Selected topics from Numerical Analysis with code and explanations in some cases.

Numerical_Analysis_Ipython_Notebook.ipynb

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      "# Numerical Analysis #\n",
      "\n",
      "This is an IPython Notebook where I have implemented most of the algorithms we covered in the course. Most are also explained, with some motivation and sometimes derivations.\n",
      "\n",
      "##Topics##\n",
      "\n",
      "Things in **bold** are not in the text/course, but I thought they were interesting so I included them.\n",
      "\n",
      "### Taylor series approximation ###\n",
      "* Taylor series approximation\n",
      "\n",
      "### Root finding ###\n",
      "* Bisection method\n",
      "* Fixed point iteration\n",
      "* Newton's method\n",
      "* Secant method\n",
      "* Method of false position\n",
      "  \n",
      "### Function interpolation ###\n",
      "* Lagrange polynomial\n",
      "* Neville's method\n",
      "* Divided differences\n",
      "* **Radial basis function interpolation**\n",
      "* **Nearest neighbor interpolation**\n",
      "\n",
      "### Solving Linear Systems ###\n",
      "* Gaussian Elimination\n",
      "* Partial pivoting\n",
      "* Scaled partial pivoting\n",
      "* Jacobi method\n",
      "* Gauss-Sidel method\n",
      "* LU decomposition\n",
      "* LDL decomposition\n",
      "* Cholesky decomposition\n",
      "* Power method\n",
      "* **Newton's method for linear least squares**\n",
      "    \n",
      "### Numerical Calculus ###\n",
      "\n",
      "* Numerical Differentiation\n",
      "* Richardson Extrapolation\n"
     ]
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     "source": [
      "There are some explainations and derivations scattered throughout this notebook. Usually I derived something to better understand it, and I skipped derivations for most linear algebra algorithms because it was a pain to LaTeX. \n",
      "\n",
      "###Derivations & Explainations###\n",
      "\n",
      "* [Fixed point iteration](#Fixed-Point-Method)\n",
      "* [Newton's method](#Newton's-Method)\n",
      "* [Secant Method](#The-Secant-Method)\n",
      "* [Method of False Position](#The-method-of-false-position)\n",
      "* [Lagrange interpolating polynomial](#Lagrange-interpolating-polynomial)\n",
      "* [Cubic spline interpolation](#Cubic-spline-interpolation)\n",
      "* [Solving a linear system with Newton's method](#Solving-a-linear-system-with-Newton's-method)\n",
      "* [Richardson Extrapolation](#Richardson-extrapolation)\n"
     ]
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     "source": [
      "## Polynomial (Taylor series) approximation ##\n",
      "Let $f(x) = sin(x)$. "
     ]
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     "input": [
      "%pylab inline\n",
      "f = lambda x: sin(x)\n",
      "xvals = arange(-5, 5, 0.05)\n",
      "plot(xvals, f(xvals))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
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        "[<matplotlib.lines.Line2D at 0x9e81ef0>]"
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       "text": [
        "<matplotlib.figure.Figure at 0x9e58ba8>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's approximate $f$ with a linear function at 0. Using a Taylor series, we know this approximation is given by $f \\approx f(x) + f'(x)x$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_d1 = lambda x: cos(x)\n",
      "f_approx_linear = lambda x: f(0) + f_d1(0)*x\n",
      "plot(xvals, f(xvals))\n",
      "plot(xvals, f_approx_linear(xvals))\n",
      "plot(0.0, f(0.0), 'b.', markersize=10)\n",
      "legend(['function', 'approx'])\n",
      "ylim([-2,2])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MGCAnh4Zvqivk2ygo6gRTkhcBuRwYPRr49FOhI5EOqugl7LvveBvYY8eoqq/K\nwT9UGLDLE6mzktGyMSV6McjM5BuUJCUB1DGlalTRm7hRo4D8fODQIaEjEbfgzVHoVS+YkryI2NkB\nY8fyLTOJ9qiil7ht2/gilJMnqaqvyI7DKrx5yBOq95Nhb0OJXkxyc3kr4zNn+EVaUjGq6AnefBO4\nd4+anVWEMSB0WxT6Nw6mJC9CtrbA1KlANbrwkuegit4E7N0LfPABcO4cUKuW0NGIx+afVJhw0hOZ\nHySjaT1K9GJ0+zZv1qdU8uqelEcVPQEADBrEG0Vt3ix0JOLx6BEwa0cU/JoHU5IXMWtrYM4c4KOP\nhI7EuFFFbyJOnAD8/XkfkRdffP7xUrfqWxWmJ3kiKywZTepQohez+/eBNm2oBXdlqKInat2786Xl\nK1cKHYnwSkuBDw9EYUSrYEryRuCll/iWg7SvrOaoojchSUmAQsGr+vr1hY5GOJ+sUWFeOq/mG1tR\nojcGpaVAu3a8UOnTR+hoxIUqevIUV1dgyBDgk0+EjkQ4RUVAxOEovNU2mJK8ETE3BxYt4uP1ZWVC\nR2N8qKI3MY/7iFy4ADRvLnQ0hjdjgQpfMT7TppElJXpjwhjQowcwaRIwcaLQ0YiHXiv6vLw8+Pr6\nok2bNujbty8KCgoqPM7BwQEdOnSAh4cHutKVFMHJ5cDkycC8eUJHYngZGcCai1GY7BFMSd4IyWTA\nZ5/xGThFRUJHY1w0rujnzJmDRo0aYc6cOYiOjkZ+fj4WL15c7rhWrVrh9OnTsHnOljFU0RvOnTuA\nszPfX7ZLF6GjMZzhQSockHvi+hyq5o3ZW2/xrQcXLhQ6EnHQa0W/Z88eBAQEAAACAgKwa9euSo+l\nBC4u9erx8c6ZM/nbYVNw6hQQVxSFqV5UzRu7qCh+UTY9XehIjIfGiT43Nxe2/7aVs7W1RW5uboXH\nyWQy9OnTB507d8batWs1PR3RsQkTgAcPgO+/FzoS/WMMeGeeCrJ2sfjgtVlCh0O01KIFb41A0y2r\nz7yqO319fZGTk1Pu6x9//PFTt2UyGWSVdMw6fvw4mjVrhps3b8LX1xfOzs7w9vau8Njw8HD15wqF\nAgqF4jnhE02ZmQGff8771r/+OmBpKXRE+rNrF5DSNAqh3aial4q5c/kiqoQE01tEpVQqoazh9nEa\nj9E7OztDqVSiadOmyM7Oho+PD/7+++8qHxMREYE6dergvffeKx8IjdELwt+fj9c/8TdWUu7fB9p6\nqVAwiu9a/ksJAAAPxklEQVQeRYleOtavB775hu+3YGbCE8X1Okbv5+eHjRs3AgA2btyIoUOHljvm\n3r17KCwsBADcvXsXP//8M9q3b6/pKYkefPIJsGKFdMc7Fy8GLHyiEPoKVfNSExDA59T/m4ZIFTSu\n6PPy8jBy5Ehcv34dDg4O2L59O+rXr4+srCxMnjwZ+/btw7Vr1zBs2DAAQGlpKd566y2EVTKwRhW9\ncMLDgb/+An74QehIdCslBejiqwKCPXFlOlXzUnT6NDBwIF/13bCh0NEIozq5kxZMERQXA+3b8w1K\nBg8WOhrdYIwngIIeIfB5xQaRvSOFDonoybRpQEkJsGaN0JEIg1ogkGp58UXgq6+A0FDg7l2ho9GN\nHTuAq/+ocNkiFrO60UwbKfvf//jGOvHxQkciXpToCQCgd2/A21saF2Xz84Hp0wHXKVEI7kRj81JX\nvz6wZAkwZQqv7El5NHRD1G7cANzcgLg43tLYWE2cCDyqp8JeO08kh9LYvClgjDfs69IFWLBA6GgM\ni4ZuSI00aQIsXcpnMzx4IHQ0mjl4kG87Z66gat6UyGR8jH7FCuD8eaGjER+q6MlTGAOGDQNcXIBI\nI7t+WVjI35F8vEKFGZeomjdF69YBq1YBf/zBWxubApp1QzSSmwu4u/MVpa+8InQ01RcYyDc/r/V6\nCGxeopk2pogxoH9/3s74//5P6GgMozq500T+5pGasLXlb4EDAoAzZwArK6Ejer7t2/kKyV1KFby3\nxCI5NFnokIgAZDK+YrZTJz7BoHt3oSMSB6roSaUmTOD/btggZBTPp1Lxi3D79wPf5FA1T4A9e4AZ\nM4DEROlvm0lDN0Qrd+/yBDpnzn9JX2xKS4FevfhCr1FTVPD8msbmCRcaCty8yTu0VtJzURIo0ROt\nXbzINxRXKvnmzGIzezbfFvHAAWDqfqrmyX/u3+fXmCZP5klfqmiMnmitXTvg00+B4cP5ykMxvQ3+\n9lt+wTghAUi/o0JsEo3Nk/+89BKwcyfQrRufjWXKXc+poifVMn06cOkSHwe3sBA6Gt7Mqn9/4PBh\n3qcnZC9V86Riv/7Ktx+MjwccHISORvdo6IboTGkp36BELgdWrxZ2zDM9HXj1Vb5R9IgRgKqAxuZJ\n1b74gs/G+f13wNpa6Gh0i1bGEp0xN+cXtU6c4EM5Qrl1C+jbl+93O2IE/1rUMVoFS6o2fTrv5TRk\nCHDvntDRGB5V9KRGMjKA117jU9emTzfsuQsL+Qybvn2Bx7tZUjVPquvRI7425J9/+LWd2rWFjkg3\naOiG6IVKBfj4ALNmGW42Q0EBr8batXt66IjG5klNPHzI3wnWrg1s2QK88ILQEWmPhm6IXrRsyS+C\nLl3Kh3H0/fc5Kwvo2ZN31Fy16r8kryrgM22o3zypLgsLYNs2vgXhwIHA7dtCR2QYlOiJRhwc+IWt\n777jPWb01e3y0iV+4XX0aODzz5/eBJrG5okmXnwRiI0FnJ15AZGZKXRE+keJnmhMLuf9Ze7c4X1F\nMjJ099yMATEx/ALaggVAWNjTM32omifaqFWL93MaMwbo3BnYt0/oiPRL40QfGxuLdu3aoVatWjhz\n5kylx8XFxcHZ2RlOTk6Ijo7W9HREpKyseHXUvz/g4cG3JHz0SLvnzMvjL8ClS/mK3IraL1A1T7Ql\nkwFz5/KhnHfeAaZOlc5Wms/SONG3b98eO3fuRM+ePSs9pqysDKGhoYiLi0NSUhK2bt2KS5cuaXpK\nIlJmZsBHH/GkvGEDX4Goyf6d9+/z+c7OznwTlD//5Csan0XVPNGlnj2Bs2f5rC5nZ2DjRr5upCYY\n4wWKWGmc6J2dndGmTZsqj0lISEDr1q3h4OAACwsL+Pv7Y/fu3Zqekohcu3bA8ePAuHHAqFF8Gubm\nzXwP18owxnvVfPgh8PLL/CLvL7/whP/SSxU/hqp5omv16/Pf1a1b+cIqJyc+hffq1aofl5vLZ4F5\negIhIYaJVRN67XWTmZkJuVyuvm1vb48//vhDn6ckAqtVizeRmjCBt4rduJG/LXZ05NV5s2Z85sOd\nO0BaGm9l8MILfFerX38FXF2rfv7H1Tz1tCH60KMHcOQIfze5fj2//eKLfFiyRQugTh2guJhfwD1/\nnl+XGjgQWLwY8PUVOvrKVZnofX19kZOTU+7rkZGRGDJkyHOfXFbDdfLh4eHqzxUKBRSm3IXIyFlY\n8EZow4fzF8bZs8DffwM5OfxtcZMmfC7+F18ArVpVv6UCVfPEELp04R+rVgHJyTypZ2YCRUVAvXo8\n8b//PtChg+G3LFQqlVAqlTV6TJUhHjp0SJt4YGdnh/T0dPXt9PR02NvbV3r8k4meSMeLL/J2sdpu\nS0jVPDE0mYyP2zs7Cx3Jf54tgiMiIp77GJ1Mr6xsVVbnzp1x5coVpKWloaSkBNu2bYOfn58uTklM\nEFXzhGhG40S/c+dOyOVyxMfHY9CgQRgwYAAAICsrC4MGDQIAmJubY8WKFejXrx9cXV0xatQouLi4\n6CZyYlJopg0hmqNeN8QoUE8bQipGO0wRSaCxeUK0Qy0QiOjR2Dwh2qGKnogaVfOEaI8qeiJqVM0T\noj2q6IloUTVPiG5QRU9Ei6p5QnSDKnoiSlTNE6I7VNETUaJqnhDdoYqeiA5V84ToFlX0RHSomidE\nt6iiJ6JC1TwhukcVPREVquYJ0T2q6IloUDVPiH5QRU9Eg6p5QvSDKnoiClTNE6I/VNETUaBqnhD9\noYqeCI6qeUL0iyp6Ijiq5gnRL40TfWxsLNq1a4datWrhzJkzlR7n4OCADh06wMPDA127dtX0dESi\naC9YQvRP46Gb9u3bY+fOnQgODq7yOJlMBqVSCRsbG01PRSSMqnlC9E/jRO/s7FztY2nTb1IRGpsn\nxDD0PkYvk8nQp08fdO7cGWvXrtX36YgRoWqeEMOosqL39fVFTk5Oua9HRkZiyJAh1TrB8ePH0axZ\nM9y8eRO+vr5wdnaGt7e3ZtESyaBqnhDDqTLRHzp0SOsTNGvWDADQuHFjvPHGG0hISKg00YeHh6s/\nVygUUCgUWp+fiBNV84RoRqlUQqlU1ugxMqblALqPjw+WLFmCTp06lbvv3r17KCsrQ926dXH37l30\n7dsXCxYsQN++fcsHIpPRWL6JUBWo4Pm1J5JDkynRE6Kl6uROjcfod+7cCblcjvj4eAwaNAgDBgwA\nAGRlZWHQoEEAgJycHHh7e6Njx47w8vLC4MGDK0zyxLRQNU+IYWld0esKVfSmgap5QnRLrxU9IZqg\nap4Qw6NeN8RgaKYNIcKgip4YDFXzhAiDKnpiEFTNEyIcquiJQVA1T4hwqKInekfVPCHCooqe6B1V\n84QIiyp6oldUzRMiPKroiV5RNU+I8KiiJ3pD1Twh4kAVPdEbquYJEQeq6IleUDVPiHhQRU/0gqp5\nQsSDKnqic1TNEyIuVNETnaNqnhBxoYqe6BRV84SID1X0RKeomidEfKiiJzpD1Twh4kQVPdEZquYJ\nESeNE/37778PFxcXuLu7Y9iwYbh9+3aFx8XFxcHZ2RlOTk6Ijo7WOFAibo+r+VndZgkdCiHkGRon\n+r59++LixYs4d+4c2rRpg6ioqHLHlJWVITQ0FHFxcUhKSsLWrVtx6dIlrQI2VkqlUugQ9EapVEq6\nmpfy/x1A358p0DjR+/r6wsyMP9zLywsZGRnljklISEDr1q3h4OAACwsL+Pv7Y/fu3ZpHa8Sk/Mu2\nK26XpKt5Kf/fAfT9mQKdjNGvX78eAwcOLPf1zMxMyOVy9W17e3tkZmbq4pRERI5dPybZap4QKahy\n1o2vry9ycnLKfT0yMhJDhgwBAHz88ceoXbs2xowZU+44mUymozCJWKkKVLh48yLiusUJHQohpDJM\nCzExMax79+7s/v37Fd5/8uRJ1q9fP/XtyMhItnjx4gqPdXR0ZADogz7ogz7oowYfjo6Oz83VMsYY\ngwbi4uLw3nvv4ciRI2jUqOK37KWlpWjbti1+/fVXNG/eHF27dsXWrVvh4uKiySkJIYRoQOMx+mnT\npqGoqAi+vr7w8PDA1KlTAQBZWVkYNGgQAMDc3BwrVqxAv3794OrqilGjRlGSJ4QQA9O4oieEEGIc\nRLUy9ssvv4SLiwvc3Nwwd+5cocPRi6VLl8LMzAx5eXlCh6JT1V1AZ2ykvOAvPT0dPj4+aNeuHdzc\n3LB8+XKhQ9K5srIyeHh4qCePSElBQQFGjBgBFxcXuLq6Ij4+vvKDa34JVj8OHz7M+vTpw0pKShhj\njN24cUPgiHTv+vXrrF+/fszBwYH9888/QoejUz///DMrKytjjDE2d+5cNnfuXIEj0l5paSlzdHRk\nqamprKSkhLm7u7OkpCShw9KZ7OxslpiYyBhjrLCwkLVp00ZS3x9jjC1dupSNGTOGDRkyROhQdG78\n+PFs3bp1jDHGHj58yAoKCio9VjQV/erVqxEWFgYLCwsAQOPGjQWOSPdmzZqFTz75ROgw9KI6C+iM\njdQX/DVt2hQdO3YEANSpUwcuLi7IysoSOCrdycjIwP79+zFp0iQwiY1Q3759G0ePHkVgYCAAfj3U\n2tq60uNFk+ivXLmC33//Ha+88goUCgVOnToldEg6tXv3btjb26NDhw5Ch6J3lS2gMzamtOAvLS0N\niYmJ8PLyEjoUnXn33Xfx6aefqgsQKUlNTUXjxo0xceJEeHp6YvLkybh3716lxxu0TXFlC7A+/vhj\nlJaWIj8/H/Hx8fjzzz8xcuRIXLt2zZDhaa2q7y8qKgo///yz+mvGWGFou4DO2JjKgr+ioiKMGDEC\nX3zxBerUqSN0ODqxd+9eNGnSBB4eHpJsgVBaWoozZ85gxYoV6NKlC2bOnInFixdj4cKFFT/AMKNJ\nz9e/f3+mVCrVtx0dHdmtW7cEjEh3Lly4wJo0acIcHByYg4MDMzc3Zy1btmS5ublCh6ZTz1tAZ2xq\nsuDPWJWUlLC+ffuyZcuWCR2KToWFhTF7e3vm4ODAmjZtyiwtLdm4ceOEDktnsrOzmYODg/r20aNH\n2aBBgyo9XjSJ/quvvmLz589njDGWnJzM5HK5wBHpjxQvxh44cIC5urqymzdvCh2Kzjx8+JC9/PLL\nLDU1lT148EByF2MfPXrExo0bx2bOnCl0KHqlVCrZ4MGDhQ5D57y9vVlycjJjjLEFCxawOXPmVHqs\naHaYCgwMRGBgINq3b4/atWtj06ZNQoekN1IcEpg2bRpKSkrg6+sLAOjWrRtWrVolcFTaeXLBX1lZ\nGYKCgiS14O/48ePYsmULOnToAA8PDwBAVFQU+vfvL3BkuifF19yXX36Jt956CyUlJXB0dERMTEyl\nx9KCKUIIkTjpXY4mhBDyFEr0hBAicZToCSFE4ijRE0KIxFGiJ4QQiaNETwghEkeJnhBCJI4SPSGE\nSNz/A7YRsQoHEfVyAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9e68f98>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's generalize this to approximations at any point."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_approx_linear = lambda x_0, x: f(x_0) + f_d1(x_0)*(x-x_0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we can approximate $f$ at several points."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(xvals, f(xvals))\n",
      "plot(xvals, f_approx_linear(-2, xvals))\n",
      "plot(xvals, f_approx_linear(1, xvals))\n",
      "plot(-2, f(-2), 'b.', markersize=10)\n",
      "plot(1, f(1), 'b.', markersize=10)\n",
      "legend(['f','approx at -2', 'approx at 1'])\n",
      "ylim([-2,2])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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GDA0ZS08XOhLVJ0nulOqKPjo6GhYWFjAzM4OOjg68vLwQFhZW7JgjR45g+PDh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\nAz3EgCybAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa0449b0>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Since generalization is fun, let's make this work for an arbitrary degree polynomial approximation, $P_n$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def factorial(n):\n",
      "    return 1 if n==0 else n*factorial(n-1)\n",
      "\n",
      "def P_n(derivatives, x_0, x):\n",
      "    #finite taylor series, baby\n",
      "    return sum( f_i(x_0)*((x-x_0)**n)/factorial(n) for n,f_i in enumerate(derivatives))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "That should do it."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "derivatives = [lambda x: sin(x), lambda x: cos(x), lambda x: -sin(x), lambda x: -cos(x)]\n",
      "plot(xvals, f(xvals))\n",
      "plot(xvals, P_n(derivatives, x_0=0.0, x=xvals))\n",
      "plot(0.0, f(0.0), 'b.', markersize=10)\n",
      "ylim([-2,2])\n",
      "legend(['function', 'degree 3 approx at 0.5'])\n",
      "show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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xT28/Hk66w3coVapnXQ8tXFpg7+29iAqKkuqYb78FmjUDBg8GXF2VGx+Rj1yp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\nIYRoOUr0hBCi5f4PKsLgeggplIkAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa02c978>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can see that the linear approximation is a special case of the taylor series,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "derivatives = [lambda x: sin(x), lambda x: cos(x)]\n",
      "plot(xvals, f(xvals))\n",
      "plot(xvals, P_n(derivatives, x_0=0.5, x=xvals))\n",
      "plot(0.5, f(0.5), 'b.', markersize=10)\n",
      "ylim([-2,2])\n",
