pymor.algorithms.samdp

Module Contents

Functions

samdp	Compute the dominant pole triplets and residues of the transfer function of an LTI system.
_twosided_rqi	Refine an initial guess for an eigentriplet of the matrix pair (A, E).
_select_max_eig	Compute poles sorted from largest to smallest residual.

pymor.algorithms.samdp.samdp(A, E, B, C, nwanted, init_shifts=None, which='NR', tol=1e-10, imagtol=1e-06, conjtol=1e-08, dorqitol=0.0001, rqitol=1e-10, maxrestart=100, krestart=20, rqi_maxiter=10, seed=0)

Compute the dominant pole triplets and residues of the transfer function of an LTI system.

This function uses the subspace accelerated dominant pole (SAMDP) algorithm as described in [RM06] in Algorithm 2 in order to compute dominant pole triplets and residues of the transfer function

$$H(s)=C(sE-A{)}^{-1}B$$

of an LTI system. It is possible to take advantage of prior knowledge about the poles by specifying shift parameters, which are injected after a new pole has been found.

Note

Pairs of complex conjugate eigenvalues are always returned together. Accordingly, the number of returned poles can be equal to nwanted + 1.

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.

nwanted

The number of dominant poles that should be computed.

init_shifts

A NumPy array containing shifts which are injected after a new pole has been found.

which

A string specifying the strategy by which the dominant poles and residues are selected. Possible values are:

• 'NR': select poles with largest norm(residual) / abs(Re(pole))

• 'NS': select poles with largest norm(residual) / abs(pole)

• 'NM': select poles with largest norm(residual)

tol

Tolerance for the residual of the poles.

imagtol

Relative tolerance for imaginary parts of pairs of complex conjugate eigenvalues.

conjtol

Tolerance for the residual of the complex conjugate of a pole.

dorqitol

If the residual is smaller than dorqitol the two-sided Rayleigh quotient iteration is executed.

rqitol

Tolerance for the relative change of a pole in the two-sided Rayleigh quotient iteration.

maxrestart

The maximum number of restarts.

krestart

Maximum dimension of search space before performing a restart.

rqi_maxiter

Maximum number of iterations for the two-sided Rayleigh quotient iteration.

seed

Random seed which is used for computing the initial shift and random restarts.

Returns

poles

A 1D NumPy array containing the computed dominant poles.

residues

A 3D NumPy array of shape (len(poles), len(C), len(B)) containing the computed residues.

rightev

A VectorArray containing the right eigenvectors of the computed poles.

leftev

A VectorArray containing the left eigenvectors of the computed poles.

pymor.algorithms.samdp._twosided_rqi(A, E, x, y, theta, init_res, imagtol, rqitol, maxiter)

Refine an initial guess for an eigentriplet of the matrix pair (A, E).

Parameters

A

The Operator A from the LTI system.

E

The Operator E from the LTI system.

x

Initial guess for right eigenvector of matrix pair (A, E).

y

Initial guess for left eigenvector of matrix pair (A, E).

theta

Initial guess for eigenvalue of matrix pair (A, E).

init_res

Residual of initial guess.

imagtol

Relative tolerance for imaginary parts of pairs of complex conjugate eigenvalues.

rqitol

Convergence tolerance for the relative change of the pole.

maxiter

Maximum number of iteration.

Returns

x

Refined right eigenvector of matrix pair (A, E).

y

Refined left eigenvector of matrix pair (A, E).

theta

Refined eigenvalue of matrix pair (A, E).

residual

Residual of the computed triplet.

pymor.algorithms.samdp._select_max_eig(H, G, X, V, B, C, which)

Compute poles sorted from largest to smallest residual.

Parameters

H

The NumPy array H from the SAMDP algorithm.

G

The NumPy array G from the SAMDP algorithm.

X

A VectorArray describing the orthogonal search space used in the SAMDP algorithm.

V

A VectorArray describing the orthogonal search space used in the SAMDP algorithm.

B

The VectorArray B from the corresponding LTI system modified by deflation.

C

The VectorArray C from the corresponding LTI system modified by deflation.

which

A string that indicates which poles to select. See samdp.

Returns

poles

A NumPy array containing poles sorted according to the chosen strategy.

rightevs

A NumPy array containing the right eigenvectors of the computed poles.

leftevs

A NumPy array containing the left eigenvectors of the computed poles.

residue

A 1D NumPy array containing the norms of the residues.