def test_adjoint():
Sq = MatrixSymbol('Sq', n, n)
assert Adjoint(A).shape == (m, n)
assert Adjoint(A * B).shape == (l, n)
assert adjoint(Adjoint(A)) == A
assert isinstance(Adjoint(Adjoint(A)), Adjoint)
assert conjugate(Adjoint(A)) == Transpose(A) == Adjoint(A).conjugate()
assert transpose(Adjoint(A)) == Adjoint(
Transpose(A)) == Transpose(A).adjoint()
assert Adjoint(eye(3)).doit() == Adjoint(eye(3)).doit(deep=False) == eye(3)
assert Adjoint(Integer(5)).doit() == Integer(5)
assert Adjoint(Matrix([[1, 2], [3, 4]])).doit() == Matrix([[1, 3], [2, 4]])
assert adjoint(trace(Sq)) == conjugate(trace(Sq))
assert trace(adjoint(Sq)) == conjugate(trace(Sq))
assert Adjoint(Sq)[0, 1] == conjugate(Sq[1, 0])
assert Adjoint(A * B).doit() == Adjoint(B) * Adjoint(A)
assert Adjoint(C + D).doit() == Adjoint(C) + Adjoint(D)
def test_DiracDelta():
assert DiracDelta(1) == 0
assert DiracDelta(5.1) == 0
assert DiracDelta(-pi) == 0
assert DiracDelta(5, 7) == 0
assert DiracDelta(nan) == nan
assert DiracDelta(0).func is DiracDelta
assert DiracDelta(x).func is DiracDelta
assert adjoint(DiracDelta(x)) == DiracDelta(x)
assert adjoint(DiracDelta(x - y)) == DiracDelta(x - y)
assert conjugate(DiracDelta(x)) == DiracDelta(x)
assert conjugate(DiracDelta(x - y)) == DiracDelta(x - y)
assert transpose(DiracDelta(x)) == DiracDelta(x)
assert transpose(DiracDelta(x - y)) == DiracDelta(x - y)
assert DiracDelta(x).diff(x) == DiracDelta(x, 1)
assert DiracDelta(x, 1).diff(x) == DiracDelta(x, 2)
assert DiracDelta(x).is_simple(x) is True
assert DiracDelta(3 * x).is_simple(x) is True
assert DiracDelta(x**2).is_simple(x) is False
assert DiracDelta(sqrt(x)).is_simple(x) is False
assert DiracDelta(x).is_simple(y) is False
assert DiracDelta(x * y).simplify(x) == DiracDelta(x) / abs(y)
assert DiracDelta(x * y).simplify(y) == DiracDelta(y) / abs(x)
assert DiracDelta(x**2 * y).simplify(x) == DiracDelta(x**2 * y)
assert DiracDelta(y).simplify(x) == DiracDelta(y)
assert DiracDelta((x - 1)*(x - 2)*(x - 3)).simplify(x) == \
DiracDelta(x - 3)/2 + DiracDelta(x - 2) + DiracDelta(x - 1)/2
pytest.raises(ArgumentIndexError, lambda: DiracDelta(x).fdiff(2))
pytest.raises(ValueError, lambda: DiracDelta(x, -1))
def test_heaviside():
x, y = symbols('x, y', extended_real=True)
z = Symbol('z')
assert Heaviside(0) == 0.5
assert Heaviside(-5) == 0
assert Heaviside(1) == 1
assert Heaviside(nan) == nan
assert Heaviside(x).is_real
assert Heaviside(z).is_real is None
assert adjoint(Heaviside(x)) == Heaviside(x)
assert adjoint(Heaviside(x - y)) == Heaviside(x - y)
assert conjugate(Heaviside(x)) == Heaviside(x)
assert conjugate(Heaviside(x - y)) == Heaviside(x - y)
assert transpose(Heaviside(x)) == Heaviside(x)
assert transpose(Heaviside(x - y)) == Heaviside(x - y)
assert Heaviside(x).diff(x) == DiracDelta(x)
assert Heaviside(z + I).is_Function is True
assert Heaviside(I * z).is_Function is True
pytest.raises(ArgumentIndexError, lambda: Heaviside(x).fdiff(2))
pytest.raises(ValueError, lambda: Heaviside(I))
pytest.raises(ValueError, lambda: Heaviside(2 + 3 * I))
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
p = Piecewise((A * B**2, x > 0), (A**2 * B, True))
assert p.adjoint() == \
Piecewise((adjoint(A*B**2), x > 0), (adjoint(A**2*B), True))
assert p.conjugate() == \
Piecewise((conjugate(A*B**2), x > 0), (conjugate(A**2*B), True))
assert p.transpose() == \
Piecewise((transpose(A*B**2), x > 0), (transpose(A**2*B), True))
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
p = Piecewise((A*B**2, x > 0), (A**2*B, True))
assert p.adjoint() == \
Piecewise((adjoint(A*B**2), x > 0), (adjoint(A**2*B), True))
assert p.conjugate() == \
Piecewise((conjugate(A*B**2), x > 0), (conjugate(A**2*B), True))
assert p.transpose() == \
Piecewise((transpose(A*B**2), x > 0), (transpose(A**2*B), True))
def test_Trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
pytest.raises(ShapeError, lambda: Trace(C))
assert trace(eye(3)) == 3
assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15
assert adjoint(Trace(A)) == trace(Adjoint(A))
assert conjugate(Trace(A)) == trace(Adjoint(A))
assert transpose(Trace(A)) == Trace(A)
assert isinstance(A / Trace(A), MatrixExpr)
