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# Fit Specifying a Function to Compute the Jacobian

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

    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

    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']))

Out:

    "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:

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

Total running time of the script: ( 0 minutes 0.159 seconds)

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