{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Taking Derivatives with Vandermonde Matrices"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\n",
        "import numpy.linalg as la\n",
        "import matplotlib.pyplot as pt"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Here are a few functions:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 205,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "if 1:\n",
        "    def f(x):\n",
        "        return np.sin(5*x)\n",
        "    def df(x):\n",
        "        return 5*np.cos(5*x)\n",
        "elif 0:\n",
        "    gamma = 0.15\n",
        "    def f(x):\n",
        "        return np.sin(1/(gamma+x))\n",
        "    def df(x):\n",
        "        return -np.cos(1/(gamma+x))/(gamma+x)**2\n",
        "else:\n",
        "    def f(x):\n",
        "        return np.abs(x-0.5)\n",
        "    def df(x):\n",
        "        # Well...\n",
        "        return -1 + 2*(x<=0.5).astype(np.float)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 206,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f560bd55e48>]"
            ]
          },
          "execution_count": 206,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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kz0iemS00s6Vmlog4YmSS+BmpYmZ/N7MP8u9FtwAx087MnjKztWa2ZB/XFK9u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\nmfOjdAAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f560bd950b8>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "x_01 = np.linspace(0, 1, 1000)\n",
        "pt.plot(x_01, f(x_01))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 246,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([ 0.  ,  0.25,  0.5 ,  0.75,  1.  ])"
            ]
          },
          "execution_count": 246,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "degree = 4\n",
        "h = 1\n",
        "\n",
        "nodes = 0.5 + np.linspace(-h/2, h/2, degree+1)\n",
        "nodes"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Build the gen. Vandermonde matrix and find the coefficients:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 247,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "V = np.array([\n",
        "    nodes**i\n",
        "    for i in range(degree+1)\n",
        "]).T"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 256,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "coeffs = la.solve(V, f(nodes))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Evaluate the interpolant:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 249,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "x_0h = 0.5+np.linspace(-h/2, h/2, 1000)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 250,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "interp_0h = 0*x_0h\n",
