], 1: [0, 1, 0, 0], 2: [0, 0, 1, 0], 3: [0, 0, 0, 1]},
1: {1: [0, 0, 1, 0], 2: [0, 0, 0, 1], 3: [-1, -1, -1, -1]},
2: {2: [-1, -1, -1, -1], 3: [1, 0, 0, 0]},
3: {3: [0, 1, 0, 0]}}
# We get the table we expect:
assert M == exp
# And all entries are of expected type:
assert all(is_int(c) for u in M for v in M[u] for c in M[u][v])
def test_PowerBasis_represent():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
col = to_col([1, 2, 3, 4])
a = A(col)
assert A.represent(a) == col
b = A(col, denom=2)
raises(ClosureFailure, lambda: A.represent(b))
def test_PowerBasis_element_from_poly():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
f = Poly(1 + 2*x)
g = Poly(x**4)
h = Poly(0, x)
assert A.element_from_poly(f).coeffs == [1, 2, 0, 0]
assert A.element_from_poly(g).coeffs == [-1, -1, -1, -1]
assert A.element_from_poly(h).coeffs == [0, 0, 0, 0]
def test_PowerBasis_element__conversions():
k = QQ.cyclotomic_field(5)
L = QQ.cyclotomic_field(7)
B = PowerBasis(k)
# ANP --> PowerBasisElement
a = k([QQ(1, 2), QQ(1, 3), 5, 7])
e = B.element_from_ANP(a)
assert e.coeffs == [42, 30, 2, 3]
assert e.denom == 6
# PowerBasisElement --> ANP
assert e.to_ANP() == a
# Cannot convert ANP from different field
d = L([QQ(1, 2), QQ(1, 3), 5, 7])
raises(UnificationFailed, lambda: B.element_from_ANP(d))
# AlgebraicNumber --> PowerBasisElement
alpha = k.to_alg_num(a)
eps = B.element_from_alg_num(alpha)
assert eps.coeffs == [42, 30, 2, 3]
assert eps.denom == 6
# PowerBasisElement --> AlgebraicNumber
assert eps.to_alg_num() == alpha
# Cannot convert AlgebraicNumber from different field
delta = L.to_alg_num(d)
raises(UnificationFailed, lambda: B.element_from_alg_num(delta))
# When we don't know the field:
C = PowerBasis(k.ext.minpoly)
# Can convert from AlgebraicNumber:
eps = C.element_from_alg_num(alpha)
assert eps.coeffs == [42, 30, 2, 3]
assert eps.denom == 6
# But can't convert back:
raises(StructureError, lambda: eps.to_alg_num())
def test_Submodule_repr():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ), denom=3)
assert repr(B) == 'Submodule[[2, 0, 0, 0], [0, 2, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]]/3'
def test_Submodule_reduced():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = A.submodule_from_matrix(6 * DomainMatrix.eye(4, ZZ), denom=3)
D = C.reduced()
assert D.denom == 1 and D == C == B
def test_Submodule_discard_before():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
B.compute_mult_tab()
C = B.discard_before(2)
assert C.parent == B.parent
assert B.is_sq_maxrank_HNF() and not C.is_sq_maxrank_HNF()
assert C.matrix == B.matrix[:, 2:]
assert C.mult_tab() == {0: {0: [-2, -2], 1: [0, 0]}, 1: {1: [0, 0]}}
def test_Submodule_QQ_matrix():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = A.submodule_from_matrix(6 * DomainMatrix.eye(4, ZZ), denom=3)
assert C.QQ_matrix == B.QQ_matrix
def test_Submodule_represent():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
a0 = A(to_col([6, 12, 18, 24]))
a1 = A(to_col([2, 4, 6, 8]))
a2 = A(to_col([1, 3, 5, 7]))
b1 = B.represent(a1)
assert b1.flat() == [1, 2, 3, 4]
c0 = C.represent(a0)
assert c0.flat() == [1, 2, 3, 4]
Y = A.submodule_from_matrix(DomainMatrix([
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
], (3, 4), ZZ).transpose())
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
z0 = Z(to_col([1, 2, 3, 4, 5, 6]))
raises(ClosureFailure, lambda: Y.represent(A(3)))
raises(ClosureFailure, lambda: B.represent(a2))
raises(ClosureFailure, lambda: B.represent(z0))
def test_Submodule_is_compat_submodule():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
