Optimization_using_SciPy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "dfd173a2-1839-4795-89cf-8bd9ae1afc38",
   "metadata": {},
   "source": [
    "# Optimization using SciPy"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c488c628-0795-4473-9ea1-5c985cda15d3",
   "metadata": {},
   "source": [
    "## Simple Optimization (Minimum and Maximum)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "278ed706-7126-4895-8650-881cfbfd15c1",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "c12d79db-4c49-495f-9bf8-dfa47d478abb",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "0a257080-b8ef-4de3-9f57-b571175065e2",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d782dd6d-cece-4882-a84a-49da86fbba6b",
   "metadata": {},
   "source": [
    "- Define a function: $f(x)=x^2 + 10sin(x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "55233f78-0094-4e13-9791-824143c4a8e5",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x**2 + 10*np.sin(x)  # Objective Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ed8aeb5b-a13d-4c7d-adde-1d4dca30357f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-10. ,  -9.6,  -9.2,  -8.8,  -8.4,  -8. ,  -7.6,  -7.2,  -6.8,\n",
       "        -6.4,  -6. ,  -5.6,  -5.2,  -4.8,  -4.4,  -4. ,  -3.6,  -3.2,\n",
       "        -2.8,  -2.4,  -2. ,  -1.6,  -1.2,  -0.8,  -0.4,   0. ,   0.4,\n",
       "         0.8,   1.2,   1.6,   2. ,   2.4,   2.8,   3.2,   3.6,   4. ,\n",
       "         4.4,   4.8,   5.2,   5.6,   6. ,   6.4,   6.8,   7.2,   7.6,\n",
       "         8. ,   8.4,   8.8,   9.2,   9.6,  10. ])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.linspace(-10, 10, 51)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "12abff68-31b5-4fd3-9dca-3e82a5115858",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([105.44021111,  93.90326781,  82.41110086,  71.59082807,\n",
       "        62.01401092,  54.10641753,  48.08080328,  43.90332136,\n",
       "        41.29886649,  39.79450795,  38.79415498,  37.67266638,\n",
       "        35.87454656,  33.00164609,  28.87602074,  23.56802495,\n",
       "        17.38520443,  10.82374143,   4.4901185 ,  -0.99463181,\n",
       "        -5.09297427,  -7.43573603,  -7.88039086,  -6.53356091,\n",
       "        -3.73418342,   0.        ,   4.05418342,   7.81356091,\n",
       "        10.76039086,  12.55573603,  13.09297427,  12.51463181,\n",
       "        11.1898815 ,   9.65625857,   8.53479557,   8.43197505,\n",
       "         9.84397926,  13.07835391,  18.20545344,  25.04733362,\n",
       "        33.20584502,  42.12549205,  51.18113351,  59.77667864,\n",
       "        67.43919672,  73.89358247,  79.10598908,  83.28917193,\n",
       "        86.86889914,  90.41673219,  94.55978889])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = f(x)\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "d757ceb7-7cb3-407d-a48d-a2b9eb588314",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 400x320 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5,4), dpi=80)\n",
    "plt.plot(x, y)\n",
    "plt.grid('on')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "bb39950d-5dde-46a5-98a4-dfa5804be38f",
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.optimize as opt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d01fef9e-5c78-4930-8ff2-3f3c03e67d00",
