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

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

.. _sphx_glr_examples_example_fit_with_inequality.py:


Fit Using Inequality Constraint
===============================

Sometimes specifying boundaries using ``min`` and ``max`` are not sufficient,
and more complicated (inequality) constraints are needed. In the example below
the center of the Lorentzian peak is constrained to be between 0-5 away from
the center of the Gaussian peak.

See also: https://lmfit.github.io/lmfit-py/constraints.html#using-inequality-constraints


.. code-block:: default

    import matplotlib.pyplot as plt
    import numpy as np

    from lmfit import Minimizer, Parameters, report_fit
    from lmfit.lineshapes import gaussian, lorentzian


    def residual(pars, x, data):
        model = (gaussian(x, pars['amp_g'], pars['cen_g'], pars['wid_g']) +
                 lorentzian(x, pars['amp_l'], pars['cen_l'], pars['wid_l']))
        return model - data









Generate the simulated data using a Gaussian and Lorentzian line shape:


.. code-block:: default

    np.random.seed(0)
    x = np.linspace(0, 20.0, 601)

    data = (gaussian(x, 21, 6.1, 1.2) + lorentzian(x, 10, 9.6, 1.3) +
            np.random.normal(scale=0.1, size=x.size))








Create the fitting parameters and set an inequality constraint for ``cen_l``.
First, we add a new fitting  parameter ``peak_split``, which can take values
between 0 and 5. Afterwards, we constrain the value for ``cen_l`` using the
expression to be ``'peak_split+cen_g'``:


.. code-block:: default

    pfit = Parameters()
    pfit.add(name='amp_g', value=10)
    pfit.add(name='amp_l', value=10)
    pfit.add(name='cen_g', value=5)
    pfit.add(name='peak_split', value=2.5, min=0, max=5, vary=True)
    pfit.add(name='cen_l', expr='peak_split+cen_g')
    pfit.add(name='wid_g', value=1)
    pfit.add(name='wid_l', expr='wid_g')

    mini = Minimizer(residual, pfit, fcn_args=(x, data))
    out = mini.leastsq()
    best_fit = data + out.residual








Performing a fit, here using the ``leastsq`` algorithm, gives the following
fitting results:


.. code-block:: default

    report_fit(out.params)





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

 Out:

 .. code-block:: none

    [[Variables]]
        amp_g:       21.2722837 +/- 0.05138760 (0.24%) (init = 10)
        amp_l:       9.46504191 +/- 0.05445416 (0.58%) (init = 10)
        cen_g:       6.10496394 +/- 0.00334614 (0.05%) (init = 5)
        peak_split:  3.52163535 +/- 0.01004618 (0.29%) (init = 2.5)
        cen_l:       9.62659929 +/- 0.01066173 (0.11%) == 'peak_split+cen_g'
        wid_g:       1.21434950 +/- 0.00327315 (0.27%) (init = 1)
        wid_l:       1.21434950 +/- 0.00327315 (0.27%) == 'wid_g'
    [[Correlations]] (unreported correlations are < 0.100)
        C(amp_g, wid_g)      =  0.620
        C(amp_g, peak_split) =  0.380
        C(peak_split, wid_g) =  0.344
        C(amp_g, amp_l)      = -0.295
        C(amp_l, cen_g)      = -0.276
        C(amp_g, cen_g)      =  0.194
        C(amp_l, wid_g)      = -0.165
        C(cen_g, wid_g)      =  0.155




and figure:


.. code-block:: default

    plt.plot(x, data, 'bo')
    plt.plot(x, best_fit, 'r--', label='best fit')
    plt.legend(loc='best')
    plt.show()



.. image:: /examples/images/sphx_glr_example_fit_with_inequality_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_inequality.py:61: 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  0.094 seconds)


.. _sphx_glr_download_examples_example_fit_with_inequality.py:


.. only :: html

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



  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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


.. only:: html

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

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