      "legend(['function', 'linear approx at 0.5'])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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54MED3kk03+zzs2GbPBEj+lCRV4RGjYRVWTdu5J1EizA5ZGVlMRsbGxYVFcWk\nUilzdnZm4eHhuY45ceIE69y5M2OMseDgYObm5pbvc8kZRSstXcpYv368U2i2ay+usRrLarGq1VJZ\nQgLvNJrj/n3GzM0ZS0nhnUT9Fad2ytWiDwkJQf369WFtbQ19fX0MGDAAR48ezXXMsWPHMHToUACA\nm5sb3r17h8TERHlOS/7fhAnApUtCFw5RvByWg6mnp8L+ZQBGDzOAuTnvRJrD0REQi4H163kn0Q5y\nFfq4uDhYWVnJbltaWiIuLq7IY2JjY+U5Lfl/BgbAjBmAry/vJJpp3/19+JSejdBdgzFjBu80mmf+\nfGDVKmFDcaJcevI8WFTMfdPYFxcKCnqc72cVSywWQywWlzaa1hg3Dli5Erh7V9jRhyhGWmYaZp2f\nBden+9HlBx1Uq8Y7keaxtwfatQPWrRO2HSTFI5FIIJFISvQYuQp9rVq1EBMTI7sdExMDyy/Gnn15\nTGxsLGrVqpXv8/lS07TEKlUCZs4UWvV//sk7jeZYcW0FXExa4eqiVgikyWlKM38+0Lq10A1pZMQ7\njXr4shHs5+dX5GPk6rpp2rQpnjx5gujoaEilUhw4cABeXl65jvHy8sLu3bsBAMHBwahatSrMqbNT\noX74QVjV8vZt3kk0Q+yHWKy5sQaVri7FuHGAqSnvRJrLzk4Yarl2Le8kmk3ucfSnTp2Ct7c3srOz\nMXLkSMyePRubN28GAIwZMwYAMHHiRAQFBcHAwACBgYFo3Lhx3iA0jl4u69YJk6j++ot3EvX3/ZHv\nUTnHCr+P8cfjx4CJCe9Emu3JE+Drr4Vx9VWq8E6jfmiZYi2Sng7Y2gJ//CGsJ0JKJyQuBD3290Db\nBxGwsTJEMT4VEwUYNkzYZ2HBAt5J1A8Vei2zaZOwjPGpU7yTqCfGGFrtaAWvWqOxYvBwREYCVavy\nTqUdnj4F3NyE1r2xMe806oXWutEyI0YA4eHA9eu8k6inAw8OID0rHQ9+G4rJk6nIlyUbG6B7d+Dn\nn3kn0UzUotcwW7cCBw8CZ87wTqJePmV+gt16O/g32YOpvVpTfzEHUVFAs2ag6yIlRC16LTRsmHBR\n68oV3knUy6rrq9C8VnMEbW6NKVOoyPNQty7Qq5cwL4QoFrXoNVBgoLBW/YULvJOoh/iUeDTa2Aj7\n2tzEd13rIjKSxnTz8vy5sEFJRARgZsY7jXqgi7FaKitLGJ+8bZuwnggp3PCjw2FuYI7n25egUSOa\npcnbuHFQjyqJAAAXCklEQVTCJ6olS3gnUQ9U6LXY7t3A9u2ARAIUc6UKrXQ7/ja67euGP9tFwKuj\nESIjAUND3qm0W0yMsJzHw4dA9eq806g+6qPXYoMGAQkJ1H1TGMYYvE97Y2GbhVgZYIRp06jIqwIr\nK2DgQGD5ct5JNAcVeg2lpydMPpk/H6APSvk7FH4IKRkpaKo3HJcvC+utENUwezawYwdAK5orBhV6\nDda/P5CcDJw9yzuJ6knPSofPOR/83PFnLJinCx8f2tZOldSqBQweLGyZSeRHffQa7sABYRLK9evU\nV/+5JX8vQUhcCGbUPox+/YQZmRUq8E5FPpeYKCxlfOcOUKcO7zSqi/roCfr2BdLSaLGzzyWkJmDF\ntRVY1n455swRlnimIq96zM2B8eNB6w0pALXotcDx48CsWUBYGKCryzsNf6OOjYJxBWN4YDkmTwb+\n+Ue4pkFUz/v3wmJ9EonQuid5UYueAAC6dhUWitqzh3cS/kJfhuL44+OY883/MGcOsHAhFXlVVqUK\n4OMD/O9/vJOoN2rRa4lr14ABA4R1RLS1m4Ixhja72mCg40CYRI3BkiXChi061NxRaZ8+AQ0a0BLc\nBaEWPZH5+mthavn69byT8HPk0REkfUrC0EYjMW+eMKKDirzqq1hRGCZMM5ZLj1r0WiQ8XFgS4fFj\n7VuCNyMrA/Yb7LGl2xZEX2yHvXuFyWQ0Ekk9ZGUBDg5CQ6V9e95pVAu16Eku9vaApyewbBnvJGVv\n7Y21cKzuCLfq7TB/vrCOChV59aGnByxaJPTXZ2fzTqN+qEWvZf5dR+T+fcDCgneaspGYmgiHDQ64\nPvI69q61RWQk8OuvvFORkmIM+OYbYNQoYPhw3mlUh1IXNUtKSkL//v3x/PlzWFtb4/fff0fVfPoD\nrK2tYWRkBF1dXejr6yMkJKTUYYlizJolTEYJDOSdpGyM+WsMDMoZ4EeHVXB2BkJDgdq1eacipXHj\nhrBmfUQEULky7zSqQamF3sfHB2ZmZvDx8cHSpUuRnJyMJfmsK1q3bl3cvn0bJkVsGUOFvux8+CAs\nY3z0qLCjjyYLSwhDh70d8GjCI3iPNYalJeDvzzsVkcd33wlbD/70E+8kqkGphd7Ozg6XLl2Cubk5\nEhISIBaL8ejRozzH1a1bF7du3YKpqancYYni7NghLGP899+a21fNGEO73e3Qx74PmovGw8tLaAnS\nCpXq7cULwNUVuHtXWOlS2yn1YmxiYiLMzc0BAObm5kgsYJk5kUiE9u3bo2nTpti6dWtpT0cUbNgw\nICMD2L+fdxLlORZxDIkfEzG68Q+YNk1oAVKRV3+1awtLI9Bwy+IrdE6gh4cHEhIS8vzc/4vPviKR\nCKICmoVXr15FzZo18fr1a3h4eMDOzg7u7u75Huvr6yv7XiwWQ0zbIymNjg6werWwbn337kClSrwT\nKZY0W4rpZ6djfZf1OH5MD8nJdAFPk8ycKUyiCgnRvklUEokEEomkRI+Rq+tGIpGgRo0aePnyJdq0\naZNv183n/Pz8ULlyZUybNi1vEOq64WLAAKG//rO/sRph1fVVuBB1AQd7HoejI7B5M42/1jQ7dgjb\nZf79t3ZPfFNq142Xlxd27doFANi1axd69OiR55i0tDSkpKQAAD5+/IgzZ87AycmptKckSrBsGbBu\nnTDsUlO8/vgai/9ejBUdVmDJEmFGMBV5zTN0qDCm/v/LECmEXMMr+/XrhxcvXuQaXhkfH4/Ro0fj\nxIkTePbsGXr16gUAyMrKwnfffYfZBXSsUYueH19fYQXHQ4d4J1GM8SfGQ19HH5Ns16BFC+GinaUl\n71REGW7fBrp0EWZ9FzHeQ2PR5uCkWNLTAScnYYOSbt14p5HPP6/+QdtdbfFwwiMM7m2Cdu2A6dN5\npyLKNGkSIJUK3XPaiAo9Kbbz54GRI4EHD9R3Sz3GGDrs7QCvBl6wiJ2EBQuEyVH6+ryTEWV6905Y\n3uPwYaBFC95pyh6tdUOKrV07wN1dvS/KnnhyArEfYtHfZiwmTwY2baIirw2qVgVWrAB++EFo2ZO8\nqEVPZF69AhwdgaAg4QKmOpFmS+G00QmrO67G7wGdUbky8MsvvFORssKYsGBfs2bAggW805St4tRO\n2luHyFSvDqxcKYxmuHULKF+ed6Li23hzI+oZ14POs86QSIRF24j2EImEPnoXF6BnT6BRI96JVAu1\n6EkujAmLRjVsCAQE8E5TPG/T3sJuvR1O9r2EPt/aY9s2wMODdyrCw/btwIYNwuJn2rJFJF2MJaWS\nmAg4OwN//qkeF7cmnZwEBoa0Q+ugqwvQShvaizGgUydhOeN583inKRt0MZaUirm5MIlq6FDg40fe\naQoX/joc+x/sh8s7X/z9tzBElGgvkUiYMbt+vbBPMhFQi54UaNgw4b87d/JMUbjOv3ZGM+OO2DTM\nGydPAk2b8k5EVMGxY8CUKcLwWk3fNpNa9EQu69cLi0apaqE/9eQUniU9w4Wl4zF9OhV58h8vL6Br\nV2DMGKE7R9tRi54U6sEDYUNxiUTYnFlVZGZnotGmRrB7sRxpd7vh1CntXtiK5PXpk3CNafRoYOJE\n3mmUh4ZXErk5OADLlwO9ewPBwarzMXjz7c3Q+2iFe390xc0QKvIkr4oVgSNHgJYthfkh2rzqObXo\nSbFMngw8fAicPMl/tmnSpyTUX20HtvMCLh9yBC2ISgpz/ryw/WBwMGBtzTuN4lEfPVGYVauAcuWE\nBaR4/z2ecfwnSMN6Y6s/FXlStHbthN2ouncH3r/nnYYPatGTYktJAVq1AgYPBnx8+GS4/uQR3He4\nY75pOOZPr8YnBFE7jAmNlHv3hCU+NGlHNZowRRQuNhb49lth6NrkyWV77pQUwGpmNzQ1bYNzC/Pu\nUkZIYXJyhLkhb98KkwHLleOdSDGo64YonKUlcOGCsN/sunVld95374CWg0+DmUbgxIJJZXdiojF0\ndITJVPr6Qp99RgbvRGWHCj0psTp1hGK/cqUwIkfZH8Ti4wH3b7PwstGP2DloBcrraUhTjJQ5fX3g\nwAFhC8IuXbSnz54KPSkVa2vg8mXgt9+AESOU1zp6+FC4LlCv71a41K+BHnZeyjkR0RoVKgAHDwJ2\ndkDr1kBcHO9EykeFnpSalRXw99/Ahw/CyIbYWMU9N2NAYKCwGcqMee8QXMEXqzqugkgkUtxJiNbS\n1RW6HgcNEmZUnzjBO5FylbrQHzx4EA4ODtDV1cWdO3cKPC4oKAh2dnawtbXF0qVLS3s6oqIMDITW\nUadOgKursKtTTo58z5mUJPwCrlwpzMh9Wmshun/VHc41nBWSmRBAWABt5kyhK2fCBGD8eNVfxK+0\nSl3onZyccOTIEbRu3brAY7KzszFx4kQEBQUhPDwc+/btw8OHD0t7SqKidHSA//1PKMo7dwozEIOD\nS/48nz4Ba9YIH6mrVwdu3gTK1XyMXWG7sLDNQgWnJkTQujVw964wqsvODti1C8jKKtlzMCY0UFRV\nqQu9nZ0dGjRoUOgxISEhqF+/PqytraGvr48BAwbg6NGjpT0lUXEODsDVq8CQIUD//sIwzD17gOTk\ngh/DmLAb1Jw5QL16wkXec+eEgl+xIjDj7Az4tPKBeWXzsnshROtUrSq8V/ftE0bm2NoC/v7A06eF\nPy4xEdi4Udh6c+zYsslaGkpd6yYuLg5WVlay25aWlrhx44YyT0k409UVFpEaNkxYKnbXLuFjsY2N\nsN5IzZrCyIcPH4DoaOD2bWHLwl69hKnq9vb/Pde5Z+fwz6t/8Huf33m9HKJlvvkGuHRJ+DS5Y4dw\nu0IFoVuydm2gcmUgPV24gHvvnnBdqksXYMkS1d7VrNBC7+HhgYSEhDw/DwgIgKenZ5FPXtILZ76+\nvrLvxWIxxNq8CpGa09cXFkLr3Vv4xbh7F3j0CEhIED4WV68OtGkjtNzr1hX6Sz+XnZONH0//iOUe\ny1FeT402ryUaoVkz4WvDBiAiQijqcXFAaipgZCQU/hkzhL1py3rLQolEAolEUqLHFBrx7Nmz8uRB\nrVq1EBMTI7sdExMDS0vLAo//vNATzVGhgrBcbEm2Jdweuh0mFU3Q066n8oIRUgSRSOi3t7PjneQ/\nXzaC/fz8inyMQoZXFjT9tmnTpnjy5Amio6MhlUpx4MABeHnROGhSuPfp7zH/4nwaTkmIgpS60B85\ncgRWVlYIDg5G165d0blzZwBAfHw8unbtCgDQ09PDunXr0LFjR9jb26N///5o2LChYpITjeV/xR9d\nbbuicc3GvKMQohFoUTOiUiKTItFiWwvcH3cfNQ1r8o5DiMqjRc2I2vE564NpLadRkSdEgWgrQaIy\nLkZdRGhCKH7r/RvvKIRoFGrRE5WQnZONqaenYln7ZaigV4F3HEI0ChV6ohJ23t0Jw/KG6GPfh3cU\nQjQOdd0Q7j5kfMD/Lv4Pfw38i4ZTEqIE1KIn3C2+shgdbTqiqUVT3lEI0UjUoidcRSVHYcudLbg/\n7j7vKIRoLGrRE658zvlgaoupsDC04B2FEI1FLXrCzeXnlxESF4LdPXbzjkKIRqMWPeEih+Vg6ump\nWNp+KSrqV+QdhxCNRoWecLE7bDfK65ZHf4f+vKMQovGo64aUuVRpKuZemIvD/Q7TcEpCygC16EmZ\nW/L3ErSt2xZulm68oxCiFahFT8rU83fPsfHWRoSNDeMdhRCtQS16UqZmnpuJyc0nw9Ko4J3GCCGK\nRS16UmauvriKqzFXsaP7Dt5RCNEq1KInZSKH5cD7tDeWtFuCSvqVeMchRKtQoSdl4td7v0JHpIOB\nTgN5RyFE65S60B88eBAODg7Q1dXFnTt3CjzO2toajRo1gqurK5o3b17a0xE19lH6EbPPz8bqjquh\nI6K2BSFlrdR99E5OTjhy5AjGjBlT6HEikQgSiQQmJialPRVRc8uuLkPrOq3R0qol7yiEaKVSF3o7\nO7tiH0ubfmuvF+9fYN3NdQgdE8o7CiFaS+mfo0UiEdq3b4+mTZti69atyj4dUTGzz8/GhGYTULtK\nbd5RCNFahbboPTw8kJCQkOfnAQEB8PT0LNYJrl69ipo1a+L169fw8PCAnZ0d3N3dS5eWqJXrMddx\nKfoSNnfbzDsKIVqt0EJ/9uxZuU9Qs2ZNAEC1atXQs2dPhISEFFjofX19Zd+LxWKIxWK5z0/4+Hd1\nSv+2/qhcrjLvOIRoDIlEAolEUqLHKGTCVEF98GlpacjOzoahoSE+fvyIM2fOYMGCBQU+z+eFnqi3\n/f/sR1ZOFoY4D+EdhRCN8mUj2M/Pr8jHlLqP/siRI7CyskJwcDC6du2Kzp07AwDi4+PRtWtXAEBC\nQgLc3d3h4uICNzc3dOvWDR06dCjtKYmaSMtMw6xzs7C6Ew2nJEQViJiKDIkRiUQ0OkdD/HTpJzx4\n/QAH+hzgHYUQjVec2klr3RCFiv0QizU31uD2D7d5RyGE/D/6XE0Uas75ORjbZCysq1rzjkII+X/U\noicKExIXgnPPziFiYgTvKISQz1CLnigEYwxTT0/ForaLYFjekHccQshnqNAThfj9we/4lPkJQ52H\n8o5CCPkCdd0QuX3K/ASfcz7Y3WM3dHV0ecchhHyBWvREbquur0Izi2b41vpb3lEIIfmgFj2RS3xK\nPFYFr8LN0Td5RyGEFIBa9EQucy/MxejGo1HPuB7vKISQAlCLnpTa7fjbCIoMouGUhKg4atGTUmGM\nwfu0N34S/wSj8ka84xBCCkGFnpTKHw//wIeMDxjhOoJ3FEJIEajrhpRYelY6ZpydgR1eO2g4JSFq\ngFr0pMRWB6+GSw0XtKnbhncUQkgxUIuelEhCagJWXFuB4FHBvKMQQoqJ1qMnJTLq2CgYVzDG8g7L\neUchhIDWoycKFvoyFMcfH6fhlISoGeqjJ8Xy7+qUvmJfVKlQhXccQkgJUKEnxfLnoz/x9tNbjGo8\nincUQkgJlbrQz5gxAw0bNoSzszN69eqF9+/f53tcUFAQ7OzsYGtri6VLl5Y6KOEnIysD089Ox6oO\nq6CnQ719hKibUhf6Dh064MGDBwgLC0ODBg2wePHiPMdkZ2dj4sSJCAoKQnh4OPbt24eHDx/KFVhd\nSSQS3hFKbe2NtXCo5gAPG49871fn11Yc9PrUm6a/vuIodaH38PCAjo7wcDc3N8TGxuY5JiQkBPXr\n14e1tTX09fUxYMAAHD16tPRp1Zi6vtkSUxOx9OpSrOiwosBj1PW1FRe9PvWm6a+vOBTSR79jxw50\n6dIlz8/j4uJgZWUlu21paYm4uDhFnJKUkfkX5+N75+/RwLQB7yiEkFIqtMPVw8MDCQkJeX4eEBAA\nT09PAIC/vz/KlSuHQYMG5TlOJBIpKCbh4V7iPfwZ8SceTXjEOwohRB5MDoGBgezrr79mnz59yvf+\n69evs44dO8puBwQEsCVLluR7rI2NDQNAX/RFX/RFXyX4srGxKbJWl3pmbFBQEKZNm4ZLly7BzMws\n32OysrLw1Vdf4fz587CwsEDz5s2xb98+NGzYsDSnJIQQUgql7qOfNGkSUlNT4eHhAVdXV4wfPx4A\nEB8fj65duwIA9PT0sG7dOnTs2BH29vbo378/FXlCCCljKrPWDSGEEOVQqZmxv/zyCxo2bAhHR0fM\nnDmTdxylWLlyJXR0dJCUlMQ7ikIVdwKdutHkCX8xMTFo06YNHBwc4OjoiLVr1/KOpHDZ2dlwdXWV\nDR7RJO/evUOfPn3QsGFD2NvbIzi4kBVlS34JVjkuXLjA2rdvz6RSKWOMsVevXnFOpHgvXrxgHTt2\nZNbW1uzt27e84yjUmTNnWHZ2NmOMsZkzZ7KZM2dyTiS/rKwsZmNjw6KiophUKmXOzs4sPDycdyyF\nefnyJQsNDWWMMZaSksIaNGigUa+PMcZWrlzJBg0axDw9PXlHUbjvv/+ebd++nTHGWGZmJnv37l2B\nx6pMi37jxo2YPXs29PX1AQDVqlXjnEjxfvzxRyxbtox3DKUozgQ6daPpE/5q1KgBFxcXAEDlypXR\nsGFDxMfHc06lOLGxsTh58iRGjRqlcUugv3//HleuXMGIEcJWnnp6eqhSpeDFBlWm0D958gSXL19G\nixYtIBaLcevWLd6RFOro0aOwtLREo0aNeEdRuoIm0KkbbZrwFx0djdDQULi5ufGOojBTp07F8uXL\nZQ0QTRIVFYVq1aph+PDhaNy4MUaPHo20tLQCjy/TFaoKmoDl7++PrKwsJCcnIzg4GDdv3kS/fv3w\n7Nmzsownt8Je3+LFi3HmzBnZz9SxhSHvBDp1oy0T/lJTU9GnTx+sWbMGlStX5h1HIY4fP47q1avD\n1dVVI5dAyMrKwp07d7Bu3To0a9YM3t7eWLJkCX766af8H1A2vUlF69SpE5NIJLLbNjY27M2bNxwT\nKc79+/dZ9erVmbW1NbO2tmZ6enqsTp06LDExkXc0hSpqAp26KcmEP3UllUpZhw4d2M8//8w7ikLN\nnj2bWVpaMmtra1ajRg1WqVIlNmTIEN6xFObly5fM2tpadvvKlSusa9euBR6vMoV+06ZNbP78+Ywx\nxiIiIpiVlRXnRMqjiRdjT506xezt7dnr1695R1GYzMxMVq9ePRYVFcUyMjI07mJsTk4OGzJkCPP2\n9uYdRakkEgnr1q0b7xgK5+7uziIiIhhjjC1YsID5+PgUeKzKLC4+YsQIjBgxAk5OTihXrhx2797N\nO5LSaGKXwKRJkyCVSuHhISxl3LJlS2zYsIFzKvl8PuEvOzsbI0eO1KgJf1evXsXevXvRqFEjuLq6\nAgAWL16MTp06cU6meJr4O/fLL7/gu+++g1QqhY2NDQIDAws8liZMEUKIhtO8y9GEEEJyoUJPCCEa\njgo9IYRoOCr0hBCi4ajQE0KIhqNCTwghGo4KPSGEaDgq9IQQouH+D62kbWX8quedAAAAAElFTkSu\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9e6cd68>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's see if we can make a better approximation..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "derivatives = [lambda x: sin(x), lambda x: cos(x), lambda x: -sin(x), lambda x: -cos(x)] * 3\n",
      "plot(xvals, f(xvals))\n",
      "plot(xvals, P_n(derivatives, x_0=0.5, x=xvals))\n",
      "plot(0.5, f(0.5), 'b.', markersize=10)\n",
      "ylim([-2,2])\n",
      "legend(['function', 'degree 11 approx at 0.5'])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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PFtnbPnXqFC5fvgyRSITZs2ejbdu2xf61859BgwZhx44duH37NvLy8jBjxgy0adMGFhYW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\nJ4QQFUeFnhBCVNz/AXr7HZtyUjaHAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa02b5c0>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What happens when a function has only a finite number of derivatives? I would expect that the taylor series would *exactly* equal the function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x: 0.1*x**4 - 3*x**3 + 0.1*x**2 - 5*x + 1\n",
      "xvals = arange(-3, 3, 0.05)\n",
      "plot(xvals, f(xvals))\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BeSV8oc+3dOlS1q1bx6ZNm1qoIu9q7L+/YBEZGXnagILy8vLTlhUR/3fs2DEm\nTpzIrbfeyrhx4+wuxycuvfRSRo8ezQcffEB6enqDj/HL5p2SkpL623l5eaSkpNhYjffl5+fz9NNP\nk5eXR5s2bewux6esIBkR3L9/f0pKSti7dy81NTWsXLmSsWPH2l2WNJJlWWRnZ5OQkMDMmTPtLser\nDh06xJEjRwD47rvvePvtt8+fmS3Tt3xxJk6caPXr189KSkqyJkyYYFVXV9tdklfFxMRYPXv2tJKT\nk63k5GRrxowZdpfkVW+88Ybl8XisNm3aWG632xoxYoTdJXnFunXrrLi4OCs6OtqaN2+e3eV41aRJ\nk6yIiAirdevWlsfjsZYsWWJ3SV61ZcsWy+VyWUlJSfX/361fv97usrxi165dVkpKipWUlGQlJiZa\nTz311Hkfr8lZIiIO4pfNOyIi4hsKfRERB1Hoi4g4iEJfRMRBFPoiIg6i0BcRcRCFvoiIgyj0RUQc\n5P8B9BKUJUepUJ0AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa02c518>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "derivatives = [f,\n",
      "               lambda x: 0.1*4*x**3 - 3*3*x**2 + 0.1*2*x - 5,\n",
      "               lambda x: 0.1*4*3*x**2 - 3*3*2*x + 0.1*2,\n",
      "               lambda x: 0.1*4*3*2*x - 3*3*2,\n",
      "               lambda x: 0.1*4*3*2,\n",
      "               lambda x: 0]\n",
      "\n",
      "plot(xvals, f(xvals))\n",
      "plot(xvals, P_n(derivatives, x_0=0.1, x=xvals))\n",
      "legend(['function', 'approximation'])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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3XDI4CKVBGVqUd0EH+xcQ2SGEA+ZoOZY+ET2V39OysWr/b9j7zz5cqEpBpbwQ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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x9e649e8>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Roundoff Error #\n",
      "\n",
      "Computer arithmetic is approximate."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "0.1 + 0.2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "0.30000000000000004"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "0.1*0.1/0.1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": [
        "0.10000000000000002"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from decimal import Decimal\n",
      "Decimal(0.275)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "Decimal('0.27500000000000002220446049250313080847263336181640625')"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This section is pretty boring. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# The bisection method #\n",
      "\n",
      "Aka, binary search over the reals. Let's approximate $\\pi$ using the bisection method. First we need a function which is zero at $\\pi$, and then we can apply a root-finding method.\n",
      "\n",
      "$f(x) = xsin(x)$ should do."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xvals = arange(-6, 0, 0.01)\n",
      "f = lambda x: x*sin(x)\n",
      "plot(xvals, f(xvals))\n",
      "plot(-pi, f(-pi), 'b.', markersize=10)\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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EVqzw37VWrDDX2LXLzDZR8RaR8lIHfgaLFpml91ddBZMnmzMnfWHHDhg71uxV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\nB5M8YKYrUITjAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa028978>"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is zero at $-pi$, so that's fine. Let's do root-finding."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def root_find(f, a, b, tol, verbose=False):\n",
      "    \"\"\"\n",
      "    Root finding with the bisection method\n",
      "    \"\"\"\n",
      "\n",
      "    f_a = f(a)\n",
      "    f_b = f(b)\n",
      "    if verbose:\n",
      "        print 'x_left =', a, 'x_right =', b, 'f(x_left) =', f_a, 'f(x_right) =', f_b, 'x_mid =', (a+b)/2, 'f(x_mid) =', f((a+b)/2)\n",
      "    if abs(f_a-f_b) < tol:\n",
      "        return (a+b)/2\n",
      "    f_mid = f((a+b)/2)\n",
      "    if f_mid > 0:\n",
      "        return root_find(f, a, (a+b)/2, tol, verbose=verbose)\n",
      "    else:\n",
      "        return root_find(f, (a+b)/2, b, tol, verbose=verbose)\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 266
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "root_find(f, -4.0, -2.0, 1e-14)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 17,
