def test_normalize_diagonal_contraction():
tp = CodegenArrayTensorProduct(M, N, P, Q)
expr = CodegenArrayContraction(CodegenArrayDiagonal(tp, (1, 3, 4)), (0, 3))
result = CodegenArrayDiagonal(
CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (0, 6)),
(0, 2, 3))
assert expr == result
expr = CodegenArrayContraction(CodegenArrayDiagonal(tp, (0, 1, 2, 3, 7)),
(1, 2, 3))
result = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q),
(0, 1, 2, 3, 5, 6, 7))
assert expr == result
expr = CodegenArrayContraction(CodegenArrayDiagonal(tp, (0, 2, 6, 7)),
(1, 2, 3))
result = CodegenArrayDiagonal(CodegenArrayContraction(tp, (3, 4, 5)),
(0, 2, 3, 4))
assert expr == result
td = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P, Q), (0, 3))
expr = CodegenArrayContraction(td, (2, 1), (0, 4, 6, 5, 3))
result = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q),
(0, 1, 3, 5, 6, 7), (2, 4))
assert expr == result
def test_codegen_recognize_matrix_expression():
expr = CodegenArrayElementwiseAdd(M, CodegenArrayPermuteDims(M, [1, 0]))
assert recognize_matrix_expression(expr) == M + Transpose(M)
expr = M[i,j] + N[i,j]
p1, p2 = _codegen_array_parse(expr)
assert recognize_matrix_expression(p1) == M + N
expr = M[i,j] + N[j,i]
p1, p2 = _codegen_array_parse(expr)
assert recognize_matrix_expression(p1) == M + N.T
expr = M[i,j]*N[k,l] + N[i,j]*M[k,l]
p1, p2 = _codegen_array_parse(expr)
assert recognize_matrix_expression(p1) == CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(M, N),
CodegenArrayTensorProduct(N, M))
expr = (M*N*P)[i, j]
p1, p2 = _codegen_array_parse(expr)
assert recognize_matrix_expression(p1) == M*N*P
expr = Sum(M[i,j]*(N*P)[j,m], (j, 0, k-1))
p1, p2 = _codegen_array_parse(expr)
assert recognize_matrix_expression(p1) == M*N*P
expr = Sum((P[j, m] + P[m, j])*(M[i,j]*N[m,n] + N[i,j]*M[m,n]), (j, 0, k-1), (m, 0, k-1))
p1, p2 = _codegen_array_parse(expr)
assert recognize_matrix_expression(p1) == M*P*N + M*P.T*N + N*P*M + N*P.T*M
def test_array_expr_zero_array():
za1 = ZeroArray(k, l, m, n)
zm1 = ZeroMatrix(m, n)
za2 = ZeroArray(k, m, m, n)
zm2 = ZeroMatrix(m, m)
zm3 = ZeroMatrix(k, k)
assert CodegenArrayTensorProduct(M, N, za1) == ZeroArray(k, k, k, k, k, l, m, n)
assert CodegenArrayTensorProduct(M, N, zm1) == ZeroArray(k, k, k, k, m, n)
assert CodegenArrayContraction(za1, (3,)) == ZeroArray(k, l, m)
assert CodegenArrayContraction(zm1, (1,)) == ZeroArray(m)
assert CodegenArrayContraction(za2, (1, 2)) == ZeroArray(k, n)
assert CodegenArrayContraction(zm2, (0, 1)) == 0
assert CodegenArrayDiagonal(za2, (1, 2)) == ZeroArray(k, n, m)
assert CodegenArrayDiagonal(zm2, (0, 1)) == ZeroArray(m)
assert CodegenArrayPermuteDims(za1, [2, 1, 3, 0]) == ZeroArray(m, l, n, k)
assert CodegenArrayPermuteDims(zm1, [1, 0]) == ZeroArray(n, m)
assert CodegenArrayElementwiseAdd(za1) == za1
assert CodegenArrayElementwiseAdd(zm1) == ZeroArray(m, n)
tp1 = CodegenArrayTensorProduct(MatrixSymbol("A", k, l), MatrixSymbol("B", m, n))
assert CodegenArrayElementwiseAdd(tp1, za1) == tp1
tp2 = CodegenArrayTensorProduct(MatrixSymbol("C", k, l), MatrixSymbol("D", m, n))
assert CodegenArrayElementwiseAdd(tp1, za1, tp2) == CodegenArrayElementwiseAdd(tp1, tp2)
assert CodegenArrayElementwiseAdd(M, zm3) == M
assert CodegenArrayElementwiseAdd(M, N, zm3) == CodegenArrayElementwiseAdd(M, N)
def test_contraction_tp_additions():
a = CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(M, N),
CodegenArrayTensorProduct(N, M)
)
tp = CodegenArrayTensorProduct(P, a, Q)
expr = CodegenArrayContraction(tp, (3, 4))
expected = CodegenArrayTensorProduct(
P,
CodegenArrayElementwiseAdd(
CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2)),
CodegenArrayContraction(CodegenArrayTensorProduct(N, M), (1, 2)),
),
Q
)
assert expr == expected
assert recognize_matrix_expression(expr) == CodegenArrayTensorProduct(P, M*N + N*M, Q)
expr = CodegenArrayContraction(tp, (1, 2), (3, 4), (5, 6))
result = CodegenArrayContraction(
CodegenArrayTensorProduct(
P,
CodegenArrayElementwiseAdd(
CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2)),
CodegenArrayContraction(CodegenArrayTensorProduct(N, M), (1, 2)),
