def solve_pos_ricc_lrcf(A,
E,
B,
C,
R=None,
S=None,
trans=False,
options=None):
"""Compute an approximate low-rank solution of a positive Riccati equation.
See :func:`pymor.algorithms.riccati.solve_pos_ricc_lrcf` for a
general description.
This function uses `slycot.sb02md` (if E and S are `None`),
`slycot.sb02od` (if E is `None` and S is not `None`) and
`slycot.sg03ad` (if E is not `None`), which are dense solvers.
Therefore, we assume all |Operators| and |VectorArrays| can be
converted to |NumPy arrays| using
:func:`~pymor.algorithms.to_matrix.to_matrix` and
:func:`~pymor.vectorarrays.interfaces.VectorArrayInterface.to_numpy`.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the positive Riccati
equation is transposed.
options
The solver options to use (see
:func:`pos_ricc_lrcf_solver_options`).
Returns
-------
Z
Low-rank Cholesky factor of the positive Riccati equation
solution, |VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, S, trans)
options = _parse_options(options, pos_ricc_lrcf_solver_options(),
'slycot', None, False)
if options['type'] != 'slycot':
raise ValueError(
f"Unexpected positive Riccati equation solver ({options['type']})."
)
if R is None:
R = np.eye(len(C) if not trans else len(B))
return solve_ricc_lrcf(A, E, B, C, -R, S, trans, options)
def solve_pos_ricc_lrcf(A, E, B, C, R=None, trans=False, options=None):
"""Compute an approximate low-rank solution of a positive Riccati equation.
See :func:`pymor.algorithms.riccati.solve_pos_ricc_lrcf` for a
general description.
This function uses `pymess.dense_nm_gmpcare`.
Parameters
----------
A
The non-parametric |Operator| A.
E
The non-parametric |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
trans
Whether the first |Operator| in the Riccati equation is
transposed.
options
The solver options to use (see
:func:`pos_ricc_lrcf_solver_options`).
Returns
-------
Z
Low-rank Cholesky factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, trans)
options = _parse_options(options, pos_ricc_lrcf_solver_options(),
'pymess_dense_nm_gmpcare', None, False)
if options['type'] == 'pymess_dense_nm_gmpcare':
X = _call_pymess_dense_nm_gmpare(A,
E,
B,
C,
R,
trans=trans,
options=options['opts'],
plus=True,
method_name='solve_pos_ricc_lrcf')
Z = _chol(X)
else:
raise ValueError(
f'Unexpected positive Riccati equation solver ({options["type"]}).'
)
return A.source.from_numpy(Z.T)
def solve_pos_ricc_lrcf(A, E, B, C, R=None, S=None, trans=False, options=None):
"""Compute an approximate low-rank solution of a positive Riccati equation.
See :func:`pymor.algorithms.riccati.solve_pos_ricc_lrcf` for a
general description.
This function uses `slycot.sb02md` (if E and S are `None`),
`slycot.sb02od` (if E is `None` and S is not `None`) and
`slycot.sg03ad` (if E is not `None`), which are dense solvers.
Therefore, we assume all |Operators| and |VectorArrays| can be
converted to |NumPy arrays| using
:func:`~pymor.algorithms.to_matrix.to_matrix` and
:func:`~pymor.vectorarrays.interfaces.VectorArrayInterface.to_numpy`.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the positive Riccati
equation is transposed.
options
The solver options to use (see
:func:`pos_ricc_lrcf_solver_options`).
Returns
-------
Z
Low-rank Cholesky factor of the positive Riccati equation
solution, |VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, S, trans)
options = _parse_options(options, pos_ricc_lrcf_solver_options(), 'slycot', None, False)
if options['type'] != 'slycot':
raise ValueError(f"Unexpected positive Riccati equation solver ({options['type']}).")
if R is None:
R = np.eye(len(C) if not trans else len(B))
return solve_ricc_lrcf(A, E, B, C, -R, S, trans, options)
def solve_pos_ricc_lrcf(A, E, B, C, R=None, S=None, trans=False, options=None):
"""Compute an approximate low-rank solution of a positive Riccati equation.
See :func:`pymor.algorithms.riccati.solve_pos_ricc_lrcf` for a
general description.