       "text": [
        "-3.141592653589794"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Nice! We can reduce the tolerance,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print root_find(f, -4.0, -2.0, 1e-9)\n",
      "print -pi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "-3.14159265358\n",
        "-3.14159265359\n"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Cool. How about a polynomial, like $x^2 - 2$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x: x**2 - 2\n",
      "xvals = arange(0,2, 0.01)\n",
      "plot(xvals, f(xvals))\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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yySUsWbKE5ORknxTva3V1UFUFO3eagD/+UVNjplEmJUHv3ibg09LggguC7yKs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\nkj1TfoeA1AAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa059198>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print root_find(f, 0., 2., 10e-9)\n",
      "print sqrt(2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "1.41421356145\n",
        "1.41421356237\n"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "How about just finding a minimum?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def find_min(f, a, b, tol):\n",
      "    f_a, f_b = f(a), f(b)\n",
      "    mid = (a+b)/2\n",
      "    f_mid = f(mid)   \n",
      "    \n",
      "    if abs(f_a-f_b) < tol:\n",
      "        return min( [f_a, f_b, f_mid] )\n",
      "        \n",
      "    max_cmp = max( [f_a, f_b, f_mid] )\n",
      "    if max_cmp == f_b:\n",
      "        return find_min(f, a, mid, tol)\n",
      "    elif max_cmp == f_a:\n",
      "        return find_min(f, mid, b, tol)\n",
      "    else:\n",
      "        print 'your function has multiple minima on this interval.'\n",
      "        "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "find_min(f, -1., 1., 10e-3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 22,
       "text": [
        "-2.0"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print find_min(sin, -pi, 0.0, 10e-9)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "-1.0\n"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xvals = arange(-pi, 0, 0.01)\n",
      "plot(xvals, sin(xvals))\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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5AZw27d0LTJoEbN4MNGggOw2Zi8WfNOeFF4CXXhIPEnn2j7acOyfG91etAtq0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\nB8TiT0TkgFj8iYgc0P8Hagyn4tdTGPsAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3b06438>"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Fixed Point Method"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "A *fixed point* of a function $f$ is a point $x$ such that $f(x) = x$. With functions of one real variable, this is an intersection with the line $y=x$, and more generally it is where $f$ intersects the identity function $I(x) = x$.\n",
      "\n",
      "Fair enough. \n",
      "\n",
      "How do we know if a fixed point exists?\n",
      "\n",
      "**Theorem** \n",
      "\n",
      "If $f \\in C[a, b]$ and $\\forall x \\in [a,b]$, $f(x) \\in [a,b] $, then $f$ has at least one fixed point in $[a,b]$."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "What does this say, intuitively, and how might we show that it is true? \n",
      "\n",
      "Intuitively, since we bound the inputs and outputs with the same domain, we're looking at a *square* in the one-dimensional case, and a hypercube in the general case. The identity function $I(x) = x$ passes through the diagonal of the square, because $I(a) = a$ and $I(b) = b$. So our statement is really that if we can bound the function by a square, then the function must pass through the diagonal. \n",
      "\n",
      "A proof follows."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We know that $a \\le f(a) \\le b$, and $a \\le f(b) \\le b$.\n",
      "\n",
      "Define $g(x) = f(x) - x$. Note that $g(a) = f(a) - a \\le 0$, and $g(b) = f(b) - b \\ge 0$. \n",
      "\n",
      "By the intermediate value theroem, we can conlude that $\\exists p : g(p) = 0$\n",
      "\n",
      "At this point, $g(p) = 0 = f(p) - p$, so $f(p) = p$ and we're done."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Fixed Point Iteration ###\n",
      "\n",
      "The idea here is to find a fixed point, starting from an initial guess. If we guess some point, $p_0$, then we can improve our guess to $p_1$ by letting $p_1 = g(p_0)$. If $p_0$ is a fixed point, then $p_0 = g(p_0) = p_1$ and we have converged. One important note is that we are only guarenteed to reduce our error $|p - p_i|$ if the derivative of $g$ is between -1 and 1. Otherwise applying $g$ to $p_0$ will take us further than we were from the fixed point. \n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x: sin(x)\n",
      "p_0 = 0.5\n",
      "xs = arange(-2, 2, 0.1)\n",
      "plot(xs, f(xs))\n",
      "plot(xs, xs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": [
        "[<matplotlib.lines.Line2D at 0xa366128>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MbL7Gr1AoMHHiRIwdOxZvvPGGrXdnc0z5ROTsOkz8Wq0W1dXVrX6elZWFKVOm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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa3663c8>"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We see that there is a fixed point at 0. Assuming we start with 0.5, how would the iteration proceed?\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p_i = 0.5\n",
      "for i in range(10):\n",
      "    p_i = f(p_i)\n",
      "    print 'iteration', i, 'point', p_i"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "iteration 0 point 0.479425538604\n",
        "iteration 1 point 0.461269555033\n",
        "iteration 2 point 0.445085336847\n",
        "iteration 3 point 0.430534904682\n",
        "iteration 4 point 0.417356952803\n",
        "iteration 5 point 0.40534569446\n",
        "iteration 6 point 0.394336465428\n",
        "iteration 7 point 0.384195664011\n",
        "iteration 8 point 0.374813558009\n",
        "iteration 9 point 0.366099037028\n"
       ]
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It looks like we're getting there. Let's run it for longer."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p_i = 0.5\n",
      "for i in range(1000):\n",
      "    p_i = f(p_i)\n",
      "    if i % 100 == 0: print 'iteration', i, 'point', p_i"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "iteration 0 point 0.479425538604\n",
        "iteration 100 point 0.161958606796\n",
        "iteration 200 point 0.11819383448\n",
        "iteration 300 point 0.0975932008598\n",
        "iteration 400 point 0.0850081570339\n",
        "iteration 500 point 0.076302909762\n",
        "iteration 600 point 0.0698213726171\n",
        "iteration 700 point 0.0647536556891\n",
        "iteration 800 point 0.0606506083178\n",
        "iteration 900 point 0.0572404402683\n"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So it looks like we start off making good progress and then slow down quite a bit."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Fixed point iteration is pretty weak. Let's look at something more powerful."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Newton's Method"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "Newton's method is a root-finding technique based on a degree 1 taylor series approximation. Intuitively, we approximate the function $f$ with its degree 1 Taylor series, which is linear. This approximation is given by\n",
      "\n",
      "$f(x) \\approx f(x_0) + f'(x_0)(x-x_0)$ for some $x_0$ near $x$.\n",
      "\n",
      "Now if our goal is to find $x$ such that $f(x) = 0$, we just set the right side equal to zero.\n",
      "\n",
      "$0 = f(x_0) + f'(x_0)(x-x_0)$\n",
      "\n",
      "$-f(x_0) = f'(x_0)(x-x_0)$\n",
      "\n",
      "$-\\frac{f(x_0)}{f'(x_0)} = x - x_0$\n",
      "\n",
      "$-\\frac{f(x_0)}{f'(x_0)} + x_0 = x$\n",
      "\n",
      "And now we know how to take some initial guess, $x_0$, and obtain a better guess, $x$. Furthermore, since we're using a Taylor series, we can  bound the error as $O(x^2)$, assuming our initial guess is close to the solution. \n",
      "\n",
      "Implementing this should be easy."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def newton(f, f_prime, x_0, tol):\n",
      "    #find a root with error tolerance tol\n",
      "    print x_0\n",
      "    x = x_0 - f(x_0)/f_prime(x_0)\n",
      "    if abs(f(x)) <= tol:\n",
      "        return x\n",
      "    return newton(f, f_prime, x, tol)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now let's test it out."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x: x**2\n",
      "df = lambda x: 2*x\n",
      "print newton(f, df, 0.5, 1e-5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.5\n",
        "0.25\n",
        "0.125\n",
        "0.0625\n",
        "0.03125\n",
        "0.015625\n",
        "0.0078125\n",
        "0.00390625\n",
        "0.001953125\n"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So it figured out that $x^2$ has a zero at $x=0$. We can also compute $\\pi$ by desiging a function with a zero at $\\pi$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x: sin(x)\n",
      "df = lambda x: cos(x)\n",
      "print newton(f, df, 3.0, 1e-12)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "3.0\n",
        "3.14254654307\n",
        "3.1415926533\n",
        "3.14159265359\n"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Wow! Newton's method converged quickly. In fact, the bounds on the Taylor approximation give us a *quadratic* convergence guarantee (if we start close enough).\n",
      "\n",
      "What if, however, we don't know $f'$?"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The Secant Method"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "A motivation for introducing the secant method is the requirement that Newton's method has for the derivative of the function we wish to find a root of. In some applications, finding $f'$ may not be feasable or tractable. \n",
      "\n",
      "Looking at what $f'$ means, we see by definition\n",
      "\n",
      "$f'(x_0) = \\lim_{x \\to x_0} \\frac{f(x) - f(x_0)}{x-x_0}$\n",
      "\n",
      "If we assume that $x_1$ is close to $x_0$ then we can approximate the limit,\n",
      "\n",
      "$f'(x_0) \\approx \\frac{f(x_1) - f(x_0)}{x_1-x_0}$\n",
      "\n",
      "Now plugging this approximation into Newton's method,\n",
      "\n",
      "$x = x_0 - \\frac{f(x_0)}{f'(x_0)}$\n",
      "\n",
      "$x = x_0 - \\frac{f(x_0)}{\\frac{f(x_1) - f(x_0)}{x_1-x_0}}$\n",
      "\n",
      "$x = x_0 - \\frac{f(x_0)(x_1-x_0)}{f(x_1)-f(x_0)}$\n",
      "\n",
      "Which requires only the function $f$. \n",
      "\n",
      "Note that this equation requires two values, $x_0$ and $x_1$. These can be found in different ways. One way is using $x_1 = x_0 + \\epsilon$ for some very small $\\epsilon$. This gives us a good approximation to the derivative, at the cost of twice as much computation.\n",
      "\n",
      "Another method, which is used in the text, is to use previous values of $x_0$ as $x_1$. In other words, we define the recurrence relation for our estimates as:\n",
      "\n",
      "$x_n = x_{n-1} - \\frac{f(x_{n-1})(x_{n-1}-x_{n-2})}{f(x_{n-1}) - f(x_{n-2})}$\n",
      "\n",
      "Using this method the right way can enable us to only need one function evaluation per step. Let's code it up."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def secant_method(f, f_x_0, x_0, x_1, tol):\n",
      "    f_x_1 = f(x_1) #for efficiency\n",
      "    x = x_1 - f_x_1*(x_1-x_0)/(f_x_1-f_x_0)\n",
      "    if abs(x-x_1) <= tol:\n",
      "        return x\n",
      "    print x\n",
      "    return secant_method(f, f_x_1, x_1, x, tol)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And our usual test of computing $\\pi$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x: sin(x)\n",
      "x_left = 3.0\n",
      "x_right = 4.0\n",
      "print secant_method(f, f(x_left), x_left, x_right, 1e-12)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "3.15716279248\n",
        "3.13945909822\n",
        "3.14159272798\n",
        "3.14159265359\n",
        "3.14159265359\n"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Not bad. One fault of the secant method is that it **isn't guaranteed to converge**. This can be improved by a neat trick, where we keep our best bound. \n",
      "\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The method of false position"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "\n",
      "The method of false position is a modification to the secant method, which guarantees convergence. It operates very similar to the bisection method, but uses additional information to speed up convergence.\n",
      "\n",
      "The idea is to consider a bound, $x_l$ on the left and $x_r$ on the right. We could use the bisection method to try out the point $x_m = \\frac{x_l + x_r}{2}$, but we can do better using what we learned from the secant method. In the method of false position, we draw a line from $(x_l, f(x_l))$ to $(x_r, f(x_r))$, and use the intersection of this line as $x_m$, the midpoint. \n",
      "\n",
      "Put another way, we approximate the function over the interval with a line such that the approximation is equal to the function at the endpoints. Here's one example of the approximation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_l, x_r = -0.7, 1\n",
      "f = lambda x: x**3\n",
      "xs = arange(-0.7, 1, 0.01)\n",
      "plot(xs, f(xs))\n",
      "plot([x_l, x_r], [f(x_l), f(x_r)])\n",
      "plot([-1, 1], [0,0])\n",
