UT3Tangent.probe_derivatives
============================

.. py:method:: t3toolbox.uniform_manifold.UT3Tangent.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.Tuple[NDArray, ...]:           # len=d, ith elm_shape=(order+1,)+W+K+C+(Ni,)


   Symmetric directional derivatives of :py:meth:`probe`, in one repeated direction ``P`` (``pp``):
   the forward Riemannian Jacobian derivatives ``y_i^(t) = d^t/ds^t [probe(X + s P)]_i|_0`` for
   ``t=0..order`` (order 0 is :py:meth:`probe`; the bare ``𝒥``, no gauge ``Π``). ``ww``/``pp`` share the
   sample stack ``W``; stacks ``order + W + K + C``. Uniform mirror of
   :py:meth:`~t3toolbox.manifold.T3Tangent.probe_derivatives`.

   No *numerical* precondition (gauge-invariant, any frame). **Structural precondition** (hard error):
   ``P`` (``pp``) shares the sample stack ``W`` and mode dims of ``X`` (``ww``).

   .. 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.uniform_tucker_tensor_train as ut3
   >>> import t3toolbox.uniform_manifold as ut3m
   >>> np.random.seed(0)
   >>> x = ut3.UniformTuckerTensorTrain.from_t3(t3.TuckerTensorTrain.randn((10, 11, 12), (5, 6, 4), (1, 2, 3, 1)))
   >>> v = ut3m.UNIFORM_COREWISE.randn(ut3m.UNIFORM_MANIFOLD.frame(x))
   >>> 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]

   ``P`` must match ``X``'s sample stack and mode dims (structural, raises):

   >>> v.probe_derivatives(ww, (np.random.randn(10), np.random.randn(11), np.random.randn(99)), 3)
   ... # doctest: +IGNORE_EXCEPTION_DETAIL
   Traceback (most recent call last):
       ...
   ValueError