def test_minpoly_fraction_field():
assert minimal_polynomial(1/x)(y) == x*y - 1
assert minimal_polynomial(1/(x + 1))(y) == x*y + y - 1
assert minimal_polynomial(sqrt(x))(y) == y**2 - x
assert minimal_polynomial(sqrt(x), method='groebner')(y) == y**2 - x
assert minimal_polynomial(sqrt(x + 1))(y) == y**2 - x - 1
assert minimal_polynomial(sqrt(x)/x)(y) == x*y**2 - 1
assert minimal_polynomial(sqrt(2)*sqrt(x))(y) == y**2 - 2 * x
assert minimal_polynomial(sqrt(2) + sqrt(x))(y) == \
y**4 - 2*x*y**2 - 4*y**2 + x**2 - 4*x + 4
assert minimal_polynomial(sqrt(2) + sqrt(x), method='groebner')(y) == \
y**4 - 2*x*y**2 - 4*y**2 + x**2 - 4*x + 4
assert minimal_polynomial(cbrt(x))(y) == y**3 - x
assert minimal_polynomial(cbrt(x) + sqrt(x))(y) == \
y**6 - 3*x*y**4 - 2*x*y**3 + 3*x**2*y**2 - 6*x**2*y - x**3 + x**2
assert minimal_polynomial(sqrt(x)/z)(y) == z**2*y**2 - x
assert minimal_polynomial(sqrt(x)/(z + 1))(y) == z**2*y**2 + 2*z*y**2 + y**2 - x
assert minimal_polynomial(1/x) == PurePoly(x*y - 1, y)
assert minimal_polynomial(1/(x + 1)) == PurePoly((x + 1)*y - 1, y)
assert minimal_polynomial(sqrt(x)) == PurePoly(y**2 - x, y)
assert minimal_polynomial(sqrt(x) / z) == PurePoly(z**2*y**2 - x, y)
# this is (sqrt(1 + x**3)/x).integrate(x).diff(x) - sqrt(1 + x**3)/x
a = sqrt(x)/sqrt(1 + x**(-3)) - sqrt(x**3 + 1)/x + 1/(x**Rational(5, 2) *
(1 + x**(-3))**Rational(3, 2)) + 1/(x**Rational(11, 2)*(1 + x**(-3))**Rational(3, 2))
assert minimal_polynomial(a)(y) == y
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(exp(x)))
def test_reduce_inequalities_general():
assert reduce_inequalities(sqrt(2) * x >= 1) == (sqrt(2) / 2 <= x)
assert reduce_inequalities(PurePoly(x + 1, x) > 0) == (Integer(-1) < x)
# issue sympy/sympy#10196
assert reduce_inequalities(x**2 >= 0)
assert reduce_inequalities(x**2 < 0) is false
def test_sympyissue_18874():
e = [sqrt(2) + sqrt(5), sqrt(2)]
assert primitive_element(e) == (PurePoly(x**4 - 46 * x**2 + 169),
[1, 2], [[0, QQ(-20, 39), 0,
QQ(1, 39)],
[0, QQ(59, 78), 0,
QQ(-1, 78)]])
def test_primitive_element():
assert primitive_element([sqrt(2)]) == (PurePoly(x**2 - 2), [1], [[1, 0]])
assert (primitive_element([sqrt(2), sqrt(3)]) ==
(PurePoly(x**4 - 10*x**2 + 1), [1, 1], [[QQ(+1, 2), 0, -QQ(9, 2), 0],
[QQ(-1, 2), 0, QQ(11, 2), 0]]))
A = QQ.algebraic_field(sqrt(2))
assert (primitive_element([sqrt(2), sqrt(3)], domain=A) ==
(PurePoly(x**2 - 2*sqrt(2)*x - 1, x, domain=A), [1, 1],
[[A.unit], [1, -A.unit]]))
assert (primitive_element([sqrt(2), sqrt(2 + sqrt(2))], domain=A) ==
(PurePoly(x**2 - 2*sqrt(2)*x - sqrt(2), x, domain=A), [1, 1],
[[A.unit], [1, -A.unit]]))
A = QQ.algebraic_field(sqrt(2) + sqrt(3))
assert (primitive_element([sqrt(2), sqrt(3)], domain=A) ==
(PurePoly(x - sqrt(2) - sqrt(3), x, domain=A), [1, 1],
[[A([QQ(1, 2), 0, -QQ(9, 2), 0])], [A([-QQ(1, 2), 0, QQ(11, 2), 0])]]))
pytest.raises(ValueError, lambda: primitive_element([]))
# issue sympy/sympy#13849
assert (primitive_element([sqrt(2), sqrt(2) + sqrt(5)]) ==
(PurePoly(x**4 - 76*x**2 + 4), [1, 2], [[QQ(1, 12), 0, QQ(-37, 6), 0],
[QQ(-1, 24), 0, QQ(43, 12), 0]]))
# issue sympy/sympy#14117
assert (primitive_element([I*sqrt(2*sqrt(2) + 3), I*sqrt(-2*sqrt(2) + 3), I]) ==
(PurePoly(x**4 + 54*x**2 + 81), [1, 2, 4], [[QQ(1, 3), 0], [QQ(1, 27), 0, 2, 0],
[QQ(-1, 54), 0, QQ(-5, 6), 0]]))
def test_solve_poly_inequality():
assert psolve(Poly(0, x), '==') == [S.Reals]
assert psolve(Poly(1, x), '==') == [S.EmptySet]
assert psolve(PurePoly(x + 1, x), '>') == [Interval(-1, oo, True, True)]
pytest.raises(ValueError, lambda: psolve(x, '=='))
pytest.raises(ValueError, lambda: psolve(Poly(x, x), '??'))
