def solve_lyap_dense(A, E, B, trans=False, options=None):
"""Compute the solution of a Lyapunov equation.
See :func:`pymor.algorithms.lyapunov.solve_lyap_dense` for a
general description.
This function uses `slycot.sb03md` (if `E is None`) and
`slycot.sg03ad` (if `E is not None`), which are based on the
Bartels-Stewart algorithm.
Parameters
----------
A
The operator A as a 2D |NumPy array|.
E
The operator E as a 2D |NumPy array| or `None`.
B
The operator B as a 2D |NumPy array|.
trans
Whether the first operator in the Lyapunov equation is
transposed.
options
The solver options to use (see
:func:`lyap_dense_solver_options`).
Returns
-------
X
Lyapunov equation solution as a |NumPy array|.
"""
_solve_lyap_dense_check_args(A, E, B, trans)
options = _parse_options(options, lyap_dense_solver_options(), 'slycot_bartels-stewart', None, False)
if options['type'] == 'slycot_bartels-stewart':
n = A.shape[0]
C = -B.dot(B.T) if not trans else -B.T.dot(B)
trana = 'T' if not trans else 'N'
dico = 'C'
job = 'B'
if E is None:
U = np.zeros((n, n))
X, scale, sep, ferr, _ = slycot.sb03md(n, C, A, U, dico, job=job, trana=trana)
_solve_check(A.dtype, 'slycot.sb03md', sep, ferr)
else:
fact = 'N'
uplo = 'L'
Q = np.zeros((n, n))
Z = np.zeros((n, n))
_, _, _, _, X, scale, sep, ferr, _, _, _ = slycot.sg03ad(dico, job, fact, trana, uplo,
n, A, E,
Q, Z, C)
_solve_check(A.dtype, 'slycot.sg03ad', sep, ferr)
X /= scale
else:
raise ValueError(f"Unexpected Lyapunov equation solver ({options['type']}).")
return X
def solve_lyap_dense(A, E, B, trans=False, options=None):
"""Compute the solution of a Lyapunov equation.
See :func:`pymor.algorithms.lyapunov.solve_lyap_dense` for a
general description.
This function uses `slycot.sb03md` (if `E is None`) and
`slycot.sg03ad` (if `E is not None`), which are based on the
Bartels-Stewart algorithm.
Parameters
----------
A
The operator A as a 2D |NumPy array|.
E
The operator E as a 2D |NumPy array| or `None`.
B
The operator B as a 2D |NumPy array|.
trans
Whether the first operator in the Lyapunov equation is
transposed.
options
The solver options to use (see
:func:`lyap_dense_solver_options`).
Returns
-------
X
Lyapunov equation solution as a |NumPy array|.
"""
_solve_lyap_dense_check_args(A, E, B, trans)
options = _parse_options(options, lyap_dense_solver_options(), 'slycot_bartels-stewart', None, False)
if options['type'] == 'slycot_bartels-stewart':
n = A.shape[0]
C = -B.dot(B.T) if not trans else -B.T.dot(B)
trana = 'T' if not trans else 'N'
dico = 'C'
job = 'B'
if E is None:
U = np.zeros((n, n))
X, scale, sep, ferr, _ = slycot.sb03md(n, C, A, U, dico, job=job, trana=trana)
_solve_check(A.dtype, 'slycot.sb03md', sep, ferr)
else:
fact = 'N'
uplo = 'L'
Q = np.zeros((n, n))
Z = np.zeros((n, n))
_, _, _, _, X, scale, sep, ferr, _, _, _ = slycot.sg03ad(dico, job, fact, trana, uplo,
n, A, E,
Q, Z, C)
_solve_check(A.dtype, 'slycot.sg03ad', sep, ferr)
X /= scale
else:
raise ValueError(f"Unexpected Lyapunov equation solver ({options['type']}).")
return X
def lyap(A,Q,C=None,E=None):
""" X = lyap(A,Q) solves the continuous-time Lyapunov equation
A X + X A^T + Q = 0
where A and Q are square matrices of the same dimension.
Further, Q must be symmetric.
X = lyap(A,Q,C) solves the Sylvester equation
A X + X Q + C = 0
where A and Q are square matrices.