   "metadata": {},
   "source": [
    "- Find a minimum of the function f(x) using the BFGS(Broyden-Fletcher-Goldfarb-Shanno) method, starting from a initial point x0 = 0. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "43022f06-4469-4299-b7b4-130fe5e7c65e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: -7.945823\n",
      "         Iterations: 5\n",
      "         Function evaluations: 12\n",
      "         Gradient evaluations: 6\n"
     ]
    }
   ],
   "source": [
    "x0 = 0\n",
    "[xopt, fopt, gopt, Bopt, func_calls, grad_calls, warnflg] = \\\n",
    "        opt.fmin_bfgs(f,  # Objective function to be minimized.\n",
    "        x0=x0,  # Initial guess.\n",
    "        maxiter=2000,  # Maximum number of iterations to perform.\n",
    "        full_output=True)  # Return fopt, func_calls, grad_calls, and warnflag in addition to xopt.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "4db61e9c-2eb3-4ef9-bf3e-887842039daa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-1.30644012]), -7.945823375615215)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xopt, fopt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "fe27dc93-cbf1-4a9d-aa1b-524620b03086",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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NtTqKz7hdxCISA9wOPGxcA6nGmFIR6QGMAxa6Fs0B+ohIlrvr9CUR4doxaVScqtdHKCnlZRWn6vloZxlT+ieS1j3a6jg+44k94v5ABfCQiKwTkZUicjHQBzhsjGkEcJV0ARBwd+64ZnQqIpCzocjqKEoFtbc3FdPQZJgzro/VUXxK3D0bQETGAOuBW4wxL4nIaGAZcAXwkjFmYLNl1wIPGmM+afEe84H5Z76OiYlJzcnJ6XCW2tpaoqKiOveNnMOjG5vYf8Lwu6l2YsLPPW7lzSwd5U9ZwL/yaJbWWZXlkfWNlJyCP0y1E2EXS7O0pbN5Zs6cWWyMaf00EGOMWx9AEtAE2JtNywPmAFVAmGuaAKVA1rneMzU11XRGbm5up17XHv9cV2jSFywxL6/KtzxLR/lTFmP8K49maZ0VWQqOnTLpC5aY77+6wfIsZ9PZPECRaaPz3B6aMMYcBT4GLgMQkX5AP+ALYAMw17Xoda4g+9xdpxVmDkuhS7idxTo8oZRXvLf1MADfGBFcT2huD0+dNXE38CMR2Qq8BdxljCkG7gLuEpE9wIPAPA+tz+diIsOYNSyFDQWVHDx6yuo4SgWdJVsOExsVxnkDgve+w23xSBEbYw4YY6YbY4YbY0YaY3Jc03cbYyYbYwYYY8YZY7Z6Yn1WuXaMc3hH94qV8qz8o6fYWnyCS4ekEBlmtzqOz+mVdR0wuX8iKXFRLN5QjMOhlzwr5SlLzwxLjOxlcRJraBF3gN0mXDM6leLKGtbmV1gdR6mgsWTLYbp1CWdq/9AblgAt4g67bozzah8dnlDKM/YfqWbn4SpmDk0hIiw0Kyk0v2s3ZPeMZURaN97bWkpNvd4wXil3Ld0S2sMSoEXcKdeOTqW6rpEPd5RaHUWpgLdkSwkJMRFMzgyNO621Rou4E64c2Zswm/DP9To8oZQ79pSdZE9ZNTOHpQT1U5rPJXS/czckdo3kkiE9Wbn3KLtKq6yOo1TAWnJmWGJ46A5LgBZxp91zYX8AHv90v8VJlApMxhiWbCkhqWskE0N4WAK0iDttRFo85w9IZsmWEg4cqbY6jlIBZ1fpSQ4cOcXlw1Owh8gN4NuiReyG712UhTHwxHLdK1aqo5ZsKQH+9RScUKZF7IbxGQlM6JfAmxuLKTp+2uo4SgUM57DEYXrGRTI+I8HqOJbTInbT9y7KotFheGrFAaujKBUwtpdUcejYaS4f3itknkt3NlrEbpqWlcTItG68tq6Q8qpaq+MoFRBytznPwZ81TIclQIvYbSLCvRdlU9/o4JmVulesVHvkbi8lqWskY9O7Wx3FL2gRe8DFg3owKCWWhasLqDhVb3UcpfzavvKT7Cuv5rKhPUP+bIkztIg9wGYTvjs9i5qGJp7/4qDVcZTya2eGJWYOS7E4if/QIvaQy4f3IjMphhe+yOdETYPVcZTyW7nbS4mLCmNSiF/E0ZwWsYfYbcI9F/bnZF0jT63Q84qVak1hxWm2FVcxY0hPwkP43hIt6ZbwoGtGpzKkVxxPrtjP/hP6BA+lWvpgu2tYYqgOSzSnRexB4XYbj90wijC7jed3NnGqrtHqSEr5ldxtpURH2Dl/QLLVUfyKFrGHDegZy4MzB3GkBn69dKfVcZTyG+VVtawvOM70gT2ICg+9B4SejRaxF9w6JYNB3YVX1xbw8c4yq+Mo5Rc+3FGGMXCZni3xb7SIvcBmE24dZCMuKowFOVs4Wl1ndSSlLPfB9lIi7DamD9RhiZa0iL2ke5Twq2uGcbS6np8s3ooxevBOha7K0/Ws2n+MadlJxEaFWx3H72gRe9HVo1K5cmRvlu0o4411+lglFbo+2llOo8Po2RJt0CL2sl9fPYyUuCh+8e521uVXWB1HKUvkbivFbhNmDOlpdRS/pEXsZd2iw/nLjaMxwLefXfPVeZRKhYpTdY18tvcIE/slkBATYXUcv6RF7AMT+iWw6M5JdI0M456F61m