assert (solve_poly_inequalities(
((Poly(x**2 - 3), '>'),
(Poly(-x**2 + 1), '>'))) == Union(Interval(-oo, -sqrt(3), True, True),
Interval(-1, 1, True, True),
Interval(sqrt(3), oo, True, True)))
def test_solve_poly_inequality():
assert psolve(Integer(0).as_poly(x), '==') == [S.ExtendedReals]
assert psolve(Integer(1).as_poly(x), '==') == [S.EmptySet]
assert psolve(PurePoly(x + 1, x), '>') == [Interval(-1, oo, True, False)]
pytest.raises(ValueError, lambda: psolve(x, '=='))
pytest.raises(ValueError, lambda: psolve(x.as_poly(), '??'))
assert (solve_poly_inequalities(
(((x**2 - 3).as_poly(), '>'),
((-x**2 + 1).as_poly(),
'>'))) == Union(Interval(-oo, -sqrt(3), False, True),
Interval(-1, 1, True, True),
Interval(sqrt(3), oo, True, False)))
def test_minpoly_domain():
F = QQ.algebraic_field(sqrt(2))
assert minimal_polynomial(sqrt(2), domain=F) == PurePoly(x - sqrt(2), x, domain=F)
assert minimal_polynomial(sqrt(8), domain=F)(x) == x - 2*sqrt(2)
assert minimal_polynomial(sqrt(Rational(3, 2)), domain=F)(x) == 2*x**2 - 3
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(y, domain=QQ))
# issue sympy/sympy#14494
F = QQ.algebraic_field(I)
assert minimal_polynomial(I, domain=F)(x) == x - I
F = QQ.algebraic_field(sqrt(3)*I)
assert minimal_polynomial(exp(I*pi/3), domain=F)(x) == 2*x - sqrt(3)*I - 1
def test_solve_poly_inequality():
assert psolve(Poly(0, x), '==') == [S.Reals]
assert psolve(Poly(1, x), '==') == [S.EmptySet]
assert psolve(PurePoly(x + 1, x), ">") == [Interval(-1, oo, True, True)]
pytest.raises(ValueError, lambda: psolve(x, '=='))
pytest.raises(ValueError, lambda: psolve(Poly(x, x), '??'))
def test_reduce_inequalities_general():
assert reduce_inequalities(Ge(sqrt(2)*x, 1)) == And(sqrt(2)/2 <= x, x < oo)
assert reduce_inequalities(PurePoly(x + 1, x) > 0) == And(Integer(-1) < x, x < oo)
def test_minimal_polynomial():
assert minimal_polynomial(-7)(x) == x + 7
assert minimal_polynomial(-1)(x) == x + 1
assert minimal_polynomial(+0)(x) == x
assert minimal_polynomial(+1)(x) == x - 1
assert minimal_polynomial(+7)(x) == x - 7
assert minimal_polynomial(Rational(1, 3),
method='groebner')(x) == 3 * x - 1
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(pi, method='groebner'))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(sin(sqrt(2)), method='groebner'))
pytest.raises(NotAlgebraic,
lambda: minimal_polynomial(2**pi, method='groebner'))
pytest.raises(ValueError, lambda: minimal_polynomial(1, method='spam'))
assert minimal_polynomial(sqrt(2))(x) == x**2 - 2
assert minimal_polynomial(sqrt(5))(x) == x**2 - 5
assert minimal_polynomial(sqrt(6))(x) == x**2 - 6
assert minimal_polynomial(2 * sqrt(2))(x) == x**2 - 8
assert minimal_polynomial(3 * sqrt(5))(x) == x**2 - 45
assert minimal_polynomial(4 * sqrt(6))(x) == x**2 - 96
assert minimal_polynomial(2 * sqrt(2) + 3)(x) == x**2 - 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) + 6)(x) == x**2 - 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) + 7)(x) == x**2 - 14 * x - 47
assert minimal_polynomial(2 * sqrt(2) - 3)(x) == x**2 + 6 * x + 1
assert minimal_polynomial(3 * sqrt(5) - 6)(x) == x**2 + 12 * x - 9
assert minimal_polynomial(4 * sqrt(6) - 7)(x) == x**2 + 14 * x - 47
assert minimal_polynomial(sqrt(1 + sqrt(6)))(x) == x**4 - 2 * x**2 - 5
assert minimal_polynomial(sqrt(I + sqrt(6)))(x) == x**8 - 10 * x**4 + 49
