m248Week_11_Root_Finding.ipynb

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    ">![image](JMUlogo.png)\n",
    ">\n",
    "> # Math 248 Computers and Numerical Algorithms\n",
    "> # Hala Nelson\n",
    "> # Week 11: Finding Roots of Nonlinear Functions\n",
    "___"
   ]
  },
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    "## Methods to find the roots of continuous functions: The goal is to numerically solve the nonlinear equation $f(x)=0$.\n",
    "\n",
    "![image](Roots1.JPG)\n",
    "\n",
    "Most of root finding methods create a sequence $\\{x_0,x_1,\\dots\\}$ that hopefully converges to the desired root $x_r$ of a continuous function $f$ (point $x_r$ such that $f(x_r)=0$ or where the graph of $f$ meets the $x$-axis). A good numerical root finding method is one that: 1. Converges to the root. 2. Reasonably accurate. 3. Is not computationally expensive. 4. Converges in a reasonable amount of time. These four points are not independent from each other.\n",
    "\n",
    "## Method 1- Newton's: Evaluates both $f$ and $f'$ at the point $x_n$. So $f'$ better be available and well behaved. \n",
    "\n",
    "Keep in mind that it is usually computationally expensive to compute the derivative numerically. So this method is good when the formula for the derivative is already available.\n",
    "\n",
    "- **Formula**: Start at $x_0$. Then iterate: $$x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_{n})}$$ \n",
    "- **Graphically**: $x_{n+1}$ is the point of intersection of the $x$-axis with the tangent to the graph of $f$ at the point $(x_n,f(x_n))$ (see the image below).\n",
    "- **Derivation**: \n",
    "\n",
    "The equation of the tangent to the graph of $f$ at the point $(x_n,f(x_n))$ is $y=f(x_n)+f'(x_n)(x-x_n)$. \n",
    "\n",
    "To find the point where this tangent line meets the $x$-axis, set $y=0$ and solve for $x$. \n",
    "\n",
    "This will give the next point in the sequence $x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_{n})}$. \n",
    "\n",
    "The idea is that it is much easier to solve the linear equation $f(x_n)+f'(x_n)(x-x_n)=0$ than the nonlinear equation $f(x)=0$. Therefore, at each step, we *linearize* the nonlinear equation then solve for the root. \n",
    "\n",
    "![image](Roots2.JPG)\n",
    "\n",
    "- **Proof of quadratic convergence of Newton's method under the right conditions on $f$, $f'$ and $f''$**: \n",
    "\n",
    "Start at $x_0$ in an interval $[x_r-b,x_r+b]$, with $b>0$, and assume $f$, $f'$, and $f''$ are all continuous on this interval. Also, assume that $f'(x)\\neq 0$ on this interval.\n",
    "\n",
    "Using Taylor's theorem near the point $x_n$, we have $$f(x)=f(x_n)+f'(x_n)(x-x_n)+\\frac{f''(\\xi_n)}{2!}(x-x_n)^2.$$ \n",
    "\n",
    "Let $x=x_r$, then $f(x_r)=0$, and after some algebra we get \n",
    "$$\\begin{aligned}\n",
    "|x_{n+1}-x_r|&=\\frac{|f''(\\xi_n)|}{2|f'(x_n)|}|x_n-x_r|^2,\\\\\n",
    "&=\\gamma|x_n-x_r|^2,\n",
    "\\end{aligned}$$\n",
    "where $\\xi_n\\in(x_n,x_r)$, and $\\gamma=\\frac{|f''(\\xi_n)|}{2|f'(x_n)|}$. \n",
    "\n",
    "Hence, if $\\frac{|f''(\\xi_n)|}{|f'(x_n)|}$ is bounded, or $$M=\\sup_{x\\in[x_r-b,x_r+b]}\\frac{|f''(x)|}{2|f'(x)|}<\\infty$$  \n",
    "\n",
    "(great when the slope $|f'(x_n)|$ is large, terrible when it is near zero), and if $|x_n-x_r|$ is small ($x_n$ close enough to the root), then $|x_{n+1}-x_r|$ is much smaller than $|x_{n}-x_r|$, and the method converges rapidly (quadratically fast) to the root. \n",
    "\n",
    "- Think of all the cases when Newton's method fails to converge or fails to capture the root.\n",
    "\n",
    "## Method 2- Secant: Similar to Newton's but avoids using the derivative by approximating with a nearby slope.  \n",