# Some easy simplifications
assert trace(Identity(5)) == 5
assert trace(ZeroMatrix(5, 5)) == 0
assert trace(2 * A * B) == 2 * Trace(A * B)
assert trace(A.T) == trace(A)
i, j = symbols('i j')
F = FunctionMatrix(3, 3, Lambda((i, j), i + j))
assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2)
pytest.raises(TypeError, lambda: Trace(1))
assert Trace(A).arg is A
assert str(trace(A)) == str(Trace(A).doit())
def test_kronecker_delta():
i, j = symbols('i j')
k = Symbol('k', nonzero=True)
assert KroneckerDelta(1, 1) == 1
assert KroneckerDelta(1, 2) == 0
assert KroneckerDelta(k, 0) == 0
assert KroneckerDelta(x, x) == 1
assert KroneckerDelta(x**2 - y**2, x**2 - y**2) == 1
assert KroneckerDelta(i, i) == 1
assert KroneckerDelta(i, i + 1) == 0
assert KroneckerDelta(0, 0) == 1
assert KroneckerDelta(0, 1) == 0
assert KroneckerDelta(i + k, i) == 0
assert KroneckerDelta(i + k, i + k) == 1
assert KroneckerDelta(i + k, i + 1 + k) == 0
assert KroneckerDelta(i, j).subs({i: 1, j: 0}) == 0
assert KroneckerDelta(i, j).subs({i: 3, j: 3}) == 1
assert KroneckerDelta(i, j)**0 == 1
for n in range(1, 10):
assert KroneckerDelta(i, j)**n == KroneckerDelta(i, j)
assert KroneckerDelta(i, j)**-n == 1 / KroneckerDelta(i, j)
assert KroneckerDelta(i, j).is_integer is True
assert adjoint(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
assert conjugate(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
assert transpose(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
# to test if canonical
assert KroneckerDelta(i, j) == KroneckerDelta(j, i)
def test_kronecker_delta():
i, j = symbols('i j')
k = Symbol('k', nonzero=True)
assert KroneckerDelta(1, 1) == 1
assert KroneckerDelta(1, 2) == 0
assert KroneckerDelta(k, 0) == 0
assert KroneckerDelta(x, x) == 1
assert KroneckerDelta(x**2 - y**2, x**2 - y**2) == 1
assert KroneckerDelta(i, i) == 1
assert KroneckerDelta(i, i + 1) == 0
assert KroneckerDelta(0, 0) == 1
assert KroneckerDelta(0, 1) == 0
assert KroneckerDelta(i + k, i) == 0
assert KroneckerDelta(i + k, i + k) == 1
assert KroneckerDelta(i + k, i + 1 + k) == 0
assert KroneckerDelta(i, j).subs({i: 1, j: 0}) == 0
assert KroneckerDelta(i, j).subs({i: 3, j: 3}) == 1
assert KroneckerDelta(i, j)**0 == 1
for n in range(1, 10):
assert KroneckerDelta(i, j)**n == KroneckerDelta(i, j)
assert KroneckerDelta(i, j)**-n == 1/KroneckerDelta(i, j)
assert KroneckerDelta(i, j).is_integer is True
assert adjoint(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
assert conjugate(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
assert transpose(KroneckerDelta(i, j)) == KroneckerDelta(i, j)
# to test if canonical
assert KroneckerDelta(i, j) == KroneckerDelta(j, i)
def test_DiracDelta():
i = Symbol('i', nonzero=True)
j = Symbol('j', positive=True)
k = Symbol('k', negative=True)
assert DiracDelta(1) == 0
assert DiracDelta(5.1) == 0
assert DiracDelta(-pi) == 0
assert DiracDelta(5, 7) == 0
assert DiracDelta(i) == 0
assert DiracDelta(j) == 0
assert DiracDelta(k) == 0
assert DiracDelta(nan) == nan
assert isinstance(DiracDelta(0), DiracDelta)
assert isinstance(DiracDelta(x), DiracDelta)
assert adjoint(DiracDelta(x)) == DiracDelta(x)
assert adjoint(DiracDelta(x - y)) == DiracDelta(x - y)
assert conjugate(DiracDelta(x)) == DiracDelta(x)
assert conjugate(DiracDelta(x - y)) == DiracDelta(x - y)
assert transpose(DiracDelta(x)) == DiracDelta(x)
assert transpose(DiracDelta(x - y)) == DiracDelta(x - y)
assert DiracDelta(x).diff(x) == DiracDelta(x, 1)
assert DiracDelta(x, 1).diff(x) == DiracDelta(x, 2)
assert DiracDelta(x).is_simple(x) is True
assert DiracDelta(3*x).is_simple(x) is True
assert DiracDelta(x**2).is_simple(x) is False
assert DiracDelta(sqrt(x)).is_simple(x) is False
assert DiracDelta(x).is_simple(y) is False
assert DiracDelta(x*y).simplify(x) == DiracDelta(x)/abs(y)
assert DiracDelta(x*y).simplify(y) == DiracDelta(y)/abs(x)
assert DiracDelta(x**2*y).simplify(x) == DiracDelta(x**2*y)
assert DiracDelta(y).simplify(x) == DiracDelta(y)
assert DiracDelta((x - 1)*(x - 2)*(x - 3)).simplify(x) == \
DiracDelta(x - 3)/2 + DiracDelta(x - 2) + DiracDelta(x - 1)/2
