Week3/NewtonsMethod.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "7e4eb54b",
   "metadata": {},
   "source": [
    "# Newton's Method\n",
    "\n",
    "The goal of this notebook is to show some examples of using Python to implement Newton's Method, which attempts to approximate a root of a given function."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "be013896",
   "metadata": {},
   "source": [
    "## Applying at a good point\n",
    "\n",
    "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/O_HhCf9s3F8\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen></iframe>\n",
    "\n",
    "Newton's Method is what's called an *iterative* method, in that we start with a value, and then iteratively generate new values from the previous value.  How well Newton's Method works depends greatly on the value we start with.  In this section, we will consider an example where Newton's Method works well.\n",
    "\n",
    "* Let $f(x) = x^3 - 2x + 2$.  Starting at $x_0 = -1.5$, apply Newton's method three times to estimate a root of $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "555b8967",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6b0b79e-ce60-4c8f-a6a8-aba085b75628",
   "metadata": {},
   "source": [
    "The goal of Newton's Method is to approximate a root of a given function $f$.  To apply Newton's Method, we need both $f$ and the derivative of $f$.  In this case, we are just going to explicitly write out the derivative of $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "eaafc3ee",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambda x: x**3 - 2*x + 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "59329e52",
   "metadata": {},
   "outputs": [],
   "source": [
    "df = lambda x: 3*x**2 - 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c16ced3a-27ab-470b-9915-53b133be1961",
   "metadata": {},
   "source": [
    "The iterative formula for Newton's Method is $z \\leadsto z - f(z)/f'(z)$.  At each stage of Newton's Method, we have some approximate root $z$, and for the next stage we will use $z - f(z)/f'(z)$ as our approximate root.  We define this function in the next cell."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d2fef738",
   "metadata": {},
   "outputs": [],
   "source": [
    "newt = lambda z: z - f(z)/df(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f8777548-9136-4255-a979-83bd0ad42adf",
   "metadata": {},
   "source": [
    "We want to apply Newton's method three times, starting at $x_0 = -1.5$.  We will hold the successive values in a length-4 NumPy array `arr`.  The zeroth element in `arr` will be `-1.5`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "1aec37dc",
   "metadata": {},
   "outputs": [],
   "source": [
    "z = -1.5\n",
    "\n",
    "reps = 3\n",
    "arr = np.zeros(reps+1)\n",
    "arr[0] = z"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e026b23e-c115-40ed-9a8a-b166156943b5",
   "metadata": {},
   "source": [
    "The zeros in `arr` are just place-holders."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "936d4998",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.5,  0. ,  0. ,  0. ])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b6b4c716-3a1e-4b1c-aa5e-7f05987d1fcc",
   "metadata": {},
   "source": [
    "Because we have already defined `newt` above, it is very easy to fill in the subsequent values produced by Newton's Method.  (I don't see a way to find these numbers using, for example, list comprehension in place of the for loop.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "bd94bf79",
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(reps):\n",
    "    arr[i+1] = newt(arr[i])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5b410159-9e19-407e-aefb-610c59d4d267",
   "metadata": {},
   "source": [