    "\n",
    "- **Method**: Start with two points $x_0$, $x_1$, then iterate: $$x_{n+2}=x_{n+1}-f(x_{n+1})\\frac{x_{n+1}-x_n}{f(x_{n+1})-f(x_n)}$$ \n",
    "\n",
    "- This method **converges linearly** $$|x_{n+1}-x_r|=C|x_{n}-x_r|, 0<C<1.$$\n",
    "\n",
    "- **Graphical explanation and program**\n",
    "\n",
    "![image](Roots3.JPG)\n",
    "![image](Roots4.JPG)\n",
    "\n",
    "## Method 3- Bisection: \n",
    "\n",
    "- This method works for any continuous function where one knows the function assumes two values of opposite signs. This means there must be a root in between them (this uses the intermediate value theorem for continuous functions). \n",
    "\n",
    "- **Method**: Start with an interval $[a,b]$ where you know $f(a)$ and $f(b)$ have different signs (if they don't, then choose a different interval where they do). Cut the interval in half ($d=\\frac{a+b}{2}$). If $f(a)$ and $f(d)$ have different signs, then the root is there and update your interval to $[a,d]$. Else, the root is in $[d,b]$ and update to this interval. This means that at each step you are *closing in* on the root and cutting the distance to it by at least a half. Stop when you *have closed in on it enough*, after you reach a certain tolerance (when $np.absolute(a-b)<10^{-15}$).   \n",
    "\n",
    "- This method **converges linearly** ($|x_{n+1}-x_r|=C|x_{n}-x_r|$, $0<C<1$). \n",
    "\n",
    "- **Graphical explanation and program (see examples below).**\n",
    "\n",
    "## Relationship between Root Finding and Fixed Point Finding (previous week material)\n",
    "\n",
    "**A fixed point search can always be reformulated as a root search**: Searching for a fixed point of a continuous function $g(x)$ is equivalent to searching for a root for the function $f(x)=g(x)-x$.\n",
    "\n",
    "**A root search sometimes can be reformulated as a fixed point search and sometimes cannot**: This depends on whether you can separate $x$ a put it alone on one side of the equation. For example: $e^x+\\cos x=0$ cannot be reformulated as a fixed point search, but $x+e^x-\\cos x=0$ can.\n",
    "___"
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   "source": [
    "# Assignment\n",
    "\n",
    "## Exercise 1\n",
    "\n",
    "Research: Can Newton's method be extended to higher dimensions? Explain."
   ]
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   "source": [
    "## Exercise 2 \n",
    "\n",
    "Research: There are so many root finding methods out there. Some are more useful or efficient than others, but it all depends on the use case. The interplay is always between accuracy, speed of convergence, and computational cost. List six different root-finding methods and write one sentence describing each."
   ]
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   "source": [
    "## Exercise 3\n",
    "\n",
    "Explain, with numbers, what it means that Newton's method converges quadratically fast to the root (of course under the right conditions on $f$, $f'$, $f''$ and the location of the starting point $x_0$). "
   ]
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   "source": [
    "*Answer is in the iPad lecture notes on Canvas.*"
   ]
  },
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   "source": [
    "## Exercise 4\n",
    "\n",
    "Write a program that finds root(s) of the continuous function $$f(x)=e^x-3x^2$$ using each of the following methods:\n",
    "\n",
    "1. Newton's: start at $x_0$, then $x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}$\n",
    "2. Bisection\n",
    "\n",
    "Note that the roots that need to be captured are: $x_1\\sim -0.46$, $x_2\\sim 0.91$, $x_3\\sim 3.73$."