pytest.raises(ArgumentIndexError, lambda: DiracDelta(x).fdiff(2))
pytest.raises(ValueError, lambda: DiracDelta(x, -1))
def test_adjoint():
assert adjoint(A * B) == Adjoint(B) * Adjoint(A)
assert adjoint(2 * A * B) == 2 * Adjoint(B) * Adjoint(A)
assert adjoint(2 * I * C) == -2 * I * Adjoint(C)
M = Matrix(2, 2, [1, 2 + I, 3, 4]).as_immutable()
MA = Matrix(2, 2, [1, 3, 2 - I, 4])
assert adjoint(M) == MA
assert adjoint(2 * M) == 2 * MA
assert adjoint(MatMul(2, M)) == MatMul(2, MA).doit()
def test_levicivita():
assert Eijk(1, 2, 3) == LeviCivita(1, 2, 3)
assert LeviCivita(1, 2, 3) == 1
assert LeviCivita(1, 3, 2) == -1
assert LeviCivita(1, 2, 2) == 0
i, j, k = symbols('i j k')
assert LeviCivita(i, j, k) == LeviCivita(i, j, k, evaluate=False)
assert LeviCivita(i, j, i) == 0
assert LeviCivita(1, i, i) == 0
assert LeviCivita(i, j, k).doit() == (j - i) * (k - i) * (k - j) / 2
assert LeviCivita(1, 2, 3, 1) == 0
assert LeviCivita(4, 5, 1, 2, 3) == 1
assert LeviCivita(4, 5, 2, 1, 3) == -1
assert LeviCivita(i, j, k).is_integer is True
assert adjoint(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
assert conjugate(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
assert transpose(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
def test_levicivita():
assert Eijk(1, 2, 3) == LeviCivita(1, 2, 3)
assert LeviCivita(1, 2, 3) == 1
assert LeviCivita(1, 3, 2) == -1
assert LeviCivita(1, 2, 2) == 0
i, j, k = symbols('i j k')
assert LeviCivita(i, j, k) == LeviCivita(i, j, k, evaluate=False)
assert LeviCivita(i, j, i) == 0
assert LeviCivita(1, i, i) == 0
assert LeviCivita(i, j, k).doit() == (j - i)*(k - i)*(k - j)/2
assert LeviCivita(1, 2, 3, 1) == 0
assert LeviCivita(4, 5, 1, 2, 3) == 1
assert LeviCivita(4, 5, 2, 1, 3) == -1
assert LeviCivita(i, j, k).is_integer is True
assert adjoint(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
assert conjugate(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
assert transpose(LeviCivita(i, j, k)) == LeviCivita(i, j, k)
def test_conjugate_transpose():
x = Symbol('x')
assert conjugate(transpose(x)) == adjoint(x)
assert transpose(conjugate(x)) == adjoint(x)
assert adjoint(transpose(x)) == conjugate(x)
assert transpose(adjoint(x)) == conjugate(x)
assert adjoint(conjugate(x)) == transpose(x)
assert conjugate(adjoint(x)) == transpose(x)
class Symmetric(Expr):
def _eval_adjoint(self):
return
def _eval_conjugate(self):
return
def _eval_transpose(self):
return self
x = Symmetric()
assert conjugate(x) == adjoint(x)
assert transpose(x) == x
def test_conjugate_transpose():
x = Symbol('x')
assert conjugate(transpose(x)) == adjoint(x)
assert transpose(conjugate(x)) == adjoint(x)
assert adjoint(transpose(x)) == conjugate(x)
assert transpose(adjoint(x)) == conjugate(x)
assert adjoint(conjugate(x)) == transpose(x)
assert conjugate(adjoint(x)) == transpose(x)
class Symmetric(Expr):
def _eval_adjoint(self):
return
def _eval_conjugate(self):
return
def _eval_transpose(self):
return self
x = Symmetric()
assert conjugate(x) == adjoint(x)
assert transpose(x) == x
def test_adjoint():
a = Symbol('a', antihermitian=True)
b = Symbol('b', hermitian=True)
assert adjoint(a) == -a
assert adjoint(I*a) == I*a
assert adjoint(b) == b
assert adjoint(I*b) == -I*b
assert adjoint(a*b) == -b*a
assert adjoint(I*a*b) == I*b*a
x, y = symbols('x y')
assert adjoint(adjoint(x)) == x
assert adjoint(x + y) == adjoint(x) + adjoint(y)
assert adjoint(x - y) == adjoint(x) - adjoint(y)
assert adjoint(x * y) == adjoint(x) * adjoint(y)
assert adjoint(x / y) == adjoint(x) / adjoint(y)
assert adjoint(-x) == -adjoint(x)
x, y = symbols('x y', commutative=False)
assert adjoint(adjoint(x)) == x
assert adjoint(x + y) == adjoint(x) + adjoint(y)
assert adjoint(x - y) == adjoint(x) - adjoint(y)
assert adjoint(x * y) == adjoint(y) * adjoint(x)
assert adjoint(x / y) == 1 / adjoint(y) * adjoint(x)
assert adjoint(-x) == -adjoint(x)
def test_array_permutedims():
pytest.raises(TypeError, lambda: permutedims(1, (1, 1)))
sa = symbols('a0:144')
m1 = Array(sa[:6], (2, 3))
assert permutedims(m1, (1, 0)) == transpose(m1)
assert m1.tomatrix().T == permutedims(m1, (1, 0)).tomatrix()
assert m1.tomatrix().T == transpose(m1).tomatrix()