),
Q
), (1, 2), (3, 4))
assert expr == result
assert recognize_matrix_expression(expr) == P*(M*N + N*M)*Q
def test_codegen_array_recognize_matrix_mul_lines():
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M), (0, 1))
assert recognize_matrix_expression(cg) == Trace(M)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 1),
(2, 3))
assert recognize_matrix_expression(cg) == Trace(M) * Trace(N)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 3),
(1, 2))
assert recognize_matrix_expression(cg) == Trace(M * N)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 2),
(1, 3))
assert recognize_matrix_expression(cg) == Trace(M * N.T)
cg = parse_indexed_expression((M * N * P)[i, j])
assert recognize_matrix_expression(cg) == M * N * P
cg = parse_matrix_expression(M * N * P)
assert recognize_matrix_expression(cg) == M * N * P
cg = parse_indexed_expression((M * N.T * P)[i, j])
assert recognize_matrix_expression(cg) == M * N.T * P
cg = parse_matrix_expression(M * N.T * P)
assert recognize_matrix_expression(cg) == M * N.T * P
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (1, 2),
(5, 6))
assert recognize_matrix_expression(cg) == [M * N, P * Q]
expr = -2 * M * N
elem = expr[i, j]
cg = parse_indexed_expression(elem)
assert recognize_matrix_expression(cg) == -2 * M * N
def test_codegen_array_diagonal():
cg = CodegenArrayDiagonal(M, (1, 0))
assert cg == CodegenArrayDiagonal(M, (0, 1))
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P), (4, 1),
(2, 0))
assert cg == CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P),
(1, 4), (0, 2))
def test_contraction_permutation_mix():
Me = M.subs(k, 3).as_explicit()
Ne = N.subs(k, 3).as_explicit()
cg1 = CodegenArrayContraction(CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), Permutation([0, 2, 1, 3])), (2, 3))
cg2 = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 3))
assert cg1 == cg2
assert recognize_matrix_expression(cg2) == M*N.T
cge1 = tensorcontraction(permutedims(tensorproduct(Me, Ne), Permutation([0, 2, 1, 3])), (2, 3))
cge2 = tensorcontraction(tensorproduct(Me, Ne), (1, 3))
assert cge1 == cge2
cg1 = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), Permutation([0, 1, 3, 2]))
cg2 = CodegenArrayTensorProduct(M, CodegenArrayPermuteDims(N, Permutation([1, 0])))
assert cg1 == cg2
assert recognize_matrix_expression(cg1) == [M, N.T]
assert recognize_matrix_expression(cg2) == [M, N.T]
cg1 = CodegenArrayContraction(
CodegenArrayPermuteDims(
CodegenArrayTensorProduct(M, N, P, Q), Permutation([0, 2, 3, 1, 4, 5, 7, 6])),
(1, 2), (3, 5)
)
cg2 = CodegenArrayContraction(
CodegenArrayTensorProduct(M, N, P, CodegenArrayPermuteDims(Q, Permutation([1, 0]))),
(1, 5), (2, 3)
)
assert cg1 == cg2
assert recognize_matrix_expression(cg1) == [M*P.T*Trace(N), Q.T]
assert recognize_matrix_expression(cg2) == [M*P.T*Trace(N), Q.T]
cg1 = CodegenArrayContraction(
CodegenArrayPermuteDims(
CodegenArrayTensorProduct(M, N, P, Q), Permutation([1, 0, 4, 6, 2, 7, 5, 3])),
(0, 1), (2, 6), (3, 7)
)
cg2 = CodegenArrayPermuteDims(
CodegenArrayContraction(
CodegenArrayTensorProduct(M, P, Q, N),
(0, 1), (2, 3), (4, 7)),
[1, 0]
)
assert cg1 == cg2
cg1 = CodegenArrayContraction(
CodegenArrayPermuteDims(
CodegenArrayTensorProduct(M, N, P, Q), Permutation([1, 0, 4, 6, 7, 2, 5, 3])),
(0, 1), (2, 6), (3, 7)
)
cg2 = CodegenArrayPermuteDims(
CodegenArrayContraction(
CodegenArrayTensorProduct(CodegenArrayPermuteDims(M, [1, 0]), N, P, Q),
(0, 1), (3, 6), (4, 5)
),
Permutation([1, 0])
)
assert cg1 == cg2
def test_arrayexpr_derivatives1():
res = array_derive(X, X)
assert res == CodegenArrayPermuteDims(CodegenArrayTensorProduct(I, I),
[0, 2, 1, 3])
cg = CodegenArrayTensorProduct(A, X, B)
res = array_derive(cg, X)
assert res == CodegenArrayPermuteDims(
CodegenArrayTensorProduct(I, A, I, B), [0, 4, 2, 3, 1, 5, 6, 7])
def test_parsing_of_matrix_expressions():
expr = M*N
assert _parse_matrix_expression(expr) == CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2))
expr = Transpose(M)
assert _parse_matrix_expression(expr) == CodegenArrayPermuteDims(M, [1, 0])
expr = M*Transpose(N)
