Full rank Cholesky updates?

[pnkraemer]

In ProbNum we use full-rank Cholesky updates (see the utility function probnum.statespace.cholesky_update) for square-root forward transitions.

Do you want the utility functions in here? For starters, we could merge the code (https://github.com/probabilistic-numerics/probnum/blob/master/src/probnum/filtsmooth/statespace/discrete_transition_utils.py)

def cholesky_update(
    S1: np.ndarray, S2: typing.Optional[np.ndarray] = None
) -> np.ndarray:
    r"""Compute Cholesky update/factorization :math:`C C^\top = S_1 S_1^\top + S_2 S_2^\top`.
    This can be used in various ways.
    For example, :math:`S_1` and :math:`S_2` do not need to be Cholesky factors; any matrix square-root is sufficient.
    As long as :math:`C C^\top = S_1 S_1^\top + S_2 S_2^\top` is well-defined (and admits a Cholesky-decomposition),
    :math:`S_1` and :math:`S_2` do not even have to be square.
    """
    # doc might need a bit more explanation in the future
    # perhaps some doctest or so?
    if S2 is not None:
        stacked_up = np.vstack((S1.T, S2.T))
    else:
        stacked_up = np.vstack(S1.T)
    upper_sqrtm = np.linalg.qr(stacked_up, mode="r")
    return triu_to_positive_tril(upper_sqrtm)


def triu_to_positive_tril(triu_mat: np.ndarray) -> np.ndarray:
    r"""Change an upper triangular matrix into a valid lower Cholesky factor.
    Transpose, and change the sign of the diagonals to '+' if necessary.
    The name of the function is leaned on `np.triu` and `np.tril`.
    """
    tril_mat = triu_mat.T
    d = np.sign(np.diag(tril_mat))
    d[d == 0] = 1.0
    with_pos_diag = tril_mat @ np.diag(d)
    return with_pos_diag

with its respective tests (subject to interface changes to remain compliant with your current implementation of rank-1 updates) as soon as ProbNum has an explicit dependency on cholupdates anyway.

This could then be used inside random_variables as well as inside statespace, and optimisation of this functionality can be outsourced to optimising cholupdates.
Note that the cholesky_update function above already handles your Issue #13 in a basic way (more optimized code seems possible, but can be a future project?!).