lyapunov#

Bases: LyapunovSolver

A solver for the Lyapunov equation using iterative doubling.

Parameters:

• max_iterations (int, default: 10 ) –

  The maximum number of iterations to perform.

• convergence_rel_tol (float, default: 1e-05 ) –

  The required relative accuracy for convergence.

• convergence_abs_tol (float, default: 1e-08 ) –

  The required absolute accuracy for convergence. Either convergence_rel_tol or convergence_abs_tol must be satisfied.

Methods:

• __call__ –

  Computes the solution of the discrete-time Lyapunov equation.

__call__(a: NDArray, q: NDArray, contact: str) -> NDArray

Bases: ABC

Solver interface for the discrete-time Lyapunov equation.

The discrete-time Lyapunov equation is defined as:

Methods:

• __call__ –

  Computes the solution of the discrete-time Lyapunov equation.

__call__(a: NDArray, q: NDArray, contact: str) -> NDArray

The equation is give by:

$$\mathbf{x} = \mathbf{q} + \mathbf{a} \mathbf{x}  \mathbf{a}^H$$

Bases: BaseBoundarySystem

A lyapunov system solver with memoization and system reduction.

The lyapyunov equation to be solved is given by:

$$\mathbf{x} = \mathbf{q} + \mathbf{a} \mathbf{x}  \mathbf{a}^H$$

Parameters:

• boundary_solver (LyapunovSolver) –

  The lyapunov solver to be memoized.

• cache_compressor (object, default: None ) –

  An object with 'compress' and 'decompress' methods to handle cache compression. If None, no compression is applied.

• num_ref_iterations (int, default: 2 ) –

  The number of fixed-point iterations to refine the solution. Default is 2.

• relative_tol (float, default: 0.2 ) –

  The relative tolerance for convergence. Default is 0.2.

• absolute_tol (float, default: 1e-06 ) –

  The absolute tolerance for convergence. Default is 1e-6.

• warning_threshold (float, default: 0.1 ) –

  The threshold for issuing a warning about high residuals. Default is 0.1.

• memoization_mode (str, default: 'auto' ) –

  The memoization mode. Can be 'off', 'auto', 'force-after-first', or 'force'. Default is 'auto'.

• agreement_threshold (float, default: 0.999 ) –

  The threshold for agreement across MPI ranks to consider a memoized solution valid. Default is 0.999.

Bases: SystemReducer

A lyapunov system reducer.

The lyapunov system can exhibit sparsity, which can be exploited to reduce the size of the system. Either zero rows or zero cols in the system matrix can be reduced, depending on which is more prevalent. Utilizing both would be possible but is not implemented yet, as it would require more complex bookkeeping of the reduced system. In addition, such systems do currently not occure in quatrex.

Parameters:

• reduce_sparsity (bool, default: True ) –

  Whether to reduce the sparsity of the system matrix. If sparsity of any obc is changed during runtime, then the cache needs to be invalidated. Default is True.

• assume_constant_sparsity (bool, default: True ) –

  Whether to assume that the sparsity pattern of the system matrix remains constant during runtime. If True, the sparsity pattern is only computed once. Default is True.

Methods:

• contract_system –

  Contract the boundary system to a reduced system.

• expand_solution –

  Expand the solution from the reduced system to the full system.

• expand_residuals –

  Expand the residuals from the reduced system to the full system.

contract_system(boundary_system: tuple[NDArray, ...]) -> tuple[NDArray, ...]

expand_solution(boundary_system: tuple[NDArray, ...], reduced_system: tuple[NDArray, ...], reduced_solution: NDArray) -> NDArray

expand_residuals(boundary_system: tuple[NDArray, ...], reduced_system: tuple[NDArray, ...], rel_residuals: NDArray, abs_residuals: NDArray) -> tuple[NDArray, NDArray]

Bases: LyapunovSolver

A solver for the Lyapunov equation by using the matrix spectrum.

Parameters:

• num_ref_iterations (int, default: 2 ) –

  The number of refinement iterations to perform.

• eig_compute_location (str, default: 'numpy' ) –

  The location where to compute the eigenvalues and eigenvectors. Can be either "numpy" or "cupy" or "nvmath".

• use_pinned_memory (bool, default: True ) –

  Whether to use pinnend memory if cupy is used. Default is True.

Methods:

• __call__ –

  Computes the solution of the discrete-time Lyapunov equation.

__call__(a: NDArray, q: NDArray, contact: str) -> NDArray