lagrange_shape_fun.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Polynomial Interpolation using Lagrange shape functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the function to interpolate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "  '''The function to be interpolated\n",
    "\n",
    "  :param x: independent variable\n",
    "  :type x: double or np.array of doubles\n",
    "  :return: the value of f evaluated at x\n",
    "  :rtype: double or np.array\n",
    "  '''\n",
    "  return 1.0/(1.0+x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct the k-th Lagrange shape function from the nodes x coordinates at the x locations (np.array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lagrange_sf(x,k,node_x):\n",
    "  '''Evaluates the generic Lagrange shape function\n",
    "\n",
    "  :param x: the locations where to evaluate the shape function\n",
    "  :type x: double or np.array\n",
    "  :param k: the function to be evaluated from 0 to len(node_x)-1. This is 1.0 at x_k and zero at the other points.\n",
    "  :type k: int\n",
    "  :param node_x: the x locations where the values of f are known. \n",
    "  :type node_x: list or np.array containing double x locations\n",
    "  :raises Exception: Generic exception when k has an invalid range\n",
    "  :return: the Lagrange shape functions evaluated at x.\n",
    "  :rtype: double, list or np.array, depending on x and node_x.\n",
    "  '''\n",
    "  # Polynomial order\n",
    "  n = len(node_x)-1\n",
    "  if(k < 0)or(k > n):\n",
    "    raise Exception(\"ERROR: invalid k\")\n",
    "  else:\n",
    "    # Construct Lagrange shape functions    \n",
    "    res = np.ones(len(x))\n",
    "    for j in range(n+1):\n",
    "      if(j == k):\n",
    "        continue\n",
    "      else:\n",
    "        res *= (x-node_x[j])/(node_x[k]-node_x[j])\n",
    "  return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the $k$-th Lagrange shape function, observing that it is equalt to $1$ at $x_{k}$.\n",
    "It is also equal to $0$ at $x_{i},\\,\\forall\\,i\\ne k$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "node_x = np.array([0,1.0,2.0])\n",
