def test_solve_lin_sys_2x2_one():
domain, x1, x2 = ring("x1,x2", QQ)
eqs = [x1 + x2 - 5,
2*x1 - x2]
sol = {x1: QQ(5, 3), x2: QQ(10, 3)}
_sol = solve_lin_sys(eqs, domain)
assert _sol == sol and all(isinstance(s, domain.dtype) for s in _sol)
def test_PolyElement___call__():
R, x = ring("x", ZZ)
f = 3*x + 1
assert f(0) == 1
assert f(1) == 4
pytest.raises(ValueError, lambda: f())
pytest.raises(ValueError, lambda: f(0, 1))
pytest.raises(CoercionFailed, lambda: f(QQ(1, 7)))
R, x, y = ring("x,y", ZZ)
f = 3*x + y**2 + 1
assert f(0, 0) == 1
assert f(1, 7) == 53
Ry = R.drop(x)
assert f(0) == Ry.y**2 + 1
assert f(1) == Ry.y**2 + 4
pytest.raises(ValueError, lambda: f())
pytest.raises(ValueError, lambda: f(0, 1, 2))
pytest.raises(CoercionFailed, lambda: f(1, QQ(1, 7)))
pytest.raises(CoercionFailed, lambda: f(QQ(1, 7), 1))
pytest.raises(CoercionFailed, lambda: f(QQ(1, 7), QQ(1, 7)))
def test_PolyElement_evaluate():
R, x = ring("x", ZZ)
f = x**3 + 4*x**2 + 2*x + 3
r = f.evaluate(x, 0)
assert r == 3 and not isinstance(r, PolyElement)
pytest.raises(CoercionFailed, lambda: f.evaluate(x, QQ(1, 7)))
R, x, y, z = ring("x,y,z", ZZ)
f = (x*y)**3 + 4*(x*y)**2 + 2*x*y + 3
r = f.evaluate(x, 0)
assert r == 3 and isinstance(r, R.drop(x).dtype)
r = f.evaluate([(x, 0), (y, 0)])
assert r == 3 and isinstance(r, R.drop(x, y).dtype)
r = f.evaluate(y, 0)
assert r == 3 and isinstance(r, R.drop(y).dtype)
r = f.evaluate([(y, 0), (x, 0)])
assert r == 3 and isinstance(r, R.drop(y, x).dtype)
r = f.evaluate([(x, 0), (y, 0), (z, 0)])
assert r == 3 and not isinstance(r, PolyElement)
pytest.raises(CoercionFailed, lambda: f.evaluate([(x, 1), (y, QQ(1, 7))]))
pytest.raises(CoercionFailed, lambda: f.evaluate([(x, QQ(1, 7)), (y, 1)]))
pytest.raises(CoercionFailed, lambda: f.evaluate([(x, QQ(1, 7)), (y, QQ(1, 7))]))
def test_FracElement___mul__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f*g == g*f == 1/(x*y)
assert x*F.ring.gens[0] == F.ring.gens[0]*x == x**2
F, x, y = field("x,y", ZZ)
assert x*3 == 3*x
assert x*QQ(3, 7) == QQ(3, 7)*x == 3*x/7
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)
assert dict(f.numerator) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1}
assert dict(f.denominator) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = ((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)
assert dict(f.numerator) == {(1, 1, 0, 0): u + 1, (0, 0, 0, 0): 1}
assert dict(f.denominator) == {(0, 0, 1, 0): v - 1, (0, 0, 0, 1): -u*v, (0, 0, 0, 0): -1}
def test_PolyElement_gcd():
R, x, y = ring("x,y", QQ)
f = x**2/2 + x + QQ(1, 2)
g = x/2 + QQ(1, 2)
assert f.gcd(g) == x + 1
def test_FracElement___sub__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f - g == (-x + y)/(x*y)
assert x - F.ring.gens[0] == F.ring.gens[0] - x == 0
F, x, y = field("x,y", ZZ)
assert x - 3 == -(3 - x)
assert x - QQ(3, 7) == -(QQ(3, 7) - x) == (7*x - 3)/7
