.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "examples/ex02_numeric_variation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_examples_ex02_numeric_variation.py>`
        to download the full example code.

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

.. _sphx_glr_examples_ex02_numeric_variation.py:


Numeric calculus of variation
-----------------------------
Each Tensor has four attributes: the (real) tensor array and the (hyper-dual)
variational arrays. To obtain the :math:`12` - component of the gradient and the
:math:`1223` - component of the hessian, a tensor has to be created with the appropriate
small-changes of the tensor components (dual arrays).

.. GENERATED FROM PYTHON SOURCE LINES 9-37

.. code-block:: Python

    import numpy as np

    import tensortrax as tr
    from tensortrax import Tensor, Δ, Δδ, f, δ
    from tensortrax.math import trace

    δF_12 = np.array(
        [
            [0, 1, 0],
            [0, 0, 0],
            [0, 0, 0],
        ],
        dtype=float,
    )

    ΔF_23 = np.array(
        [
            [0, 0, 0],
            [0, 0, 1],
            [0, 0, 0],
        ],
        dtype=float,
    )

    x = np.eye(3) + np.arange(9).reshape(3, 3) / 10
    F = Tensor(x=x, δx=δF_12, Δx=ΔF_23, Δδx=None)
    I1_C = trace(F.T @ F)








.. GENERATED FROM PYTHON SOURCE LINES 38-40

The function as well as the gradient and hessian components are accessible with
helpers.

.. GENERATED FROM PYTHON SOURCE LINES 40-44

.. code-block:: Python

    ψ = f(I1_C)
    P_12 = δ(I1_C)
    A_1223 = Δδ(I1_C)








.. GENERATED FROM PYTHON SOURCE LINES 45-47

To obtain full gradients and hessians of scalar-valued functions in one function call,
``tensortrax`` provides helpers (decorators) which handle the multiple function calls.

.. GENERATED FROM PYTHON SOURCE LINES 47-53

.. code-block:: Python

    fun = lambda F: trace(F.T @ F)

    func = tr.function(fun)(x)
    grad = tr.gradient(fun)(x)
    hess = tr.hessian(fun)(x)








.. GENERATED FROM PYTHON SOURCE LINES 54-55

For tensor-valued functions, use ``jacobian()`` instead of ``gradient()``.

.. GENERATED FROM PYTHON SOURCE LINES 55-58

.. code-block:: Python

    fun = lambda F: F.T @ F
    jac = tr.jacobian(fun)(x)








.. GENERATED FROM PYTHON SOURCE LINES 59-60

Evaluate the gradient- as well as the hessian-vector(s)-product.

.. GENERATED FROM PYTHON SOURCE LINES 60-63

.. code-block:: Python

    gvp = tr.gradient_vector_product(fun)(x, δx=x)
    hvp = tr.hessian_vector_product(fun)(x, δx=x)
    hvsp = tr.hessian_vectors_product(fun)(x, δx=x, Δx=x)








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

   **Total running time of the script:** (0 minutes 0.003 seconds)


.. _sphx_glr_download_examples_ex02_numeric_variation.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: ex02_numeric_variation.ipynb <ex02_numeric_variation.ipynb>`

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

      :download:`Download Python source code: ex02_numeric_variation.py <ex02_numeric_variation.py>`

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

      :download:`Download zipped: ex02_numeric_variation.zip <ex02_numeric_variation.zip>`


.. only:: html

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

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