4+pDVkZTymeW7j1Df6NB7S5yFFrGPjEiLJ+eeKaR1j+a/3trGox/u1gN4KiTkuv4XeOkQLeK2aBH7UEZSDDn3TGF4ajf+8sk+HszZSmOTw+pYSnlNbUMTn+wsY3TfeFK6RVkdx295rIhFZJ6IGBG5xvV1DxHJFZG9IrJNRM731LoCWXJsJIvunMT5A5J5bV0hNz6zmvWHjlsdSymvWLn3KKfqm5ilwxJn5ZEiFpEM4DvA6maTHwFWG2OygXnAKyKiJxACMZFhPHfLOG6enM76Q8e57okvue2FPLYVn7A6mlIe9f62w4A+Eulc3C5iEbEBzwLfA5pfQjYHeBLAGJMHlAAXuLu+YBFut/HLq4exbP4FfGNELz7ZVc43/vo5//mP9ewtO2l1PKXcVt/oYNmOMoalxtEnIdrqOH5N3D1gJCIPALHGmJ+JyHLgMWAlUGKMiWy23OtArjHm7628x3xg/pmvY2JiUnNycjqcpba2lqgo/xiH6miWomrDOwcdbD5qEGBgd2F8D2F0shAT7t7jZPxpu4B/5dEsrfNElm3HHPx1i4NrMm3MSu/8Pp8/bRfofJ6ZM2cWG2PSWp1pjOn0BzAMWAWEu75eDlwDJAJ1LZZ9HbitPe+bmppqOiM3N7dTr/OGzmbZWHDc3PlSnsl+6D2TvmCJ6f+TpWbe82vN4g2Fpqqm3qdZvMWf8miW1nkiy4/f2GzSFywx+8tPWp7FkzqbBygybXReWIdr/evOAzKAvSICkAI8DfwMaBSRFGPMmSsYMoACN9cX9Eb1ieepm8ZRVdvAh9vLeHdzCSv2HOGTXeVEhdu4fFgv5ozvw8R+Cbi2uVJ+p7HJwYc7ShmUEktmcler4/g9t4rYGPME8MSZr88MTRhj3hKRicDdwM9FZDyQCqxwZ32hJC4qnNlj05g9No1j1XW8v62UxRuKWLyxmMUbi8lIjOZb4/owe2waPeP8579tSgGsOVjB8dMN3DIlw+ooAcHdPeKzWQC8LCJ7gXpgrjFGn6rZCYldI5k7KZ25k9LZV36S19cVsXhDEb//YDd//HA3V49K5aHLB5McG3nuN1PKB86cLXH5cD1boj08WsTGmAubfV4GXOrJ91eQ1SOWhy4fzI8uG8jHO8tZuPoQb24s5uOdZSyYNYgbx/fFZtMhC2WdJofhg+1lZCbHkN1DhyXaQ6+sC1Dhdhszh6Xw8u0TeHLuGKIjwnj4zW1c9+SX7CipsjqeCmHrDx3nyMk6Lh/WS49jtJMWcYATEWYO68VH91/A7dP6sbmwkiv/9jn/s3QHtQ1NVsdTIejMsITe5Kf9tIiDRNfIMH76jSG8c+80hqV245mVB/nOS+u0jJVPORyG3G2l9E2IZmjvOKvjBAwt4iAzLLUbi++Zwo0T+rJy71HueHEd9U16lzflG5uLKjl8opZZw1J0WKIDvHnWhLKI3Sb8zzXDsNtg4eoCjhwVLp7RSHSE/nUr73p/m/OyAR2W6BjdIw5SNpvwq6uHccvkdHZXGuY9n8epukarY6kgZozh/W2H6d0tilF94q2OE1C0iIOYiPDzq4ZyUZqw5mAF857Po1rLWHnJ9pIqCitquEyHJTpMizjIiQhzsmzcMa0fa/MruO35PBr0ZvTKC5Zu1VtedpYWcQgQER6+YjC3TE5nbX4Ff/l4r9WRVJBxOAzvbCqhd7coxqV3tzpOwNEiDhHOMh7C0N5xPP7pPtblV1gdSQWRdYeOU1xZw5WjeuuVnZ2gRRxCIsJs/PmGUYTbbfzw9U2crNVbfyjPeGtTMQDfHJ1qcZLApEUcYrJ6xPJfVwymsKKGn7+zw+o4KgjUNzpYuuUwg1JiGZSiF3F0hhZxCJo7KZ3pA5PJ2VDE0i2HrY6jAtzy3eWcqGng6lG6N9xZWsQhSET43eyRJMZE8NCbWyk9UWt1JBXA3t5UAsBVo3pbnCRwaRGHqOTYSB65bgQnahq4/41NOBx6GbTquKraBj7aWcbEfgmkxnexOk7A0iIOYZcM6cmNE/ryxb5jvLgq3+o4KgDlbiulrtHBNXqQzi1axCHup98YTGp8F/60bA+Vp+utjqMCzNubiomwO5+lqDpPizjERUeE8eOZA6mqbeTxT/dZHUcFkLKqWr7cf4wLBybTLTrc6jgBTYtYceWI3gxLjePFLw9RWHHa6jgqQLy7uQRj9NxhT9AiVthswkOzBlPf5OAPH+62Oo4KEG9uLCY2Kozpg3pYHSXgaRErAKZkJTF9YDJvbypha9EJq+MoP7ev/CTbS6qYNSyFqHC71XECnhax+sqDswZjE/jNezsxRk9nU217a6Pz3OFr9CIOj9AiVl8ZmBLL7LFprDpwjOV7jlgdR/kpYwxvbSomJS6KiZmJVscJClrE6mvmXzKQqHAbj7y3iya9yEO1YtX+YxQdr+GqUb2x653WPEKLWH1NSrco7piWye6yk+SsL7I6jvJDL67KRwT+Y0Jfq6MEDS1i9W/uuiCThJgI/rhsNzX1TVbHUX6k6Phplu0o48IByWQkxVgdJ2hoEat/ExsVzn0XZVFWVcerawusjqP8yMLVBTgM3Dwlw+ooQUWLWLXqhgl9SeoaybMrD1DfqM+4U1Db0MSivAL6JcVwQXay1XGCihaxalVUuJ3bpmVQcqKWt11PX1Ch7Z1NJVSebuCmSen6OCQP0yJWbZo7KZ3YyDCeXLFfb5MZ4owxvPBlPtERdmaPS7M6TtBxu4hFJEpE3hKRPSKyWUSWiUiWa14PEckVkb0isk1Eznc/svKVuKhw5k5OZ/+RUyzbWWZ1HGWh9YeOs+NwFdeNSSMuSm/w42me2iN+GhhojBkJvA0865r+CLDaGJMNzANeERH9Wwwg86ZmEBFm4/+W79er7ULYC1/mA3DLlHRrgwQpt4vYGFNrjHnP/Ou3dDWQ4fp8DvCka7k8oAS4wN11Kt/pERvFnHFpbC6sZNWBY1bHURYoq6old1spU7MSyeoRa3WcoCSe3ssRkZeBCuCXQIkxJrLZvNeBXGPM31u8Zj4w/8zXMTExqTk5OR1ed21tLVFRUZ2N7lHBlOVIjeGnq5sY1F34wSj3b/ASTNvGk/w1yzsHmlh6yHDPMBujkn1/WMmftgt0Ps/MmTOLjTGtD7AbYzz2ATwErAKigUSgrsX814HbzvU+qamppjNyc3M79TpvCLYs9726waQvWGK2FFb6RR5P0SytO5OltqHRjP3Vh2bqIx+bxiaHpVn8RWfzAEWmjc7z2D9vIvIAcC0wyxhz2hhzDGgUkZRmi2UAeoVAALr7gv4APLFCn+IRSt7fWsrR6npumpSu95XwIo8UsWto4UbgEmNMZbNZbwB3u5YZD6QCKzyxTuVbg3vFMX1gMu9vK+XAkWqr4ygfcDgMT392gKhwG9eP72N1nKDmidPX0oA/AvHApyKySUTWuGYvAKaIyF7gBWCuMabB3XUqa/zn9CyMgadWHLA6ivKBnA1F7Dhcxc2TM4iPjrA6TlALc/cNjDFFQKv/ZzHGlAGXursO5R/GZyQwLr07b24s5v7LBtAj1n8OoCjPqm00/P6D3STERHDvRVlWxwl6emWd6pA7zutHfZODhat1qD+YfVDgoPxkHT+8ZIBewOEDWsSqQy4ZkkKfhC4sXH2I2ga9RWYwKq6sYVmhYUDPrtyoY8M+oUWsOsRuE+ZN6UfFqXre2qg3AwpGv8vdRYMDHr5iCGF2rQhf0K2sOmzO+D7ERobx3OcH9bLnILOx4DhvbyphWIJwwQC91aWvaBGrDusaGcYNE/qwt7yaz/YetTqO8hBjDL9asgO7TZidpdXgS7q1VafcMiUDm8CzK/VUtmDx7pbDbCioZO7EvvSK0Ys3fEmLWHVKWvdoZg3vxcq9R9ldetLqOH6vocnB7tKT7D9S7Zf3dq5taOKR93YSFxXGD2YMsDpOyHH7PGIVum6f1o+lWw7z988P8tvZI6yO4zcamhysy3fev3en62NvWTX1Tc5HTsVE2BnauxvDUrsxPC2O4anx9E+27kGcxhh+/s52Sk7U8l9XDKZ7jF684WtaxKrTxvTtzpi+8by5qZgfzRxIUtfIc78oyK3Lr+DhN7exu+xf/0tIiYtialYig3vF0dDkYGvxCbYXV7E2v+KrZUakdWNCnIOLmxw+P1PhsY/2siivkPOyk7hFHwpqCS1i5Zbbp2Xy3Vc28I/VBXx/RrbVcSxz/FQ9v83dxaK8QiLsNu6+oD/nZScxuFccCa3sYTochoKK02wtPsGX+4/x5sYithQ5yP3Dcm6f1o854/oQE+n9X8+Fqw/x54/3Mjy1G0/MHUu4nq5mCS1i5ZbLhvYkNb4LL6/O564LMokKd/9+xYHEGMM/1xfxv+/vouJUPVOzEvnV1cPITO561tfZbEJGUgwZSTFcObI3P7psID97+WO+ONLEL97dwWMf7eXmyencMS2TbtHeubItd1sp//32NtITo3l+3ni6+qD4Vev0nz/lljC7jXlTMzhaXc87m0qsjuNTx0/Vc+Mzq/nRP7dgE/jzDaNYePvEc5ZwaxJiIvhGPxtfPngRv75mGN2jw/nrJ/s473ef8H/L91FT79mrGPPyK7hv0UYSYiJ46bYJOqxkMS1i5bY54/vQNTKMZ1Ye8MszAryh8nQ9c59bw+oDFdw4oS8fz7+Qq0elIuLeaV9R4XbmTkrn4/sv5NE5I4nrEs7vcndz/u8/5eVV+dQ3OtzOvrv0JLe/kEe4TXj+1gmkJ1p3oFA5aRErt8VFhfPtiX3ZW17Np7vL217QGDi0it7ly+HQKufXAehETQM3PbeW7SVVzL9kAP977XCPDx/YbcK1Y9L45P4L+eXVQzEGfvr2di5+dDn/WHOIqtqO3022tqGJF7/M59vPruZ0fRNP3jSW4WndPJpbdY4OCimPmDe1H3//4iBPrdjPxdEHoOIAJGRC30kgApUF8PK1UHmIwcYG+c9CfDrctBji+1odv92qahu4+bk1bC0+wfcvzua+i717gDIizMbNkzOYPTaN57/I56kV+3n4zW38askOLh/eiznj+jCxX8JZ98RP1TXyjzWHePqzgxytriOpawR/vXE052XrJcz+QotYeURKtyhuHWLn+t234XjxKLawCGhqcJbt3BxYeB1UHATT+K8fuoqDzunfXessaz93sraBW/6+ls1FJ7h3ehY/8OFZItERYXx3eha3TMlg6ZYSXssrZPGGYhZvKCYjMZpvjk6jd3wUUeF214eNqHA7aw9W8OzKAxw/3UBKXBQ/u3IIN4zvS5eI0Dqo6u+0iJVnGMMDR36CTcqwORxQ7/qvc8VBeOEKqC4D09jiNY1wPB8KVkP6ZJ9H7ojqukZufT6PjQWV3HNhf+6/dIDb48Gd0TUyjOvH9+X68X3ZV36S19cVsXhDEX/6aE+br0nr3oUHLhvI7LFpRIZpAfsjLWLlGQWriTxZCNLiYJJphKoSsLcxhmoLdw5j+HERNzY5uOvldaw/dJy7zs/kx5cNtKSEW8rqEctDlw/mR5cNZGNBJdV1DdQ2OKhtaPrqz+TYSGYOS9Hzg/2cFrHyjIoDzrJtqv/3ebZwcLRxcMnR4BxL9mN/+HAPX+w7xn9M7MuDswb5RQk3F263MaFfgtUxlBv0n0nlGQmZzjHh1pgmiO0F0uLffQmD7hnOA3p+6oPtpTy5Yj+j+sTzsyuH+F0Jq+CgRaw8o+8k54G51so2oR/c+p7zT3sEjbZIsEdAYibMXey3B+ryj57igdc30z06nP/79hgdX1Veo0MTyjNEnKeivXwtpvIQNU02wmgkLKEftrmLIb4P3JsHBavZ+dk7DD//qn+d2uaHauqbuHvheqrrG3nptgn0ju9idSQVxLSIlefE94V785CC1WzKy+PR9Q1cMfqbzIt3PYBSBNInU9KjiuF+fHDOGMPDb21lV+lJ7r9kgJ5vq7xOhyaUZ7nKdszV3+Vg9Aie/Tyfhib3L8v1pVfXOs/RnT4wme9Oz7I6jgoBWsTKK6LC7dw6JYPiyhpy1hdZHafdthad4OfvbCetexf+dP0obDb/HDpRwUWLWHnNrVMzSIiJ4LGP9lLb4Nm7h3nDqbpG7lu0EYAn544lPlqfVKF8Q4tYeU1sVDjfnZ5FaVUtL686ZHWcc/rFu9s5ePQUC2YNYliq3gxH+Y4WsfKqb0/sS2p8Fx5fvq9TdwzzlaVbDvP6uiLOH5DMPH1ckPIxLWLlVVHhdr4/I5vK0w0889kBq+O0qriyhp8s3kJiTAR/+NYIHRdWPuf1IhaRbBH5UkT2iEieiAz19jqVf7l2dCpZPbry3OcHOXKyzuo4X+Mwhh8u2kRVbSO//9YIesRGWR1JhSBf7BE/BTxtjBkA/BZ4wQfrVH4kzG7jgUsHcrq+icc/3Wd1nK/JPWRYm1/BrVMyuGhQT6vjqBDl1SIWkR7AOGCha1IO0EdE9OTMEHPZ0J6M7BPPP9Yc4miNfzyZY0PBcd7NdzCwZywPzhpkdRwVwry9R9wHOGyM80a0xhgDFACB80gG5REiwoLLBtLQZHj3oPUXeJyoaeAHizZhE/jLjaND7unTyr+I8eJzw0RkLPCKMWZgs2lrgQeNMZ80mzYfmH/m65iYmNScnJwOr6+