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = (u*v - x)/(y - u*v)
assert dict(f.numerator) == {(1, 0, 0, 0): -1, (0, 0, 0, 0): u*v}
assert dict(f.denominator) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): -u*v}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = (u*v - x)/(y - u*v)
assert dict(f.numerator) == {(1, 0, 0, 0): -1, (0, 0, 0, 0): u*v}
assert dict(f.denominator) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): -u*v}
Fuv, u, v = field("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Fuv)
f = u - x
assert dict(f) == {(0, 0, 0): u, (1, 0, 0): -Fuv.one}
def test_FracElement___add__():
F, x, y = field("x,y", QQ)
f, g = 1/x, 1/y
assert f + g == g + f == (x + y)/(x*y)
z = symbols('z')
pytest.raises(TypeError, lambda: x + z)
assert x + F.ring.gens[0] == F.ring.gens[0] + x == 2*x
F, x, y = field("x,y", ZZ)
assert x + 3 == 3 + x
assert x + QQ(3, 7) == QQ(3, 7) + x == (7*x + 3)/7
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = (u*v + x)/(y + u*v)
assert dict(f.numerator) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
assert dict(f.denominator) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = (u*v + x)/(y + u*v)
assert dict(f.numerator) == {(1, 0, 0, 0): 1, (0, 0, 0, 0): u*v}
assert dict(f.denominator) == {(0, 1, 0, 0): 1, (0, 0, 0, 0): u*v}
def test_modular():
assert solve_congruence(*list(zip([3, 4, 2], [12, 35, 17]))) == (1719,
7140)
assert solve_congruence(*list(zip([3, 4, 2], [12, 6, 17]))) is None
assert solve_congruence(*list(zip([3, 4, 2], [13, 7, 17]))) == (172, 1547)
assert solve_congruence(*list(zip([-10, -3, -15], [13, 7, 17]))) == (172,
1547)
assert solve_congruence(
*list(zip([-10, -3, 1, -15], [13, 7, 7, 17]))) is None
assert solve_congruence(
*list(zip([-10, -5, 2, -15], [13, 7, 7, 17]))) == (835, 1547)
assert solve_congruence(
*list(zip([-10, -5, 2, -15], [13, 7, 14, 17]))) == (2382, 3094)
assert solve_congruence(
*list(zip([-10, 2, 2, -15], [13, 7, 14, 17]))) == (2382, 3094)
assert solve_congruence(*list(zip((1, 1, 2), (3, 2, 4)))) is None
pytest.raises(
ValueError,
lambda: solve_congruence(*list(zip([3, 4, 2], [12.1, 35, 17]))))
assert integer_rational_reconstruction(ZZ(2), 3, ZZ) == QQ(-1)
assert integer_rational_reconstruction(ZZ(21), 33, ZZ) == QQ(-1)
assert integer_rational_reconstruction(ZZ(-21), 17, ZZ) == QQ(-4)
assert integer_rational_reconstruction(ZZ(17), 333, ZZ) is None
assert integer_rational_reconstruction(ZZ(49), 335, ZZ) == QQ(8, 7)
def test_Domain__algebraic_field():
alg = ZZ.algebraic_field(sqrt(3))
assert alg.minpoly == Poly(x**2 - 3)
assert alg.domain == QQ
assert alg.from_expr(sqrt(3)).denominator == 1
assert alg.from_expr(2 * sqrt(3)).denominator == 1
assert alg.from_expr(sqrt(3) / 2).denominator == 2
assert alg([QQ(7, 38), QQ(3, 2)]).denominator == 38
alg = QQ.algebraic_field(sqrt(2))
assert alg.minpoly == Poly(x**2 - 2)
assert alg.domain == QQ
