tv_apply_derivatives
====================

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

   .. code-block:: python

      def tv_apply_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
      ) -> NDArray:                               # apply-derivative jets, shape=(order+1,)+W+K+C


   Symmetric derivatives of applying a tangent vector in all modes, in one repeated direction.

   The all-modes Riemannian analog of :py:func:`tv_probe_derivatives` (and the derivative analog
   of :py:func:`apply.tv_apply`): returns ``y^(t) = d^t/ds^t [apply(v, W + s P)]|_0`` for
   ``t=0..order``, where ``v`` is the tangent vector ``(frame, variation)``. A single left-to-right pass
   (frame mu via P, perturbation sigma via Q) to the terminal carry, bond summed. Stacks ``order + W +
   K + C`` (sample stack ``W``, tangent stack ``K``, frame stack ``C``). Verified against
   :py:func:`dense_apply_derivatives` on the densified tangent.

   .. seealso:: :py:obj:`t3_apply_derivatives`, :py:obj:`tv_probe_derivatives`, :py:obj:`apply.tv_apply`