    "In good situations, the values produced by Newton's Method will be getting gradually closer to an actual root of $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "86d983e4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.5       , -1.84210526, -1.77282692, -1.76930129])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1348a117-d49d-434a-92b3-c92d743f0b6d",
   "metadata": {},
   "source": [
    "As a reality check, let's see if `newt(arr[2])` is actually `arr[3]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "0761be7e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.7693012925504912"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newt(-1.77282692)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8de3d12c-acb3-4acb-97d2-27c9b516ca3b",
   "metadata": {},
   "source": [
    "To get a sense for what Newton's Method is doing, let's plot $y = f(x)$.  We will use one of the pre-defined styles mentioned in a previous section."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "2b243cc6",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "plt.style.use('seaborn-darkgrid')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "78ef0b18-adf8-46e1-95c6-f15adb5ea64e",
   "metadata": {},
   "source": [
    "Looking at the graph, it is clear that this cubic polynomial has only one root, and it looks to be at approximately $x \\approx -1.8$.  You should imagine starting at $(-1.5, f(-1.5))$, drawing the tangent line to $y = f(x)$ at that point, and finding the root of the tangent line (i.e., finding where the tangent line crosses the $x$-axis).  According to our implementation above, the tangent line should cross the $x$-axis at approximately $x = -1.842$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "d4ad7e32",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1118ca2c0>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "x = np.linspace(-2,2,1000)\n",
    "ax.plot(x, f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3e60b6f0-ad63-4606-82ed-48433f9953bb",
   "metadata": {},
   "source": [
    "Let's actually apply `f` to the elements in `arr` and see if the outputs are getting closer and closer to 0.  (One of the reasons we made `arr` as a NumPy array, as opposed to a list, is so that we could make this evaluation using `f(arr)`.  That syntax would not work if `arr` were a list.)  In this case, `f(arr[3])` is $-0.00006605...$, so by the third iteration of Newton's Method, we have indeed gotten very close to an actual root of $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "ca43db3d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.62500000e+00, -5.66700685e-01, -2.61909903e-02, -6.60651520e-05])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(arr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9568c1b8",
   "metadata": {},
   "source": [
    "## Applying at a worse point\n",
    "\n",
    "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/0ZhC0g1ax1M\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen></iframe>\n",
    "\n",
    "As mentioned above, the success of Newton's Method depends heavily on the starting point.  In this section, we will see some of what can go wrong if we do not begin at a good point.\n",
    "\n",
    "* Let $f(x) = x^3 - 2x + 2$.  Starting at $x_0 = -0.5$, apply Newton's method three times to estimate a root of $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "95af61d7",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('seaborn-darkgrid')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3564ab08-a3ff-4139-ad6d-b48c66450c97",
   "metadata": {},
   "source": [
    "We will use the same setup as in the previous ssection."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "0b2c1265",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambda x: x**3 - 2*x + 2\n",
    "df = lambda x: 3*x**2 - 2\n",
    "\n",
