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

    Click :ref:`here <sphx_glr_download_examples_example_fit_with_derivfunc.py>` to download the full example code
.. rst-class:: sphx-glr-example-title

.. _sphx_glr_examples_example_fit_with_derivfunc.py:


Fit Specifying a Function to Compute the Jacobian
=================================================

Specifying an analytical function to calculate the Jacobian can speed-up the
fitting procedure.


.. code-block:: default

    import matplotlib.pyplot as plt
    import numpy as np

    from lmfit import Minimizer, Parameters


    def func(pars, x, data=None):
        a, b, c = pars['a'], pars['b'], pars['c']
        model = a * np.exp(-b*x) + c
        if data is None:
            return model
        return model - data


    def dfunc(pars, x, data=None):
        a, b = pars['a'], pars['b']
        v = np.exp(-b*x)
        return np.array([v, -a*x*v, np.ones(len(x))])


    def f(var, x):
        return var[0] * np.exp(-var[1]*x) + var[2]


    params = Parameters()
    params.add('a', value=10)
    params.add('b', value=10)
    params.add('c', value=10)

    a, b, c = 2.5, 1.3, 0.8
    x = np.linspace(0, 4, 50)
    y = f([a, b, c], x)
    data = y + 0.15*np.random.normal(size=x.size)

    # fit without analytic derivative
    min1 = Minimizer(func, params, fcn_args=(x,), fcn_kws={'data': data})
    out1 = min1.leastsq()
    fit1 = func(out1.params, x)

    # fit with analytic derivative
    min2 = Minimizer(func, params, fcn_args=(x,), fcn_kws={'data': data})
    out2 = min2.leastsq(Dfun=dfunc, col_deriv=1)
    fit2 = func(out2.params, x)








Comparison of fit to exponential decay with/without analytical derivatives
to model = a*exp(-b*x) + c


.. code-block:: default

    print('''
    "true" parameters are: a = %.3f, b = %.3f, c = %.3f

    ==============================================
    Statistic/Parameter|   Without   | With      |
    ----------------------------------------------
    N Function Calls   |   %3i       |   %3i     |
    Chi-square         |   %.4f    |   %.4f  |
       a               |   %.4f    |   %.4f  |
       b               |   %.4f    |   %.4f  |
       c               |   %.4f    |   %.4f  |
    ----------------------------------------------
    ''' % (a, b, c,
           out1.nfev, out2.nfev,
           out1.chisqr, out2.chisqr,
           out1.params['a'], out2.params['a'],
           out1.params['b'], out2.params['b'],
           out1.params['c'], out2.params['c']))





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

 Out:

 .. code-block:: none


    "true" parameters are: a = 2.500, b = 1.300, c = 0.800

    ==============================================
    Statistic/Parameter|   Without   | With      |
    ----------------------------------------------
    N Function Calls   |    44       |    14     |
    Chi-square         |   0.8974    |   0.8974  |
       a               |   2.4327    |   2.4327  |
       b               |   1.1946    |   1.1946  |
       c               |   0.7669    |   0.7669  |
    ----------------------------------------------





and the best-fit to the synthetic data (with added noise) is the same for
both methods:


.. code-block:: default

    plt.plot(x, data, 'ro')
    plt.plot(x, fit1, 'b')
    plt.plot(x, fit2, 'k')
    plt.show()



.. image:: /examples/images/sphx_glr_example_fit_with_derivfunc_001.png
    :class: sphx-glr-single-img


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

 Out:

 .. code-block:: none

    /Users/Newville/Codes/lmfit-py/examples/example_fit_with_derivfunc.py:81: UserWarning: Matplotlib is currently using agg, which is a non-GUI backend, so cannot show the figure.
      plt.show()





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

   **Total running time of the script:** ( 0 minutes  0.078 seconds)


.. _sphx_glr_download_examples_example_fit_with_derivfunc.py:


.. only :: html

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



  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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


.. only:: html

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

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