pymor.algorithms.lradi

Module Contents

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

Return previously computed shifts.

pymor.algorithms.lradi.lyap_lrcf_solver_options(lradi_tol=1e-10, lradi_maxiter=500, lradi_shifts='projection_shifts', projection_shifts_init_maxiter=20, projection_shifts_subspace_columns=6, wachspress_large_ritz_num=50, wachspress_small_ritz_num=25, wachspress_tol=1e-10)

Return available Lyapunov solvers with default options.

Parameters:

• lradi_tol – See solve_lyap_lrcf.

• lradi_maxiter – See solve_lyap_lrcf.

• lradi_shifts – See solve_lyap_lrcf.

• projection_shifts_init_maxiter – See projection_shifts_init.

• projection_shifts_subspace_columns – See projection_shifts.

• wachspress_large_ritz_num – See wachspress_shifts_init.

• wachspress_small_ritz_num – See wachspress_shifts_init.

• wachspress_tol – See wachspress_shifts_init.

Returns:

A dict of available solvers with default solver options.

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

Find further projection shifts.

Uses Galerkin projection on spaces spanned by LR-ADI iterates. See [Kurschner16], 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.projection_shifts_init(A, E, B, shift_options)

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 [Kurschner16], 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.solve_lyap_lrcf(A, E, B, trans=False, cont_time=True, options=None)

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

See

• pymor.algorithms.lyapunov.solve_cont_lyap_lrcf

for a general description.

This function uses the low-rank ADI iteration as described in Algorithm 4.3 in [Kurschner16].

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.

• cont_time – Whether the continuous- or discrete-time Lyapunov equation is solved. Only the continuous-time case is implemented.

• 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.wachspress_shifts_init(A, E, B, shift_options)

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.