assert _parse_matrix_expression(expr) == CodegenArrayContraction(CodegenArrayTensorProduct(M, CodegenArrayPermuteDims(N, [1, 0])), (1, 2))
def test_codegen_array_recognize_matrix_mul_lines():
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M), (0, 1))
assert recognize_matrix_expression(cg) == Trace(M)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 1),
(2, 3))
assert recognize_matrix_expression(cg) == Trace(M) * Trace(N)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 3),
(1, 2))
assert recognize_matrix_expression(cg) == Trace(M * N)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 2),
(1, 3))
assert recognize_matrix_expression(cg) == Trace(M * N.T)
cg = parse_indexed_expression((M * N * P)[i, j])
assert recognize_matrix_expression(cg) == M * N * P
cg = parse_matrix_expression(M * N * P)
assert recognize_matrix_expression(cg) == M * N * P
cg = parse_indexed_expression((M * N.T * P)[i, j])
assert recognize_matrix_expression(cg) == M * N.T * P
cg = parse_matrix_expression(M * N.T * P)
assert recognize_matrix_expression(cg) == M * N.T * P
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, P, Q), (1, 2),
(5, 6))
assert recognize_matrix_expression(cg) == CodegenArrayTensorProduct(
M * N, P * Q)
expr = -2 * M * N
elem = expr[i, j]
cg = parse_indexed_expression(elem)
assert recognize_matrix_expression(cg) == -2 * M * N
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
cg = CodegenArrayPermuteDims(
CodegenArrayContraction(
CodegenArrayTensorProduct(
a,
CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(b, c),
CodegenArrayTensorProduct(c, b),
)), (2, 4)), [0, 1, 3, 2])
assert recognize_matrix_expression(cg) == a * (b.T * c + c.T * b)
za = ZeroArray(m, n)
assert recognize_matrix_expression(za) == ZeroMatrix(m, n)
cg = CodegenArrayTensorProduct(3, M)
assert recognize_matrix_expression(cg) == 3 * M
def test_normalize_diagonal_permutedims():
tp = CodegenArrayTensorProduct(M, Q, N, P)
expr = CodegenArrayDiagonal(
CodegenArrayPermuteDims(tp, [0, 1, 2, 4, 7, 6, 3, 5]), (2, 4, 5), (6, 7),
(0, 3))
result = CodegenArrayDiagonal(tp, (2, 6, 7), (3, 5), (0, 4))
assert expr == result
tp = CodegenArrayTensorProduct(M, N, P, Q)
expr = CodegenArrayDiagonal(CodegenArrayPermuteDims(tp, [0, 5, 2, 4, 1, 6, 3, 7]), (1, 2, 6), (3, 4))
result = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, P, N, Q), (3, 4, 5), (1, 2))
assert expr == result
def test_recognize_expression_implicit_mul():
cg = CodegenArrayTensorProduct(a, b)
assert recognize_matrix_expression(cg) == a*b.T
cg = CodegenArrayTensorProduct(a, I, b)
assert recognize_matrix_expression(cg) == a*b.T
cg = CodegenArrayContraction(CodegenArrayTensorProduct(I, I), (1, 2))
assert recognize_matrix_expression(cg) == I
cg = CodegenArrayPermuteDims(CodegenArrayTensorProduct(I, Identity(1)), [0, 2, 1, 3])
assert recognize_matrix_expression(cg) == I
def test_codegen_array_parse():
expr = M[i, j]
assert _codegen_array_parse(expr) == (M, (i, j))
expr = M[i, j] * N[k, l]
assert _codegen_array_parse(expr) == (CodegenArrayTensorProduct(M, N),
(i, j, k, l))
expr = M[i, j] * N[j, k]
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(
CodegenArrayTensorProduct(M, N), (1, 2)), (i, k, j))
expr = Sum(M[i, j] * N[j, k], (j, 0, k - 1))
assert _codegen_array_parse(expr) == (CodegenArrayContraction(
CodegenArrayTensorProduct(M, N), (1, 2)), (i, k))
expr = M[i, j] + N[i, j]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(M, N),
(i, j))
expr = M[i, j] + N[j, i]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(
M, CodegenArrayPermuteDims(N, Permutation([1, 0]))), (i, j))
expr = M[i, j] + M[j, i]
assert _codegen_array_parse(expr) == (CodegenArrayElementwiseAdd(
M, CodegenArrayPermuteDims(M, Permutation([1, 0]))), (i, j))
expr = (M * N * P)[i, j]
assert _codegen_array_parse(expr) == (CodegenArrayContraction(
CodegenArrayTensorProduct(M, N, P), (1, 2), (3, 4)), (i, j))
expr = expr.function # Disregard summation in previous expression
ret1, ret2 = _codegen_array_parse(expr)
assert ret1 == CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N, P),
(1, 2), (3, 4))
assert str(ret2) == "(i, j, _i_1, _i_2)"
