def quad_over_lin_canon(expr, real_args, imag_args, real2imag):
"""Convert quad_over_lin to real.
"""
if imag_args[0] is None:
matrix = real_args[0]
else:
matrix = bmat([real_args[0], imag_args[0]])
return expr.copy([matrix, real_args[1]]), None
def hermitian_canon(expr, real_args, imag_args, real2imag):
"""Canonicalize functions that take a Hermitian matrix.
"""
if imag_args[0] is None:
matrix = real_args[0]
else:
if real_args[0] is None:
real_args[0] = np.zeros(imag_args[0].shape)
matrix = bmat([[real_args[0], -imag_args[0]],
[imag_args[0], real_args[0]]])
return expr.copy([matrix]), None
def matrix_frac_canon(expr, real_args, imag_args, real2imag):
"""Convert matrix_frac to real.
"""
if real_args[0] is None:
real_args[0] = np.zeros(imag_args[0].shape)
if imag_args[0] is None:
imag_args[0] = np.zeros(real_args[0].shape)
vec = vstack([at_least_2D(real_args[0]), at_least_2D(imag_args[0])])
if real_args[1] is None:
real_args[1] = np.zeros(imag_args[1].shape)
elif imag_args[1] is None:
imag_args[1] = np.zeros(real_args[1].shape)
matrix = bmat([[real_args[1], -imag_args[1]], [imag_args[1],
real_args[1]]])
return expr.copy([vec, matrix]), None
def matrix_frac_canon(expr, args):
X = args[0] # n by m matrix.
P = args[1] # n by n matrix.
if len(X.shape) == 1:
X = reshape(X, (X.shape[0], 1))
n, m = X.shape
T = Variable((m, m), symmetric=True)
M = bmat([[P, X], [X.T, T]])
# ^ a matrix with Schur complement T - X.T*P^-1*X.
constraints = [PSD(M)]
if not P.is_symmetric():
ut = upper_tri(P)
lt = upper_tri(P.T)
constraints.append(ut == lt)
return trace(T), constraints
def quad_canon(expr, real_args, imag_args, real2imag):
"""Convert quad_form to real.
"""
if imag_args[0] is None:
vec = real_args[0]
matrix = real_args[1]
elif real_args[0] is None:
vec = imag_args[0]
matrix = real_args[1]
else:
vec = vstack([at_least_2D(real_args[0]), at_least_2D(imag_args[0])])
if real_args[1] is None:
real_args[1] = np.zeros(imag_args[1].shape)
elif imag_args[1] is None:
imag_args[1] = np.zeros(real_args[1].shape)
matrix = bmat([[real_args[1], -imag_args[1]],
[imag_args[1], real_args[1]]])
matrix = psd_wrap(matrix)
return expr.copy([vec, matrix]), None