This function uses `pymess.dense_nm_gmpcare`.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the Riccati equation is
transposed.
options
The solver options to use (see
:func:`pos_ricc_lrcf_solver_options`).
Returns
-------
Z
Low-rank Cholesky factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, S, trans)
options = _parse_options(options, pos_ricc_lrcf_solver_options(), 'pymess_dense_nm_gmpcare', None, False)
if options['type'] == 'pymess_dense_nm_gmpcare':
X = _call_pymess_dense_nm_gmpare(A, E, B, C, R, S, trans=trans, options=options['opts'], plus=True)
Z = _chol(X)
else:
raise ValueError(f'Unexpected positive Riccati equation solver ({options["type"]}).')
return A.source.from_numpy(Z.T)
def solve_ricc_lrcf(A,
E,
B,
C,
R=None,
S=None,
trans=False,
options=None,
default_solver=None):
"""Compute an approximate low-rank solution of a Riccati equation.
See :func:`pymor.algorithms.riccati.solve_ricc_lrcf` for a
general description.
This function uses `pymess.dense_nm_gmpcare` and `pymess.lrnm`.
For both methods,
:meth:`~pymor.vectorarrays.interface.VectorArray.to_numpy`
and
:meth:`~pymor.vectorarrays.interface.VectorSpace.from_numpy`
need to be implemented for `A.source`.
Additionally, since `dense_nm_gmpcare` is a dense solver, it
expects :func:`~pymor.algorithms.to_matrix.to_matrix` to work
for A and E.
If the solver is not specified using the options or
default_solver arguments, `dense_nm_gmpcare` is used for small
problems (smaller than defined with
:func:`~pymor.algorithms.lyapunov.mat_eqn_sparse_min_size`) and
`lrnm` for large problems.
Parameters
----------
A
The non-parametric |Operator| A.
E
The non-parametric |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the Riccati equation is
transposed.
options
The solver options to use (see
:func:`ricc_lrcf_solver_options`).
default_solver
Default solver to use (pymess_lrnm,
pymess_dense_nm_gmpcare).
If `None`, chose solver depending on dimension `A`.
Returns
-------
Z
Low-rank Cholesky factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, S, trans)
if default_solver is None:
default_solver = 'pymess_lrnm' if A.source.dim >= mat_eqn_sparse_min_size(
) else 'pymess_dense_nm_gmpcare'
options = _parse_options(options, ricc_lrcf_solver_options(),
default_solver, None, False)
if options['type'] == 'pymess_dense_nm_gmpcare':
X = _call_pymess_dense_nm_gmpare(A,
E,
B,
C,
R,
S,
trans=trans,
options=options['opts'],
plus=False,
method_name='solve_ricc_lrcf')
Z = _chol(X)
elif options['type'] == 'pymess_lrnm':
if S is not None:
raise NotImplementedError
if R is not None:
import scipy.linalg as spla
Rc = spla.cholesky(R) # R = Rc^T * Rc
Rci = spla.solve_triangular(Rc, np.eye(
Rc.shape[0])) # R^{-1} = Rci * Rci^T
if not trans:
C = C.lincomb(Rci.T) # C <- Rci^T * C = (C^T * Rci)^T
else:
B = B.lincomb(Rci.T) # B <- B * Rci
opts = options['opts']
opts.type = pymess.MESS_OP_NONE if not trans else pymess.MESS_OP_TRANSPOSE
eqn = RiccatiEquation(opts, A, E, B, C)
Z, status = pymess.lrnm(eqn, opts)
relres = status.res2_norm / status.res2_0
if relres > opts.adi.res2_tol:
logger = getLogger('pymor.bindings.pymess.solve_ricc_lrcf')
logger.warning(
f'Desired relative residual tolerance was not achieved '
f'({relres:e} > {opts.adi.res2_tol:e}).')
else:
raise ValueError(
f'Unexpected Riccati equation solver ({options["type"]}).')
return A.source.from_numpy(Z.T)
def solve_ricc(A, E=None, B=None, Q=None, C=None, R=None, G=None,
trans=False, options=None, default_solver='pymess'):
"""Find a factor of the solution of a Riccati equation
Returns factor :math:`Z` such that :math:`Z Z^T` is
approximately the solution :math:`X` of a Riccati equation
.. math::
A^T X E + E^T X A - E^T X B R^{-1} B^T X E + Q = 0.