X = lyap(A,Q,None,E) solves the generalized continuous-time
Lyapunov equation
A X E^T + E X A^T + Q = 0
where Q is a symmetric matrix and A, Q and E are square matrices
of the same dimension. """
# Make sure we have access to the right slycot routines
try:
from slycot import sb03md
except ImportError:
raise ControlSlycot("can't find slycot module 'sb03md'")
try:
from slycot import sb04md
except ImportError:
raise ControlSlycot("can't find slycot module 'sb04md'")
# Reshape 1-d arrays
if len(shape(A)) == 1:
A = A.reshape(1,A.size)
if len(shape(Q)) == 1:
Q = Q.reshape(1,Q.size)
if C != None and len(shape(C)) == 1:
C = C.reshape(1,C.size)
if E != None and len(shape(E)) == 1:
E = E.reshape(1,E.size)
# Determine main dimensions
if size(A) == 1:
n = 1
else:
n = size(A,0)
if size(Q) == 1:
m = 1
else:
m = size(Q,0)
# Solve standard Lyapunov equation
if C==None and E==None:
# Check input data for consistency
if shape(A) != shape(Q):
raise ControlArgument("A and Q must be matrices of identical \
sizes.")
if size(A) > 1 and shape(A)[0] != shape(A)[1]:
raise ControlArgument("A must be a quadratic matrix.")
if size(Q) > 1 and shape(Q)[0] != shape(Q)[1]:
raise ControlArgument("Q must be a quadratic matrix.")
if not (asarray(Q) == asarray(Q).T).all():
raise ControlArgument("Q must be a symmetric matrix.")
# Solve the Lyapunov equation by calling Slycot function sb03md
try:
X,scale,sep,ferr,w = sb03md(n,-Q,A,eye(n,n),'C',trana='T')
except ValueError(ve):
if ve.info < 0:
e = ValueError(ve.message)
e.info = ve.info
elif ve.info == n+1:
e = ValueError("The matrix A and -A have common or very \
close eigenvalues.")
e.info = ve.info
else:
e = ValueError("The QR algorithm failed to compute all \
the eigenvalues (see LAPACK Library routine DGEES).")
e.info = ve.info
raise e
# Solve the Sylvester equation
elif C != None and E==None:
# Check input data for consistency
if size(A) > 1 and shape(A)[0] != shape(A)[1]:
raise ControlArgument("A must be a quadratic matrix.")
if size(Q) > 1 and shape(Q)[0] != shape(Q)[1]:
raise ControlArgument("Q must be a quadratic matrix.")
if (size(C) > 1 and shape(C)[0] != n) or \
(size(C) > 1 and shape(C)[1] != m) or \
(size(C) == 1 and size(A) != 1) or (size(C) == 1 and size(Q) != 1):
raise ControlArgument("C matrix has incompatible dimensions.")
# Solve the Sylvester equation by calling the Slycot function sb04md
try:
X = sb04md(n,m,A,Q,-C)
except ValueError(ve):
if ve.info < 0:
e = ValueError(ve.message)
e.info = ve.info
elif ve.info > m:
e = ValueError("A singular matrix was encountered whilst \
solving for the %i-th column of matrix X." % ve.info-m)
e.info = ve.info
else:
e = ValueError("The QR algorithm failed to compute all the \
eigenvalues (see LAPACK Library routine DGEES).")
e.info = ve.info
raise e
# Solve the generalized Lyapunov equation
elif C == None and E != None:
# Check input data for consistency
if (size(Q) > 1 and shape(Q)[0] != shape(Q)[1]) or \
(size(Q) > 1 and shape(Q)[0] != n) or \
(size(Q) == 1 and n > 1):
raise ControlArgument("Q must be a square matrix with the same \
dimension as A.")
if (size(E) > 1 and shape(E)[0] != shape(E)[1]) or \
(size(E) > 1 and shape(E)[0] != n) or \
(size(E) == 1 and n > 1):
raise ControlArgument("E must be a square matrix with the same \
dimension as A.")
if not (asarray(Q) == asarray(Q).T).all():
raise ControlArgument("Q must be a symmetric matrix.")
