T3Tangent.probe_derivatives
===========================

.. py:method:: t3toolbox.manifold.T3Tangent.probe_derivatives(ww, pp, order)

   .. code-block:: python

      def probe_derivatives(
              self,
              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,)
              order:  int,                    # highest derivative order
      ) -> typ.Sequence[NDArray]:             # len=d, elm_shape=(order+1,)+W+K+C+(Ni,)


   Symmetric directional derivatives of probing this tangent vector, in one repeated direction.

   Returns, for each mode ``i``, the stack ``y_i^(t) = d^t/ds^t [probe(X + s P)]_i|_0`` for
   ``t=0..order`` -- the derivative analogue of :py:meth:`probe`, obtained by perturbing every probe
   vector in the same direction ``P``. Index ``0`` is the ordinary :py:meth:`probe`. Stacks ``order
   + W + K + C`` (order outermost; sample stack ``W``, tangent stack ``K``, frame stack ``C``). The
   points ``X`` (``ww``) and the perturbations ``P`` (``pp``) must share the sample stack ``W``.

   See ``docs/symmetric_probe_derivatives.tex``.

   .. seealso:: :py:obj:`probe`, :py:obj:`apply_derivatives`, :py:obj:`probe_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)
   >>> ww = (np.random.randn(10), np.random.randn(11), np.random.randn(12))
   >>> pp = (np.random.randn(10), np.random.randn(11), np.random.randn(12))
   >>> zj = v.probe_derivatives(ww, pp, 3)
   >>> print([z.shape for z in zj])           # (order+1,) + (Ni,)
   [(4, 10), (4, 11), (4, 12)]
   >>> print([bool(np.allclose(z[0], z0)) for z, z0 in zip(zj, v.probe(ww))])  # order 0 == probe
   [True, True, True]