tv_probe_derivatives
====================

.. py:function:: t3toolbox.backend.sampling_derivatives.tv_probe_derivatives(ww, pp, variation, frame, order)

   .. code-block:: python

      def tv_probe_derivatives(
              ww:         typ.Sequence[NDArray],  # probe vectors X,        len=d, elm_shape=W+(Ni,)
              pp:         typ.Sequence[NDArray],  # perturbation vectors P, len=d, elm_shape=W+(Ni,)
              variation:  typ.Tuple[
                  typ.Sequence[NDArray],          # var_tucker_cores dU. len=d, elm_shape=K+C+(nOi,Ni)
                  typ.Sequence[NDArray],          # var_tt_cores     dG. len=d, elm_shape=K+C+(rLi,nUi,rRi)
              ],                                  # = T3Variations.data
              frame:       typ.Tuple[
                  typ.Sequence[NDArray],          # up_tucker_cores  U. len=d
                  typ.Sequence[NDArray],          # down_tt_cores    O. len=d
                  typ.Sequence[NDArray],          # left_tt_cores    P. len=d
                  typ.Sequence[NDArray],          # right_tt_cores   Q. len=d
              ],                                  # = T3Frame.data = (up, down, left, right) = (U, O, P, Q)
              order:      int,                    # highest derivative order K
      ) -> typ.Tuple[NDArray, ...]:               # z_jets. len=d, elm_shape=(order+1,)+W+K+C+(Ni,)


   Symmetric derivatives of probing a tangent vector, in one repeated direction (Riemannian J^(s)).

   The probe-derivative analog of :py:func:`probing.tv_probe`: returns, for each mode ``i``, the
   stack ``y_i^(t) = d^t/ds^t [J^(s) v]_i (X + s P)|_0`` for ``t=0..K``, where ``v`` is the tangent
   vector represented by ``(frame, variation)``. Index ``0`` is the ordinary tangent probe.

   Identical structure to :py:func:`t3_probe_derivatives` but on the tangent calculus: a frame sweep
   (frame cores) and a variation sweep. Full ``order + W + K + C`` stacking (base-inner, mirroring
   tv_probe): sample stack ``W``, tangent stack ``K`` (a batch of tangents sharing the frame, on
   the variation cores), frame stack ``C``; any may be empty.

   The symmetric-derivative formulation is not in the published T4S paper; verified against
   :py:func:`dense_probe_derivatives` on the densified tangent. (Project write-up in preparation.)

   .. seealso:: :py:obj:`t3_probe_derivatives`, :py:obj:`compute_sigma_jets`, :py:obj:`compute_tau_jets`, :py:obj:`compute_deta_jets`, :py:obj:`assemble_tangent_z_jets`, :py:obj:`t3toolbox.backend.probing.tv_probe`