pymor.algorithms.gram_schmidt

Module Contents

pymor.algorithms.gram_schmidt.gram_schmidt(A, product=None, return_R=False, atol=1e-13, rtol=1e-13, offset=0, reiterate=True, reiteration_threshold=0.9, check=True, check_tol=0.001, copy=True)[source]

Orthonormalize a VectorArray using the modified Gram-Schmidt algorithm.

Parameters:

  • A – The VectorArray which is to be orthonormalized.

  • product – The inner product Operator w.r.t. which to orthonormalize. If None, the Euclidean product is used.

  • return_R – If True, the R matrix from QR decomposition is returned.

  • atol – Vectors of norm smaller than atol are removed from the array.

  • rtol – Relative tolerance used to detect linear dependent vectors (which are then removed from the array).

  • offset – Assume that the first offset vectors are already orthonormal and start the algorithm at the offset + 1-th vector.

  • reiterate – If True, orthonormalize again if the norm of the orthogonalized vector is much smaller than the norm of the original vector.

  • reiteration_threshold – If reiterate is True, re-orthonormalize if the ratio between the norms of the orthogonalized vector and the original vector is smaller than this value.

  • check – If True, check if the resulting VectorArray is really orthonormal.

  • check_tol – Tolerance for the check.

  • copy – If True, create a copy of A instead of modifying A in-place.

Returns:

  • Q – The orthonormalized VectorArray.

  • R – The upper-triangular/trapezoidal matrix (if return_R is True).

pymor.algorithms.gram_schmidt.gram_schmidt_biorth(V, W, product=None, reiterate=True, reiteration_threshold=0.1, check=True, check_tol=0.001, copy=True)[source]

Biorthonormalize a pair of VectorArrays using the biorthonormal Gram-Schmidt process.

See Algorithm 1 in [BKohlerS11].

Note that this algorithm can be significantly less accurate compared to orthogonalization, in particular, when V and W are almost orthogonal.

Parameters:

  • V – The VectorArrays which are to be biorthonormalized.

  • W – The VectorArrays which are to be biorthonormalized.

  • product – The inner product Operator w.r.t. which to biorthonormalize. If None, the Euclidean product is used.

  • reiterate – If True, orthonormalize again if the norm of the orthogonalized vector is much smaller than the norm of the original vector.

  • reiteration_threshold – If reiterate is True, re-orthonormalize if the ratio between the norms of the orthogonalized vector and the original vector is smaller than this value.

  • check – If True, check if the resulting VectorArray is really orthonormal.

  • check_tol – Tolerance for the check.

  • copy – If True, create a copy of V and W instead of modifying V and W in-place.

Returns:

The biorthonormalized |VectorArrays|.