def test_matrix_derivative_non_matrix_result():
# This is a 4-dimensional array:
assert A.diff(A) == ArrayDerivative(A, A)
assert A.T.diff(A) == ArrayDerivative(A.T, A)
assert (2*A).diff(A) == ArrayDerivative(2*A, A)
assert MatAdd(A, A).diff(A) == ArrayDerivative(MatAdd(A, A), A)
assert (A + B).diff(A) == ArrayDerivative(A + B, A) # TODO: `B` can be removed.
def diff(self, *args, **kwargs):
"""
Calculate the derivative of each element in the array.
Examples
========
>>> from sympy import ImmutableDenseNDimArray
>>> from sympy.abc import x, y
>>> M = ImmutableDenseNDimArray([[x, y], [1, x*y]])
>>> M.diff(x)
[[1, 0], [0, y]]
"""
from sympy.tensor.array.array_derivatives import ArrayDerivative
kwargs.setdefault('evaluate', True)
return ArrayDerivative(self.as_immutable(), *args, **kwargs)
def _matrix_derivative(expr, x):
from sympy.tensor.array.array_derivatives import ArrayDerivative
lines = expr._eval_derivative_matrix_lines(x)
parts = [i.build() for i in lines]
from sympy.codegen.array_utils import recognize_matrix_expression
parts = [[recognize_matrix_expression(j).doit() for j in i] for i in parts]
def _get_shape(elem):
if isinstance(elem, MatrixExpr):
return elem.shape
return 1, 1
def get_rank(parts):
return sum([j not in (1, None) for i in parts for j in _get_shape(i)])
ranks = [get_rank(i) for i in parts]
rank = ranks[0]
def contract_one_dims(parts):
if len(parts) == 1:
return parts[0]
else:
p1, p2 = parts[:2]
if p2.is_Matrix:
p2 = p2.T
if p1 == Identity(1):
pbase = p2
elif p2 == Identity(1):
pbase = p1
else:
pbase = p1 * p2
if len(parts) == 2:
return pbase
else: # len(parts) > 2
if pbase.is_Matrix:
raise ValueError("")
return pbase * Mul.fromiter(parts[2:])
if rank <= 2:
return Add.fromiter([contract_one_dims(i) for i in parts])
return ArrayDerivative(expr, x)
def test_mixed_deriv_mixed_expressions():
expr = 3*Trace(A)
assert expr.diff(A) == 3*Identity(k)
expr = k
deriv = expr.diff(A)
assert isinstance(deriv, ZeroMatrix)
assert deriv == ZeroMatrix(k, k)
expr = Trace(A)**2
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(A)*A
# TODO: this is not yet supported:
assert expr.diff(A) == ArrayDerivative(expr, A)
expr = Trace(Trace(A)*A)
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(Trace(Trace(A)*A)*A)
assert expr.diff(A) == (3*Trace(A)**2)*Identity(k)
def test_matrix_derivative_trivial_cases():
# Cookbook example 33:
# TODO: find a way to represent a four-dimensional zero-array:
assert X.diff(A) == ArrayDerivative(X, A)
def test_array_derivative_construction():
d = ArrayDerivative(x, m, evaluate=False)
assert d.shape == (2, 2)
expr = d.doit()
assert isinstance(expr, MatrixCommon)
assert expr.shape == (2, 2)
d = ArrayDerivative(m, m, evaluate=False)
assert d.shape == (2, 2, 2, 2)
expr = d.doit()
assert isinstance(expr, NDimArray)
assert expr.shape == (2, 2, 2, 2)
d = ArrayDerivative(m, x, evaluate=False)
assert d.shape == (2, 2)
expr = d.doit()
assert isinstance(expr, MatrixCommon)
assert expr.shape == (2, 2)
d = ArrayDerivative(M, N, evaluate=False)
assert d.shape == (4, 3, 3, 2)
expr = d.doit()
assert isinstance(expr, ArrayDerivative)
assert expr.shape == (4, 3, 3, 2)
d = ArrayDerivative(M, (N, 2), evaluate=False)
assert d.shape == (4, 3, 4, 3, 3, 2)
expr = d.doit()
assert isinstance(expr, ArrayDerivative)
assert expr.shape == (4, 3, 4, 3, 3, 2)
d = ArrayDerivative(M.as_explicit(), (N.as_explicit(), 2), evaluate=False)
assert d.doit().shape == (4, 3, 4, 3, 3, 2)
expr = d.doit()
assert isinstance(expr, ArrayDerivative)
assert expr.shape == (4, 3, 4, 3, 3, 2)
def test_matrix_derivatives_of_traces():
