{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Interpolation Error\n",
    "\n",
    "Copyright (C) 2020 Andreas Kloeckner\n",
    "\n",
    "<details>\n",
    "<summary>MIT License</summary>\n",
    "Permission is hereby granted, free of charge, to any person obtaining a copy\n",
    "of this software and associated documentation files (the \"Software\"), to deal\n",
    "in the Software without restriction, including without limitation the rights\n",
    "to use, copy, modify, merge, publish, distribute, sublicense, and/or sell\n",
    "copies of the Software, and to permit persons to whom the Software is\n",
    "furnished to do so, subject to the following conditions:\n",
    "\n",
    "The above copyright notice and this permission notice shall be included in\n",
    "all copies or substantial portions of the Software.\n",
    "\n",
    "THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n",
    "IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n",
    "FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\n",
    "AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n",
    "LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\n",
    "OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN\n",
    "THE SOFTWARE.\n",
    "</details>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import numpy.linalg as la\n",
    "import matplotlib.pyplot as pt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's fix a function to interpolate:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "if 1:\n",
    "    def f(x):\n",
    "        return np.exp(1.5*x)\n",
    "elif 0:\n",
    "    def f(x):\n",
    "        return np.sin(20*x)\n",
    "else:\n",
    "    def f(x):\n",
    "        return (x>=0.5).astype(np.int).astype(np.float)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f1b5180f240>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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ezBqb2WQzW2Jmi8zstiTHtDKz9Wa2IPHxh+yUK1K8Zs6EFi3gyCPDYwW8pCKVkfw24Hfu\nXmpm+wPzzWyiuy8rc9w0d2+f+RJFitu2baEt88QTMGAAXHBB1BVJdVJhyLv7F8AXiccbzWwp0Ago\nG/KW+fJEitvf/w5XXAG1a4e9Zxo3jroiqW4qdeHVzA4FmgNzkzzd0swWmtlrZtYsA7WJFLVhw8Li\npo4dYdIkBbxUTcoXXhOtmtFAT3ffWObp+UATd99sZq2BMcBRyV6npKTkh8exWIxYLFbJkkUK24YN\n0KMHLFgQ7sHaokXUFUmuxeNx4vF4Rl7L3L3ig