Solver algorithms
=================

Reference for the analysis types and backend algorithms
femorph-solver ships.  The :doc:`user-guide
</user-guide/solving/index>` section walks through how to drive
each solver; this page lists the underlying algorithm and its
sources.

.. toctree::
    :maxdepth: 1
    :caption: Analysis types

    static
    modal
    cyclic
    transient

.. toctree::
    :maxdepth: 1
    :caption: Backends

    linear_backends
    eigen_backends

Quick reference table
---------------------

.. list-table::
    :header-rows: 1
    :widths: 20 26 30 24

    * - Analysis
      - Entry point
      - Algorithm
      - Backend chain
    * - :doc:`static`
      - :meth:`Model.solve <femorph_solver.Model.solve>`
      - Sparse direct factor + back-substitute
      - :doc:`linear_backends`
    * - :doc:`modal`
      - :meth:`Model.modal_solve <femorph_solver.Model.modal_solve>`
      - Shift-invert Lanczos / block-Davidson / LOBPCG
      - :doc:`eigen_backends` × :doc:`linear_backends`
    * - :doc:`cyclic`
      - :meth:`Model.cyclic_modal_solve <femorph_solver.Model.cyclic_modal_solve>`
      - Per-harmonic real 2n-augmented Lanczos
        (Grimes-Lewis-Simon 1994)
      - :doc:`eigen_backends` × :doc:`linear_backends`
    * - :doc:`transient`
      - :meth:`Model.transient_solve` (roadmap) /
        modal-superposition (TA-14)
      - Newmark-β / generalised-α (direct);
        modal-superposition (decoupled SDOFs)
      - :doc:`linear_backends`

How the dispatch chains plug together
-------------------------------------

* :doc:`linear_backends` — one chain
  (Pardiso → CHOLMOD → MUMPS → UMFPACK → SuperLU)
  used by every analysis that factors a sparse matrix.
* :doc:`eigen_backends` — modal / cyclic-modal pick
  ARPACK / PRIMME / LOBPCG.  The shift-invert path inside the
  Lanczos / block-Davidson loop calls back into the linear
  chain, so the same Pardiso / CHOLMOD factor accelerates
  every shift-invert matrix-vector product.

See the per-chapter cite blocks for full algorithmic
references.

.. seealso::

    :doc:`/user-guide/solving/index` — the user-facing how-to
    for each entry point.