{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "aquatic-special",
   "metadata": {},
   "outputs": [],
   "source": [
    "# In this Python notebook we attempt to evaluate numerical derivatives of a\n",
    "# list of data.\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "recorded-status",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-10,  -9,  -8,  -7,  -6,  -5,  -4,  -3,  -2,  -1,   0,   1,   2,\n",
       "          3,   4,   5,   6,   7,   8,   9,  10]),\n",
       " array([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First, we introduce the 'diff' function.  diff(x) will take list x and\n",
    "# create a new list whose elements are equal to the difference between\n",
    "# adjacent elements in x.  For example, let's make x a list from -10 to 10 in\n",
    "# steps of 1.\n",
    "xdata = np.arange(-10, 11, 1)\n",
    "dx = np.diff(xdata)\n",
    "xdata, dx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "honest-florist",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "length of x: 21\n",
      "length of dx: 20\n"
     ]
    }
   ],
   "source": [
    "# In this case, dx is a list of ones.  Note that, because it is determined\n",
    "# from a difference, the list dx is one element shorter than the list x.\n",
    "print('length of x:', len(xdata))\n",
    "print('length of dx:', len(dx))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "entertaining-curtis",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Now let's make y = x^2 and plot y vs x.  The result is clearly a\n",
    "# quadratic.\n",
    "y = xdata**2\n",
    "plt.plot(xdata, y, 'go', fillstyle = 'none');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "immune-visit",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "dy/dx: [-19. -17. -15. -13. -11.  -9.  -7.  -5.  -3.  -1.   1.   3.   5.   7.\n",
      "   9.  11.  13.  15.  17.  19.]\n"
     ]
    }
   ],
   "source": [
    "# To evaluate the derivative of y with respect to x, we need to determine\n",
    "# the change in y over the change in x.\n",
    "dydx = np.diff(y)/np.diff(xdata)\n",
    "print('dy/dx:', dydx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "presidential-burst",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "length of x1: 20\n"
     ]
    }
   ],
   "source": [
    "# We should expect dydx vs x to be a straigt line of slope 2.  To generate\n",
    "# the plot, remember that we have to reduce the length of x by 1.\n",
    "x1 = xdata[1:len(xdata)]\n",
    "print('length of x1:', len(x1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "assigned-woman",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The best-fit parameters are: (1) slope = 2.0 and (2) intercept -0.9999999999999997\n"
     ]
    }
   ],
   "source": [
    "# We will fit the data to confirm the slope is about right.\n",
    "def linearFunc(x, slope, intercept):\n",
    "    y = slope*x + intercept\n",
    "    return y\n",
    "from scipy.optimize import curve_fit\n",
    "a_fit, cov = curve_fit(linearFunc, x1, dydx)\n",
    "print('The best-fit parameters are: (1) slope =', a_fit[0], 'and (2) intercept',\\\n",
    "      a_fit[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "difficult-judges",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the numerical derivative and our fit line.\n",
    "plt.plot(x1, dydx, 'o', fillstyle = 'none')\n",
    "fitFcn = np.poly1d(a_fit)\n",
    "plt.plot(x1, fitFcn(x1), 'k-');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "dominant-philip",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Notice from the fit that the slope is indeed 2, but the y-intercept is 1\n",
    "# instead of the expected zero.  This is an artifact of taking derivatives\n",
    "# of a discrete set of the data.  We can improve our results if we reduced\n",
    "# the spacing between the x data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "happy-handling",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Just for fun, let's add some noise to our data.  Let's suppose that the\n",
    "# uncertainty in x is 10% and that dy = 2*x*dx.  'np.random.uniform()' \n",
    "# generates a random number uniformly distributed between 0 and 1.  \n",
    "# You can confirm for yourself that a + (b - a)*np.random.uniform() generates \n",
    "# a random number between a and b.\n",
    "xN = []\n",
    "yN = []\n",
    "for k in range(len(xdata)):\n",
    "    xN = xN + [xdata[k] + 0.1*xdata[k]*(-1 + (1 - (-1))*np.random.uniform())]\n",
    "    yN = yN + [xdata[k]**2 + 2*xdata[k]*0.1*xdata[k]*(-1 + (1 - (-1))*np.random.uniform())]\n",
    "plt.plot(xN, yN, 'go', fillstyle = 'none')\n",
    "dyNdx = np.diff(yN)/np.diff(xN)\n",
    "xN1 = xN[1:len(xN)]\n",
    "plt.figure()\n",
    "plt.plot(xN1, dyNdx, 'o', fillstyle = 'none');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "linear-archives",
   "metadata": {},
   "outputs": [],
   "source": [
    "# As you might have anticipated, taking the ratio of the differences between a\n",
    "# pair of noisey datasets results in even more fluctuations.  Obtaining\n",
    "# clean derivatives from discrete datasets requires precision measurements."
   ]
  }
 ],
 "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": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}