def gram_schmidt_basis_extension(basis, U, U_ind=None, product=None, copy_basis=True, copy_U=True):
'''Extend basis using Gram-Schmidt orthonormalization.
Parameters
----------
basis
The basis to extend.
U
The new basis vectors.
U_ind
Indices of the new basis vectors in U.
product
The scalar product w.r.t. which to orthonormalize; if None, the l2-scalar
product on the coefficient vector is used.
copy_basis
If copy_basis is False, the old basis is extended in-place.
copy_U
If copy_U is False, the new basis vectors are removed from U.
Returns
-------
The new basis.
Raises
------
ExtensionError
Gram-Schmidt orthonormalization fails. Usually this is the case when U
is not linearily independent from the basis. However this can also happen
due to rounding errors ...
'''
if basis is None:
basis = NumpyVectorArray(np.zeros((0, U.dim)))
basis_length = len(basis)
new_basis = basis.copy() if copy_basis else basis
new_basis.append(U, o_ind=U_ind, remove_from_other=(not copy_U))
gram_schmidt(new_basis, offset=len(basis), product=product)
if len(new_basis) <= basis_length:
raise ExtensionError
return new_basis
def trivial_basis_extension(basis, U, U_ind=None, copy_basis=True, copy_U=True):
'''Trivially extend basis by just adding the new vector.
We check that the new vector is not already contained in the basis, but we do
not check for linear independence.
Parameters
----------
basis
The basis to extend.
U
The new basis vector.
U_ind
Indices of the new basis vectors in U.
copy_basis
If copy_basis is False, the old basis is extended in-place.
copy_U
If copy_U is False, the new basis vectors are removed from U.
Returns
-------
The new basis.
Raises
------
ExtensionError
Is raised if U is already contained in basis.
'''
if basis is None:
basis = NumpyVectorArray(np.zeros((0, U.dim)))
if np.any(U.almost_equal(basis, ind=U_ind)):
raise ExtensionError
new_basis = basis.copy() if copy_basis else basis
new_basis.append(U, o_ind=U_ind, remove_from_other=(not copy_U))
return new_basis