assert m1.tomatrix().C == conjugate(m1).tomatrix()
assert m1.tomatrix().H == adjoint(m1).tomatrix()
assert m1.tomatrix().T == m1.transpose().tomatrix()
assert m1.tomatrix().C == m1.conjugate().tomatrix()
assert m1.tomatrix().H == m1.adjoint().tomatrix()
pytest.raises(ValueError, lambda: permutedims(m1, (0, )))
pytest.raises(ValueError, lambda: permutedims(m1, (0, 0)))
pytest.raises(ValueError, lambda: permutedims(m1, (1, 2, 0)))
# Some tests with random arrays:
dims = 6
shape = [random.randint(1, 5) for i in range(dims)]
elems = [random.random() for i in range(tensorproduct(*shape))]
ra = Array(elems, shape)
perm = list(range(dims))
# Randomize the permutation:
random.shuffle(perm)
# Test inverse permutation:
assert permutedims(permutedims(ra, perm), _af_invert(perm)) == ra
# Test that permuted shape corresponds to action by `Permutation`:
assert permutedims(ra, perm).shape == tuple(Permutation(perm)(shape))
z = Array.zeros(4, 5, 6, 7)
assert permutedims(z, (2, 3, 1, 0)).shape == (6, 7, 5, 4)
assert permutedims(z, [2, 3, 1, 0]).shape == (6, 7, 5, 4)
assert permutedims(z, Permutation([2, 3, 1, 0])).shape == (6, 7, 5, 4)
po = Array(sa, [2, 2, 3, 3, 2, 2])
pytest.raises(ValueError, lambda: permutedims(po, (1, 1)))
pytest.raises(ValueError, lambda: po.transpose())
pytest.raises(ValueError, lambda: po.adjoint())
assert permutedims(po, reversed(range(po.rank()))) == Array(
[[[[[[sa[0], sa[72]], [sa[36], sa[108]]],
[[sa[12], sa[84]], [sa[48], sa[120]]],
[[sa[24], sa[96]], [sa[60], sa[132]]]],
[[[sa[4], sa[76]], [sa[40], sa[112]]],
[[sa[16], sa[88]], [sa[52], sa[124]]],
[[sa[28], sa[100]], [sa[64], sa[136]]]],
[[[sa[8], sa[80]], [sa[44], sa[116]]],
[[sa[20], sa[92]], [sa[56], sa[128]]],
[[sa[32], sa[104]], [sa[68], sa[140]]]]],
[[[[sa[2], sa[74]], [sa[38], sa[110]]],
[[sa[14], sa[86]], [sa[50], sa[122]]],
[[sa[26], sa[98]], [sa[62], sa[134]]]],
[[[sa[6], sa[78]], [sa[42], sa[114]]],
[[sa[18], sa[90]], [sa[54], sa[126]]],
[[sa[30], sa[102]], [sa[66], sa[138]]]],
[[[sa[10], sa[82]], [sa[46], sa[118]]],
[[sa[22], sa[94]], [sa[58], sa[130]]],
[[sa[34], sa[106]], [sa[70], sa[142]]]]]],
[[[[[sa[1], sa[73]], [sa[37], sa[109]]],
[[sa[13], sa[85]], [sa[49], sa[121]]],
[[sa[25], sa[97]], [sa[61], sa[133]]]],
[[[sa[5], sa[77]], [sa[41], sa[113]]],
[[sa[17], sa[89]], [sa[53], sa[125]]],
[[sa[29], sa[101]], [sa[65], sa[137]]]],
[[[sa[9], sa[81]], [sa[45], sa[117]]],
[[sa[21], sa[93]], [sa[57], sa[129]]],
[[sa[33], sa[105]], [sa[69], sa[141]]]]],
[[[[sa[3], sa[75]], [sa[39], sa[111]]],
[[sa[15], sa[87]], [sa[51], sa[123]]],
[[sa[27], sa[99]], [sa[63], sa[135]]]],
[[[sa[7], sa[79]], [sa[43], sa[115]]],
[[sa[19], sa[91]], [sa[55], sa[127]]],
[[sa[31], sa[103]], [sa[67], sa[139]]]],
[[[sa[11], sa[83]], [sa[47], sa[119]]],
[[sa[23], sa[95]], [sa[59], sa[131]]],
[[sa[35], sa[107]], [sa[71], sa[143]]]]]]])
assert permutedims(po,
(1, 0, 2, 3, 4, 5)) == Array([[[[[[sa[0], sa[1]],
[sa[2], sa[3]]],
[[sa[4], sa[5]],
[sa[6], sa[7]]],
[[sa[8], sa[9]],
[sa[10], sa[11]]]],
[[[sa[12], sa[13]],
[sa[14], sa[15]]],
[[sa[16], sa[17]],
[sa[18], sa[19]]],
[[sa[20], sa[21]],
[sa[22], sa[23]]]],
[[[sa[24], sa[25]],
[sa[26], sa[27]]],
[[sa[28], sa[29]],
[sa[30], sa[31]]],
[[sa[32], sa[33]],
[sa[34], sa[35]]]]],
[[[[sa[72], sa[73]],
[sa[74], sa[75]]],
[[sa[76], sa[77]],
[sa[78], sa[79]]],
[[sa[80], sa[81]],
[sa[82], sa[83]]]],
[[[sa[84], sa[85]],
[sa[86], sa[87]]],
[[sa[88], sa[89]],
[sa[90], sa[91]]],
[[sa[92], sa[93]],
[sa[94], sa[95]]]],
[[[sa[96], sa[97]],
[sa[98], sa[99]]],
[[sa[100], sa[101]],
[sa[102], sa[103]]],
[[sa[104], sa[105]],
[sa[106],
sa[107]]]]]],
[[[[[sa[36], sa[37]],
[sa[38], sa[39]]],
[[sa[40], sa[41]],
[sa[42], sa[43]]],
[[sa[44], sa[45]],
[sa[46], sa[47]]]],
[[[sa[48], sa[49]],
[sa[50], sa[51]]],
[[sa[52], sa[53]],
[sa[54], sa[55]]],
[[sa[56], sa[57]],
[sa[58], sa[59]]]],
[[[sa[60], sa[61]],
[sa[62], sa[63]]],
[[sa[64], sa[65]],
[sa[66], sa[67]]],
[[sa[68], sa[69]],
[sa[70], sa[71]]]]],
[[[[sa[108], sa[109]],
[sa[110], sa[111]]],
[[sa[112], sa[113]],
[sa[114], sa[115]]],
[[sa[116], sa[117]],
[sa[118], sa[119]]]],
[[[sa[120], sa[121]],
[sa[122], sa[123]]],