    "node_y = f(node_x)\n",
    "\n",
    "x = np.linspace(0,2,100)\n",
    "\n",
    "plt.plot(node_x,[0,0,0],'ro')\n",
    "plt.plot(node_x,[1,1,1],'bo')\n",
    "plt.plot(x,lagrange_sf(x,2,node_x))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now construct the complete Lagrange interpolating polynomial, as a linear combination of shape functions and values of $f$ at $x_{0},x_{1},\\dots,x_{n}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lagrange_int(x,node_x,node_y):\n",
    "  '''Constructs the Lagrange polynomial interpolant\n",
    "\n",
    "  :param x: the x locations where the interpolanting polynomial needs to be evaluated\n",
    "  :type x: double or np.array\n",
    "  :param node_x: the x coordinates of the known locations. \n",
    "  :type node_x: list or np.array\n",
    "  :param node_y: the y coordinates of the known locations. \n",
    "  :type node_y: list or np.array\n",
    "  :return: the Lagrange polynomial evaluated at x\n",
    "  :rtype: list or np.array\n",
    "  '''\n",
    "  n = len(node_x)-1\n",
    "  res = np.zeros(len(x))\n",
    "  for j in range(n+1):\n",
    "    res += lagrange_sf(x,j,node_x)*node_y[j]\n",
    "  return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare the interpolating polynomial and the original function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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w4OLiAsCyZcuYM2cO/fv3JyYmpsbtAlq35PMZA5k3PpD9Jy9w57+289HuU5TJlHdCWB0J9CboL3/5C6tWrSI1NZWwsDCee+458vLyqm1vNCimDmrPxmeH0NvfjXlrE/j1e7v54VxuA1YthKhvEuhNkFKKBx54gMTERKZOncobb7zBX//615tu187NgY+m9uHNB0JJyS5gXMQu/rHxBwpLZHYkIayBjKFbgaioKAIDA3F1dSUuLg5HR0c6dOhQ4zYX84tZsC6Rzw+m4efuwPyJQQwOsL6Ju4WwNjKGbuUGDRqEq6srALNmzSIwMJB58+Zx5cqVardxdbTjjUkhrJjWD4NSTPlgP8+uOkzmZTnFUYimSgLdyqxYsYJ7772XV199lW7duvH555/XeO76wE4erJ8zmGeGB/Bt3FmG/3Mbn+xNwSQHTYVociTQrYy3tzcrVqxg69atODk5MWnSJFauXFnjNva2Rn4/sjPr5wyhu5cz//dVPPcu3k38mZwGqloIURdkDN2KlZaWEhkZyUMPPYSdnR179uyhY8eOtGrVqtpttNZ8FXOGBd8mciG/mEcH+PP7OzvjZG/bgJULIaojFxYJysrK6NSpE9nZ2bz44ovMmTMHe3v7atvnFJTwxndJfLIvBXfHZrx4V1cmhnrLlaZCWJgcFBUYjUY2bNjA0KFDmTt3Ll26dOGTTz7BZDJV2d7ZwZa/TgxizcxBeLvY87vVsTzw/l45d12IRkwC/TbSpUsX1q5dy+bNm/Hw8GDKlCls2rSpxm2CfVz439ODeO3eHhzLuMxdEbt45esEcq7IZBpCNDYy5HKbMplMbNiwgTFjxqCU4pNPPqFbt2707t272m0u5hfzj++SWLk/FXdHO/40qiu/7u2DwSDDMEI0FBlDFzUqKSkhICCAlJQUJk2axKuvvkrXrl2rbR9/Jod5axM4mHKREB9n5k3oTi9f1wasWIjbl4yhixrZ2toSGxvLX/7yF9atW0f37t2ZOnUqp0+frrJ9kLczn88YwL/uD+FsTiH3vrub36+O4XxuYQNXLoSoTPbQxTUyMjJYuHAhixcvZs+ePYSGhqK1rvbslryiUt7ZmswHO09iY1Q8PbQj0wZ3wN625tv6CiF+GRlyEbV28eLFq7cTeOKJJ7C3t2fu3Lm0a9euyvYp2fn8bV0iGxPO4+3SnBfGdmNsjzZymqMQdUyGXESt/RTmWmvs7OxYsmQJnTp14umnnyY1NfWG9n7ujrw/JYwV0/rR0t6GmSsOMem9PcScvtTQpQtx2zIr0JVSo5VSSUqpZKXU3GraDFVKxSilEpRS2+u2TGEpSikWL15McnIyU6dOZenSpXTs2JHVq1dX2X5gJw++fWYwC+/twansAia+E8WcVYdJu1jQwJULcfu56ZCLUsoIHANGAmnAAeAhrfXRSm1cgN3AaK11qlKqldY6o6bXlSGXpik1NZU33niDuXPn4uXlRWxsLAaDgR49etzQNq+olMXbklm68yQaeHxQe54e1lFuIyDELbjVIZe+QLLW+oTWuhhYBdx9XZuHgS+11qkANwtz0XT5+voSERGBl5cXAC+88ALBwcGMHz+e3bt3X9O2RTMbnhvVla1/HMq44La8t/1H7vj7Vv4bdZLi0qqvUBVC/HLmBLo3UPn8tbSKZZV1BlyVUtuUUgeVUo9W9UJKqelKqWilVHRmZuYvq1g0KsuXL+eVV15h9+7dDBo0iPDw8BuuPvVyac6/7g/lm9nhdGvrxCtfH2XEv7bzdWy63KZXiDpkTqBXdZrC9f8V2gC9gbuAUcBflFKdb9hI6yVa6zCtdZinp8yOYw3c3d156aWXSE1NJSIigrS0tKsTV5eUlFBY+PO56UHezkRO68eyqX1wsDMye+VhJr4bRVRylqXKF8KqmBPoaUDlc9V8gPQq2mzQWudrrbOAHUBI3ZQomgJHR0dmz55NcnIys2bNAmDlypX4+fnx6quvkpFRPgqnlGJol1Z8+8xg/jkphOy8YiYv3ceUD/YRlyb3XxfiVpgT6AeAAKVUe6WUHfAgsPa6NmuAwUopG6WUA9APSKzbUkVTYGNjc/W2vF26dCEsLIx58+bh6+vLE088QVxcHABGg+K+3j5s/sMd/N9d3Yg7k8P4t3cxM/IQP2bmWfIjCNFkmXVhkVJqLPAmYAQ+1FovUErNANBav1fR5jlgKmAClmqt36zpNeUsl9tHYmIiERERfPTRR3Tu3JnDhw/fcMFRbmEJS3ecYOmukxSVmrivlzfPDA/Ax9XBQlUL0TjJlaKiUcjOzubMmTMEBweTk5PD4MGDeeSRR3jiiSdwd3cHIPNyEe9uSyZyb/nFSw/1bcfMYZ1o5VT9ZBxC3E7kSlHRKLi7uxMcHAyU3zPGzc2NP//5z/j4+PD4449z4MABPFs2Y9747mx7bij39fbmk32pDPnHVhZ8e5TsvCILfwIhGjfZQxcWFR8fzzvvvMPHH39Mfn4+SUlJdO788wlSKdn5vLX5OF8dPoO9rZFHB/gzfUgH3BztLFi1EJYjQy6i0cvNzWXjxo1MmjQJgN/+9rcYDAaeeuopgoODSc7II2Lzcb4+ko6DrZHfDPRn2mAJdnH7kSEX0eg5OTldDXOtNSUlJXzwwQeEhITQr18/tq1dxd/GB/Dds0MY1rUVi7f/yODXt7Bw/Q8yFCNEBdlDF41WdnY2H3/8MUuXLiUhIYHnn3+ev/3tb5hMJo5n5PHO1h/5+kg69jZGHunvy5ODO8jBU2H1ZMhFNGlaa/bu3Uu7du3w8fFh3bp1/OEPf+Cxxx5j8F2T+Cwhl7Wx6RgNiof6tGP6HR3xdmlu6bKFqBcy5CKaNKUUAwYMwMfHBwB7e3vc3d2ZO3cug0MCSProBf7QNY+JIW1ZsT+VO/6+lec+i5ULlMRtR/bQRZN1/Phxli9fzkcffQTAqVOnOJdbxBvfxLAu6RJFpSbGBLVhxh0dCfZxsXC1QtQNGXIRVs1kMpGamoq/vz+lpaV4e3vT3MWT7vfO5pSNL/klJgZ1cmfGHR0J7+Qh0+KJJk2GXIRVMxgM+Pv7A+Xj7f/+978J7OjLxn/MJPHv92H/w3riUrOZ8sF+7orYxZqYM5SWyf3YhfWRPXRhtc6fP8+nn35KZGQkr8xfwCXnABZ9/wNpuSW0aWnHtCEdeaBPO1rKDEqiCZEhFyEqLFz4Oq9+8CVOfe/Fvl0QzZSJSb29+O2IQLxdmhMZF8mLm18kNScVX2dfFgxfwOQeky1dthBXSaALUUl8fDyffvopKzfu5lKb3jh0GYSNjQ2BPkVsP/8ylw1HrrZ1sHVgyfglEuqi0ZBAF6IKWmtiY2PZF3+cbLcgluyMB+1AUXYSuflrKWgbBU6l+Dn7cerZU5YuVwhAAl0Isxj+0hzHpOE4uY/H1tmH0svZ5J3cwGXX9ZRFXrR0eUIANQe6TUMXI0Rj5evempTu35Kn12F/sRdOZeNxCZ6Ms36AZ1Ye5s72zdi9NpKJE+8mLCwMg0FOEhONi/RIISosGL4AB1sHUJpCt4NktH2ZC47PMKhrKVuTMpj11UmWnWvL8OnzaNe+EzNmzGDdunUUFxdbunQhANlDF+Kqnw58XnuWy6tM7nEfBcWlrI1J57+7HEhq1R6DqZivErax7PHZnEnYj7u7OwkJCbi5udG2bVsLfxJxu5IxdCFqQWvN4dOXiNybyjdH0ikqNRHazoWH+/ry3otPsvX7jfTq1Ytx48YxduxYwsLCMBqNli5bWBE5KCpEPbhUUMwXh86wcn8qyRl5NLdR+BuyyNq3hkObvsJkMjFu3Di+/vproHwSDycnJwtXLZo6CXQh6pHWmoMpF1mxP5V1cWcpLDER4OlAF9sLhLdrxoP3jCM3NxdPT09CQ0MZNWoUo0aNol+/ftjYyKinqB0JdCEaSG5hCWtj0ll94DRxZ3KwNSqGd23NqC7OHNmwko0b1rNv3z5MJhPOzs4sX76cCRMmoLWWm4YJs8hpi0I0ECd7Wx7p78cj/f344Vwun0Wn8dXhM2xIOIdHi/5MeP4+/hXgxOm4vWzcuOHqhNirV6/m//7v/xg5ciQjRoxg2LBhuLm5WfjTiKZG9tCFqGclZSa2JWXy+cHTbPkhg5IyTZfWLbmnlzd3h3rR1rk5mzdv5q233mLbtm1cvnwZpRS9evViy5YtODk5yR68uEqGXIRoJC7mF/NN3Fn+dyiNQ6mXUAr6tXfj7lBvxgS1wdFWsX//fjZt2kRCQgKffvopAI899hgnT57kV7/6FcOGDaNfv340a9bMwp9GWIIEuhCN0KmsfNbGpvNVzBlOZOZja1QMCfBkfIgXIwJb06LZzyOiCxcu5LPPPuPw4cNorWnevDmPPvoo7733HgBlZWVyeuRtQgJdiEZMa038mVy+PpLO17HpnM0ppJmNgWFdWjEupC2/6toKB7vycL948SI7duxg69at+Pj48Mc//pHS0lJ8fHwIDAxkyJAh3HHHHfTr1w8HBwcLfzJRHyTQhWgiTCZNdMpFvj2Szrr4c2ReLsLetjzcx/QoD/fKe+5Qfn77yy+/zPbt24mJicFkMmFra0tERAQzZsyguLiYvLw8OchqJSTQhWiCykyaA6cusD7uLOvjz5FxuQg7GwPhnTwY3b0NIwJb4+Zod802OTk5REVFsXPnTu655x769u3Lpk2bGDlyJEFBQQwaNIjw8HDCw8Px8/OTA61NkAS6EE2cyaQ5mHqRDfHn2BB/jjOXrmBQEObnxp3dWzMysDV+7o5VbnvixAlWrFjBrl272LNnD7m5uQAcPHiQXr16cezYMS5dukRoaCh2dnZVvoZoPCTQhbAiWmsS0nP5LuEc3x09zw/nLgPQqVULRnRrzYhurejp64rRcOPed1lZGfHx8URFRfHkk09ia2vLnDlziIiIwN7enrCwMPr378+AAQOYOHGi3CK4EbrlQFdKjQbeAozAUq31wmra9QH2Ag9orT+v6TUl0IWoG6nZBXyfeJ7NiefZf/ICpSaNi4Mtd3T25FddWzE4wPOGoZnKzp07R1RUFHv27GH37t0cPHgQd3d3zpw5g1KKRYsWUVxcTL9+/ejVq5ccbLWwWwp0pZQROAaMBNKAA8BDWuujVbT7HigEPpRAF6Lh5RaWsONYJlt+yGB7UibZ+cUoBSE+Lgzt4sngAE9CfJyxMVa/511UVERKSsrVq1iHDh3K9u3bATAajQQHBzNp0iSef/55ALnoqYHd6qX/fYFkrfWJihdbBdwNHL2u3WzgC6DPLdQqhLgFTva2jAv2YlywF2UmzZG0S2w/lsm2pEze2nycNzcdx8nehkGdPAgP8CC8k8cNY+/NmjW7GuYA27Zt49y5c+zfv599+/axb98+srKyADCZTPj6+tKxY0f69OlDnz59CAsLo0OHDhLyFmDOHvqvgdFa62kVz6cA/bTWsyq18QZWAL8CPgC+qWoPXSk1HZgO4Ovr2zslJaWuPocQ4iYu5hcT9WMWO45lsvN4FmdzCgFo59acQR09GNDRnQEd3WnV0t7s18zLy+P555/nwIEDxMTEUFRUBMDLL7/MvHnzyM/PZ926dYSFheHv7y8hXwdudQ+9qn+B6/8KvAn8WWtdVtM/mNZ6CbAEyodczHhvIUQdcXW0u7r3rrXmRFY+UclZ7Dqexbq4s6w6cBqAjp6O9O/gTr8O7vRr70Zrp+oDvkWLFixatAiAkpIS4uPjiY6OJiysPG8OHjzI/fffX/7+rq706tWLXr168eSTTxIQEFDPn/j2Y84e+gDgZa31qIrnzwNorV+r1OYkPwe/B1AATNdaf1Xd68oYuhCNR5lJczQ9l6gfs9h3IpsDpy6SV1QKgJ+7A3393ejj70Zvf1c6eDiavaddXFxMXFwcBw8evPqIi4tj27ZtDBgwgLVr1/L3v/+dnj170rNnT0JDQ+nevbvcp6YGt3pQ1Ibyg6LDgTOUHxR9WGudUE37ZVQz5FKZBLoQjVdpmYmE9FwOnLrAvpMXOHDqApcKSgBwd7Sjp68rvfxc6O3rSrCPC83tzL+PTHFxMUajEaPReDXQY2NjycvLA8DGxoZTp07h7e3NoUOHuHjxIiEhIXh4eNTLZ21qbmnIRWtdqpSaBWyk/LTFD305jG4AAA2ISURBVLXWCUqpGRXr36vTaoUQFmdjNBDSzoWQdi5MG9wBk0lzIiuPA6cuEn3qIodSL7Ip8TwARoOiS+uWhLRzIbSdM8E+LgS0alHtmTSVL16aMGECEyZMwGQy8eOPPxITE0N8fDxeXl4ARERE8NFHHwHg5eVFSEgIPXv2ZP78+Sil5Ayb68iFRUKIX+RCfjGHUy8Sc/rS1cflwvJhGntbA4Ftnejh7Ux3b2eCvJwJaN0C2xpOl6xKVlYWhw8f5siRI8TGxhIbG4tSipiYGADGjRvH6dOn6dGjx9VHcHAwPj4+df55Gwu5UlQIUe9MJs2p7HyOpOUQm3aJ+DM5JKTnUlBcBoCd0UBA6xYEtnWia1snurVpSde2TjVe9FT1+5iuXsH6+uuvs2PHDo4cOUJaWhoAw4cPZ9OmTQDMmzcPd3d3goKCCAoKolWrVnX4iS1DAl0IYRFlFSEffyaHxLOXOXo2l6PpuWTlFV1t49HCjs6tW9K5dUs6tWpBp1YtCGjVAjdHu1oNp1y6dIn4+HiUUgwaNIiysjK8vLzIyMj4+b08PPjd737HCy+8gNaa7du3061bN1q1atVkhm5kTlEhhEUYDYqOni3o6NmCu0N/Xp55uYikc5f54Vwux85f5tj5PD6LPk1+xd48gHNzWzp4OtLBowXtPRzw93DE390RP3cHWtrb3vBeLi4uhIeH//zeRiPnzp3j3LlzJCQkXH34+/sDcObMGYYNGwaAm5sbgYGBBAYG8pvf/IaBAwdiMpkAmtT9bGQPXQjRKGitOZtTSHJGHscz8jiRmceJzHxOZOVxPrfomrZujnb4ujnQzs0BH9fmtHN1wNu1Od4u9ni5NL86IUhNrly5wq5du0