..
   Copyright 2021 IRT Saint-Exupéry, https://www.irt-saintexupery.com

   This work is licensed under the Creative Commons Attribution-ShareAlike 4.0
   International License. To view a copy of this license, visit
   http://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative
   Commons, PO Box 1866, Mountain View, CA 94042, USA.

.. _gen_linear_solver_algos:

Linear solvers
==============

.. warning::
   Some capabilities may require the :ref:`installation <installation>` of |g| with :ref:`all its features <dependencies>`
   and some others may depend on :ref:`plugins <plugins>`.

.. warning::
   All the features of the wrapped libraries may not be exposed through |g|.


.. note::
   The algorithm settings can be passed to a function of the form

   .. code-block:: python

      function(..., settings_model: Base | None = None, **settings: Any)

   either one by one:

   .. code-block:: python

      function(..., setting_name_1=setting_name_1, setting_name_2=setting_name_2, ...)

   or using the argument name ``"settings"`` and the Pydantic model associated with the algorithm:

   .. code-block:: python

      settings = AlgorithmSettings(setting_name_1=setting_name_1, setting_name_2=setting_name_2, ...)
      function(..., settings_model=settings)





.. raw:: html

    <style>
    th {
      cursor: pointer;
    }

    th, td {
      text-align: center;
    }

    tr:hover {
        border: 1px solid black;
    }

    tr:nth-child(even) {
      background-color: #f2f2f2
    }
    </style>

    <script>
    function sortTable(n) {
      var table, rows, switching, i, x, y, shouldSwitch, dir, switchcount = 0;
      table = document.getElementById("myTable");
      switching = true;
      dir = "asc";
      while (switching) {
        switching = false;
        rows = table.rows;
        for (i = 1; i < (rows.length - 1); i++) {
          shouldSwitch = false;
          x = rows[i].getElementsByTagName("TD")[n];
          y = rows[i + 1].getElementsByTagName("TD")[n];
          if (dir == "asc") {
            if (x.innerHTML.toLowerCase() > y.innerHTML.toLowerCase()) {
              shouldSwitch= true;
              break;
            }
          } else if (dir == "desc") {
            if (x.innerHTML.toLowerCase() < y.innerHTML.toLowerCase()) {
              shouldSwitch = true;
              break;
            }
          }
        }
        if (shouldSwitch) {
          rows[i].parentNode.insertBefore(rows[i + 1], rows[i]);
          switching = true;
          switchcount ++;
        } else {
          if (switchcount == 0 && dir == "asc") {
            dir = "desc";
            switching = true;
          }
        }
      }
    }
    </script>



.. _BICG_options:

BICG
----



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import BICG_Settings`



BI-Conjugate Gradient



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.bicg.html.





































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


.. raw:: html

    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


.. raw:: html

    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _BICGSTAB_options:

BICGSTAB
--------



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import BICGSTAB_Settings`



Bi-Conjugate Gradient STABilized



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.bicgstab.html.





































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


.. raw:: html

    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


.. raw:: html

    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _CG_options:

CG
--



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import CG_Settings`



Conjugate Gradient



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.cg.html.





































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


.. raw:: html

    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


.. raw:: html

    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _CGS_options:

CGS
---



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import CGS_Settings`



Conjugate Gradient Squared



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.cgs.html.





































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


.. raw:: html

    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


.. raw:: html

    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _DEFAULT_options:

DEFAULT
-------



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import LGMRES_Settings`



Default solver (LGMRES)



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.lgmres.html.



































































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**inner_m** : *<class 'int'>, optional*

Number of inner GMRES iterations per each outer iteration.


By default it is set to 30.


.. raw:: html

    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


.. raw:: html

    </li>

    <li>

**outer_k** : *<class 'int'>, optional*

Number of vectors to carry between inner GMRES iterations.


By default it is set to 3.


.. raw:: html

    </li>

    <li>

**outer_v** : *list[tuple[numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]], numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]]]], optional*

List of tuples `(v, Av)` used to augment the Krylov subspace.


By default it is set to [].


.. raw:: html

    </li>

    <li>

**prepend_outer_v** : *<class 'bool'>, optional*

Whether to put `outer_v` augmentation vectors before Krylov iterates.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


.. raw:: html

    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**store_outer_Av** : *<class 'bool'>, optional*

Whether A @ v should be additionnaly stored in `outer_v` list.


By default it is set to True.


.. raw:: html

    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _GCROT_options:

GCROT
-----



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import GCROT_Settings`



Generalized Conjugate Residual with Optimal Truncation



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.gcrotmk.html.



































































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**CU** : *typing.Optional[collections.abc.Iterable[tuple[numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]], numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]]]]], optional*

List of tuples `(c, u)` required to form the matrices C and U.

