pymor.algorithms.bernoulli

Module Contents

Functions

solve_bernoulli	Compute a solution factor of a Bernoulli equation.
bernoulli_stabilize	Compute Bernoulli stabilizing feedback.

pymor.algorithms.bernoulli.solve_bernoulli(A, E, B, trans=False, maxiter=100, after_maxiter=3, tol=1e-08)

Compute a solution factor of a Bernoulli equation.

Returns a matrix \(Y\) with identical dimensions to the matrix \(A\) such that \(X = Y Y^H\) is an approximate solution of a (generalized) algebraic Bernoulli equation:

• if trans is True

  \[A^H X E + E^H X A - E^H X B B^H X E = 0.\]

• if trans is False

  \[A X E^H + E X A^H - E X B^H B X E^H = 0.\]

This function is based on [BBQOrti07].

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 to solve transposed or standard Bernoulli equation.

maxiter

The maximum amount of iterations.

after_maxiter

The number of iterations which are to be performed after tolerance is reached. This will improve the quality of the solution in cases where the iterates which are used by the stopping criterion stagnate prematurely.

tol

Tolerance for stopping criterion based on relative change of iterates.

Returns

Y

The solution factor as a NumPy array.

pymor.algorithms.bernoulli.bernoulli_stabilize(A, E, B, ast_spectrum, trans=False)

Compute Bernoulli stabilizing feedback.

Returns a matrix \(K\) that stabilizes the spectrum of the matrix pair \((A, E)\):

• if trans is True the spectrum of

  \[(A - B K, E)\]

  contains the eigenvalues of \((A, E)\) where anti-stable eigenvalues have been mirrored on the imaginary axis.

• if trans is False the spectrum of

  \[(A - K B, E)\]

  contains the eigenvalues of \((A, E)\) where anti-stable eigenvalues have been mirrored on the imaginary axis.

See e.g. [BBQOrti07].

Parameters

A

The Operator A.

E

The Operator E.

B

The operator B as a VectorArray.

ast_spectrum

Tuple (lev, ew, rev) where ew contains the anti-stable eigenvalues and lev and rev are VectorArrays representing the eigenvectors.

trans

Indicates which stabilization to perform.

Returns

K

The stabilizing feedback as a VectorArray.