assert minimal_polynomial(2 * I +
sqrt(2 + I))(x) == x**4 + 4 * x**2 + 8 * x + 37
assert minimal_polynomial(sqrt(2) + sqrt(3))(x) == x**4 - 10 * x**2 + 1
assert minimal_polynomial(sqrt(2) + sqrt(3) +
sqrt(6))(x) == x**4 - 22 * x**2 - 48 * x - 23
e = 1 / sqrt(sqrt(1 + sqrt(3)) - 4)
assert minimal_polynomial(e) == minimal_polynomial(e, method='groebner')
assert minimal_polynomial(e)(x) == (222 * x**8 + 240 * x**6 + 94 * x**4 +
16 * x**2 + 1)
a = 1 - 9 * sqrt(2) + 7 * sqrt(3)
assert minimal_polynomial(
1 / a)(x) == 392 * x**4 - 1232 * x**3 + 612 * x**2 + 4 * x - 1
assert minimal_polynomial(
1 / sqrt(a))(x) == 392 * x**8 - 1232 * x**6 + 612 * x**4 + 4 * x**2 - 1
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(oo))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(2**y))
pytest.raises(NotAlgebraic, lambda: minimal_polynomial(sin(1)))
assert minimal_polynomial(sqrt(2))(x) == x**2 - 2
assert minimal_polynomial(sqrt(2)) == PurePoly(x**2 - 2)
assert minimal_polynomial(sqrt(2), method='groebner') == PurePoly(x**2 - 2)
a = sqrt(2)
b = sqrt(3)
assert minimal_polynomial(b)(x) == x**2 - 3
assert minimal_polynomial(a) == PurePoly(x**2 - 2)
assert minimal_polynomial(b) == PurePoly(x**2 - 3)
assert minimal_polynomial(sqrt(a)) == PurePoly(x**4 - 2)
assert minimal_polynomial(a + 1) == PurePoly(x**2 - 2 * x - 1)
assert minimal_polynomial(sqrt(a / 2 +
17))(x) == 2 * x**4 - 68 * x**2 + 577
assert minimal_polynomial(sqrt(b / 2 +
17))(x) == 4 * x**4 - 136 * x**2 + 1153
# issue diofant/diofant#431
K = QQ.algebraic_field(sqrt(2))
theta = K.to_expr(K([17, Rational(1, 2)]))
assert minimal_polynomial(theta)(x) == 2 * x**2 - 68 * x + 577
K = QQ.algebraic_field(RootOf(x**7 + x - 1, 3))
theta = K.to_expr(K([1, 0, 0, 2, 1]))
ans = minimal_polynomial(theta)(x)
assert ans == (x**7 - 7 * x**6 + 19 * x**5 - 27 * x**4 + 63 * x**3 -
115 * x**2 + 82 * x - 147)
assert minimal_polynomial(theta, method='groebner')(x) == ans
K = QQ.algebraic_field(RootOf(x**5 + 5 * x - 1, 2))
theta = K.to_expr(K([1, -1, 1]))
ans = (x**30 - 15 * x**28 - 10 * x**27 + 135 * x**26 + 330 * x**25 -
705 * x**24 - 150 * x**23 + 3165 * x**22 - 6850 * x**21 +
7182 * x**20 + 3900 * x**19 + 4435 * x**18 + 11970 * x**17 -
259725 * x**16 - 18002 * x**15 + 808215 * x**14 - 200310 * x**13 -
647115 * x**12 + 299280 * x**11 - 1999332 * x**10 + 910120 * x**9 +
2273040 * x**8 - 5560320 * x**7 + 5302000 * x**6 - 2405376 * x**5 +
1016640 * x**4 - 804480 * x**3 + 257280 * x**2 - 53760 * x + 1280)
assert minimal_polynomial(sqrt(theta) + root(theta, 3),
method='groebner')(x) == ans
K1 = QQ.algebraic_field(RootOf(x**3 + 4 * x - 15, 1))
K2 = QQ.algebraic_field(RootOf(x**3 - x + 1, 0))
theta = sqrt(1 + 1 / (K1.to_expr(K1([1, 0, 1])) + 1 /
(sqrt(3) + K2.to_expr(K2([-1, 2, 1])))))
ans = (2262264837876687263 * x**36 - 38939909597855051866 * x**34 +
315720420314462950715 * x**32 - 1601958657418182606114 * x**30 +
5699493671077371036494 * x**28 - 15096777696140985506150 * x**26 +
30847690820556462893974 * x**24 - 49706549068200640994022 * x**22 +
64013601241426223813103 * x**20 - 66358713088213594372990 * x**18 +
55482571280904904971976 * x**16 - 37309340229165533529076 * x**14 +
20016999328983554519040 * x**12 - 8446273798231518826782 * x**10 +
2738866994867366499481 * x**8 - 657825125060873756424 * x**6 +
110036313740049140508 * x**4 - 11416087328869938298 * x**2 +
