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)

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)

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, 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.