root_finding.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Add some physical examples to motivate root finding. Just using algebraic examples up to this point."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Solving Algebraic Equations: Root-finding Methods"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solving a linear equation (e.g. $y=mx+b$) is one of the first topic one encounters in any sort of introductory algebra course. This of course can be done by hand quite easily. One of course then moves onto more sophisticated equations involving terms like $x^2,x^3,\\ldots$. For example, the equation $$x^2+10x+9=0$$ can be solved using the familiar quadratic formula $x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$ where $a=1$, $b=10$, and $c=9$. Applying the quadratic formula, we see that $x=-1,-9$. When we plot the function $f(x)=x^2+10x+9$, we can see that $f=0$ at $x=-1$ and $x=-9$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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OieBRTEyYssVORXVm8sSmnDyhYYlgS6odTxYBO2RyR+GSJA1RBAcAlwI3AHtk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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112f99590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(-15,5,1000)\n",
    "x_zeros=[-1,-9]\n",
    "fig = plt.figure(figsize=(10,10))\n",
    "ax = fig.gca()\n",
    "ax.plot(x,0*x,'--k',linewidth=2.0)\n",
    "ax.plot(x,x**2+10*x+9,linewidth=2.0)\n",
    "ax.plot(x_zeros,[0,0],'ro',markersize=10)\n",
    "ax.set_xlabel(r'$x$',fontsize=22)\n",
    "ax.set_ylabel(r'$f(x)$',fontsize=22)\n",
    "ax.set_title(r'$f(x)=x^2+10x+9$',fontsize=22)\n",
    "ax.set_ylim([-20,20])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These types of equations, often referred to as polynomials, can also be solved through factorization. For example, the equation $$x^3+6x^2+11x+6=0$$ can be rewritten as $$(x+1)(x+2)(x+3)=0$$ such that $x=-1,-2,-3$ are all roots of the equation. Again, let's plot the function and the roots to check our answer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8PpQoSdKoMfhNU0TMBeZ2fJ1h2LItIjYAjsrMzSZ4bC3gxszMiNga+ElmbjDBeW7ZJklS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\nWtfdM4usSgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113493410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x=np.linspace(-4,0,1000)\n",
    "x_zeros=[-1,-2,-3]\n",
    "fig = plt.figure(figsize=(10,10))\n",
    "ax = fig.gca()\n",
    "ax.plot(x,0*x,'--k',linewidth=2.0)\n",
    "ax.plot(x,x**3+6*x**2+11*x+6,linewidth=2.0)\n",
    "ax.plot(x_zeros,[0,0,0],'ro',markersize=10)\n",
    "ax.set_xlabel(r'$x$',fontsize=22)\n",
    "ax.set_ylabel(r'$f(x)$',fontsize=22)\n",
    "ax.set_title(r'$f(x)=x^3+6x^2+11x+6$',fontsize=22)\n",
    "ax.set_ylim([-2,2])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note also that finding the value of $x$ where $f(x)=0$ is the just as easy as finding the value of $x$ where $f(x)=a$, where $a$,a constant, is some value of $f$. This is because $f(x)=a$ can be rewritten as $f(x)-a=0$. Thus, we can rewrite $f(x)=a$ as a root-finding problem $g(x)=0$ where $g(x)=f(x)-a$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far, we've only considered the case of finding roots of polynomials. In these cases, we could always find the roots using either the quadratic formula (when our leading term was $x^2$) or a factorization method when possible. But what if our function is more complicated, say $f(x)=2x^2\\sin(x)+\\ln(x)$, or better yet, what if our function is unknown altogether? Often, we resort to **root finding algorithms** where we make a first guess about the root of our function and then repeatedly refine our guess until we have a reasonably accurate estimate of the root."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Bisection Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the most popular root-finding methods is known as the **bisection method**. In order to apply the bisection method, there are two important requirements:\n",
    "\n",
    "1. the function $f$ must be _monotonically increasing (or decreasing)_ between $x=a$ and $x=b$, \n",
    "2. between $f(x=a)$ and $f(x=b)$, $f(x)=0$ for some value of $x=x_0$ such that $a<x_0<b$.\n",
    "\n",