551322649782053543)
assert minimal_polynomial(theta)(x) == ans
a = sqrt(2) / 3 + 7
f = 81*x**8 - 2268*x**6 - 4536*x**5 + 22644*x**4 + 63216*x**3 - \
31608*x**2 - 189648*x + 141358
assert minimal_polynomial(sqrt(a) + sqrt(sqrt(a)))(x) == f
assert minimal_polynomial(a**Rational(
3, 2))(x) == 729 * x**4 - 506898 * x**2 + 84604519
K = QQ.algebraic_field(RootOf(x**3 + x - 1, 0))
a = K.to_expr(K([0, 1]))
assert minimal_polynomial(1 / a**2)(x) == x**3 - x**2 - 2 * x - 1
# issue sympy/sympy#5994
eq = (-1 / (800 * sqrt(
Rational(-1, 240) + 1 /
(18000 * cbrt(Rational(-1, 17280000) + sqrt(15) * I / 28800000)) +
2 * cbrt(Rational(-1, 17280000) + sqrt(15) * I / 28800000))))
assert minimal_polynomial(eq)(x) == 8000 * x**2 - 1
ex = 1 + sqrt(2) + sqrt(3)
mp = minimal_polynomial(ex)(x)
assert mp == x**4 - 4 * x**3 - 4 * x**2 + 16 * x - 8
ex = 1 / (1 + sqrt(2) + sqrt(3))
mp = minimal_polynomial(ex)(x)
assert mp == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = cbrt(expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3))
mp = minimal_polynomial(p)(x)
assert mp == x**8 - 8 * x**7 - 56 * x**6 + 448 * x**5 + 480 * x**4 - 5056 * x**3 + 1984 * x**2 + 7424 * x - 3008
p = expand((1 + sqrt(2) - 2 * sqrt(3) + sqrt(7))**3)
mp = minimal_polynomial(p)(x)
assert mp == x**8 - 512 * x**7 - 118208 * x**6 + 31131136 * x**5 + 647362560 * x**4 - 56026611712 * x**3 + 116994310144 * x**2 + 404854931456 * x - 27216576512
assert minimal_polynomial(-sqrt(5) / 2 - Rational(1, 2) +
(-sqrt(5) / 2 - Rational(1, 2))**2)(x) == x - 1
a = 1 + sqrt(2)
assert minimal_polynomial((a * sqrt(2) + a)**3)(x) == x**2 - 198 * x + 1
p = 1 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, method='groebner')(x) == 8 * x**4 - 16 * x**3 + 4 * x**2 + 4 * x - 1
p = 2 / (1 + sqrt(2) + sqrt(3))
assert minimal_polynomial(
p, method='groebner')(x) == x**4 - 4 * x**3 + 2 * x**2 + 4 * x - 2
assert minimal_polynomial(1 + sqrt(2) * I,
method='groebner')(x) == x**2 - 2 * x + 3
assert minimal_polynomial(1 / (1 + sqrt(2)) + 1,
method='groebner')(x) == x**2 - 2
assert minimal_polynomial(sqrt(2) * I + I * (1 + sqrt(2)),
method='groebner')(x) == x**4 + 18 * x**2 + 49
assert minimal_polynomial(exp_polar(0))(x) == x - 1
def test_sparse_matrix():
def sparse_eye(n):
return SparseMatrix.eye(n)
def sparse_zeros(n):
return SparseMatrix.zeros(n)
# creation args
pytest.raises(TypeError, lambda: SparseMatrix(1, 2))
pytest.raises(ValueError, lambda: SparseMatrix(2, 2, (1, 3, 4, 5, 6)))
a = SparseMatrix(((1, 0), (0, 1)))
assert SparseMatrix(a) == a
a = MutableSparseMatrix([])
b = MutableDenseMatrix([1, 2])
assert a.row_join(b) == b
assert a.col_join(b) == b
assert type(a.row_join(b)) == type(a)
assert type(a.col_join(b)) == type(a)
# test element assignment
a = SparseMatrix(((1, 0), (0, 1)))
a[3] = 4
assert a[1, 1] == 4
a[3] = 1
a[0, 0] = 2
assert a == SparseMatrix(((2, 0), (0, 1)))
a[1, 0] = 5
assert a == SparseMatrix(((2, 0), (5, 1)))
a[1, 1] = 0
assert a == SparseMatrix(((2, 0), (5, 0)))
assert a._smat == {(0, 0): 2, (1, 0): 5}
# test_multiplication
a = SparseMatrix((
(1, 2),
(3, 1),
(0, 6),
))
b = SparseMatrix((
(1, 2),
(3, 0),
))
c = a * b
assert c[0, 0] == 7
assert c[0, 1] == 2
assert c[1, 0] == 6
assert c[1, 1] == 6
assert c[2, 0] == 18
assert c[2, 1] == 0
c = b * x
assert isinstance(c, SparseMatrix)
assert c[0, 0] == x
assert c[0, 1] == 2 * x