[[sa[124], sa[125]],
[sa[126], sa[127]]],
[[sa[128], sa[129]],
[sa[130], sa[131]]]],
[[[sa[132], sa[133]],
[sa[134], sa[135]]],
[[sa[136], sa[137]],
[sa[138], sa[139]]],
[[sa[140], sa[141]],
[sa[142],
sa[143]]]]]]])
assert permutedims(po,
(0, 2, 1, 4, 3, 5)) == Array([[[[[[sa[0], sa[1]],
[sa[4], sa[5]],
[sa[8], sa[9]]],
[[sa[2], sa[3]],
[sa[6], sa[7]],
[sa[10], sa[11]]]],
[[[sa[36], sa[37]],
[sa[40], sa[41]],
[sa[44], sa[45]]],
[[sa[38], sa[39]],
[sa[42], sa[43]],
[sa[46], sa[47]]]]],
[[[[sa[12], sa[13]],
[sa[16], sa[17]],
[sa[20], sa[21]]],
[[sa[14], sa[15]],
[sa[18], sa[19]],
[sa[22], sa[23]]]],
[[[sa[48], sa[49]],
[sa[52], sa[53]],
[sa[56], sa[57]]],
[[sa[50], sa[51]],
[sa[54], sa[55]],
[sa[58], sa[59]]]]],
[[[[sa[24], sa[25]],
[sa[28], sa[29]],
[sa[32], sa[33]]],
[[sa[26], sa[27]],
[sa[30], sa[31]],
[sa[34], sa[35]]]],
[[[sa[60], sa[61]],
[sa[64], sa[65]],
[sa[68], sa[69]]],
[[sa[62], sa[63]],
[sa[66], sa[67]],
[sa[70], sa[71]]]]]],
[[[[[sa[72], sa[73]],
[sa[76], sa[77]],
[sa[80], sa[81]]],
[[sa[74], sa[75]],
[sa[78], sa[79]],
[sa[82], sa[83]]]],
[[[sa[108], sa[109]],
[sa[112], sa[113]],
[sa[116], sa[117]]],
[[sa[110], sa[111]],
[sa[114], sa[115]],
[sa[118], sa[119]]]]],
[[[[sa[84], sa[85]],
[sa[88], sa[89]],
[sa[92], sa[93]]],
[[sa[86], sa[87]],
[sa[90], sa[91]],
[sa[94], sa[95]]]],
[[[sa[120], sa[121]],
[sa[124], sa[125]],
[sa[128], sa[129]]],
[[sa[122], sa[123]],
[sa[126], sa[127]],
[sa[130], sa[131]]]]],
[[[[sa[96], sa[97]],
[sa[100], sa[101]],
[sa[104], sa[105]]],
[[sa[98], sa[99]],
[sa[102], sa[103]],
[sa[106], sa[107]]]],
[[[sa[132], sa[133]],
[sa[136], sa[137]],
[sa[140], sa[141]]],
[[sa[134], sa[135]],
[sa[138], sa[139]],
[sa[142],
sa[143]]]]]]])
po2 = po.reshape(4, 9, 2, 2)
assert po2 == Array([[[[sa[0], sa[1]], [sa[2], sa[3]]],
[[sa[4], sa[5]], [sa[6], sa[7]]],
[[sa[8], sa[9]], [sa[10], sa[11]]],
[[sa[12], sa[13]], [sa[14], sa[15]]],
[[sa[16], sa[17]], [sa[18], sa[19]]],
[[sa[20], sa[21]], [sa[22], sa[23]]],
[[sa[24], sa[25]], [sa[26], sa[27]]],
[[sa[28], sa[29]], [sa[30], sa[31]]],
[[sa[32], sa[33]], [sa[34], sa[35]]]],
[[[sa[36], sa[37]], [sa[38], sa[39]]],
[[sa[40], sa[41]], [sa[42], sa[43]]],
[[sa[44], sa[45]], [sa[46], sa[47]]],
[[sa[48], sa[49]], [sa[50], sa[51]]],
[[sa[52], sa[53]], [sa[54], sa[55]]],
[[sa[56], sa[57]], [sa[58], sa[59]]],
[[sa[60], sa[61]], [sa[62], sa[63]]],
[[sa[64], sa[65]], [sa[66], sa[67]]],
[[sa[68], sa[69]], [sa[70], sa[71]]]],
[[[sa[72], sa[73]], [sa[74], sa[75]]],
[[sa[76], sa[77]], [sa[78], sa[79]]],
[[sa[80], sa[81]], [sa[82], sa[83]]],
[[sa[84], sa[85]], [sa[86], sa[87]]],
[[sa[88], sa[89]], [sa[90], sa[91]]],
[[sa[92], sa[93]], [sa[94], sa[95]]],
[[sa[96], sa[97]], [sa[98], sa[99]]],
[[sa[100], sa[101]], [sa[102], sa[103]]],
[[sa[104], sa[105]], [sa[106], sa[107]]]],
[[[sa[108], sa[109]], [sa[110], sa[111]]],
[[sa[112], sa[113]], [sa[114], sa[115]]],
[[sa[116], sa[117]], [sa[118], sa[119]]],
[[sa[120], sa[121]], [sa[122], sa[123]]],
[[sa[124], sa[125]], [sa[126], sa[127]]],
[[sa[128], sa[129]], [sa[130], sa[131]]],
[[sa[132], sa[133]], [sa[134], sa[135]]],
[[sa[136], sa[137]], [sa[138], sa[139]]],
[[sa[140], sa[141]], [sa[142], sa[143]]]]])
assert permutedims(po2, (3, 2, 0, 1)) == Array(
[[[[
sa[0], sa[4], sa[8], sa[12], sa[16], sa[20], sa[24], sa[28], sa[32]
],
[
sa[36], sa[40], sa[44], sa[48], sa[52], sa[56], sa[60], sa[64],
sa[68]
],
[
sa[72], sa[76], sa[80], sa[84], sa[88], sa[92], sa[96], sa[100],
sa[104]
],
[
sa[108], sa[112], sa[116], sa[120], sa[124], sa[128], sa[132],
sa[136], sa[140]
]],
[[
sa[2], sa[6], sa[10], sa[14], sa[18], sa[22], sa[26], sa[30],
sa[34]
],
[
sa[38], sa[42], sa[46], sa[50], sa[54], sa[58], sa[62], sa[66],
sa[70]
],
[
sa[74], sa[78], sa[82], sa[86], sa[90], sa[94], sa[98], sa[102],
sa[106]
],
[
sa[110], sa[114], sa[118], sa[122], sa[126], sa[130], sa[134],
sa[138], sa[142]
]]],
[[[
sa[1], sa[5], sa[9], sa[13], sa[17], sa[21], sa[25], sa[29],
sa[33]
],
[
sa[37], sa[41], sa[45], sa[49], sa[53], sa[57], sa[61], sa[65],
sa[69]
],
[
sa[73], sa[77], sa[81], sa[85], sa[89], sa[93], sa[97], sa[101],
sa[105]
],
[
sa[109], sa[113], sa[117], sa[121], sa[125], sa[129], sa[133],
sa[137], sa[141]