    "Let's consider what both of these conditions actually mean. Consider a function $f$ that is _monotonically increasing (or decreasing)_ from $x=a$ to $x=b$. This is just a fancy way of saying that as we move from $a$ to $b$, $f$ **only** increases (or **only** decreases). For example, the above function ($f(x)=x^3+6x^2+11x+6$) is neither monotonically increasing or decreasing on the interval shown, $x\\in[-4,0]$. However, from $x=-4$ to $x=-2-\\sqrt{3}/3$, the function increases, from $x=-2-\\sqrt{3}/3$ to $x=-2+\\sqrt{3}/3$ the function decreases, and then from $x=-2+\\sqrt{3}/3$ to $x=0$, the function increases. Thus, when applying our bisection method, we must use it on each of these intervals individually. Lets consider the middle interval, $x\\in[-2-\\sqrt{3}/3,-2+\\sqrt{3}/3]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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C/6Q6Qrg5MIHqCOHRVIXw7wWjSZLUOBbBNrMIjk4E7wB2AbagOmX8HNUq473cwk6SpO6w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\nqVAGQUmSpEL9P6UQzzKxC2F8AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11394a6d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xzoom = np.linspace(-2.0-np.sqrt(3.0)/3.0,-2.0+np.sqrt(3.0)/3.0,1000)\n",
    "def cubic_func(x):\n",
    "    return x**3+6*x**2+11*x+6\n",
    "fig = plt.figure(figsize=(10,10))\n",
    "ax = fig.gca()\n",
    "ax.plot(xzoom,0*xzoom,'--k',linewidth=2.0)\n",
    "ax.plot(xzoom,cubic_func(xzoom),linewidth=2.0)\n",
    "ax.set_xlabel(r'$x$',fontsize=22)\n",
    "ax.set_ylabel(r'$f(x)$',fontsize=22)\n",
    "ax.set_title(r'$f(x)=x^3+6x^2+11x+6$',fontsize=22)\n",
    "ax.set_ylim([-.5,.5])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's call $a=-2.3$ and $b=-1.5$. Looking at the above figure, we can see that between $f(a)$ and $f(b)$, $f=0$ (i.e. the blue curve intersects the black dotted line). Thus, we have also satisfied the second condition for applying the bisection method to find the roots of our function. Now that we've shown that conditions for applying the bisection method are satisfied, how exactly does the method work?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Input interval endpoints $a,b$ for a function $f$ which satisfies the above conditions.\n",
    "2. Calculate first guess of root $c=a+(b-a)/2$ and evaluate $f(c)$.\n",
    "3. Calculate $f(a)\\times f(c)$,\n",
    "\n",
    "   * If $f(a)\\times f(c)>0$ (i.e. $c$ is on the same side of the root as $a$), set $a=c$ and update $c$.\n",
    "   * If $f(a)\\times f(c)<0$ (i.e. $c$ is on the same side of the root as $b$), set $b=c$ and update $c$.\n",
    "   \n",
    "4. Evalute $f(c)$.\n",
    "5. Repeat steps 3 and 4 for a finite number of steps or until $f(c)<\\epsilon$, where $\\epsilon$ is some specified error.\n",
    "\n",
    "Now let's write a function that performs the bisection method which accepts the bounds on our interval $a,b$ and our function $f$. Note that we can write our bisection method function in such a way that it is not specific to the kind of function to which we apply it. This will allow us to reuse this piece of code for several different functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def bisection(a,b,f,options):\n",
    "    \"\"\"Bisection method for interval (a,b) as applied to a monotonically increasing (or decreasing) function f on (a,b)\"\"\"\n",
    "    \n",
    "    #First guess for root value\n",
    "    c = a + (b - a)/2\n",
    "    \n",
    "    #Evaluate function at first guess\n",
    "    f_guess = f(c)\n",
    "    \n",
    "    #Initialize the counter\n",
    "    count = 0\n",
    "    \n",
    "    #Initialize lists of endpoints and roots\n",
    "    a_list = [a]\n",
    "    b_list = [b]\n",
    "    c_list = [c]\n",
    "    \n",
    "    #Begin iteration\n",
    "    while(count<options['max_count'] and np.fabs(f_guess)>options['tol']):\n",
    "        #Output iteration and endpoints\n",
    "        print \"Iteration %d, Endpoints (%f,%f)\"%(count,a,b)\n",
    "        \n",
    "        #Check relative sign\n",
    "        if f_guess*f(a)>0:\n",
    "            #advance left endpoint\n",
    "            a=c\n",
    "        elif f_guess*f(a)<0:\n",
    "            #advance right endpoint\n",
    "            b=c\n",
    "        else:\n",
    "            #case where f(c)==0\n",
    "            print \"f(c) == 0 exactly. Breaking out of loop.\"\n",
    "            break\n",
    "            \n",
    "        #update root value guess\n",
    "        c = a + (b-a)/2\n",
    "        \n",
