def w_s_update(u, S):
    """Computes the updates for the proxy of the derivative of u for smoothing

    Parameters
    ----------
    u : np.array of shape (None,)
        Beamlet radiation distribution u
    S : np.array of shape (None, None)   
    Returns
    -------
    w_next : list of np.array of shape (None,)
        updates for generalized OAR auxilary dose distributions

    
    """
    w_next = S.dot(u)
    w_next[w_next>0] = 0
    return w_next