2tpaoKP8Y49MsrXtsUxO7jjv46fgwUrtac1DMYQxPbHWw5Zhhdr8mLsnwjycY+9Pfk2ZpW2fzzJw5s9gYk9bqTGOM1z6AHkAVEOb6WoBSIOtsr0tNTTWdkZub26nXeYNmad3mwuMmfcESc9Nza4zD4bAkw98+2WvSFywx3391g3n//fctydAaf/p70ixt62weoMi00XleHZowxpQDG4C5rknXucL41xEb5TMj0uKZnCJ8tucIC9cU+Hz9K/ce4Y8f7mZQSiy/uXa43l9Y+QVfnDVxF3CXiOwBHgTm+WCdyo9dn22jT0IXfr1kB/vKT/psvcWVNdz36kZiIsJ4Yu5YoiP05oPKP3i9iI0xu40xk40xA4wx44wxW729TuXfuoQJf5ozioYmB/e9uom6Ru/fh6KusYn/XLie46cb+OOckfRLivH6OpVqL72yTlliXEYC916UzY7DVTz6YdtPIPaUX7y7g81FJ/jPC/tz6dAUr69PqY7QIlaWue+iLEb1iefplQf4ct9Rr63ntbwCXllTwNSsRO6/dOC5X6CUj2kRK8uE2W38+YZRRIfbmf/6ZipPt/IEaDe9nlfIg4u3khrfhb/cMBq73kdC+SEtYmWp9MQYfn7VUEqrannoza1nTnv0iJdW5fPjnC2kxndh0Z2TSOzqH+cLK9WSFrGy3OyxaVw+PIX3tpbyxw/34HC4X8ZPf7af/357O5lJMbxx92T6JER7IKlS3qFFrCwnIvzmm8MZlhrH3z7dx50vr+dkJ+9dbIzhLx/v5Tfv7WJAz64sumsSvbrpE5iVf9MiVn4hPjqCf949hW+OTuWjnWVc8/gXHDhS3aH3MMbw+w928+iyPQztHceiOyfrbS1VQNAiVn4jKtzOo3NG8l9XDObg0VNc/bcv+GRX2Tlf19DkYPGGImY+tpL/W76f0X3jeeU7k0iI0WfOqcCglxYpvyIi3HFeJoNS4rj31Q3c/uI67jw/k6n9k8hMjqF3ty5fPUHjVF0ji/IKeW7lAUpO1BITYeeOaf34wSUD6BqpP9oqcOhPq/JL07KTePfeaXznpXU8teIAT61wPmYpKtxGRmIMfROiWXOwghM1DSR1jeBHlw1k7sR0ukWHW5xcqY7TIlZ+q09CNG/fO5V1+cc5cKSa/UdOceDoKfaXV7NsZxnpCdH8eOZArhuTpvcTVgFNi1j5tcgwO1OzkpialfS16XWNTUTYbXr3NBUUtIhVQIoM0z1gFTz0rAmllLKYFrFSSllMi1gppSymRayUUhbTIlZKKYtpESullMW0iJVSymJaxEopZTHx5BMRPEVE6oAjnXhpV6Bj9070Hs3SNn/Ko1lap1na1tk8ycaYVh8T45dF3FkiUmSMSbM6B2iWs/GnPJqldZqlbd7Io0MTSillMS1ipZSyWLAV8aNWB2hGs7TNn/JoltZplrZ5PE9QjRErpVQgCrY9YqWUCjhaxEopZbGAKmIRuUJE1otInYg81mKeTUT+KiL7RWSfiNx7lvfJFpEvRWSPiOSJyFAPZHtcRDY1+6gVkfvaWHa5iBxstuwP3V1/i/f/uYgcafb+/zjLsh7fFi3e/z4R2SYiW0Vki4jMPcuyXtku7f0eReR2Ednr+hl6RkQ8+gA8EYkSkbdcOTaLyDIRyWpluQwRaWrx89Tfk1lc68kXkd3N1nF9G8t5e7sktvhe94hIo4gktFjOa9tFRP7i2h5GREY1m97u3w+3tpMxJmA+gAHASODXwGMt5t0MfAzYgQTgEDC0jff5BLjV9flsIM/DOVOAGiCljfnLgWu8uJ1+3nL7nGVZb2+Li4Furs/7AEeB/r7cLu35HoF+QInr706Ad4DvejhHFHA5/zo2cy+wvJXlMoBKb/18NFtPPjDqHMt4fbu0ss4HgHd9uV2A84G0ltukvb8f7m6ngNojNsbsMcZsBhpbmX098IwxpskYUwG8BtzYciER6QGMAxa6JuUAfVrbM3HDLcAHxphSD76nx/liWxhjPjbGnHB9XgiU4ixkn+jA9zgbeMcYU2qcv1lP0srPjzuMMbXGmPdc7w+wGme5+DOvb5dW3A485+V1fI0x5jNjTFHzaR38/XBrOwVUEZ9DX5x7wWfku6a11Ac4bIxpBHBttII2lu2s2zj3D9Ijrv+uvyYimR5c9xnfcv339xMRmd7GMr7YFl8RkRlAdyDvLIt5eru093ts78+PJ30feLuNeTGu/wpvEJH/FhFvPaTvJdf2fk5EkluZ79PtIiJTcP6MLGljEV9tF+jY74db28mvilhEVonI0TY+fLYX5U42ETkPiAXeO8vb3WSMGQSMAFbS9g9dZ7M8CWQYY0YCPwVeE5H0jn7PHspyZrnhwPPA9caYU228nVvbJZCIyENAFvCTVmYfBlKNMeOBGcB5wP1eiHG+MWYEMAbnkNGLXlhHR90OvHSm/Frw1XbxOb96irMxZrIbLy8A0oFVrq8zXNNaKgR6iUiYMaZRRATnv1ytLduZbLcDLxpjms7yXoWuPw3wNxH5g4gkGmOOtWcFHdlOxpgvRGQjzv9iHWoxu1PboqNZRGQIzlK9zRjz+Vney63t0ob2fo8FQPMDPxmtLOMRIvIAcC0wwxhzuuV8Y0wdUO76vEJE/g78B/A7T+YwxhS4/mwQ58HvPa0s5svt0hWYA4xvbb6vtkszHfn9cGs7+dUesZveAL4jInbX0dbrcY4Tf40xphzYAJw5en8dUGSM2eduABGJwzlW9PezLBMmIj2bfX0dUOZm2bRcR1qzz7OBUcDWlst5c1s0W/9gnP87uNMYs+wsy3llu3Tge8wBrhKRFNcv3N3AInfW3RoRmY9z7PASY0xlG8v0OHPEXUQicZb2Rg/niBGR+GaTbmxjHT7ZLi7XA5uNMbtam+mL7dJcB38/3NtO3jgC6a0PnEfgi4Aq4KTr86tc8+zA48ABYD/w/Wavuwp4ttnXA3HuOe8B1gHDPZTvTmBFK9PHAe+5Po9xrXMrsBnnmR4jPbydXgS2AZuA9cBsX2+LZu+/DDjuynLm4zJfbpe2vkfg2TM/P66vv+P62dmPc4w/3MPbIg0wrvc/sy3WuOb9Erjb9fm1rr+/zcB24K9ApIezZOIssS2ubf42zuEsn2+XZuv5EpjXYppPtgvwFM4+aQTKgH3n+v3w5HbSS5yVUspiwTQ0oZRSAUmLWCmlLKZFrJRSFtMiVkopi2kRK6WUxbSIlVLKYlrESillMS1ipZSymBaxUkpZ7P8Be6O/hs1Qjq0AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 400x320 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5,4), dpi=80)\n",
    "plt.plot(x,y)\n",
    "plt.plot([x0, xopt],[f(x0), fopt], 'o')\n",
    "plt.grid('on')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c420906-6659-41dc-96a2-bdd42b54eacd",
   "metadata": {},
   "source": [
    "- What if we start from x0 = 6 ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "3a4a92a8-ac67-4a35-83b2-77cd9999ecc6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 8.315586\n",
      "         Iterations: 7\n",
      "         Function evaluations: 16\n",
      "         Gradient evaluations: 8\n"
     ]
    }
   ],
   "source": [
    "x0 = 6 \n",
    "[xopt, fopt, gopt, Bopt, func_calls, grad_calls, warnflg] = \\\n",
    "        opt.fmin_bfgs(f,  # Objective function to be minimized.\n",
    "        x0=x0,  # Initial guess.\n",
    "        maxiter=2000,  # Maximum number of iterations to perform.\n",
    "        full_output=True)  # Return fopt, func_calls, grad_calls, and warnflag in addition to xopt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "9a44df29-90f2-4b98-9a78-c5c208d585ff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([3.83746709]), 8.315585579477458)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xopt, fopt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "15ea4fb3-d2b7-41c3-8b02-4a9f60b92626",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 400x320 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5,4), dpi=80)\n",
    "plt.plot(x, y)\n",
    "plt.plot([x0, xopt], [f(x0), fopt], 'o')\n",
    "plt.grid('on')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "39e18abb-6c6c-4db3-98c8-a5b905f3bc85",
   "metadata": {},
   "source": [
    "- Find the global minimum using the basinhopping method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "4fa88ea0-2f6f-4a67-8de7-f92d06abd2fd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "                        fun: -7.9458233756152845\n",
       " lowest_optimization_result:       fun: -7.9458233756152845\n",
       " hess_inv: array([[0.0858477]])\n",
       "      jac: array([1.1920929e-07])\n",
       "  message: 'Optimization terminated successfully.'\n",
       "     nfev: 12\n",
       "      nit: 5\n",
       "     njev: 6\n",
       "   status: 0\n",
       "  success: True\n",
       "        x: array([-1.30644001])\n",
       "                    message: ['requested number of basinhopping iterations completed successfully']\n",
       "      minimization_failures: 0\n",
       "                       nfev: 1218\n",
       "                        nit: 100\n",
       "                       njev: 609\n",
       "                          x: array([-1.30644001])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res = \\\n",
    "    opt.basinhopping(f, \n",
    "                     x0=x0, \n",
    "                     T=1,  # The “temperature” parameter\n",
    "                     stepsize=2)  # Initial step size\n",
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "8124f70f-43e9-4fc8-8dcd-62fa3102f936",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 400x320 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5,4), dpi=80)\n",
    "plt.plot(x, y)\n",
    "plt.plot([x0, res.x], [f(x0), res.fun], 'o')\n",