alg = QQ.algebraic_field(sqrt(2), sqrt(3))
assert alg.minpoly == Poly(x**4 - 10 * x**2 + 1)
assert alg.domain == QQ
assert alg.is_nonpositive(alg([-1, 1])) is True
assert alg.is_nonnegative(alg([2, -1])) is True
assert alg(1).numerator == alg(1)
assert alg.from_expr(sqrt(3) / 2).numerator == alg.from_expr(2 * sqrt(3))
assert alg.from_expr(sqrt(3) / 2).denominator == 4
pytest.raises(DomainError, lambda: AlgebraicField(ZZ, sqrt(2)))
assert alg.characteristic == 0
assert alg.is_RealAlgebraicField is True
assert int(alg(2)) == 2
assert int(alg.from_expr(Rational(3, 2))) == 1
pytest.raises(TypeError, lambda: int(alg([1, 1])))
alg = QQ.algebraic_field(I)
assert alg.algebraic_field(I) == alg
assert alg.is_RealAlgebraicField is False
alg = QQ.algebraic_field(sqrt(2)).algebraic_field(sqrt(3))
assert alg.minpoly == Poly(x**2 - 3, x, domain=QQ.algebraic_field(sqrt(2)))
# issue sympy/sympy#14476
assert QQ.algebraic_field(Rational(1, 7)) is QQ
alg = QQ.algebraic_field(sqrt(2)).algebraic_field(I)
assert alg.from_expr(2 * sqrt(2) + I / 3) == alg(
[alg.domain(1) / 3, alg.domain(2 * sqrt(2))])
alg2 = QQ.algebraic_field(sqrt(2))
assert alg2.from_expr(sqrt(2)) == alg2.convert(alg.from_expr(sqrt(2)))
eq = -x**3 + 2 * x**2 + 3 * x - 2
rs = roots(eq, multiple=True)
alg = QQ.algebraic_field(rs[0])
assert alg.ext_root == RootOf(eq, 2)
alg1 = QQ.algebraic_field(I)
alg2 = QQ.algebraic_field(sqrt(2)).algebraic_field(I)
assert alg1 != alg2
def test_dmp_from_diofant():
assert dmp_from_diofant([Integer(1), Integer(2)], 0, ZZ) == [ZZ(1), ZZ(2)]
assert dmp_from_diofant([Rational(1, 2), Integer(3)], 0,
QQ) == [QQ(1, 2), QQ(3, 1)]
assert dmp_from_diofant([[Integer(1), Integer(2)], [Integer(0)]], 1,
ZZ) == [[ZZ(1), ZZ(2)], []]
assert dmp_from_diofant([[Rational(1, 2), Integer(2)]], 1,
QQ) == [[QQ(1, 2), QQ(2, 1)]]
def test_dmp_clear_denoms():
R0, X = ring('x', QQ)
R1 = R0.domain.ring.poly_ring('x')
assert R0.dmp_clear_denoms(0) == (1, 0)
assert R0.dmp_clear_denoms(1) == (1, 1)
assert R0.dmp_clear_denoms(7) == (1, 7)
assert R0.dmp_clear_denoms(QQ(7, 3)) == (3, 7)
assert R0.dmp_clear_denoms(3 * X**2 + X) == (1, 3 * X**2 + X)
assert R0.dmp_clear_denoms(X**2 + X / 2) == (2, 2 * X**2 + X)
assert R0.dmp_clear_denoms(3 * X**2 + X,
convert=True) == (1, 3 * R1.x**2 + R1.x)
assert R0.dmp_clear_denoms(X**2 + X / 2,
convert=True) == (2, 2 * R1.x**2 + R1.x)
assert R0.dmp_clear_denoms(X / 2 + QQ(1, 3)) == (6, 3 * X + 2)
assert R0.dmp_clear_denoms(X / 2 + QQ(1, 3),
convert=True) == (6, 3 * R1.x + 2)
assert R0.dmp_clear_denoms(3 * X**2 + X,
convert=True) == (1, 3 * R1.x**2 + R1.x)
assert R0.dmp_clear_denoms(X**2 + X / 2,
convert=True) == (2, 2 * R1.x**2 + R1.x)
R0, a = ring('a', EX)
assert R0.dmp_clear_denoms(3 * a / 2 + Rational(9, 4)) == (4, 6 * a + 9)
assert R0.dmp_clear_denoms(7) == (1, 7)
assert R0.dmp_clear_denoms(sin(x) / x * a) == (x, a * sin(x))