D = C.submodule_from_matrix(5 * DomainMatrix.eye(4, ZZ))
assert B.is_compat_submodule(C) is True
assert B.is_compat_submodule(A) is False
assert B.is_compat_submodule(D) is False
def test_Submodule_eq():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = A.submodule_from_matrix(6 * DomainMatrix.eye(4, ZZ), denom=3)
assert C == B
def test_Submodule_add():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(DomainMatrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
], (2, 4), ZZ).transpose(), denom=6)
C = A.submodule_from_matrix(DomainMatrix([
[0, 10, 0, 0],
[0, 0, 7, 0],
], (2, 4), ZZ).transpose(), denom=15)
D = A.submodule_from_matrix(DomainMatrix([
[20, 0, 0, 0],
[ 0, 20, 0, 0],
[ 0, 0, 14, 0],
], (3, 4), ZZ).transpose(), denom=30)
assert B + C == D
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
Y = Z.submodule_from_gens([Z(0), Z(1)])
raises(TypeError, lambda: B + Y)
def test_Submodule_mul():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
C = A.submodule_from_matrix(DomainMatrix([
[0, 10, 0, 0],
[0, 0, 7, 0],
], (2, 4), ZZ).transpose(), denom=15)
C1 = A.submodule_from_matrix(DomainMatrix([
[0, 20, 0, 0],
[0, 0, 14, 0],
], (2, 4), ZZ).transpose(), denom=3)
C2 = A.submodule_from_matrix(DomainMatrix([
[0, 0, 10, 0],
[0, 0, 0, 7],
], (2, 4), ZZ).transpose(), denom=15)
C3_unred = A.submodule_from_matrix(DomainMatrix([
[0, 0, 100, 0],
[0, 0, 0, 70],
[0, 0, 0, 70],
[-49, -49, -49, -49]
], (4, 4), ZZ).transpose(), denom=225)
C3 = A.submodule_from_matrix(DomainMatrix([
[4900, 4900, 0, 0],
[4410, 4410, 10, 0],
[2107, 2107, 7, 7]
], (3, 4), ZZ).transpose(), denom=225)
assert C * 1 == C
assert C ** 1 == C
assert C * 10 == C1
assert C * A(1) == C2
assert C.mul(C, hnf=False) == C3_unred
assert C * C == C3
assert C ** 2 == C3
def test_Submodule_reduce_element():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.whole_submodule()
b = B(to_col([90, 84, 80, 75]), denom=120)
C = B.submodule_from_matrix(DomainMatrix.eye(4, ZZ), denom=2)
b_bar_expected = B(to_col([30, 24, 20, 15]), denom=120)
b_bar = C.reduce_element(b)
assert b_bar == b_bar_expected
C = B.submodule_from_matrix(DomainMatrix.eye(4, ZZ), denom=4)
b_bar_expected = B(to_col([0, 24, 20, 15]), denom=120)
b_bar = C.reduce_element(b)
assert b_bar == b_bar_expected
C = B.submodule_from_matrix(DomainMatrix.eye(4, ZZ), denom=8)
b_bar_expected = B(to_col([0, 9, 5, 0]), denom=120)
b_bar = C.reduce_element(b)
assert b_bar == b_bar_expected
a = A(to_col([1, 2, 3, 4]))
raises(NotImplementedError, lambda: C.reduce_element(a))
C = B.submodule_from_matrix(DomainMatrix([
[5, 4, 3, 2],
[0, 8, 7, 6],
[0, 0,11,12],
[0, 0, 0, 1]
], (4, 4), ZZ).transpose())
raises(StructureError, lambda: C.reduce_element(b))
def test_is_HNF():
M = DM([
[3, 2, 1],
[0, 2, 1],
[0, 0, 1]
], ZZ)
M1 = DM([
[3, 2, 1],
[0, -2, 1],
[0, 0, 1]
], ZZ)
M2 = DM([
[3, 2, 3],
[0, 2, 1],
[0, 0, 1]
], ZZ)
assert is_sq_maxrank_HNF(M) is True
assert is_sq_maxrank_HNF(M1) is False
assert is_sq_maxrank_HNF(M2) is False
def test_make_mod_elt():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
col = to_col([1, 2, 3, 4])
eA = make_mod_elt(A, col)
eB = make_mod_elt(B, col)
assert isinstance(eA, PowerBasisElement)
assert not isinstance(eB, PowerBasisElement)
def test_ModuleElement_repr():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 2, 3, 4]), denom=2)
assert repr(e) == '[1, 2, 3, 4]/2'
def test_ModuleElement_reduced():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([2, 4, 6, 8]), denom=2)
f = e.reduced()