]],
[[
sa[3], sa[7], sa[11], sa[15], sa[19], sa[23], sa[27], sa[31],
sa[35]
],
[
sa[39], sa[43], sa[47], sa[51], sa[55], sa[59], sa[63], sa[67],
sa[71]
],
[
sa[75], sa[79], sa[83], sa[87], sa[91], sa[95], sa[99], sa[103],
sa[107]
],
[
sa[111], sa[115], sa[119], sa[123], sa[127], sa[131], sa[135],
sa[139], sa[143]
]]]])
def test_adjoint():
a = Symbol('a', antihermitian=True)
b = Symbol('b', hermitian=True)
assert adjoint(a) == -a
assert adjoint(I*a) == I*a
assert adjoint(b) == b
assert adjoint(I*b) == -I*b
assert adjoint(a*b) == -b*a
assert adjoint(I*a*b) == I*b*a
x, y = symbols('x y')
assert adjoint(adjoint(x)) == x
assert adjoint(x + y) == adjoint(x) + adjoint(y)
assert adjoint(x - y) == adjoint(x) - adjoint(y)
assert adjoint(x * y) == adjoint(x) * adjoint(y)
assert adjoint(x / y) == adjoint(x) / adjoint(y)
assert adjoint(-x) == -adjoint(x)
x, y = symbols('x y', commutative=False)
assert adjoint(adjoint(x)) == x
assert adjoint(x + y) == adjoint(x) + adjoint(y)
assert adjoint(x - y) == adjoint(x) - adjoint(y)
assert adjoint(x * y) == adjoint(y) * adjoint(x)
assert adjoint(x / y) == 1 / adjoint(y) * adjoint(x)
assert adjoint(-x) == -adjoint(x)
def test_array_permutedims():
pytest.raises(TypeError, lambda: permutedims(1, (1, 1)))
sa = symbols('a0:144')
m1 = Array(sa[:6], (2, 3))
assert permutedims(m1, (1, 0)) == transpose(m1)
assert m1.tomatrix().T == permutedims(m1, (1, 0)).tomatrix()
assert m1.tomatrix().T == transpose(m1).tomatrix()
assert m1.tomatrix().C == conjugate(m1).tomatrix()
assert m1.tomatrix().H == adjoint(m1).tomatrix()
assert m1.tomatrix().T == m1.transpose().tomatrix()
assert m1.tomatrix().C == m1.conjugate().tomatrix()
assert m1.tomatrix().H == m1.adjoint().tomatrix()
pytest.raises(ValueError, lambda: permutedims(m1, (0,)))
pytest.raises(ValueError, lambda: permutedims(m1, (0, 0)))
pytest.raises(ValueError, lambda: permutedims(m1, (1, 2, 0)))
# Some tests with random arrays:
dims = 6
shape = [random.randint(1, 5) for i in range(dims)]
elems = [random.random() for i in range(tensorproduct(*shape))]
ra = Array(elems, shape)
perm = list(range(dims))
# Randomize the permutation:
random.shuffle(perm)
# Test inverse permutation:
assert permutedims(permutedims(ra, perm), _af_invert(perm)) == ra
# Test that permuted shape corresponds to action by `Permutation`:
assert permutedims(ra, perm).shape == tuple(Permutation(perm)(shape))
z = NDimArray.zeros(4, 5, 6, 7)
assert permutedims(z, (2, 3, 1, 0)).shape == (6, 7, 5, 4)
assert permutedims(z, [2, 3, 1, 0]).shape == (6, 7, 5, 4)
assert permutedims(z, Permutation([2, 3, 1, 0])).shape == (6, 7, 5, 4)
po = Array(sa, [2, 2, 3, 3, 2, 2])
pytest.raises(ValueError, lambda: permutedims(po, (1, 1)))
pytest.raises(ValueError, lambda: po.transpose())
pytest.raises(ValueError, lambda: po.adjoint())
assert permutedims(po, reversed(range(po.rank()))) == Array(
[[[[[[sa[0], sa[72]], [sa[36], sa[108]]], [[sa[12], sa[84]], [sa[48], sa[120]]], [[sa[24],
sa[96]], [sa[60], sa[132]]]],
[[[sa[4], sa[76]], [sa[40], sa[112]]], [[sa[16],
sa[88]], [sa[52], sa[124]]],
[[sa[28], sa[100]], [sa[64], sa[136]]]],
[[[sa[8],
sa[80]], [sa[44], sa[116]]], [[sa[20], sa[92]], [sa[56], sa[128]]], [[sa[32],
sa[104]], [sa[68], sa[140]]]]],
[[[[sa[2], sa[74]], [sa[38], sa[110]]], [[sa[14],
sa[86]], [sa[50], sa[122]]], [[sa[26], sa[98]], [sa[62], sa[134]]]],
[[[sa[6],
sa[78]], [sa[42], sa[114]]], [[sa[18], sa[90]], [sa[54], sa[126]]], [[sa[30],
sa[102]], [sa[66], sa[138]]]],
[[[sa[10], sa[82]], [sa[46], sa[118]]], [[sa[22],
sa[94]], [sa[58], sa[130]]],
[[sa[34], sa[106]], [sa[70], sa[142]]]]]],
[[[[[sa[1],
sa[73]], [sa[37], sa[109]]], [[sa[13], sa[85]], [sa[49], sa[121]]], [[sa[25],
sa[97]], [sa[61], sa[133]]]],
[[[sa[5], sa[77]], [sa[41], sa[113]]], [[sa[17],
sa[89]], [sa[53], sa[125]]],
[[sa[29], sa[101]], [sa[65], sa[137]]]],
[[[sa[9],
sa[81]], [sa[45], sa[117]]], [[sa[21], sa[93]], [sa[57], sa[129]]], [[sa[33],
sa[105]], [sa[69], sa[141]]]]],
[[[[sa[3], sa[75]], [sa[39], sa[111]]], [[sa[15],
sa[87]], [sa[51], sa[123]]], [[sa[27], sa[99]], [sa[63], sa[135]]]],
[[[sa[7],
sa[79]], [sa[43], sa[115]]], [[sa[19], sa[91]], [sa[55], sa[127]]], [[sa[31],