    "        #evaluate function at current guess\n",
    "        f_guess = f(c)\n",
    "        \n",
    "        #save a,b,c values\n",
    "        a_list.append(a)\n",
    "        b_list.append(b)\n",
    "        c_list.append(c)\n",
    "        \n",
    "        #update the counter\n",
    "        count = count + 1\n",
    "        \n",
    "    #print final zero approximation\n",
    "    print \"Value of the function at approximate zero is f(c=%f)=%f\"%(c,f_guess)\n",
    "    \n",
    "    #return lists of a,b,c\n",
    "    return a_list,b_list,c_list"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have an algorithm for our bisection method, let's apply it to the above function for our chosen endpoints $a=-2.3$ and $b=-1.5$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0, Endpoints (-2.500000,-1.900000)\n",
      "Iteration 1, Endpoints (-2.200000,-1.900000)\n",
      "Iteration 2, Endpoints (-2.050000,-1.900000)\n",
      "Iteration 3, Endpoints (-2.050000,-1.975000)\n",
      "Iteration 4, Endpoints (-2.012500,-1.975000)\n",
      "Iteration 5, Endpoints (-2.012500,-1.993750)\n",
      "Iteration 6, Endpoints (-2.003125,-1.993750)\n",
      "Iteration 7, Endpoints (-2.003125,-1.998437)\n",
      "Iteration 8, Endpoints (-2.000781,-1.998437)\n",
      "Iteration 9, Endpoints (-2.000781,-1.999609)\n",
      "Iteration 10, Endpoints (-2.000195,-1.999609)\n",
      "Iteration 11, Endpoints (-2.000195,-1.999902)\n",
      "Iteration 12, Endpoints (-2.000049,-1.999902)\n",
      "Iteration 13, Endpoints (-2.000049,-1.999976)\n",
      "Iteration 14, Endpoints (-2.000012,-1.999976)\n",
      "Iteration 15, Endpoints (-2.000012,-1.999994)\n",
      "Iteration 16, Endpoints (-2.000003,-1.999994)\n",
      "Iteration 17, Endpoints (-2.000003,-1.999998)\n",
      "Iteration 18, Endpoints (-2.000001,-1.999998)\n",
      "Iteration 19, Endpoints (-2.000001,-2.000000)\n",
      "Iteration 20, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 21, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 22, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 23, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 24, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 25, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 26, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 27, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 28, Endpoints (-2.000000,-2.000000)\n",
      "Iteration 29, Endpoints (-2.000000,-2.000000)\n",
      "Value of the function at approximate zero is f(c=-2.000000)=-0.000000\n"
     ]
    }
   ],
   "source": [
    "opt = {'max_count':100,'tol':1.0e-10}\n",
    "alist,blist,clist = bisection(-2.5,-1.9,cubic_func,opt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus, we see that our bisection method performs very well for our cubic function on the given interval. This provides a nice check of our method as we can analytically verify (by hand) that our root on this interval is $x=-2$. Now let's apply this method over the two remaining intervals to find our other zeros. For the leftmost interval, we'll choose $a=-5,b=-2-\\sqrt{3}/3$ and for the rightmost interval, we'll choose $a=-2+\\sqrt{3}/3,b=4$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0, Endpoints (-5.000000,-2.577350)\n",
      "Iteration 1, Endpoints (-3.788675,-2.577350)\n",
      "Iteration 2, Endpoints (-3.183013,-2.577350)\n",
      "Iteration 3, Endpoints (-3.183013,-2.880181)\n",
      "Iteration 4, Endpoints (-3.031597,-2.880181)\n",
      "Iteration 5, Endpoints (-3.031597,-2.955889)\n",
      "Iteration 6, Endpoints (-3.031597,-2.993743)\n",
      "Iteration 7, Endpoints (-3.012670,-2.993743)\n",
      "Iteration 8, Endpoints (-3.003207,-2.993743)\n",
      "Iteration 9, Endpoints (-3.003207,-2.998475)\n",
      "Iteration 10, Endpoints (-3.000841,-2.998475)\n",
      "Iteration 11, Endpoints (-3.000841,-2.999658)\n",
      "Iteration 12, Endpoints (-3.000249,-2.999658)\n",
      "Iteration 13, Endpoints (-3.000249,-2.999954)\n",
      "Iteration 14, Endpoints (-3.000101,-2.999954)\n",
      "Iteration 15, Endpoints (-3.000028,-2.999954)\n",
      "Iteration 16, Endpoints (-3.000028,-2.999991)\n",
      "Value of the function at approximate zero is f(c=-3.000000)=0.000000\n"
     ]
    }
   ],
   "source": [
    "opt['tol'] = 1.0e-6\n",