If E in `None`, it is taken to be the identity matrix.
Q can instead be given as C^T * C. In this case, Q needs to be
`None`, and C not `None`.
B * R^{-1} B^T can instead be given by G. In this case, B and R
need to be `None`, and G not `None`.
If R and G are `None`, then R is taken to be the identity
matrix.
If trans is `True`, then the dual Riccati equation is solved
.. math::
A X E^T + E X A^T - E X C^T R^{-1} C X E^T + Q = 0,
where Q can be replaced by B * B^T and C^T * R^{-1} * C by G.
This uses the `pymess` package, in particular its `care` and
`lrnm` methods.
Operators Q, R, and G are not supported,
Both methods can be used for large-scale problems.
The restrictions are:
- `care` needs access to all matrix data, i.e., it expects
:func:`~pymor.algorithms.to_matrix.to_matrix` to work for
A, E, B, and C,
- `lrnm` needs access to the data of the operators B and C,
i.e., it expects
:func:`~pymor.algorithms.to_matrix.to_matrix` to work for
B and C.
Parameters
----------
A
The |Operator| A.
B
The |Operator| B or `None`.
E
The |Operator| E or `None`.
Q
The |Operator| Q or `None`.
C
The |Operator| C or `None`.
R
The |Operator| R or `None`.
G
The |Operator| G or `None`.
trans
If the dual equation needs to be solved.
options
The |solver_options| to use (see
:func:`ricc_solver_options`).
default_solver
The solver to use when no `options` are specified (pymess,
pymess_care, pymess_lrnm).
Returns
-------
Z
Low-rank factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, Q, C, R, G, trans)
options = _parse_options(options, ricc_solver_options(), default_solver, None, False)
if options['type'] == 'pymess':
if A.source.dim >= PYMESS_MIN_SPARSE_SIZE:
options = dict(options, type='pymess_lrnm') # do not modify original dict!
else:
options = dict(options, type='pymess_care') # do not modify original dict!
if options['type'] == 'pymess_care':
if Q is not None or R is not None or G is not None:
raise NotImplementedError
A_mat = to_matrix(A, format='dense') if A.source.dim < PYMESS_MIN_SPARSE_SIZE else to_matrix(A)
if E is not None:
E_mat = to_matrix(E, format='dense') if A.source.dim < PYMESS_MIN_SPARSE_SIZE else to_matrix(E)
else:
E_mat = None
B_mat = to_matrix(B, format='dense') if B else None
C_mat = to_matrix(C, format='dense') if C else None
if not trans:
Z = pymess.care(A_mat, E_mat, B_mat, C_mat)
else:
if E is None:
Z = pymess.care(A_mat.T, None, C_mat.T, B_mat.T)
else:
Z = pymess.care(A_mat.T, E_mat.T, C_mat.T, B_mat.T)
elif options['type'] == 'pymess_lrnm':
if Q is not None or R is not None or G is not None:
raise NotImplementedError
opts = options['opts']
if not trans:
opts.type = pymess.MESS_OP_TRANSPOSE
else:
opts.type = pymess.MESS_OP_NONE
eqn = RiccatiEquation(opts, A, E, B, C)
Z, status = pymess.lrnm(eqn, opts)
Z = A.source.from_numpy(np.array(Z).T)
return Z
def solve_ricc_lrcf(A, E, B, C, R=None, S=None, trans=False, options=None):
"""Compute an approximate low-rank solution of a Riccati equation.
See :func:`pymor.algorithms.riccati.solve_ricc_lrcf` for a
general description.
This function uses `slycot.sb02md` (if E and S are `None`),
`slycot.sb02od` (if E is `None` and S is not `None`) and
`slycot.sg03ad` (if E is not `None`), which are dense solvers.
Therefore, we assume all |Operators| and |VectorArrays| can be
converted to |NumPy arrays| using
:func:`~pymor.algorithms.to_matrix.to_matrix` and
:func:`~pymor.vectorarrays.interfaces.VectorArrayInterface.to_numpy`.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the Riccati equation is
transposed.
options
The solver options to use (see
:func:`ricc_lrcf_solver_options`).
Returns
-------
Z
Low-rank Cholesky factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, S, trans)
options = _parse_options(options, ricc_lrcf_solver_options(), 'slycot',
None, False)
if options['type'] != 'slycot':
raise ValueError(
f"Unexpected Riccati equation solver ({options['type']}).")