sa[103]], [sa[67], sa[139]]]],
[[[sa[11], sa[83]], [sa[47], sa[119]]], [[sa[23],
sa[95]], [sa[59], sa[131]]],
[[sa[35], sa[107]], [sa[71], sa[143]]]]]]])
assert permutedims(po, (1, 0, 2, 3, 4, 5)) == Array(
[[[[[[sa[0], sa[1]], [sa[2], sa[3]]], [[sa[4], sa[5]], [sa[6], sa[7]]], [[sa[8], sa[9]], [sa[10],
sa[11]]]],
[[[sa[12], sa[13]], [sa[14], sa[15]]], [[sa[16], sa[17]], [sa[18],
sa[19]]], [[sa[20], sa[21]], [sa[22], sa[23]]]],
[[[sa[24], sa[25]], [sa[26],
sa[27]]], [[sa[28], sa[29]], [sa[30], sa[31]]], [[sa[32], sa[33]], [sa[34],
sa[35]]]]],
[[[[sa[72], sa[73]], [sa[74], sa[75]]], [[sa[76], sa[77]], [sa[78],
sa[79]]], [[sa[80], sa[81]], [sa[82], sa[83]]]],
[[[sa[84], sa[85]], [sa[86],
sa[87]]], [[sa[88], sa[89]], [sa[90], sa[91]]], [[sa[92], sa[93]], [sa[94],
sa[95]]]],
[[[sa[96], sa[97]], [sa[98], sa[99]]], [[sa[100], sa[101]], [sa[102],
sa[103]]],
[[sa[104], sa[105]], [sa[106], sa[107]]]]]],
[[[[[sa[36], sa[37]], [sa[38], sa[39]]], [[sa[40], sa[41]], [sa[42], sa[43]]],
[[sa[44], sa[45]], [sa[46], sa[47]]]],
[[[sa[48], sa[49]], [sa[50], sa[51]]], [[sa[52], sa[53]], [sa[54], sa[55]]],
[[sa[56], sa[57]], [sa[58], sa[59]]]],
[[[sa[60], sa[61]], [sa[62], sa[63]]], [[sa[64], sa[65]], [sa[66], sa[67]]],
[[sa[68], sa[69]], [sa[70], sa[71]]]]],
[[[[sa[108], sa[109]], [sa[110], sa[111]]],
[[sa[112], sa[113]], [sa[114], sa[115]]], [[sa[116], sa[117]], [sa[118], sa[119]]]],
[[[sa[120], sa[121]], [sa[122], sa[123]]], [[sa[124], sa[125]], [sa[126], sa[127]]],
[[sa[128], sa[129]], [sa[130], sa[131]]]],
[[[sa[132], sa[133]], [sa[134], sa[135]]], [[sa[136], sa[137]], [sa[138], sa[139]]],
[[sa[140], sa[141]], [sa[142], sa[143]]]]]]])
assert permutedims(po, (0, 2, 1, 4, 3, 5)) == Array(
[[[[[[sa[0], sa[1]], [sa[4], sa[5]], [sa[8], sa[9]]], [[sa[2], sa[3]], [sa[6], sa[7]], [sa[10],
sa[11]]]],
[[[sa[36], sa[37]], [sa[40], sa[41]], [sa[44], sa[45]]], [[sa[38],
sa[39]], [sa[42], sa[43]], [sa[46], sa[47]]]]],
[[[[sa[12], sa[13]], [sa[16],
sa[17]], [sa[20], sa[21]]], [[sa[14], sa[15]], [sa[18], sa[19]], [sa[22],
sa[23]]]],
[[[sa[48], sa[49]], [sa[52], sa[53]], [sa[56], sa[57]]], [[sa[50],
sa[51]], [sa[54], sa[55]], [sa[58], sa[59]]]]],
[[[[sa[24], sa[25]], [sa[28],
sa[29]], [sa[32], sa[33]]], [[sa[26], sa[27]], [sa[30], sa[31]], [sa[34],
sa[35]]]],
[[[sa[60], sa[61]], [sa[64], sa[65]], [sa[68], sa[69]]], [[sa[62],
sa[63]], [sa[66], sa[67]], [sa[70], sa[71]]]]]],
[[[[[sa[72], sa[73]], [sa[76],
sa[77]], [sa[80], sa[81]]], [[sa[74], sa[75]], [sa[78], sa[79]], [sa[82],
sa[83]]]],
[[[sa[108], sa[109]], [sa[112], sa[113]], [sa[116], sa[117]]], [[sa[110],
sa[111]], [sa[114], sa[115]],
[sa[118], sa[119]]]]],
[[[[sa[84], sa[85]], [sa[88],
sa[89]], [sa[92], sa[93]]], [[sa[86], sa[87]], [sa[90], sa[91]], [sa[94],
sa[95]]]],
[[[sa[120], sa[121]], [sa[124], sa[125]], [sa[128], sa[129]]], [[sa[122],
sa[123]], [sa[126], sa[127]],
[sa[130], sa[131]]]]],
[[[[sa[96], sa[97]], [sa[100],
sa[101]], [sa[104], sa[105]]], [[sa[98], sa[99]], [sa[102], sa[103]], [sa[106],
sa[107]]]],
[[[sa[132], sa[133]], [sa[136], sa[137]], [sa[140], sa[141]]], [[sa[134],
sa[135]], [sa[138], sa[139]],
[sa[142], sa[143]]]]]]])
po2 = po.reshape(4, 9, 2, 2)
assert po2 == Array([[[[sa[0], sa[1]], [sa[2], sa[3]]], [[sa[4], sa[5]], [sa[6], sa[7]]], [[sa[8], sa[9]], [sa[10], sa[11]]], [[sa[12], sa[13]], [sa[14], sa[15]]], [[sa[16], sa[17]], [sa[18], sa[19]]], [[sa[20], sa[21]], [sa[22], sa[23]]], [[sa[24], sa[25]], [sa[26], sa[27]]], [[sa[28], sa[29]], [sa[30], sa[31]]], [[sa[32], sa[33]], [sa[34], sa[35]]]], [[[sa[36], sa[37]], [sa[38], sa[39]]], [[sa[40], sa[41]], [sa[42], sa[43]]], [[sa[44], sa[45]], [sa[46], sa[47]]], [[sa[48], sa[49]], [sa[50], sa[51]]], [[sa[52], sa[53]], [sa[54], sa[55]]], [[sa[56], sa[57]], [sa[58], sa[59]]], [[sa[60], sa[61]], [sa[62], sa[63]]], [[sa[64], sa[65]], [sa[66], sa[67]]], [[sa[68], sa[69]], [sa[70], sa[71]]]], [[[sa[72], sa[73]], [sa[74], sa[75]]], [[sa[76], sa[77]], [sa[78], sa[79]]], [[sa[80], sa[81]], [sa[82], sa[83]]], [[sa[84], sa[85]], [sa[86], sa[87]]], [[sa[88], sa[89]], [sa[90], sa[91]]], [[sa[92], sa[93]], [sa[94], sa[95]]], [[sa[96], sa[97]], [sa[98], sa[99]]], [[sa[100], sa[101]], [sa[102], sa[103]]], [[sa[104], sa[105]], [sa[106], sa[107]]]], [[[sa[108], sa[109]], [sa[110], sa[111]]], [[sa[112], sa[113]], [sa[114], sa[115]]], [[sa[116], sa[117]], [sa[118], sa[119]]], [[sa[120], sa[121]], [sa[122], sa[123]]], [[sa[124], sa[125]], [sa[126], sa[127]]], [[sa[128], sa[129]], [sa[130], sa[131]]], [[sa[132], sa[133]], [sa[134], sa[135]]], [[sa[136], sa[137]], [sa[138], sa[139]]], [[sa[140], sa[141]], [sa[142], sa[143]]]]])