assert c[1, 0] == 3 * x
assert c[1, 1] == 0
c = 5 * b
assert isinstance(c, SparseMatrix)
assert c[0, 0] == 5
assert c[0, 1] == 2 * 5
assert c[1, 0] == 3 * 5
assert c[1, 1] == 0
# test_power
A = SparseMatrix([[2, 3], [4, 5]])
assert (A**5)[:] == [6140, 8097, 10796, 14237]
A = SparseMatrix([[2, 1, 3], [4, 2, 4], [6, 12, 1]])
assert (A**3)[:] == [290, 262, 251, 448, 440, 368, 702, 954, 433]
# test_creation
a = SparseMatrix([[x, 0], [0, 0]])
m = a
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
b = SparseMatrix(2, 2, [x, 0, 0, 0])
m = b
assert m.cols == m.rows
assert m.cols == 2
assert m[:] == [x, 0, 0, 0]
assert a == b
S = sparse_eye(3)
del S[1, :]
assert S == SparseMatrix([[1, 0, 0], [0, 0, 1]])
S = sparse_eye(3)
del S[:, 1]
assert S == SparseMatrix([[1, 0], [0, 0], [0, 1]])
S = SparseMatrix.eye(3)
S[2, 1] = 2
S.col_swap(1, 0)
assert S == SparseMatrix([[0, 1, 0], [1, 0, 0], [2, 0, 1]])
S.row_swap(0, 1)
assert S == SparseMatrix([[1, 0, 0], [0, 1, 0], [2, 0, 1]])
S.col_swap(0, 1)
assert S == SparseMatrix([[0, 1, 0], [1, 0, 0], [0, 2, 1]])
S.row_swap(0, 2)
assert S == SparseMatrix([[0, 2, 1], [1, 0, 0], [0, 1, 0]])
S.col_swap(0, 2)
assert S == SparseMatrix([[1, 2, 0], [0, 0, 1], [0, 1, 0]])
a = SparseMatrix(1, 2, [1, 2])
b = a.copy()
c = a.copy()
assert a[0] == 1
del a[0, :]
assert a == SparseMatrix(0, 2, [])
del b[:, 1]
assert b == SparseMatrix(1, 1, [1])
# test_determinant
assert SparseMatrix(1, 1, [0]).det() == 0
assert SparseMatrix([[1]]).det() == 1
assert SparseMatrix(((-3, 2), (8, -5))).det() == -1
assert SparseMatrix(((x, 1), (y, 2 * y))).det() == 2 * x * y - y
assert SparseMatrix(((1, 1, 1), (1, 2, 3), (1, 3, 6))).det() == 1
assert SparseMatrix(((3, -2, 0, 5), (-2, 1, -2, 2), (0, -2, 5, 0),
(5, 0, 3, 4))).det() == -289
assert SparseMatrix(((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12),
(13, 14, 15, 16))).det() == 0
assert SparseMatrix(((3, 2, 0, 0, 0), (0, 3, 2, 0, 0), (0, 0, 3, 2, 0),
(0, 0, 0, 3, 2), (2, 0, 0, 0, 3))).det() == 275
assert SparseMatrix(((1, 0, 1, 2, 12), (2, 0, 1, 1, 4), (2, 1, 1, -1, 3),
(3, 2, -1, 1, 8), (1, 1, 1, 0, 6))).det() == -55
assert SparseMatrix(((-5, 2, 3, 4, 5), (1, -4, 3, 4, 5), (1, 2, -3, 4, 5),
(1, 2, 3, -2, 5), (1, 2, 3, 4, -1))).det() == 11664
assert SparseMatrix(((2, 7, -1, 3, 2), (0, 0, 1, 0, 1), (-2, 0, 7, 0, 2),
(-3, -2, 4, 5, 3), (1, 0, 0, 0, 1))).det() == 123
# test_slicing
m0 = sparse_eye(4)
assert m0[:3, :3] == sparse_eye(3)
assert m0[2:4, 0:2] == sparse_zeros(2)
m1 = SparseMatrix(3, 3, lambda i, j: i + j)
assert m1[0, :] == SparseMatrix(1, 3, (0, 1, 2))
assert m1[1:3, 1] == SparseMatrix(2, 1, (2, 3))
m2 = SparseMatrix([[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11],
[12, 13, 14, 15]])
assert m2[:, -1] == SparseMatrix(4, 1, [3, 7, 11, 15])
assert m2[-2:, :] == SparseMatrix([[8, 9, 10, 11], [12, 13, 14, 15]])
assert SparseMatrix([[1, 2], [3, 4]])[[1], [1]] == Matrix([[4]])
# test_submatrix_assignment
m = sparse_zeros(4)
m[2:4, 2:4] = sparse_eye(2)
assert m == SparseMatrix([(0, 0, 0, 0), (0, 0, 0, 0), (0, 0, 1, 0),
(0, 0, 0, 1)])
assert len(m._smat) == 2
m[:2, :2] = sparse_eye(2)
assert m == sparse_eye(4)
m[:, 0] = SparseMatrix(4, 1, (1, 2, 3, 4))
assert m == SparseMatrix([(1, 0, 0, 0), (2, 1, 0, 0), (3, 0, 1, 0),
(4, 0, 0, 1)])
m[:, :] = sparse_zeros(4)
assert m == sparse_zeros(4)
