.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/optimization/least_squares/plot_01_levenberg_marquardt.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_optimization_least_squares_plot_01_levenberg_marquardt.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_optimization_least_squares_plot_01_levenberg_marquardt.py:


Levenberg-Marquardt: fitting a damped oscillation
=======================================================

Fits a four-parameter damped cosine to noisy data. The residuals depend
nonlinearly on the decay rate and frequency, so the fit needs an
iterative method; Levenberg-Marquardt blends gradient descent (far from
the solution) with Gauss-Newton (close to it).

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.optimization import levenberg_marquardt








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Noisy data from a known model
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-32

.. code-block:: Python


    rng = np.random.default_rng(0)
    t = np.linspace(0.0, 10.0, 120)
    true = np.array([2.0, 0.3, 1.5, 0.4])  # amplitude, decay, frequency, phase


    def model(p, t):
        return p[0] * np.exp(-p[1] * t) * np.cos(p[2] * t + p[3])


    y = model(true, t) + 0.05 * rng.normal(size=t.size)








.. GENERATED FROM PYTHON SOURCE LINES 33-35

Fit from a rough initial guess
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-49

.. code-block:: Python


    x0 = [1.0, 0.1, 1.3, 0.0]
    result = levenberg_marquardt(lambda p: model(p, t) - y, x0)
    print(f"true parameters:   {true}")
    print(f"fitted parameters: {result.x.round(4)}")
    print(f"residual cost 0.5*||r||^2 = {result.cost:.4f} after {result.nfev} evaluations")

    fig, ax = plt.subplots()
    ax.plot(t, y, ".", color="0.5", label="data")
    ax.plot(t, model(x0, t), "--", label="initial guess")
    ax.plot(t, model(result.x, t), label="Levenberg-Marquardt fit")
    ax.set_xlabel("t")
    ax.legend()
    ax.set_title("Nonlinear least squares (Levenberg 1944, Marquardt 1963)")



.. image-sg:: /api/gallery/optimization/least_squares/images/sphx_glr_plot_01_levenberg_marquardt_001.png
   :alt: Nonlinear least squares (Levenberg 1944, Marquardt 1963)
   :srcset: /api/gallery/optimization/least_squares/images/sphx_glr_plot_01_levenberg_marquardt_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    true parameters:   [2.  0.3 1.5 0.4]
    fitted parameters: [2.0297 0.3011 1.5011 0.4084]
    residual cost 0.5*||r||^2 = 0.1332 after 8 evaluations

    Text(0.5, 1.0, 'Nonlinear least squares (Levenberg 1944, Marquardt 1963)')




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.031 seconds)


.. _sphx_glr_download_api_gallery_optimization_least_squares_plot_01_levenberg_marquardt.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_levenberg_marquardt.ipynb <plot_01_levenberg_marquardt.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_levenberg_marquardt.py <plot_01_levenberg_marquardt.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_levenberg_marquardt.zip <plot_01_levenberg_marquardt.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_