assert permutedims(po2, (3, 2, 0, 1)) == Array([[[[sa[0], sa[4], sa[8], sa[12], sa[16], sa[20], sa[24], sa[28], sa[32]], [sa[36], sa[40], sa[44], sa[48], sa[52], sa[56], sa[60], sa[64], sa[68]], [sa[72], sa[76], sa[80], sa[84], sa[88], sa[92], sa[96], sa[100], sa[104]], [sa[108], sa[112], sa[116], sa[120], sa[124], sa[128], sa[132], sa[136], sa[140]]], [[sa[2], sa[6], sa[10], sa[14], sa[18], sa[22], sa[26], sa[30], sa[34]], [sa[38], sa[42], sa[46], sa[50], sa[54], sa[58], sa[62], sa[66], sa[70]], [sa[74], sa[78], sa[82], sa[86], sa[90], sa[94], sa[98], sa[102], sa[106]], [sa[110], sa[114], sa[118], sa[122], sa[126], sa[130], sa[134], sa[138], sa[142]]]], [[[sa[1], sa[5], sa[9], sa[13], sa[17], sa[21], sa[25], sa[29], sa[33]], [sa[37], sa[41], sa[45], sa[49], sa[53], sa[57], sa[61], sa[65], sa[69]], [sa[73], sa[77], sa[81], sa[85], sa[89], sa[93], sa[97], sa[101], sa[105]], [sa[109], sa[113], sa[117], sa[121], sa[125], sa[129], sa[133], sa[137], sa[141]]], [[sa[3], sa[7], sa[11], sa[15], sa[19], sa[23], sa[27], sa[31], sa[35]], [sa[39], sa[43], sa[47], sa[51], sa[55], sa[59], sa[63], sa[67], sa[71]], [sa[75], sa[79], sa[83], sa[87], sa[91], sa[95], sa[99], sa[103], sa[107]], [sa[111], sa[115], sa[119], sa[123], sa[127], sa[131], sa[135], sa[139], sa[143]]]]])
def test_adjoint():
assert adjoint(A).is_commutative is False
assert adjoint(A * A) == adjoint(A)**2
assert adjoint(A * B) == adjoint(B) * adjoint(A)
assert adjoint(A * B**2) == adjoint(B)**2 * adjoint(A)
assert adjoint(A * B -
B * A) == adjoint(B) * adjoint(A) - adjoint(A) * adjoint(B)
assert adjoint(A + I * B) == adjoint(A) - I * adjoint(B)
assert adjoint(X) == X
assert adjoint(-I * X) == I * X
assert adjoint(Y) == -Y
assert adjoint(-I * Y) == -I * Y
assert adjoint(X) == conjugate(transpose(X))
assert adjoint(Y) == conjugate(transpose(Y))
assert adjoint(X) == transpose(conjugate(X))
assert adjoint(Y) == transpose(conjugate(Y))
def test_adjoint():
assert adjoint(A).is_commutative is False
assert adjoint(A * A) == adjoint(A)**2
assert adjoint(A * B) == adjoint(B) * adjoint(A)
assert adjoint(A * B**2) == adjoint(B)**2 * adjoint(A)
assert adjoint(A * B -
B * A) == adjoint(B) * adjoint(A) - adjoint(A) * adjoint(B)
assert adjoint(A + I * B) == adjoint(A) - I * adjoint(B)
assert adjoint(2**x) == 2**adjoint(x)
assert adjoint(x**pi) == adjoint(x**pi, evaluate=False)
def test_adjoint():
x, y = symbols('x y')
assert adjoint(adjoint(x)) == x
assert adjoint(x + y) == adjoint(x) + adjoint(y)
assert adjoint(x - y) == adjoint(x) - adjoint(y)
assert adjoint(x * y) == adjoint(x) * adjoint(y)
assert adjoint(x / y) == adjoint(x) / adjoint(y)
assert adjoint(-x) == -adjoint(x)
x, y = symbols('x y', commutative=False)
assert adjoint(adjoint(x)) == x
assert adjoint(x + y) == adjoint(x) + adjoint(y)
assert adjoint(x - y) == adjoint(x) - adjoint(y)
assert adjoint(x * y) == adjoint(y) * adjoint(x)
assert adjoint(x / y) == 1 / adjoint(y) * adjoint(x)
assert adjoint(-x) == -adjoint(x)
def test_adjoint():
assert adjoint(A).is_commutative is False
assert adjoint(A*A) == adjoint(A)**2
assert adjoint(A*B) == adjoint(B)*adjoint(A)
assert adjoint(A*B**2) == adjoint(B)**2*adjoint(A)
assert adjoint(A*B - B*A) == adjoint(B)*adjoint(A) - adjoint(A)*adjoint(B)
assert adjoint(A + I*B) == adjoint(A) - I*adjoint(B)
assert adjoint(X) == X
assert adjoint(-I*X) == I*X
assert adjoint(Y) == -Y
assert adjoint(-I*Y) == -I*Y
assert adjoint(X) == conjugate(transpose(X))
assert adjoint(Y) == conjugate(transpose(Y))
assert adjoint(X) == transpose(conjugate(X))
assert adjoint(Y) == transpose(conjugate(Y))
assert adjoint(2**x) == 2**adjoint(x)
assert adjoint(x**pi) == adjoint(x**pi, evaluate=False)
