def solve_discrete_lyapunov(A, B, max_it=50, method="doubling"):
r"""
Computes the solution to the discrete lyapunov equation
.. math::
AXA' - X + B = 0
X is computed by using a doubling algorithm. In particular, we
iterate to convergence on X_j with the following recursions for j =
1, 2,... starting from X_0 = B, a_0 = A:
.. math::
a_j = a_{j-1} a_{j-1}
X_j = X_{j-1} + a_{j-1} X_{j-1} a_{j-1}'
Parameters
----------
A : array_like(float, ndim=2)
An n x n matrix as described above. We assume in order for
convergence that the eigenvalues of A have moduli bounded by
unity
B : array_like(float, ndim=2)
An n x n matrix as described above. We assume in order for
convergence that the eigenvalues of A have moduli bounded by
unity
max_it : scalar(int), optional(default=50)
The maximum number of iterations
method : string, optional(default="doubling")
Describes the solution method to use. If it is "doubling" then
uses the doubling algorithm to solve, if it is "bartels-stewart"
then it uses scipy's implementation of the Bartels-Stewart
approach.
Returns
========
gamma1: array_like(float, ndim=2)
Represents the value V
"""
if method == "doubling":
A, B = list(map(np.atleast_2d, [A, B]))
alpha0 = A
gamma0 = B
diff = 5
n_its = 1
while diff > 1e-15:
alpha1 = alpha0.dot(alpha0)
gamma1 = gamma0 + np.dot(alpha0.dot(gamma0), alpha0.conjugate().T)
diff = np.max(np.abs(gamma1 - gamma0))
alpha0 = alpha1
gamma0 = gamma1
n_its += 1
if n_its > max_it:
msg = "Exceeded maximum iterations {}, check input matrics"
raise ValueError(msg.format(n_its))
elif method == "bartels-stewart":
gamma1 = sp_solve_discrete_lyapunov(A, B)
else:
msg = "Check your method input. Should be doubling or bartels-stewart"
raise ValueError(msg)
return gamma1
def solve_discrete_lyapunov(A, B, max_it=50, method="doubling"):
r"""
Computes the solution to the discrete lyapunov equation
.. math::
AXA' - X + B = 0
X is computed by using a doubling algorithm. In particular, we
iterate to convergence on X_j with the following recursions for j =
1, 2,... starting from X_0 = B, a_0 = A:
.. math::
a_j = a_{j-1} a_{j-1}
X_j = X_{j-1} + a_{j-1} X_{j-1} a_{j-1}'
Parameters
----------
A : array_like(float, ndim=2)
An n x n matrix as described above. We assume in order for
convergence that the eigenvalues of A have moduli bounded by
unity
B : array_like(float, ndim=2)
An n x n matrix as described above. We assume in order for
convergence that the eigenvalues of A have moduli bounded by
unity
max_it : scalar(int), optional(default=50)
The maximum number of iterations
method : string, optional(default="doubling")
Describes the solution method to use. If it is "doubling" then
uses the doubling algorithm to solve, if it is "bartels-stewart"
then it uses scipy's implementation of the Bartels-Stewart
approach.
Returns
========
gamma1: array_like(float, ndim=2)
Represents the value V
"""
if method=="doubling":
A, B = list(map(np.atleast_2d, [A, B]))
alpha0 = A
gamma0 = B
diff = 5
n_its = 1
while diff > 1e-15:
alpha1 = alpha0.dot(alpha0)
gamma1 = gamma0 + np.dot(alpha0.dot(gamma0), alpha0.conjugate().T)
diff = np.max(np.abs(gamma1 - gamma0))
alpha0 = alpha1
gamma0 = gamma1
n_its += 1
if n_its > max_it:
msg = "Exceeded maximum iterations {}, check input matrics"
raise ValueError(msg.format(n_its))
elif method=="bartels-stewart":
gamma1 = sp_solve_discrete_lyapunov(A, B)
else:
msg = "Check your method input. Should be doubling or bartels-stewart"
raise ValueError(msg)
return gamma1