R0, X, Y = ring('x y', QQ)
R1 = R0.domain.ring.poly_ring('x', 'y')
assert R0.dmp_clear_denoms(0) == (1, 0)
assert R0.dmp_clear_denoms(1) == (1, 1)
assert R0.dmp_clear_denoms(7) == (1, 7)
assert R0.dmp_clear_denoms(QQ(7, 3)) == (3, 7)
assert R0.dmp_clear_denoms(3 * X**2 + X) == (1, 3 * X**2 + X)
assert R0.dmp_clear_denoms(X**2 + X / 2) == (2, 2 * X**2 + X)
assert R0.dmp_clear_denoms(3 * X**2 + X,
convert=True) == (1, 3 * R1.x**2 + R1.x)
assert R0.dmp_clear_denoms(X**2 + X / 2,
convert=True) == (2, 2 * R1.x**2 + R1.x)
R0, a, b = ring('a b', EX)
assert R0.dmp_clear_denoms(3 * a / 2 + Rational(9, 4)) == (4, 6 * a + 9)
assert R0.dmp_clear_denoms(7) == (1, 7)
assert R0.dmp_clear_denoms(sin(x) / x * b) == (x, b * sin(x))
def test_dup_sturm():
R, x = ring("x", QQ)
assert R.dup_sturm(5) == [1]
assert R.dup_sturm(x) == [x, 1]
f = x**3 - 2 * x**2 + 3 * x - 5
assert R.dup_sturm(f) == [
f, 3 * x**2 - 4 * x + 3, -10 * x / 9 + QQ(13, 3), -QQ(3303, 100)
]
def test_dup_euclidean_prs():
R, x = ring("x", QQ)
f = x**8 + x**6 - 3 * x**4 - 3 * x**3 + 8 * x**2 + 2 * x - 5
g = 3 * x**6 + 5 * x**4 - 4 * x**2 - 9 * x + 21
assert R.dup_euclidean_prs(f, g) == [
f, g, -5 * x**4 / 9 + x**2 / 9 - QQ(1, 3),
-117 * x**2 / 25 - 9 * x + QQ(441, 25),
233150 * x / 19773 - QQ(102500, 6591), -QQ(1288744821, 543589225)
]
def test_dup_count_complex_roots_implicit():
R, x = ring("x", ZZ)
f = (x**2 + 1) * (x - 1) * (x + 1) * x
assert R.dup_count_complex_roots(f) == 5
assert R.dup_count_complex_roots(f, sup=(0, 0)) == 3
assert R.dup_count_complex_roots(f, inf=(0, 0)) == 3
assert R.dup_count_complex_roots(f, inf=QQ(-2), sup=QQ(-1)) == 1
def test_Domain_convert():
assert QQ.convert(10e-52) == QQ(
1684996666696915,
1684996666696914987166688442938726917102321526408785780068975640576)
R, x = ring("x", ZZ)
assert ZZ.convert(x - x) == 0
assert ZZ.convert(x - x, R) == 0
F3 = FF(3)
assert F3.convert(Float(2.0)) == F3.dtype(2)
assert F3.convert(PythonRational(2, 1)) == F3.dtype(2)
pytest.raises(CoercionFailed, lambda: F3.convert(PythonRational(1, 2)))
assert F3.convert(2.0) == F3.dtype(2)
pytest.raises(CoercionFailed, lambda: F3.convert(2.1))
assert RR.convert(CC(1)) == RR(1)
pytest.raises(CoercionFailed, lambda: RR.convert(CC(1, 2)))
assert QQ.convert(ALG(1), ALG) == QQ(1)
pytest.raises(CoercionFailed, lambda: QQ.convert(ALG([1, 1]), ALG))
assert ZZ.convert(ALG(1), ALG) == ZZ(1)
pytest.raises(CoercionFailed, lambda: ZZ.convert(ALG([1, 1]), ALG))
assert EX.convert(ALG([1, 1]), ALG) == sqrt(2) + sqrt(3) + 1
ALG2 = QQ.algebraic_field(sqrt(2))
a2 = ALG2.convert(sqrt(2))
a = ALG.convert(a2, ALG2)
assert a.rep.to_dense() == [QQ(1, 2), 0, -QQ(9, 2), 0]
assert RR.convert(a) == RR(1.4142135623730951)
assert CC.convert(a) == CC(1.4142135623730951)
assert ZZ_python.convert(3.0) == ZZ_python.dtype(3)
pytest.raises(CoercionFailed, lambda: ZZ_python.convert(3.2))
assert CC.convert(QQ_python(1, 2)) == CC(0.5)