assert f.denom == 1 and f == e
def test_ModuleElement_reduced_mod_p():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([20, 40, 60, 80]))
f = e.reduced_mod_p(7)
assert f.coeffs == [-1, -2, -3, 3]
def test_ModuleElement_from_int_list():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
c = [1, 2, 3, 4]
assert ModuleElement.from_int_list(A, c).coeffs == c
def test_ModuleElement_len():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(0)
assert len(e) == 4
def test_ModuleElement_column():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(0)
col1 = e.column()
assert col1 == e.col and col1 is not e.col
col2 = e.column(domain=FF(5))
assert col2.domain.is_FF
def test_ModuleElement_QQ_col():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 2, 3, 4]), denom=1)
f = A(to_col([3, 6, 9, 12]), denom=3)
assert e.QQ_col == f.QQ_col
def test_ModuleElement_to_ancestors():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
D = C.submodule_from_matrix(5 * DomainMatrix.eye(4, ZZ))
eD = D(0)
eC = eD.to_parent()
eB = eD.to_ancestor(B)
eA = eD.over_power_basis()
assert eC.module is C and eC.coeffs == [5, 0, 0, 0]
assert eB.module is B and eB.coeffs == [15, 0, 0, 0]
assert eA.module is A and eA.coeffs == [30, 0, 0, 0]
a = A(0)
raises(ValueError, lambda: a.to_parent())
def test_ModuleElement_compatibility():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
C = B.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
D = B.submodule_from_matrix(5 * DomainMatrix.eye(4, ZZ))
assert C(0).is_compat(C(1)) is True
assert C(0).is_compat(D(0)) is False
u, v = C(0).unify(D(0))
assert u.module is B and v.module is B
assert C(C.represent(u)) == C(0) and D(D.represent(v)) == D(0)
u, v = C(0).unify(C(1))
assert u == C(0) and v == C(1)
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
raises(UnificationFailed, lambda: C(0).unify(Z(1)))
def test_ModuleElement_eq():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 2, 3, 4]), denom=1)
f = A(to_col([3, 6, 9, 12]), denom=3)
assert e == f
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
assert e != Z(0)
assert e != 3.14
def test_ModuleElement_equiv():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 2, 3, 4]), denom=1)
f = A(to_col([3, 6, 9, 12]), denom=3)
assert e.equiv(f)
C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
g = C(to_col([1, 2, 3, 4]), denom=1)
h = A(to_col([3, 6, 9, 12]), denom=1)
assert g.equiv(h)
assert C(to_col([5, 0, 0, 0]), denom=7).equiv(QQ(15, 7))
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
raises(UnificationFailed, lambda: e.equiv(Z(0)))
assert e.equiv(3.14) is False
def test_ModuleElement_add():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
e = A(to_col([1, 2, 3, 4]), denom=6)
f = A(to_col([5, 6, 7, 8]), denom=10)
g = C(to_col([1, 1, 1, 1]), denom=2)
assert e + f == A(to_col([10, 14, 18, 22]), denom=15)
assert e - f == A(to_col([-5, -4, -3, -2]), denom=15)
assert e + g == A(to_col([10, 11, 12, 13]), denom=6)
assert e + QQ(7, 10) == A(to_col([26, 10, 15, 20]), denom=30)
assert g + QQ(7, 10) == A(to_col([22, 15, 15, 15]), denom=10)
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
raises(TypeError, lambda: e + Z(0))
raises(TypeError, lambda: e + 3.14)
def test_ModuleElement_mul():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
e = A(to_col([0, 2, 0, 0]), denom=3)
f = A(to_col([0, 0, 0, 7]), denom=5)
g = C(to_col([0, 0, 0, 1]), denom=2)
h = A(to_col([0, 0, 3, 1]), denom=7)
assert e * f == A(to_col([-14, -14, -14, -14]), denom=15)
assert e * g == A(to_col([-1, -1, -1, -1]))