hISCAxMZHExEQSEhJYtGgRDz/8MHv37mX48OF07dqVrl270q1bN7p06cKwYcMsesaNDLkIIZq0guJSTmUVkJKdT8qFAlKyC0i9kE/axSukX7pCSdm1Oebc3Ja2zva0cbanjZM9rZzsae3UDM8WzfBs2QyPip/2tteebqm1RmuNwWAgKSmJd999l6SkJBITE0lNTQXKb4Vwxx13sGHDBiIiIujSpcvVR+fOnfHy8qp2+CYyLpIXN79Iak4qvs6+LBi+gMk9Jtfqu5AhFyFEk+ZgZ0OglxOBXk43rCszac7nFnLmUnm4p128wrmcQs7mFHI25woJFWP2Ve27OtgZcXO0w83RDlcHO1wdbHFxsMOpuS3Oze0In/JHxtjb0tLeBhtdQkb6adp17kRuYQmX8wo4e/Ys27dvp6Cg4OprJicn07FjR9atW8fevXvp3LkzAQEBxJbG8rvtv6OgpLxtSk4K07+eDlDrUK+O7KELIaxeaZmJrLxiMi8XkZlXSOblIrLyirmQX/7Izi/mUkExFwuKuVRQcvX0y5uxMSjsbY3YGkDpUnRpCR5urtjZGDiXfob0M2loUxmYTGitwajBV1Ngs4M8m+8B8HP249Szp8z+LLKHLoS4rdkYDeXDL872gPNN25eZNHmFpeQWlpBbWEJeYSmXC0spKCkjv6iU/KJSCkvKuFJSxpViE0WlZRSVmigqNVFaZqKkzIR7iw507uBHfkEB+QVXSEg/CiYFSgE/H9RNzUmtu89ZZ68khBBWwmhQODvY4uxw49k0v5T/m7NIybnxDrO+zr519h5N53wcIYRowhYMX4CD7bWzPTnYOrBg+II6ew8JdCGEaACTe0xmyfgl+Dn7oVD4OfuxZPySOjsgCnJQVAghmpSaDorKHroQQlgJCXQhhLASEuhCCGElJNCFEMJKSKALIYSVkEAXQggrIYEuhBBWQgJdCCGshAS6EEJYCQl0IYSwEhLoQghhJSTQhRDCSkigCyGElZBAF0IIKyGBLoQQVkICXQghrIQEuhBCWAkJdCGEsBJmBbpSarRSKkkplayUmlvF+slKqSMVj91KqZC6L1UIIURNbhroSikj8A4wBggEHlJKBV7X7CRwh9Y6GPgrsKSuCxVCCFEzc/bQ+wLJWusTWutiYBVwd+UGWuvdWuuLFU/3Aj51W6YQQoibMSfQvYHTlZ6nVSyrzhPA+qpWKKWmK6WilVLRmZmZ5lcphBDipswJdFXFMl1lQ6WGUR7of65qvdZ6idY6TGsd5unpaX6VQgghbsrGjDZpQLtKz32A9OsbKaWCgaXAGK11dt2UJ4QQwlzm7KEfAAKUUu2VUnbAg8Dayg2UUr7Al8AUrfWxui9TCCHEzdx0D11rXaqUmgVsBIzAh1rrBKXUjIr17wEvAe7Au0opgFKtdVj9lS2EEOJ6Susqh8PrXVhYmI6OjrbIewshRFOllDpY3Q6zXCkqhBBWQgJdCCGshAS6EEJYCQl0IYSwEhLoQghhJSTQhRDCSkigCyGElZBAF0IIKyGBLoQQVkICXQghrIQEuhBCWAkJdCGEsBIS6EIIYSUk0IUQwkpIoAshhJWQQBdCCCshgS6EEFZCAl0IIayEBLoQQlgJCXQhhLASEuhCCGElJNCFEMJKSKALIYSVkEAXQggrIYEuhBBWQgJdCCGshAS6EEJYCQl0IYSwEhLoQghhJSTQhRDCSkigCyGElTAr0JVSo5VSSUqpZKXU3CrWK6VURMX6I0qpXnVfKhAZCf7+YDCU/4yMrJe3Ebcx6WOiPtV3/9Ja1/gAjMCPQAfADogFAq9rMxZYDyigP7DvZq/bu3dvXSuffKK1g4PW8PPDwaF8uRB1QfqYqE911L+AaF1Nrqry9dVTSg0AXtZaj6p4/nzFH4LXKrV5H9imtV5Z8TwJGKq1Plvd64aFheno6Gjz//L4+0NKyo3L/fzg1CnzX0eI6kgfE/WpjvqXUuqg1jqsqnXmDLl4A6crPU+rWFbbNiilpiulopVS0ZmZmWa8dSWpqbVbLkRtSR8T9akB+pc5ga6qWHb9br05bdBaL9Fah2mtwzw9Pc2p72e+vrVbLkRtSR8T9akB+pc5gZ4GtKv03AdI/wVtbs2