CC01 = ComplexField(tol=0.1)
assert CC.convert(CC01(0.3)) == CC(0.3)
assert RR.convert(complex(2 + 0j)) == RR(2)
pytest.raises(CoercionFailed, lambda: RR.convert(complex(2 + 3j)))
assert ALG.convert(EX(sqrt(2)), EX) == ALG.from_expr(sqrt(2))
pytest.raises(CoercionFailed, lambda: ALG.convert(EX(sqrt(5)), EX))
pytest.raises(CoercionFailed, lambda: ALG2.convert(ALG.unit))
def test_dmp_ground_monic():
R, x = ring('x', ZZ)
assert R.dmp_ground_monic(3 * x**2 + 6 * x + 9) == x**2 + 2 * x + 3
pytest.raises(ExactQuotientFailed,
lambda: R.dmp_ground_monic(3 * x**2 + 4 * x + 5))
R, x = ring('x', QQ)
assert R.dmp_ground_monic(0) == 0
assert R.dmp_ground_monic(1) == 1
assert R.dmp_ground_monic(7 * x**2 + x + 21) == x**2 + x / 7 + 3
assert R.dmp_ground_monic(3 * x**2 + 4 * x +
2) == x**2 + 4 * x / 3 + QQ(2, 3)
R, x, y = ring('x y', ZZ)
assert R.dmp_ground_monic(3 * x**2 + 6 * x + 9) == x**2 + 2 * x + 3
pytest.raises(ExactQuotientFailed,
lambda: R.dmp_ground_monic(3 * x**2 + 4 * x + 5))
R, x, y = ring('x y', QQ)
assert R.dmp_ground_monic(0) == 0
assert R.dmp_ground_monic(1) == 1
assert R.dmp_ground_monic(7 * x**2 + x + 21) == x**2 + x / 7 + 3
def test_PolyElement_compose():
R, x = ring("x", ZZ)
f = x**3 + 4*x**2 + 2*x + 3
r = f.compose(x, 0)
assert r == 3 and isinstance(r, R.dtype)
assert f.compose(x, x) == f
assert f.compose(x, x**2) == x**6 + 4*x**4 + 2*x**2 + 3
pytest.raises(CoercionFailed, lambda: f.compose(x, QQ(1, 7)))
R, x, y, z = ring("x,y,z", ZZ)
f = x**3 + 4*x**2 + 2*x + 3
r = f.compose(x, 0)
assert r == 3 and isinstance(r, R.dtype)
r = f.compose([(x, 0), (y, 0)])
assert r == 3 and isinstance(r, R.dtype)
r = f.compose({x: 0, y: 0})
assert r == 3 and isinstance(r, R.dtype)
pytest.raises(ValueError, lambda: f.compose("spam"))
r = (x**3 + 4*x**2 + 2*x*y*z + 3).compose(x, y*z**2 - 1)
q = (y*z**2 - 1)**3 + 4*(y*z**2 - 1)**2 + 2*(y*z**2 - 1)*y*z + 3
assert r == q and isinstance(r, R.dtype)
def test_dup_count_complex_roots_1():
R, x = ring("x", ZZ)
f = x - 1
assert R.dup_count_complex_roots(f, a, b) == 1
assert R.dup_count_complex_roots(f, c, d) == 1
f = -f
assert R.dup_count_complex_roots(f, a, b) == 1
assert R.dup_count_complex_roots(f, c, d) == 1
f = x + 1
assert R.dup_count_complex_roots(f, a, b) == 1
assert R.dup_count_complex_roots(f, c, d) == 0
R, x = ring("x", QQ)
f = x - QQ(1, 2)
assert R.dup_count_complex_roots(f, c, d) == 1
R, x = ring("x", EX)
pytest.raises(DomainError, lambda: R.dup_count_complex_roots(x))
def test_PolyElement_terms():
R, x, y, z = ring("x,y,z", QQ)
terms = (x**2/3 + y**3/4 + z**4/5).terms()
assert terms == [((2, 0, 0), QQ(1, 3)), ((0, 3, 0), QQ(1, 4)), ((0, 0, 4), QQ(1, 5))]
R, x, y = ring("x,y", ZZ)
f = x*y**7 + 2*x**2*y**3
assert f.terms() == f.terms(lex) == f.terms('lex') == [((2, 3), 2), ((1, 7), 1)]
assert f.terms(grlex) == f.terms('grlex') == [((1, 7), 1), ((2, 3), 2)]
