T3Tangent.entries_derivatives
=============================

.. py:method:: t3toolbox.manifold.T3Tangent.entries_derivatives(index, pp, order)

   .. code-block:: python

      def entries_derivatives(
              self,
              index:  NDArray,                # int, shape=(d,)+W -- grid points
              pp:     typ.Sequence[NDArray],  # perturbation vectors P, len=d, elm_shape=W+(Ni,)
              order:  int,                    # highest derivative order
      ) -> NDArray:                           # shape=(order+1,)+W+K+C


   Symmetric directional derivatives of this tangent's entries at ``index``, in direction ``P``.

   The Taylor data of the tangent's multilinear extension at grid corner ``index``, in direction
   ``P``: ``y^(t) = d^t/ds^t apply(e_{index} + s P)|_0`` for ``t=0..order``. Index ``0`` is the
   ordinary :py:meth:`entries`. Stacks ``order + W + K + C``. ``index`` and ``P`` share ``W``.

   .. seealso:: :py:obj:`entries`, :py:obj:`apply_derivatives`, :py:obj:`entries_derivatives_transpose`

   .. rubric:: Examples

   >>> import numpy as np
   >>> import t3toolbox.tucker_tensor_train as t3
   >>> import t3toolbox.frame_variations_format as bvf
   >>> import t3toolbox.manifold as t3m
   >>> np.random.seed(0)
   >>> x = t3.TuckerTensorTrain.randn((10, 11, 12), (5, 6, 4), (1, 2, 3, 1))
   >>> frame, variations = bvf.t3_orthogonal_representations(x)
   >>> v = t3m.T3Tangent(frame, variations)
   >>> index = np.array([3, 5, 7])
   >>> pp = (np.random.randn(10), np.random.randn(11), np.random.randn(12))
   >>> yj = v.entries_derivatives(index, pp, 3)
   >>> print(yj.shape)
   (4,)
   >>> print(bool(np.allclose(yj[0], v.entries(index))))   # order 0 == entries
   True