    "plt.grid('on')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3ce1da10-066e-4015-aaea-9fc0ddcd359a",
   "metadata": {},
   "source": [
    "## Solving Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "feea8649-b195-45ee-b462-0b990e7b7e0d",
   "metadata": {},
   "source": [
    "- Solve the equation $f(x)=x^2 + 10sin(x)=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "574bd99e-0148-4923-be90-80487e004f7a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = 1\n",
    "xopt = opt.fsolve(f, x0)  # Find a root with initial guess x0\n",
    "xopt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "aa019ad7-644c-420b-a1e2-173918ba861e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 400x320 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5,4), dpi=80)\n",
    "plt.plot(x, y)\n",
    "plt.plot([x0, xopt], [f(x0), f(xopt)], 'o')\n",
    "plt.grid('on')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "fff41731-d397-442d-87fe-b8c70da63325",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2.47948183])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = -3\n",
    "xopt = opt.fsolve(f, x0)  # Find a root with initial guess x0\n",
    "xopt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "d86c105b-94fb-4808-bd43-b2cdf405c339",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 400x320 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5, 4), dpi=80)\n",
    "plt.plot(x, y)\n",
    "plt.plot([x0, xopt], [f(x0), f(xopt)], 'o')\n",
    "plt.grid('on')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e25bfd3b-cb87-41b8-a25c-e5cb759245ec",
   "metadata": {},
   "source": [
    "## Curve Fitting"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa392f06-bd48-4cc3-a2e4-d556aeda641f",
   "metadata": {},
   "source": [
    "- Curve fitting is a procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the offsets('the residuals') of the points from the curve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "df4cfafa-c5ad-4bbb-9a29-3488d79dcda9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-10. ,  -9.6,  -9.2,  -8.8,  -8.4,  -8. ,  -7.6,  -7.2,  -6.8,\n",
       "        -6.4,  -6. ,  -5.6,  -5.2,  -4.8,  -4.4,  -4. ,  -3.6,  -3.2,\n",
       "        -2.8,  -2.4,  -2. ,  -1.6,  -1.2,  -0.8,  -0.4,   0. ,   0.4,\n",
       "         0.8,   1.2,   1.6,   2. ,   2.4,   2.8,   3.2,   3.6,   4. ,\n",
       "         4.4,   4.8,   5.2,   5.6,   6. ,   6.4,   6.8,   7.2,   7.6,\n",
       "         8. ,   8.4,   8.8,   9.2,   9.6,  10. ])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "a210fe86-5a30-4aeb-b7b0-d3d89e0c1e0e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([105.44021111,  93.90326781,  82.41110086,  71.59082807,\n",
       "        62.01401092,  54.10641753,  48.08080328,  43.90332136,\n",
       "        41.29886649,  39.79450795,  38.79415498,  37.67266638,\n",
       "        35.87454656,  33.00164609,  28.87602074,  23.56802495,\n",
       "        17.38520443,  10.82374143,   4.4901185 ,  -0.99463181,\n",
       "        -5.09297427,  -7.43573603,  -7.88039086,  -6.53356091,\n",
       "        -3.73418342,   0.        ,   4.05418342,   7.81356091,\n",
       "        10.76039086,  12.55573603,  13.09297427,  12.51463181,\n",
       "        11.1898815 ,   9.65625857,   8.53479557,   8.43197505,\n",