    "aleft,bleft,cleft=bisection(-5.0,-2.0-np.sqrt(3.0)/3.0,cubic_func,opt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0, Endpoints (-1.422650,4.000000)\n",
      "Iteration 1, Endpoints (-1.422650,1.288675)\n",
      "Iteration 2, Endpoints (-1.422650,-0.066987)\n",
      "Iteration 3, Endpoints (-1.422650,-0.744819)\n",
      "Iteration 4, Endpoints (-1.083734,-0.744819)\n",
      "Iteration 5, Endpoints (-1.083734,-0.914276)\n",
      "Iteration 6, Endpoints (-1.083734,-0.999005)\n",
      "Iteration 7, Endpoints (-1.041370,-0.999005)\n",
      "Iteration 8, Endpoints (-1.020187,-0.999005)\n",
      "Iteration 9, Endpoints (-1.009596,-0.999005)\n",
      "Iteration 10, Endpoints (-1.004301,-0.999005)\n",
      "Iteration 11, Endpoints (-1.001653,-0.999005)\n",
      "Iteration 12, Endpoints (-1.000329,-0.999005)\n",
      "Iteration 13, Endpoints (-1.000329,-0.999667)\n",
      "Iteration 14, Endpoints (-1.000329,-0.999998)\n",
      "Iteration 15, Endpoints (-1.000164,-0.999998)\n",
      "Iteration 16, Endpoints (-1.000081,-0.999998)\n",
      "Iteration 17, Endpoints (-1.000040,-0.999998)\n",
      "Iteration 18, Endpoints (-1.000019,-0.999998)\n",
      "Iteration 19, Endpoints (-1.000008,-0.999998)\n",
      "Iteration 20, Endpoints (-1.000003,-0.999998)\n",
      "Iteration 21, Endpoints (-1.000001,-0.999998)\n",
      "Value of the function at approximate zero is f(c=-1.000000)=-0.000000\n"
     ]
    }
   ],
   "source": [
    "aright,bright,cright=bisection(-2.0+np.sqrt(3.0)/3.0,4.0,cubic_func,opt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our bisection method is able to calculate the zeros of our cubic function in a reasonable number of iterations. Now that we've shown that our bisection algorithm works, let's apply it to a more useful case: an unknown function where we can't use a simple method like the quadratic formula or factorization to find the zeros."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x and y data for unknown function saved in  example_data/root_finding_unknown_function.txt\n",
      "Name of unknown function is sine_integral_func\n"
     ]
    }
   ],
   "source": [
    "%run make_unknown_function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "unknowns = np.loadtxt('example_data/root_finding_unknown_function.txt')\n",
    "x_unknown = unknowns[0,:]\n",
    "y_unknown = unknowns[1,:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's plot our unknown function to determine where it is monotonically increasing/decreasing. We'll make smart choices about our intervals by \"bracketing\" our function at $(0.5,2.9)$ and $(3.5,6)$. Note that we are free to make any choice when it comes to our intervals provided that on $(a,b)$, $f$ is either monotonically increasing or decreasing."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a1 = 0.5\n",
    "b1 = 2.9\n",
    "a2 = 3.5\n",
    "b2 = 6.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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2Un+J4EnAzpRAt3rLW3OAHwO/yuTfU/vOwQp28Mj/pjpR47nAK4BXAhu0nPogcAJwBOVn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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114f3ead0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(10,10))\n",
    "ax = fig.gca()\n",
    "ax.plot(x_unknown,y_unknown,linewidth=2.0)\n",
    "ax.plot(x_unknown,0*x_unknown,'--k',linewidth=2.0)\n",
    "ax.plot([a1,a2],[y_unknown[np.where(x_unknown<a1)[0][-1]],y_unknown[np.where(x_unknown<a2)[0][-1]]],'ko',marker=r'$[$',markersize=25)\n",
    "ax.plot([b1,b2],[y_unknown[np.where(x_unknown<b1)[0][-1]],y_unknown[np.where(x_unknown<b2)[0][-1]]],'ko',marker=r'$]$',markersize=25)\n",
    "ax.set_xlabel(r'$x$',fontsize=22)\n",
    "ax.set_ylabel(r'$f(x)$',fontsize=22)\n",
    "ax.set_title(r'Unknown Function',fontsize=22)\n",
    "ax.set_xlim([x_unknown[0],x_unknown[-1]])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can apply our bisection method to this new unknown function to try to find the roots of this equation. Note that here we cannot use our quadratic method or factorization; the form of the function has been defined when we ran the script `make_unknown_function.py`. It is only important to know the form of the function so that we can feed it to the bisection method; suffice it to say that is complicated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 0, Endpoints (0.500000,2.900000)\n",
      "Iteration 1, Endpoints (1.700000,2.900000)\n",