expr = KroneckerDelta(i, j) * M[i, k]
assert _codegen_array_parse(expr) == (M, ({i, j}, k))
expr = KroneckerDelta(i, j) * KroneckerDelta(j, k) * M[i, l]
assert _codegen_array_parse(expr) == (M, ({i, j, k}, l))
expr = KroneckerDelta(j, k) * (M[i, j] * N[k, l] + N[i, j] * M[k, l])
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(
CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(M, N),
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N),
Permutation(0, 2)(1, 3))),
(1, 2)), (i, l, frozenset({j, k})))
expr = KroneckerDelta(j, m) * KroneckerDelta(
m, k) * (M[i, j] * N[k, l] + N[i, j] * M[k, l])
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(
CodegenArrayElementwiseAdd(
CodegenArrayTensorProduct(M, N),
CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N),
Permutation(0, 2)(1, 3))),
(1, 2)), (i, l, frozenset({j, m, k})))
expr = KroneckerDelta(i, j) * KroneckerDelta(j, k) * KroneckerDelta(
k, m) * M[i, 0] * KroneckerDelta(m, n)
assert _codegen_array_parse(expr) == (M, ({i, j, k, m, n}, 0))
expr = M[i, i]
assert _codegen_array_parse(expr) == (CodegenArrayDiagonal(M,
(0, 1)), (i, ))
def test_codegen_array_contraction_indices_types():
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (0, 1))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 0), (0, 1)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(0, 1)]
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 1), (1, 0)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(1, 2)]
cg = CodegenArrayContraction(CodegenArrayTensorProduct(M, M, N), (1, 4), (2, 5))
indtup = cg._get_contraction_tuples()
assert indtup == [[(0, 1), (2, 0)], [(1, 0), (2, 1)]]
assert cg._contraction_tuples_to_contraction_indices(cg.expr, indtup) == [(1, 4), (2, 5)]
def _(expr: Inverse, x: Expr):
mat = expr.I
dexpr = array_derive(mat, x)
tp = CodegenArrayTensorProduct(-expr, dexpr, expr)
mp = CodegenArrayContraction(tp, (1, 4), (5, 6))
pp = CodegenArrayPermuteDims(mp, [1, 2, 0, 3])
return pp
def _(expr: CodegenArrayTensorProduct, x: Expr):
args = expr.args
addend_list = []
for i, arg in enumerate(expr.args):
darg = array_derive(arg, x)
if darg == 0:
continue
args_prev = args[:i]
args_succ = args[i + 1:]
shape_prev = reduce(operator.add, map(get_shape, args_prev), ())
shape_succ = reduce(operator.add, map(get_shape, args_succ), ())
addend = CodegenArrayTensorProduct(*args_prev, darg, *args_succ)
tot1 = len(get_shape(x))
tot2 = tot1 + len(shape_prev)
tot3 = tot2 + len(get_shape(arg))
tot4 = tot3 + len(shape_succ)
perm = [i for i in range(tot1, tot2)] + \
[i for i in range(tot1)] + [i for i in range(tot2, tot3)] + \
[i for i in range(tot3, tot4)]
addend = CodegenArrayPermuteDims(addend, _af_invert(perm))
addend_list.append(addend)
if len(addend_list) == 1:
return addend_list[0]
elif len(addend_list) == 0:
return S.Zero
else:
return CodegenArrayElementwiseAdd(*addend_list)
def test_nested_permutations():
cg = CodegenArrayPermuteDims(CodegenArrayPermuteDims(M, (1, 0)), (1, 0))
assert cg == M
times = 3
plist1 = [list(range(6)) for i in range(times)]
plist2 = [list(range(6)) for i in range(times)]
for i in range(times):
random.shuffle(plist1[i])
random.shuffle(plist2[i])
plist1.append([2, 5, 4, 1, 0, 3])
plist2.append([3, 5, 0, 4, 1, 2])
plist1.append([2, 5, 4, 0, 3, 1])
plist2.append([3, 0, 5, 1, 2, 4])
plist1.append([5, 4, 2, 0, 3, 1])
plist2.append([4, 5, 0, 2, 3, 1])
Me = M.subs(k, 3).as_explicit()
Ne = N.subs(k, 3).as_explicit()
Pe = P.subs(k, 3).as_explicit()
cge = tensorproduct(Me, Ne, Pe)
for permutation_array1, permutation_array2 in zip(plist1, plist2):
p1 = Permutation(permutation_array1)
p2 = Permutation(permutation_array2)
cg = CodegenArrayPermuteDims(
CodegenArrayPermuteDims(
CodegenArrayTensorProduct(M, N, P),
p1),
p2
)
result = CodegenArrayPermuteDims(
CodegenArrayTensorProduct(M, N, P),
p2*p1
)
assert cg == result
# Check that `permutedims` behaves the same way with explicit-component arrays:
result1 = permutedims(permutedims(cge, p1), p2)
result2 = permutedims(cge, p2*p1)
assert result1 == result2
def tensorproduct(*args):
"""
Tensor product among scalars or array-like objects.