   ]
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     "text": [
      "When the starting point is 2, Newton's method converges to 0.910007572488709\n",
      "When the starting point is 10, Newton's method converges to 3.733079028632814\n",
      "When the starting point is -5, Newton's method converges to -0.45896226753694847\n"
     ]
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\n",
      "text/plain": [
       "<Figure size 1800x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1800x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1800x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Newton's Method\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "def f(x):\n",
    "    return np.exp(x)-3*x**2\n",
    "\n",
    "def f_prime(x):\n",
    "    return np.exp(x)-6*x\n",
    "\n",
    "# let's start at 2\n",
    "x=[2]\n",
    "n=50\n",
    "for i in range(1,n):\n",
    "    a=x[len(x)-1] # the last element of x\n",
    "    x.append(a-f(a)/f_prime(a))\n",
    "print(f\"When the starting point is {x[0]}, Newton's method converges to {x[len(x)-1]}\")\n",
    "# Let's visualize the sequence\n",
    "# Set the figure\n",
    "fig,subs=plt.subplots(nrows=1,ncols=2,figsize=(25,7))\n",
    "\n",
    "# plot the function and the convergence of Newton's sequence on the x-axis\n",
    "subs[0].grid()\n",
    "subs[0].axhline(y=0, color='k') # shows the x-axis\n",
    "subs[0].axvline(x=0, color='k') # shows the v-axis\n",
    "subs[0].set_title(\"Newton's method\", fontsize='25')\n",
    "x1=np.linspace(-2,5,100)\n",
    "subs[0].plot(x1,f(x1),'b')\n",
    "subs[0].plot(x[len(x)-1],0,'k.',markersize=20)\n",
    "subs[0].plot(x,np.zeros(n,),'r.',markersize=12)\n",
    "\n",
    "# plot the sequence against its index\n",
    "subs[1].plot(x,'k.')\n",
    "\n",
    "# let's start at 10\n",
    "x=[10]\n",
    "n=50\n",
    "for i in range(1,n):\n",
    "    a=x[len(x)-1] # the last element of x\n",
    "    x.append(a-f(a)/f_prime(a))\n",
    "print(f\"When the starting point is {x[0]}, Newton's method converges to {x[len(x)-1]}\")\n",
    "# Let's visualize the sequence\n",
    "# Set the figure\n",
    "fig,subs=plt.subplots(nrows=1,ncols=2,figsize=(25,7))\n",
    "\n",
    "# plot the function and the convergence of Newton's sequence on the x-axis\n",
    "subs[0].grid()\n",
    "subs[0].axhline(y=0, color='k') # shows the x-axis\n",
    "subs[0].axvline(x=0, color='k') # shows the v-axis\n",
    "subs[0].set_title(\"Newton's method\", fontsize='25')\n",
    "x1=np.linspace(-2,5,100)\n",
    "subs[0].plot(x1,f(x1),'b')\n",
    "subs[0].plot(x[len(x)-1],0,'k.',markersize=20)\n",
    "subs[0].plot(x,np.zeros(n,),'r.',markersize=12)\n",
    "\n",
    "# plot the sequence against its index\n",
    "subs[1].plot(x,'k.')\n",
    "\n",
    "# let's start at -5\n",
    "x=[-5]\n",
    "n=50\n",
    "for i in range(1,n):\n",
    "    a=x[len(x)-1] # the last element of x\n",
    "    x.append(a-f(a)/f_prime(a))\n",
    "print(f\"When the starting point is {x[0]}, Newton's method converges to {x[len(x)-1]}\")\n",
    "# Let's visualize the sequence\n",
    "# Set the figure\n",
    "fig,subs=plt.subplots(nrows=1,ncols=2,figsize=(25,7))\n",
    "# plot the function and the convergence of Newton's sequence on the x-axis\n",
    "subs[0].grid()\n",
    "subs[0].axhline(y=0, color='k') # shows the x-axis\n",
    "subs[0].axvline(x=0, color='k') # shows the y-axis\n",
    "subs[0].set_title(\"Newton's method\", fontsize='25')\n",
    "x1=np.linspace(-2,5,100)\n",
    "subs[0].plot(x1,f(x1),'b')\n",
    "subs[0].plot(x[len(x)-1],0,'k.',markersize=20)\n",
    "subs[0].plot(x,np.zeros(n,),'r.',markersize=12)\n",
    "\n",
    "# plot the sequence against its index\n",
    "subs[1].plot(x,'k.')\n",
    "\n",
    "# For the bisection method, see last exercise below. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 5\n",
    "\n",
    "Write a program that outputs a graph that explains how the secant method works. This is not about programing the secant method itself, just plotting a function and the secant lines explaining the idea behind the secant method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(1.96078431372549, 0.25, 'x4')"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x1008 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Let's work on the function $f(x)=x^2-3$ \n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "# define the function\n",
