Finite-section SISO AAK/Nehari certificate
==========================================

.. admonition:: Tutorial goal

   Use Schmidt-pair identities to certify the finite AAK/Nehari target and attach the rational candidate.

.. 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
-------

This is the first tutorial whose structure is deliberately AAK/Nehari-shaped.  It builds a
finite Hankel matrix from an exact rational anticausal tail, computes the first neglected
Schmidt pair, checks the singular-vector identities, and attaches the finite Nehari/rational
candidate for the same rank.

The tutorial is intentionally finite-section.  It is closer to the AAK/Nehari construction
than the earlier rank sweeps because it verifies the Schmidt-pair equations directly, but it
is still not advertised as a full infinite-dimensional Hardy-space solver.

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

For target rank ``r``, the finite certificate reports the first neglected singular value

.. math::

   \sigma_{r+1}.

It also checks the finite Schmidt-pair equations

.. math::

   H v_{r+1}=\sigma_{r+1}u_{r+1},\qquad
   H^T u_{r+1}=\sigma_{r+1}v_{r+1}.

On an exact rank-three rational tail, the rank-three certificate should recover the stable
poles and produce residuals near machine precision.

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

Look for small Schmidt-pair residuals, rank-3 pole recovery, tiny rational error, and poles inside the unit disk.

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

.. code-block:: bash

   python examples/aak_siso_certificate_demo.py

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

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