{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Convolutional Cubic Splines\n",
    "\n",
    "$C^2$-continuous cubic splines through evenly spaced data points can be created by convolving the data points with a $C^2$-continuous piecewise cubic kernel, characterized as follows:\n",
    "\n",
    "\\begin{split}\n",
    " y(0) &= 1\\\\\n",
    " y(x) &= 0,\\text{ for all integer }x \\neq 0\\\\\n",
    " y'(x) &= sgn(x) * 3(\\sqrt3-2)^{|x|},\\text{ for all integer }x\\\\\n",
    "\\end{split}\n",
    "which implies\n",
    "\n",
    " $$y''(x) = -6\\sqrt 3(\\sqrt3-2)^{|x|},\\text{ for all integer }x \\neq 0$$\n",
    "\n",
    "The double-sided exponential allows the convolution to be performed extremely efficiently.\n",
    "\n",
    "Any of the standard boundary conditions can then be applied by adjusting the start and end derivatives appropritately, and propagating the change to the rest of the curve.\n",
    "\n",
    "The following function calculates \"natural\" cubic splines (with $y'' = 0$ at start and end) using this technique, which is much easiser than the way everyone is taught!\n",
    "\n",
    "Use it however you like.\n",
    "\n",
    "Cheers,\n",
    "\n",
    "Matt Timmermans"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "#given an array of Y values at consecutive integral x abscissas,\n",
    "#return array of corresponding derivatives to make a natural cubic spline\n",
    "def naturalSpline(ys):\n",
    "    vs = [0.0] * len(ys)\n",
    "    if (len(ys) < 2):\n",
    "        return vs\n",
    "    \n",
    "    DECAY = math.sqrt(3)-2;\n",
    "    endi = len(ys)-1\n",
    "\n",
    "    # make convolutional spline\n",
    "    S = 0.0;E = 0.0\n",
    "    for i in range(len(Y)):\n",
    "        vs[i]+=S;vs[endi-i]+=E;\n",
    "        S=(S+3.0*ys[i])*DECAY;\n",
    "        E=(E-3.0*ys[endi-i])*DECAY;\n",
    "\n",
    "    #Natural Boundaries\n",
    "    S2 = 6.0*(ys[1]-ys[0]) - 4.0*vs[0] - 2.0*vs[1]\n",
    "    E2 = 6.0*(ys[endi-1]-ys[endi]) + 4.0*vs[endi] + 2.0*vs[endi-1]\n",
    "    # A = dE2/dE = -dS2/dS,  B = dE2/dS = -dS2/dS\n",
    "    A = 4.0+2.0*DECAY\n",
    "    B = (4.0*DECAY+2.0)*(DECAY**(len(ys)-2))\n",
    "    DEN = A*A - B*B\n",
    "    S = (A*S2 + B*E2) / DEN\n",
    "    E = (-A*E2 - B*S2) / DEN\n",
    "    for i in range(len(ys)):\n",
    "        vs[i]+=S;vs[endi-i]+=E\n",
    "        S*=DECAY;E*=DECAY\n",
    "    return vs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "[<matplotlib.lines.Line2D at 0x4db5160>]"