8z2AsYDE9z9sRSO/xg4wd3Xlvm6p/J+IsVqxozQ\nnmnTBh5+GPbdN+qKJB+YGe5epZZ4qiP5QcCS8gLezBq4+6rE4xMJvzzWJjtWRH5s61a491549tnw\noYurkikVhryZnQp0ARaZ2ULAgV5AE8DdvT/QycxuArYC3wKXZK9kkcLy4YfQpUvYWKy0FBo2jLoi\nKSQptWsy9mZq14j8wB2efhr++MfwceutUENr0CWJXLRrRCSDPv007Pu+YUPow2vfGckWjRtEcsg9\n3FD7hBOgVauwclUBL9mkkbxIjqxaBTfeGO7aNHEiNG8edUVSDDSSF8mBl1+GX/86jNrnzVPAS+5o\nJC+SRevWhQuqb78Nr7wCJ58cdUVSbDSSF8mScePguOPCTT0WLlTASzQ0khfJsK++gp49Ye5ceOEF\n0M4dEiWN5EUyxB1Gjgyj94YN4d13FfASPY3kRTLg88/DpmLvvx967y1bRl2RSKCRvEga3OH558PM\nmV/+MvTeFfCSTzSSF6miTz6B7t1h9WrNe5f8pZG8SCVt3w6PPw7/8i9h1ercuQp4yV8ayYtUQmkp\n3HBD2Od9+nRtSSD5TyN5kRRs3Ai33w7nnQc33QRTpijgpXpQyItUYPz4cFF19WpYvBiuuQZMt62X\nakLtGpFyrFwZFjWVlsLAgXDWWVFXJFJ5GsmLlLF9Ozz5ZJgWecwxsGiRAl6qL43kRXaxYEHoudeq\nBVOnQrNmUVckkh6N5EWAtWvDitU2bcLcdwW8FAqFvBS1HTtCv71Zs3AxdckSuPZa3WtVCofaNVK0\n5s+Hm28O4f7663D88VFXJJJ5Gq9I0Vm7NvTd27YNrZmZMxXwUrgU8lI0duyAAQPCjJmaNWHp0jDn\nXa0ZKWQVtmvMrDEwBGgIbAeedfd+SY7rB7QGNgFXu3tphmsVqbLZs8Oc9732gv/+b2jRIuqKRHIj\nlTHMNuB37t4MOBm42cx2W9BtZq2BI9y9KdAdeDrjlYpUwaefQpcu0Lkz3HILzJihgJfiUmHIu/sX\n34/K3X0jsBRoVOawDoTRPu4+FzjQzBpkuFaRlG3eDH36hN0hDz8cli2Dq65Sa0aKT6Vm15jZoUBz\nYG6ZpxoBn+7y+f8lvrYqjdpEKs0dXnwR7rwz3Dh7wQJo0iTqqkSik3LIm9n+wGigZ2JEv9vTSb7F\nk71OSUnJD49jsRgx3QRTMmTevNB337IFhg2D3/wm6opEqiYejxOPxzPyWuaeNIt3P8hsL2A8MMHd\nH0vy/NPAFHd/KfH5MqCVu68qc5yn8n4ilbFyJdx1F0yaBH/6E3TtGmbPiBQKM8Pdq7T3aaodykHA\nkmQBnzAWuCpRTEtgfdmAF8m0b76B3r3huOPgZz8Lffdrr1XAi+wqlSmUpwJdgEVmtpDQhukFNAHc\n3fu7++tm1sbMPiRMobwmm0VLcdu6Ncx3v/fesDvk/Plw6KFRVyWSn1Jq12TszdSukTS4w6uvhouq\njRrBQw/BCSdEXZVI9qXTrtHeNVItzJkDd9wB69dD375w/vm6O5NIKjRrWPLahx+GhUydOoUtCEpL\noXVrBbxIqhTykpdWrYLbboOWLcMK1Q8+0EVVkapQyEteWbcO7r575w07liyBXr1g332jrUukulLI\nS17YtAnuvx+OOgq++CKsVO3XD+rXj7oykepNIS+R2rIFnngCmjYN/fYZM8KdmrQVgUhmaHaNRGLb\nNnjhBSgpgWOPhdde0+6QItmgkJec2rED/vY3+OMf4Z/+KQT9aadFXZVI4VLIS07s2BEWMvXpE2bI\nPPoonHuupkKKZJtCXrKqbLjfdx+0a6dwF8kVhbxkhcJdJD8o5CWjFO4i+UUhLxmhcBfJTwp5Scu2\nbTBqVFjItPfeCneRfKOQlyrZsgWGDIEHH4SGDcP/amdIkfyjkJdK2bQJ+veHv/wFfvlLeO453UtV\nJJ8p5CUl69bBk0+G/WROPz3033XDDpH8p71rZI9WrQp3YjryyLC3+9SpMHq0Al6kulDIS1LLl0OP\nHnD00bBxY7iP6vPPwzHHRF2ZiFSGQl52M2sWXHghnHIK1KsHS5eGXSJ1o2yR6kk9eWH7dhg7Fv78\n57CX++9+B0OHwn77RV2ZiKRLIV/ENm+GwYPhkUfCqP2OO+C3v9Ut9kQKiUK+CH35ZZgp89e/hnuo\nDhoUtvvVHHeRwlNhT97MBprZKjN7t5znW5nZejNbkPj4Q+bLlExYvBhuuCHcYu/zz8NMmVdfDfPc\nFfAihSmVkfxzwOPAkD0cM83d22emJMmk7dvDXZceeyzcFPumm2DZMmjQIOrKRCQXKgx5d59hZhXd\ncVPjwDyzYUNYjfr446Hf3rMnXHwx1KoVdWUikkuZmkLZ0swWmtlrZtYsQ68pVfDBB3DrrXDYYTBn\nTri93ttvwxVXKOBFilEmLrzOB5q4+2Yzaw2MAY4q7+CSkpIfHsdiMWKxWAZKKG7uMHFiaMm88w5c\nfz28+y40bhx1ZSJSFfF4nHg8npHXMnev+KDQrhnn7r9K4diPgRPcfW2S5zyV95PUrFsXVqE+/TTU\nqRNaMpddBvvsE3VlIpJJZoa7V6ktnupI3iin725mDdx9VeLxiYRfHD8KeMmcefPC9MdXXoE2bWDg\nQDj1VM2QEZEfqzDkzWw4EAN+amYrgN5ALcDdvT/QycxuArYC3wKXZK/c4rV5M7z4Ygj3tWuhe3d4\n/32oXz/qykQkn6XUrsnYm6ldU2nLloV2zNChYT+ZHj3gvPOghnYdEikauWjXSA59911YpPTXv4a5\n7d26wYIF0KSiiawiImUo5PPI0qWhvz50aNjS96abwl4ymvooIlWlkI/Ypk0wciQMGAAffwxdu8KM\nGdC0adSViUghUE8+Au5hPvuAATBqVNgc7LrroHVr2HvvqKsTkXyjnnw1sXYtDBsWwn3jxtBrX7QI\nGjWKujIRKVQayWfZtm3w5pth3/YJE8K89uuug1hMM2REJDXpjOQV8lmyaBEMGRJG7o0bh177pZfC\nT38adWUiUt2oXZMnvvwShg8Po/bVq+HKK+Gtt3TzaxGJjkbyadqyBcaPD8E+bRpccEEYtZ9xhm6j\nJyKZoZF8jrnD3LlhPvtLL8Fxx4VgHzYM6taNujoRkZ0U8pWwZElox7z4Iuy1V9ij/Z134NBDo65M\nRCQ5hXwFPvkERowI4b5mTbh4OmoUtGihXR9FJP+pJ5/El1+GIB8+PGwQdtFFcPnl4YbXmvYoIrmm\nKZQZ8M03MGZMaMXMmhXms19+OZx7rvaOEZFoKeSr6JtvwsyY0aPDgqXTTw/B3r497Ldf1NWJiAQK\n+Ur4+usQ7KNGweTJ4Y5KnTtDhw5Qr16kpYmIJKWQr8CGDTBuXAj2KVPCiL1TpxDsP/lJzssREakU\nhXwS69fD2LGhFROPQ6tWYcTevj0cdFBOShARyQiFfMKqVWHE/sorMH16WHXauXNYhXrggVl7WxGR\nrCrqkP/wwzArZswYWLw43P+0Qwdo1w4OOCCjbyUiEomiCnl3mD9/Z7CvWRNaMB07wplnQu3aGSpW\nRCRPFHzIb90KU6eGUH/1Vdh333Dv044d4cQTtUBJRApbVjcoM7OBQDtglbv/qpxj+gGtgU3A1e5e\nWpVidrVuHbzxRuixT5gARx0VQn3SJDj66HRfXUSkOKQyBn4OOK+8J82sNXCEuzcFugNPV6UQd3jv\nPXjwwTDFsUmTsKvjb34TbsAxZw7ceWfhBHw8Ho+6hLyhc7GTzsVOOheZUWHIu/sMYN0eDukADEkc\nOxc40MwapPLm334bRuk33wyHHQZt28KKFSHMv58pc+ONhXkPVP0A76RzsZPOxU46F5mRiV0oGwGf\n7vL5/yW+tirZwZ99Bq+9FladTp0KzZuHcB8/Ho49Vjs7iohkUiZCPlksl3t19de/htatwx4xgwdr\nKwERkWxKaXaNmTUBxiW78GpmTwNT3P2lxOfLgFbu/qORvJnlz+5kIiLVSLZv/2ckH7EDjAVuBl4y\ns5bA+mQBD1UvUkREqiaVKZTDgRjwUzNbAfQGagHu7v3d/XUza2NmHxKmUF6TzYJFRCR1OV0MJSIi\nuZWVtaJmdr6ZLTOzD8zsP5I8X8vMRpjZcjObbWaHZKOOfJDCufh3M3vPzErNbJKZ/TyKOnOhonOx\ny3GdzGyHmR2fy/pyKZVzYWYXJ342FpnZC7muMVdS+DfyczObbGYLEv9OWkdRZ7aZ2UAzW2Vm7+7h\nmH6J3Cw1s+YpvbC7Z/SD8IvjQ6AJsDdQChxd5pibgKcSjy8BRmS6jnz4SPFctALqJB7fWMznInHc\n/sBUYBZwfNR1R/hzcSQwHzgg8fnBUdcd4bl4BuieeHwM8HHUdWfpXJwGNAfeLef51sBriccnAXNS\ned1sjORPBJa7+yfuvhUYQVgwtasOwODE49HAWVmoIx9UeC7cfaq7/yPx6RzCGoNClMrPBcB9wIPA\nllwWl2OpnIvrgSfd/WsAd/8qxzXmSirnYgfw/Z6yBxHW4hQcz9LC02yEfNnFUZ/x4+D64Rh33w6s\nN7NCnDGfyrnYVTdgQlYrik6F5yLx52djd389l4VFIJWfi6OAX5jZDDObZWblbi1SzaVyLvoAV5rZ\np8B44NYc1ZZvylt4ukeZWAxVViqLo8oeY0mOKQQpLxQzsyuAEwjtm0K0x3NhZgb0BbpW8D2FIJWf\ni70ILZvTgUOA6WZ27Pcj+wKSyrm4DHjO3fsmpmm/AByb9cryT6UWnn4vGyP5zwg/lN9rDKwsc8yn\nwM8BzKwmoe+4pz9TqqtUzgVmdjZwF3BB4k/WQlTRuahL+IcbN7OPgZbAqwV68TWVn4vPgFfdfYe7\n/x14H2iam/JyKpVz0Q0YCeDuc4A6ZnZwbsrLK5+RyM2EpHlSVjZCfh5wpJk1MbNawKWEBVO7GsfO\nEVtnYHIW6sgHFZ4LM2tB2LmzvbuviaDGXNnjuXD3r929vrsf7u6HEa5PXODuCyKqN5tS+TcyBjgT\nIBFoTYGPclplbqRyLj4BzgYws2OA2gV8jaKihadXAVS08HRXGW/XuPt2M7sFmEj4JTLQ3ZeaWR9g\nnruPBwYCQ81sObCG8H9swUnxXDwE7AeMSrQsPnH3jtFVnR0pnovdvoUCbdekci7c/Q0zO9fM3gO2\nAbcX4l+7Kf5c3A48a2b/TrgI27X8V6y+srXwVIuhREQKmG6cJyJSwBTyIiIFTCEvIlLAFPIiIgVM\nIS8iUsAU8iIiBUwhLyJSwBTyIiIF7P8BN6Xn/aCLDaMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f1b540aa4e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_01 = np.linspace(0, 1, 1000)\n",
    "pt.plot(x_01, f(x_01))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And let's fix some parameters. Note that the interpolation interval is just $[0,h]$, not $[0,1]$!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  1.])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "degree = 1\n",
    "h = 1\n",
    "\n",
    "nodes = 0.5 + np.linspace(-h/2, h/2, degree+1)\n",
    "nodes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now build the Vandermonde matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "V = np.array([\n",
    "    nodes**i\n",
    "    for i in range(degree+1)\n",
    "]).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.,  0.],\n",
       "       [ 1.,  1.]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now find the interpolation coefficients as `coeffs`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "#clear\n",
    "coeffs = la.solve(V, f(nodes))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are some points. Evaluate the interpolant there:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "x_0h = 0.5+np.linspace(-h/2, h/2, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [],
   "source": [
    "#clear\n",
    "interp_0h = 0*x_0h\n",
    "for i in range(degree+1):\n",