CBeDgcO0yB4fy5ULUBeljoj41QP8yJ9APAAFKqfZKKTvgQWDtdW3WAo9WnO3SH8ipafz8F5k8GZYsKR9vUqr855Il5cuFqAvSx0R9aoD+ddODogBKqbHAm5Sf8fKh1nqBUmoGgNb6PaWUAt4GRgMFwFStdY1HPGt9UFQIIUSNB0VtzHkBrfU6YN11y96r9LsGZt5KkUIIIW6NXCkqhBBWQgJdCCGshAS6EEJYCQl0IYSwEmad5VIvb6xUJlDFdbBm8QCy6rCcutJY64LGW5vUVTtSV+1YY11+Wusqr8y0WKDfCqVUdHWn7VhSY60LGm9tUlftSF21c7vVJUMuQghhJSTQhRDCSjTVQF9i6QKq0VjrgsZbm9RVO1JX7dxWdTXJMXQhhBA3aqp76EIIIa4jgS6EEFai0QX6rUxIfbNt67muyRX1HFFK7VZKhVRad0opFaeUilFK1ektJs2oa6hSKqfivWOUUi+Zu2091/VcpZrilVJlSim3inX1+X19qJTKUErFV7PeUv3rZnVZqn/drC5L9a+b1dXg/Usp1U4ptVUplaiUSlBKzamiTf32r+omG7XEg1uYkNqcbeu5roGAa8XvY6g0UTZwCvCw0Pc1FPjml2xbn3Vd1348sKW+v6+K1x4C9ALiq1nf4P3LzLoavH+ZWVeD9y9z6rJE/wLaAr0qfm8JHGvo/Gpse+h9gWSt9QmtdTGwCrj7ujZ3A8t1ub2Ai1KqrZnb1ltdWuvdWuuLFU/3Uj5rU327lc9s0e/rOg8BK+vovWuktd4BXKihiSX6103rslD/Muf7qo5Fv6/rNEj/0lqf1Vofqvj9MpDIjXMr12v/amyBfisTUps1UXU91lXZE5T/Ff6JBr5TSh1USk2vo5pqU9cApVSsUmq9Uqp7Lbetz7pQSjlQPjHKF5UW19f3ZQ5L9K/aaqj+Za6G7l9ms1T/Ukr5Az2Bfdetqtf+ZdYEFw3oViakNmui6l/I7NdWSg2j/D+48EqLB2mt05VSrYDvlVI/VOxhNERdhyi/90OeKp956isgwMxt67Oun4wHorTWlfe26uv7Mocl+pfZGrh/mcMS/as2Grx/KaVaUP4H5Fmtde71q6vYpM76V2PbQ7+VCanrc6Jqs15bKRUMLAXu1lpn/7Rca51e8TMD+B/l/3vVIHVprXO11nkVv68DbJVSHuZsW591VfIg1/3vcD1+X+awRP8yiwX6101ZqH/VRoP2L6WULeVhHqm1/rKKJvXbv+r6wMAtHlSwAU4A7fn5wED369rcxbUHFfabu2091+ULJAMDr1vuCLSs9PtuYHQD1tWGny8g6wukVnx3Fv2+Kto5Uz4O6tgQ31el9/Cn+oN8Dd6/zKyrwfuXmXU1eP8ypy5L9K+Kz70ceLOGNvXav+rsy63Df6SxlB8d/hF4sWLZDGBGpS/tnYr1cUBYTds2YF1LgYtATMUjumJ5h4p/nFggwQJ1zap431jKD6YNrGnbhqqr4vljwKrrtqvv72slcBYooXyv6IlG0r9uVpel+tfN6rJU/6qxLkv0L8qHwTRwpNK/09iG7F9y6b8QQliJxjaGLoQQ4heSQBdCCCshgS6EEFZCAl0IIayEBLoQQlgJCXQhhLASEuhCCGEl/h8O7EjTdBU7qwAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "node_x = np.array([0.0,1.0,2.0])\n",
    "node_y = f(node_x)\n",
    "\n",
    "x = np.linspace(0,2,100)\n",
    "\n",
    "plt.plot(node_x,[0,0,0],'ro')\n",
    "plt.plot(node_x,[1,1,1],'bo')\n",
    "plt.plot(node_x,node_y,'go')\n",
    "plt.plot(x,f(x),'k--')\n",
    "plt.plot(x,lagrange_int(x,node_x,node_y))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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