<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def _bounding_ellipsoids(points, ell, pointvol=0., vol_dec=0.5, vol_check=2.):
 """ Internal method used to compute a set of bounding ellipsoids when a bounding ellipsoid for the entire set has already been calculated. Parameters points : `~numpy.ndarray` with shape (npoints, ndim) A set of coordinates. ell : Ellipsoid The bounding ellipsoid containing :data:`points`. pointvol : float, optional Volume represented by a single point. When provided, used to set a minimum bound on the ellipsoid volume as `npoints * pointvol`. Default is `0.`. vol_dec : float, optional The required fractional reduction in volume after splitting an ellipsoid in order to to accept the split. Default is `0.5`. vol_check : float, optional The factor used to when checking whether the volume of the original bounding ellipsoid is large enough to warrant more trial splits via `ell.vol > vol_check * npoints * pointvol`. Default is `2.0`. Returns ------- ells : list of :class:`Ellipsoid` objects List of :class:`Ellipsoid` objects used to bound the collection of points. Used to initialize the :class:`MultiEllipsoid` object returned in :meth:`bounding_ellipsoids`. """


    npoints, ndim = points.shape

    # Starting cluster centers are initialized using the major-axis
    # endpoints of the original bounding ellipsoid.
    p1, p2 = ell.major_axis_endpoints()
    start_ctrs = np.vstack((p1, p2))  # shape is (k, ndim) = (2, ndim)

    # Split points into two clusters using k-means clustering with k=2.
    try:
        with warnings.catch_warnings():
            warnings.simplefilter("ignore")
            k2_res = kmeans2(points, k=start_ctrs, iter=10, minit='matrix',
                             check_finite=False)
        labels = k2_res[1]  # cluster identifier ; shape is (npoints,)

        # Get points in each cluster.
        points_k = [points[labels == k, :] for k in (0, 1)]

        # If either cluster has less than ndim+1 points, the bounding ellipsoid
        # will be ill-constrained. Reject the split and simply return the
        # original ellipsoid bounding all the points.
        if points_k[0].shape[0] < 2 * ndim or points_k[1].shape[0] < 2 * ndim:
            return [ell]

        # Bounding ellipsoid for each cluster, possibly enlarged
        # to a minimum volume.
        ells = [bounding_ellipsoid(points_j, pointvol=pointvol)
                for points_j in points_k]

        # If the total volume decreased by a factor of `vol_dec`, we accept
        # the split into subsets. We then recursively split each subset.
        if ells[0].vol + ells[1].vol < vol_dec * ell.vol:
            return (_bounding_ellipsoids(points_k[0], ells[0],
                                         pointvol=pointvol, vol_dec=vol_dec,
                                         vol_check=vol_check) +
                    _bounding_ellipsoids(points_k[1], ells[1],
                                         pointvol=pointvol, vol_dec=vol_dec,
                                         vol_check=vol_check))

        # Otherwise, see if the total ellipsoid volume is larger than the
        # minimum volume by a factor of `vol_check`. If it is, this indicates
        # that there may be more than 2 clusters and we should try to
        # subdivide further.
        if ell.vol > vol_check * npoints * pointvol:
            out = (_bounding_ellipsoids(points_k[0], ells[0],
                                        pointvol=pointvol, vol_dec=vol_dec,
                                        vol_check=vol_check) +
                   _bounding_ellipsoids(points_k[1], ells[1],
                                        pointvol=pointvol, vol_dec=vol_dec,
                                        vol_check=vol_check))

            # Only accept the split if the volume decreased significantly.
            if sum(e.vol for e in out) < vol_dec * ell.vol:
                return out
    except:
        pass

    # Otherwise, we are happy with the single bounding ellipsoid.
    return [ell]