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) == 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)
def test_hadamard_power():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
B = MatrixSymbol('B', m, n)
C = MatrixSymbol('C', m, p)
assert hadamard_power(A, 1) == A
assert isinstance(hadamard_power(A, 2), HadamardPower)
assert hadamard_power(A, n).T == hadamard_power(A.T, n)
assert hadamard_power(A, n)[0, 0] == A[0, 0]**n
assert hadamard_power(m, n) == m**n
raises(ValueError, lambda: hadamard_power(A, A))
# Testing printer:
assert str(hadamard_power(A, n)) == "A.**n"
assert str(hadamard_power(A, 1+n)) == "A.**(n + 1)"
assert str(hadamard_power(A*B.T, 1+n)) == "(A*B.T).**(n + 1)"
def test_derivatives_of_hadamard_expressions():
# Hadamard Product
expr = hadamard_product(a, x, b)
assert expr.diff(x) == DiagonalizeVector(hadamard_product(b, a))
expr = a.T * hadamard_product(A, X, B) * b
assert expr.diff(X) == DiagonalizeVector(a) * hadamard_product(
B, A) * DiagonalizeVector(b)
# Hadamard Power
expr = hadamard_power(x, 2)
assert expr.diff(x).doit() == 2 * DiagonalizeVector(x)
expr = hadamard_power(x.T, 2)
assert expr.diff(x).doit() == 2 * DiagonalizeVector(x)
expr = hadamard_power(x, S.Half)
assert expr.diff(x) == S.Half * DiagonalizeVector(
hadamard_power(x, -S.Half))
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 * hadamard_power(a.T * X * b, -S.Half) * b.T
def test_derivatives_of_hadamard_expressions():
# Hadamard Product
expr = hadamard_product(a, x, b)
assert expr.diff(x) == DiagonalizeVector(hadamard_product(b, a))
expr = a.T*hadamard_product(A, X, B)*b
assert expr.diff(X) == DiagonalizeVector(a)*hadamard_product(B, A)*DiagonalizeVector(b)
# Hadamard Power
expr = hadamard_power(x, 2)
assert expr.diff(x).doit() == 2*DiagonalizeVector(x)
expr = hadamard_power(x.T, 2)
assert expr.diff(x).doit() == 2*DiagonalizeVector(x)
expr = hadamard_power(x, S.Half)
assert expr.diff(x) == S.Half*DiagonalizeVector(hadamard_power(x, -S.Half))
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*hadamard_power(a.T*X*b, -S.Half)*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 test_hadamard_power():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
assert hadamard_power(A, 1) == A
assert isinstance(hadamard_power(A, 2), HadamardPower)
assert hadamard_power(A, n).T == hadamard_power(A.T, n)
assert hadamard_power(A, n)[0, 0] == A[0, 0]**n
assert hadamard_power(m, n) == m**n
raises(ValueError, lambda: hadamard_power(A, A))
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