# Make sure we have access to the write slicot routine
try:
from slycot import sg03ad
except ImportError:
raise ControlSlycot("can't find slycot module 'sg03ad'")
# Solve the generalized Lyapunov equation by calling Slycot
# function sg03ad
try:
A,E,Q,Z,X,scale,sep,ferr,alphar,alphai,beta = \
sg03ad('C','B','N','T','L',n,A,E,eye(n,n),eye(n,n),-Q)
except ValueError(ve):
if ve.info < 0 or ve.info > 4:
e = ValueError(ve.message)
e.info = ve.info
elif ve.info == 1:
e = ValueError("The matrix contained in the upper \
Hessenberg part of the array A is not in \
upper quasitriangular form")
e.info = ve.info
elif ve.info == 2:
e = ValueError("The pencil A - lambda * E cannot be \
reduced to generalized Schur form: LAPACK \
routine DGEGS has failed to converge")
e.info = ve.info
elif ve.info == 4:
e = ValueError("The pencil A - lambda * E has a \
degenerate pair of eigenvalues. That is, \
lambda_i = lambda_j for some i and j, where \
lambda_i and lambda_j are eigenvalues of \
A - lambda * E. Hence, the equation is \
singular; perturbed values were \
used to solve the equation (but the matrices \
A and E are unchanged)")
e.info = ve.info
raise e
# Invalid set of input parameters
else:
raise ControlArgument("Invalid set of input parameters")
return X
def dlyap(A, Q, C=None, E=None):
""" dlyap(A,Q) solves the discrete-time Lyapunov equation
:math:`A X A^T - X + Q = 0`
where A and Q are square matrices of the same dimension. Further
Q must be symmetric.
dlyap(A,Q,C) solves the Sylvester equation
:math:`A X Q^T - X + C = 0`
where A and Q are square matrices.
dlyap(A,Q,None,E) solves the generalized discrete-time Lyapunov
equation
:math:`A X A^T - E X E^T + Q = 0`
where Q is a symmetric matrix and A, Q and E are square matrices
of the same dimension. """
# Make sure we have access to the right slycot routines
if sb03md is None:
raise ControlSlycot("can't find slycot module 'sb03md'")
if sb04qd is None:
raise ControlSlycot("can't find slycot module 'sb04qd'")
if sg03ad is None:
raise ControlSlycot("can't find slycot module 'sg03ad'")
# Reshape 1-d arrays
if len(shape(A)) == 1:
A = A.reshape(1, A.size)
if len(shape(Q)) == 1:
Q = Q.reshape(1, Q.size)
if C is not None and len(shape(C)) == 1:
C = C.reshape(1, C.size)
if E is not None and len(shape(E)) == 1:
E = E.reshape(1, E.size)
# Determine main dimensions
if size(A) == 1:
n = 1
else:
n = size(A, 0)
if size(Q) == 1:
m = 1
else:
m = size(Q, 0)
# Solve standard Lyapunov equation
if C is None and E is None:
# Check input data for consistency
if shape(A) != shape(Q):
raise ControlArgument("A and Q must be matrices of identical \
sizes.")
if size(A) > 1 and shape(A)[0] != shape(A)[1]:
raise ControlArgument("A must be a quadratic matrix.")
if size(Q) > 1 and shape(Q)[0] != shape(Q)[1]:
raise ControlArgument("Q must be a quadratic matrix.")
if not _is_symmetric(Q):
raise ControlArgument("Q must be a symmetric matrix.")
# Solve the Lyapunov equation by calling the Slycot function sb03md
try:
X, scale, sep, ferr, w = \
sb03md(n, -Q, A, eye(n, n), 'D', trana='T')
except ValueError as ve:
if ve.info < 0:
e = ValueError(ve.message)
e.info = ve.info
else:
e = ValueError("The QR algorithm failed to compute all the \
eigenvalues (see LAPACK Library routine DGEES).")
e.info = ve.info
raise e
# Solve the Sylvester equation
elif C is not None and E is None:
# Check input data for consistency
if size(A) > 1 and shape(A)[0] != shape(A)[1]:
raise ControlArgument("A must be a quadratic matrix")
if size(Q) > 1 and shape(Q)[0] != shape(Q)[1]:
raise ControlArgument("Q must be a quadratic matrix")
if (size(C) > 1 and shape(C)[0] != n) or \
(size(C) > 1 and shape(C)[1] != m) or \
(size(C) == 1 and size(A) != 1) or (size(C) == 1 and size(Q) != 1):
raise ControlArgument("C matrix has incompatible dimensions")
# Solve the Sylvester equation by calling Slycot function sb04qd
try:
X = sb04qd(n, m, -A, asarray(Q).T, C)
except ValueError as ve:
if ve.info < 0:
e = ValueError(ve.message)
e.info = ve.info
elif ve.info > m:
e = ValueError("A singular matrix was encountered whilst \
solving for the %i-th column of matrix X." % ve.info - m)
e.info = ve.info
else:
e = ValueError("The QR algorithm failed to compute all the \
eigenvalues (see LAPACK Library routine DGEES)")
e.info = ve.info
raise e
# Solve the generalized Lyapunov equation
elif C is None and E is not None:
# Check input data for consistency
if (size(Q) > 1 and shape(Q)[0] != shape(Q)[1]) or \
(size(Q) > 1 and shape(Q)[0] != n) or \
(size(Q) == 1 and n > 1):
raise ControlArgument("Q must be a square matrix with the same \
dimension as A.")