    "def f(x):\n",
    "    return x**2-3\n",
    "\n",
    "# let's start at the two points (x0,f(x0)) and (x1,f(x1))\n",
    "x0=0\n",
    "x1=4\n",
    "\n",
    "# equation of the secant line through points (x0,f(x0)) and (x1,f(x1))\n",
    "def secant(x):\n",
    "    return f(x0)+((f(x1)-f(x0))/(x1-x0))*(x-x0)\n",
    "\n",
    "# The point of intesection between the secant line and the x-axis is:\n",
    "x2= -f(x0)*((x1-x0)/(f(x1)-f(x0)))+x0\n",
    "\n",
    "# Set the figure\n",
    "plt.figure(figsize=(14,14))\n",
    "plt.axis('equal') # the scale on the x-axis is the same as the y-axis\n",
    "plt.xlim(-8,8)\n",
    "plt.ylim(-4,15)\n",
    "plt.title('The Secant Method', fontsize=20)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.axhline(y=0, color='k') # shows the x-axis\n",
    "plt.axvline(x=0, color='k') # shows the y-axis\n",
    "# show the grid\n",
    "plt.grid()\n",
    "\n",
    "# plot the function\n",
    "xx=np.linspace(-8,8,100)\n",
    "plt.plot(xx,f(xx),'b')\n",
    "\n",
    "# plot the secant line\n",
    "plt.plot(xx,secant(xx),'g')\n",
    "\n",
    "# Highlight the important points\n",
    "plt.plot(x0,f(x0),'r.', markersize='15')\n",
    "plt.plot(x1,f(x1),'r.', markersize='15')\n",
    "plt.plot(x2,0,'g*', markersize='15')\n",
    "\n",
    "# Now repeat the same process for the points (x1,f(x1)) and (x2,f(x2))\n",
    "\n",
    "# equation of the secant line through points (x1,f(x1)) and (x2,f(x2))\n",
    "def secant(x):\n",
    "    return f(x1)+((f(x2)-f(x1))/(x2-x1))*(x-x1)\n",
    "# The point of intesection between the new secant line and the x-axis is:\n",
    "x3= -f(x1)*((x2-x1)/(f(x2)-f(x1)))+x1\n",
    "\n",
    "# plot the new secant line\n",
    "plt.plot(xx,secant(xx),'k')\n",
    "\n",
    "# Highlight the important points\n",
    "plt.plot(x0,0,'r*', markersize='15')\n",
    "plt.plot(x1,0,'r*', markersize='15')\n",
    "plt.plot(x2,f(x2),'r.', markersize='15')\n",
    "plt.plot(x3,0,'k*', markersize='15')\n",
    "\n",
    "# Now repeat the same process for the points (x2,f(x2)) and (x3,f(x3))\n",
    "\n",
    "# equation of the secant line through points (x2,f(x2)) and (x3,f(x3))\n",
    "def secant(x):\n",
    "    return f(x2)+((f(x3)-f(x2))/(x3-x2))*(x-x2)\n",
    "# The point of intesection between the new secant line and the x-axis is:\n",
    "x4= -f(x2)*((x3-x2)/(f(x3)-f(x2)))+x2\n",
    "\n",
    "# plot the new secant line\n",
    "plt.plot(xx,secant(xx),'m')\n",
    "\n",
    "# Highlight the important points\n",
    "plt.plot(x3,f(x3),'r.', markersize='15')\n",
    "plt.plot(x4,0,'m*', markersize='15')\n",
    "\n",
    "# Highlight the root this seems to be converging to:\n",
    "r=np.sqrt(3)\n",
    "plt.plot(r,0,'b*', markersize='17')\n",
    "\n",
    "plt.annotate('x0',(0.05,0.25), fontsize='17')\n",
    "plt.annotate('x1',(3.85,0.25), fontsize='17')\n",
    "plt.annotate('x2',(x2,0.25), fontsize='17', color='g')\n",
    "plt.annotate('x3',(x3,0.25), fontsize='17')\n",
    "plt.annotate('x4',(x4,0.25), fontsize='17',color='m')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 6\n",
    "\n",
    "Write a program that approximates $\\sqrt{7}$ numerically using Secant's method applied to an appropriate function (what easy continuous function $f(x)$ has root equal to $\\sqrt{7}$?). After how many steps did Secant's method converge?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "When the starting points are 2, and 10, Secant's method converges to 2.6457513110645907\n",
      "[2, 10, 2.25, 2.4081632653061225, 2.6659364731653885, 2.644806167400881, 2.645747718746274, 2.645751311706349, 2.6457513110645903, 2.6457513110645907, 2.6457513110645907]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fdd1a144550>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1800x504 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def f(x):\n",
    "    return x**2-7\n",
    "\n",
    "\n",
    "x=[2,10]\n",
    "n=10\n",
    "for i in range(1,n):\n",
    "    a=x[len(x)-2] # the element before the last element of the sequence x\n",
