{
  "cells": [
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "%matplotlib inline"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "\nFit Specifying a Function to Compute the Jacobian\n=================================================\n\nSpecifying an analytical function to calculate the Jacobian can speed-up the\nfitting procedure.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import matplotlib.pyplot as plt\nimport numpy as np\n\nfrom lmfit import Minimizer, Parameters\n\n\ndef func(pars, x, data=None):\n    a, b, c = pars['a'], pars['b'], pars['c']\n    model = a * np.exp(-b*x) + c\n    if data is None:\n        return model\n    return model - data\n\n\ndef dfunc(pars, x, data=None):\n    a, b = pars['a'], pars['b']\n    v = np.exp(-b*x)\n    return np.array([v, -a*x*v, np.ones(len(x))])\n\n\ndef f(var, x):\n    return var[0] * np.exp(-var[1]*x) + var[2]\n\n\nparams = Parameters()\nparams.add('a', value=10)\nparams.add('b', value=10)\nparams.add('c', value=10)\n\na, b, c = 2.5, 1.3, 0.8\nx = np.linspace(0, 4, 50)\ny = f([a, b, c], x)\ndata = y + 0.15*np.random.normal(size=x.size)\n\n# fit without analytic derivative\nmin1 = Minimizer(func, params, fcn_args=(x,), fcn_kws={'data': data})\nout1 = min1.leastsq()\nfit1 = func(out1.params, x)\n\n# fit with analytic derivative\nmin2 = Minimizer(func, params, fcn_args=(x,), fcn_kws={'data': data})\nout2 = min2.leastsq(Dfun=dfunc, col_deriv=1)\nfit2 = func(out2.params, x)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Comparison of fit to exponential decay with/without analytical derivatives\nto model = a*exp(-b*x) + c\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "print('''\n\"true\" parameters are: a = %.3f, b = %.3f, c = %.3f\n\n==============================================\nStatistic/Parameter|   Without   | With      |\n----------------------------------------------\nN Function Calls   |   %3i       |   %3i     |\nChi-square         |   %.4f    |   %.4f  |\n   a               |   %.4f    |   %.4f  |\n   b               |   %.4f    |   %.4f  |\n   c               |   %.4f    |   %.4f  |\n----------------------------------------------\n''' % (a, b, c,\n       out1.nfev, out2.nfev,\n       out1.chisqr, out2.chisqr,\n       out1.params['a'], out2.params['a'],\n       out1.params['b'], out2.params['b'],\n       out1.params['c'], out2.params['c']))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "and the best-fit to the synthetic data (with added noise) is the same for\nboth methods:\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "plt.plot(x, data, 'ro')\nplt.plot(x, fit1, 'b')\nplt.plot(x, fit2, 'k')\nplt.show()"
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.7.6"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}