Examples
========
>>> from sympy.tensor.array import tensorproduct, Array
>>> from sympy.abc import x, y, z, t
>>> A = Array([[1, 2], [3, 4]])
>>> B = Array([x, y])
>>> tensorproduct(A, B)
[[[x, y], [2*x, 2*y]], [[3*x, 3*y], [4*x, 4*y]]]
>>> tensorproduct(A, x)
[[x, 2*x], [3*x, 4*x]]
>>> tensorproduct(A, B, B)
[[[[x**2, x*y], [x*y, y**2]], [[2*x**2, 2*x*y], [2*x*y, 2*y**2]]], [[[3*x**2, 3*x*y], [3*x*y, 3*y**2]], [[4*x**2, 4*x*y], [4*x*y, 4*y**2]]]]
Applying this function on two matrices will result in a rank 4 array.
>>> from sympy import Matrix, eye
>>> m = Matrix([[x, y], [z, t]])
>>> p = tensorproduct(eye(3), m)
>>> p
[[[[x, y], [z, t]], [[0, 0], [0, 0]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[x, y], [z, t]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[0, 0], [0, 0]], [[x, y], [z, t]]]]
"""
from sympy.tensor.array import SparseNDimArray, ImmutableSparseNDimArray
if len(args) == 0:
return S.One
if len(args) == 1:
return _arrayfy(args[0])
from sympy.codegen.array_utils import _CodegenArrayAbstract, CodegenArrayTensorProduct
from sympy.tensor.array.expressions.array_expressions import _ArrayExpr
from sympy import MatrixSymbol
if any(
isinstance(arg, (_ArrayExpr, _CodegenArrayAbstract, MatrixSymbol))
for arg in args):
return CodegenArrayTensorProduct(*args)
if len(args) > 2:
return tensorproduct(tensorproduct(args[0], args[1]), *args[2:])
# length of args is 2:
a, b = map(_arrayfy, args)
if not isinstance(a, NDimArray) or not isinstance(b, NDimArray):
return a * b
if isinstance(a, SparseNDimArray) and isinstance(b, SparseNDimArray):
lp = len(b)
new_array = {
k1 * lp + k2: v1 * v2
for k1, v1 in a._sparse_array.items()
for k2, v2 in b._sparse_array.items()
}
return ImmutableSparseNDimArray(new_array, a.shape + b.shape)
product_list = [i * j for i in Flatten(a) for j in Flatten(b)]
return ImmutableDenseNDimArray(product_list, a.shape + b.shape)
def test_special_matrices():
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
expr = a.T*b
elem = expr[0, 0]
cg = parse_indexed_expression(elem)
assert cg == CodegenArrayContraction(CodegenArrayTensorProduct(a, b), (0, 2))
assert recognize_matrix_expression(cg) == a.T*b
def _(expr: ElementwiseApplyFunction, x: Expr):
assert get_rank(expr) == 2
assert get_rank(x) == 2
fdiff = expr._get_function_fdiff()
dexpr = array_derive(expr.expr, x)
tp = CodegenArrayTensorProduct(ElementwiseApplyFunction(fdiff, expr.expr),
dexpr)
td = CodegenArrayDiagonal(tp, (0, 4), (1, 5))
return td
def test_recognize_diagonalized_vectors():
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a), (1, 2))
assert recognize_matrix_expression(cg) == A*a
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, B), (1, 2, 4))
assert recognize_matrix_expression(cg) == A*DiagonalizeVector(a)*B
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, B), (0, 2, 4))
assert recognize_matrix_expression(cg) == A.T*DiagonalizeVector(a)*B
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, b, a.T, B), (0, 2, 4, 7, 9))
assert recognize_matrix_expression(cg).doit() == A.T*DiagonalizeVector(a)*DiagonalizeVector(b)*DiagonalizeVector(a)*B.T
def test_codegen_extra():
if not np:
skip("NumPy not installed")
M = MatrixSymbol("M", 2, 2)
N = MatrixSymbol("N", 2, 2)
P = MatrixSymbol("P", 2, 2)
Q = MatrixSymbol("Q", 2, 2)
ma = np.matrix([[1, 2], [3, 4]])
mb = np.matrix([[1, -2], [-1, 3]])
mc = np.matrix([[2, 0], [1, 2]])
md = np.matrix([[1, -1], [4, 7]])
cg = CodegenArrayTensorProduct(M, N)
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.einsum(ma, [0, 1], mb, [2, 3])).all()
cg = CodegenArrayElementwiseAdd(M, N)
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == ma + mb).all()
cg = CodegenArrayElementwiseAdd(M, N, P)
f = lambdify((M, N, P), cg, 'numpy')
assert (f(ma, mb, mc) == ma + mb + mc).all()
cg = CodegenArrayElementwiseAdd(M, N, P, Q)
f = lambdify((M, N, P, Q), cg, 'numpy')
assert (f(ma, mb, mc, md) == ma + mb + mc + md).all()
cg = CodegenArrayPermuteDims(M, [1, 0])
f = lambdify((M, ), cg, 'numpy')
assert (f(ma) == ma.T).all()
cg = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N), [1, 2, 3, 0])