      "scatter([x_l, 0.0, x_r], [f(x_l), f(0.0), f(x_r)])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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4LPkzJqROYOmzS2ng18DcQkUEKHxuqsUvV4wbZ59ied68/4R+Xn4ery17jXWH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\ni1Hwi4hYzP8H5/ag+j9z8soAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa022240>"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Obviously this example is a simple one, but it illustrates how we can use our bounds and the function itself to do better than use the midpoint each time. Let's modify our bisection code to use this. First we need to derive a formula for the line passing through the endpoints.\n",
      "\n",
      "This line, $l(x)$, must satisfy\n",
      "\n",
      "$l(x_r) = f(x_r)$\n",
      "\n",
      "and\n",
      "\n",
      "$l(x_l) = f(x_l)$\n",
      "\n",
      "Using the general form for a line, $l(x) = ax + b$, \n",
      "\n",
      "$ax_r + b = f(x_r)$\n",
      "\n",
      "$ax_l + b = f(x_l)$\n",
      "\n",
      "We can subtract the second from the first, \n",
      "\n",
      "$ax_r + b - ax_l - b = f(x_r) - f(x_l)$\n",
      "\n",
      "$a(x_r - x_l) = (f(x_r)-f(x_l))$\n",
      "\n",
      "$a = \\frac{f(x_r) - f(x_l)}{x_r - x_l}$\n",
      "\n",
      "We just plug this result into one of the equations to get $b$,\n",
      "\n",
      "$\\frac{f(x_r) - f(x_l)}{x_r - x_l} x_r + b = f(x_r)$\n",
      "\n",
      "$b = f(x_r) - \\frac{f(x_r) - f(x_l)}{x_r - x_l} x_r$\n",
      "\n",
      "And now we know $l(x)$,\n",
      "\n",
      "$l(x) = \\frac{f(x_r) - f(x_l)}{x_r - x_l} x + f(x_r) - \\frac{f(x_r) - f(x_l)}{x_r - x_l} x_r$\n",
      "\n",
      "We don't need the entire line, however. We just need to solve for when it's zero.\n",
      "\n",
      "$l(x) = 0 = \\frac{f(x_r) - f(x_l)}{x_r - x_l} x + f(x_r) - \\frac{f(x_r) - f(x_l)}{x_r - x_l} x_r$\n",
      "\n",
      "Solving this for $x$, we get\n",
      "\n",
      "$x = x_r - \\frac{f(x_r)(x_r - x_l)}{f(x_r) - f(x_l)}$\n",
      "\n",
      "This is a recurrence relation in terms of the left and right bounds, $x_l$ and $x_r$. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def false_position(f, x_l, x_r, tol):\n",
      "    assert sign(f(x_l)) == -1 and sign(f(x_r)) == 1, 'convergence not guarenteed'\n",
      "    print 'left:', x_l, 'right:', x_r\n",
      "    \n",
      "    #root finding with error tolerance tol\n",
      "    x_m = x_r - f(x_r)*(x_r - x_l)/(f(x_r)-f(x_l))\n",
      "    \n",
      "    #base condition\n",
      "    if abs(f(x_m)) <= tol:\n",
      "        return x_m\n",
      "    \n",
      "    #recurrence relation\n",
      "    if sign(f(x_m)) == -1:\n",
      "        return false_position(f, x_m, x_r, tol)\n",
      "    else:\n",
      "        return false_position(f, x_l, x_m, tol)\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And our test"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x: x**2 - 2\n",
      "x_left = 0.0\n",
      "x_right = 2.0\n",
      "print false_position(f, x_left, x_right, 1e-12)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "left: 0.0 right: 2.0\n",
        "left: 1.0 right: 2.0\n",
        "left: 1.33333333333 right: 2.0\n",
        "left: 1.4 right: 2.0\n",
        "left: 1.41176470588 right: 2.0\n",
        "left: 1.41379310345 right: 2.0\n",
        "left: 1.41414141414 right: 2.0\n",
        "left: 1.41420118343 right: 2.0\n",
        "left: 1.41421143847 right: 2.0\n",
        "left: 1.41421319797 right: 2.0\n",
        "left: 1.41421349985 right: 2.0\n",
        "left: 1.41421355165 right: 2.0\n",
        "left: 1.41421356053 right: 2.0\n",
        "left: 1.41421356206 right: 2.0\n",
        "left: 1.41421356232 right: 2.0\n",
        "left: 1.41421356236 right: 2.0\n",
        "left: 1.41421356237 right: 2.0\n",
        "1.41421356237\n"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This shows us a common failure of the false-position method where only one endpoint converges. There are ways to fix this, but are not covered in the text."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Interpolation #\n",
      "\n",
      "Let's look at polynomial interpolation. The first thing to note is that given $n+1$ points, we can construct a polynomial of degree $n$ or lower going through all the points. Intuition is enough for me, but I think we could prove it fairly easily by using the fact that $x^p$ and $x^{p+1}$ are linearly independent and considering fitting each point an axis. Another way of saying it, we could turn this into a problem with $n$ equations and $n$ unknowns, which has a unique solution."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Lagrange interpolating polynomial"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "The idea here is pretty straightforward, and we rely heavily on the fact that we can make the polynomial zero out at certain places. In other words, if the polynomial is:\n",
      "\n",
      "$(x-x_1)(x-x_2)(x-x_3)$ \n",
      "\n",
      "Then it is zero at $x_1$, $x_2$, and $x_3$. \n",
      "\n",
      "### The Setup ###\n",
      "\n",
      "We desire some polynomial, $P(x)$ of degree $n$ such that for $x_1 ... x_{n+1}$, $P(x_i) = f(x_i)$ for any function $f$. Another way to think about this is to just let $y_i = f(x_i)$ and to find a polynomial through all the points.\n",
      "\n",
      "How can we make sure that this polynomial fits all the points? We let the polynomial be a sum of terms, where all but one of the terms will drop out (equal zero) at each point $x_1 ... x_{n+1}$. The term which remains will reduce to $f(x_i)$.\n",
      "\n",
      "\n",
      "Looking at one point, $x_i$, how will we get the other terms to vanish? Simple - just multiply them by $(x-x_i)$. Now if we apply this logic to the other terms, we see that the $i$th term in the sum will be the product of $(x-x_j)$ for $j \\neq i$. Let's formalize what we have so far.\n",
      "\n",
      "$P(x) = \\sum_{i=1}^{n}{F_i(x)}$\n",
      "\n",
      "And each term $F_i(x)$ is given by \n",
      "\n",
      "$F_i(x) = \\prod_{j \\neq i}^{n}{(x-x_j)}$\n",
      "\n",
      "Now that we have taken care of the zeroing, we need each term, $F_i(x)$ to equal 1 if $x = x_i$. Once we have this, we'll just multiply each term in the summation by the value that it has to equal.\n",
      "\n",
      "After staring at it for a few seconds, the solution appears:\n",
      "\n",
      "$F_i(x) = \\prod_{j \\neq i}^{n}{ \\frac{(x- x_j)}{(x_i-x_j)} } $\n",
      "\n",
      "Plugging in $x = x_i$, all the terms cancel and we are left with 1. The intution for $F_i(x)$ is that it's an *indicator function*, which is one whenever we're at point $x_i$ and is zero whenever we're at another point $x_j$. Putting it all together,\n",
      "\n",
      "$P(x) = \\sum_{i=1}^{n}{ \\prod_{j \\neq i}^{n}{ \\frac{(x- x_j)}{(x_i-x_j)} f(x_i) }} $\n",
      "\n",
      "To save computation, we don't need $f(x_i)$ inside the inner loop:\n",
      "\n",
      "$P(x) = \\sum_{i=1}^{n}{ f(x_i) \\prod_{j \\neq i}^{n}{ \\frac{(x- x_j)}{(x_i-x_j)} }} $\n",
      "\n",
      "\n",
      "Let's implement it and see how it works."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def poly_interpolate(x_values, y_values, x_new):\n",
      "    #consider y = f(x). \n",
      "    n = x_values.shape[0]\n",
      "    \n",
      "    #This is the indicator function discussed above\n",
      "    F_i = lambda x,i: prod([ (x_new - x[j])/(x[i]-x[j]) for j in xrange(n) if j != i])\n",
      "    \n",
      "    #We return a weighted sum of the indicator functions centered at each point.\n",
      "    return sum( F_i(x_values, i)*y_i for i, y_i in enumerate(y_values))\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's do a few examples."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_values = hstack((arange(-5, -1, 0.7), arange(1, 5, 0.7)))\n",
      "y_values = sin(x_values)\n",
      "plot(arange(-7, 7, 0.1), sin(arange(-7, 7, 0.1)))\n",
      "plot(arange(-7,7,0.1), [poly_interpolate(x_values, y_values, xi) for xi in arange(-7,7,.1)])\n",
      "scatter(x_values, y_values)\n",
      "legend(['actual', 'interpolation'])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 37,
       "text": [
        "<matplotlib.legend.Legend at 0xa384e10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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MZej5ax2akGZ7mROx15WOYpQ2/BVKYEXj77Z5oqlLZ375O1zpGEYpNhbu34f6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\nvSRJkon7P1M2MkiDpX5vAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa386b70>"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Wow, it looks spot on in this interval. How does it do outside the range where there were points?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_test = arange(-8, 8, 0.1)\n",
      "y_interpolated = [poly_interpolate(x_values, y_values, x_new) for x_new in x_test]\n",
      "plot(x_test, y_interpolated)\n",
      "plot(x_test, sin(x_test))\n",
      "legend(['interpolated', 'actual'])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 38,
       "text": [
        "<matplotlib.legend.Legend at 0xa632fd0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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+ABU2J6ERjXC6m4BA2wQM9B+M6QOHYGg/t3a1xttjwwbgz38GNm3SLeFpDCYf\n7N4fesPBxgEymQxaoUWNqgZ3Gu4AAFzsXeDr7IsgtyDIXeUIcgtCmGcY+vj0QYBLgEEm1KLOUV51\nB+nHf8OP5/Jx7no+rtTlo1J7GbWyG1DbVQC21YDKBTJYw0pYQ2h1N3tobe8AtjWAyhk2Gld0bQyA\nO4Lh7xyMKJ9gDA6Lwti+sfB27yrxV0itpdUKHPrlIn44ex6/XLmAi5UFuFZ/AdVW19BocxNa+5uA\n1gZWamdAyAD830OmgdauEoAMVg3usGn0hItWju52IZC79kRE955Ijo2DIj4A9vbSZsPhw7pumd27\njbMAkMkH+7U71/RzDQOAk52TbgpbC7jDk1qvXtWIsqrbuFutQU2dBtY2WtjaCng4uaCbszO6OrD1\n/bjQagWuV9WgrFK3RoFWCGi1AjbW1gj0coOnq4NZDLndvRt4+WXgxx+BkBDDHtvkg12iUxMRGd3n\nnwOffaabd8rBgJf8GOxERBKaNg2Qy3WjZgyFwU5EJKEbN3Szym7fDgwYYJhjWvxi1kREpszLS7dc\n50sv6Zbw7CwMdiIiI5oyBfDzA77+uvPOya4YIiIj+/lnYOpUID8f6NLBdTjYFUNEZAL699dNW7Bm\nTeecjy12IqJO8OuvwJNP6lZr68jwR7bYiYhMRO/eQEICsGWL8c/FYCci6iSvvaa7ccnYGOxERJ1k\n7FjdnO7Hjxv3PAx2IqJOYm0NvPIK8MUXxj0PL54SEXWi69d1qzQVFrZvUQ5ePCUiMjHe3sCwYcDO\nncY7B4OdiKiTTZ0KpKUZ7/jsiiEi6mR37gD+/sDly23vjmFXDBGRCXJxAYYPN153DIOdiEgC06YZ\nrzuGXTFERBJob3cMu2KIiEyUiwswZAiwf7/hj81gJyKSSHIycOCA4Y/LYCciksi9YDd0rzSDnYhI\nIlFRuiXzCgoMe1wGOxGRRGQyYNQow3fHGCXYU1NT4e/vj/j4eMTHxyM9Pd0YpyEiMnvG6Gc3ynDH\npUuXwtnZGW+99VbLJ+ZwRyIilJUBkZHAjRuAjc2jt5d0uCNDm4jo0Xx8gIAAICfHcMc0WrCvXr0a\ncXFxmDt3Lqqqqox1GiIiszdsGHD0qOGO1+6umOTkZJSVlTV5f9myZRgwYAC8vLwAAO+99x5KS0vx\n5ZdfPnhimQxLlizRv1YoFFAoFO0phYjIrH3zDfCf/+gev6dUKqFUKvWvly5d+sgeEaNPKVBUVISU\nlBScOnV1AE+sAAALOklEQVTqwROzj52ICABw6RLwxBPA1au6kTIPI1kfe2lpqf759u3bERsba4zT\nEBFZhOBgQKMBSkoMc7xWXINtu0WLFuHkyZOQyWQIDg7GmjVrjHEaIiKLIJMBAwcCx44BgYEGOB5n\ndyQikt7KlUB5OfDJJw/fjrM7EhGZiXstdkNgi52IyATU1OgWur51C7C3b3k7ttiJiMyEoyMQHg7k\n5nb8WAx2IiIT0b8/kJ3d8eMw2ImITESfPsAvv3T8OAx2IiIT0bs38OuvHT8OL54SEZmI6mrdBdQ7\nd1qe6ZEXT4mIzIiTE+DnB+Tnd+w4DHYiIhNiiO4YBjsRkQmJi+v4BVQGOxGRCWGwExFZmLg4dsUQ\nEVmUoCDg7l3g5s32H4PBTkRkQmSyjl9AZbATEZmYjvazM9iJiExMZCRw7lz792ewExGZmPDwjt2k\nxGAnIjIx4eHA+fPt35/BTkRkYvz8gNu3dXPGtAeDnYjIxFhZAb16tb87hsFORGSCOtLPzmAnIjJB\nHelnZ7ATEZkgBjsRkYUJC2t/VwxXUCIiMkF37gC+vrp5Y2Sy//8+V1AiIjJTLi6AszNw9Wrb92Ww\nExGZqPb2szPYiYhMVHv72dsd7GlpaYiOjoa1tTVyc3Mf+GzFihXo1asXIiIisH///vaegojosdbp\nLfbY2Fhs374dQ4cOfeD9s2fPYsuWLTh79izS09Px2muvQavVtvc0RESPrZ49gcLCtu/X7mCPiIhA\nWFhYk/d37tyJGTNmwNbWFnK5HKGhocjOzm7vaYiIHltyOVBU1Pb9DN7Hfu3aNfj7++tf+/v742p7\nLusSET3mgoN1wd7WkeE2D/swOTkZZWVlTd5fvnw5UlJSWn0S2f2DMImIqFXc3HRj2CsrAQ+P1u/3\n0GA/cOBAmwvx8/NDSUmJ/vWVK1fg5+fX7Lapqan65wqFAgqFos3nIyKyZN26KbF4sRI9erR+nw7f\neTp8+HB89NFH6Nu3LwDdxdOZM2ciOzsbV69exahRo1BQUNCk1c47T4mIHm3yZOD554FnntG9Nuqd\np9u3b0dAQACysrIwfvx4jB07FgAQFRWFadOmISoqCmPHjsXnn3/OrhgionZqzwVUzhVDRGTC/vEP\noKAAWL1a95pzxRARmbng4LaPZWewExGZMHbFEBFZmN9P38uuGCIiM+fiAtjbAxUVrd+HwU5EZOLa\n2h3DYCciMnEMdiIiCyOXt21kDIOdiMjE3ZsMrLUeOlcMERFJLz4eqK5u/fYc7khEZEY43JGI6DHE\nYCcisjAMdiIiC8NgJyKyMAx2IiILw2AnIrIwDHYiIgvDYCcisjAMdiIiC8NgJyKyMAx2IiILw2An\nIrIwDHYiIgvDYCcisjAMdiIiC8NgJyKyMAx2IiILw2AnIrIw7Q72tLQ0REdHw9raGrm5ufr3i4qK\n4ODggPj4eMTHx+O1114zSKFERNQ67Q722NhYbN++HUOHDm3yWWhoKPLy8pCXl4fPP/+8QwVKTalU\nSl1Cq7BOw2KdhmUOdZpDja3V7mCPiIhAWFiYIWsxSebyw2adhsU6Dcsc6jSHGlvLKH3shYWFiI+P\nh0KhwI8//miMUxARUQtsHvZhcnIyysrKmry/fPlypKSkNLuPr68vSkpK4O7ujtzcXEyaNAlnzpyB\ns7OzYSomIqKHEx2kUCjEiRMn2vx5SEiIAMAHH3zwwUcbHiEhIY/M5Ye22FtLCKF/XlFRAXd3d1hb\nW+PSpUu4cOECevbs2WSfgoICQ5yaiIh+p9197Nu3b0dAQACysrIwfvx4jB07FgBw+PBhxMXFIT4+\nHlOnTsWaNWvg5uZmsIKJiOjhZOL+5jYREZk9Se88zc7ORr9+/RAfH4+kpCTk5ORIWc5DrV69GpGR\nkYiJicGiRYukLuehPv74Y1hZWeHWrVtSl9Kst99+G5GRkYiLi8PTTz+N27dvS12SXnp6OiIiItCr\nVy/8z//8j9TlNKukpATDhw9HdHQ0YmJisGrVKqlLeiiNRoP4+PgWB1yYgqqqKkyZMgWRkZGIiopC\nVlaW1CU1a8WKFYiOjkZsbCxmzpyJhoaG5jds/2XTjhs2bJhIT08XQgixd+9eoVAopCynRZmZmWLU\nqFFCpVIJIYS4fv26xBW17PLly2L06NFCLpeLmzdvSl1Os/bv3y80Go0QQohFixaJRYsWSVyRjlqt\nFiEhIaKwsFCoVCoRFxcnzp49K3VZTZSWloq8vDwhhBB3794VYWFhJlnnPR9//LGYOXOmSElJkbqU\nFr3wwgviyy+/FEII0djYKKqqqiSuqKnCwkIRHBws6uvrhRBCTJs2TXz11VfNbitpi71Hjx761lpV\nVRX8/PykLKdFX3zxBRYvXgxbW1sAgJeXl8QVteytt97CBx98IHUZD5WcnAwrK92vXv/+/XHlyhWJ\nK9LJzs5GaGgo5HI5bG1tMX36dOzcuVPqsprw8fFBnz59AABOTk6IjIzEtWvXJK6qeVeuXMHevXsx\nb968BwZZmJLbt2/jhx9+wJw5cwAANjY2cHV1lbiqplxcXGBra4va2lqo1WrU1ta2mJmSBvvKlSux\ncOFCBAYG4u2338aKFSukLKdFFy5cwJEjRzBgwAAoFAocP35c6pKatXPnTvj7+6N3795Sl9Jq//rX\nvzBu3DipywAAXL16FQEBAfrX/v7+uHr1qoQVPVpRURHy8vLQv39/qUtp1p/+9Cd8+OGH+v/ITVFh\nYSG8vLzw0ksvISEhAS+//DJqa2ulLqsJDw8PfV76+vrCzc0No0aNanZbgwx3fJiWbnJatmwZVq1a\nhVWrVmHy5MlIS0vDnDlzcODAAWOX1KyH1alWq1FZWYmsrCzk5ORg2rRpuHTpkgRVPrzOFStWYP/+\n/fr3pGwhtebmtmXLlsHOzg4zZ87s7PKaJZPJpC6hTaqrqzFlyhT84x//gJOTk9TlNLF79254e3sj\nPj7epG/XV6vVyM3NxWeffYakpCQsWLAAK1euxN/+9jepS3vAxYsX8emnn6KoqAiurq6YOnUq/v3v\nf+O5555runHn9RA15ezsrH+u1WqFi4uLhNW0bMyYMUKpVOpfh4SEiIqKCgkraurUqVPC29tbyOVy\nIZfLhY2NjQgKChLl5eVSl9as9evXi0GDBom6ujqpS9E7duyYGD16tP718uXLxcqVKyWsqGUqlUo8\n+eST4pNPPpG6lBYtXrxY+Pv7C7lcLnx8fETXrl3FrFmzpC6ridLSUiGXy/Wvf/jhBzF+/HgJK2re\n5s2bxdy5c/WvN2zYIF577bVmt5X076PQ0FAcPnwYAJCZmWmyk4pNmjQJmZmZAID8/HyoVCp4enpK\nXNWDYmJiUF5ejsLCQhQWFsLf3x+5ubnw9vaWurQm0tPT8eGHH2Lnzp3o0qWL1OXoJSYm4sKFCygq\nKoJKpcKWLVswceJEqctqQgiBuXPnIioqCgsWLJC6nBYtX74cJSUlKCwsxObNmzFixAhs2LBB6rKa\n8PHxQUBAAPLz8wEAGRkZiI6OlriqpiIiIpCVlYW6ujoIIZCRkYGoqKhmtzV6V8zDrF27Fq+//joa\nGhrg4OCAtWvXSllOi+bMmYM5c+YgNjYWdnZ2JvnL+Xum3K3wxhtvQKVSITk5GQAwcOBAk5je2cbG\nBp999hlGjx4NjUaDuXPnIjIyUuqymjh69Cg2btyI3r17Iz4+HoBuGNyYMWMkruzhTPl3cvXq1Xju\nueegUqkQEhKC9evXS11SE3FxcXjhhReQmJgIKysrJCQkYP78+c1uyxuUiIgsjOleqiYionZhsBMR\nWRgGOxGRhWGwExFZGAY7EZGFYbATEVkYBjsRkYVhsBMRWZj/By94yj1QrKy+AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa04d780>"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ah, so we can see that it's only accurate where data were available. Let's run another example, *with noise*. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_values = arange(-5, 5, 0.3)\n",