expr = Trace(A)*A
assert expr.diff(A) == ArrayDerivative(Trace(A)*A, A)
assert expr[i, j].diff(A[m, n]).doit() == (
KDelta(i, m)*KDelta(j, n)*Trace(A) +
KDelta(m, n)*A[i, j]
)
## First order:
# Cookbook example 99:
expr = Trace(X)
assert expr.diff(X) == Identity(k)
assert expr.rewrite(Sum).diff(X[m, n]).doit() == KDelta(m, n)
# Cookbook example 100:
expr = Trace(X*A)
assert expr.diff(X) == A.T
assert expr.rewrite(Sum).diff(X[m, n]).doit() == A[n, m]
# Cookbook example 101:
expr = Trace(A*X*B)
assert expr.diff(X) == A.T*B.T
assert expr.rewrite(Sum).diff(X[m, n]).doit().dummy_eq((A.T*B.T)[m, n])
# Cookbook example 102:
expr = Trace(A*X.T*B)
assert expr.diff(X) == B*A
# Cookbook example 103:
expr = Trace(X.T*A)
assert expr.diff(X) == A
# Cookbook example 104:
expr = Trace(A*X.T)
assert expr.diff(X) == A
# Cookbook example 105:
# TODO: TensorProduct is not supported
#expr = Trace(TensorProduct(A, X))
#assert expr.diff(X) == Trace(A)*Identity(k)
## Second order:
# Cookbook example 106:
expr = Trace(X**2)
assert expr.diff(X) == 2*X.T
# Cookbook example 107:
expr = Trace(X**2*B)
assert expr.diff(X) == (X*B + B*X).T
expr = Trace(MatMul(X, X, B))
assert expr.diff(X) == (X*B + B*X).T
# Cookbook example 108:
expr = Trace(X.T*B*X)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 109:
expr = Trace(B*X*X.T)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 110:
expr = Trace(X*X.T*B)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 111:
expr = Trace(X*B*X.T)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 112:
expr = Trace(B*X.T*X)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 113:
expr = Trace(X.T*X*B)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 114:
expr = Trace(A*X*B*X)
assert expr.diff(X) == A.T*X.T*B.T + B.T*X.T*A.T
# Cookbook example 115:
expr = Trace(X.T*X)
assert expr.diff(X) == 2*X
expr = Trace(X*X.T)
assert expr.diff(X) == 2*X
# Cookbook example 116:
expr = Trace(B.T*X.T*C*X*B)
assert expr.diff(X) == C.T*X*B*B.T + C*X*B*B.T
# Cookbook example 117:
expr = Trace(X.T*B*X*C)
assert expr.diff(X) == B*X*C + B.T*X*C.T
# Cookbook example 118:
expr = Trace(A*X*B*X.T*C)
assert expr.diff(X) == A.T*C.T*X*B.T + C*A*X*B
# Cookbook example 119:
expr = Trace((A*X*B + C)*(A*X*B + C).T)
assert expr.diff(X) == 2*A.T*(A*X*B + C)*B.T
# Cookbook example 120:
# TODO: no support for TensorProduct.
# expr = Trace(TensorProduct(X, X))
# expr = Trace(X)*Trace(X)
# expr.diff(X) == 2*Trace(X)*Identity(k)
# Higher Order
# Cookbook example 121:
expr = Trace(X**k)
#assert expr.diff(X) == k*(X**(k-1)).T
# Cookbook example 122:
expr = Trace(A*X**k)
#assert expr.diff(X) == # Needs indices
# Cookbook example 123:
expr = Trace(B.T*X.T*C*X*X.T*C*X*B)
assert expr.diff(X) == C*X*X.T*C*X*B*B.T + C.T*X*B*B.T*X.T*C.T*X + C*X*B*B.T*X.T*C*X + C.T*X*X.T*C.T*X*B*B.T
# Other
# Cookbook example 124:
expr = Trace(A*X**(-1)*B)
assert expr.diff(X) == -Inverse(X).T*A.T*B.T*Inverse(X).T
# Cookbook example 125:
expr = Trace(Inverse(X.T*C*X)*A)
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == - X.inv().T*A.T*X.inv()*C.inv().T*X.inv().T - X.inv().T*A*X.inv()*C.inv()*X.inv().T
# Cookbook example 126:
expr = Trace((X.T*C*X).inv()*(X.T*B*X))
assert expr.diff(X) == -2*C*X*(X.T*C*X).inv()*X.T*B*X*(X.T*C*X).inv() + 2*B*X*(X.T*C*X).inv()
# Cookbook example 127:
expr = Trace((A + X.T*C*X).inv()*(X.T*B*X))
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == B*X*Inverse(A + X.T*C*X) - C*X*Inverse(A + X.T*C*X)*X.T*B*X*Inverse(A + X.T*C*X) - C.T*X*Inverse(A.T + (C*X).T*X)*X.T*B.T*X*Inverse(A.T + (C*X).T*X) + B.T*X*Inverse(A.T + (C*X).T*X)
