def test_derivatives_of_hadamard_expressions():
# Hadamard Product
expr = hadamard_product(a, x, b)
assert expr.diff(x) == DiagMatrix(hadamard_product(b, a))
expr = a.T * hadamard_product(A, X, B) * b
assert expr.diff(X) == HadamardProduct(a * b.T, A, B)
# Hadamard Power
expr = hadamard_power(x, 2)
assert expr.diff(x).doit() == 2 * DiagMatrix(x)
expr = hadamard_power(x.T, 2)
assert expr.diff(x).doit() == 2 * DiagMatrix(x)
expr = hadamard_power(x, S.Half)
assert expr.diff(x) == S.Half * DiagMatrix(
hadamard_power(x, Rational(-1, 2)))
expr = hadamard_power(a.T * X * b, 2)
assert expr.diff(X) == 2 * a * a.T * X * b * b.T
expr = hadamard_power(a.T * X * b, S.Half)
assert expr.diff(X) == a / (2 * sqrt(a.T * X * b)) * b.T
def test_matrix_derivative_by_scalar():
assert A.diff(i) == ZeroMatrix(k, k)
assert (A*(X + B)*c).diff(i) == ZeroMatrix(k, 1)
assert x.diff(i) == ZeroMatrix(k, 1)
assert (x.T*y).diff(i) == ZeroMatrix(1, 1)
assert (x*x.T).diff(i) == ZeroMatrix(k, k)
assert (x + y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, 2).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, i).diff(i).dummy_eq(
HadamardProduct(x.applyfunc(log), HadamardPower(x, i)))
assert hadamard_product(x, y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_product(i*OneMatrix(k, 1), x, y).diff(i) == hadamard_product(x, y)
assert (i*x).diff(i) == x
assert (sin(i)*A*B*x).diff(i) == cos(i)*A*B*x
assert x.applyfunc(sin).diff(i) == ZeroMatrix(k, 1)
assert Trace(i**2*X).diff(i) == 2*i*Trace(X)
mu = symbols("mu")
expr = (2*mu*x)
assert expr.diff(x) == 2*mu*Identity(k)
def _eval_derivative(self, x):
from sympy.matrices.expressions.hadamard import hadamard_product
dexpr = self.expr.diff(x)
fdiff = self._get_function_fdiff()
return hadamard_product(dexpr,
ElementwiseApplyFunction(fdiff, self.expr))
def identify_hadamard_products(expr: tUnion[ArrayContraction, ArrayDiagonal]):
editor: _EditArrayContraction = _EditArrayContraction(expr)
map_contr_to_args: tDict[FrozenSet, List[_ArgE]] = defaultdict(list)
map_ind_to_inds: tDict[Optional[int], int] = defaultdict(int)
for arg_with_ind in editor.args_with_ind:
for ind in arg_with_ind.indices:
map_ind_to_inds[ind] += 1
if None in arg_with_ind.indices:
continue
map_contr_to_args[frozenset(arg_with_ind.indices)].append(arg_with_ind)
k: FrozenSet[int]
v: List[_ArgE]
for k, v in map_contr_to_args.items():
make_trace: bool = False
if len(k) == 1 and next(iter(k)) >= 0 and sum(
[next(iter(k)) in i for i in map_contr_to_args]) == 1:
# This is a trace: the arguments are fully contracted with only one
# index, and the index isn't used anywhere else:
make_trace = True
first_element = S.One
elif len(k) != 2:
# Hadamard product only defined for matrices:
continue
if len(v) == 1:
# Hadamard product with a single argument makes no sense:
continue
for ind in k:
if map_ind_to_inds[ind] <= 2:
# There is no other contraction, skip:
continue
def check_transpose(x):
x = [i if i >= 0 else -1 - i for i in x]
return x == sorted(x)
# Check if expression is a trace:
if all([map_ind_to_inds[j] == len(v) and j >= 0
for j in k]) and all([j >= 0 for j in k]):
# This is a trace
make_trace = True
first_element = v[0].element
if not check_transpose(v[0].indices):
first_element = first_element.T
hadamard_factors = v[1:]
else:
hadamard_factors = v
# This is a Hadamard product:
hp = hadamard_product(*[
i.element if check_transpose(i.indices) else Transpose(i.element)
for i in hadamard_factors
])
hp_indices = v[0].indices
if not check_transpose(hadamard_factors[0].indices):
hp_indices = list(reversed(hp_indices))
if make_trace:
hp = Trace(first_element * hp.T)._normalize()
hp_indices = []
editor.insert_after(v[0], _ArgE(hp, hp_indices))
for i in v:
editor.args_with_ind.remove(i)
return editor.to_array_contraction()
