wacm_formulas_fibbonacci.ipynb

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  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# imports\n",
      "import math\n",
      "import matplotlib.pyplot as plt\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Phi = (1+ math.sqrt(5))/2\n",
      "phi = 1/Phi\n",
      "print 'phi: ' + str(phi)\n",
      "print 'Phi: ' + str(Phi)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "phi: 0.609690881047\n",
        "Phi: 1.6401754251\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def fibonacci(seed1, seed2, n):   \n",
      "    '''\n",
      "    compute n terms of the sequence defined by\n",
      "    the recurrence relation x_{n} = x_{n-1} + x_{n-2}\n",
      "    with x_1 = seed1 and x_2 = seed2\n",
      "    '''\n",
      "    fibs = [seed1, seed2]\n",
      "    for i in range(n): fibs += [fibs[-1] + fibs[-2]]\n",
      "    return fibs"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Phi approximation\n",
      "\n",
      "# params\n",
      "seed1 = 10\n",
      "seed2 = 14\n",
      "iters = 20\n",
      "\n",
      "# compute sequence\n",
      "fibs = fibonacci(seed1, seed2, iters)\n",
      "\n",
      "# computer Phi approximates as x_{n-1} / x_{n-2}\n",
      "def approximate_Phi(x_nminus2, x_nminus1): return float(x_nminus1)/float(x_nminus2) if x_nminus2 != 0 else 0.0\n",
      "Phi_approximation_list = [approximate_Phi(x_nminus2, x_nminus1) for (x_nminus1, x_nminus2) in zip(fibs[1:], fibs)]\n",
      "\n",
      "# computer approximation errors\n",
      "approximation_errors = [Phi-Phi_approximate for Phi_approximate in Phi_approximation_list]\n",
      "\n",
      "# print sequence, Phi, er\n",
      "print '\\nsequence' +'\\t' + 'Phi approximate' + '\\t\\t' + 'approximation error\\n'\n",
      "for (fib, Phi_approximation, approximation_error) in zip(fibs,Phi_approximation_list, approximation_errors):\n",
      "    print \"{:.0f}\".format(fib) +'\\t\\t' + \"{:10.12f}\".format(Phi_approximation) + '\\t\\t' + \"{:+10.12f}\".format(approximation_error)\n",
      "\n",
      "# plot approximations\n",
      "print '\\n\\nPhi approximates:\\n'\n",
      "plt.figure()\n",
      "plt.plot(Phi_approximation_list, c='r')\n",
      "plt.axhline(Phi, c='b')\n",
      "plt.show()\n",
      "\n",
      "# plot errors\n",
      "print '\\n\\napproximation errors:\\n'\n",
      "plt.figure()\n",
      "plt.plot(approximation_errors, c='r')\n",
      "plt.show()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "sequence\tPhi approximate\t\tapproximation error\n",
        "\n",
        "10\t\t1.400000000000\t\t+0.240175425099\n",
        "14\t\t1.714285714286\t\t-0.074110289187\n",
        "24\t\t1.583333333333\t\t+0.056842091766\n",
        "38\t\t1.631578947368\t\t+0.008596477731\n",
        "62\t\t1.612903225806\t\t+0.027272199293\n",
        "100\t\t1.620000000000\t\t+0.020175425099\n",
        "162\t\t1.617283950617\t\t+0.022891474482\n",
        "262\t\t1.618320610687\t\t+0.021854814412\n",
        "424\t\t1.617924528302\t\t+0.022250896797\n",
        "686\t\t1.618075801749\t\t+0.022099623350\n",
        "1110\t\t1.618018018018\t\t+0.022157407081\n",
        "1796\t\t1.618040089087\t\t+0.022135336012\n",
        "2906\t\t1.618031658637\t\t+0.022143766462\n",
        "4702\t\t1.618034878775\t\t+0.022140546324\n",
        "7608\t\t1.618033648791\t\t+0.022141776308\n",
        "12310\t\t1.618034118603\t\t+0.022141306496\n",
        "19918\t\t1.618033939151\t\t+0.022141485949\n",
        "32228\t\t1.618034007695\t\t+0.022141417404\n",
        "52146\t\t1.618033981513\t\t+0.022141443586\n",
        "84374\t\t1.618033991514\t\t+0.022141433585\n",
        "136520\t\t1.618033987694\t\t+0.022141437405\n",
        "\n",
        "\n",
        "Phi approximates:\n",
        "\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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a30navumJrxYKGdMabNgw8JWpIiI3mNfrxev1jnj7mMLe4XCQnZ3N2LFjGTt2LA899BAf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\nRK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x105e2f8d0>"
       ]
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "\n",
        "approximation errors:\n",
        "\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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55Zd5jcYpEomopqZGc+bM0YMPPhi9P+HXZ0LrlS+xnTt3RmbPnh2ZNWtW5Cc/\n+Um6m2Nbv//97yOlpaWR0tLSyM0338xzmYSqqqpIbm5uxOl0RlwuV2Tbtm2RY8eORZYsWRIpLCyM\nlJeXR06cOJHuZtrG0Odz69atkdWrV0fmzp0bKSkpiXzjG9+I9PT0pLuZtvDOO+9EHA5HpLS0NDJv\n3rzIvHnzIq+++mrCr08WnQEAxu8wEQDg0iEMAACEAQCAMAAAiDAAAIgwAACIMAAAiDAAAEj6fytC\nQMsPH8MwAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x105cba190>"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}