      "Iteration 2, Endpoints (1.700000,2.300000)\n",
      "Iteration 3, Endpoints (1.700000,2.000000)\n",
      "Iteration 4, Endpoints (1.850000,2.000000)\n",
      "Iteration 5, Endpoints (1.925000,2.000000)\n",
      "Iteration 6, Endpoints (1.925000,1.962500)\n",
      "Iteration 7, Endpoints (1.925000,1.943750)\n",
      "Iteration 8, Endpoints (1.925000,1.934375)\n",
      "Iteration 9, Endpoints (1.925000,1.929688)\n",
      "Iteration 10, Endpoints (1.925000,1.927344)\n",
      "Iteration 11, Endpoints (1.926172,1.927344)\n",
      "Iteration 12, Endpoints (1.926172,1.926758)\n",
      "Iteration 13, Endpoints (1.926172,1.926465)\n",
      "Iteration 14, Endpoints (1.926318,1.926465)\n",
      "Iteration 15, Endpoints (1.926392,1.926465)\n",
      "Value of the function at approximate zero is f(c=1.926447)=-0.000001\n",
      "Iteration 0, Endpoints (3.500000,6.000000)\n",
      "Iteration 1, Endpoints (4.750000,6.000000)\n",
      "Iteration 2, Endpoints (4.750000,5.375000)\n",
      "Iteration 3, Endpoints (4.750000,5.062500)\n",
      "Iteration 4, Endpoints (4.750000,4.906250)\n",
      "Iteration 5, Endpoints (4.828125,4.906250)\n",
      "Iteration 6, Endpoints (4.867188,4.906250)\n",
      "Iteration 7, Endpoints (4.886719,4.906250)\n",
      "Iteration 8, Endpoints (4.886719,4.896484)\n",
      "Iteration 9, Endpoints (4.891602,4.896484)\n",
      "Iteration 10, Endpoints (4.891602,4.894043)\n",
      "Iteration 11, Endpoints (4.892822,4.894043)\n",
      "Iteration 12, Endpoints (4.893433,4.894043)\n",
      "Iteration 13, Endpoints (4.893738,4.894043)\n",
      "Iteration 14, Endpoints (4.893738,4.893890)\n",
      "Iteration 15, Endpoints (4.893814,4.893890)\n",
      "Value of the function at approximate zero is f(c=4.893833)=0.000001\n"
     ]
    }
   ],
   "source": [
    "a1list,b1list,c1list = bisection(a1,b1,sine_integral_func,opt)\n",
    "a2list,b2list,c2list = bisection(a2,b2,sine_integral_func,opt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's plot the zeros on top of our plot of the unknown function to get a sense of how good our guess is."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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6vGYZ7CZgsFNT6nV1ZwFPBo4AXue/iCX1irpUyu6UBf5Xq5tvB74DfD2TvzVV\n2zAz2E3AYKemRPBt4M2U2+M9038FS+pFEcwGdgHexcLr8BI4FvgaZZh2QUPlDR2D3QQMdmpCBK8H\nDqdcV7d5Jhc1XJIkTSiCp1MWO3418Ki6+UrgEOAwZ9N2nsFuAgY7dVsE6wPnA0sBu2dyaMMlSdKU\nRPA4yjDtHsCadfODwC+Bg7GL1zEGuwkY7NRNdUjjDGAzXK9OUp+r97Z+ASXgvQRYvO66itLF+65d\nvPYy2E3AYKduiuAzwEeAa4CnZnJbwyVJUltEsCrluuG3AmvVzfOB31LWxftVJvc1VN7AMNhNwGCn\nbolgO+D/KBcdb5fJ6c1WJEntV7t42wN7Ai8GZtVdt1NGKr4H/N7Riukx2E3AYKduiGBF4AJgdeB/\nM/lYwyVJUsfVJVNeBbweeEbLrr9RAt4RLpsyNQa7CRjs1GkRBHAU8DLK/WC3yWRes1VJUndFsDEl\n4L0WWLVl17nAT4CjMrm6gdL6isFuAgY7dVoEbwEOBe4EnpbJlQ2XJEmNqUO1z6Xc2eKlwNItu89i\nYcj7RwPlTVtEHAxsWF9elpl7dOg8BrtFMdipk+qtec6j3DLs9Zl8v+GSJKlnRLAksAOwG2VW7ZyW\n3b8HfkGZdPHXBsqbkpZgtyZwTWZu16HzGOwWxWCnTolgFnA6sCXwI+A1XiwsSWOrt1nciRLyXgQs\n2bL7Msoaeb8CzsxkfvcrnJyImAtgsGuIwU6dEsFHgM8A1wGbZHJrwyVJUl+IYGlgR0oX70XAii27\nb6Lczuw44ORMbu5+heMz2DXMYKdOiOAplAuCZwM7ZHJ8wyVJUl+qox9bATtTrslbr2V3Un7XnlAf\nf8jkga4X2cJg1zCDndotgkcBZwNPBb6VydsaLkmSBkJdZeCJlJD3AmBrFt6zFuAu4BTgZMqlMBd0\ne9jWYNcwg53aLYJPA3tTboz91EzuargkSRpIESwFbEMJeS8AnjTqkDsot3E8HTgNODeT+ztbk8Gu\nUQY7tVMEW1BmcgWwrXeXkKTuiWB1yl0vtgWeA6w76pD7KEO357Q8/tbOiW0Gu4YZ7NQudUbXeZTp\n7l/K5IMNlyRJQ60GvefUxzbAxmMcdgsLw955wEWUsDetIVyDXcMMdmqXCPYH3gX8BXiGN7uWpN4S\nwWOAzYHNWh6PH+PQeym/yy9qeVwKXJfJgkWfw2DXKIOd2iGCbYG5wHxgi0z+2GxFkqSJ1MkYq7Mw\n7D0F2KSnOEDNAAAgAElEQVRuG8t9lPvcXg5c0fJ8DXBtJvcZ7BpmsNNM1SHYC4D1gU9l8vGGS5Ik\nzUAEK1CGbTdpeWwIPG6Ct94EWywB998P5/+Eso7pjWX7wx63ZPLg9Goz2C2SwU4zFcEXgQ8AFwNP\nb3oNJUlSZ0SwLOUf8esDG7R8vQawGjALtqtHz53o426nLM8y8ri75fke4MGWx/yFX8deBrtFMNhp\nJiLYDPhDffmsTM5ush5JUjMiWAx4HKxwLCzxaPjnQZSw99iWx2NanqeZPYKxcsusadYtqaoLEX8b\nWAz4sqFOkoZXnVTxr4jb7gTuzORr4x0bweLAcsBSwNJjPM8BFqfktVmjvv7cWJ9psJNm7iOU6y7+\nBl5XJ0manLqUyi31MSURYwe7xWZalDTMIngy8NH68q2Z3NNkPZKk4Wawk6apttC/A8wGDsqc+ApZ\nSZI6yWAnTd97KOsdXQvs1XAtkiQZ7KTpiGBd4FP15Z6Z3NFkPZIkgcFOmrK6Qvk3gCWBH2ZyXMMl\nSZIEGOyk6dgNeCFwK/DehmuRJOkhBjtpCiJYHtivvtwrk383WY8kSa0MdtLUfBZ4PHAGZUasJEk9\nw2AnTVIEWwJ7Uu7Tt2ddXVySpIdExMERMRdYs4nzG+ykSYgoa9VR7un3xUwubrgkSVJvuwa4rNsn\njczs9jl7UkTkWDfTlQAi+CDwBeAq4MneYUKS1KTxcovBrjLYaTwRrA1cTLkZ846Z/LbZiiRJw268\n3OJQrLQIdc26Aymh7khDnSSpl9mxq+zYaSwR7AL8HLgd2CiTfzZckiRJduykqYpgDgvXrPuooU6S\n1OsMdtL4PkKZrn4+8K2Ga5EkaUIOxVYOxapVBOtTJkw8Ctgqk983XJIkSQ9xKFaapDphYn9KqDvc\nUCdJ6hd27Co7dhoRwc7ALykTJp6QyQ0NlyRJ0sPYsZMmIYIlWThh4uOGOklSPzHYSQ+3F7AOcBHw\njYZrkSRpShyKrRyKVQTrAH8BlgC2zeS0hkuSJGlMDsVKE9uPEuqOMNRJkvqRHbvKjt1wi2An4Fjg\nTsqECRcjliT1LDt20jgieBTw1fpyX0OdJKlfGewk+B9gQ+Ay4GsN1yJJ0rQ5FFs5FDucIngscAWw\nHPDiTI5tuCRJkibkUKw0tk9SQt0JwHEN1yJJ0ozYsavs2A2fCDYBzgcSeGomFzdckiRJk2LHTmpR\n7wf7Vcr/A98y1EmSBoEdu8qO3XBpuR/srcAGmdzccEmSJE2aHTupqsubfLm+3MdQJ0kaFAY7DaN3\nAusDlwLfbLgWSZLaxqHYyqHY4RDB44DLgWWBnTL5TcMlSZI0ZQ7FSsUnKaHut4Y6SdKgsWNX2bEb\nfBE8GbiAsrzJJplc0nBJkiRNix07Cb7AwuVNDHWSpIFjx66yYzfYIngecBJwJ7BeJjc2XJIkSdNm\nx05DK4LFgC/Wl58z1EmSBpXBTsPgNcCmwHXAfg3XIklSxxjsNNAiWBL4dH35/zK5p8l6JEnqJIOd\nBt07gTWAC4HvN1yLJEkd5eSJyskTgyeCxwB/A5YDdsjk+IZLkiSpLZw8oWH0MUqoO9FQJ0kaBnbs\nKjt2gyWC9YBLgFnApplc0HBJkiS1jR07DZvPArOB7xnqJEnDwo5dZcducESwJXAmcB+wQSbXNlyS\nJEltZcdOQyGCoNw6DOCrhjpJ0jCxY1fZsRsMEewEHAvcTLl12O0NlyRJUtvZsdPAq7cO+0x9+RlD\nnSRp2BjsNEh2A54KXAt8o+FaJEnqOoOdBkIEs4FP1Zf7ZHJfk/VIktQEg50GxVuA9YC/Aoc3XIsk\nSY1w8kTl5In+FcEcyq3DVgZ2zeSnDZckSVJHOXlCg+xdlFD3R+BnDdciSVJj7NhVduz6UwQrAFcC\nywPbZ3JSwyVJktRxduw0qPaihLr/M9RJkoadHbvKjl3/iWAVyrV1SwJbZHJ2wyVJktQVduw0iD5G\nCXU/N9RJkmTH7iF27PpLBOsDl1D+cfLkTC5puCRJkrrGjp0Gzb7ALOBwQ50kSYUdu8qOXf+IYGPg\nIuBBYINM/t5wSZIkdZUdOw2SfYAADjHUSZK0kB27yo5df4jgacB5wP3Aeplc13BJkiR1nR07DYp9\n6vM3DXWSJD2cHbvKjl3vi+CZwDnAPcC6mdzQcEmSJDXCjp0GwSfr84GGOkmSHsmOXWXHrrdF8Czg\n98BdwDqZ3NRwSZIkNcaOnfrdSLduf0OdJEljs2NX2bHrXRFsA5wK3E7p1t3acEmSJDVqvNwyq4li\n2iEiVgSOBNYCrgZekZm3jXHc1cAdwHxgXmZu3sUyNUMRBPCp+vIrhjpJksbXz0OxHwZOzMwNgZPr\n67EksF1mbmqo60vPA7YBbgX2b7gWSZJ6Wj8Hu52Bw+vXhwP/uYhjHWLtQ7VbN3Jt3Rczub3JeiRJ\n6nV9e41dRNyamSvUrwO4ZeT1qOOupFybNR84KDMPGefzvMaux0SwI3AccCNl3bq7Gi5JkqSe0JfX\n2EXEicDKY+z6aOuLzMyIGC+hbpWZ/4yIlYATI+LSzDy93bWqvUZ16z5vqJMkaWI9Hewyc/vx9kXE\nDRGxcmb+KyJWAf49zmf8sz7fGBFHA5sDYwa7iNin5eXczJw73do1YzsCzwRuAL7ZcC2SJDUqIrYD\ntpvwuD4eiv0CcHNmfj4iPgwsn5kfHnXMHGDxzLwzIpYCTgD2zcwTxvg8h2J7RO3WnQlsAXwgky83\nXJIkST1lvNzSz8FuReAnwJq0LHcSEasCh2TmiyJiXeDn9S2zgCMy87PjfJ7BrkdE8ELgt5Rr69bJ\n5O6GS5IkqacMXLBrN4Ndb6jdujOAZwEfyuQLDZckSVLPMdhNwGDXGyJ4HnAScDOwtpMmJEl6JO8V\nq55Xu3WfqC+/bKiTJGlq7NhVduyaF8F2wCmUu0ysnckdzVYkSVJvsmOnfvDx+vxVQ50kSVNnx66y\nY9esCJ4DnEa5S8jamdzWcEmSJPUsO3bqdSPduv0MdZIkTY8du8qOXXMieDZliZM7gbUyubXhkiRJ\n6ml27NTLRrp1BxjqJEmaPjt2lR27ZkSwBfAH4C7KtXU3N1ySJEk9z46detXH6vOBhjpJkmbGjl1l\nx677IngGcC5wD6Vbd2PDJUmS1Bfs2KkX7V2fv2mokyRp5uzYVXbsuiuCJwEXA/cD62Tyz4ZLkiSp\nb9ixU6/5cH3+jqFOkqT2sGNX2bHrngjWAS6vLzfI5Kom65Ekqd/YsVMv+SCwOPBDQ50kSe1jx66y\nY9cdEawCXAU8Gtg4k780XJIkSX3Hjp16xXspoe5oQ50kSe1lx66yY9d5EawI/B1YGtg8k3MaLkmS\npL5kx0694B2UUHeioU6SpPazY1fZseusCJamdOtWBLbL5NSGS5IkqW/ZsVPT9qSEut8DpzVciyRJ\nA8mOXWXHrnMiWAK4ElgFeHEmxzZckiRJfc2OnZr0BkqouxA4ruFaJEkaWAY7dVQEs4AP1ZefycQW\nsSRJHWKwU6ftBqwDXAH8tOFaJEkaaAY7dUwEiwEfqS8/l8n8JuuRJGnQGezUSS8GNgauBb7fcC2S\nJA08g506aa/6/JVMHmi0EkmShoDLnVQud9JeEWwF/A64DVgzkzsbLkmSpIHhcifqtpFu3dcNdZIk\ndYcdu8qOXftEsBHwF+B+YK1Mbmi4JEmSBoodO3XTB+vzdw11kiR1jx27yo5de0SwGnAVsDjwhEyu\naLgkSZIGzni5ZVYbPnh9YG1gOeBRwN2U5S0uz0yvrRo+7wFmA0cZ6iRJ6q4pd+wiYjnglcAuwNbA\nnHEOTeDPwDHADzLz0hnU2XF27GYuguWBa4BlgM0zOafhkiRJGkgzvsYuIpaMiH2BvwFvpoS21wJP\np3TslgWWAFYDNgG2B44CNgPOiohjImLDGf53qLftSQl1pxjqJEnqvkl17CJiU+A7wB+AL2bmlVM6\nScQywNuAt9f3f30atXaUHbuZiWAJyrV1KwM7ZHJ8wyVJkjSwxsstEwa7iHgW8BngDZl5zQyLWAI4\nALgpM/eeyWe1m8FuZiJ4K3AIcAGwaSbOypEkqUNmMhS7E7DDTEMdQGbel5l7AFdFxJNm+nnqDREs\nxsIlTr5gqJMkqRkud1LZsZu+CHYBfg78HVg/kwcbLkmSpIHWkQWKI2KViHhfRGwwk89R/4oggA/V\nl1821EmS1JwZdewi4nuUmbEXZOamLdvfBDwB+Exm3jHjKrvAjt30RLANcCpwC7BmJnc3XJIkSQOv\nUwsU3wbsDg+/piozv1uXNjk0IvbKzKtneB71rr3q84GGOkmSmjXTjt1BwN6ZefM4+x8DfDkz3zjt\nk3SJHbupi2BjynqG9wJrZXJjwyVJkjQUOtWx2wv4dkScB/wW+FO2JMXMvDkiFszwHOpd76/P3zXU\nSZLUvJl27F4G/JByb1CA24HfUa65OpdyN4p3Zub2M6yz4+zYTU0EK1Nmwc4GNvS+sJIkdU+nOna7\nU24Zdi/l1mL/ATwXeFHdfyPw8hmeQ73pHcCjgKMNdZIk9YaZBrvLM/PCka+BIwEiYnVgB0rA+8cM\nz6EeE8FSlFvEAXy5yVokSdJCM1rHDsga4h6+MfPazDyUshTKJ2Z4DvWeNwArAmcBv2+4FkmSVM20\nY7cvZfLE/pl5SuuOiPhf4AHgnhmeQz0kgsWB99aXX/b2YZIk9Y4ZdezqMie7AZtExGtH7X4BsA/l\nOiwNjpcA6wNXA0c3W4okSWrVsXvFRsRKwLbAbzKz5xeudVbs5ERwGvAc4L2Z7Nd0PZIkDaPxckvH\ngl2/MdhNLILNgLMpy9qskcmdDZckSdJQGi+3TDgUGxGPiYg5HShorXZ/pjpuZEHigw11kiT1nslc\nYzcL+G5EPL5dJ42IXYGPtOvz1HkRrEVZk/BB4ICGy5EkSWOYMNhl5g3A3sDPIuINETHt4cqIWD0i\nvgnsTFngVv3j3cDiwJGZXNt0MZIk6ZEmNSs2M/9GWWx4G+CyiNg7Ip42mZAXEUtHxI4R8V3gT8CF\nmfm6zHxwRpWrayJYDnhrfemCxJIk9ahJr2OXmbcDb4mIpwMfBP4fMD8izgGuBW6jXFT/KMritSsC\n6wBPodxa7NvAxpnpzeL7z+7AMsApmZzXdDGSJGlsk54VWydQzMvMefX1ssDzgWcDGwGrAUsB8ykh\n72rgQuB04IzMXNDu4tvJWbFji2A2cCWwOvDiTI5tuCRJkobejJY7iYjVgPMpd5HYJDPvaH+JzTLY\njS2CVwNHAJcCG2fS0wFdkqRhMO3lTqpdgP0pw3EPLX0SEfu2pzz1ogiChUucfMVQJ0lSb5tssLsT\nuB9YKTP/1bJ9p/aXpB6yLfB0yjWS32+4FkmSNIHJTp74CfB74ON1ssTZwB8py19ocL2vPn89k/sa\nrUSSJE1oKpMnlgbeA+wKPBkYGde9DbgAOI9yHd55wF8yc37bq+0gr7F7uAg2BP5K6dSumcm/Gy5J\nkiRVbb1XbEQsBzwD+AFwErAp8AQWdgDvB/5M6eqdAJyQmXdNr/TuMNg9XARfoywifWgmuzddjyRJ\nWqitwa7lQ8/JzM3q14+mdPI2BZ5GCX5PAZak3IbqF8A+mfmXaZ+wgwx2C0WwPGVtwqWATTL5c8Ml\nSZKkFuPllkkvUDyOw0a+yMz7KR26P7acdHHKGnebAc8CfhsR787Mo2d4XnXWWyih7mRDnSRJ/WNG\nHbspnShiXeAB4MuZuVtXTjoFduyKCGYBVwBrAS/J5NcNlyRJkkaZ6Tp27fB74CxgXhfPqanbmRLq\nrgCOa7gWSZI0Bd0Mdt8AbgZ+3sVzaureU5/3d0FiSZL6S9eGYnudQ7EQwTOAc4E7gNUzubPhkiRJ\n0hh6YShWve/d9flQQ50kSf3Hjl017B27CFYGrqHcTWT9TK5quCRJkjQOO3aayNuA2cAvDHWSJPUn\nO3bVMHfsIliC0q1bCdg2k9MaLkmSJC2CHTstyqsooe484PSGa5EkSdNksBtyEQQLlzjZLxNbuJIk\n9SmHYqthHYqNYDvgFOAGYK1M7m+2IkmSNBGHYjWekW7dNw11kiT1Nzt21TB27CJYD7iccpu3NTO5\noeGSJEnSJNix01jeCQTwQ0OdJEn9z45dNWwduwiWBa4FlgE2zeT8hkuSJEmTZMdOo72JEupONdRJ\nkjQYDHZDKILFgXfVl/s1WYskSWofg91wehGwLnAVcEzDtUiSpDYx2A2nkW7dgZnMb7QSSZLUNk6e\nqIZl8kQETwIuBu4BVsvktoZLkiRJU+TkCY14R33+vqFOkqTBYseuGoaOXQTLAdcBSwGbZPLnhkuS\nJEnTYMdOAG+khLpTDHWSJA0eg92QiGAxFg7Dfq3JWiRJUmcY7IbHC4H1gWtwiRNJkgaSwW54vLM+\nfyOTBxutRJIkdYSTJ6pBnjwRwQbAZcB9wOqZ3NxwSZIkaQacPDHc/qc+/9BQJ0nS4LJjVw1qxy6C\npSlLnCwLbJrJ+Q2XJEmSZsiO3fB6PSXU/c5QJ0nSYDPYDbAIApc4kSRpaBjsBtvzgI2A64GjG65F\nkiR1mMFusI0scfKtTOY1WokkSeo4J09UgzZ5IoJ1gL8B84A1M7mh4ZIkSVKbOHli+LwdCOAnhjpJ\nkoZD3wa7iNg1Ii6OiPkR8fRFHLdDRFwaEZdHxIe6WWNTIpgDvKW+dNKEJElDom+DHXARsAtw2ngH\nRMTiwIHADsCTgFdFxEbdKa9RrwZWAM7O5Oymi5EkSd0xq+kCpiszLwWIWORlcZsDV2Tm1fXYHwMv\nBS7pdH1NqUucjEyasFsnSdIQ6eeO3WSsBvyj5fW1ddsgew7wFODfwFEN1yJJkrqopzt2EXEisPIY\nu/bOzGMm8RHDOOV3pFt3UCb3N1qJJEnqqp4Odpm5/Qw/4jpgjZbXa1C6dmOKiH1aXs7NzLkzPH9X\nRbAG5brDB4FvNVyOJElqk4jYDthuouN6OthNwXgX2p0LbBARa1PuvrAb8KrxPiQz92l3YV22J7A4\ncGQm1zddjCRJao/abJo78joiPjHWcX17jV1E7BIR/wC2BI6NiN/U7atGxLEAmfkg5V6pxwN/AY7M\nzIGcOBHBo4Hd68uvN1mLJElqhneeqPr9zhMRvAr4IWUZmKdmDuX1hZIkDQXvPDH43l6fv26okyRp\nONmxq/qtYxcRBwMbller3gTXvQy4E1g1k7saLE2SJHWYHbvBsyGwZvlykyfVbYcb6iRJGl6DMit2\nWF0D+VLKsi4A32iyGEmS1Cw7dv3v9cBSwCmZg3urNEmSNDGDXf8bmTRht06SpCFnsOtrj10eeCJl\n8eVfNlyMJElqmMGur626Wv3i4EzmNVqKJElqnMGuby3xaFj+sZT7wh7SdDWSJKl5Bru+tfoq9Ra5\nR3tfWEmSBAa7vhTBbHjcKvWlkyYkSRJgsOtXu8DsR8G99wCnNl2MJEnqDQa7/lSXOLnhOu8LK0mS\nRhjs+kwETwa2hQXz4bobmq5HkiT1DoNd/3lbebrpBnhwfrOlSJKkXmKw6yMRLEO5hRhwrTNhJUnS\nw8xqugBNyeuApYHT4E6vrZMkSQ9jx65PRBB4X1hJkrQIBrv+sQ2wMXADcHTDtUiSpB5ksOsfI926\ngzN5oNFKJElSTzLY9YEIVgH+C5gPHNxwOZIkqUcZ7PrD7pSJLr/K5Nqmi5EkSb3JYNfjyn1h2bO+\n/HqTtUiSpN5msOt9OwOrAn8F/q/hWiRJUg8z2PW+h5Y48b6wkiRpUQx2PSyCJwLPBe4BvjfGIWtG\nxNyIcEKFJEnyzhM9bo/6/KNMbhu177JuFyNJknpbZDq6BxARmZnRdB0jIlgSuA5YAXhmJn9suCRJ\nktQjxsstDsX2rl0poe5cQ50kSZoMg13velt9/lajVUiSpL7hUGzVS0OxETwNOA+4HVgtk7sbLkmS\nJPUQh2L7y8iCxN8z1EmSpMmyY1f1SscugmWA64GlgSdncnHDJUmSpB5jx65/vIYS6k431EmSpKkw\n2PWQCAL47/rSSROSJGlKHIqtemEoNoItgTOBm4DVM7m/yXokSVJvcii2P4x0675jqJMkSVNlx65q\numMXwYqUO00sAayfyd+aqkWSJPU2O3a97w2UUHeCoU6SJE2Hwa4HOGlCkiS1g8GuN2wHbEhZv+6Y\nZkuRJEn9ymDXG0a6dYdk8mCjlUiSpL7l5ImqqckTETweuJYSstfK5Npu1yBJkvqLkyd611uAWcAx\nhjpJkjQTBrsGRbA4sEd9+c0ma5EkSf3PYNesFwJrAVcBJzZciyRJ6nMGu2aNTJo4KJMFjVYiSZL6\nnpMnqm5PnohgTUqnbj7lvrD/7ta5JUlSf3PyRO/ZnfL9/5mhTpIktYMdu6qbHbsIZgPXACsD22Zy\nWjfOK0mSBoMdu96yMyXUXQKc3nAtkiRpQBjsmvHQfWEzsWUqSZLawqHYqltDsRFsAFwG3Ausmslt\nnT6nJEkaLA7F9o496/OPDHWSJKmd7NhV3ejYRbAEcB2wIrB5Jud08nySJGkw2bHrDS+nhLo/Aec2\nXIskSRowBrvuctKEJEnqGIdiq04PxUawCXAhcAewWiZ3depckiRpsDkU27yRbt33DXWSJKkT7NhV\nnezYRbAUcD2wLPCUTC7qxHkkSdJwsGPXrFdQQt2ZhjpJktQpBrvuGFm77uBGq5AkSQPNodiqU0Ox\nETwVOB+4nXKniXvafQ5JkjRcHIptzu71+fuGOkmS1El27KpOdOycNCFJkjrBjl0zRiZN/MFQJ0mS\nOs1g11l71OeDGq1CkiQNBYdiq3YPxUbwFOACnDQhSZLazKHY7hvp1v3AUCdJkrrBjl3Vzo5dBHOA\nf+KkCUmS1AF27LprN5w0IUmSusxg1xkjw7DeaUKSJHWNQ7FVu4ZinTQhSZI6zaHY7nHShCRJaoQd\nu6odHbtRkyaemsmFbSlOkiSphR277mi904ShTpIkdZXBrr32rM9OmpAkSV3nUGw106HYlkkTd1Am\nTdzdtuIkSZJaOBTbebvX5+8b6iRJUhPs2FUz6djVSRPXA8sBT8vkgrYWJ0mS1MKO3f9v7+5D9bzv\nOo5/vms6XbpqcQ6z2oADN9E6tZ3MYRgNcxvxgbnB6izIcELoH04FQWYrbBGGoIJuKMOxtWNjdU7q\nKtbuoSl2o/0nbTRZuz7oFANJ181iRaxT6+zXP+4rbZKePLX3Ode5f+f1+ud+yMU5X25CeOf63b/r\nWl8/l0XUHRB1AMBchN1yHL923YdmnQIA2NIsxU6e61JsVV6V5L7YNAEAbBBLsevnxDtNiDoAYDbO\n2E2eyxk7myYAgDk4Y7c+bJoAADYNYff8HF+GdacJAGB2lmIn57sUa9MEADAXS7HLZ9MEALCpOGM3\nOZ8zdjZNAABzcsZuua6OTRMAwCYj7J4bmyYAgE3HUuzkXJdiq/KDSe6PTRMAwEwsxS6PTRMAwKbk\njN3kXM7YTZsmHklySWyaAABm4ozdclydRdTdI+oAgM1G2J