Finite-section AAK/Nehari reduction on a non-exact tail
=======================================================

.. admonition:: Tutorial goal

   Use the high-level finite AAK/Nehari reducer on a stable tail with a small non-rational residual.

.. note::

   New to the terminology? See the :doc:`lattice DSP concept map <../../algorithms/concept_map>` and the :doc:`causality/data-use guide <../../theory/causality_and_data_use>` for how online, offline, block, and MIMO examples should be read.

Context
-------

Exact rational-tail validation is necessary but too clean.  This tutorial adds a small
deterministic residual to a stable rank-three tail and then calls the high-level
``finite_aak_reduce_tail`` helper.  The helper evaluates candidate ranks, selects the first
stable rational model that meets user-chosen tolerances, and attaches a finite Schmidt-pair
certificate for the selected rank.

The tutorial is intentionally finite-section.  It shows how the current package behaves on
non-exact data while keeping the scope finite-dimensional.

Key idea and equations
----------------------

The observed tail is

.. math::

   \gamma_n = \sum_{i=1}^3 c_i p_i^n + \epsilon_n,
   \qquad |p_i|<1,

where ``epsilon_n`` is a small deterministic residual.  For each candidate rank ``r``, the
helper reports

.. math::

   \sigma_{r+1},\qquad
   \frac{\|\gamma-\widehat\gamma_r\|_2}{\|\gamma\|_2},\qquad
   \frac{\|\gamma-g_r\|_2}{\|\gamma\|_2},\qquad
   \max_i |p_i|.

The selected candidate is the first rank satisfying the supplied error and pole-radius
tolerances.

How to read the result
----------------------

Look for a selected stable rank, a rational error below tolerance, and Schmidt-pair residuals near numerical precision for the selected finite Hankel certificate.

Run command
-----------

.. code-block:: bash

   python examples/finite_aak_noisy_tail_demo.py

Source code
-----------

.. literalinclude:: ../../../examples/finite_aak_noisy_tail_demo.py
   :language: python
   :linenos: