.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/calculus/autodiff/plot_01_reverse_mode_gradients.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_calculus_autodiff_plot_01_reverse_mode_gradients.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_calculus_autodiff_plot_01_reverse_mode_gradients.py:


Reverse-mode automatic differentiation for multivariable gradients
=========================================================================

A minimal backpropagation-style engine: one forward pass builds a
computation graph, then one backward pass computes the gradient with
respect to every input in a single traversal -- the technique underlying
modern deep-learning frameworks, applied here to a small analytic
function.

.. GENERATED FROM PYTHON SOURCE LINES 13-17

.. code-block:: Python

    import math

    from mathematicskit.calculus.systems.autodiff import Variable, gradient








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Gradient of a simple multivariable function
---------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-31

.. code-block:: Python


    f = lambda x, y: x * x * y + y.sin() * x
    x0, y0 = 2.0, 0.5
    grad = gradient(f, [x0, y0])
    print(f"gradient at ({x0}, {y0}): {grad}")

    # Closed-form check: df/dx = 2xy + sin(y), df/dy = x^2 + x*cos(y)
    expected_dx = 2.0 * x0 * y0 + math.sin(y0)
    expected_dy = x0**2 + x0 * math.cos(y0)
    print(f"closed-form gradient:      [{expected_dx}, {expected_dy}]")





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

 .. code-block:: none

    gradient at (2.0, 0.5): [2.479425538604203, 5.7551651237807455]
    closed-form gradient:      [2.479425538604203, 5.7551651237807455]




.. GENERATED FROM PYTHON SOURCE LINES 32-34

Shared subexpressions accumulate gradient contributions correctly
------------------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 34-39

.. code-block:: Python


    x = Variable(3.0)
    y = x * x + x  # used twice: y = x^2 + x, dy/dx = 2x + 1
    y.backward()
    print(f"d(x^2 + x)/dx at x=3: {x.grad} (expected {2 * 3.0 + 1})")




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

 .. code-block:: none

    d(x^2 + x)/dx at x=3: 7.0 (expected 7.0)





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

   **Total running time of the script:** (0 minutes 0.001 seconds)


.. _sphx_glr_download_api_gallery_calculus_autodiff_plot_01_reverse_mode_gradients.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_reverse_mode_gradients.ipynb <plot_01_reverse_mode_gradients.ipynb>`

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

      :download:`Download Python source code: plot_01_reverse_mode_gradients.py <plot_01_reverse_mode_gradients.py>`

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

      :download:`Download zipped: plot_01_reverse_mode_gradients.zip <plot_01_reverse_mode_gradients.zip>`


.. only:: html

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

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