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.transpose(np.einsum(ma, [0, 1], mb, [2, 3]),
(1, 2, 3, 0))).all()
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N), (1, 2))
f = lambdify((M, N), cg, 'numpy')
assert (f(ma, mb) == np.diagonal(np.einsum(ma, [0, 1], mb, [2, 3]),
axis1=1,
axis2=2)).all()
def test_codegen_array_shape():
expr = CodegenArrayTensorProduct(M, N, P, Q)
assert expr.shape == (k, k, k, k, k, k, k, k)
Z = MatrixSymbol("Z", m, n)
expr = CodegenArrayTensorProduct(M, Z)
assert expr.shape == (k, k, m, n)
expr2 = CodegenArrayContraction(expr, (0, 1))
assert expr2.shape == (m, n)
expr2 = CodegenArrayDiagonal(expr, (0, 1))
assert expr2.shape == (m, n, k)
exprp = CodegenArrayPermuteDims(expr, [2, 1, 3, 0])
assert exprp.shape == (m, k, n, k)
expr3 = CodegenArrayTensorProduct(N, Z)
expr2 = CodegenArrayElementwiseAdd(expr, expr3)
assert expr2.shape == (k, k, m, n)
# Contraction along axes with discordant dimensions:
raises(ValueError, lambda: CodegenArrayContraction(expr, (1, 2)))
# Also diagonal needs the same dimensions:
raises(ValueError, lambda: CodegenArrayDiagonal(expr, (1, 2)))
def test_parsing_of_matrix_expressions():
expr = M * N
assert parse_matrix_expression(expr) == CodegenArrayContraction(
CodegenArrayTensorProduct(M, N), (1, 2))
expr = Transpose(M)
assert parse_matrix_expression(expr) == CodegenArrayPermuteDims(M, [1, 0])
expr = M * Transpose(N)
assert parse_matrix_expression(expr) == CodegenArrayContraction(
CodegenArrayTensorProduct(M, CodegenArrayPermuteDims(N, [1, 0])),
(1, 2))
expr = 3 * M * N
res = parse_matrix_expression(expr)
rexpr = recognize_matrix_expression(res)
assert expr == rexpr
expr = 3 * M + N * M.T * M + 4 * k * N
res = parse_matrix_expression(expr)
rexpr = recognize_matrix_expression(res)
assert expr == rexpr
expr = Inverse(M) * N
rexpr = recognize_matrix_expression(parse_matrix_expression(expr))
assert expr == rexpr
expr = M**2
rexpr = recognize_matrix_expression(parse_matrix_expression(expr))
assert expr == rexpr
expr = M * (2 * N + 3 * M)
res = parse_matrix_expression(expr)
rexpr = recognize_matrix_expression(res)
assert expr.expand() == rexpr.doit()
expr = Trace(M)
result = CodegenArrayContraction(M, (0, 1))
assert parse_matrix_expression(expr) == result
def test_codegen_array_contraction_construction():
cg = CodegenArrayContraction(A)
assert cg == A
s = Sum(A[i] * B[i], (i, 0, 3))
cg = parse_indexed_expression(s)
assert cg == CodegenArrayContraction(CodegenArrayTensorProduct(A, B),
(0, 1))
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, B), (1, 0))
assert cg == CodegenArrayContraction(CodegenArrayTensorProduct(A, B),
(0, 1))
expr = M * N
result = CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2))
assert parse_matrix_expression(expr) == result
elem = expr[i, j]
assert parse_indexed_expression(elem) == result
expr = M * N * M
result = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, M),
(1, 2), (3, 4))
assert parse_matrix_expression(expr) == result
elem = expr[i, j]
result = CodegenArrayContraction(CodegenArrayTensorProduct(M, M, N),
(1, 4), (2, 5))
cg = parse_indexed_expression(elem)
cg = cg.sort_args_by_name()
assert cg == result
def test_arrayexpr_derivatives1():
res = array_derive(X, X)
assert res == CodegenArrayPermuteDims(CodegenArrayTensorProduct(I, I),
[0, 2, 1, 3])
cg = CodegenArrayTensorProduct(A, X, B)
res = array_derive(cg, X)
assert res == CodegenArrayPermuteDims(
CodegenArrayTensorProduct(I, A, I, B), [0, 4, 2, 3, 1, 5, 6, 7])
cg = CodegenArrayContraction(X, (0, 1))
res = array_derive(cg, X)
assert res == CodegenArrayContraction(CodegenArrayTensorProduct(I, I),
(1, 3))
cg = CodegenArrayDiagonal(X, (0, 1))
res = array_derive(cg, X)
assert res == CodegenArrayDiagonal(CodegenArrayTensorProduct(I, I), (1, 3))
cg = ElementwiseApplyFunction(sin, X)
res = array_derive(cg, X)
assert res.dummy_eq(
CodegenArrayDiagonal(
CodegenArrayTensorProduct(ElementwiseApplyFunction(cos, X), I, I),
(0, 3), (1, 5)))
def test_codegen_array_doit():
M = MatrixSymbol("M", 2, 2)
N = MatrixSymbol("N", 2, 2)
P = MatrixSymbol("P", 2, 2)
Q = MatrixSymbol("Q", 2, 2)
M = M.as_explicit()
N = N.as_explicit()
P = P.as_explicit()
Q = Q.as_explicit()