      "y_values = sin(x_values) + uniform(-.2, .2, x_values.shape) #add random noise\n",
      "scatter(x_values, y_values)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 39,
       "text": [
        "<matplotlib.collections.PathCollection at 0xa8aa400>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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29mgUFZ3EtWupiI09jrVr11p6VyzinlfdhIaG4sKFC3csnz9/PiIiIu77y/m/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\ndRMyWYfLVQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa3905f8>"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's fit it."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_test = arange(-5, 5, 0.1)\n",
      "y_interpolated = [poly_interpolate(x_values, y_values, x_new) for x_new in x_test]\n",
      "plot(x_test, y_interpolated)\n",
      "plot(x_test, sin(x_test))\n",
      "legend(['interpolated', 'actual'])\n",
      "ylim([-10, 10])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 40,
       "text": [
        "(-10, 10)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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50zt3wvr1rlOXz8hQe6obNio89frPNJ38Pl8e/4zo4Gje7vA6vZv3xt3NnZwc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\nLuwbPJWBU+EMpMdecSU7vTN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       "text": [
        "<matplotlib.figure.Figure at 0xa025588>"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This shows how this interpolation method can easily fail in real-life problems. If you have a machine learning background, you should realize that what this polynomial interpolation technique is doing is **overfitting**. If that means nothing to you - no worries. I've got some other IPython notebooks on machine learning."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Neville's method##\n",
      "\n",
      "Neville's (sp) method is another way to generate the polynomial approximation, but has a nice recursive form.\n",
      "\n",
      "Define $P_{j,j}(x) = y_j$, which is the base case of the recusion. The rule for composing two previous polynomial approximations is:\n",
      "\n",
      "$P_{i,j}(x) = \\frac{ (x-x_i)P_{i+1,j}(x) - (x-x_j)P_{i,j-1}(x) } { x_j - x_i } $\n",
      "\n",
      "Implementation should be nice."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def neville_poly(xvalues, yvalues, x, i=None, j=None):\n",
      "    if i is None and j is None: #first iteration\n",
      "        i, j = 0, yvalues.shape[0]-1\n",
      "    \n",
      "    if i != j: #descend\n",
      "        P_iplus1_j = neville_poly(xvalues, yvalues, x, i+1, j)\n",
      "        P_i_jminus1 = neville_poly(xvalues, yvalues, x, i, j-1)\n",
      "        #combine them\n",
      "        return (P_iplus1_j*(x-xvalues[i]) - P_i_jminus1*(x-xvalues[j])) / (xvalues[j] - xvalues[i])\n",
      "    else:\n",
      "        #base case\n",
      "        return yvalues[i]\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_values = hstack((arange(-6, -1, 0.7), arange(1, 5, 0.7)))\n",
      "y_values = sin(x_values)\n",
      "plot(arange(-6, 5, 0.2), [neville_poly(x_values, y_values, x) for x in arange(-6,5,.2)])\n",
      "scatter(x_values, y_values)\n",
      "legend(['approximation'])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 42,
       "text": [
        "<matplotlib.legend.Legend at 0xa890470>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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5P6NT66Zdu3aIj49HYmIiCgsLsXnzZvTv3/+ZY/r374/169cDAKKjo1G7dm04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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xae59fd0>"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Divided Difference Method##\n",
      "\n",
      "Neville's method is very wasteful of computation. In the divided difference method, we do the same amount of computation but the result is the coefficients on the polynomial; we only have to run it once.\n",
      "\n",
      "Since $P(x)$ is a polynomial, it can be written in the form:\n",
      "\n",
      "$P(x) = a_0 + a_1(x-x_0) + a_2(x-x_0)(x-x_1) + ...$\n",
      "\n",
      "or\n",
      "\n",
      "$P(x) = a_0 + \\sum_{k=1}^{n}(a_k) \\prod_{j=0}^{k-1} (x-x_j) $\n",
      "\n",
      "\n",
      "The divided difference method is a recursive formula for determining the coefficients $a_i$. \n",
      "\n",
      "Define $f[x_i] = f(x_i)$ (base case) and $f[x_i ... x_j] = \\frac{ f[x_{i+1} ... x_j] - f[x_i ... x_{j-1}]} { x_j - x_i } $\n",
      "\n",
      "This should be easy to implement."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "class lagrangepoly(object):\n",
      "    def __init__(self, x, a):\n",
      "        #the vector x is the points,\n",
      "        #the vector a is the coefficients\n",
      "        self.x = x\n",
      "        self.a = a\n",
      "    \n",
      "    def apply(self, x_new):\n",
      "        val = 0.0\n",
      "        for k,coef in enumerate(self.a):\n",
      "            val += coef*prod([(x_new - x_j) for j,x_j in enumerate(self.x) if j <= k-1])\n",
      "        return val\n",
      "    \n",
      "def divided_diff_coefs(xvalues, yvalues):\n",
      "    #return the coefficients for the lagrange approximating polynomial\n",
      "    #let's do it iteratively for a change\n",
      "    N = xvalues.shape[0]\n",
      "    coef = {(i,i):yvalues[i] for i in range(N)}\n",
      "    for delta in range(1, N):\n",
      "        for i in range(0, N-delta):\n",
      "            coef[i,i+delta] = (coef[i+1,i+delta] - coef[i,i+delta-1]) / (xvalues[i+delta] - xvalues[i])\n",
      "    #ok, now construct the ones we care about\n",
      "    c = zeros((N,))\n",
      "    for j in range(N):\n",
      "        c[j] = coef[0,j]\n",
      "    return c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_values = hstack((arange(-6, -1, 0.7), arange(1, 5, 0.7)))\n",
      "y_values = sin(x_values)\n",
      "poly = lagrangepoly(x_values, divided_diff_coefs(x_values,y_values))\n",
      "plot(arange(-7, 6, 0.2), [poly.apply(x) for x in arange(-7,6,.2)])\n",
      "scatter(x_values, y_values)\n",
      "legend(['approximation'])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 44,
       "text": [
        "<matplotlib.legend.Legend at 0xaed4eb8>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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5opMQTWhV6C0tLbF8+XKEhobC19cXr7zyCnx8fLB69WqsXr0aABAWFoZWrVrB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\nn5hfkVcb09DFixfZ5cuXmVw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       "text": [
        "<matplotlib.figure.Figure at 0xa384710>"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 44
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Interpolation with Radial Basis Functions ##\n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The 'indicator function' viewpoint is a valuable one for interpolation. The idea, as we discussed before, is to make the interpolation a sum of terms,\n",
      "\n",
      "$P(x) = \\sum_{i=1}^{n}{y_i F(i, x)}$\n",
      "\n",
      "Where $F(i, x)$ is $1$ if $x = x_i$ and $F(i, x) = 0$ if $x = x_j$, where $x_j$ is another of the points in our provided set. In polynomial interpolation, we did this by carefully constructing a polynomial which would zero out at all the other points. Another way to look at it is to say that the indicator, $F(i, x)$ should only be nonzero if $x$ is close to $x_i$. To achieve, this, we'll use *radial basis functions*.\n",
      "\n",
      "Here's what the plot of an RBF looks like"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def rbf(x, center, width):\n",
      "    return exp( -norm(x - center)**2 / width)\n",
      "\n",
      "x = arange(-5, 5, 0.1)\n",
      "y = [rbf(xi, 0.0, 0.1) for xi in x]\n",
      "plot(x, y)\n",
      "title('Radial basis function centered at 0')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": [
        "<matplotlib.text.Text at 0xaef6ac8>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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sUx1RM9irtRIQYDqJWlVl+30TtUSDrRVNEz699ptvvsH777+PvXv31rk+PT3d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\nBjkRkZtjkBMRuTkGORGRm/t/g/ENTYD0ysQAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa8b2940>"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can make it wider, and move the center."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = [rbf(xi, 2.0, 0.5) for xi in x]\n",
      "plot(x, y)\n",
      "title('Radial basis function centered at 2')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 46,
       "text": [
        "<matplotlib.text.Text at 0xb2ec1d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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yd3QCPN+KEjHIiRRCLSPy4GDpnOTV1XJXQmYMciKFUEuQazTS6Ww5c0U5GORE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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa8ae320>"
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So now our interpolation method is pretty straightforward. We'll just use a weighted combination of radial basis functions centered at each point we have. In other words,\n",
      "\n",
      "$P(x) = \\sum_{i=1}^{n} { F(x, x_i)}$, where $F(x, x_i) = e^{-||x-x_i||/w}$, and $w$ is the width parameter."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def rbf_interpolate(x_new, x_values, y_values, width):\n",
      "    #interpolate using radial basis functions\n",
      "    return sum( rbf(x_new, x_i, width)*y_i for x_i, y_i in zip(x_values, y_values))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 47
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And a test,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = arange(-5, 5, 0.8)\n",
      "y = sin(x)\n",
      "scatter(x, y)\n",
      "x_new = arange(-5, 5, 0.1)\n",
      "y_interp = [rbf_interpolate(xi, x, y, 0.1) for xi in x_new]\n",
      "plot(x_new, y_interp)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 48,
       "text": [
        "[<matplotlib.lines.Line2D at 0xb301908>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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DgLVr1Y/RXTHRe4nkZFmjz82t+9ysLKCoCBg+/PpjrijfLFgA3HuvOsn+3Dng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\n+39WfjyJ5qQQJgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb301630>"
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we make each rbf function have a larger width, then the interpolation is more smooth,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = arange(-5, 5, 0.8)\n",
      "y = sin(x)\n",
      "scatter(x, y)\n",
      "x_new = arange(-5, 5, 0.1)\n",
      "y_interp = [rbf_interpolate(xi, x, y, 0.3) for xi in x_new]\n",
      "plot(x_new, y_interp)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 49,
       "text": [
        "[<matplotlib.lines.Line2D at 0xb3166a0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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jQ4cOBpO5zoMP6kk3Tz9tOolwxPr1MGYMyOW+v7h0wlTXrl2ZN28eAPPmzaN7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\niJY7K7rcAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb316940>"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This method is far more robust under noise,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = arange(-5, 5, 0.5)\n",
      "y = sin(x) + uniform(-.2, .2, x.shape)\n",
      "scatter(x, y)\n",
      "x_new = arange(-5, 5, 0.1)\n",
      "y_interp = [rbf_interpolate(xi, x, y, 0.1) for xi in x_new]\n",
      "plot(x_new, y_interp)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 50,
       "text": [
        "[<matplotlib.lines.Line2D at 0xb624908>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SFkvkwt1dDd/ExelOkjdHuRELUuhdQlgY3Huva96UffzxTnh7fwIsBX7F2/tN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\nrkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb624c88>"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Nearest-Neighbor Interpolation ##\n",
      "\n",
      "Again, this is a method not covered in the text, but it's easy to understand and practical. The idea here is that given a new point, $x$, we just approximate $f(x)$ with $f(x_i)$, for the $x_i$ which is closest to $x$. Simple."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def nn_interpolate(x_new, x_values, y_values):\n",
      "    #interpolate based on nearest-neighbor\n",
      "    closest_x_index = argmin(abs(x_values - x_new))\n",
      "    return y_values[closest_x_index]\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And an example"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = arange(-5, 5, 0.8)\n",
      "y = sin(x)\n",
      "scatter(x, y)\n",
      "x_new = arange(-5, 5, 0.1)\n",
      "y_interp = [nn_interpolate(xi, x, y) for xi in x_new]\n",
      "plot(x_new, y_interp)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 52,
       "text": [
        "[<matplotlib.lines.Line2D at 0xb301cc0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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hw/1gNC4H4AOD4fd4550n8OKLlxAREdFmOzodcOutLinZITaDPjU1FRUVFW0e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\nGPRERCrHoCciUjkGPRGRyjHoiYhU7v8DXMhlf9GcG0QAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb6130f0>"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Cool. This one is nice because it will work okay in noisy situations."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Cubic spline interpolation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "Time to bite the bullet and grind this one out. When interpolating with a cubic spline, we set the following conditions:\n",
