def solve_ricc_lrcf(
A,
E,
B,
C,
R=None,
S=None,
trans=False,
options=None,
default_sparse_solver_backend=_DEFAULT_RICC_LRCF_SPARSE_SOLVER_BACKEND,
default_dense_solver_backend=_DEFAULT_RICC_LRCF_DENSE_SOLVER_BACKEND):
"""Compute an approximate low-rank solution of a Riccati equation.
Returns a low-rank Cholesky factor :math:`Z` such that :math:`Z Z^T`
approximates the solution :math:`X` of a (generalized)
continuous-time algebraic Riccati equation:
- if trans is `False`
.. math::
A X E^T + E X A^T
- (E X C^T + S) R^{-1} (E X C^T + S)^T
+ B B^T = 0.
- if trans is `True`
.. math::
A^T X E + E^T X A
- (E^T X B + S) R^{-1} (E^T X B + S)^T
+ C^T C = 0.
If E is None, it is taken to be identity, and similarly for R.
If S is None, it is taken to be zero.
We assume:
- A and E are real |Operators|,
- B, C and S are real |VectorArrays| from `A.source`,
- R is a real |NumPy array|,
- (E, A, B, C) is stabilizable and detectable,
- R is symmetric positive definite, and
- :math:`B B^T - S R^{-1} S^T` (:math:`C^T C - S R^{-1} S^T`) is
positive semi-definite trans is `False` (`True`).
For large-scale problems, we additionally assume that `len(B)` and
`len(C)` are small.
If the solver is not specified using the options argument, a solver
backend is chosen based on availability in the following order:
- for sparse problems (minimum size specified by
:func:`~pymor.algorithms.lyapunov.mat_eqn_sparse_min_size`)
1. `pymess` (see :func:`pymor.bindings.pymess.solve_ricc_lrcf`),
- for dense problems (smaller than
:func:`~pymor.algorithms.lyapunov.mat_eqn_sparse_min_size`)
1. `pymess` (see :func:`pymor.bindings.pymess.solve_ricc_lrcf`),
2. `slycot` (see :func:`pymor.bindings.slycot.solve_ricc_lrcf`),
3. `scipy` (see :func:`pymor.bindings.scipy.solve_ricc_lrcf`).
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:`pymor.bindings.scipy.ricc_lrcf_solver_options`,
- :func:`pymor.bindings.slycot.ricc_lrcf_solver_options`,
- :func:`pymor.bindings.pymess.ricc_lrcf_solver_options`.
default_sparse_solver_backend
Default sparse solver backend to use (pymess).
default_dense_solver_backend
Default dense solver backend to use (pymess, slycot, scipy).
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 options:
solver = options if isinstance(options, str) else options['type']
backend = solver.split('_')[0]
else:
if A.source.dim >= mat_eqn_sparse_min_size():
backend = default_sparse_solver_backend
else:
backend = default_dense_solver_backend
if backend == 'scipy':
from pymor.bindings.scipy import solve_ricc_lrcf as solve_ricc_impl
elif backend == 'slycot':
from pymor.bindings.slycot import solve_ricc_lrcf as solve_ricc_impl
elif backend == 'pymess':
from pymor.bindings.pymess import solve_ricc_lrcf as solve_ricc_impl
else:
raise ValueError(f'Unknown solver backend ({backend}).')
return solve_ricc_impl(A, E, B, C, R, S, trans=trans, options=options)
def solve_pos_ricc_lrcf(
A,
E,
B,
C,
R=None,
S=None,
trans=False,
options=None,
default_dense_solver_backend=_DEFAULT_RICC_LRCF_DENSE_SOLVER_BACKEND):
"""Compute an approximate low-rank solution of a positive Riccati equation.
Returns a low-rank Cholesky factor :math:`Z` such that :math:`Z Z^T`
approximates the solution :math:`X` of a (generalized) positive
continuous-time algebraic Riccati equation:
- if trans is `False`
.. math::
A X E^T + E X A^T
+ (E X C^T + S) R^{-1} (E X C^T + S)^T
+ B B^T = 0.
- if trans is `True`
.. math::
A^T X E + E^T X A
+ (E^T X B + S) R^{-1} (E^T X B + S)^T
+ C^T C = 0.
If E is None, it is taken to be identity, and similarly for R.
If S is None, it is taken to be zero.
If the solver is not specified using the options argument, a solver
backend is chosen based on availability in the following order:
1. `pymess` (see :func:`pymor.bindings.pymess.solve_pos_ricc_lrcf`),
2. `slycot` (see :func:`pymor.bindings.slycot.solve_pos_ricc_lrcf`),
3. `scipy` (see :func:`pymor.bindings.scipy.solve_pos_ricc_lrcf`).
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:`pymor.bindings.scipy.pos_ricc_lrcf_solver_options`,
- :func:`pymor.bindings.slycot.pos_ricc_lrcf_solver_options`,
- :func:`pymor.bindings.pymess.pos_ricc_lrcf_solver_options`.
default_dense_solver_backend
Default dense solver backend to use (pymess, slycot, scipy).
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)
if options:
solver = options if isinstance(options, str) else options['type']
backend = solver.split('_')[0]
else:
backend = default_dense_solver_backend
if backend == 'scipy':
from pymor.bindings.scipy import solve_pos_ricc_lrcf as solve_ricc_impl
elif backend == 'slycot':
from pymor.bindings.slycot import solve_pos_ricc_lrcf as solve_ricc_impl
elif backend == 'pymess':
from pymor.bindings.pymess import solve_pos_ricc_lrcf as solve_ricc_impl
else:
raise ValueError(f'Unknown solver backend ({backend}).')
return solve_ricc_impl(A, E, B, C, R, S, trans=trans, options=options)