If ``None`` start from empty matrices.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**discard_C** : *<class 'bool'>, optional*

Whether to discard the C-vectors at the end.


By default it is set to True.


.. raw:: html

    </li>

    <li>

**k** : *<class 'int'>, optional*

Number of vectors to carry between inner FGMRES iterations.

If ``None`` use the same value as ``m``.


By default it is set to 30.


.. raw:: html

    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**m** : *<class 'int'>, optional*

Number of inner FGMRES iterations per each outer iteration.


By default it is set to 30.


.. raw:: html

    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


.. raw:: html

    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


.. raw:: html

    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**truncate** : *typing.Literal['oldest', 'smallest'], optional*

The vectors from the previous cycle to drop.


By default it is set to oldest.


.. raw:: html

    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _GMRES_options:

GMRES
-----



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import GMRES_Settings`



Generalized Minimum RESidual



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.gmres.html.











































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


.. raw:: html

    </li>

    <li>

**restart** : *<class 'int'>, optional*

Number of iterations between restarts.


By default it is set to 20.


.. raw:: html

    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


.. raw:: html

    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>



.. _LGMRES_options:

LGMRES
------



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import LGMRES_Settings`



Loose Generalized Minimum RESidual



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.lgmres.html.



































































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


.. raw:: html

    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


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    </li>

    <li>

**inner_m** : *<class 'int'>, optional*

Number of inner GMRES iterations per each outer iteration.


By default it is set to 30.


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    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


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    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


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    </li>

    <li>

**outer_k** : *<class 'int'>, optional*

Number of vectors to carry between inner GMRES iterations.


By default it is set to 3.


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    </li>

    <li>

**outer_v** : *list[tuple[numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]], numpy.ndarray[typing.Any, numpy.dtype[numpy.number[typing.Any]]]]], optional*

List of tuples `(v, Av)` used to augment the Krylov subspace.


By default it is set to [].


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    </li>

    <li>

**prepend_outer_v** : *<class 'bool'>, optional*

Whether to put `outer_v` augmentation vectors before Krylov iterates.


By default it is set to False.


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    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


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    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


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    </li>

    <li>

**store_outer_Av** : *<class 'bool'>, optional*

Whether A @ v should be additionnaly stored in `outer_v` list.


By default it is set to True.


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    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


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    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


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    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


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    </li>

    </ul>
    </dd>
    </dl>



.. _TFQMR_options:

TFQMR
-----



Module: :class:`gemseo.algos.linear_solvers.scipy_linalg.scipy_linalg`


:code:`from gemseo.settings.linear_solvers import BaseSciPyLinalgSettingsBase`



Transpose-Free Quasi-Minimal Residual



More details about the algorithm and its settings on https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.tfqmr.html.





































































.. raw:: html

    <dl class="field-list simple">
    <dt class="field-odd">Optional settings</dt>
    <dd class="field-odd"><ul class="simple">


    <li>

**atol** : *<class 'float'>, optional*

The absolute tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 0.0.


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    </li>

    <li>

**callback** : *typing.Optional[typing.Annotated[collections.abc.Callable, WithJsonSchema(json_schema={}, mode=None)]], optional*

The user-supplied function to call after each iteration.

It is called as callback(xk), where xk is the current solution vector.
If ``None``, no function is called.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**M** : *typing.Union[scipy.sparse.linalg._interface.LinearOperator, numpy.ndarray, scipy.sparse._base.sparray, NoneType], optional*

The preconditioner approximating the inverse of A.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**maxiter** : *<class 'int'>, optional*

Maximum number of iterations.


By default it is set to 1000.


.. raw:: html

    </li>

    <li>

**rtol** : *<class 'float'>, optional*

The relative tolerance.

Algorithm stops if norm(b - A @ x) <= max(rtol*norm(b), atol).


By default it is set to 1e-12.


.. raw:: html

    </li>

    <li>

**save_when_fail** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**store_residuals** : *<class 'bool'>, optional*

Whether to store the residual norms at each iterations.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**use_ilu_precond** : *<class 'bool'>, optional*

Whether to use an incomplete LU factorization as preconditioner.


By default it is set to False.


.. raw:: html

    </li>

    <li>

**x0** : *typing.Optional[numpy.ndarray], optional*

Starting guess for the solution.

If ``None``, start from a matrix of zeros.


By default it is set to None.


.. raw:: html

    </li>

    </ul>
    </dd>
    </dl>