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from sympy.abc import x
from sympy.core.numbers import (I, Rational)
from sympy.core.singleton import S
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.polys import Poly, cyclotomic_poly
from sympy.polys.domains import FF, QQ
from sympy.polys.matrices import DomainMatrix, DM
from sympy.polys.matrices.exceptions import DMRankError
from sympy.polys.numberfields.utilities import (
AlgIntPowers, coeff_search, extract_fundamental_discriminant,
isolate, supplement_a_subspace,
)
from sympy.printing.lambdarepr import IntervalPrinter
from sympy.testing.pytest import raises
def test_AlgIntPowers_01():
T = Poly(cyclotomic_poly(5))
zeta_pow = AlgIntPowers(T)
raises(ValueError, lambda: zeta_pow[-1])
for e in range(10):
a = e % 5
if a < 4:
c = zeta_pow[e]
assert c[a] == 1 and all(c[i] == 0 for i in range(4) if i != a)
else:
assert zeta_pow[e] == [-1] * 4
def test_AlgIntPowers_02():
T = Poly(x**3 + 2*x**2 + 3*x + 4)
m = 7
theta_pow = AlgIntPowers(T, m)
for e in range(10):
computed = theta_pow[e]
coeffs = (Poly(x)**e % T + Poly(x**3)).rep.to_list()[1:]
expected = [c % m for c in reversed(coeffs)]
assert computed == expected
def test_coeff_search():
C = []
search = coeff_search(2, 1)
for i, c in enumerate(search):
C.append(c)
if i == 12:
break
assert C == [[1, 1], [1, 0], [1, -1], [0, 1], [2, 2], [2, 1], [2, 0], [2, -1], [2, -2], [1, 2], [1, -2], [0, 2], [3, 3]]
def test_extract_fundamental_discriminant():
# To extract, integer must be 0 or 1 mod 4.
raises(ValueError, lambda: extract_fundamental_discriminant(2))
raises(ValueError, lambda: extract_fundamental_discriminant(3))
# Try many cases, of different forms:
cases = (
(0, {}, {0: 1}),
(1, {}, {}),
(8, {2: 3}, {}),
(-8, {2: 3, -1: 1}, {}),
(12, {2: 2, 3: 1}, {}),
(36, {}, {2: 1, 3: 1}),
(45, {5: 1}, {3: 1}),
(48, {2: 2, 3: 1}, {2: 1}),
(1125, {5: 1}, {3: 1, 5: 1}),
)
for a, D_expected, F_expected in cases:
D, F = extract_fundamental_discriminant(a)
assert D == D_expected
assert F == F_expected
def test_supplement_a_subspace_1():
M = DM([[1, 7, 0], [2, 3, 4]], QQ).transpose()
# First supplement over QQ:
B = supplement_a_subspace(M)
assert B[:, :2] == M
assert B[:, 2] == DomainMatrix.eye(3, QQ).to_dense()[:, 0]
# Now supplement over FF(7):
M = M.convert_to(FF(7))
B = supplement_a_subspace(M)
assert B[:, :2] == M
# When we work mod 7, first col of M goes to [1, 0, 0],
# so the supplementary vector cannot equal this, as it did
# when we worked over QQ. Instead, we get the second std basis vector:
assert B[:, 2] == DomainMatrix.eye(3, FF(7)).to_dense()[:, 1]
def test_supplement_a_subspace_2():
M = DM([[1, 0, 0], [2, 0, 0]], QQ).transpose()
with raises(DMRankError):
supplement_a_subspace(M)
def test_IntervalPrinter():
ip = IntervalPrinter()
assert ip.doprint(x**Rational(1, 3)) == "x**(mpi('1/3'))"
assert ip.doprint(sqrt(x)) == "x**(mpi('1/2'))"
def test_isolate():
assert isolate(1) == (1, 1)
assert isolate(S.Half) == (S.Half, S.Half)
assert isolate(sqrt(2)) == (1, 2)
assert isolate(-sqrt(2)) == (-2, -1)
assert isolate(sqrt(2), eps=Rational(1, 100)) == (Rational(24, 17), Rational(17, 12))
assert isolate(-sqrt(2), eps=Rational(1, 100)) == (Rational(-17, 12), Rational(-24, 17))
raises(NotImplementedError, lambda: isolate(I))
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