m[:, :] = ((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16))
assert m == SparseMatrix(
((1, 2, 3, 4), (5, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
m[:2, 0] = [0, 0]
assert m == SparseMatrix(
((0, 2, 3, 4), (0, 6, 7, 8), (9, 10, 11, 12), (13, 14, 15, 16)))
# test_reshape
m0 = sparse_eye(3)
assert m0.reshape(1, 9) == SparseMatrix(1, 9, (1, 0, 0, 0, 1, 0, 0, 0, 1))
m1 = SparseMatrix(3, 4, lambda i, j: i + j)
assert m1.reshape(4, 3) == \
SparseMatrix([(0, 1, 2), (3, 1, 2), (3, 4, 2), (3, 4, 5)])
assert m1.reshape(2, 6) == \
SparseMatrix([(0, 1, 2, 3, 1, 2), (3, 4, 2, 3, 4, 5)])
# test_applyfunc
m0 = sparse_eye(3)
assert m0.applyfunc(lambda x: 2 * x) == sparse_eye(3) * 2
assert m0.applyfunc(lambda x: 0) == sparse_zeros(3)
# test_LUdecomp
testmat = SparseMatrix([[0, 2, 5, 3], [3, 3, 7, 4], [8, 4, 0, 2],
[-2, 6, 3, 4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L * U).permuteBkwd(p) - testmat == sparse_zeros(4)
testmat = SparseMatrix([[6, -2, 7, 4], [0, 3, 6, 7], [1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L * U).permuteBkwd(p) - testmat == sparse_zeros(4)
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L * U).permuteBkwd(p) - M == sparse_zeros(3)
# test_LUsolve
A = SparseMatrix([[2, 3, 5], [3, 6, 2], [8, 3, 6]])
B = SparseMatrix(3, 1, [3, 7, 5])
b = A * B
soln = A.LUsolve(b)
assert soln == B
A = SparseMatrix([[0, -1, 2], [5, 10, 7], [8, 3, 4]])
B = SparseMatrix(3, 1, [-1, 2, 5])
b = A * B
soln = A.LUsolve(b)
assert soln == B
# test_inverse
A = sparse_eye(4)
assert A.inv() == sparse_eye(4)
assert A.inv(method='CH') == sparse_eye(4)
assert A.inv(method='LDL') == sparse_eye(4)
A = SparseMatrix([[2, 3, 5], [3, 6, 2], [7, 2, 6]])
Ainv = SparseMatrix(Matrix(A).inv())
assert A * Ainv == sparse_eye(3)
assert A.inv(method='CH') == Ainv
assert A.inv(method='LDL') == Ainv
A = SparseMatrix([[2, 3, 5], [3, 6, 2], [5, 2, 6]])
Ainv = SparseMatrix(Matrix(A).inv())
assert A * Ainv == sparse_eye(3)
assert A.inv(method='CH') == Ainv
assert A.inv(method='LDL') == Ainv
# test_cross
v1 = Matrix(1, 3, [1, 2, 3])
v2 = Matrix(1, 3, [3, 4, 5])
assert v1.cross(v2) == Matrix(1, 3, [-2, 4, -2])
assert v1.norm(2)**2 == 14
# conjugate
a = SparseMatrix(((1, 2 + I), (3, 4)))
assert a.C == SparseMatrix([[1, 2 - I], [3, 4]])
# mul
assert a * Matrix(2, 2, [1, 0, 0, 1]) == a
assert a + Matrix(2, 2, [1, 1, 1, 1]) == SparseMatrix([[2, 3 + I], [4, 5]])
assert a * 0 == Matrix([[0, 0], [0, 0]])
# col join
assert a.col_join(sparse_eye(2)) == SparseMatrix([[1, 2 + I], [3, 4],
[1, 0], [0, 1]])
A = SparseMatrix(ones(3))
B = eye(3)
assert A.col_join(B) == Matrix([[1, 1, 1], [1, 1, 1], [1, 1, 1], [1, 0, 0],
[0, 1, 0], [0, 0, 1]])
# row join
A = SparseMatrix(((1, 0, 1), (0, 1, 0), (1, 1, 0)))
B = Matrix(((1, 0, 0), (0, 1, 0), (0, 0, 1)))
assert A.row_join(B) == Matrix([[1, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0],
[1, 1, 0, 0, 0, 1]])
# symmetric
assert not a.is_symmetric(simplify=False)
assert sparse_eye(3).is_symmetric(simplify=False)
# test_cofactor
assert sparse_eye(3) == sparse_eye(3).cofactorMatrix()
test = SparseMatrix([[1, 3, 2], [2, 6, 3], [2, 3, 6]])
assert test.cofactorMatrix() == \
SparseMatrix([[27, -6, -6], [-12, 2, 3], [-3, 1, 0]])
test = SparseMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert test.cofactorMatrix() == \
SparseMatrix([[-3, 6, -3], [6, -12, 6], [-3, 6, -3]])