expr = CodegenArrayTensorProduct(M, N, P, Q)
assert expr.doit() == tensorproduct(M, N, P, Q)
expr2 = CodegenArrayContraction(expr, (0, 1))
assert expr2.doit() == tensorcontraction(tensorproduct(M, N, P, Q), (0, 1))
expr2 = CodegenArrayDiagonal(expr, (0, 1))
#assert expr2 = ... # TODO: not implemented
expr = CodegenArrayTensorProduct(M, N)
exprp = CodegenArrayPermuteDims(expr, [2, 1, 3, 0])
assert exprp.doit() == permutedims(tensorproduct(M, N), [2, 1, 3, 0])
expr = CodegenArrayElementwiseAdd(M, N)
assert expr.doit() == M + N
def test_nested_array_elementwise_add():
cg = CodegenArrayContraction(
CodegenArrayElementwiseAdd(CodegenArrayTensorProduct(M, N),
CodegenArrayTensorProduct(N, M)), (1, 2))
result = CodegenArrayElementwiseAdd(
CodegenArrayContraction(CodegenArrayTensorProduct(M, N), (1, 2)),
CodegenArrayContraction(CodegenArrayTensorProduct(N, M), (1, 2)))
assert cg == result
cg = CodegenArrayDiagonal(
CodegenArrayElementwiseAdd(CodegenArrayTensorProduct(M, N),
CodegenArrayTensorProduct(N, M)), (1, 2))
result = CodegenArrayElementwiseAdd(
CodegenArrayDiagonal(CodegenArrayTensorProduct(M, N), (1, 2)),
CodegenArrayDiagonal(CodegenArrayTensorProduct(N, M), (1, 2)))
assert cg == result
def test_permute_tensor_product():
cg1 = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N, P, Q),
Permutation([2, 3, 1, 0, 5, 4, 6, 7]))
cg2 = CodegenArrayTensorProduct(N, CodegenArrayPermuteDims(M, [1, 0]),
CodegenArrayPermuteDims(P, [1, 0]), Q)
assert cg1 == cg2
# TODO: reverse operation starting with `CodegenArrayPermuteDims` and getting down to `bb`...
cg1 = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N, P, Q),
Permutation([2, 3, 4, 5, 0, 1, 6, 7]))
cg2 = CodegenArrayTensorProduct(N, P, M, Q)
assert cg1 == cg2
cg1 = CodegenArrayPermuteDims(CodegenArrayTensorProduct(M, N, P, Q),
Permutation([2, 3, 4, 6, 5, 7, 0, 1]))
assert cg1.expr == CodegenArrayTensorProduct(N, P, Q, M)
assert cg1.permutation == Permutation([0, 1, 2, 4, 3, 5, 6, 7])
cg1 = CodegenArrayContraction(
CodegenArrayPermuteDims(CodegenArrayTensorProduct(N, Q, Q, M),
[2, 1, 5, 4, 0, 3, 6, 7]), [1, 2, 6])
cg2 = CodegenArrayPermuteDims(
CodegenArrayContraction(CodegenArrayTensorProduct(Q, Q, N, M),
(3, 5, 6)), [0, 2, 3, 1, 4])
assert cg1 == cg2
cg1 = CodegenArrayContraction(
CodegenArrayContraction(
CodegenArrayContraction(
CodegenArrayContraction(
CodegenArrayPermuteDims(
CodegenArrayTensorProduct(N, Q, Q, M),
[2, 1, 5, 4, 0, 3, 6, 7]), [1, 2, 6]), [1, 3, 4]),
[1]), [0])
cg2 = CodegenArrayContraction(CodegenArrayTensorProduct(M, N, Q, Q),
(0, 3, 5), (1, 4, 7), (2, ), (6, ))
assert cg1 == cg2
def test_recognize_diagonalized_vectors():
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
X = MatrixSymbol("X", k, k)
x = MatrixSymbol("x", k, 1)
I1 = Identity(1)
I = Identity(k)
# Check matrix recognition over trivial dimensions:
cg = CodegenArrayTensorProduct(a, b)
assert recognize_matrix_expression(cg) == a * b.T
cg = CodegenArrayTensorProduct(I1, a, b)
assert recognize_matrix_expression(cg) == a * I1 * b.T
# Recognize trace inside a tensor product:
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, B, C), (0, 3),
(1, 2))
assert recognize_matrix_expression(cg) == Trace(A * B) * C
# Transform diagonal operator to contraction:
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(A, a), (1, 2))
assert cg.transform_to_product() == CodegenArrayContraction(
CodegenArrayTensorProduct(A, DiagMatrix(a)), (1, 2))
assert recognize_matrix_expression(cg) == A * DiagMatrix(a)
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(a, b), (0, 2))
assert cg.transform_to_product() == CodegenArrayContraction(
CodegenArrayTensorProduct(DiagMatrix(a), b), (0, 2))
assert recognize_matrix_expression(cg).doit() == DiagMatrix(a) * b
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(A, a), (0, 2))
assert cg.transform_to_product() == CodegenArrayContraction(
CodegenArrayTensorProduct(A, DiagMatrix(a)), (0, 2))
assert recognize_matrix_expression(cg) == A.T * DiagMatrix(a)
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(I, x, I1), (0, 2),
(3, 5))
assert cg.transform_to_product() == CodegenArrayContraction(