      "\n",
      "1. Each spline is a polynomial of degree 3 (duh)\n",
      "2. Each spline agrees with the function values at two points.\n",
      "3. At each point, the two splines which agree must have the same first and second derivatives\n",
      "4. (natural) At the endpoints, the second derivatives are zero.\n",
      "\n",
      "This gives us enough unknowns to solve for all the coefficients in the splines.\n",
      "\n",
      "Derivation:\n",
      "\n",
      "Let $x_0 ... x_n$ be the points and $y_0 ... y_n$ be the function values. We have (n-1) splines, each with 4 parameters. Each spline is of the form:\n",
      "\n",
      "$S_j(x) = a_j + b_j(x-x_j) + c_j(x-x_j)^2 + d_j(x-x_j)^3$\n",
      "\n",
      "So obviously each $a_j = y_j$.\n",
      "\n",
      "Let's assume we know the slope of the spline $s_j'$ at each point. Also let $\\Delta x_j = x_{j+1} - x_{j}$.\n",
      "\n",
      "Given the slopes at each point, \n",
      "\n",
      "$S_j'(x_j) = s_j' = b_j$\n",
      "\n",
      "And furthermore, since $s_{j+1}' = S_j'(x_{j+1}) = S_{j+1}'(x_{j+1})$, \n",
      "\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$b_j + 2 c_j(\\Delta x) + 3 d_j(\\Delta x)^2 = b_{j+1}$\n",
      "\n",
      "And due to the continuity of second derivatives,\n",
      "\n",
      "$2 c_j + 6 d_j (\\Delta x) = 2 c_{j+1}$ (and by simplifiying), $c_j + 3 d_j (\\Delta x) = c_{j+1}$\n",
      "\n",
      "(Some magic happens)\n",
      "\n",
      "And a tridiagonal linear system emerges from the aether. $Ax = b$,\n",
      "\n",
      "where $x$ is the vector of derivatives at each point, which we wish to determine. $A$ is the $(n+1) \\times (n+1)$ tridiagonal matrix with elements given by \n",
      "\n",
      "$A_{0,0} = A_{n+1,n+1} = 1$\n",
      "\n",
      "$A_{i,i} = 2(\\Delta x_{i-1} + \\Delta x_i) = 2(x_{i+1} - x_{i-1})$\n",
      "\n",
      "$A_{i,i-1} = \\Delta x_{i-1} = (x_i - x_{i-1})$ \n",
      "\n",
      "$A_{i,i+1} = \\Delta x_i = (x_{i+1} - x_i)$ for $i=1...N$. All other elements of $A$ are zero.\n",
      "\n",
      "The vector $b$ is given by \n",
      "\n",
      "$b_0 = b_{n+1} = 0$\n",
      "\n",
      "$b_i = \\frac{3}{\\Delta x_i} (y_{i+1} - y_i) - \\frac{3}{\\Delta x_{i-1}}(y_i - y_{i-1})$ for $i=1...n$\n",
      "\n",
      "Which I think looks nicer written as \n",
      "\n",
      "$b_i = \\frac{3\\Delta y_i}{\\Delta x_i} - \\frac{3\\Delta y_{i-1}}{\\Delta x_{i-1}}$ for $i=1...n$\n",
      "\n",
      "\n",
      "From this equation the coefficients for the splines can be derived. It's in the code below.\n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "class cubicpoly(object):\n",
      "    def __init__(self, x_j, x_jplus1, a, b, c, d):\n",
      "        #a,b,c,d - 0,1,2,3 degree coefficients\n",
      "        #x_j , x_j+1 are points\n",
      "        self.x_j = x_j\n",
      "        self.x_jplus1 = x_jplus1\n",
      "        self.a, self.b, self.c, self.d = a,b,c,d\n",
      "        \n",
      "    def apply(self, x):\n",
      "        return self.a + self.b*(x-self.x_j) + self.c*(x-self.x_j)**2 + self.d*(x-self.x_j)**3\n",
      "    \n",
      "    def __str__(self):\n",
      "        return 'a:' + str(self.a) + ' b:' + str(self.b) + ' c:' + str(self.c) + ' d:' + str(self.d)\n",
      "    \n",
      "def eval_spline(spline, x):\n",
      "    #determine which interval to use...\n",
      "    for poly in spline:\n",
      "        if poly.x_j <= x <= poly.x_jplus1:\n",
      "            return poly.apply(x)\n",
      "        \n",
      "    #if it didn't fit, then predict with one of the endpoints...\n",
      "    if x < spline[0].x_j:\n",
      "        return spline[0].apply(x)\n",
      "    elif x > spline[-1].x_jplus1:\n",
      "        return spline[-1].apply(x)\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 53
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "That should do it for evaluation. Let's build a random spline and see what it looks like."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "spline = []\n",
      "for i in range(5):\n",
      "    spline.append( cubicpoly(i, i+1, randn(1), randn(1), randn(1), randn(1)))\n",
      "    \n",
      "x = arange(-2, 2, 0.1)\n",
      "y = [eval_spline(spline, xi) for xi in x]\n",
      "plot(x,y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 54,
       "text": [
        "[<matplotlib.lines.Line2D at 0xbdb0a20>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0xb841dd8>"
       ]
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "It's pretty obvious that this spline disobeys our rules for interpolation (which is ok, it's random). Let's write the code to fit it."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def fit_cubic_spline(x, y):\n",
      "    #x and y are vectors in this case.\n",
      "    #set up a system of equations\n",
      "    N = x.shape[0]-1\n",
      "    A = zeros((N+1,N+1))\n",
      "    b = zeros((N+1,))\n",
      "    #set the values of A. It's tridiagonal.\n",
      "    #first and last rows\n",
      "    A[0,0] = 1\n",
      "    A[N,N] = 1\n",
      "    #tridiagonal values\n",
      "    for i in range(1, N):\n",
      "        A[i,i] = 2*(-x[i-1]+x[i+1])\n",
      "        A[i,i-1] = x[i] - x[i-1]\n",
      "        A[i,i+1] = x[i+1] - x[i]\n",
      "        b[i] = (3.0/(x[i+1]-x[i]))*(y[i+1]-y[i]) - (3.0/(x[i]-x[i-1]))*(y[i]-y[i-1])\n",
      "    \n",
      "    \n",
      "    #solve the system of equations...\n",
      "    c = solve(A, b)\n",
      "\n",
      "    #now construct the splines.\n",
      "    spline = []\n",
      "    for i in range(N):\n",
      "        a_i = y[i] #spline must agree with points!\n",
      "        b_i = (x[i+1]-x[i])**-1 * (y[i+1]-y[i]) - (x[i+1]-x[i])/3.0*(2*c[i]+c[i+1])\n",
      "        c_i = c[i]\n",
      "        d_i = (c[i+1] - c[i])/(3.0*(x[i+1]-x[i]))\n",
      "        spline.append(cubicpoly(x[i], x[i+1], a_i, b_i, c_i, d_i))\n",
      "    return spline\n",
      "    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 55
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = arange(-5,5,0.6)\n",
      "y = sin(x)-0.5*cos(2*x)\n",
      "scatter(x,y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 56,
       "text": [
        "<matplotlib.collections.PathCollection at 0xbea2518>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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o5Ofno6CgAEajEdu2bUNiYuI9YxITE5GWlgYAyM7ORvv27eHr66tks0TkoEwmE4YMScCk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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb84f9b0>"
       ]
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s = fit_cubic_spline(x,y)\n",
      "xnew = arange(-6,6,.1)\n",
      "yh = [eval_spline(s, xi) for xi in xnew]\n",
      "scatter(x,y)\n",
      "plot(xnew,yh)\n",
      "plot(xnew, sin(xnew)-.5*cos(2*xnew))\n",
      "legend(['spline', 'actual'])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 57,
       "text": [
        "<matplotlib.legend.Legend at 0xbed35f8>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ancViMebNm4eTJ0/Czs4OAwYMQEREBNzc3B7bbtiwYYiKipIqqCpQphOxDznq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\nEELUHBV6QghRc1ToCSFEzVGhJ4QQNff/tX6x9ueS4voAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xbddc400>"
       ]
      }
     ],
     "prompt_number": 57
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Solutions to systems of linear equations ##\n",
      "\n",
      "This will cover chapter 6 and maybe chapter 7. \n",
      "\n",
      "\n",
      "### Gaussian Elimination ###\n",
      "\n",
      "The first algorithm produced in the text. Each row is used to eliminate one variable from the other rows. Implementation should be relatively straightforward:\n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def gauss_elim(A):\n",
      "    #assume A is an augmented matrix ( [A | b])\n",
      "    for i in range(A.shape[0]): #for each row\n",
      "        for j in range(i+1, A.shape[0]): #for each following row\n",
      "            multiplier = A[j,i]/A[i,i]\n",
      "            A[j,:] = A[j,:] - multiplier*A[i,:]\n",
      "    return A\n",
      "            "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 58
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "let's try it out."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[1, 3],[2,4]], float)\n",
      "print gauss_elim(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 1.  3.]\n",
        " [ 0. -2.]]\n"
       ]
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And a bigger matrix"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[1,2,4],[2,1,4],[3,4,2]], float)\n",
      "print gauss_elim(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 1.          2.          4.        ]\n",
        " [ 0.         -3.         -4.        ]\n",
        " [ 0.          0.         -7.33333333]]\n"
       ]
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To actually solve systems we'll need to backward substitute, so let's write some code to do that"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def back_sub(A):\n",
      "    #A must be an augmented matrix in row-echelon form\n",
      "    assert A.shape[0] == A.shape[1]-1, 'A should be square | augmented'\n",
      "    x = zeros(A.shape[0])\n",
      "    for i in reversed(range(A.shape[0])):\n",
      "        terms = sum( A[i,j]*x[j] for j in range(i+1,A.shape[0]) )\n",
      "        x[i] = (A[i,-1] - terms) / A[i,i]\n",
      "    return x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 61
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Some handy code for augmentation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def augment(A, b):\n",
      "    return hstack((A, b.reshape((-1,1))))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 62
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[1,2,4],[2,1,4],[3,4,2]], float)\n",
      "b  = array([2,5,1])\n",
      "A_aug = augment(A, b)\n",
      "G = gauss_elim(A_aug)\n",
      "x = back_sub(G)\n",
      "print x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 1.63636364 -1.36363636  0.77272727]\n"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And now if it worked, $Ax = b$ should hold."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print dot(A, x), b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 2.  5.  1.] [2 5 1]\n"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Cool."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = eye(3)\n",
      "A_aug = augment(A, array([1,2,3]))\n",
      "B = gauss_elim(A_aug)\n",
      "x = back_sub(B)\n",
      "print x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 1.  2.  3.]\n"
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can wrap these together for a matsolve function"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matsolve(A, b):\n",
      "    A_aug = augment(A, b)\n",
      "    return back_sub(gauss_elim(A_aug))\n",
      "\n",
      "A = randn(20, 20)\n",
      "b = arange(20)\n",
      "x = matsolve(A, b) #solve the matrix\n",
      "print dot(A, x) #if it worked, then this should equal b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[  0.   1.   2.   3.   4.   5.   6.   7.   8.   9.  10.  11.  12.  13.  14.\n",
        "  15.  16.  17.  18.  19.]\n"
       ]
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Cool. Let's try the numerically unstable example in the text."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[.003, 59.14],[5.291, -6.130]])\n",
      "b = array([59.17, 46.78])\n",
      "print matsolve(A, b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 10.   1.]\n"
       ]
      }
     ],
     "prompt_number": 67
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "No problems here. Still, let's examine pivoting strategies."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Partial Pivoting ### \n",
      "\n",
      "To prevent numerical problems, we can swap rows to prevent the denominator of our multiplier from becoming large. When we're going though each row, we'll just swap the current row with the one which has the largest norm component in the pivot column."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def gauss_elim_partial_pivot(A):\n",
      "    #assume A is an augmented matrix ( [A | b])\n",
      "    for i in range(A.shape[0]): #for each row\n",
      "        #pivoting - swap this row with another\n",
      "        row_ind = argmax(abs(A[i:,i]))\n",
      "        #swap\n",
      "        A[[row_ind+i, i],:] = A[[i,row_ind+i],:]\n",
      "        \n",
      "        for j in range(i+1, A.shape[0]): #for each following row\n",
      "            multiplier = A[j,i]/A[i,i]\n",
      "            A[j,:] = A[j,:] - multiplier*A[i,:]\n",
      "    return A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 68
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[1, 3],[2,4]], float)\n",
      "print gauss_elim_partial_pivot(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 2.  4.]\n",
        " [ 0.  1.]]\n"
       ]
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's confirm that this works"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matsolve(A, b):\n",
      "    A_aug = augment(A, b)\n",
      "    return back_sub(gauss_elim_partial_pivot(A_aug))\n",
      "\n",
      "A = randn(20, 20)\n",
      "b = arange(20)\n",
      "x = matsolve(A, b) #solve the matrix\n",
      "print dot(A, x) #if it worked, then this should equal b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ -3.99680289e-15   1.00000000e+00   2.00000000e+00   3.00000000e+00\n",
        "   4.00000000e+00   5.00000000e+00   6.00000000e+00   7.00000000e+00\n",
        "   8.00000000e+00   9.00000000e+00   1.00000000e+01   1.10000000e+01\n",