R, x, y = ring("x,y", ZZ, grlex)
f = x*y**7 + 2*x**2*y**3
assert f.terms() == f.terms(grlex) == f.terms('grlex') == [((1, 7), 1), ((2, 3), 2)]
assert f.terms(lex) == f.terms('lex') == [((2, 3), 2), ((1, 7), 1)]
def test_PolyElement_coeffs():
R, x, y, z = ring("x,y,z", QQ)
coeffs = (x**2/3 + y**3/4 + z**4/5).coeffs()
assert coeffs == [QQ(1, 3), QQ(1, 4), QQ(1, 5)]
R, x, y = ring("x,y", ZZ)
f = x*y**7 + 2*x**2*y**3
assert f.coeffs() == f.coeffs(lex) == f.coeffs('lex') == [2, 1]
assert f.coeffs(grlex) == f.coeffs('grlex') == [1, 2]
R, x, y = ring("x,y", ZZ, grlex)
f = x*y**7 + 2*x**2*y**3
assert f.coeffs() == f.coeffs(grlex) == f.coeffs('grlex') == [1, 2]
assert f.coeffs(lex) == f.coeffs('lex') == [2, 1]
def _do_test_benchmark_minimal_polynomial():
R, x, y, z = ring("x,y,z", QQ, lex)
F = [x**3 + x + 1, y**2 + y + 1, (x + y) * z - (x**2 + y)]
G = [x + 155*z**5/2067 - 355*z**4/689 + 6062*z**3/2067 - 3687*z**2/689 + 6878*z/2067 - QQ(25, 53),
y + 4*z**5/53 - 91*z**4/159 + 523*z**3/159 - 387*z**2/53 + 1043*z/159 - QQ(308, 159),
z**6 - 7*z**5 + 41*z**4 - 82*z**3 + 89*z**2 - 46*z + 13]
assert groebner(F, R) == G
def test_arithmetics():
assert ZZ.rem(ZZ(2), ZZ(3)) == 2
assert ZZ.div(ZZ(2), ZZ(3)) == (0, 2)
assert QQ.rem(QQ(2, 3), QQ(4, 7)) == 0
assert QQ.div(QQ(2, 3), QQ(4, 7)) == (QQ(7, 6), 0)
assert QQ_python.factorial(QQ_python(7, 2)) == 6
assert CC.gcd(CC(1), CC(2)) == 1
assert CC.lcm(CC(1), CC(2)) == 2
assert EX(Rational(2, 3)).numerator == 2
assert EX(Rational(2, 3)).denominator == 3
assert abs(EX(-2)) == 2
assert -EX(2) == -2
assert 2 + EX(3) == EX(3) + 2 == 5
assert 2 - EX(3) == EX(2) - 3 == -1
assert 2 * EX(3) == EX(3) * 2 == 6
assert 2 / EX(3) == EX(2) / 3 == EX(Rational(2, 3))
assert EX(2)**2 == 4
pytest.raises(TypeError, lambda: EX(2) + object())
pytest.raises(TypeError, lambda: EX(2) - object())
pytest.raises(TypeError, lambda: EX(2) * object())
pytest.raises(TypeError, lambda: EX(2)**object())
pytest.raises(TypeError, lambda: EX(2) / object())
F11 = FF(11)
assert +F11(2) == F11(2)
assert F11(5) + 7 == 7 + F11(5) == F11(1)
assert F11(5) - 7 == 5 - F11(7) == F11(9)
assert F11(5) * 7 == 7 * F11(5) == F11(2)
assert F11(5) / 9 == 5 / F11(9) == F11(3)
assert F11(4) % 9 == 4 % F11(9) == F11(4)
pytest.raises(TypeError, lambda: F11(2) + object())
pytest.raises(TypeError, lambda: F11(2) - object())
pytest.raises(TypeError, lambda: F11(2) * object())
pytest.raises(TypeError, lambda: F11(2)**object())
pytest.raises(TypeError, lambda: F11(2) / object())
pytest.raises(TypeError, lambda: F11(2) % object())
pytest.raises(TypeError, lambda: object() % F11(2))
def test_dmp_quo_ground():
R, x = ring('x', ZZ)
pytest.raises(ZeroDivisionError,
lambda: R.dmp_quo_ground(x**2 + 2 * x + 3, ZZ(0)))
f = 3 * x**2 + 2
assert R.dmp_quo_ground(f, ZZ(2)) == x**2 + 1
R, x = ring('x', QQ)
f = 3 * x**2 + 2
assert R.dmp_quo_ground(f, QQ(2)) == 3 * x**2 / 2 + 1