assert e * h == A(to_col([-2, -2, -2, 4]), denom=21)
assert e * QQ(6, 5) == A(to_col([0, 4, 0, 0]), denom=5)
assert (g * QQ(10, 21)).equiv(A(to_col([0, 0, 0, 5]), denom=7))
assert e // QQ(6, 5) == A(to_col([0, 5, 0, 0]), denom=9)
U = Poly(cyclotomic_poly(7, x))
Z = PowerBasis(U)
raises(TypeError, lambda: e * Z(0))
raises(TypeError, lambda: e * 3.14)
raises(TypeError, lambda: e // 3.14)
raises(ZeroDivisionError, lambda: e // 0)
def test_ModuleElement_div():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
e = A(to_col([0, 2, 0, 0]), denom=3)
f = A(to_col([0, 0, 0, 7]), denom=5)
g = C(to_col([1, 1, 1, 1]))
assert e // f == 10*A(3)//21
assert e // g == -2*A(2)//9
assert 3 // g == -A(1)
def test_ModuleElement_pow():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
C = A.submodule_from_matrix(3 * DomainMatrix.eye(4, ZZ))
e = A(to_col([0, 2, 0, 0]), denom=3)
g = C(to_col([0, 0, 0, 1]), denom=2)
assert e ** 3 == A(to_col([0, 0, 0, 8]), denom=27)
assert g ** 2 == C(to_col([0, 3, 0, 0]), denom=4)
assert e ** 0 == A(to_col([1, 0, 0, 0]))
assert g ** 0 == A(to_col([1, 0, 0, 0]))
assert e ** 1 == e
assert g ** 1 == g
def test_ModuleElement_mod():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 15, 8, 0]), denom=2)
assert e % 7 == A(to_col([1, 1, 8, 0]), denom=2)
assert e % QQ(1, 2) == A.zero()
assert e % QQ(1, 3) == A(to_col([1, 1, 0, 0]), denom=6)
B = A.submodule_from_gens([A(0), 5*A(1), 3*A(2), A(3)])
assert e % B == A(to_col([1, 5, 2, 0]), denom=2)
C = B.whole_submodule()
raises(TypeError, lambda: e % C)
def test_PowerBasisElement_polys():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 15, 8, 0]), denom=2)
assert e.numerator(x=zeta) == Poly(8 * zeta ** 2 + 15 * zeta + 1, domain=ZZ)
assert e.poly(x=zeta) == Poly(4 * zeta ** 2 + QQ(15, 2) * zeta + QQ(1, 2), domain=QQ)
def test_PowerBasisElement_norm():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
lam = A(to_col([1, -1, 0, 0]))
assert lam.norm() == 5
def test_PowerBasisElement_inverse():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
e = A(to_col([1, 1, 1, 1]))
assert 2 // e == -2*A(1)
assert e ** -3 == -A(3)
def test_ModuleHomomorphism_matrix():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
phi = ModuleEndomorphism(A, lambda a: a ** 2)
M = phi.matrix()
assert M == DomainMatrix([
[1, 0, -1, 0],
[0, 0, -1, 1],
[0, 1, -1, 0],
[0, 0, -1, 0]
], (4, 4), ZZ)
def test_ModuleHomomorphism_kernel():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
phi = ModuleEndomorphism(A, lambda a: a ** 5)
N = phi.kernel()
assert N.n == 3
def test_EndomorphismRing_represent():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
R = A.endomorphism_ring()
phi = R.inner_endomorphism(A(1))
col = R.represent(phi)
assert col.transpose() == DomainMatrix([
[0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -1, -1, -1, -1]
], (1, 16), ZZ)
B = A.submodule_from_matrix(DomainMatrix.zeros((4, 0), ZZ))
S = B.endomorphism_ring()
psi = S.inner_endomorphism(A(1))
col = S.represent(psi)
assert col == DomainMatrix([], (0, 0), ZZ)
raises(NotImplementedError, lambda: R.represent(3.14))
def test_find_min_poly():
T = Poly(cyclotomic_poly(5, x))
A = PowerBasis(T)
powers = []
m = find_min_poly(A(1), QQ, x=x, powers=powers)
assert m == Poly(T, domain=QQ)
assert len(powers) == 5
# powers list need not be passed
m = find_min_poly(A(1), QQ, x=x)
assert m == Poly(T, domain=QQ)
B = A.submodule_from_matrix(2 * DomainMatrix.eye(4, ZZ))
raises(MissingUnityError, lambda: find_min_poly(B(1), QQ))
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