    "    b=x[len(x)-1] # the last element of the sequence x\n",
    "    x.append(b-f(b)*(b-a)/(f(b)-f(a)))\n",
    "print(f\"When the starting points are {x[0]}, and {x[1]}, Secant's method converges to {x[len(x)-1]}\")\n",
    "\n",
    "print(x)\n",
    "\n",
    "# plot everything\n",
    "fig,subs=plt.subplots(nrows=1,ncols=2,figsize=(25,7))\n",
    "\n",
    "subs[0].grid()\n",
    "subs[0].axhline(y=0, color='k')\n",
    "subs[0].axvline(x=0, color='k')\n",
    "subs[0].set_title(\"Secant method\", fontsize='25')\n",
    "x1=np.linspace(-5,10,100)\n",
    "subs[0].plot(x1,f(x1),'b')\n",
    "subs[0].plot(x[len(x)-1],0,'k.',markersize=20)\n",
    "subs[0].plot(x,np.zeros(len(x)),'r.',markersize=12)\n",
    "\n",
    "# plot the sequence against its index\n",
    "subs[1].plot(x,'k.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 7\n",
    "\n",
    "Write a program that uses the bisection method to approximate the positive root of $$f(x)=x^2-3$$ to within $10^{-15}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The left endpoint of the interval is now 1.7320508075688767\n",
      "The right endpoint of the interval is now 1.7320508075688776\n",
      "The midpoint of the interval is now 1.7320508075688776\n",
      "The approximation of the root using the bisection method is 1.7320508075688776\n",
      "We have the correct root for this function, it is $\\sqrt{3}=$ 1.7320508075688772\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7ffca38eb700>]"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1008x1008 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Bisection Method: Let's find the positive root of $f(x)=x^2-3$\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# define the function\n",
    "def f(x):\n",
    "    return x**2-3;\n",
    "#  define the starting inverval [a,b]\n",
    "a=0;\n",
    "b=4;\n",
    "\n",
    "while np.absolute(a-b)>10**(-15):\n",
    "    if f(a)*f(b)<0:\n",
    "        d=(a+b)/2\n",
    "        if f(a)*f(d)<0:\n",
    "            b=d\n",
    "        else:\n",
    "            a=d\n",
    "    else:\n",
    "        print('Choose a different interval that contains the root.')\n",
    "        break\n",
    "print('The left endpoint of the interval is now',a)\n",
    "print('The right endpoint of the interval is now',b)\n",
    "print('The midpoint of the interval is now',d)\n",
    "print('The approximation of the root using the bisection method is',d)\n",
    "print(\"We have the correct root for this function, it is $\\sqrt{3}=$\",np.sqrt(3))\n",
    "\n",
    "# Let's plot everything\n",
    "# Set the figure\n",
    "plt.figure(figsize=(14,14))\n",
    "plt.axis('equal') # the scale on the x-axis is the same as the y-axis\n",
    "plt.xlim(-5,5)\n",
    "plt.ylim(-3,3)\n",
    "plt.title('The Bisection Method', fontsize=20)\n",
    "plt.xlabel('x')\n",
    "plt.ylabel('y')\n",
    "plt.axhline(y=0, color='k') # shows the x-axis\n",
    "plt.axvline(x=0, color='k') # shows the y-axis\n",
    "# show the grid\n",
    "plt.grid()\n",
    "\n",
    "# plot the function\n",
    "xx=np.linspace(-8,8,100)\n",
    "plt.plot(xx,f(xx),'b')\n",
    "\n",
    "# Highlight the important points\n",
    "plt.plot(0,0,'m.', markersize='17')\n",
    "plt.plot(4,0,'m.', markersize='17')\n",
    "plt.plot(2,0,'r.', markersize='17')\n",
    "plt.plot(1,0,'r.', markersize='17')\n",
    "plt.plot(1.5,0,'g.', markersize='17')\n",
    "plt.plot(1.75,0,'y.', markersize='30')\n",
    "\n",
    "# label the points using annotate from matplotlib.pyplot\n",
    "plt.annotate('a1',(0.05,-0.3), fontsize='20')\n",
    "plt.annotate('b1',(3.85,-0.3), fontsize='20')\n",
    "plt.annotate('d1',(1.85,-0.3), fontsize='20')\n",
    "plt.annotate('a2',(0.85,0.25), fontsize='20')\n",
    "plt.annotate('b2',(1.85,0.25), fontsize='20')\n",
    "plt.annotate('d2',(1.35,0.25), fontsize='20')\n",
    "plt.annotate('d3 is very close to the root', xy=(1.75, 0), xytext=(1.9,-1.5),\n",
    "             arrowprops=dict(facecolor='y', shrink=0.05),fontsize='16')\n",
    "\n",
    "# Highlight the root this seems to be converging to:\n",
    "r=np.sqrt(3)\n",
    "plt.plot(r,0,'b*', markersize='17')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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