A_source = A.source
A = to_matrix(A, format='dense')
E = to_matrix(E, format='dense') if E else None
B = B.to_numpy().T
C = C.to_numpy()
S = S.to_numpy().T if S else None
n = A.shape[0]
dico = 'C'
if E is None:
if S is None:
if not trans:
A = A.T
G = C.T.dot(C) if R is None else slycot.sb02mt(
n, C.shape[0], C.T, R)[-1]
else:
G = B.dot(B.T) if R is None else slycot.sb02mt(
n, B.shape[1], B, R)[-1]
Q = B.dot(B.T) if not trans else C.T.dot(C)
X, rcond = slycot.sb02md(n, A, G, Q, dico)[:2]
_ricc_rcond_check('slycot.sb02md', rcond)
else:
m = C.shape[0] if not trans else B.shape[1]
p = B.shape[1] if not trans else C.shape[0]
if R is None:
R = np.eye(m)
if not trans:
A = A.T
B, C = C.T, B.T
X, rcond = slycot.sb02od(n,
m,
A,
B,
C,
R,
dico,
p=p,
L=S,
fact='C')[:2]
_ricc_rcond_check('slycot.sb02od', rcond)
else:
jobb = 'B'
fact = 'C'
uplo = 'U'
jobl = 'Z' if S is None else 'N'
scal = 'N'
sort = 'S'
acc = 'R'
m = C.shape[0] if not trans else B.shape[1]
p = B.shape[1] if not trans else C.shape[0]
if R is None:
R = np.eye(m)
if S is None:
S = np.empty((n, m))
if not trans:
A = A.T
E = E.T
B, C = C.T, B.T
out = slycot.sg02ad(dico, jobb, fact, uplo, jobl, scal, sort, acc,
n, m, p, A, E, B, C, R, S)
X = out[1]
rcond = out[0]
_ricc_rcond_check('slycot.sg02ad', rcond)
return A_source.from_numpy(_chol(X).T)
def solve_ricc_lrcf(A, E, B, C, R=None, S=None, trans=False, options=None):
"""Compute an approximate low-rank solution of a Riccati equation.
See :func:`pymor.algorithms.riccati.solve_ricc_lrcf` for a
general description.
This function uses `slycot.sb02md` (if E and S are `None`),
`slycot.sb02od` (if E is `None` and S is not `None`) and
`slycot.sg03ad` (if E is not `None`), which are dense solvers.
Therefore, we assume all |Operators| and |VectorArrays| can be
converted to |NumPy arrays| using
:func:`~pymor.algorithms.to_matrix.to_matrix` and
:func:`~pymor.vectorarrays.interfaces.VectorArrayInterface.to_numpy`.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the Riccati equation is
transposed.
options
The solver options to use (see
:func:`ricc_lrcf_solver_options`).
Returns
-------
Z
Low-rank Cholesky factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, S, trans)
options = _parse_options(options, ricc_lrcf_solver_options(), 'slycot', None, False)
if options['type'] != 'slycot':
raise ValueError(f"Unexpected Riccati equation solver ({options['type']}).")
