# pymor.algorithms.lyapunov¶

## Module Contents¶

### Functions¶

 mat_eqn_sparse_min_size Returns minimal size for which a sparse solver will be used by default. solve_lyap_lrcf Compute an approximate low-rank solution of a Lyapunov equation. _solve_lyap_lrcf_check_args solve_lyap_dense Compute the solution of a Lyapunov equation. _solve_lyap_dense_check_args _chol Cholesky decomposition.

### Attributes¶

pymor.algorithms.lyapunov._DEFAULT_LYAP_LRCF_SPARSE_SOLVER_BACKEND[source]
pymor.algorithms.lyapunov._DEFAULT_LYAP_LRCF_DENSE_SOLVER_BACKEND[source]
pymor.algorithms.lyapunov._DEFAULT_LYAP_DENSE_SOLVER_BACKEND[source]
pymor.algorithms.lyapunov.mat_eqn_sparse_min_size(value=1000)[source]

Returns minimal size for which a sparse solver will be used by default.

pymor.algorithms.lyapunov.solve_lyap_lrcf(A, E, B, trans=False, options=None, default_sparse_solver_backend=_DEFAULT_LYAP_LRCF_SPARSE_SOLVER_BACKEND, default_dense_solver_backend=_DEFAULT_LYAP_LRCF_DENSE_SOLVER_BACKEND)[source]

Compute an approximate low-rank solution of a Lyapunov equation.

Returns a low-rank Cholesky factor $$Z$$ such that $$Z Z^T$$ approximates the solution $$X$$ of a (generalized) continuous-time algebraic Lyapunov equation:

• if trans is False and E is None:

$A X + X A^T + B B^T = 0,$
• if trans is False and E is an Operator:

$A X E^T + E X A^T + B B^T = 0,$
• if trans is True and E is None:

$A^T X + X A + B^T B = 0,$
• if trans is True and E is an Operator:

$A^T X E + E^T X A + B^T B = 0.$

We assume A and E are real Operators, E is invertible, and all the eigenvalues of (A, E) all lie in the open left half-plane. Operator B needs to be given as a VectorArray from A.source, and for large-scale problems, we assume len(B) is small.

If the solver is not specified using the options argument, a solver backend is chosen based on availability in the following order:

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.

trans

Whether the first Operator in the Lyapunov equation is transposed.

options

The solver options to use. See:

• pymor.bindings.pymess.lyap_lrcf_solver_options.

default_sparse_solver_backend

Default sparse solver backend to use (pymess, lradi).

default_dense_solver_backend

Default dense solver backend to use (pymess, slycot, scipy).

Returns

Z

Low-rank Cholesky factor of the Lyapunov equation solution, VectorArray from A.source.

pymor.algorithms.lyapunov._solve_lyap_lrcf_check_args(A, E, B, trans)[source]
pymor.algorithms.lyapunov.solve_lyap_dense(A, E, B, trans=False, options=None, default_solver_backend=_DEFAULT_LYAP_DENSE_SOLVER_BACKEND)[source]

Compute the solution of a Lyapunov equation.

Returns the solution $$X$$ of a (generalized) continuous-time algebraic Lyapunov equation:

• if trans is False and E is None:

$A X + X A^T + B B^T = 0,$
• if trans is False and E is an Operator:

$A X E^T + E X A^T + B B^T = 0,$
• if trans is True and E is None:

$A^T X + X A + B^T B = 0,$
• if trans is True and E is an Operator:

$A^T X E + E^T X A + B^T B = 0.$

We assume A and E are real NumPy arrays, E is invertible, and that no two eigenvalues of (A, E) sum to zero (i.e., there exists a unique solution X).

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 pymor.bindings.pymess.solve_lyap_dense)

2. slycot (see pymor.bindings.slycot.solve_lyap_dense)

3. scipy (see pymor.bindings.scipy.solve_lyap_dense)

Parameters

A

The matrix A as a 2D NumPy array.

E

The matrix E as a 2D NumPy array or None.

B

The matrix B as a 2D NumPy array.

trans

Whether the first operator in the Lyapunov equation is transposed.

options

The solver options to use. See:

• pymor.bindings.slycot.lyap_dense_solver_options,

• pymor.bindings.pymess.lyap_dense_solver_options.

default_solver_backend

Default solver backend to use (pymess, slycot, scipy).

Returns

X

Lyapunov equation solution as a NumPy array.

pymor.algorithms.lyapunov._solve_lyap_dense_check_args(A, E, B, trans)[source]
pymor.algorithms.lyapunov._chol(A)[source]

Cholesky decomposition.

This implementation uses SVD to compute the Cholesky factor (can be used for singular matrices).

Parameters

A

Symmetric positive semidefinite matrix as a NumPy array.

Returns

L

Cholesky factor of A (in the sense that L * L^T approximates A).