    "    interp_0h += coeffs[i] * x_0h**i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now plot the interpolant with the function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f1b517bfa20>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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4kuBTUlLYsmVL0M/rFkvyxpjA27ABrr8epkxxJjW51D2zbt06Zs+eTfny5YN+brdYkjfG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\nq+RN4CxY4FTvhw4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AvndzywEvZ2fz3KZN/PPVV23GqTHGBJBPo2tEJAaYnteNVxEZCcxX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\nglmSN8aYCPb/AVSEE+/blf8mAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f1b517d1f60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pt.plot(x_01, f(x_01), \"--\", color=\"gray\", label=\"$f$\")\n",
    "pt.plot(x_0h, interp_0h, color=\"red\", label=\"Interpolant\")\n",
    "pt.plot(nodes, f(nodes), \"or\")\n",
    "pt.legend(loc=\"best\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Also plot the error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "jupyter": {
     "outputs_hidden": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Max error: 0.633384\n"
     ]
    },
    {
     "data": {
      "image/png": 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bb5YrIpJtmjXzkTbPPpue10uk9F9Vsb/WLfbi4uIf7hcUFFBQUJDw7z78MDRv\n7pMKRESy1VlnwT33+AqVVSkpKaGkpCQpr1Xj6Boz6woUhxAKyx8PBEIIYWgVx/4V+EcI4blqzlXr\n0TUbVpkcNw46dqzVKUREMsKqVb4H9fTpvtxBTVI9uqYUaGtmrc2sIVAEbG49tTrsf1K9W2+Fo49W\ngheR7Ne4MfToAU89lfrXSmicvJkVAnfhHwojQgi3mNkQoDSEMMbMDgKeB3YAVgFfhhD2r+I8tWrJ\nL14MP/+5r1HTsuUW/7qISMZ57TWfHFVaWvOxOT8Z6pxzPLlrXLyI5Ip16zyvTZoEe++9+WNzehXK\nqVPhlVdgwCZTsEREsleDBnDaaalfmTKjk3wIvnzB4MHQtGnsaEREkuv002HkSM91qZLRSX7cOPj8\nczj//NiRiIgk38EHw5o1XrFIlYxN8uvWwZVX+mptW7KQj4hItjCDnj1TW7LJ2CT/l7/42jT/+7+x\nIxERSZ3TT/ckv359as6fkUn+m2+guNiXL7CUjLoXEckMHTrATjv5KJtUyMgkP2wYHHEEdO4cOxIR\nkdTb0JpPhYwbJ//ZZ3DAAd4R0apVmgITEYno00/hoIN8oEnDhps+n1Pj5K+9Fvr0UYIXkfyx556+\nNterryb/3Bk1bmXaNHjpJZgzJ3YkIiLptaFkc+yxyT1vRpVrjjrKF+353e/SFpKISEb48kto3x6+\n+AKaNPnxczlRrnn9da9LXXhh7EhERNJvt92gUydfxiWZMiLJr18PAwfCjTfCVlvFjkZEJI7TTkv+\n8sMZkeSffhrq1YNTTokdiYhIPCed5P2S33+fvHNGT/Jr1sDVV8Mtt2jik4jkt1139flBL7+cvHNG\nT/IPPQRt28Lhh8eOREQkvlNP9epGskQdXfPtt75YvvZtFRFxS5b4mPmKo2yydnTNHXd4C14JXkTE\nJbtkE60lv3Qp7Luv72/Ypk3aQhARyXj33w9vvukbikCW7vHar593tN51V9peXkQkKyxdCvvss7Fk\nU5ckH2VZg/nz4fHHYebMGK8uIpLZmjeHAw/0ks2JJ9btXFFq8tdeC5de6v8QERHZ1KmnJmdiVELl\nGjMrBO7EPxRGhBCGVnq+IfAocCCwDPhtCGFhFecJU6YEfvMbmDsXtt227v8AEZFcVLFks/XWKRxd\nY2b1gHuAo4EOQE8za1/psPOA/4QQ9sY/DG6t7nyDBsE11yjBl5SUxA4hY+habKRrsVG+X4uKJZu6\nSKRc0wXzb/VmAAAEVElEQVSYG0JYEEJYC4wCelQ6pgfwt/L7zwDdqzvZvHlwwQW1CTW35PsbuCJd\ni410LTbStUhOySaRJN8CWFTh8eLyn1V5TAihDFhhZjtVdbIbb6x65xMREfmxE0/0yaJ1kUiSr6oO\nVLmQX/kYq+IYwD+ZRESkZrvuWvfJojV2vJpZV6A4hFBY/nggECp2vprZuPJjJptZfeCLEMImY2fM\nLH2D8kVEckgqx8mXAm3NrDXwBVAE9Kx0zD+As4HJwKnAhGQGKSIitVNjkg8hlJlZX+BVNg6hnGlm\nQ4DSEMIYYATwmJnNBZbjHwQiIhJZWpc1EBGR9ErJjFczKzSzWWY2x8wGVPF8QzMbZWZzzewdM2uV\nijgyQQLXor+ZzTCzD8zsNTPbI0ac6VDTtahw3Clmtt7MOqczvnRK5FqY2Wnl743pZvb3dMeYLgn8\njexhZhPMbEr538kxMeJMNTMbYWZLzOzDzRzzp/K8+YGZJdYlG0JI6g3/4JgHtAa2Aj4A2lc65mLg\nvvL7vwVGJTuOTLgleC26AY3L71+Uz9ei/LhtgTeAt4HOseOO+L5oC7wPbFf+uFnsuCNeiweAPuX3\n9wU+iR13iq7FIUBH4MNqnj8GGFt+/2Dg3UTOm4qWfFInT2W5Gq9FCOGNEMKq8ofvsukchFyRyPsC\n4HpgKLA6ncGlWSLX4gLg3hDCSoAQwrI0x5guiVyL9cB25fd3AD5LY3xpE0J4C/hqM4f0wJePIYQw\nGdjezHat6bypSPJJnTyV5RK5FhWdB9Rx6kPGqvFalH/9bBlCeCmdgUWQyPtiH6Cdmb1lZm+b2dFp\niy69ErkWQ4CzzGwRMAb4fZpiyzSVr9VnJNAoTMVSw0mdPJXlErkWfqDZmfgCb91SGlE8m70WZmbA\ncHwo7uZ+Jxck8r5ogJdsfg20AiaZWYcNLfscksi16An8NYQwvHzezt/xdbTyTcL5pKJUtOQX42/K\nDVoCn1c6ZhGwB0D55KntQgib+5qSrRK5FpjZEcAg4Ljyr6y5qKZr0RT/wy0xs0+ArsCLOdr5msj7\nYjHwYghhfQjhU2A2sHd6wkurRK7FecBTACGEd4HGZtYsPeFllMWU581yVeaTylKR5H+YPFW+BHER\nMLrSMRsmT8FmJk/lgBqvhZl1Au4Hjg8hLI8QY7ps9lqEEFaGEJqHENqEEPbC+yeOCyFMiRRvKiXy\nN/ICcDhAeULbG5if1ijTI5FrsQA4AsDM9gUa5XAfhVH9N9jRQC/4YSWCFSGEJTWdMOnlmqDJUz9I\n8FrcCmwDPF1eslgQQjghXtSpkeC1+NGvkKPlmkSuRQjhFTM7ysxmAOuAK3Lx226C74srgIfMrD/e\nCXt29WfMXmY2EigAdjazhcBgoCG+jMyDIYSXzOw3ZjYP+A44N6Hzlg/HERGRHBRl+z8REUkPJXkR\nkRymJC8iksOU5EVEcpiSvIhIDlOSFxHJYUryIiI5TEleRCSH/T+9wH3CZf+0bgAAAABJRU5ErkJg\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f1b5170b2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "error = interp_0h - f(x_0h)\n",
    "pt.plot(x_0h, error)\n",
    "print(\"Max error: %g\" % np.max(np.abs(error)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* What does the error look like? (Approximately)\n",
    "* How will the error react if we shrink the interval?\n",
    "* What will happen if we increase the polynomial degree?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}