       "         9.84397926,  13.07835391,  18.20545344,  25.04733362,\n",
       "        33.20584502,  42.12549205,  51.18113351,  59.77667864,\n",
       "        67.43919672,  73.89358247,  79.10598908,  83.28917193,\n",
       "        86.86889914,  90.41673219,  94.55978889])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "dd93c471-2487-44b5-bef5-c58cf3e2c169",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 400x320 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5, 4), dpi=80)\n",
    "plt.plot(x, y, 'x')\n",
    "plt.grid('on')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "8e8da9b3-fdae-4eb1-bf51-e6fb726585e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([104.79007818,  92.84737189,  91.16546145,  70.19916045,\n",
       "        60.49216398,  49.65453058,  46.66886681,  39.63909326,\n",
       "        40.88801752,  40.33926062,  33.7432906 ,  38.1071468 ,\n",
       "        27.43961785,  23.26491219,  34.9461266 ,  24.18860581,\n",
       "        13.87410455,  15.83893475,   2.2110732 ,  -0.68390617,\n",
       "        -3.14719515,  -6.30062629, -12.23529856, -10.18447733,\n",
       "        -6.55951127,   1.54848861,  -5.53267414,   3.24199201,\n",
       "        11.62199903,  16.16692355,  11.25225384,  10.87851279,\n",
       "        11.05741697,   4.49652931,  19.13882957,  10.324653  ,\n",
       "        10.87823016,   2.14254584,  22.77515694,  21.65289278,\n",
       "        29.66178561,  44.14429197,  56.83300904,  60.06967082,\n",
       "        63.24926928,  74.9758168 ,  73.44161391,  76.93229318,\n",
       "        90.35069653,  97.74216377,  93.07723233])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_noise = f(x) + 5*np.random.randn(x.size)  # Add some random noise\n",
    "y_noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "ec4ea7fd-41a1-43c6-8299-b15af2d255cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 400x320 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5, 4), dpi=80)\n",
    "plt.plot(x, y, 'x')\n",
    "plt.plot(x, y_noise, 'o')\n",
    "plt.grid('on')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c994b0f1-7921-4c26-9e52-d9d3508fc0ed",
   "metadata": {},
   "source": [
    "- Fit a function $ f2(x)= ax^2+bsin(x)$ by determining the optimal values of the coefficients a and b."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "a0860ab5-4e9e-4fe7-b277-e5df9b972f0a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f2(x, a, b):\n",
    "    return a*x**2 + b*np.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "6ed32a46-3a76-4d5a-851b-125254ffb3ec",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.99232992, 9.62116576])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "guess = [2, 2]  # Initial guess for a and b, respectively\n",
    "params, cov = opt.curve_fit(f2, x, y_noise, guess)\n",
    "params"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d0ae92a5-6b82-4acf-9b0d-3fe2dd382c50",
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b = params[0], para\n",
    "b = params[1]\n",
    "y_fit = f2(x, a, b)\n",
    "y_fit"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [conda env:datascience] *",
   "language": "python",
   "name": "conda-env-datascience-py"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}