# pymor.algorithms.lradi¶

## Module Contents¶

### Functions¶

 lyap_lrcf_solver_options Return available Lyapunov solvers with default options. solve_lyap_lrcf Compute an approximate low-rank solution of a Lyapunov equation. projection_shifts_init Find starting projection shifts. projection_shifts Find further projection shifts. wachspress_shifts_init Compute optimal shifts for symmetric matrices. cycle_shifts Return previously computed shifts.

Return available Lyapunov solvers with default options.

Parameters

projection_shifts_init_maxiter
projection_shifts_init_seed
projection_shifts_subspace_columns
wachspress_large_ritz_num
wachspress_small_ritz_num
wachspress_tol

Returns

A dict of available solvers with default solver options.

pymor.algorithms.lradi.solve_lyap_lrcf(A, E, B, trans=False, options=None)[source]

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

See pymor.algorithms.lyapunov.solve_lyap_lrcf for a general description.

This function uses the low-rank ADI iteration as described in Algorithm 4.3 in .

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 lyap_lrcf_solver_options).

Returns

Z

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

pymor.algorithms.lradi.projection_shifts_init(A, E, B, shift_options)[source]

Find starting projection shifts.

Uses Galerkin projection on the space spanned by the right-hand side if it produces stable shifts. Otherwise, uses a randomly generated subspace. See , pp. 92-95.

Parameters

A

The Operator A from the corresponding Lyapunov equation.

E

The Operator E from the corresponding Lyapunov equation.

B

The VectorArray B from the corresponding Lyapunov equation.

shift_options

The shift options to use (see lyap_lrcf_solver_options).

Returns

shifts

A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lradi.projection_shifts(A, E, V, Z, prev_shifts, shift_options)[source]

Find further projection shifts.

Uses Galerkin projection on spaces spanned by LR-ADI iterates. See , pp. 92-95.

Parameters

A

The Operator A from the corresponding Lyapunov equation.

E

The Operator E from the corresponding Lyapunov equation.

V

A VectorArray representing the currently computed iterate.

Z

A VectorArray representing the current approximate solution.

prev_shifts

A NumPy array containing the set of all previously used shift parameters.

shift_options

The shift options to use (see lyap_lrcf_solver_options).

Returns

shifts

A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lradi.wachspress_shifts_init(A, E, B, shift_options)[source]

Compute optimal shifts for symmetric matrices.

This method computes optimal shift parameters for the LR-ADI iteration based on Wachspress’ method which is discussed in [LW02]. This implementation assumes that $$A$$ and $$E$$ are both real and symmetric.

Parameters

A

The Operator A from the corresponding Lyapunov equation.

E

The Operator E from the corresponding Lyapunov equation.

B

The VectorArray B from the corresponding Lyapunov equation.

shift_options

The shift options to use (see lyap_lrcf_solver_options).

Returns

shifts

A NumPy array containing a set of stable shift parameters.

pymor.algorithms.lradi.cycle_shifts(A, E, V, Z, prev_shifts, shift_options)[source]

Return previously computed shifts.