..
   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>`.

.. 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>



.. _BDF_options:

BDF
---



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


:code:`from gemseo.settings.ode import BDF_Settings`



Implicit multi-step variable-order (1 to 5) method based on a backward differentiation formula for the derivative approximation



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













































.. raw:: html

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


    <li>

**atol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *typing.Optional[typing.Annotated[float, Gt(gt=0)]], optional*

The initial step size.

If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**jac_sparsity** : *typing.Optional[gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[float]]], optional*

Sparsity structure of the Jacobian matrix.


By default it is set to None.


.. raw:: html

    </li>

    <li>

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

The maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**rtol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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



.. _DOP853_options:

DOP853
------



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


:code:`from gemseo.settings.ode import DOP853_Settings`



Explicit Runge-Kutta method of order 8



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







































.. raw:: html

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


    <li>

**atol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *typing.Optional[typing.Annotated[float, Gt(gt=0)]], optional*

The initial step size.

If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

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

The maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**rtol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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



.. _LSODA_options:

LSODA
-----



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


:code:`from gemseo.settings.ode import LSODA_Settings`



Adams/BDF method with automatic stiffness detection/switching



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

























































.. raw:: html

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


    <li>

**atol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *typing.Optional[typing.Annotated[float, Gt(gt=0)]], optional*

The initial step size.

If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**lband** : *typing.Optional[int], optional*

The lower boundary of the bandwidth of the Jacobian.

The SciPy documentation does not mention what happens if ``None``.


By default it is set to None.


.. raw:: html

    </li>

    <li>

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

The maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

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

Minimum allowed step size.


By default it is set to 0.


.. raw:: html

    </li>

    <li>

**rtol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

    <li>

**uband** : *typing.Optional[int], optional*

The upper boundary of the bandwidth of the Jacobian.

The SciPy documentation does not mention what happens if ``None``.


By default it is set to None.


.. raw:: html

    </li>

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



.. _RK23_options:

RK23
----



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


:code:`from gemseo.settings.ode import RK23_Settings`



Explicit Runge-Kutta method of order 3(2)



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







































.. raw:: html

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


    <li>

**atol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *typing.Optional[typing.Annotated[float, Gt(gt=0)]], optional*

The initial step size.

If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

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

The maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**rtol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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



.. _RK45_options:

RK45
----



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


:code:`from gemseo.settings.ode import RK45_Settings`



Explicit Runge-Kutta method of order 5(4)



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







































.. raw:: html

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


    <li>

**atol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *typing.Optional[typing.Annotated[float, Gt(gt=0)]], optional*

The initial step size.

If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

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

The maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**rtol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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



.. _Radau_options:

Radau
-----



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


:code:`from gemseo.settings.ode import Radau_Settings`



Implicit Runge-Kutta method of the Radau IIA type of order 5



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













































.. raw:: html

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


    <li>

**atol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The absolute tolerance.


By default it is set to 1e-06.


.. raw:: html

    </li>

    <li>

**first_step** : *typing.Optional[typing.Annotated[float, Gt(gt=0)]], optional*

The initial step size.

If ``None``, let the algorithm choose.


By default it is set to None.


.. raw:: html

    </li>

    <li>

**jac_sparsity** : *typing.Optional[gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[float]]], optional*

Sparsity structure of the Jacobian matrix.


By default it is set to None.


.. raw:: html

    </li>

    <li>

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

The maximum allowed step size.


By default it is set to inf.


.. raw:: html

    </li>

    <li>

**rtol** : *typing.Union[typing.Annotated[float, Gt(gt=0)], gemseo.utils.pydantic_ndarray._NDArrayPydantic[typing.Any, numpy.dtype[typing.Annotated[float, Gt(gt=0)]]]], optional*

The relative tolerance.


By default it is set to 0.001.


.. raw:: html

    </li>

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