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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BrXTihuG9Gd5uHNEB7pC7D8Oe2bB3HhxdCk6e0PYqSJqgHbBjkI4UouE11AryQFVVCxroWkII0XAsVTD3Zq3v8cQvwCybh0TzZbOpbDtxlh/25bBsfy5ZhRUYFOge7ccN3SMY1i74txvsghO1MeQpOL5WK8E48C3s+j/wDIHE8VrNclB72fQqGoyUWAghxOVY+ghk74RrvwLfGL2jEaLBqarKzoxCFu/O4fu9OZwqrsTJZKBvrD/3DYljSJsgfN2d/vwTGYwQM0AbV74Kh3/QkuVN78OGtyGwrdYFI3ECeIfZ94sSzZ6iqqp9L6Ao6cBZQAU+UFX1w189Pg2YBhAUFNR51qxZdo3nYjblWKiuqsTd1QWDAkYFDIqCUQGjAZyN4GpScDYquJjAyQBKM/8rtrS0FA+P+msA39TJfNVNY5mvoFMptDn0OifDr+FYq5t1i6OxzJejkPmqm4vNl6qqnCi2seWUlc05Fk5XqpgUSAww0j3YRHKgEVdT/fyeNFcXE5C/jqDcFLyLD6OiUOjTjrzA/uQH9MJidqz/S/n+qhs952vgwIHbf6/0tyES5FBVVbMVRQkElgP3qKq65mLv26VLF3Xbtm12jedioh9dQl2mwaCAu5MJTxcTvh5O+Lo74+fuhJ+7E74e2stALxdCvV0J9XHB08Vsv+B1kpKSwoABA/QOo9GQ+aqbRjFfuQfg48EQ2hFuWvSXDkGoL41ivhyIzFfd/DxfqqpyOLeExbtzWLwnm+OnyzEZFPrE+TMqKZShbYPwdrXz77szx7SDSPbMhjNpYHSG1sO1Eoy4oVrXDJ3J91fd6DlfiqL8boJs92d0VVWza1/mKYqyEOgGXDRB1suKB/uzadMWOnbugtWmYrHZsNhULFaVGquN8morZVUWyqstlFZZa19aKK6wcKasijNl1RzLL+VMWTXl1dbffH5PZxMhPi6E+rgS6uNKlJ8b0f4eRPu7E+HrhpPJoMNXLYT4y6pKYM5N4OQB4z/VNTkWwt7OVtqYvjqNhTuyOJxbgkGBnq38uL1/K0a0C6bFpZRP1BffGBjwL+j/MGTv0Eow9s6Dg4vAxQfajdWS5fDuWj9mIf4iuz6rK4riDhhUVS2pfX0Y8LQ9r/lXtArwIMPTQNtQr8v+XBXVVk6XVZFbXEV2YQXZhRXkFFWSVVhBTlEFuzMKOVtec+79DQqE+7oR7e9OjL8HCSGetA3xIjbQAxez7NwVwuGoKnx7t7aSdfMi7ahdIZqY0ioLS/edYuHOTDakVqByiI4RPjx9VTuuaB9CgKfOK7WKAi07a2PYM3AsRVtV3j0Lts8AnwitVjlpEgTE6xuraJTsvewRBCysrdc1AV+pqrrUztfUlauTkTAnN8JauNE5ssVF36ewvJr0grJz41hBGen5ZWw6dprKGhsARoNCjL87CSFetAnxpF2oNx3CfPB2a3rlGkI0Kpunw4FvtB33UX30jkaIemOx2lifdpoFOzJZtj+Xihor4b6ujGll5v6xvYn2d9c7xIszmrXyirih2t2dQ0u0leV1r8PaVyEkWUuU24+TP2jFJbNrgqyq6jEg2Z7XaIx83JzoGOFEx4gLE2irTeXE6TIOnSrhYE4xB3NK2HnyLN/tzj73PtH+7nQI9zk32oR4SYmGEA3l5GZY9jjEXwm979M7GiEum6qqHMgpZuGOLL7dnU1+SRVeLibGdmrJNR1b0jmyBatXr3bc5PjXnD0h+VptlOTCvvnayvKP/9Z+dmMGaMlywihwlo104vdJ4ZwDMRoUYgI8iAnwYGRiyLm3F1fWsC+ziF2Zhew6Wcj61AIW7swCwMloIDncmx4xfnSP9qNTpA9uTvLfKkS9K82HuVO09lJXvyf9WEWjdqqokm92ZZ2rKzYbFQbEB3JNx5YMahOIs6kJlPh5BkHPv2sj/3BtvfIcWHg7mN0g4UotWY4ZKPsIxG/Id0Qj4OViplesP71i/QHtL/6cokp2ZxSyM6OQzelneC8ljbd/SsVkUEgK86Z7jF9t0uwrtcxCXC6bFRZMhfLTMHU5uProHZEQdVZlsbL8QC5zt2Wy9mg+NhU6Rvjwv6vaMSoptGE32zW0gHgY/B8Y+BhkbNYS5X0LYO9ccPPXyi+SJkHLTvLHrwAkQW6UFEU51xHjitqV5tIqC9uOn2HTsTNsTj/NR2uO8X5KGs4mA91j/OgX58+A+ABaBXg0+x7OQtRZygvaJqAxb2v1jEI0Ivuyipi7LYNvd2dTWF5DqLcLdw2M5ZpOYY2ndKK+GAwQ2VMbI16A1BVaCcb2z2DLB+DbSkuUkybIwT/NnCTITYSHs4kB8YEMiA8EoKzKwpbjZ1hzJJ81R/J5ZslBnllykJY+rvRr7c+A+ED6xQXg6iSry0L8oaPLYc1L0OFG6HST3tEIcUnOllXzza4s5mzL5GBOMU4mA8PbBTOhcxi9Y/0xGmShBJOzVmaRcCVUFGqt4vbMgZTnIeU5COumndzX7hpw99M7WtHAJEFuotydTQyMD2RgbcKccaacNUe1ZPm73Tl8vSUDF7OBvnEBDGsbxOBLPQpUiOak8CQsuA2CEuHKV/SORog/ZLHaWHu0gLnbM1h+IJcaq0piS2/+d1U7xiS3lC5If8TVR/sDuNNNUJSp9VbeMxu+/4d2nHzsEC1Zbn0FOLnpHa1oAJIgNxPhvm7c0D2SG7pHUmO1sSX9DMv2n2LZgVyWH8jFoEC3aF+GtQ1mRPtgQn1c9Q5ZCH1ZqrTDQGxWmPg5mOVnQjimY/mlzN2eyYIdmeQWV+Hr7sRNPaOY0CWMhODL7+/f7HiHQZ/7tXFqn5Yo750LR5aCkye0HaMly1F9wSB3YZsqSZCbIbPRQO9Yf3rH+vPkmHbsyypm2YFTLNufy9OLD/D04gN0jWrBmORQrkgMwd9D/6M7hWhwSx+F7J0w6Uvwa6V3NEJcoLLGytJ9p/hqy0m2pJ/BaFAY0DqAp8aEMyghUNp/1pfg9toY8iQcX6eVYBz4FnZ9CZ4hkDgeEidCcKJs7mtiJEFu5hRFITHMm8Qwbx4aFk96QRmLd2ezaHc2//l2P08s2k/vWH9GJ4cyvF0w3q5yi040A3vmwLZPoNc90GaU3tEIcc7R3BK+3pLBgp2ZFJbXEOnnxr9GJDCuU0sCvVz0Dq/pMhghpr82rnxFW03eMwc2vQ8b3oaANtqqcuIEvSMV9UQSZHGBaH937hkcxz2D4zh0qpjvdmfz3e4cHp63h8cX7mNo2yDGdwlDVVW9QxXCPvIOwnf3QUQvGPyk3tEIQWWNle/35vD1lpNsPX4Ws1FheLtgru8WQY8YPwyy4a5hmV2h3VhtlJ2GAwu1ZHnlU7DyKTp4twfPadD2KmkJ2YhJgix+V0KwFwnBXvxjWDy7M4v4ZmcW3+7KYsneHHycFa6rOsSEzmHEBMhpRKKJqCqB2ZPByQMmzJDDA4SuDp8q4estJ1mwI5PiSgvR/u78e2QC4zqF4Selb47B3Q+6TtXGmWOwdx5Om2bAd/dqG/xaD9faxsUN07pmiEZDnv3Fn1IU5dzR1o+OTOCng3lMX7abD1an8X5KGp0jWzCpSzijk0OlbZxovFQVFt0DZ9LgpkXgGax3RKIZqqi2sqR2tXj7ibM4GQ2MaB/Mdd0i6BHjK33sHZlvDPR/mC22rgxo7a2tKu+bBwe/AxdvbcU5aRKE99D6MQuHJgmyqBNnk5ErEkNwPX2Ytp16sGBnFnO3ZfDw/D38b8kBxnUK48YekcQGyqqyaGQ2fwD7F2qbcaL76h2NaGaO5Jbw5aYTLNiZRUmlhZgAdx6/sg3XdAqTFpyNjaJoJ/K17ATDntEOGdozW0uYt38G3hHaQSSJEyEwQe9oxe+QBFn8ZYFeLtzRvxW394th24mzzNx4gi83n+CzDcfpGePHjT0iGdYuCLNR/lIWDi5jCyx7DOJHQq/79I5GNBM1Vhs/7j/FzI0n2Jx+BieTgZG1q8XdomW1uEkwmiBuiDaqSuHQEi1ZXvc6rH0VgpO0VeXE8XLXysFIgiwum6IodI3ypWuULwWlbZmzLYMvN53krq92EODpzA3dI7ixR6S0ixOOqawA5k4Br5Zw9Xty61PY3amiSr7acpJZW06SV1JFuK8rj16RwIQu4bJa3JQ5e0DyJG2U5ML+BVqyvOwxWP4fiO6vJcttRoGzp97RNnuSIIt65e/hzN8HxHJ7v1asPpLHzI0neGPFUd5LSWNcp5bc0juauCD5wRcOwmaF+VO1JHnqcnBtoXdEoolSVZWNx04zc+MJlh3IxaaqDIwPZHKPSPq1DpCjn5sbzyDocac28o/A3jlasvzNHbDYVTv+OmkStBoIRmmvqgdJkIVdGA0KgxKCGJQQRGpeKZ+uT2f+9ky+3pLBgPgApvaJoXesn9xCFPpa/SIcWwWj34KQZL2jEU1QSWUNC3ZkMXPTCVLzSvFxMzO1TzQ3dI8kwk+OLBZAQGsY9DgMfAwyNmuJ8r4F2gY/Nz9oP05Lllt2lsNIGpAkyMLuYgM9eG5sIv8YFs+Xm07w+cYT3PjJZhKCPbmtbwxjOoRKnbJoeEdXwOqXoMMN0OkmvaMRTcyhU8XM3HiChTuzKK+2khzuwysTkhmVFIKLWbr9iItQFIjooY0RL0LqCi1Z3v45bPlQ65KRNEk7jERO97Q7SZBFg/F1d+KewXFM6x/Dol3ZfLIunYfm7ub1FUe4o38rxncOk18comEUnoQFUyGoHYx8RVZlRL2wWG38uD+XzzccZ8vxMzibDIxJDmVyz0iSwuTACFEHJidIGKmNyiI4sEhLllNegJTnIayrliy3Gwvu/npH2yRJgiwanLPJyIQu4YzvHMaqw3m8/VMqj3+zj7dWHmVavxiu7x6Bm5N8awo7sVTBnJu1+uOJX4CT3OYWl6ewvJqvt2Qwc+NxsosqifB147GRbRjfOYwWsulOXC4Xb+g0WRtFmbB3ntYy7vt/wNJHoNVg7Zjr+JHyfFaPJAsRulEUrU55YHwgG9NO886qVJ5ZcpD3UtK4pXcUN/WKwstFNieIevbjY5C9AybOlNuU4rIcyS1hxvrjLNyZSWWNjV6t/HjqqvYMSgiUTXfCPrzDoM/92ji1r3Zz31w4+qN2AmibMVqyHN0PDHJH9nJIgix0pygKvWL96RXrz/YTZ3l3VSqvLDvCB2uOMbVPDLf0icJTEmVRH/bMha0fQc+7oe0YvaMRjZDNprLqcB4z1h9nXWoBziYDYzu2ZErvKBKCvfQOTzQnwe21MfgJOLFeK8E4sAh2fwUewVpv5aSJWq9lKSOrM0mQhUPpHNmCT6d0ZV9WEW//dJTXVxzhsw3p3NG/FTf1jJKjrMVfl3cIvrsXInpqp+UJUQelVRbmbcvgsw3HOX66nCAvZ/45PJ7rukVI72KhL4NRWzGO7qftqTiyVCvB2PwBbHwHAhK0RDlxAvhE6B1toyEJsnBI7Vt688HkLuzJLOTVZUd4/odDfLwunbsHxnJtt3CcTZIoizqoKoE5k7VbkONnSF9RcclOni7nsw3Hmbstg5IqCx0jfHhwWDxXtA+W7jvC8ZhdtY177cZC+RnYv1BLllc+rY3I3lqy3PYq6fv+JyRBFg4tKcyHz2/pxpb0M7yy7DBPLNrPh2uOce/gWMZ1CsMkv6DEn1FVWHQvnE6FmxaBV4jeEQkHp6oqm9PP8PHadFYeysWoKIxMDOFvvaPoGCFJhWgk3Hyh663aOJNeu7lvNnx3H3z/T2g9HBInai9NctLtr0mCLBqFbtG+zJ7Wg3WpBbyy7Aj/mr+XD9cc49Er2jC4TaAcOCJ+35YPtSNdBz8B0X31jkY4MIvVxg/7TvHR2mPsySyihZuZuwbEcmOPSIK9XfQOT4i/zjca+v8T+v0Dsndqq8r75sPB77QuGW2v1trGRfQEgyw8gSTIohFRFIW+cQH0ifVn2YFcXlx6iKlfbKNHjC+PjWxLYpi33iEKR5OxVeta0XoE9L5f72iEgyqtsjB7awafrksnq7CCaH93nh3bnnGdpDe7aGIUBVp20sawZyA9RUuW986FHZ+Dd7hWq5w0CQIT9I5WV5Igi0ZHURSGtwtmUEIgs7ac5I0VRxn9zjqu7hDKP4bHE9ZC+kAKoOw0zDFOQ4kAACAASURBVJ0CXqEwdrqsiojfOFVUyYwN6Xy1+SQllRa6RfnyxOi2DGkThEHatImmzmiC2CHaqCqFw99rJRjr34B1r2ndL5ImaUddN8PSNEmQRaNlNhqY3DOKqzu2ZPrqND5em873+05xS+9o/j6wlfRQbs5sVu2kvLJ8uHWZbEYRF8gosfHgnF0s2pWNTVW5on0IU/tGS32xaL6cPbTNe0kToTQP9i3QkuVlj8Hy/2gdMpImQZvR4Oypd7QNQhJk0eh5upj55/AEbugeySvLDvPBmjTmbMvg4eHxTOwSLitBzdHqlyDtJxj9JoR20Dsa4QBUVWXt0QI+WnuMtUcrcHM6xY09Irm1TzThvnLXSYhzPAKhxx3aKDiqlWDsmQ3f3AmLH9SOv06aBK0GNemOQJIgiyYj1MeV1yZ24Jbe0Tz13X4eWbCXr7ac5Kkx7WRlqDk5ugJWvwjJ10Onm/WORuis2mJj0e5sPl57jEOnSgj0dGZ8azP/uXYg3m5N95e7EPXCPw4GPQYD/w0ZW7REef8CbYOfm59WfpE0CVp2bnKHkUiCLJqc9i29mXN7Txbtzua57w8y9r0NjO8cxr9GJBDgKa1smrTCDK20IrAtXPlqk3vCFpeurMrCrK0ZfLz2GDlFlcQHefLy+CTGdAhl47q1khwLUReKAhHdtTHiBUhbqSXL2z/XOgW1iNYS5aSJ4NdK72jrhSTIoklSFIWrOrRkcJsg3vkplU/WHePHfae4b0gcN/eKkgb/TZGlGubeDFYLTPwCnOS2eXN0tqyazzYc5/ONxyksr6F7tC/PXZPIgNYB0g5SiPpgcoL4K7RRWaS1itszW7tzt/oFaNmldnPfNeDur3e0f5kkyKJJ83A28cgVCUzsEsZT3x3gmSUHmb01g2eubk/3GD+9wxP1adljkLUdJs4E/1i9oxENLLuwgo/XpvP1lpNU1FgZ0iaIOwe0onOklFcJYTcu3tDxRm0UZcG+eVrN8g//hKWPaB0ykiZC/MhGt2hh9wRZUZQRwJuAEfhYVdUX7H1NIX4tJsCDz/7WlZUH83jyu/1M+nATk7qE8+jIBHzcnPQOT1yuvfO023w974a2Y/SORjSg1LxSpq9O45udWajAVR1CuaN/K1oHNY+d9kI4DO+W0Ps+beTu/6W/8vwfwclD64CRNBGi+4PB8fuL2zVBVhTFCLwLDAUyga2KoixSVfWAPa8rxMUoisKQtkH0jvXnzZVH+WjtMVYczOXxUW24ukNLuf3aWOUd0o6SjugJQ57UOxrRQHZlFPJ+SirLDuTibDJwY49IpvaNlj7oQjiCoHYw9CntBNMT67USjAPfwu6vwSMYEsdryXJwkt6R/i57ryB3A1JVVT0GoCjKLOAqwLES5E3Tcanw0TsK0UBcnYw8ckUCV3UI5d8L9/LA7N3M257JM1cnEu3vrnd4oi6qSmHOTdqtu/EzmnTLIaG1aluXWsD7KWlsSDuNl4uJewbGcnOvKPw8ZAOuEA7HYIDovtoY+QocWaqtLG/+ADa+A/7xRHh2hZI24Bmkd7QXUFRVtd8nV5TxwAhVVafW/nsy0F1V1bvPe59pwDSAoKCgzrNmzbJbPBdjri6k25a/g6pyOOFeCgJ6Nuj1G6vS0lI8PDz0DuOy2VSVlAwLc49UU2ODMa3MjIw2Y6rn3slNZb4ayiXNl6rS5uCrBOatZ3fyUxS2cNyVCHtr6t9fNlVlR66VxcdqOF5sw8dZYXiUmQHhJlxNdf9ZberzVd9kvupG5uvPmWpKCMhfT1DuanyKDrCj40sUe8c3eBwDBw7crqpql4s9Zu8V5Is9c12Qkauq+iHwIUCXLl3UAQMG2Dmki+jageJPx9N+/wvQ7XYY9j8wyWrEH0lJSUGX/ys7GATcXVzJU4sPsGBPDgdKXHh5fDKJYd71do2mNF8N4ZLma8tHkLcWBv2HDv3ubZC4HFVT/f6y2lQW78nm3VWpHMktJ9rfnRfHxXB1x5Y4m/56DWNTnS97kfmqG5mvSzUagE0/zKbHiIkO15bT3glyJhB+3r/DgGw7X7PuWkSxs+Pz9K/+CTa9BxmbYcIM8I3ROzLRQAK9XHj3+k5c0zGXfy/cy9Xvref2fjHcNyTusn4RCzvJ3AZLH4XWI6DPg3pHI+pZjdXGNzuzeC8ljfSCMuICPXjz2g6MSgrFKCdjCtGkVLoGOVxyDPZPkLcCcYqiRANZwLXA9Xa+5l+iGsww4nmI7A3f/h0+6A9j3oZ2V+sdmmhAg9sEsSzKl2eXHOC9lDSWH8jl5QnJdAiXGnWHUXYa5twMXiEwdrpW4yaahCqLlfnbs3h/dSoZZypoG+LF+zd0Yni7YDkyXgjRoOyaIKuqalEU5W7gR7Q2b5+qqrrfnte8bG1GQXAizPubdujA8dtg2DNgdtE7MtFAvF3NvDQ+mZGJITy6YC/XvLee2/rG8MDQ1riYZTVZVzYrLLgNyvLg1mXgKj1um4LKGiuzt2YwfXUaOUWVJIf78OTodgxKCJTuMkIIXdi9D7Kqqt8D39v7OvWqRST8bSmsfErbZZmxGSZ81mSOTxSXZkB8IMse6Mdz3x/igzXHWH4wl5fHJ8vBA3pa87J2xOmoNyC0o97RiMtUXm3hy00n+XDtMfJLquga1YIXxyXRN85fEmMhhK7k3uTvMTnB8Gfh2q+h8KRWcrFvvt5RiQbm6WLm+WsS+b9bu1NVY2PC9A28/OMhqi02vUNrflJXQMoLkHQtdJ6idzTiMpRU1vDuqlT6vLiKZ78/SFygB1/f1oM5t/eknxwJLYRwAHLU9J9JGAl3rIV5t2jj+HoY/pyUXDQzfeL8+fGBfjz93X7eXZXG6iP5vDGpA7GBclpXgyjKhPm3QWAbGPW6Q27oEH+upLKGGeuP8/HaYxRXWhgQH8A9g2LpHOmrd2hCCHEBSZAvhU8E/O0HWPk0bHgLMrfAhM+l5KKZ8XA28dL4ZAYlBPHvhXu58q11PHpFAjf1jJINRPZkqdY25VlrYOJM7VAQ0aiUVln4bH06H61Np6iihiFtgrh3cCxJYbL5VQjhmCRBvlRGs9YfOaoPLLwdPugHo9/UjksUzcqI9sF0ivThX/P28OR3B1h5KI+XxycT7C13Fexi2eOQtU37o9Q/Vu9oRB2UVVn4fONxPlpzjLPlNQxOCOT+Ia3rtce4EELYg9Qg11Xr4XDHOu2c8fm3wnf3QU2F3lGJBhbo6cKnU7ryzNXt2Xr8DMPfWMOSPTl6h9X07J0HWz6AHndJy8VGpLzawger0+j70ipeWnqY5HAfvrmrN59M6SrJsRCiUZAV5L/COwymLIGfnoH1b2iHFkz4DPzj9I5MNCBFUbixRyS9WvnxwOxd3PXVDtYcCeeJMW1xc5IfrcuWfxgW3Qvh3WHoU3pHIy5BRbWVLzefYPrqNApKq+kb588DQ1vTKUI6vwghGhf5Lf5XGc3aL+3I3rUlF/1h9BuQNFHvyEQDiwnwYN6dvXhjxRHeS0lj+8mzvHN9RxKCvfQOrdEyWipg9mQwu2p/fBrNeock/kBljZWvNp/k/dVp5JdU0TvWj+lDWtMlSjbfCSEaJymxuFyth2klFyFJ2gEGi+6B6nK9oxINzGw08M/hCfzfrd0pqqhhzDvrmbnpBKqq6h1a46OqtD7yHpw+CuM/Aa9QvSMSv6OyxsrnG47T76VVPL34ALEBHsy5vSdfTu0hybEQolGTFeT64N0Sbl4Mq56Fda/Vllx8DgGt9Y5MNLDesf78cF9fHpqzm/98s4/1Rwt4cVyS3mE1Lls/JihvDQx6HGIG6B2NuIgaq4052zJ4e2Uqp4or6RblyxvXdqBXK3+9QxNCiHohK8j1xWiCIU/ADfOhNBc+HAC7Z+kdldCBv4czM6Z05d8jE1hxMJeRb63l6Fmr3mE1DpnbYemjnPbtAn0e0jsa8StWm8o3O7MY/OpqHlu4j1AfF76c2p3Zt/eQ5FgI0aRIglzf4oZoJRehHbTa5G/ukpKLZshgUJjWrxXz7uyF0aDw/JZK3l2VitUmJRe/q/wMzL0ZPEM42OZ+MMjTk6NQVZVl+08x8s213D97F+7OJmZM6cr8O3vRO1aOhRZCND3yG8gevELhpkXQ75+w60v4aBDkHdI7KqGDDuE+LL63D12DjLz842Gu/2gT2YXSFvA3bDathr80FyZ+jsUsJxQ6ivWpBVz93gamzdxOjdXGO9d3ZMk9fRiYECiJsRCiyZIE2V6MJq2G8sb5UJYPHw2EXV/pHZXQgZeLmTuSnXllQjL7sooYIT2Tf2vNy5C6Aka8AC076R2NAHacPMv1H23iho83k19cyUvjklj2QD9GJYXKyZFCiCZPNunZW+xgreRiwW3wzZ2QvhaufAWc3PWOTDQgRVEY3zmMLpEtuK+2Z/Kqw2E8OaYdHs7N/Mcw7SdIeR6SJkGXW/SOptk7mFPMq8uOsOJgLv4eTjwxui3XdYvAxWzUOzQhhGgwzfw3cwPxCoGbvoXVL8LqlyBrO0z8HALb6B2ZaGBR/u7Mu6Mnb608yrurUtl6/AyvTexA58hmepBCUSbMnwoBCTDqdZBb9ro5XlDG6yuOsGh3Nh7OJv45PJ4pvaJwb+5/wAkhmiUpsWgoBiMM/DdMXggVZ+DDgbDz/0D65DY7ZqOBh4bFM2taTyxWlQnTN/Dc9weprGlmnS4s1TB3ivZy0ky5q6KTU0WVPLpgL4NfW82y/bnc2b8V6x4exF0DYyU5FkI0W/Ls19BaDdRKLuZPhW/vqi25eBWcPfSOTDSwbtG+LL2/L899f4gP1xxjxcFcXh6f3HxWk5f/BzK3yjHtOimqqGH66jQ+XZeOTVWZ3COSvw9sRaCni96hCSGE7iRB1oNnsFZyseZlSHkBsndoSUJQO70jEw3M08XM89ckcmViCP+av4fx0zcwtU80Dw2Lb9o1n/vmw+bp0P1OaDdW72ialSqLlZkbT/DOqlSKKmq4ukNLHhzamnBfN71DE0IIhyElFnoxGGHAI1qiXFmktYLb8YWUXDRTfeL8+fGBflzfLYKP1qYz8s21bEgr0Dss+8g/AovuhbBuMPRpvaNpNmw2lQU7Mhn0ymqeWXKQpDAfFt/Th9cndZDkWAghfkUSZL3F9NdKLiJ6wKJ7YME0qCrROyqhAw9nE8+OTeTLqd2x2FSu/2gzD87eRUFpld6h1Z/qMpgzGUzO2l0Tk5PeETV5qqqScjiPK99ex4NzdtPC3cyXU7vzxS3daBfqrXd4QgjhkKTEwhF4BMKNC2Dtq1q7q+wdMOFzCG6vd2RCB71j/Vn2QD/eXZXK9NVprDyUxyNXJDCpS3jj7j+rqvDd/ZB/WNus6t1S74iavD2ZhTz//SE2HjtNhK8bb13XkVGJIY37+0gIIRqArCA7CoMR+j+sncBXVaqVXGybISUXzZSL2chDw+L54b6+JAR78uiCvUz4YCMHsov1Du2v2/YJ7J0DAx/TNqsKuzleUMZdX+1gzDvrOZxbwpOj27Liwf6MSZZDPoQQ4lLICrKjie77y8Eii++H42th1Bvg4qV3ZEIHsYGezJrWg/k7snju+4Nc+fZaru0azkPD4vH3cNY7vEuXtR2WPgqxQ6HvQ3pH02QVlFbx9sqjfLn5JGajgXsHxXJbvxg8Xcx6hyaEEI2KJMiOyCNAK7lY9xqsehayd2n1miFJekcmdPDzKXxD2wTx5sqjfLHxON/tzuGeQbFM6R2Fs8nBu12Un4E5N4NHEFzzIRjkxlV9q7SovLniKB+uSaPSYuParuHcNziOQC9p2SaEEH+F/KZyVAYD9PsH3LwYasrh4yGw9WMpuWjGvN3M/Hd0W358oB/do315/odDDH1tDUv35aA66veFzaZtPC3N1U6PdPPVO6ImpcZqY+amEzy8poLXVxyhb1wAyx7ox7NjEyU5FkKIyyAryI4uqndtycU0WPIQHF8Ho9+SkotmrFWAB59M6cqaI/k8s+QAd/zfDjqE+/Dw8Hh6xfrrHd6F1r4Kqcu1w3BadtY7miZDVVV+2HeKl388THpBGa1bGPhsak86RTSTQ2aEEMLOZAW5MXD3hxvmweAn4MAi+KAf5OzWOyqhs36tA/j+3r68OC6RvOJKrv94Mzd8vImdJ8/qHZombZVWIpQ4Ebrcqnc0TYKqqmxILWDsexv4+5c7MBsVPrm5C492c5HkWAgh6pEkyI2FwQB9H4QpS8BSpZVcbPlISi6aOZPRwKSuEfz0jwH8d1RbDuWUMPa9DUz7Ypu+HS+KsmD+rRCQAKPfAEU6J1wOVVVZezSfiR9s5PqPN5NbXMlL45P44b5+DG4ThCLzK4QQ9UpKLBqbyJ5aycU3d8D3/9C6XIx5G1yk4X9z5mI2ckufaCZ2DWfGunQ+XHOMZQfWMjghkL8PjKVzZAOuLlqqYe4U7Q+5STPByb3hrt3EqKrKmqMFvLniCDtOFhLi7cLTV7VjYpfwpn0UuRBC6EwS5MbI3Q+umw0b3oKVT2vlFhM+g9COekcmdObhbOKewXHc1DOKLzYe59P16Yx7fwM9Yny5e2AcvWP97L/auPy/kLkFxs8A/zj7XquJstpUlu0/xQdrjrEro5BQbxeeubo9E7qEOX7XEiGEaAIkQW6sDAboc792RPW8W+CTYTDsGeg2TW5nC7zdzNwzOI5b+kTz9ZaTfLT2GDd+spmkMG9u6R3NyMQQnEx2qLDatwA2vw/d74D219T/52/iyqoszNmWwafr08k4U0GErxvPjU1kfOcw+/x/CSGEuChJkBu7iB5aycXCO+CHh2tLLt4BVx+9IxMOwN3ZxNS+MUzuGcm87Zl8sjad+2fv4tnvD3Jj90iu7x5BgGc9HThScBQW3QNh3WDo/+rnczYT2YUVfLHxBF9tPkFxpYXOkS14bGQbhrYNxign3wkhRIOTBLkpcPOF62bBxndg5VNal4sJM6StljjH2WTkhu6RXNc1gjVH8/lsw3FeX3GEd1elMio5hBu6R9Ipwuevl19Ul8HsyWBy1sp9TE71Gn9TZLWppBzO46vNJ1l1OA+AEe2Dmdo3RjpSCCGEzuyWICuK8iRwG5Bf+6Z/q6r6vb2u1+wZDND7XojoCfP+Bp8Mh2H/0251S8mFqGUwKAyID2RAfCBp+aV8seE4c7dnsmBHFrGBHkzsEsbYjmF1W1VWVVj8AOQfgskLwLul/b6AJiCnqII5WzOZvfUk2UWVBHg6c+eAVlzbNYJwXze9wxNCCIH9V5BfV1X1FTtfQ5wvvCvcvga++TssfUQ7WOSqd8BVVqTEhVoFePDUVe3554gEluzJZs62TJ77/hAvLT3MoIRAxncOo398wJ9vCtv2KeyZDQMfg1aDGib4Rqa4soal+07x7a4sNqSdRlWhb5w//x3dlsFtgjAbpb5YCCEciZRYNEVuvnDd17DpPa2jwPR+2m3vMCm5EL/l4WxiUtcIJnWNIDWvhLnbMpm/I4tlB3LxdDYxtF0Qo5NC6R3r/9uNYlk7tD/EYodA33/o8wU4qMoaK6uP5PPtrixWHMyj2mIj0s+NewbFMb5TGBF+sloshBCOSlHtdNBEbYnFFKAY2AY8pKrqb474UhRlGjANICgoqPOsWbPsEs+fKS0txcPDQ5dr25Nn8WHa7X8Fp+rTHIu5mcywMfVSctFU58teGtt8WWwqB05b2XrKyvZcC+UWcDNB5yATHQONtPUz4qGW0GXbQ4DKti6vYTHX3/HnjW2+flZarbI738KOPCv7CqxUWcHLCboFm+gZaiLG22CXNnuNdb70IvNVNzJfdSPzVTd6ztfAgQO3q6ra5WKPXVaCrCjKCiD4Ig89BmwCCgAV+B8QoqrqLX/0+bp06aJu27btL8dzOVJSUhgwYIAu17a7irPwzV1weAnEj4Sr3tVWmS9Dk54vO2jM81VtsbEuNZ/Fu3NYdiCX0ioLzkaY7fkGiVU7ODX+W1q261Ov12ws82WzqRzOLWF9agHLD+Sy7cRZrDaVIC9nhrQJYli7YHq38sNk5xKKxjJfjkLmq25kvupG5qtu9JwvRVF+N0G+rBILVVWHXGIAHwGLL+da4jK4toBrv4TN02HZf7QuF+M/hfBuekcmGgEnk4FBCUEMSgii2mJj2/EzVP30Eh2yt/B4zd/4v5lFRPiuonu0L91j/Oge7UtYC9cmefyxqqqcPFPO+tTTrE8rYFPaaU6XVQOQEOzJnf1bMbRtEIktvTFIezYhhGi07NnFIkRV1Zzaf44F9tnrWuISKAr0uFNLiudOgRlXwOAnoOfdWgcMIS6Bk8lAL8M+yPkIEicwrf+zxB3JZ11qAcsP5jJ3eyYAod4udIv2pUO4D4lh3rQN8cbVqfGdAFdWZWFPZhG7MgrZefIsuzIKySupAiDIy5n+rQPoFetP71g/QrxddY5WCCFEfbHnJr2XFEXpgFZicRy43Y7XEpeqZWe4fS0suhuW/wdOrIer37/skgvRTBRnw7xbwS8ORr1BhLM7N/u7c3OvKGw2lSN5JWxJP8PmY2dYn3aab3ZlA2BQIDbQg/ah3sQFedIqwJ3YQA8ifN3sXn5wKaotNjLPlnMkt4RDp0o4XDuOny7DVluFFu3vTu9YfzpF+NCzlT+tAtyb5Cq5EEIIOybIqqpOttfnFpfJ1QcmzoQtH8Kyx2F6X63kIqK73pEJR2at0e4+1FTApJngfOGmCoNBISHYi4RgL27qGYWqquQWV7E3q4i9WUXsyypifVoBC3ZmnfsYs1EhwteNsBZuhPq40tLHhVAfV06dthKYXUwLdzMt3JxwMf/11edqi40zZdXkl1RRUFpFfkkVeSWVZJ6t4MTpck6eKSenqOJcIqwoEOnrRnywJ6OTQ+kQ7kNyuA++7nL4iRBCNBfS5q25UhTofvuvSi7+C73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   "source": [
    "#\n",
    "#Plot a different natural spline, along with its 1st and 2nd derivatives, each time you run this\n",
    "#\n",
    "%run plothelp.py\n",
    "%matplotlib inline\n",
    "import random\n",
    "import numpy\n",
    "Y = [random.random()*10.0+2 for _ in range(5)]\n",
    "V = naturalSpline(Y)\n",
    "xs = numpy.linspace(0,len(Y)-1, 1000)\n",
    "plt.figure(0, figsize=(12.0,4.0))\n",
    "plt.plot(xs,[hermite_interp(Y,V,x) for x in xs])\n",
    "plt.plot(range(0,len(Y)),[Y[x] for x in range(0,len(Y))], \"bo\")\n",
    "plt.figure(1, figsize=(12.0,4.0));plt.grid(True)\n",
    "plt.plot(xs,[hermite_interp1(Y,V,x) for x in xs])\n",
    "plt.plot(xs,[hermite_interp2(Y,V,x) for x in xs])"
   ]
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   "source": [
    "## The Kernel\n",
    "\n",
    "The kernel decays quickly around $x=0$, which is why cubic splines suffer from very little \"ringing\" -- moving one point doesn't significantly affect the curve at points far away."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "[<matplotlib.lines.Line2D at 0xcd998c8>]"
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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    "#\n",
    "# Plot the kernel\n",
    "#\n",
    "DECAY = math.sqrt(3)-2;\n",
    "vs = [3*(DECAY**x) for x in range(1,7)]\n",
    "ys = [0]*len(vs) + [1] + [0]*len(vs)\n",
    "vs = [-v for v in vs[::-1]] + [0.0] + vs\n",
    "xs = numpy.linspace(0,len(ys)-1, 1000)\n",
    "plt.figure(0, figsize=(12.0,4.0));plt.grid(True);plt.ylim([-0.2,1.1]);plt.xticks(range(-5,6))\n",
    "plt.plot([x-6.0 for x in xs],[hermite_interp(ys,vs,x) for x in xs])"
