Fibonacci.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Fibonacci Numbers [[1]](https://en.wikipedia.org/wiki/Fibonacci_number)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In mathematics, the Fibonacci numbers are the numbers in the following integer sequence, called the Fibonacci sequence, and characterized by the fact that every number after the first two is the sum of the two preceding ones:\n",
    "\n",
    "$$ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \\cdots$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In mathematical terms, the sequence $F_n$ of Fibonacci numbers is defined by the recurrence relation:\n",
    "\n",
    "$$ F_{n}=F_{n-1}+F_{n-2} $$\n",
    "\n",
    "with seed values:\n",
    "$$ F_0 = 0 , F_1 = 1 $$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In matrix notation this definition is equivalent to:\n",
    "\n",
    "\\begin{eqnarray}\n",
    "    \\begin{bmatrix}\n",
    "    F_1 \\\\\n",
    "    F_0\n",
    "    \\end{bmatrix}\n",
    "    & = &\n",
    "    \\begin{bmatrix}\n",
    "    1 \\\\\n",
    "    0\n",
    "    \\end{bmatrix} \\\\\n",
    "    \\begin{bmatrix}\n",
    "    F_{n+1} \\\\\n",
    "    F_n\n",
    "    \\end{bmatrix}\n",
    "    & = &\n",
    "    \\begin{bmatrix}\n",
    "    1 & 1 \\\\\n",
    "    1 & 0\n",
    "    \\end{bmatrix}\n",
    "    \\begin{bmatrix}\n",
    "    F_n \\\\\n",
    "    F_{n-1}\n",
    "    \\end{bmatrix}\n",
    "\\end{eqnarray}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we want to compute only the $n^{th}$ Fibonacci number, then the following identity is useful:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\begin{bmatrix}\n",
    "    F_{n} \\\\\n",
    "    F_{n-1}\n",
    "    \\end{bmatrix} =\n",
    "    \\begin{bmatrix}\n",
    "    1 & 1 \\\\\n",
    "    1 & 0\n",
    "    \\end{bmatrix}^{n-1}\n",
    "    \\begin{bmatrix}\n",
    "    F_1 \\\\\n",
    "    F_0\n",
    "    \\end{bmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "By using matrix exponentiation (for instance, calculating $M^8$ as $((M^2)^2)^2$ ), $F_{n}$ can be calculated in $O(log(n))$ time complexity. This algorithm is sometimes called **fast fibonacci transform**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Follow the instructions in the next sections. Feel free to create extra cells (for instance, you can try different values for $F_1$ and $F_0$).**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1. Fast Fibonacci Transform Implementation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implement a function that returns $F_n$ as we described above (for this assignment we are not concerned about the efficiency of your implementation, i.e. you can use $M^n$ assuming octave does matrix exponentiation for you):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "A = [[1.,1.],[1.,0.]]\n",
    "A=np.matrix(A)\n",
    "B=[[1.],[0.]]\n",
    "B=np.matrix(B)\n",
    "\n",
    "\n",
    "def fib(N):\n",
    "    return ((A**(N-1))*B)[0,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2. Plot $F_{n+1} / F_n$ ratio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initialize $F_0 = 0$ and $F_1 = 1$, then plot the $\\dfrac{F_{n+1}}{F_{n}}$ values for $ 1 \\leq n \\leq 100$. As $n \\to \\infty$, we expect $\\dfrac{F_{n+1}}{F_{n}} \\to \\dfrac{\\sqrt{5}+1}{2}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAD8CAYAAACMwORRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fa7af50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "yt=[]\n",
    "\n",
    "\n",
    "for i in range(1,100):\n",
    "    yt.append(fib(i+1)/fib(i));\n",
    "\n",
    "plt.plot(range(len(yt)),yt)\n",
    "plt.show()\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3. Plot $F_{n+1} / F_n$ ratio starting with $F_0 = 2$ and $F_1 = 1 - \\sqrt{5}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Initialize $F_0 = 2$ and $F_1 = F_1 = 1 - \\sqrt{5}$, then plot the $\\dfrac{F_{n+1}}{F_{n}}$ values for $ 1 \\leq n \\leq 100$. If we would represent $\\sqrt{5}$ exactly in our floating point arithmetic, then as $n \\to \\infty$, we expect  $\\dfrac{F_{n+1}}{F_{n}} \\to \\dfrac{1 - \\sqrt{5}}{2}$, but for the very large values of $n$, this ratio unexpectedly converges to $\\dfrac{\\sqrt{5} + 1}{2}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/bugrahansahin/anaconda/lib/python2.7/site-packages/ipykernel/__main__.py:10: RuntimeWarning: divide by zero encountered in double_scalars\n",
      "/Users/bugrahansahin/anaconda/lib/python2.7/site-packages/ipykernel/__main__.py:10: RuntimeWarning: invalid value encountered in double_scalars\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXYAAAD8CAYAAABjAo9vAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nzP8DrC/2y37Zfo4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11348bd90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXQAAAD8CAYAAABn919SAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fdf7510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B=np.matrix([[1.-math.sqrt(5)],[2.]])\n",
    "\n",
    "yt=[]\n",
    "xt=[]\n",
    "\n",
    "a=B[0,0]\n",
    "b=B[1,0]\n",
    "\n",
    "for i in range(1,100):\n",
    "    yt.append(fib(i+1)/fib(i));\n",
    "    c=a+b;\n",
    "    xt.append(c/a)\n",
    "    b=a;\n",
    "    a=c;\n",
    "\n",
    "#this one gives wrong plot due to numerical overflow\n",
    "plt.plot(range(len(yt)),yt)\n",
    "plt.show()\n",
    "\n",
    "plt.plot(range(len(xt)),xt)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}