0fmyYAgE3LUuzkbEuxJ2ya+I8kL/P9\nOgBgLpZinz+bJgCATc0Zu8mZztidsmniiu4c3tDhAABO4Izd8/O2PLNpQtQBAJuSsDs3106PNk0A\nAJuWpdjJ6U5pnrJp4tLuPLHhwwEAnGC4pdiqurqqHqiq/6uqK89w3JGquq+qDlXVPc/hV+2dHj8h\n6gCAzWzb3AM8D/cneWuSD53luE6yu7sfP99fUJUXJXnH9NIyLACwqa1s2HX3w0lSdca7gB13Tget\n4fidJu61aQIA2OxWdin2PHSSO6rqYFXtPevRJzt+7bqznRUEAJjdpj5jV1X7k+xY44+u7+5bz/HH\n7OruR6vqpUn2V9XD3X3XaX7fvmdeXX0k+fNdWWya+NT5zA0AsExVtTvJ7rMdt6nDrrvfuISf8ej0\n+FhV3ZLkNUnWDLvu3nf8eVU+MD29yaYJAGBO3f2FJF84/rqq3rvWcaMsxZ7mjhG1vaounp5flORN\nWWy6OPMPO3nThGVYAGAlrGzYVdVbq+poktcmua2qPju9f2lV3TYdtiPJXVV1OMmBJH/d3befw4+3\naQIAWDkuUDw58UJ/Vbk7ya4ke7vzkXknAwA42WlvrCDsFo5/QFW5PMmX404TAMAmNdydJ9bR8Uuc\n2DQBAKwUZ+wmVdVJb0/y1Sy+X3dldw7NPBYAwLM4Y3du3pZnNk2IOgBgpQi7k107PbovLACwcizF\nTqal2MSmCQBgk7MUe+5smgAAVpKwezbLsADAShJ2Jzto0wQAsKqE3cncFxYAWFk2T0ymzRMX+34d\nALDZ2TxxDkQdALDKhB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAg\nhB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQd\nAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDA\nIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCE\nHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0A\nwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAg\nhB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQd\nAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDAIIQdAMAghB0AwCCEHQDA\nIIQdAMAgVjbsqur3q+qhqvpSVX26qr79NMftqaqHq+orVfXujZ4TAGCjrGzYJbk9yeXd/cNJ/iHJ\ndc4av7YAAAXESURBVKceUFUXJPnjJHuS/ECSa6rq+zd0Sp5WVbvnnmFkPt/15zNefz7j9eczXn9z\nfsYrG3bdvb+7n5peHkhy2RqHvSbJP3b3ke7+3yR/luRnN2pGnmX33AMMbvfcA2wBu+ceYAvYPfcA\nW8DuuQfYAnbP9YtXNuxO8UtJPrPG+9+d5OgJr49N7wEADGfb3AOcSVXtT7JjjT+6vrtvnY75rSRP\ndvefrnFcr+d8AACbSXWvbvtU1S8m2ZvkJ7r7v9f489cm2dfde6bX1yV5qrt/d41jV/eDAAC2nO6u\nU9/b1GfszqSq9iT5jSRXrRV1k4NJXlFV35Pkq0nenuSatQ5c68MBAFglq/wduz9K8uIk+6vqUFV9\nMEmq6tKqui1JuvubSd6V5PNJHkzyqe5+aK6BAQDW00ovxQIA8IxVPmO3FC5gvL6q6saq+npV3T/3\nLKOqqp1VdWdVPVBVX66qX517ptFU1bdW1YGqOjx9xvvmnmlEVXXBtAJz69yzjKiqjlTVfdNnfM/c\n84yoqi6pqpunGyg8OH3Xf2Nn2Mpn7KYLGP99kjckeSTJvUmusVy7PFX1uiRPJPl4d79q7nlGVFU7\nkuzo7sNV9eIkf5vkLf4eL1dVbe/ub1TVtiR3J/m17j4w91wjqapfT/LqJBd395vnnmc0VfXPSV7d\n3Y/PPcuoqupjSb7Y3TdO/1Zc1N3/vpEzbPUzdi5gvM66+64k/zb3HCPr7q919+Hp+RNJHkpy6bxT\njae7vzE9fWGSC5M8dYbDOU9VdVmSn0rykSQ2s60fn+06mW5t+rruvjFZfM9/o6MuEXYuYMxQph3g\nV2RxNxaWqKpeUFWHk3w9ye3dfe/cMw3mD7O40oFgXj+d5I6qOlhVe+ceZkAvT/JYVX20qv6uqj5c\nVds3eoitHnZbdx2a4UzLsDdnsUT4xNzzjKa7n+ruH8ni9oU/VlWXzz3TKKrqZ5L8S3cfijNK62lX\nd1+R5CeT/PL0VRmWZ1uSK5N8sLuvTPKfSX5zo4fY6mH3SJKdJ7zemcVZO1gpVXVhkr9I8onu/su5\n5xnZtLRyZ5I9c88ykB9P8ubpO2CfTPL6qvr4zDMNp7sfnR4fS3JLFl9HYnmOJTl2wtn8m7MIvQ21\n1cPu6QsYV9ULs7iA8V/NPBOcl6qqJDckebC73z/3PCOqqu+sqkum5y9K8sYsvsvIEnT39d29s7tf\nnuTnk/xNd79j7rlGUlXbq+ri6flFSd6UxNUKlqi7v5bkaFW9cnrrDUke2Og5VvbOE8vQ3d+squMX\nML4gyQ12Ei5XVX0yyVVJXlJVR5O8p7s/OvNYo9mV5BeS3FdVh6b3ruvuz80402heluRj0076F2Rx\nsfPPzDzTyHxNZvm+K8kti/8HZluSm7r79nlHGtKvJLlpOln0T0neudEDbOnLnQAAjGSrL8UCAAxD\n2AEADELYAQAMQtgBAAxC2AEADELYAQAMQtgBAAxC2AEADELYAQAMQtgBAAxC2AEADELYAQAMYtvc\nAwCMpKpekuS9SSrJK5J8OMkdSX4vyf8kuSTJu7v70dmGBIYl7ACWpKq+JckNSd7V3ceq6oeS3Jvk\n1iTXJnlLFqF3OMkfzDYoMCxLsQDLc22SD3T3sen1fyW5MMmh7v7XJJ3kS1mEHsDSCTuA5Xm8u+88\n4fWV0+PnkqS7b+zuK7r7Kxs/GrAVVHfPPQPAkKrqT5K8Pcl3tH9sgQ3gjB3A+nl9krtFHbBRhB3A\nOqiqy5J8b5IvnvL+O+eZCNgKhB3AElTVS6vqnqp63/TWnunx4AnHvDLJ9234cMCWIewAluOqJD+a\n5MmquijJTyd5LMm3JU9f3+59SX5ntgmB4dk8AbAEU8y9P8mTSbYn+e0kO5O8J8nRLP4jva+7j8w1\nIzA+YQcAMAhLsQAAgxB2AACDEHYAAIMQdgAAgxB2AACDEHYAAIMQdgAAgxB2AACDEHYAAIMQdgAA\ng/h/WzZK0k5WWAQAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x113dd7bd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(10,10))\n",
    "ax = fig.gca()\n",
    "ax.plot(x_unknown,y_unknown,linewidth=2.0)\n",
    "ax.plot(x_unknown,0*x_unknown,'--k',linewidth=2.0)\n",
    "ax.plot([a1,a2],[y_unknown[np.where(x_unknown<a1)[0][-1]],y_unknown[np.where(x_unknown<a2)[0][-1]]],'ko',marker=r'$[$',markersize=25)\n",
    "ax.plot([b1,b2],[y_unknown[np.where(x_unknown<b1)[0][-1]],y_unknown[np.where(x_unknown<b2)[0][-1]]],'ko',marker=r'$]$',markersize=25)\n",
    "ax.plot([c1list[-1],c2list[-1]],[sine_integral_func(c1list[-1]),sine_integral_func(c2list[-1])],'ro',markersize=10)\n",
    "ax.set_xlabel(r'$x$',fontsize=22)\n",
    "ax.set_ylabel(r'$f(x)$',fontsize=22)\n",
    "ax.set_title(r'Unknown Function',fontsize=22)\n",
    "ax.set_xlim([x_unknown[0],x_unknown[-1]])\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": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}