..
   Programmer(s): Daniel R. Reynolds @ SMU
   ----------------------------------------------------------------
   SUNDIALS Copyright Start
   Copyright (c) 2002-2024, Lawrence Livermore National Security
   and Southern Methodist University.
   All rights reserved.

   See the top-level LICENSE and NOTICE files for details.

   SPDX-License-Identifier: BSD-3-Clause
   SUNDIALS Copyright End
   ----------------------------------------------------------------

.. _SUNLinSol.SPBCGS:

The SUNLinSol_SPBCGS Module
======================================

The SUNLinSol_SPBCGS implementation of the ``SUNLinearSolver`` class performs
a Scaled, Preconditioned, Bi-Conjugate Gradient, Stabilized :cite:p:`Van:92` method;
this is an iterative linear solver that is designed to be compatible with any
``N_Vector`` implementation that supports a minimal subset of operations
(:c:func:`N_VClone()`, :c:func:`N_VDotProd()`, :c:func:`N_VScale()`,
:c:func:`N_VLinearSum()`, :c:func:`N_VProd()`, :c:func:`N_VDiv()`, and
:c:func:`N_VDestroy()`).  Unlike the SPGMR and SPFGMR algorithms,
SPBCGS requires a fixed amount of memory that does not increase with
the number of allowed iterations.


.. _SUNLinSol.SPBCGS.Usage:

SUNLinSol_SPBCGS Usage
------------------------

The header file to be included when using this module
is ``sunlinsol/sunlinsol_spbcgs.h``.  The SUNLinSol_SPBCGS module
is accessible from all SUNDIALS solvers *without*
linking to the ``libsundials_sunlinsolspbcgs`` module library.


The module SUNLinSol_SPBCGS provides the following
user-callable routines:


.. c:function:: SUNLinearSolver SUNLinSol_SPBCGS(N_Vector y, int pretype, int maxl, SUNContext sunctx)

   This constructor function creates and allocates memory for a SPBCGS
   ``SUNLinearSolver``.

   **Arguments:**
      * *y* -- a template vector.
      * *pretype* -- a flag indicating the type of preconditioning to use:

        * ``SUN_PREC_NONE``
        * ``SUN_PREC_LEFT``
        * ``SUN_PREC_RIGHT``
        * ``SUN_PREC_BOTH``

      * *maxl* -- the maximum number of linear iterations to allow.
      * *sunctx* -- the :c:type:`SUNContext` object (see :numref:`SUNDIALS.SUNContext`)

   **Return value:**
      If successful, a ``SUNLinearSolver`` object.  If either *y* is
      incompatible then this routine will return ``NULL``.

   **Notes:**
      This routine will perform consistency checks to ensure that it is
      called with a consistent ``N_Vector`` implementation (i.e. that it
      supplies the requisite vector operations).

      A ``maxl`` argument that is :math:`\le0` will result in the default
      value (5).

      Some SUNDIALS solvers are designed
      to only work with left preconditioning (IDA and IDAS) and others
      with only right preconditioning (KINSOL). While it is possible to
      configure a SUNLinSol_SPBCGS object to use any of the
      preconditioning options with these solvers, this use mode is not
      supported and may result in inferior performance.

   .. note::

      With ``SUN_PREC_RIGHT`` or ``SUN_PREC_BOTH`` the initial guess must be zero (use
      :c:func:`SUNLinSolSetZeroGuess` to indicate the initial guess is zero).


.. c:function:: SUNErrCode SUNLinSol_SPBCGSSetPrecType(SUNLinearSolver S, int pretype)

   This function updates the flag indicating use of preconditioning.

   **Arguments:**
      * *S* -- SUNLinSol_SPBCGS object to update.
      * *pretype* -- a flag indicating the type of preconditioning to use:

        * ``SUN_PREC_NONE``
        * ``SUN_PREC_LEFT``
        * ``SUN_PREC_RIGHT``
        * ``SUN_PREC_BOTH``

   **Return value:**
      * A :c:type:`SUNErrCode`


.. c:function:: SUNErrCode SUNLinSol_SPBCGSSetMaxl(SUNLinearSolver S, int maxl)

   This function updates the number of linear solver iterations to allow.

