.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/optimization/scalar_search/plot_01_golden_section.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_scalar_search_plot_01_golden_section.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_optimization_scalar_search_plot_01_golden_section.py:


Golden-section search: shrinking the bracket
==================================================

Locates the minimum of a unimodal function on an interval. Each
iteration costs one new function evaluation and shrinks the bracket by
the factor :math:`1/\varphi \approx 0.618`.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.optimization import golden_section_search








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Minimize f(x) = x^2 - sin(4x) on [-1, 2]
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-32

.. code-block:: Python



    def f(x):
        return x**2 - np.sin(4.0 * x)


    result = golden_section_search(f, -1.0, 2.0, tol=1e-8)
    print(f"minimizer x = {result.x:.8f}, f(x) = {result.fun:.8f}")
    print(f"{result.iterations} iterations, {result.nfev} function evaluations")

    widths = result.brackets[:, 1] - result.brackets[:, 0]
    print(f"successive width ratio: {widths[1] / widths[0]:.6f} (1/phi = {(np.sqrt(5) - 1) / 2:.6f})")





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

 .. code-block:: none

    minimizer x = 0.34886654, f(x) = -0.86296114
    41 iterations, 43 function evaluations
    successive width ratio: 0.618034 (1/phi = 0.618034)




.. GENERATED FROM PYTHON SOURCE LINES 33-35

The first few brackets, and the width decay
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-48

.. code-block:: Python


    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    xs = np.linspace(-1.0, 2.0, 400)
    ax1.plot(xs, f(xs), color="0.3")
    for i, (a, b) in enumerate(result.brackets[:8]):
        ax1.plot([a, b], [-1.2 - 0.12 * i] * 2, "|-", color="tab:blue")
    ax1.plot(result.x, result.fun, "o", color="tab:red")
    ax1.set_title("Brackets (top to bottom: iterations 0-7)")
    ax2.semilogy(widths, "o-", ms=3)
    ax2.set_xlabel("iteration")
    ax2.set_ylabel("bracket width")
    ax2.set_title("Width shrinks by 1/phi per step (Kiefer, 1953)")
    fig.tight_layout()



.. image-sg:: /api/gallery/optimization/scalar_search/images/sphx_glr_plot_01_golden_section_001.png
   :alt: Brackets (top to bottom: iterations 0-7), Width shrinks by 1/phi per step (Kiefer, 1953)
   :srcset: /api/gallery/optimization/scalar_search/images/sphx_glr_plot_01_golden_section_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.114 seconds)


.. _sphx_glr_download_api_gallery_optimization_scalar_search_plot_01_golden_section.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_golden_section.ipynb <plot_01_golden_section.ipynb>`

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

      :download:`Download Python source code: plot_01_golden_section.py <plot_01_golden_section.py>`

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

      :download:`Download zipped: plot_01_golden_section.zip <plot_01_golden_section.zip>`


.. only:: html

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

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