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    "## Derivation\n",
    "\n",
    "Each segment of the curve is a cubic polynomial, which has 4 unknowns: $Y = Ax^3 + Bx^2 + C +D$.\n",
    "\n",
    "The kernel consists of two *main lobe* segments (for $\\text{x in }[-1,0]$ and $\\text{x in }[0,1]$), and many *side lobe* segments.\n",
    "\n",
    "For each side lobe, the end points are set:\n",
    "\n",
    "$$\n",
    "\\begin{split}\n",
    " y_0 &= 0\\\\\n",
    " y_1 &= 0\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "There are only 2 unknowns left, so specifying the first and second derivatives at one end will fix the first and second derivatives at the other end as well.  There is a linear relationship:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "    y'_1 \\\\\n",
    "    y''_1\n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "    -2 & -\\frac{1}{2} \\\\\n",
    "    -6 & -2\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    y'_0 \\\\\n",
    "    y''_0\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "If $(y'_0,y''_0)$ is an eigenvector of this matrix, then $(y'_1,y''_1)$ will be as well, implying the same for $(y'_2,y''_2)$, etc.  All of the adjacent sidelobes will have the same shape with different amplitudes.\n",
    "\n",
    "The two eigenvectors correspond to exponentially increasing or decreasing sidelobe amplitudes, respectively:\n",
    "\n",
    "$$\n",
    "\\begin{split}\n",
    "\\begin{bmatrix}\n",
    "    -2 & -\\frac{1}{2} \\\\\n",
    "    -6 & -2\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    1 \\\\\n",
    "    2\\sqrt{3}\n",
    "\\end{bmatrix}\n",
    "&=\n",
    "\\frac{1}{\\sqrt{3}-2}\n",
    "\\begin{bmatrix}\n",
    "    1 \\\\\n",
    "    2\\sqrt{3}\n",
    "\\end{bmatrix}\n",
    "\\\\\n",
    "\\begin{bmatrix}\n",
    "    -2 & -\\frac{1}{2} \\\\\n",
    "    -6 & -2\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "    1 \\\\\n",
    "    -2\\sqrt{3}\n",
    "\\end{bmatrix}\n",
    "&=\n",
    "\\left(\\sqrt{3}-2\\right)\n",
    "\\begin{bmatrix}\n",
    "    1 \\\\\n",
    "    -2\\sqrt{3}\n",
    "\\end{bmatrix}\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "To create the kernel, then, we just calculate the main lobes to meet the side lobes with first and second derivaties along these eigenvectors.  The $C^2$-continuity requirement then forces exponentially decaying sidelobes on both sides:\n",
    "\n",
    "$$\n",
    "\\begin{gather}\n",
    "y(0)=1\\\\\n",
    "y'(0) = 0\n",
    "\\\\\n",
    "y(-1) = y(1) = 0\n",
    "\\\\\n",
    "\\frac{y''(-1)}{y'(-1)} = -\\frac{y''(1)}{y'(1)} = 2\\sqrt{3}\n",
    "\\end{gather}\n",
    "$$"
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