R, x = ring('x', ZZ)
f = 0
assert R.dmp_quo_ground(f, ZZ(3)) == 0
f = 6 * x**2 + 2 * x + 8
assert R.dmp_quo_ground(f, ZZ(1)) == f
assert R.dmp_quo_ground(f, ZZ(2)) == 3 * x**2 + x + 4
assert R.dmp_quo_ground(f, ZZ(3)) == 2 * x**2 + 2
R, x = ring('x', QQ)
f = 6 * x**2 + 2 * x + 8
assert R.dmp_quo_ground(f, QQ(1)) == f
assert R.dmp_quo_ground(f, QQ(2)) == 3 * x**2 + x + 4
assert R.dmp_quo_ground(f, QQ(7)) == 6 * x**2 / 7 + 2 * x / 7 + QQ(8, 7)
R, x, y = ring('x y', ZZ)
f = 6 * x**2 + 2 * x + 8
assert R.dmp_quo_ground(f, ZZ(1)) == f
assert R.dmp_quo_ground(f, ZZ(2)) == 3 * x**2 + x + 4
assert R.dmp_quo_ground(f, ZZ(3)) == 2 * x**2 + 2
def test_dmp_validate():
assert dmp_validate([]) == ([], 0)
assert dmp_validate([0, 0, 0, 1, 0]) == ([1, 0], 0)
assert dmp_validate([[[]]]) == ([[[]]], 2)
assert dmp_validate([[0], [], [0], [1], [0]]) == ([[1], []], 1)
pytest.raises(ValueError, lambda: dmp_validate([[0], 0, [0], [1], [0]]))
pytest.raises(TypeError,
lambda: dmp_validate([[], [0, QQ(1, 2)], [1]], ZZ))
def test_FracElement___truediv__():
F, x, y = field("x,y", QQ)
f, g = 1 / x, 1 / y
assert f / g == y / x
assert x / F.ring.gens[0] == F.ring.gens[0] / x == 1
F, x, y = field("x,y", ZZ)
assert x * 3 == 3 * x
assert x / QQ(3, 7) == (QQ(3, 7) / x)**-1 == 7 * x / 3
pytest.raises(ZeroDivisionError, lambda: x / 0)
pytest.raises(ZeroDivisionError, lambda: 1 / (x - x))
pytest.raises(ZeroDivisionError, lambda: x / (x - x))
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
f = (u * v) / (x * y)
assert dict(f.numer) == {(0, 0, 0, 0): u * v}
assert dict(f.denom) == {(1, 1, 0, 0): 1}
g = (x * y) / (u * v)
assert dict(g.numer) == {(1, 1, 0, 0): 1}
assert dict(g.denom) == {(0, 0, 0, 0): u * v}
Ruv, u, v = ring("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Ruv)
f = (u * v) / (x * y)
assert dict(f.numer) == {(0, 0, 0, 0): u * v}
assert dict(f.denom) == {(1, 1, 0, 0): 1}
g = (x * y) / (u * v)
assert dict(g.numer) == {(1, 1, 0, 0): 1}
assert dict(g.denom) == {(0, 0, 0, 0): u * v}
Fuv, u, v = field("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Fuv)
pytest.raises(TypeError, lambda: u / x)
def test_dup_gcdex():
R, x = ring("x", QQ)
f = x**4 - 2*x**3 - 6*x**2 + 12*x + 15
g = x**3 + x**2 - 4*x - 4
s = -x/5 + QQ(3, 5)
t = x**2/5 - 6*x/5 + 2
h = x + 1
assert f.half_gcdex(g) == (s, h)
assert f.gcdex(g) == (s, t, h)
f = x**4 + 4*x**3 - x + 1
g = x**3 - x + 1
s, t, h = f.gcdex(g)
S, T, H = g.gcdex(f)
assert s*f + t*g == h
assert S*g + T*f == H
f = 2*x
g = x**2 - 16
s = x/32
t = -QQ(1, 16)
h = 1
assert f.half_gcdex(g) == (s, h)
assert f.gcdex(g) == (s, t, h)
R, x = ring("x", FF(11))
assert R.zero.gcdex(R(2)) == (0, 6, 1)
assert R(2).gcdex(R(2)) == (0, 6, 1)
assert R.zero.gcdex(3*x) == (0, 4, x)
assert (3*x).gcdex(3*x) == (0, 4, x)
assert (x**2 + 8*x + 7).gcdex(x**3 + 7*x**2 + x + 7) == (5*x + 6, 6, x + 7)
def test_dmp_sqr():