if (size(E) > 1 and shape(E)[0] != shape(E)[1]) or \
(size(E) > 1 and shape(E)[0] != n) or \
(size(E) == 1 and n > 1):
raise ControlArgument("E must be a square matrix with the same \
dimension as A.")
if not _is_symmetric(Q):
raise ControlArgument("Q must be a symmetric matrix.")
# Solve the generalized Lyapunov equation by calling Slycot
# function sg03ad
try:
A, E, Q, Z, X, scale, sep, ferr, alphar, alphai, beta = \
sg03ad('D', 'B', 'N', 'T', 'L', n,
A, E, eye(n, n), eye(n, n), -Q)
except ValueError as ve:
if ve.info < 0 or ve.info > 4:
e = ValueError(ve.message)
e.info = ve.info
elif ve.info == 1:
e = ValueError("The matrix contained in the upper \
Hessenberg part of the array A is not in \
upper quasitriangular form")
e.info = ve.info
elif ve.info == 2:
e = ValueError("The pencil A - lambda * E cannot be \
reduced to generalized Schur form: LAPACK \
routine DGEGS has failed to converge")
e.info = ve.info
elif ve.info == 3:
e = ValueError("The pencil A - lambda * E has a \
pair of reciprocal eigenvalues. That is, \
lambda_i = 1/lambda_j for some i and j, \
where lambda_i and lambda_j are eigenvalues \
of A - lambda * E. Hence, the equation is \
singular; perturbed values were \
used to solve the equation (but the \
matrices A and E are unchanged)")
e.info = ve.info
raise e
# Invalid set of input parameters
else:
raise ControlArgument("Invalid set of input parameters")
return _ssmatrix(X)
def solve_lyap(A, E, B, trans=False, options=None):
"""Find a factor of the solution of a Lyapunov equation.
Returns factor :math:`Z` such that :math:`Z Z^T` is
approximately the solution :math:`X` of a Lyapunov equation (if
E is `None`).
.. math::
A X + X A^T + B B^T = 0
or generalized Lyapunov equation
.. math::
A X E^T + E X A^T + B B^T = 0.
If trans is `True`, then it solves (if E is `None`)
.. math::
A^T X + X A + B^T B = 0
or
.. math::
A^T X E + E^T X A + B^T B = 0.
This uses the `slycot` package, in particular its interfaces to
SLICOT functions `SB03MD` (for the standard Lyapunov equations)
and `SG03AD` (for the generalized Lyapunov equations).
These methods are only applicable to medium-sized dense
problems and need access to the matrix data of all operators.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The |Operator| B.
trans
If the dual equation needs to be solved.
options
The |solver_options| to use (see
:func:`lyap_solver_options`).
Returns
-------
Z
Low-rank factor of the Lyapunov equation solution, |VectorArray| from `A.source`.
"""
_solve_lyap_check_args(A, E, B, trans)
options = _parse_options(options, lyap_solver_options(), 'slycot', None, False)
assert options['type'] == 'slycot'
import slycot
A_mat = to_matrix(A, format='dense')
if E is not None:
E_mat = to_matrix(E, format='dense')
B_mat = to_matrix(B, format='dense')
n = A_mat.shape[0]
if not trans:
C = -B_mat.dot(B_mat.T)
trana = 'T'
else:
C = -B_mat.T.dot(B_mat)
trana = 'N'
dico = 'C'
if E is None:
U = np.zeros((n, n))
X, scale, _, _, _ = slycot.sb03md(n, C, A_mat, U, dico, trana=trana)
else:
job = 'B'
fact = 'N'
Q = np.zeros((n, n))
Z = np.zeros((n, n))
uplo = 'L'
X = C
_, _, _, _, X, scale, _, _, _, _, _ = slycot.sg03ad(dico, job, fact, trana, uplo, n, A_mat, E_mat,
Q, Z, X)
from pymor.bindings.scipy import chol
Z = chol(X, copy=False)
Z = A.source.from_numpy(np.array(Z).T)
return Z