# test_jacobian
L = SparseMatrix(1, 2, [x**2 * y, 2 * y**2 + x * y])
syms = [x, y]
assert L.jacobian(syms) == Matrix([[2 * x * y, x**2], [y, 4 * y + x]])
L = SparseMatrix(1, 2, [x, x**2 * y**3])
assert L.jacobian(syms) == SparseMatrix([[1, 0],
[2 * x * y**3, x**2 * 3 * y**2]])
# test_QR
A = Matrix([[1, 2], [2, 3]])
Q, S = A.QRdecomposition()
R = Rational
assert Q == Matrix([[5**R(-1, 2), (R(2) / 5) * (R(1) / 5)**R(-1, 2)],
[2 * 5**R(-1, 2), (-R(1) / 5) * (R(1) / 5)**R(-1, 2)]])
assert S == Matrix([[5**R(1, 2), 8 * 5**R(-1, 2)],
[0, (R(1) / 5)**R(1, 2)]])
assert Q * S == A
assert Q.T * Q == sparse_eye(2)
R = Rational
# test nullspace
# first test reduced row-ech form
M = SparseMatrix([[5, 7, 2, 1], [1, 6, 2, -1]])
out, tmp = M.rref()
assert out == Matrix([[1, 0, -R(2) / 23, R(13) / 23],
[0, 1, R(8) / 23, R(-6) / 23]])
M = SparseMatrix([[1, 3, 0, 2, 6, 3, 1], [-2, -6, 0, -2, -8, 3, 1],
[3, 9, 0, 0, 6, 6, 2], [-1, -3, 0, 1, 0, 9, 3]])
out, tmp = M.rref()
assert out == Matrix([[1, 3, 0, 0, 2, 0, 0], [0, 0, 0, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 1, R(1) / 3], [0, 0, 0, 0, 0, 0, 0]])
# now check the vectors
basis = M.nullspace()
assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0])
assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0])
assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0])
assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1) / 3, 1])
# test eigen
sparse_eye3 = sparse_eye(3)
assert sparse_eye3.charpoly(x) == PurePoly(((x - 1)**3))
assert sparse_eye3.charpoly(y) == PurePoly(((y - 1)**3))
# test values
M = Matrix([(0, 1, -1), (1, 1, 0), (-1, 0, 1)])
vals = M.eigenvals()
assert sorted(vals) == [-1, 1, 2]
R = Rational
M = Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
assert M.eigenvects() == [
(1, 3, [Matrix([1, 0, 0]),
Matrix([0, 1, 0]),
Matrix([0, 0, 1])])
]
M = Matrix([[5, 0, 2], [3, 2, 0], [0, 0, 1]])
assert M.eigenvects() == [(1, 1, [Matrix([R(-1) / 2,
R(3) / 2, 1])]),
(2, 1, [Matrix([0, 1, 0])]),
(5, 1, [Matrix([1, 1, 0])])]
assert M.zeros(3, 5) == SparseMatrix(3, 5, {})
A = SparseMatrix(
10, 10, {
(0, 0): 18,
(0, 9): 12,
(1, 4): 18,
(2, 7): 16,
(3, 9): 12,
(4, 2): 19,
(5, 7): 16,
(6, 2): 12,
(9, 7): 18
})
assert A.row_list() == [(0, 0, 18), (0, 9, 12), (1, 4, 18), (2, 7, 16),
(3, 9, 12), (4, 2, 19), (5, 7, 16), (6, 2, 12),
(9, 7, 18)]
assert A.col_list() == [(0, 0, 18), (4, 2, 19), (6, 2, 12), (1, 4, 18),
(2, 7, 16), (5, 7, 16), (9, 7, 18), (0, 9, 12),
(3, 9, 12)]
assert SparseMatrix.eye(2).nnz() == 2
M = SparseMatrix.eye(3) * 2
M[1, 0] = -1
M.col_op(1, lambda v, i: v + 2 * M[i, 0])
assert M == Matrix([[2, 4, 0], [-1, 0, 0], [0, 0, 2]])
M = SparseMatrix.zeros(3)
M.fill(1)
assert M == ones(3)
assert SparseMatrix(ones(0, 3)).tolist() == []
def test_RootOf___new__():
assert RootOf(x, 0) == 0
assert RootOf(x, -1) == 0
assert RootOf(x - 1, 0) == 1
assert RootOf(x - 1, -1) == 1
assert RootOf(x + 1, 0) == -1
assert RootOf(x + 1, -1) == -1
assert RootOf(x**2 + 2 * x + 3, 0) == -1 - I * sqrt(2)
assert RootOf(x**2 + 2 * x + 3, 1) == -1 + I * sqrt(2)
assert RootOf(x**2 + 2 * x + 3, -1) == -1 + I * sqrt(2)
assert RootOf(x**2 + 2 * x + 3, -2) == -1 - I * sqrt(2)
r = RootOf(x**2 + 2 * x + 3, 0, radicals=False)
assert isinstance(r, RootOf) is True
r = RootOf(x**2 + 2 * x + 3, 1, radicals=False)
assert isinstance(r, RootOf) is True