   **Arguments:**
      * *S* -- SUNLinSol_SPBCGS object to update.
      * *maxl* -- maximum number of linear iterations to allow.  Any
        non-positive input will result in the default value (5).

   **Return value:**
      * A :c:type:`SUNErrCode`



.. _SUNLinSol.SPBCGS.Description:

SUNLinSol_SPBCGS Description
-------------------------------

The SUNLinSol_SPBCGS module defines the *content* field of a
``SUNLinearSolver`` to be the following structure:

.. code-block:: c

   struct _SUNLinearSolverContent_SPBCGS {
     int maxl;
     int pretype;
     sunbooleantype zeroguess;
     int numiters;
     sunrealtype resnorm;
     int last_flag;
     SUNATimesFn ATimes;
     void* ATData;
     SUNPSetupFn Psetup;
     SUNPSolveFn Psolve;
     void* PData;
     N_Vector s1;
     N_Vector s2;
     N_Vector r;
     N_Vector r_star;
     N_Vector p;
     N_Vector q;
     N_Vector u;
     N_Vector Ap;
     N_Vector vtemp;
   };

These entries of the *content* field contain the following
information:

* ``maxl`` - number of SPBCGS iterations to allow (default is 5),

* ``pretype`` - flag for type of preconditioning to employ
  (default is none),

* ``numiters`` - number of iterations from the most-recent solve,

* ``resnorm`` - final linear residual norm from the most-recent
  solve,

* ``last_flag`` - last error return flag from an internal
  function,

* ``ATimes`` - function pointer to perform :math:`Av` product,

* ``ATData`` - pointer to structure for ``ATimes``,

* ``Psetup`` - function pointer to preconditioner setup routine,

* ``Psolve`` - function pointer to preconditioner solve routine,

* ``PData`` - pointer to structure for ``Psetup`` and ``Psolve``,

* ``s1, s2`` - vector pointers for supplied scaling matrices
  (default is ``NULL``),

* ``r`` - a ``N_Vector`` which holds the current scaled,
  preconditioned linear system residual,

* ``r_star`` - a ``N_Vector`` which holds the initial scaled,
  preconditioned linear system residual,

* ``p, q, u, Ap, vtemp`` - ``N_Vector`` used for workspace by the
  SPBCGS algorithm.




This solver is constructed to perform the following operations:

* During construction all ``N_Vector`` solver data is allocated, with
  vectors cloned from a template ``N_Vector`` that is input, and
  default solver parameters are set.

* User-facing "set" routines may be called to modify default
  solver parameters.

* Additional "set" routines are called by the SUNDIALS solver
  that interfaces with SUNLinSol_SPBCGS to supply the ``ATimes``,
  ``PSetup``, and ``Psolve`` function pointers and ``s1`` and ``s2``
  scaling vectors.

* In the "initialize" call, the solver parameters are checked
  for validity.

* In the "setup" call, any non-``NULL`` ``PSetup`` function is
  called.  Typically, this is provided by the SUNDIALS solver itself,
  that translates between the generic ``PSetup`` function and the
  solver-specific routine (solver-supplied or user-supplied).

* In the "solve" call the SPBCGS iteration is performed.  This
  will include scaling and preconditioning if those options have been
  supplied.

The SUNLinSol_SPBCGS module defines implementations of all
"iterative" linear solver operations listed in
:numref:`SUNLinSol.API`:

* ``SUNLinSolGetType_SPBCGS``

* ``SUNLinSolInitialize_SPBCGS``

* ``SUNLinSolSetATimes_SPBCGS``

* ``SUNLinSolSetPreconditioner_SPBCGS``

* ``SUNLinSolSetScalingVectors_SPBCGS``

* ``SUNLinSolSetZeroGuess_SPBCGS`` -- note the solver assumes a non-zero guess
  by default and the zero guess flag is reset to ``SUNFALSE`` after each call to
  ``SUNLinSolSolve_SPBCGS``.

* ``SUNLinSolSetup_SPBCGS``

* ``SUNLinSolSolve_SPBCGS``

* ``SUNLinSolNumIters_SPBCGS``

* ``SUNLinSolResNorm_SPBCGS``

* ``SUNLinSolResid_SPBCGS``

* ``SUNLinSolLastFlag_SPBCGS``

* ``SUNLinSolSpace_SPBCGS``

* ``SUNLinSolFree_SPBCGS``