A_source = A.source
A = to_matrix(A, format='dense')
E = to_matrix(E, format='dense') if E else None
B = B.to_numpy().T
C = C.to_numpy()
S = S.to_numpy().T if S else None
n = A.shape[0]
dico = 'C'
if E is None:
if S is None:
if not trans:
A = A.T
G = C.T.dot(C) if R is None else slycot.sb02mt(n, C.shape[0], C.T, R)[-1]
else:
G = B.dot(B.T) if R is None else slycot.sb02mt(n, B.shape[1], B, R)[-1]
Q = B.dot(B.T) if not trans else C.T.dot(C)
X, rcond = slycot.sb02md(n, A, G, Q, dico)[:2]
_ricc_rcond_check('slycot.sb02md', rcond)
else:
m = C.shape[0] if not trans else B.shape[1]
p = B.shape[1] if not trans else C.shape[0]
if R is None:
R = np.eye(m)
if not trans:
A = A.T
B, C = C.T, B.T
X, rcond = slycot.sb02od(n, m, A, B, C, R, dico, p=p, L=S, fact='C')[:2]
_ricc_rcond_check('slycot.sb02od', rcond)
else:
jobb = 'B'
fact = 'C'
uplo = 'U'
jobl = 'Z' if S is None else 'N'
scal = 'N'
sort = 'S'
acc = 'R'
m = C.shape[0] if not trans else B.shape[1]
p = B.shape[1] if not trans else C.shape[0]
if R is None:
R = np.eye(m)
if S is None:
S = np.empty((n, m))
if not trans:
A = A.T
E = E.T
B, C = C.T, B.T
out = slycot.sg02ad(dico, jobb, fact, uplo, jobl, scal, sort, acc,
n, m, p,
A, E, B, C, R, S)
X = out[1]
rcond = out[0]
_ricc_rcond_check('slycot.sg02ad', rcond)
return A_source.from_numpy(_chol(X).T)
def solve_ricc(A, E=None, B=None, Q=None, C=None, R=None, G=None, trans=False, options=None):
"""Find a factor of the solution of a Riccati equation
Returns factor :math:`Z` such that :math:`Z Z^T` is
approximately the solution :math:`X` of a Riccati equation
.. math::
A^T X E + E^T X A - E^T X B R^{-1} B^T X E + Q = 0.
If E in `None`, it is taken to be the identity matrix.
Q can instead be given as C^T * C. In this case, Q needs to be
`None`, and C not `None`.
B * R^{-1} B^T can instead be given by G. In this case, B and R
need to be `None`, and G not `None`.
If R and G are `None`, then R is taken to be the identity
matrix.
If trans is `True`, then the dual Riccati equation is solved
.. math::
A X E^T + E X A^T - E X C^T R^{-1} C X E^T + Q = 0,
where Q can be replaced by B * B^T and C^T * R^{-1} * C by G.
This uses the `slycot` package, in particular its interfaces to
SLICOT functions `SB02MD` (for the standard Riccati equations)
and `SG02AD` (for the generalized Riccati 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.
B
The |Operator| B or `None`.
E
The |Operator| E or `None`.
Q
The |Operator| Q or `None`.
C
The |Operator| C or `None`.
R
The |Operator| R or `None`.
G
The |Operator| G or `None`.
trans
If the dual equation needs to be solved.
options
The |solver_options| to use (see
:func:`ricc_solver_options`).
Returns
-------
Z
Low-rank factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, Q, C, R, G, trans)
options = _parse_options(options, ricc_solver_options(), 'slycot', None, False)
assert options['type'] == 'slycot'
import slycot
A_mat = to_matrix(A, format='dense')
B_mat = to_matrix(B, format='dense') if B else None
C_mat = to_matrix(C, format='dense') if C else None
R_mat = to_matrix(R, format='dense') if R else None
G_mat = to_matrix(G, format='dense') if G else None
Q_mat = to_matrix(Q, format='dense') if Q else None
n = A_mat.shape[0]
dico = 'C'
if E is None:
if not trans:
if G is None:
if R is None:
G_mat = B_mat.dot(B_mat.T)
else:
G_mat = slycot.sb02mt(n, B_mat.shape[1], B_mat, R_mat)[-1]
if C is not None:
Q_mat = C_mat.T.dot(C_mat)
X = slycot.sb02md(n, A_mat, G_mat, Q_mat, dico)[0]
else:
if G is None:
if R is None:
G_mat = C_mat.T.dot(C_mat)
else:
G_mat = slycot.sb02mt(n, C_mat.shape[0], C_mat.T, R_mat)[-1]
if B is not None:
Q_mat = B_mat.dot(B_mat.T)
X = slycot.sb02md(n, A_mat.T, G_mat, Q_mat, dico)[0]
else:
E_mat = to_matrix(E, format='dense') if E else None
jobb = 'B' if G is None else 'B'
fact = 'C' if Q is None else 'N'
uplo = 'U'
jobl = 'Z'
scal = 'N'
sort = 'S'
acc = 'R'
if not trans:
m = 0 if B is None else B_mat.shape[1]
p = 0 if C is None else C_mat.shape[0]
if G is not None:
B_mat = G_mat
R_mat = np.empty((1, 1))
elif R is None:
R_mat = np.eye(m)
if Q is None:
Q_mat = C_mat
L_mat = np.empty((n, m))
ret = slycot.sg02ad(dico, jobb, fact, uplo, jobl, scal, sort, acc, n, m, p,
A_mat, E_mat, B_mat, Q_mat, R_mat, L_mat)
else:
m = 0 if C is None else C_mat.shape[0]
p = 0 if B is None else B_mat.shape[1]
if G is not None:
C_mat = G_mat
R_mat = np.empty((1, 1))
elif R is None:
C_mat = C_mat.T
R_mat = np.eye(m)
else:
C_mat = C_mat.T
if Q is None:
Q_mat = B_mat.T
L_mat = np.empty((n, m))
ret = slycot.sg02ad(dico, jobb, fact, uplo, jobl, scal, sort, acc, n, m, p,
A_mat.T, E_mat.T, C_mat, Q_mat, R_mat, L_mat)
X = ret[1]
iwarn = ret[-1]
if iwarn == 1:
print('slycot.sg02ad warning: solution may be inaccurate.')
from pymor.bindings.scipy import chol
Z = chol(X, copy=False)
Z = A.source.from_numpy(np.array(Z).T)
return Z
def solve_ricc_lrcf(A, E, B, C, R=None, S=None, trans=False, options=None, default_solver=None):
"""Compute an approximate low-rank solution of a Riccati equation.
See :func:`pymor.algorithms.riccati.solve_ricc_lrcf` for a
general description.
This function uses `pymess.dense_nm_gmpcare` and `pymess.lrnm`.
For both methods,
:meth:`~pymor.vectorarrays.interfaces.VectorArrayInterface.to_numpy`
and
:meth:`~pymor.vectorarrays.interfaces.VectorSpaceInterface.from_numpy`
need to be implemented for `A.source`.
Additionally, since `dense_nm_gmpcare` is a dense solver, it
expects :func:`~pymor.algorithms.to_matrix.to_matrix` to work
for A and E.
If the solver is not specified using the options or
default_solver arguments, `dense_nm_gmpcare` is used for small
problems (smaller than defined with
:func:`~pymor.algorithms.lyapunov.mat_eqn_sparse_min_size`) and
`lrnm` for large problems.
Parameters
----------
A
The |Operator| A.
E
The |Operator| E or `None`.
B
The operator B as a |VectorArray| from `A.source`.
C
The operator C as a |VectorArray| from `A.source`.
R
The operator R as a 2D |NumPy array| or `None`.
S
The operator S as a |VectorArray| from `A.source` or `None`.
trans
Whether the first |Operator| in the Riccati equation is
transposed.
options
The solver options to use (see
:func:`ricc_lrcf_solver_options`).
default_solver
Default solver to use (pymess_lrnm,
pymess_dense_nm_gmpcare).
If `None`, chose solver depending on dimension `A`.
Returns
-------
Z
Low-rank Cholesky factor of the Riccati equation solution,
|VectorArray| from `A.source`.
"""
_solve_ricc_check_args(A, E, B, C, R, S, trans)
if default_solver is None:
default_solver = 'pymess_lrnm' if A.source.dim >= mat_eqn_sparse_min_size() else 'pymess_dense_nm_gmpcare'
options = _parse_options(options, ricc_lrcf_solver_options(), default_solver, None, False)
if options['type'] == 'pymess_dense_nm_gmpcare':
X = _call_pymess_dense_nm_gmpare(A, E, B, C, R, S, trans=trans, options=options['opts'], plus=False)
Z = _chol(X)
elif options['type'] == 'pymess_lrnm':
if S is not None:
raise NotImplementedError
if R is not None:
import scipy.linalg as spla
Rc = spla.cholesky(R) # R = Rc^T * Rc
Rci = spla.solve_triangular(Rc, np.eye(Rc.shape[0])) # R^{-1} = Rci * Rci^T
if not trans:
C = C.lincomb(Rci.T) # C <- Rci^T * C = (C^T * Rci)^T
else:
B = B.lincomb(Rci.T) # B <- B * Rci
opts = options['opts']
opts.type = pymess.MESS_OP_NONE if not trans else pymess.MESS_OP_TRANSPOSE
eqn = RiccatiEquation(opts, A, E, B, C)
Z, status = pymess.lrnm(eqn, opts)
else:
raise ValueError(f'Unexpected Riccati equation solver ({options["type"]}).')
return A.source.from_numpy(Z.T)