r = RootOf(x**2 + 2 * x + 3, -1, radicals=False)
assert isinstance(r, RootOf) is True
r = RootOf(x**2 + 2 * x + 3, -2, radicals=False)
assert isinstance(r, RootOf) is True
assert RootOf((x - 1) * (x + 1), 0, radicals=False) == -1
assert RootOf((x - 1) * (x + 1), 1, radicals=False) == 1
assert RootOf((x - 1) * (x + 1), -1, radicals=False) == 1
assert RootOf((x - 1) * (x + 1), -2, radicals=False) == -1
assert RootOf((x - 1) * (x + 1), 0, radicals=True) == -1
assert RootOf((x - 1) * (x + 1), 1, radicals=True) == 1
assert RootOf((x - 1) * (x + 1), -1, radicals=True) == 1
assert RootOf((x - 1) * (x + 1), -2, radicals=True) == -1
assert RootOf((x - 1) * (x**3 + x + 3), 0) == RootOf(x**3 + x + 3, 0)
assert RootOf((x - 1) * (x**3 + x + 3), 1) == 1
assert RootOf((x - 1) * (x**3 + x + 3), 2) == RootOf(x**3 + x + 3, 1)
assert RootOf((x - 1) * (x**3 + x + 3), 3) == RootOf(x**3 + x + 3, 2)
assert RootOf((x - 1) * (x**3 + x + 3), -1) == RootOf(x**3 + x + 3, 2)
assert RootOf((x - 1) * (x**3 + x + 3), -2) == RootOf(x**3 + x + 3, 1)
assert RootOf((x - 1) * (x**3 + x + 3), -3) == 1
assert RootOf((x - 1) * (x**3 + x + 3), -4) == RootOf(x**3 + x + 3, 0)
assert RootOf(x**4 + 3 * x**3, 0) == -3
assert RootOf(x**4 + 3 * x**3, 1) == 0
assert RootOf(x**4 + 3 * x**3, 2) == 0
assert RootOf(x**4 + 3 * x**3, 3) == 0
pytest.raises(GeneratorsNeeded, lambda: RootOf(0, 0))
pytest.raises(GeneratorsNeeded, lambda: RootOf(1, 0))
pytest.raises(PolynomialError, lambda: RootOf(Integer(0).as_poly(x), 0))
pytest.raises(PolynomialError, lambda: RootOf(Integer(1).as_poly(x), 0))
pytest.raises(PolynomialError, lambda: RootOf(x - y, 0))
pytest.raises(IndexError, lambda: RootOf(x**2 - 1, -4))
pytest.raises(IndexError, lambda: RootOf(x**2 - 1, -3))
pytest.raises(IndexError, lambda: RootOf(x**2 - 1, 2))
pytest.raises(IndexError, lambda: RootOf(x**2 - 1, 3))
pytest.raises(ValueError, lambda: RootOf(x**2 - 1, x))
pytest.raises(
NotImplementedError,
lambda: RootOf(Symbol('a', nonzero=False) * x**5 + 2 * x - 1, x, 0))
pytest.raises(
NotImplementedError, lambda:
(Symbol('a', nonzero=False) * x**5 + 2 * x - 1).as_poly(x).all_roots())
assert RootOf((x - y).as_poly(x), 0) == y
assert RootOf((x**2 - y).as_poly(x), 0) == -sqrt(y)
assert RootOf((x**2 - y).as_poly(x), 1) == sqrt(y)
assert isinstance(RootOf(x**3 - y, x, 0), RootOf)
p = Symbol('p', positive=True)
assert RootOf(x**3 - p, x, 0) == root(p, 3) * RootOf(x**3 - 1, 0)
assert RootOf(y * x**3 + y * x + 2 * y, x, 0) == -1
assert RootOf(x**3 + x + 1, 0).is_commutative is True
e = RootOf(x**2 - 4, x, 1, evaluate=False)
assert isinstance(e, RootOf)
assert e.doit() == 2
assert e.args == (x**2 - 4, x, 1)
assert e.poly == PurePoly(x**2 - 4, x)
assert e.index == 1
assert RootOf(x**7 - 0.1 * x + 1, 0) == RootOf(10 * x**7 - x + 10, 0)
e = x**7 - x
p = (x**7 - x).as_poly(modulus=7)
F7 = p.domain
assert (RootOf(p, 1) == RootOf(e, 1, modulus=7) == RootOf(
e, x, 1, modulus=7) == RootOf(p, 1, evaluate=False) == RootOf(
e, 1, domain=F7, evaluate=False))
def test_pickling_polys_polytools():
for c in (Poly, Poly(x, x), PurePoly, PurePoly(x)):
check(c)
for c in (GroebnerBasis, GroebnerBasis([x**2 - 1], x)):
check(c)
def test_solve_poly_inequality():
assert psolve(Poly(0, x), '==') == [S.Reals]
assert psolve(Poly(1, x), '==') == [S.EmptySet]
assert psolve(PurePoly(x + 1, x), ">") == [Interval(-1, oo, True, True)]