        "   1.20000000e+01   1.30000000e+01   1.40000000e+01   1.50000000e+01\n",
        "   1.60000000e+01   1.70000000e+01   1.80000000e+01   1.90000000e+01]\n"
       ]
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Woah there, it looks like we've got a little bit of numerical error around 0, but for the most part these look good."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Scaled Partial Pivoting ###\n",
      "\n",
      "We can extend this even further to account for relative magnitudes of the rows. Just define a vector of *scale factors*, one for each row. The scale factor for a row is the infinity norm of the row (max of the norms of the components). "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def gauss_elim_partial_pivot_scaled(A, verbose=False):\n",
      "    #assume A is an augmented matrix ( [A | b])\n",
      "    \n",
      "    #compute scale factors\n",
      "    s = array([norm(Ai, inf) for Ai in A])\n",
      "    if verbose: print 'scale factors:', s\n",
      "        \n",
      "    for i in range(A.shape[0]): #for each row\n",
      "        #pivoting - swap this row with another\n",
      "        #now find the \"relative\" largest\n",
      "        \n",
      "        row_ind = argmax(abs(A[i:,i])/s[i:]) #find the row to swap the current with\n",
      "        if verbose: print 'swapping row', row_ind+i, 'and row', i\n",
      "        \n",
      "        if verbose: print 'before swap\\n:', A\n",
      "        #swap\n",
      "        A[[row_ind+i, i],:] = A[[i,row_ind+i],:]\n",
      "        s[[row_ind+i,i]] = s[[i,row_ind+i]] #also swap scale factors!\n",
      "        \n",
      "        if verbose: print 'after swap\\n:', A\n",
      "        #if verbose: print A\n",
      "        \n",
      "        for j in range(i+1, A.shape[0]): #for each following row\n",
      "            multiplier = A[j,i]/A[i,i]\n",
      "            A[j,:] = A[j,:] - multiplier*A[i,:]\n",
      "            if verbose: print 'row op:', 'row', j, '=', 'row', j, '-', multiplier, '*', 'row', i\n",
      "        \n",
      "        if verbose: print 'matrix after row ops:\\n', A\n",
      "            \n",
      "    return A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 397
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matsolve(A, b):\n",
      "    A_aug = augment(A, b)\n",
      "    return back_sub(gauss_elim_partial_pivot_scaled(A_aug))\n",
      "\n",
      "A = randn(20, 20)\n",
      "b = arange(20)\n",
      "x = matsolve(A, b) #solve the matrix\n",
      "print dot(A, x) #if it worked, then this should equal b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ -3.55271368e-15   1.00000000e+00   2.00000000e+00   3.00000000e+00\n",
        "   4.00000000e+00   5.00000000e+00   6.00000000e+00   7.00000000e+00\n",
        "   8.00000000e+00   9.00000000e+00   1.00000000e+01   1.10000000e+01\n",
        "   1.20000000e+01   1.30000000e+01   1.40000000e+01   1.50000000e+01\n",
        "   1.60000000e+01   1.70000000e+01   1.80000000e+01   1.90000000e+01]\n"
       ]
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "How about a more pathalogical case?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = randn(10, 10)\n",
      "#set the diag to a very small number\n",
      "A = A - diag(A) + diag(ones(10)*1e-15)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 301
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matsolve(A, b):\n",
      "    A_aug = augment(A, b)\n",
      "    return back_sub(gauss_elim_partial_pivot_scaled(A_aug))\n",
      "\n",
      "b = arange(10)\n",
      "x = matsolve(A, b) #solve the matrix\n",
      "print dot(A, x) #if it worked, then this should equal b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[  7.10542736e-15   1.00000000e+00   2.00000000e+00   3.00000000e+00\n",
        "   4.00000000e+00   5.00000000e+00   6.00000000e+00   7.00000000e+00\n",
        "   8.00000000e+00   9.00000000e+00]\n"
       ]
      }
     ],
     "prompt_number": 302
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Nice. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Iterative Methods ##\n",
      "\n",
      "##Jacobi##\n",
      "\n",
      "To find a solution to $Ax = b$, let $A = D + R$, where $D$ is diagonal and $R$ has no diagonal elements. Thus\n",
      "\n",
      "$(D+R)x = b$\n",
      "\n",
      "$Dx = b - Rx$\n",
      "\n",
      "$x = D^{-1}(b - Rx)$, which is easily found.\n",
      "\n",
      "To put in normal form, let $T = -D^{-1}R$ and $c = D^{-1}b$, then\n",
      "\n",
      "$x^{(n)} = Tx^{(n-1)} + c$.\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def jacobi_solve(A, b, max_iter=100, tol=1e-4):\n",
      "    assert A.shape[0] == A.shape[1], 'A must be square!'\n",
      "    D,R = zeros(A.shape), zeros(A.shape)\n",
      "    for i in range(A.shape[0]):\n",
      "        for j in range(A.shape[1]):\n",
      "            if i == j:\n",
      "                D[i,j] = A[i,j]\n",
      "            else:\n",
      "                R[i,j] = A[i,j]\n",
      "    #ok, now iterate\n",
      "    x_vals = [0.0*b]\n",
      "    D_inv = inv(D)\n",
      "    T = -dot(D_inv, R)\n",
      "    C = dot(D_inv, b)\n",
      "    for i in range(max_iter):\n",
      "        #recurrence relation\n",
      "        x_vals.append( dot(T, x_vals[-1]) + C)\n",
      "        if norm(x_vals[-1]-x_vals[-2], inf)/norm(x_vals[-1], inf) < tol:\n",
      "            #done\n",
      "            return x_vals[-1]\n",
      "        print x_vals[-1]\n",
      "    return x_vals[-1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 75
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[16,3],\n",
      "           [7,-11]])\n",
      "b = array([11,13])\n",
      "x = jacobi_solve(A, b)\n",
      "print 'x =',x\n",
      "print 'Now, Ax should equal b.'\n",
      "print dot(A,x), b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0.6875     -1.18181818]\n",
        "[ 0.90909091 -0.74431818]\n",
        "[ 0.82705966 -0.60330579]\n",
        "[ 0.80061983 -0.65550749]\n",
        "[ 0.81040765 -0.67233283]\n",
        "[ 0.81356241 -0.66610422]\n",
        "[ 0.81239454 -0.66409665]\n",
        "[ 0.81201812 -0.66483984]\n",
        "[ 0.81215747 -0.66507938]\n",
        "[ 0.81220238 -0.6649907 ]\n",
        "x = [ 0.81218576 -0.66496212]\n",
        "Now, Ax should equal b.\n",
        "[ 11.00008574  12.99988361] [11 13]\n"
       ]
      }
     ],
     "prompt_number": 76
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 76
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "\n",
      "\n",
      "###Gauss-Sidel###\n",
      "\n",
      "In Gauss-Sidel, we just update our previous approximation as we go. In other words, we don't do batch updates like Jacobi. \n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def g_s_solve(A, b, max_iter=100, tol=1e-4, verbose=False):\n",
      "    assert A.shape[0] == A.shape[1], 'A must be square!'\n",
      "    D = diag(diag(A)) #matrix of diagonal elements\n",
      "    R = A - D #remainder\n",
      "    \n",
      "    if verbose: print 'A decomposed into D + R\\n', D, '\\n', R\n",
      "    \n",
      "    #ok, now iterate\n",
      "    x_vals = [0.0*b]\n",
      "    D_inv = inv(D) #inverse of a diagonal is 1 / the diagonal elements!\n",
      "    T = -dot(D_inv, R)\n",
      "    C = dot(D_inv, b)\n",
      "    \n",
      "    if verbose: print 'Linear iteration matrix T:\\n', T\n",
      "    if verbose: print 'Linear iteration vector C:\\n', C\n",
      "    \n",
      "    for i in range(max_iter):\n",
      "        #recurrence relation\n",
      "        if verbose: print 'x at iteration', i+1, x_vals[-1]\n",
      "        x_new = x_vals[-1]\n",
      "        for i in range(x_new.shape[0]):\n",
      "            if verbose: print 'updating x[', i, '], old:', x_new[i]\n",
      "            if verbose: print x_new, 'dot', T[i,:], '+', C[i], '=', dot(x_new, T[i,:])+C[i]\n",
      "            x_new[i] = (dot(T, x_new) + C)[i] #an efficient implementation wouldn't redo this work\n",
      "            if verbose: print 'x[', i, '] =', x_new[i]\n",
      "        x_vals.append( x_new )\n",
      "        \n",
      "        #convergence check\n",
      "        #if norm(x_vals[-1]-x_vals[-2], inf)/norm(x_vals[-1], inf) < tol:\n",
      "            #done\n",
      "        #    return x_vals[-1]\n",
      "        #print x_vals[-1]\n",
      "    return x_vals[-1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 322
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[16,3],\n",
      "           [7,-11]])\n",
      "b = array([11,13])\n",
      "x = g_s_solve(A, b)\n",
      "print 'x =',x\n",
      "print 'Now, Ax should equal b.'\n",
      "print dot(A,x), b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "x = [ 0.6875     -0.74431818]\n",
        "Now, Ax should equal b.\n",
        "[  8.76704545  13.        ] [11 13]\n"
       ]
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##LU decomposition##\n",
      "\n",
      "In LU decomposition, we decompose a square matrix $A$ into the product of a unit-lower matrix and an upper triangular matrix. The intuition is that each step of Gaussian Elimination is just a linear operation, and as such can be represented as a matrix. Composing these operations results in an upper triangular matrix, $U$. I.e.\n",
      "\n",
      "\n",
      "$E_1 E_2 ... E_n A$ = $U$\n",
      "\n",
      "Where $E_i$ is the $i$th row operation. Thus,\n",
      "\n",
      "$A = (E_n)^{-1} ... (E_2)^{-1} (E_1)^{-1} U$\n",
      "\n",
      "And since the inverse of a unit lower triangular matrix is just the negative of the off diagonal elements, and the multiplication of unit lower triangular matricies is lower triangular, we have:\n",
      "\n",
      "$A = LU$\n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def lu_decomp(A):\n",
      "    #very similar to gaussian elimination, but we save the row operations.\n",
      "    U = copy(A)\n",
      "    L = eye(A.shape[0])\n",
      "    \n",
      "    for i in range(A.shape[0]): #for each row\n",
      "        for j in range(i+1, A.shape[0]): #for each following row\n",
      "            multiplier = U[j,i]/U[i,i]\n",
      "            U[j,:] = U[j,:] - multiplier*U[i,:]\n",
      "            L[j,i] = multiplier\n",
      "    return L,U\n",
      "            "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 156
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[1, 3],[2,4]], float)\n",
      "L,U = lu_decomp(A)\n",
      "print L\n",
      "print U\n",
      "print 'LU should equal A\\n', dot(L,U),'\\n', A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 1.  0.]\n",
        " [ 2.  1.]]\n",
        "[[ 1.  3.]\n",
        " [ 0. -2.]]\n",
        "LU should equal A\n",
        "[[ 1.  3.]\n",
        " [ 2.  4.]] \n",
        "[[ 1.  3.]\n",
        " [ 2.  4.]]\n"
       ]
      }
     ],
     "prompt_number": 158
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = randn(5,5)\n",
      "L,U = lu_decomp(A)\n",
      "print 'LU should equal A\\n', dot(L,U)- A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "LU should equal A\n",
        "[[  0.00000000e+00   0.00000000e+00   0.00000000e+00   0.00000000e+00\n",
        "    0.00000000e+00]\n",
        " [  0.00000000e+00   0.00000000e+00  -5.55111512e-17   0.00000000e+00\n",
        "    0.00000000e+00]\n",
        " [  0.00000000e+00   0.00000000e+00   0.00000000e+00   4.44089210e-16\n",
        "    3.33066907e-16]\n",
        " [  0.00000000e+00   0.00000000e+00   1.11022302e-16  -4.44089210e-16\n",
        "    2.22044605e-16]\n",
        " [  0.00000000e+00   0.00000000e+00   5.55111512e-17   4.44089210e-16\n",
        "   -1.56125113e-16]]\n"
       ]
      }
     ],
     "prompt_number": 160
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##LDL' Decomposition##\n",
      "\n",
      "This is a decomposition method for symmetric positive-definite matricies. It decomposes\n",
      "\n",
      "$A = L D L^{T}$,\n",
      "\n",
      "with $L$ lower triangular and $D$ diagonal. The method is similar to GE, using row operations."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def ldl_decomp(A):\n",
      "    #A should be symmetric positivedefinite\n",
      "    assert array_equal(A, A.T), 'A not symmetric'\n",
      "    L = eye(A.shape[0])\n",
      "    U = copy(A)\n",
      "    for i in range(A.shape[0]): #for each row\n",
      "        for j in range(i+1, A.shape[0]): #for each following row\n",
      "            multiplier = U[j,i]/U[i,i]\n",
      "            U[j,:] = U[j,:] - multiplier*U[i,:]\n",
      "            L[j,i] = multiplier\n",
      "    \n",
      "    #turn U into unit upper triangular\n",
      "    d = diag(U).reshape((-1,1))\n",
      "    U = U / d\n",
      "    D = diag(d.ravel())\n",
      "    \n",
      "    assert array_equal(U, L.T), 'A not positive definite!' + array_str(U) + '\\n' + array_str(L)\n",
      "    return L, D    "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 200
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[3,2,1],[2,4,5],[1,5,6]],float)\n",
      "print A\n",
      "L,D = ldl_decomp(A)\n",
      "print 'L:\\n', L\n",
      "print 'D:\\n', D\n",
      "print 'LDL.T should equal A\\n', dot(dot(L,D), L.T),'\\n', A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[[ 3.  2.  1.]\n",
        " [ 2.  4.  5.]\n",
        " [ 1.  5.  6.]]\n",
        "L:\n",
        "[[ 1.          0.          0.        ]\n",
        " [ 0.66666667  1.          0.        ]\n",
        " [ 0.33333333  1.625       1.        ]]\n",
        "D:\n",
        "[[ 3.          0.          0.        ]\n",
        " [ 0.          2.66666667  0.        ]\n",
        " [ 0.          0.         -1.375     ]]\n",
        "LDL.T should equal A\n",
        "[[ 3.  2.  1.]\n",
        " [ 2.  4.  5.]\n",
        " [ 1.  5.  6.]] \n",
        "[[ 3.  2.  1.]\n",
        " [ 2.  4.  5.]\n",
        " [ 1.  5.  6.]]\n"