R, x = ring('x', ZZ)
assert R.dmp_sqr(0) == 0
assert R.dmp_sqr(2) == 4
assert R.dmp_sqr(x + 2) == x**2 + 4 * x + 4
assert R.dmp_sqr(2 * x**4 + x +
7) == 4 * x**8 + 4 * x**5 + 28 * x**4 + x**2 + 14 * x + 49
R, x = ring('x', QQ)
assert R.dmp_sqr(0) == 0
assert R.dmp_sqr(QQ(2, 3)) == QQ(4, 9)
assert R.dmp_sqr(x / 3 + QQ(2, 3)) == x**2 / 9 + 4 * x / 9 + QQ(4, 9)
R, x = ring('x', FF(7))
assert R.dmp_sqr(3 * x + 4) == 2 * x**2 + 3 * x + 2
R, x, y, z = ring('x y z', ZZ)
assert R.dmp_sqr(0) == 0
assert R.dmp_sqr(2) == 4
R, x, y, z = ring('x y z', QQ)
assert R.dmp_sqr(0) == 0
assert R.dmp_sqr(QQ(2, 3)) == QQ(4, 9)
R, x, y = ring('x y', FF(7))
assert R.dmp_sqr(3 * x + 4) == 2 * x**2 + 3 * x + 2
def test_dup_isolate_real_roots_list_QQ():
R, x = ring("x", ZZ)
f, g = x**5 - 200, x**5 - 201
assert R.dup_isolate_real_roots_list([f, g]) == \
[((QQ(75, 26), QQ(101, 35)), {0: 1}), ((QQ(309, 107), QQ(26, 9)), {1: 1})]
R, x = ring("x", QQ)
f, g = -x**5 / 200 + 1, -x**5 / 201 + 1
assert R.dup_isolate_real_roots_list([f, g]) == \
[((QQ(75, 26), QQ(101, 35)), {0: 1}), ((QQ(309, 107), QQ(26, 9)), {1: 1})]
def test_PolyElement_gcdex():
_, x = ring("x", QQ)
f, g = 2*x, x**2 - 16
s, t, h = x/32, -QQ(1, 16), 1
assert f.half_gcdex(g) == (s, h)
assert f.gcdex(g) == (s, t, h)
_, x, y = ring("x,y", QQ)
pytest.raises(MultivariatePolynomialError, lambda: (x + y).half_gcdex(x*y))
pytest.raises(MultivariatePolynomialError, lambda: (x + y).gcdex(x*y))
def test_dup_count_complex_roots_exclude():
R, x = ring("x", ZZ)
f = (x**2 + 1) * (x - 1) * (x + 1) * x
a, b = (-1, 0), (1, 1)
assert R.dup_count_complex_roots(f, a, b) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['S']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['N']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['S', 'N']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['E']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['W']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['E', 'W']) == 4
assert R.dup_count_complex_roots(f, a, b, exclude=['N', 'S', 'E',
'W']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['SW']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['SE']) == 3
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE']) == 2
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S']) == 1
assert R.dup_count_complex_roots(f, a, b, exclude=['SW', 'SE', 'S',
'N']) == 0
a, b = (0, 0), (1, 1)
assert R.dup_count_complex_roots(f, a, b, exclude=True) == 1
R, x = ring("x", QQ.algebraic_field(I))
f = x**4 + I * x**3 - x + 1
assert R.dup_count_complex_roots(f, inf=(0, 0), sup=(1, 1)) == 1
r = R.dup_isolate_complex_roots_sqf(f)
assert r == [((QQ(-201, 100), QQ(-201, 100)), (0, 0)),
((QQ(-201, 100), 0), (0, QQ(201, 100))),
((0, QQ(-201, 100)), (QQ(201, 100), 0)),
((0, 0), (QQ(201, 100), QQ(201, 100)))]
assert all(R.dup_count_complex_roots(f, inf=i, sup=s) == 1 for i, s in r)
