pymor.algorithms.eigs

Module Contents

pymor.algorithms.eigs.eigs(A, E=None, k=3, sigma=None, which='LM', b=None, l=None, maxiter=1000, tol=1e-13, imag_tol=1e-12, complex_pair_tol=1e-12, complex_evp=False, left_evp=False, shifted_system_solver=None)[source]

Approximate a few eigenvalues of a linear Operator.

Computes k eigenvalues w with corresponding eigenvectors v which solve the eigenvalue problem

\[A v_i = w_i v_i\]

or the generalized eigenvalue problem

\[A v_i = w_i E v_i\]

if E is not None.

The implementation is based on Algorithm 4.2 in [Leh95].

Parameters:

  • A – The linear Operator for which the eigenvalues are to be computed.

  • E – The linear Operator which defines the generalized eigenvalue problem.

  • k – The number of eigenvalues and eigenvectors which are to be computed.

  • sigma – If not None transforms the eigenvalue problem such that the k eigenvalues closest to sigma are computed.

  • which –

    A string specifying which k eigenvalues and eigenvectors to compute:

    • 'LM': select eigenvalues with largest magnitude

    • 'SM': select eigenvalues with smallest magnitude

    • 'LR': select eigenvalues with largest real part

    • 'SR': select eigenvalues with smallest real part

    • 'LI': select eigenvalues with largest imaginary part

    • 'SI': select eigenvalues with smallest imaginary part

  • b – Initial vector for Arnoldi iteration. Default is a random vector.

  • l – The size of the Arnoldi factorization. Default is min(n - 1, max(2*k + 1, 20)).

  • maxiter – The maximum number of iterations.

  • tol – The relative error tolerance for the Ritz estimates.

  • imag_tol – Relative imaginary parts below this tolerance are set to 0.

  • complex_pair_tol – Tolerance for detecting pairs of complex conjugate eigenvalues.

  • complex_evp – Whether to consider an eigenvalue problem with complex operators. When operators are real setting this argument to False will increase stability and performance.

  • left_evp – If set to True compute left eigenvectors else compute right eigenvectors.

  • shifted_system_solver – The Solver to use if sigma is not None.

Returns:

  • w – A 1D NumPy array which contains the computed eigenvalues.

  • v – A VectorArray which contains the computed eigenvectors.