       ]
      }
     ],
     "prompt_number": 201
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Cholesky Decomposition##\n",
      "\n",
      "A trivial change of the $LDL^T$ method gives \n",
      "\n",
      "$L D L^{T}$ = $L D^{1/2} D^{1/2} L^{T} = L_{*} L_{*}^{T}$ for $L_* = L D^{1/2}$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def cholesky_decompose(A, verbose=False):\n",
      "    #A should be symmetric positive definite\n",
      "    assert array_equal(A, A.T), 'A not symmetric'\n",
      "    \n",
      "    L = eye(A.shape[0])\n",
      "    U = copy(A)\n",
      "    for i in range(A.shape[0]): #for each row\n",
      "        if verbose: print 'current U:\\n', U\n",
      "        if verbose: print 'current L:\\n', L\n",
      "            \n",
      "        for j in range(i+1, A.shape[0]): #for each following row\n",
      "            multiplier = U[j,i]/U[i,i]\n",
      "            U[j,:] = U[j,:] - multiplier*U[i,:]\n",
      "            L[j,i] = multiplier\n",
      "            if verbose: print 'row op:', 'U row', j, '=', 'U row', j, '-', multiplier, '* U row', i\n",
      "            if verbose: print 'setting L[', j, ',', i, '] =', multiplier\n",
      "    \n",
      "    #turn U into unit upper triangular\n",
      "    d = diag(U).reshape((-1,1))\n",
      "    U = U / d\n",
      "    D = diag(d.ravel())\n",
      "    \n",
      "    if verbose: print 'matrix D:\\n', D\n",
      "        \n",
      "    assert array_equal(U, L.T), 'A not positive definite!' + array_str(U) + '\\n' + array_str(L)\n",
      "    #we're doing cholesky now\n",
      "    L = dot(L, D**0.5)\n",
      "    \n",
      "    if verbose: print '(L D**0.5) =\\n', L\n",
      "    return L"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 304
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[7,2,1],[2,6,5],[1,5,6]],float)\n",
      "L = cholesky_decompose(A)\n",
      "print 'L:\\n', L\n",
      "print 'LL.T should equal A\\n', dot(L, L.T),'\\n', A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "L:\n",
        "[[ 2.64575131  0.          0.        ]\n",
        " [ 0.75592895  2.32992949  0.        ]\n",
        " [ 0.37796447  2.02335982  1.32783956]]\n",
        "LL.T should equal A\n",
        "[[ 7.  2.  1.]\n",
        " [ 2.  6.  5.]\n",
        " [ 1.  5.  6.]] \n",
        "[[ 7.  2.  1.]\n",
        " [ 2.  6.  5.]\n",
        " [ 1.  5.  6.]]\n"
       ]
      }
     ],
     "prompt_number": 206
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Power method for the largest eigenvector/eigenvalue##\n",
      "\n",
      "We have $Ax = \\lambda x$\n",
      "\n",
      "So, $ AAx = A \\lambda x = \\lambda^2 x$\n",
      "\n",
      "And generally $A^n x = \\lambda^n x$ for an eigenvector $x$ and value $\\lambda$\n",
      "\n",
      "If we assume that A has a largest eigenvalue, $\\lambda_0$, then by eigendecomposition,\n",
      "\n",
      "$A^n x = \\lambda_0^n x + \\lambda_1^n x + ...$\n",
      "\n",
      "Since $\\lambda_0$ is largest, it dominates the right hand side for an arbitrary $x$.\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def power_method(A, max_iter=100):\n",
      "    x = ones(A.shape[0])\n",
      "    x_norm = inf \n",
      "    lambda_est = inf\n",
      "    \n",
      "    for i in range(max_iter):\n",
      "        x_next = dot(A,x)\n",
      "        x_next_norm = norm(x_next)\n",
      "        lambda_est_next = x_next_norm\n",
      "        x_next /= x_next_norm\n",
      "        if abs(lambda_est_next - lambda_est) < 1e-5:\n",
      "            return x_next, lambda_est_next\n",
      "        else:\n",
      "            x = x_next\n",
      "            x_norm = x_next_norm\n",
      "            lambda_est = lambda_est_next\n",
      "\n",
      "    return x_next, lambda_est_next"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 216
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = array([[7,2,1],[2,6,5],[1,5,6]],float)\n",
      "print power_method(A)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "(array([ 0.39899057,  0.66640463,  0.6298503 ]), 11.922709493687325)\n"
       ]
      }
     ],
     "prompt_number": 217
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eigv, eigx = eig(A)\n",
      "print eigx[:,0], eigv[0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0.39873773  0.66645864  0.62995326] 11.9227107582\n"
       ]
      }
     ],
     "prompt_number": 219
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##Symmetric power method##\n",
      "\n",
      "Not yet implemented."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "##QR Method##"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Not yet implemented.\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Solving a linear system with Newton's method"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "\n",
      "Newton's method (see above) is a way to find roots of a function. By applying it in a clever way, however, we can get an iterative method for solving linear systems.\n",
      "\n",
      "Consider the system $Ax = b$. For an iterative method, we'd like **the error**, $||Ax - b||^2$ to be lower after each iteration. In other words, we want to find a point of zero error, a zero of the error function. Since such a point may not exist, we'd like to get as close as possible, or find a minimum of the error function. A minimum of the error function occurs where the derivative is zero. Thus we'll use newton's method on the derivative.\n",
      "\n",
      "Recall that Newton's method approximated the function with it's first order polynomial expansion, and then solved that:\n",
      "\n",
      "$f(x) \\approx f(x_0) + f'(x_0)(x-x_0)$\n",
      "\n",
      "$-\\frac{f(x_0)}{f'(x_0)} + x_0 = x$\n",
      "\n",
      "So in our case the function we're approximating is the derivative of the error, which is given by:\n",
      "\n",
      "$f'(x) = \\frac{d}{dx} (Ax-b)^{T} (Ax-b)$\n",
      "\n",
      "$f'(x) = \\frac{d}{dx} (x^{T}A^{T}Ax - x^{T}A^{T}b - b^{T}Ax + b^{T}b)$\n",
      "\n",
      "$f'(x) = 2A^{T}Ax - 2A^{T}b$\n",
      "\n",
      "And we're going to need the derivative of this function with respect to $x$, so\n",
      "\n",
      "$f''(x) = A^{T}A$\n",
      "\n",
      "And now we can construct the update:\n",
      "\n",
      "$x^{(n)} = x^{(n-1)} - \\frac{f'(x^{(n-1)})}{f''(x^{(n-1)})}$\n",
      "\n",
      "Or\n",
      "\n",
      "$x^{(n)} = x^{(n-1)} - (A^{T}A)^{-1} (A^{T}Ax^{(n-1)} -A^{T}b)$\n",
      "\n",
      "Simplifying,\n",
      "\n",
      "\n",
      "$x^{(n)} = (A^{T}A)^{-1}A^{T}b$\n",
      "\n",
      "Interestingly, this has no dependence on the previous value of $x$, so it's a closed-form solution.\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def newton_solve(A,b):\n",
      "    \"\"\"Closed-form solution from doing newton's method on a linear system\"\"\"\n",
      "    return dot(inv(dot(A.T,A)), dot(A.T,b))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 136
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = randn(8,8)\n",
      "b = arange(8)\n",
      "x = newton_solve(A,b)\n",
      "print 'Ax =', around(dot(A,x),5), '\\nb = ', b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Ax = [ 0.  1.  2.  3.  4.  5.  6.  7.] \n",
        "b =  [0 1 2 3 4 5 6 7]\n"
       ]
      }
     ],
     "prompt_number": 142
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If you have a stats or machine learning background, you'll recognize this as the pseudoinverse of A, and the problem is linear least squares. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Numerical Differentiation ##\n",
      "\n",
      "Given a function $f$, we wish to approximate $f'$ and $\\int f $ numerically.\n",
      "\n",
      "The definition of each gives an idea for how an approximation can be made. Consider $f'$.\n",
      "\n",
      "$f'(x) = \\lim_{h \\to 0} \\frac{ f(x+h)-f(x) } {h}$\n",
      "\n",
      "So simply approximate $f'(x)$ by $g(x,h)$, where \n",
      "\n",
      "$g(x,h) = \\frac{ f(x+h)- f(x)}{h}$\n",
      "\n",
      "Which will be more accurate for smaller values of $h$, until roundoff errors start to kick in."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_prime = lambda f, x, h: (f(x+h)-f(x))/h\n",
      "\n",
      "xs = arange(-5, 5, .1)\n",
      "plot(xs, [f_prime(sin, xi, 0.1) for xi in xs])\n",
      "plot(xs, cos(xs))\n",
      "ylim([-1.5, 1.5])\n",
      "legend(['Approximate derivative', 'actual derivative'])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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8Y0EDLHQWclm5EZg3D2p220J5+/J4FPPQOo5R8/ODyxes8XPxl1a9EVi+HJo3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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xc4f3320>"
       ]
      }
     ],
     "prompt_number": 225
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can obtain a more accurate approximation by using the central difference,\n",
      "\n",
      "$f'(x) \\approx \\frac{ f(x+h) - f(x-h) }{2h}$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_prime_cent = lambda f, x, h: (f(x+h)-f(x-h))/2/h\n",
      "\n",
      "plot(xs, [f_prime_cent(sin, xi, 0.1) for xi in xs])\n",
      "plot(xs, cos(xs))\n",
      "legend(['Approx', 'actual'])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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RxoZ6vKNoFS8rP0Rek/KOodH+fz6v2ZdDoaJ/A3djCfZcoGOS5RF1Qwrv+hLe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\nBfi9UAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xc490828>"
       ]
      }
     ],
     "prompt_number": 224
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Comparing the error at a point,"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print 'actual', cos(.5)\n",
      "print 'forward difference, h=0.1', f_prime(sin, .5, .1)\n",
      "print 'forward difference, h=0.01', f_prime(sin, .5, .01)\n",
      "print 'forward difference, h=0.0001', f_prime(sin, .5, .0001)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "actual 0.87758256189\n",
        "forward difference, h=0.1 0.852169347908\n",
        "forward difference, h=0.01 0.87517082787\n",
        "forward difference, h=0.0001 0.877558589151\n"
       ]
      }
     ],
     "prompt_number": 226
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print 'actual', cos(.5)\n",
      "print 'central difference, h=0.1', f_prime_cent(sin, .5, .1)\n",
      "print 'central difference, h=0.01', f_prime_cent(sin, .5, .01)\n",
      "print 'central difference, h=0.0001', f_prime_cent(sin, .5, .0001)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "actual 0.87758256189\n",
        "central difference, h=0.1 0.876120655432\n",
        "central difference, h=0.01 0.877567935587\n",
        "central difference, h=0.0001 0.877582560428\n"
       ]
      }
     ],
     "prompt_number": 227
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Shows that the central difference method is much more accurate. It turns out that the central difference method results from doing a degree 2 polynomial interpolation and then taking its derivative. \n",
      "\n",
      "Because it is mentioned in the text, the formula for a 3-point derivative at an endpoint is different, because we can't use information on one side of the point. Thus we have\n",
      "\n",
      "\n",
      "$f'(x) \\approx \\frac{ -3f(x) + 4f(x+h) - f(x+2h)}{2h}$\n",
      "\n",
      "Which is the approximation on a left endpoint. The right endpoint is the same with $+h$ replaced with $-h$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f_prime_left = lambda f, x, h: (-3*f(x) + 4*f(x+h) - f(x+2*h))/2/h\n",
      "f_prime_right = lambda f, x, h: (3*f(x) - 4*f(x-h) + f(x-2*h))/2/h\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 347
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Richardson extrapolation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "Richardson Extrapolation is a sequence acceleration method which we will use for accelerating the convergence of numerical integration techniques. The idea is to combine two approximations in such a way that their errors cancel out.\n",
      "\n",
      "Consider an approximation, $A(x)$ to some true value $A^*$. Suppose that the approximation has an error term of the form:\n",
      "\n",
      "$A^{*} - A(x) = \\alpha_0 x + O(x^2)$, for some constant $\\alpha$, and this formula just says that the error decreases linearly as $x \\to 0$.\n",
      "\n",
      "Consider another evaluation of the sequence at a smaller value of $x$, $A(\\frac{x}{t})$. Just by plugging things in,\n",
      "\n",
      "$A^{*} - A(\\frac{x}{t}) = \\alpha_0 \\frac{x}{t} + O(\\frac{x^2}{t^2})$\n",
      "\n",
      "We want to combine $A(\\frac{x}{t})$ and $A(x)$ to get the $\\alpha_0 x$ term to cancel.\n",
      "\n",
      "So multiply $A(\\frac{x}{t})$ by $t$, to get\n",
      "\n",
      "$tA^{*} = tA(\\frac{x}{t}) + \\alpha_0 x + O(\\frac{x^2}{t})$\n",
      "\n",
      "Now, subtracting $A(x)$ from both sides\n",
      "\n",
      "\n",
      "$tA^{*} - A^{*} = tA(\\frac{x}{t}) + \\alpha_0 x + O(\\frac{x^2}{t}) - A(x) - \\alpha_0 x - O(x^2)$\n",
      "\n",
      "Note that the $\\alpha_0 x$ terms cancel and we have\n",
      "\n",
      "$(t-1)A^{*} =  tA(\\frac{x}{t}) + O(\\frac{x^2}{t}) - A(x) - O(x^2)$\n",
      "\n",
      "We can group the $x^2$ terms,\n",
      "\n",
      "$(t-1)A^{*} =  tA(\\frac{x}{t}) - A(x) + O(x^2)$\n",
      "\n",
      "Dividing by $(t-1)$ gives the formula:\n",
      "\n",
      "$A^{*} = \\frac{tA(\\frac{x}{t}) - A(x)}{t-1} + O(x^2)$\n",
      "\n",
      "What we have, then, is a way to increase the speed of convergence from $O(x)$ to $O(x^2)$, at the cost of an additional function evaluation. Note that this method can be extended as far as you'd like, just replace $x$ with $x^n$ and $t$ with $t^n$ in the error terms and we can still increase the convergence speed.\n"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}