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# 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

    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:

    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'`:

    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:

    report_fit(out.params)

Out:

    [[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:

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

Total running time of the script: ( 0 minutes 0.199 seconds)

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