        "for i in range(degree+1):\n",
        "    interp_0h += coeffs[i] * x_0h**i"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 251,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f560b9caf98>"
            ]
          },
          "execution_count": 251,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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ySS/gUkrlXUyMNZ/Pjh1QpYrdabySXsCllHKt0FB44glr8ZbUVLvTKAdpi18p\n5ZyMDOjaFerUgVdftTuN19EWv1LK9Xx8YOFCzKpVsGiR3WmUA/Jc+EWknIisF5H9IhItIoHZbHdY\nRPaIyE4R2Zb3qEqpQqtsWT5/4gnSRo+G3bvtTqNuwZkW/3hgvTGmLrAh835WDBBqjLnHGNPUif0p\npQqxux9+mLUREaR26wbnz9sdR+Ugz338IhIHtDHGnBKRykCMMaZ+Ftv9DNxnjDl3i9fTPn6l3Nzh\nw4c5O2QIdxUpQtH168HX1+5IHs/VffyVjDG/z9F6CqiUzXYG+ExEtovIo07sTylVyIWEhFBkxgxO\n//ILKRMm2B1HZaNITk+KyHqgchZPXbcUjzHGiEh2zfWWxpgTIlIBWC8iccaYzXmLq5Qq7Bo1acKX\n//43QaNGQYsW0L273ZHUDXIs/MaYDtk9JyKnRKSyMeakiNwGnM7mNU5k/ntGRD4CmgJZFv7IyMir\nX4eGhhKqq/0o5ZZa9uhhXdDVtSs0aAD16tkdyWPExMQQExPj1Gs408f/b+CcMeYlERkPBBpjxt+w\nTQnA1xhzSUQCgGhgsjHmpmu8tY9fKQ80Zw688grExkKpUnan8UguXXNXRMoBHwDVgcNAX2NMgohU\nAeYYYzqLSC1gRea3FAEWGWP+lc3raeFXyhM9+iicPQvLl1tj/lW+0sXWlVKFT3IytGvHF8HBrE9I\noEhyMmn+/nQcO5bWnTvbnc7t5aXw59jHr5RSTvP354vHH2ftsGH8KyPj6sMTDx0C0OJvA/27SylV\n4NYvXHhd0QeYdugQ62fNsimRd9PCr5QqcEWSk7N83DcpycVJFGjhV0q5QJq/f5aPpxcr5uIkCrTw\nK6VcoOPYsUysXfu6x54uXpwOo0fblMi76cldpVSB+/0E7vOzZuGblES6nx+df/6Z1t9/D1262JzO\n++hwTqWUPY4dg+bNrcVbeve2O43b0uGcSin3ERwMn34KHTtC1arWvD7KJbSPXylln3vugQULoEcP\nrnz3Ham6bq9LaOFXStkrIgIiIzHh4ayYM0eLvwtoV49Syn6PP07xn37iwVmzWOLnR7+hQylatKjd\nqTyWntxVShUOGRmY/v2JP3GC9cOGMXDwYPyzGf+v/uDqFbiUUir/+Pgg775L1YwMHoiKYuHChaSk\npNidyiNpV49SqvAoVgxZuZLaDzxAydq18fPzszuRR9IWv1KqcClfHomOpvKHHyILFtidxiNpi18p\nVfgEB8N9KWOtAAALUElEQVS6dRAaCuXKQbdudifyKNriV0oVTvXqWRd4jRgBmzYBoANA8ocWfqVU\n4XXffbB4MfTujdm5k/nz57N//367U7k9LfxKqcKtfXt46y0kIoKwatX49NNPiY2NtTuVW9Nx/Eop\n9/D++/DUU/y6YgWLduwgJCSEsLAwfLx8AXddbF0p5dnefRcmTCB57Vo+3LULgAEDBuDr62tzMPvo\n7JxKKc82ZAikpeEfEcHADRuIS0316qKfV1r4lVLu5c9/htRUfDp04PaNG+1O45a08Cul3M/IkZCR\nAW3awPr11tBP5TAt/Eop9zRqFBQvDm3bwtq1cNddAJw5c4ZixYpRqlQpmwMWXt59Olwp5d6GDYPX\nXoMOHWDbNgAOHz7M7NmziYuLszdbIaajepRS7i8qCh55BD78ENq04ejRo6xYsYJatWrRqVMnj57b\nX4dzKqW81+efQ//+8MYb0KcPycnJrFmzhqNHj9KjRw+Cg4PtTlggtPArpbzb7t3QpQv89a/WTYS9\ne/ciIjRo0MDudAVCC79SSv3yi7WO74MPwowZ4OHj/LXwK6UUQEIC9OgBgYHW1b4ePMJHl15USimw\nCv7atRAUBC1awMGDN22ybds21q1bR1JSkg0B7aWFXynlmfz94X//gyeegJYtrYVdrnH77beTkpLC\n66+/zldffeVV6/tqV49SyvNt3gz9+sGYMfDMM3DNjJ6nT59m06ZNHD58mPvvv5/mzZu7xYyfm6Ki\niJ45k2nR0drHr5RSWTp61BruWbIkLFgAlStf9/Tp06f54YcfCA0NRSRXddTlNkVFsW7cOKYdOoSA\n9vErpVSWqlWDL76Apk2hceObun4qVqxI27ZtC33RB6yW/qFDef7+PBd+EekjIj+ISLqINM5huzAR\niRORAyLyTF73p5RSTitSBKZOtRZ1GTECxo2DxMRbftvmzZvZsGEDZ86ccUHInKWmppJ8/rxTr+FM\ni/87oAewKbsNRMQXeB0IA24HBoiIZ15FoZRyH6Gh1sVeCQnW5G6ffZbj5vXq1SM9PZ333nuP119/\nnXXr1vHzzz+Tnp7umrzX+OnAAVKOHHHqNZzu4xeRjcDfjTHfZvFcC2CSMSYs8/54AGPMi1lsq338\nSinXW7MGHn8c2rWDF1+ESpWy3dQYw4kTJ9i/fz8HDhygf//++T4LaFJSEmfOnCExMfHmq43PnoWB\nA9l04gTrEhOZdvhwnvr4C3pa5qrA0WvuHwOaFfA+lVLKceHh8N13MGUKNGwITz9tdQH5+9+0qYhQ\npUoVqlSpQmhoaJYvl56ezty5cylVqhQlS5akZMmSlChRAj8/Pxo3vrlXPCUlhU8//ZRLly5x4cIF\nrly5QoUKFahSpcr1hX/lSmsq6sGDab16Naxbx/OzZt10rsIRORZ+EVkPVM7iqQnGmE8deP1cNeEj\nIyOvfh0aGprtgVVKqXxVujT85z/w2GPw979b4/8nTYIBA6zzArkgInTp0oWLFy9y6dIlEhMTOXfu\nXLbb+/n58ac//YnSpUsTGBhIYGDg9SeYz5yx5h3auhXef58YY4j5xz8A8G3ePE+Fv6C7epoDkdd0\n9TwLZBhjXspiW+3qUUoVDp9/DpGRcOIETJwIgwaBn59rM1y5Aq++as03NHSodVK6RImbNrNzyobs\ndrodqCMiISJSFOgHrMynfSqlVMFo184a+jlnjjXmv2ZNqyvo+PGC3/elS1bBr1sXduyALVus4p9F\n0c8rZ4Zz9hCRo0BzIEpE1mQ+XkVEogCMMWnAaGAdsBdYaozZ53xspZQqYCLW6J+NG2H1aqv137Ah\ndO1qTfyWkJB/+zIGtm+3zi3UrGkV+xUrYNkyqFMn//aTSa/cVUopR128aJ1kXbbM6g5q2tT65dCm\nDdxzj3VVsKPOnoWvv7YWi1+71lo8fsgQq1snJMThl9FpmZVSylUuXbK6g2JirH9/+MEaCtqggTUd\nRKVK1nTQv5+oTUiA06chPh6+/x4uX4YmTaBjR2vN4EaN/tg2F7TwK6WUXdLT4aefIC4OTp2ybomJ\nVjcOWFNFV6xo/VK44w5rCol8mB5CC79SSnkZXYhFKaXULWnhV0opL6OFXymlvIwWfqWU8jJa+JVS\nysto4VdKKS+jhV8ppbyMFn6llPIyWviVUsrLaOFXSikvo4VfKaW8jBZ+pZTyMlr4lVLKy2jhV0op\nL6OFXymlvIwWfqWU8jJa+JVSysto4VdKKS+jhV8ppbyMFn6llPIyWviVUsrLaOFXSikvo4VfKaW8\njBZ+pZTyMlr4lVLKy2jhV0opL6OFXymlvIwWfqWU8jJa+JVSysto4VdKKS+jhV8ppbyMFn6llPIy\neS78ItJHRH4QkXQRaZzDdodFZI+I7BSRbXndn1JKqfzhTIv/O6AHsOkW2xkg1BhzjzGmqRP78xox\nMTF2Ryg09Fj8QY/FH/RYOCfPhd8YE2eM2e/g5pLX/XgjfVP/QY/FH/RY/EGPhXNc0cdvgM9EZLuI\nPOqC/SmllMpBkZyeFJH1QOUsnppgjPnUwX20NMacEJEKwHoRiTPGbM5tUKWUUvlDjDHOvYDIRuDv\nxphvHdh2EpBojJmRxXPOBVFKKS9ljMlVd3qOLf5cyHKnIlIC8DXGXBKRAKAjMDmrbXMbXCmlVN44\nM5yzh4gcBZoDUSKyJvPxKiISlblZZWCziOwCYoFVxphoZ0MrpZTKO6e7epRSSrkXl165KyJhIhIn\nIgdE5JlstpmZ+fxuEbnHlflc6VbHQkQGZR6DPSLylYjcZUdOV3DkfZG5XRMRSRORnq7M50oOfkZC\nMy+I/F5EYlwc0WUc+IwEichaEdmVeSyG2RCzwInIPBE5JSLf5bBN7uqmMcYlN8AXOAiEAH7ALqDB\nDdtEAKszv24GbHVVPlfeHDwWLYAymV+HefOxuGa7z4FVQC+7c9v4vggEfgCCM+8H2Z3bxmMRCfzr\n9+MAnAOK2J29AI7FA8A9wHfZPJ/ruunKFn9T4KAx5rAxJhVYAjx0wzbdgAUAxphYIFBEKrkwo6vc\n8lgYY7YYY37NvBsLBLs4o6s48r4AGAMsA864MpyLOXIsBgLLjTHHAIwxZ12c0VUcORYngNKZX5cG\nzhlj0lyY0SWMNfz9Qg6b5LpuurLwVwWOXnP/WOZjt9rGEwueI8fiWsOB1QWayD63PBYiUhXrQ/9m\n5kOeemLKkfdFHaCciGzMvCjyYZelcy1HjsUcoKGIHAd2A+NclK2wyXXdzK/hnI5w9MN647BOT/yQ\nO/wziUhb4M9Ay4KLYytHjsWrwHhjjBERwXOnAHHkWPgBjYH2QAlgi4hsNcYcKNBkrufIsZgA7DLG\nhIpIbawLRO82xlwq4GyFUa7qpisLfzxQ7Zr71bB+M+W0TXDmY57GkWNB5gndOUCYMSanP/XcmSPH\n4l5giVXzCQLCRSTVGLPSNRFdxpFjcRQ4a4y5AlwRkU3A3YCnFX5HjsX9wDQAY8whEfkZqAdsd0nC\nwiPXddOVXT3bgToiEiIiRYF+wI0f3JXAEAARaQ4kGGNOuTCjq9zyWIhIdWAFMNgYc9CGjK5yy2Nh\njKlljKlpjKmJ1c8/ygOLPjj2GfkEaCUivpkXSDYD9ro4pys4cizigAcBMvu06wE/uTRl4ZDruumy\nFr8xJk1ERgPrsM7YzzXG7BORkZnPzzbGrBaRCBE5CFwGHnFVPldy5FgALwBlgTczW7qpxgOntXbw\nWHgFBz8jcSKyFtgDZABzjDEeV/gdfF/8E3hHRHZjNWKfNsacty10ARGRxUAbICjzotlJWF1+ea6b\negGXUkp5GV16USmlvIwWfqWU8jJa+JVSysto4VdKKS+jhV8ppbyMFn6llPIyWviVUsrLaOFXSikv\n8/8fyAPKscyX9AAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f560ba11588>"
            ]
          },
          "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": [
        "Now build the gen. Vandermonde matrix $V'=$`Vprime` of the derivatives:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 252,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "def monomial_deriv(i, x):\n",
        "    if i == 0:\n",
        "        return 0*x\n",
        "    else:\n",
        "        return i*nodes**(i-1)\n",
        "\n",
        "Vprime = np.array([\n",
        "    monomial_deriv(i, nodes)\n",
        "    for i in range(degree+1)\n",
        "]).T"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Compute the value of the derivative at the nodes as `fderiv`:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 257,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "fderiv = Vprime.dot(la.inv(V)).dot(f(nodes))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "And plot vs `df`, the exact derivative:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 258,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f560b9168d0>"
            ]
          },
          "execution_count": 258,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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QljoRkYOw1ImIHISlTkTkICx1IiIHYakTETnI/wdtQeOamDJ+YAAAAABJRU5E\nrkJggg==\n",
            "text/plain": [
              "<matplotlib.figure.Figure at 0x7f560b947470>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "pt.plot(x_01, df(x_01), \"--\", color=\"gray\", label=\"$f$\")\n",
        "pt.plot(nodes, fderiv, \"or\")\n",
        "pt.legend(loc=\"best\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "* Why don't we hit the values of the derivative exactly?\n",
        "* Do an accuracy study."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 255,
      "metadata": {
        "collapsed": false
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "1.85604897639\n"
          ]
        }
      ],
      "source": [
        "print(np.max(np.abs(df(nodes) - fderiv)))"
      ]
    }
  ],
  "metadata": {},
  "nbformat": 4,
  "nbformat_minor": 0
}