modified_gram_schmidt

probnum.utils.linalg.modified_gram_schmidt(v, orthogonal_basis, inner_product=None, normalize=False)

Stabilized Gram-Schmidt process.

Computes a vector \(v'\) such that \(\langle v', b_i \rangle = 0\) for all basis vectors \(b_i \in B\) in the orthogonal basis in a numerically stable fashion.

Parameters

• v (ndarray) – Vector (or stack of vectors) to orthogonalize against orthogonal_basis.

• orthogonal_basis (Iterable[ndarray]) – Orthogonal basis.

• inner_product (Union[ndarray, LinearOperator, Callable[[ndarray, ndarray], ndarray], None]) – Inner product defining orthogonality. Can be either a numpy.ndarray or a Callable defining the inner product. Defaults to the euclidean inner product.

• normalize (bool) – Normalize the output vector, s.t. \(\langle v', v' \rangle = 1\).

Returns

Orthogonalized vector.

Return type

v_orth