..
   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_ode_algos:

Ordinary differential equations 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>`.

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



.. 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;
            }
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            if (x.innerHTML.toLowerCase() < y.innerHTML.toLowerCase()) {
              shouldSwitch = true;
              break;
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          }
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        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>



.. _BDF_options:

BDF
---



Module: :class:`gemseo.algos.ode.lib_scipy_ode`


ODE solver implemented in the SciPy library.



More details about the algorithm and its options on https://docs.scipy.org/doc/scipy/reference/generated/BDF.html.


.. raw:: html

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
    <link rel="stylesheet" href="../_static/css/gemseo.css" type="text/css" />












































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


    <li>

**atol** : *float | NDArray[float], optional*

Absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *float | None, optional*

Initial step size. If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**max_step** : *float, optional*

Maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**min_step** : *float, optional*

Minimum allowed step for the "LSODA" method.


By default it is set to 0.


.. raw:: html

    </li>

    <li>

**rtol** : *float | NDArray[float], optional*

Relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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



.. _DOP853_options:

DOP853
------



Module: :class:`gemseo.algos.ode.lib_scipy_ode`


ODE solver implemented in the SciPy library.



More details about the algorithm and its options on https://docs.scipy.org/doc/scipy/reference/generated/DOP853.html.


.. raw:: html

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
    <link rel="stylesheet" href="../_static/css/gemseo.css" type="text/css" />












































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


    <li>

**atol** : *float | NDArray[float], optional*

Absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *float | None, optional*

Initial step size. If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**max_step** : *float, optional*

Maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**min_step** : *float, optional*

Minimum allowed step for the "LSODA" method.


By default it is set to 0.


.. raw:: html

    </li>

    <li>

**rtol** : *float | NDArray[float], optional*

Relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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



.. _LSODA_options:

LSODA
-----



Module: :class:`gemseo.algos.ode.lib_scipy_ode`


ODE solver implemented in the SciPy library.



More details about the algorithm and its options on https://docs.scipy.org/doc/scipy/reference/generated/LSODA.html.


.. raw:: html

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
    <link rel="stylesheet" href="../_static/css/gemseo.css" type="text/css" />
























































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


    <li>

**atol** : *float | NDArray[float], optional*

Absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *float | None, optional*

Initial step size. If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**lband** : *int | None, optional*

Lower boundary of the bandwidth for the "LSODA" method.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**max_step** : *float, optional*

Maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**min_step** : *float, optional*

Minimum allowed step for the "LSODA" method.


By default it is set to 0.


.. raw:: html

    </li>

    <li>

**rtol** : *float | NDArray[float], optional*

Relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

    <li>

**uband** : *int | None, optional*

Upper boundary of the bandwidth for the "LSODA" method.


By default it is set to None.


.. raw:: html

    </li>

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



.. _RK23_options:

RK23
----



Module: :class:`gemseo.algos.ode.lib_scipy_ode`


ODE solver implemented in the SciPy library.



More details about the algorithm and its options on https://docs.scipy.org/doc/scipy/reference/generated/RK23.html.


.. raw:: html

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
    <link rel="stylesheet" href="../_static/css/gemseo.css" type="text/css" />












































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


    <li>

**atol** : *float | NDArray[float], optional*

Absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *float | None, optional*

Initial step size. If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**max_step** : *float, optional*

Maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**min_step** : *float, optional*

Minimum allowed step for the "LSODA" method.


By default it is set to 0.


.. raw:: html

    </li>

    <li>

**rtol** : *float | NDArray[float], optional*

Relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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



.. _RK45_options:

RK45
----



Module: :class:`gemseo.algos.ode.lib_scipy_ode`


ODE solver implemented in the SciPy library.



More details about the algorithm and its options on https://docs.scipy.org/doc/scipy/reference/generated/RK45.html.


.. raw:: html

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
    <link rel="stylesheet" href="../_static/css/gemseo.css" type="text/css" />












































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


    <li>

**atol** : *float | NDArray[float], optional*

Absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *float | None, optional*

Initial step size. If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**max_step** : *float, optional*

Maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**min_step** : *float, optional*

Minimum allowed step for the "LSODA" method.


By default it is set to 0.


.. raw:: html

    </li>

    <li>

**rtol** : *float | NDArray[float], optional*

Relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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



.. _Radau_options:

Radau
-----



Module: :class:`gemseo.algos.ode.lib_scipy_ode`


ODE solver implemented in the SciPy library.



More details about the algorithm and its options on https://docs.scipy.org/doc/scipy/reference/generated/Radau.html.


.. raw:: html

    <link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
    <link rel="stylesheet" href="../_static/css/gemseo.css" type="text/css" />












































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


    <li>

**atol** : *float | NDArray[float], optional*

Absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *float | None, optional*

Initial step size. If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**max_step** : *float, optional*

Maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**min_step** : *float, optional*

Minimum allowed step for the "LSODA" method.


By default it is set to 0.


.. raw:: html

    </li>

    <li>

**rtol** : *float | NDArray[float], optional*

Relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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