CodegenArrayTensorProduct(I, DiagMatrix(x), I1), (0, 2))
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(I, x, A, B), (1, 2),
(5, 6))
assert cg.transform_to_product() == CodegenArrayDiagonal(
CodegenArrayContraction(
CodegenArrayTensorProduct(I, DiagMatrix(x), A, B), (1, 2)), (3, 4))
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(x, I1), (1, 2))
assert isinstance(cg, CodegenArrayDiagonal)
assert cg.diagonal_indices == ((1, 2), )
assert recognize_matrix_expression(cg) == x
cg = CodegenArrayDiagonal(CodegenArrayTensorProduct(x, I), (0, 2))
assert cg.transform_to_product() == CodegenArrayContraction(
CodegenArrayTensorProduct(DiagMatrix(x), I), (0, 2))
assert recognize_matrix_expression(cg).doit() == DiagMatrix(x)
cg = CodegenArrayDiagonal(x, (1, ))
assert cg == x
# Ignore identity matrices with contractions:
cg = CodegenArrayContraction(CodegenArrayTensorProduct(I, A, I, I), (0, 2),
(1, 3), (5, 7))
assert cg.split_multiple_contractions() == cg
assert recognize_matrix_expression(cg) == Trace(A) * I
cg = CodegenArrayContraction(CodegenArrayTensorProduct(Trace(A) * I, I, I),
(1, 5), (3, 4))
assert cg.split_multiple_contractions() == cg
assert recognize_matrix_expression(cg).doit() == Trace(A) * I
# Add DiagMatrix when required:
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a), (1, 2))
assert cg.split_multiple_contractions() == cg
assert recognize_matrix_expression(cg) == A * a
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, B), (1, 2, 4))
assert cg.split_multiple_contractions() == CodegenArrayContraction(
CodegenArrayTensorProduct(A, DiagMatrix(a), B), (1, 2), (3, 4))
assert recognize_matrix_expression(cg) == A * DiagMatrix(a) * B
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, B), (0, 2, 4))
assert cg.split_multiple_contractions() == CodegenArrayContraction(
CodegenArrayTensorProduct(A, DiagMatrix(a), B), (0, 2), (3, 4))
assert recognize_matrix_expression(cg) == A.T * DiagMatrix(a) * B
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, b, a.T, B),
(0, 2, 4, 7, 9))
assert cg.split_multiple_contractions() == CodegenArrayContraction(
CodegenArrayTensorProduct(A, DiagMatrix(a), DiagMatrix(b),
DiagMatrix(a), B), (0, 2), (3, 4), (5, 7),
(6, 9))
assert recognize_matrix_expression(
cg).doit() == A.T * DiagMatrix(a) * DiagMatrix(b) * DiagMatrix(a) * B.T
cg = CodegenArrayContraction(CodegenArrayTensorProduct(I1, I1, I1),
(1, 2, 4))
assert cg.split_multiple_contractions() == CodegenArrayContraction(
CodegenArrayTensorProduct(I1, I1, I1), (1, 2), (3, 4))
assert recognize_matrix_expression(cg).doit() == Identity(1)
cg = CodegenArrayContraction(CodegenArrayTensorProduct(I, I, I, I, A),
(1, 2, 8), (5, 6, 9))
assert recognize_matrix_expression(
cg.split_multiple_contractions()).doit() == A
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, C, a, B),
(1, 2, 4), (5, 6, 8))
assert cg.split_multiple_contractions() == CodegenArrayContraction(
CodegenArrayTensorProduct(A, DiagMatrix(a), C, DiagMatrix(a), B),
(1, 2), (3, 4), (5, 6), (7, 8))
assert recognize_matrix_expression(
cg) == A * DiagMatrix(a) * C * DiagMatrix(a) * B
cg = CodegenArrayContraction(
CodegenArrayTensorProduct(a, I1, b, I1, (a.T * b).applyfunc(cos)),
(1, 2, 8), (5, 6, 9))
assert cg.split_multiple_contractions().dummy_eq(
CodegenArrayContraction(
CodegenArrayTensorProduct(a, I1, b, I1, (a.T * b).applyfunc(cos)),
(1, 2), (3, 8), (5, 6), (7, 9)))
assert recognize_matrix_expression(cg).dummy_eq(
MatMul(a, I1, (a.T * b).applyfunc(cos), Transpose(I1), b.T))
cg = CodegenArrayContraction(
CodegenArrayTensorProduct(A.T, a, b, b.T, (A * X * b).applyfunc(cos)),
(1, 2, 8), (5, 6, 9))
assert cg.split_multiple_contractions().dummy_eq(
CodegenArrayContraction(
CodegenArrayTensorProduct(A.T, DiagMatrix(a), b, b.T,
(A * X * b).applyfunc(cos)), (1, 2),
(3, 8), (5, 6, 9)))
# assert recognize_matrix_expression(cg)
# Check no overlap of lines:
cg = CodegenArrayContraction(CodegenArrayTensorProduct(A, a, C, a, B),
(1, 2, 4), (5, 6, 8), (3, 7))
assert cg.split_multiple_contractions() == cg
cg = CodegenArrayContraction(CodegenArrayTensorProduct(a, b, A), (0, 2, 4),
(1, 3))
assert cg.split_multiple_contractions() == cg
