def test_dft():
n, i, j = symbols('n i j')
assert DFT(4).shape == (4, 4)
assert ask(Q.unitary(DFT(4)))
assert Abs(simplify(det(Matrix(DFT(4))))) == 1
assert DFT(n) * IDFT(n) == Identity(n)
assert DFT(n)[i, j] == exp(-2 * S.Pi * I / n)**(i * j) / sqrt(n)
def test_dft2():
assert DFT(1).as_explicit() == Matrix([[1]])
assert DFT(2).as_explicit() == 1/sqrt(2)*Matrix([[1,1],[1,-1]])
assert DFT(4).as_explicit() == Matrix([[S.Half, S.Half, S.Half, S.Half],
[S.Half, -I/2, Rational(-1,2), I/2],
[S.Half, Rational(-1,2), S.Half, Rational(-1,2)],
[S.Half, I/2, Rational(-1,2), -I/2]])
def test_issue_20275():
# XXX We use complex expansions because complex exponentials are not
# recognized by polys.domains
A = DFT(3).as_explicit().expand(complex=True)
eigenvects = A.eigenvects()
assert eigenvects[0] == (-1, 1, [Matrix([[1 - sqrt(3)], [1], [1]])])
assert eigenvects[1] == (1, 1, [Matrix([[1 + sqrt(3)], [1], [1]])])
assert eigenvects[2] == (-I, 1, [Matrix([[0], [-1], [1]])])
A = DFT(4).as_explicit().expand(complex=True)
eigenvects = A.eigenvects()
assert eigenvects[0] == (-1, 1, [Matrix([[-1], [1], [1], [1]])])
assert eigenvects[1] == (1, 2, [
Matrix([[1], [0], [1], [0]]),
Matrix([[2], [1], [0], [1]])
])
assert eigenvects[2] == (-I, 1, [Matrix([[0], [-1], [0], [1]])])
# XXX We skip test for some parts of eigenvectors which are very
# complicated and fragile under expression tree changes
A = DFT(5).as_explicit().expand(complex=True)
eigenvects = A.eigenvects()
assert eigenvects[0] == (-1, 1,
[Matrix([[1 - sqrt(5)], [1], [1], [1], [1]])])
assert eigenvects[1] == (1, 2, [
Matrix([[S(1) / 2 + sqrt(5) / 2], [0], [1], [1], [0]]),
Matrix([[S(1) / 2 + sqrt(5) / 2], [1], [0], [0], [1]])
])
def test_matrix_element_sets():
X = MatrixSymbol('X', 4, 4)
assert ask(Q.real(X[1, 2]), Q.real_elements(X))
assert ask(Q.integer(X[1, 2]), Q.integer_elements(X))
assert ask(Q.complex(X[1, 2]), Q.complex_elements(X))
assert ask(Q.integer_elements(Identity(3)))
assert ask(Q.integer_elements(ZeroMatrix(3, 3)))
from sympy.matrices.expressions.fourier import DFT
assert ask(Q.complex_elements(DFT(3)))
def mdft(n):
r"""
.. deprecated:: 1.9
Use DFT from sympy.matrices.expressions.fourier instead.
To get identical behavior to ``mdft(n)``, use ``DFT(n).as_explicit()``.
"""
from sympy.matrices.expressions.fourier import DFT
return DFT(n).as_mutable()
def test_dft_creation():
assert DFT(2)
assert DFT(0)
raises(ValueError, lambda: DFT(-1))
raises(ValueError, lambda: DFT(2.0))
raises(ValueError, lambda: DFT(2 + 1j))
n = symbols('n')
assert DFT(n)
n = symbols('n', integer=False)
raises(ValueError, lambda: DFT(n))
n = symbols('n', negative=True)
raises(ValueError, lambda: DFT(n))
from sympy.physics.quantum import TensorProduct from sympy.physics.quantum.constants import hbar # %% # Remeber this to have LaTeX rendered output in Jupyter init_printing() # %% [markdown] # ## Computation # %% Omega = Symbol(r'\Omega') omega = Symbol(r'\omega', real=True) # %% F = DFT(2).as_mutable() # %% Omega = I * omega * Matrix([[0, 1], [-1, 0]]) # %% Omega.eigenvects() # %% T = (pi / (2 * omega)**2) * F.adjoint() * Omega * F # %% T # %% T.eigenvects()
delta = Symbol(r'\delta', real=True)
Omega = Symbol(r'\Omega')
omega = Symbol(r'\omega', real=True)
# %%
hbar = 1
# %%
Delta = Rational(1,2) # 1/N, N=dimension of H_T
# %%
delta = 0.0001
# %%
# F = mdft(2) # deprecation
F = DFT(2).as_mutable()
# %%
T = (pi/omega) * Matrix([
[delta, 0],
[0, Delta+delta]
])
# %%
T
# %%
Omega = (omega**2/(pi*Delta**2))*F*T*(F.adjoint())
# %%
Omega
N = Integer(4) # %% deltaT = Rational(1, 1) # %% DeltaT = N * deltaT # %% T = diag(*Range(N)) # %% T # %% F = DFT(N) # %% F.as_explicit() # %% Dagger(F).as_explicit() # %% Omega = (Dagger(F) @ T @ F).as_explicit() # %% Omega # %% T @ Omega - Omega @ T