compute_dxi_tilde
=================

.. py:function:: t3toolbox.backend.probing.compute_dxi_tilde(sigma_tildes, tau_tildes, down_tt_cores, mus, nus)

   .. code-block:: python

      def compute_dxi_tilde(
              sigma_tildes:           typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,rR(i+1))
              tau_tildes:             typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,rLi)
              down_tt_cores:         typ.Union[typ.Sequence[NDArray], NDArray], # len=d, elm_shape=(rLi,nOi,rR(i+1))
              mus:                    typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,rLi)
              nus:                    typ.Union[typ.Sequence[NDArray], NDArray],  # len=d, elm_shape=(...,rR(i+1))
      ) -> typ.Union[typ.Sequence[NDArray], NDArray]: # dxi_tildes. len=d, elm_shape=(...,nOi)


   Adjoint-var-downward edge variables dxi_tilde.
   Used for computing the transpose of the map from a tangent vector to its probes.

   See Section 6.2.3, particularly Algorithm 8, in:
       Alger, N., Christierson, B., Chen, P., & Ghattas, O. (2026).
       "Tucker Tensor Train Taylor Series."
       arXiv preprint arXiv:2603.21141.
       `https://arxiv.org/abs/2603.21141 <https://arxiv.org/abs/2603.21141>`_