.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/optimization/test_functions/plot_01_rosenbrock_banana.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_test_functions_plot_01_rosenbrock_banana.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_optimization_test_functions_plot_01_rosenbrock_banana.py:


Rosenbrock's banana function: a narrow curved valley
==========================================================

:math:`f(x, y) = 100(y-x^2)^2 + (1-x)^2` has its minimum :math:`f=0` at
:math:`(1, 1)`, at the end of a banana-shaped valley along the parabola
:math:`y = x^2`. The walls are steep, but the floor is almost flat, so a
method without curvature information finds the valley quickly and then
crawls along it.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.optimization import GradientDescentLineSearch, rosenbrock, rosenbrock_grad, rosenbrock_hess








.. GENERATED FROM PYTHON SOURCE LINES 19-23

Ill-conditioning at the minimum
-------------------------------
The Hessian at :math:`(1, 1)` has eigenvalues about 2500 times apart:
the curvature across the valley dwarfs the curvature along it.

.. GENERATED FROM PYTHON SOURCE LINES 23-29

.. code-block:: Python


    eigs = np.linalg.eigvalsh(rosenbrock_hess(np.array([1.0, 1.0])))
    print(f"Hessian eigenvalues at (1, 1): {eigs.round(3)}, condition number {eigs[-1] / eigs[0]:.0f}")
    t = 0.0
    print(f"on the floor f({t}, {t**2}) = {rosenbrock(np.array([t, t**2])):.1f}; 0.5 above it f({t}, {t**2 + 0.5}) = {rosenbrock(np.array([t, t**2 + 0.5])):.1f}")





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

 .. code-block:: none

    Hessian eigenvalues at (1, 1): [3.990000e-01 1.001601e+03], condition number 2508
    on the floor f(0.0, 0.0) = 1.0; 0.5 above it f(0.0, 0.5) = 26.0




.. GENERATED FROM PYTHON SOURCE LINES 30-32

Gradient descent crawls along the valley floor
----------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-60

.. code-block:: Python


    x0 = np.array([-1.2, 1.0])
    gd = GradientDescentLineSearch(tol=1e-6, max_iter=5000).minimize(rosenbrock, rosenbrock_grad, x0)
    print(f"gradient descent from {x0}: {gd.iterations} iterations, converged = {gd.converged}, x = {gd.x.round(4)}")

    xs = np.linspace(-2.0, 2.0, 300)
    ys = np.linspace(-1.0, 3.0, 300)
    X, Y = np.meshgrid(xs, ys)
    Z = 100.0 * (Y - X**2) ** 2 + (1.0 - X) ** 2

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(11, 4.5))
    cs = ax1.contourf(X, Y, np.log10(1.0 + Z), levels=30, cmap="viridis")
    fig.colorbar(cs, ax=ax1, label=r"$\log_{10}(1 + f)$")
    ax1.plot(xs, xs**2, "w--", lw=1, label=r"valley floor $y = x^2$")
    ax1.plot(*gd.path.T, ".-", color="tab:orange", ms=2, lw=0.8, label="gradient descent")
    ax1.plot(1.0, 1.0, "r*", ms=14, label="minimum (1, 1)")
    ax1.set_ylim(-1.0, 3.0)
    ax1.set_xlabel("x")
    ax1.set_ylabel("y")
    ax1.legend(loc="lower right", fontsize=8)
    ax1.set_title("Rosenbrock's banana function (1960)")

    gap = np.array([rosenbrock(x) for x in gd.path])
    ax2.semilogy(gap)
    ax2.set_xlabel("iteration")
    ax2.set_ylabel(r"$f(x_k)$")
    ax2.set_title("Fast descent into the valley, then a slow crawl")
    fig.tight_layout()



.. image-sg:: /api/gallery/optimization/test_functions/images/sphx_glr_plot_01_rosenbrock_banana_001.png
   :alt: Rosenbrock's banana function (1960), Fast descent into the valley, then a slow crawl
   :srcset: /api/gallery/optimization/test_functions/images/sphx_glr_plot_01_rosenbrock_banana_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    gradient descent from [-1.2  1. ]: 5000 iterations, converged = False, x = [0.999  0.9981]





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

   **Total running time of the script:** (0 minutes 0.370 seconds)


.. _sphx_glr_download_api_gallery_optimization_test_functions_plot_01_rosenbrock_banana.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_rosenbrock_banana.ipynb <plot_01_rosenbrock_banana.ipynb>`

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

      :download:`Download Python source code: plot_01_rosenbrock_banana.py <plot_01_rosenbrock_banana.py>`

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

      :download:`Download zipped: plot_01_rosenbrock_banana.zip <plot_01_rosenbrock_banana.zip>`


.. only:: html

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

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