    "newt = lambda z: z - f(z)/df(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "a781132a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1114a84c0>]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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u4ctDR82OJoKQy2XwX6sLSY6N4v/NyjU7jt9JYQfQliO13PfWN8TYrfz5hnGMD+FbFfnboNQEnls4iqevHEmny+DBJTt5ZOlOio61mB1NBJF3thaTX1bPg3NyI2IJYCnsAPm66BgPvbeTjOQYXrpxHLmpclDtTCwWC7MHp/Lm7ZN4cHYuW4vruP7lzfzvF4do7ZBVECLdseZ2frOygPH9e3FJhLxTlcIOgK8OHeN7S3fR3xHL768bEzFngvhKtN3KrZMH8O5dk7lApbNoYxE3vrqFjTJNEtGeXrufpjYnPzxvSFgfaDyZFLafbS6q5ZH3d5LdO47fXzuWFDnrwWNpCdH8x8XDefHa0VgtFv51yU5+/OEeqhvbzI4mAmzdvmo+3lvFv8wZzOC08Fov5HSksP1oT0UD339/F/0dcbx47Rgc8eE/xxYIk7N788ZtE/nOjBw+Lazmmr9s5u1tpXS6vF8qWAS/2pYOfrV6H8PSE7h/ziCz4wSUFLafHD7azENLdtIrzs7zC0fjiIADIoEUbbdy7/Qc3rh9EiMzk/jNmkLufH0buqLR7GjCz55aU0hdq5PHLlJERdjpnpH1rw2QqsY2Hng3H4DnF46mT5LMWftLdu84XrhmNE9ePJyKhjZuX7yVFz4/KAclw9SJqZC7p2UzrE+i2XECTgrbx1o7Ovn++7uoa+3g2YWj5BLrALBYLMwf0Ye375jExXkZvLLpCDe/tpUtR2rNjiZ8qLqpnV+u6poKuXPKALPjmEIK24dchsEvPtLsrWjkiYtHhNQa1uGgV1wUP79I8cI1o3G6DO5/+xt+tWofjW1Os6MJL7kMg8dX7KW5o5MnLhkesVe+enRpulIqClgEDARigCe11st8mCsk/WnDYVYXVPPg7FzmDEk1O07EmprTmzdvn8jvvzjEm1tL+PxADT88b6j8n4SwxZuL+epwLY9eMDTs7iLjDk9/Td0C1GitzwEWAC/4LlJo+qSgij9vLOKykRlhuxZvKImLsvG9uYNZdNN4esVG8YO/7eLRD/ZQ09RudjThpt3lDfxu/SHOHZrGVaMzzY5jKk8L+x3gZyd9HNHvOQ8fbeaJjwsYnZXEoxcMjZiT+EPByMwkXr1lPPfPzOHT/dVc9/JmPtxVjmHIKYChoKHVyU+W7yEtIZqfyM+WZ4WttW7UWjcopZKAd4Gf+jZW6Gjt6ORHH+zBbrXwy0tHRNxpRqEgymbl7mk5LL51Irkp8fziowLuenUzJXWyLkkwcxkGP1+xl/L6Nv7zkuERsVbImVg83dNQSg0AlgIvaq0Xnfp4S0u7Ybd7dtcHm81KZ6fLo+f606m5DMPgh0vzeX97KS/dOpFzhpq3cHqojJnZXC6D178u4qmVBbgMePi8Idw2LSdoDmIF23idYEau59cU8tzaQh6/NI+bp2YHTa6z4U2uqChbj28jPCpspVQGsA54QGv9SXdfU1XV4PF7TocjntraZk+f7jen5lqWX84TKwu4d3o235kx0LxghM6YBYtmLPxkaT7rDxxlREYiP71wWFCc1xus4xXoXOsP1PC9pbu4ZGQGj80f1uNUSDiOV3p6Uo+F7eluxY+B3sDPlFLrjv+J83BbIanoWAu/WVPI5GwHd0/LMTuOcFNfRxy/vXIk/3lJ1wU3t/1VLrgJFvurm/jp8r2oPon8KIIWdjobHp3Wp7V+CHjIx1lChtNl8NiKvUTbrTx+kcJmlW+oUGSxWLhweB+m5vTm2U8P8MqmI6wpqOLRC4YyOVvu4m6G6qZ2Hn5vJ3FRNp66Ii9sb6brqeCYuAsxf9lYxM6yBn50/lC57DwMnLjg5nfXjMYA/uWdfJ74WFPX0mF2tIjS2tHJI0t3UtvSwW+vGklmcqzZkYKOFLabdpbV89LGwywY0YcLVPjenTkSTcnpWgXwtskDWL6rgute3szKvZVyCmAAOF0GP12+l70VjTx5iVwl3BMpbDe0dnTy2ApNWmIM/zZviNlxhB/ERtn419m5vHLzBDKSYvjJ8r088v4uyutbzY4WtlyGwZMrC/h0fw2PnDtYrkg9DSlsNzy3ppCiYy08dtEwkmLlhvPhTGUksuim8Tw0ZxCbi2q5/uUtvLG1BKesue1ThmHw9Jr9LN9VwXdm5HDDhH5mRwpqUthnaW9FA4s2HOKKUZlyQCpC2K0WbpnUnzfvmMiYfsn8du1+bvvrVrYX15kdLSwYhsGL6w/x9vZSbp7Yn3umdX+utfgHKeyz4HQZPLlyHynx0Tw4J9fsOCLA+vWK47mrR/Hry0ZQ3+rk3rd28PhHWtYl8YJhGDz32UFe3nSEq8Zk8tCcXDl97yzI+/qz8MaWYnRlI8/fMI7kWLk8NhJZLBbmDUtnem4KL20sYvHmYj4trOa7Mwdy9di+2OXUzrPmMgz++5NCluwo49pxffnBvMFS1mdJ9rDPoLi2hT9sOMzcIanMz8swO44wWVyUjQfOyeWN2yeSl5HEb9bs53aZJjlrzk4X//GRZsmOMm6b3J9/mzcYq5T1WZPCPoOn1+7HbrXwb/PkiivxDwNT4nnhmtH812UjqG3p4N63dvDDZbsprpUFpXrS0Orkofd2snx3JffNyOGBc2QaxF0yJXIan++vYf2Bozw8Z5BcICO+xWKxcN6wdGbkprB4czGvfn2Ez/bXcO24vtw9LVtWlztJSV0L31u6i6JjLfxs/jAuHxXZ61p7Sgq7B21OF79dt5/clHiuH9/X7DgiiMVF2bhneg5Xjs7k9xsO89a2EpbvruDuadlcM7Yv0fbIfiP7ddExfvLhXpwugxcWjmZStsPsSCErsr+TTmPx5mKKa1v5/rzBQbP0pghuaYkx/PTCYfz11gnkZSTxzLoDLFz0NUu/KcMZhEuA+pvLMHhp42EeeDcfR1wUi24cJ2XtJWmibpTXt7LoqyLmDU1jao6ccy3cMzQ9keevGc0LC0eTmhDNL1ft45q/dN3pJlIuvKlubOOh93by+y8Oc4FK5+WbxzMwNd7sWCFPpkS68dxnBwF4eO4gk5OIUDZ1YG+m5DhYf+Aof9hwmF98VMDLXx3hrmnZXKjSw/Kdm2EYrNhTydNr99PmdPGj84dw9ZgsObjoI1LYp9hZVs8qXcU907LJktXChJcsFgvnDE5l1qAU1hXW8McNh3lshebF9Ye4aWI/rhidSUJ0ePwYltW38vSa/Xy6v4bRWcn8/KJhDEyRvWpfCo/vFB8xDIPnPj1ASnwUt04eYHYcEUYsFgvnDk1j7pBUNhw8xmubj/DMugP8+csiFo7NYuHYrJBdTrS1o5NXNh3htc3FADw0ZxA3Tugn68T7gRT2ST7bf5RtJfU8ev4Q4qNl4XThexaLhZmDUpg5KIVdZfW8trmYVzYd4dWvjzAzN4XbZuYyOi0+JMqu3eniw90VLNpYREVDGxeodB6cnRuyv3hCgVeFrZSaCvxaaz3XN3HM43QZvPD5AXJ6x3H56Cyz44gIMDIrmf+6LI+y+lbe/6aM9/PLufe1LWQlx7AgL4OLhvchNwgP1LV0dPLBzgpe/foIFQ1tjMpK4omLhzO+fy+zo4U9jwtbKfXvwK1Ak+/imGdZfhmHjrbw1BV5si6ECKis5Fi+OyuXe6fn8HVZI4s3HuLlr4pYtLGIoekJXDS8D/OGpdHfYe5tUw/WNLNkRynLd1fQ2NbJ2L7J/OzCYUzJcchBxQDxZg97P3A18JqPspimpaOTP2w4zPh+ycweLIunC3PYbVYWjMpkev9kqpvaWa2rWLm3kuc/P8jznx8kp3ccM3JTmJmbwth+yQG53+GRYy18UlDFugNH2VVaT5TNwryhaVwzti9j+yVLUQeYxZvbHymlBgJvaq2nnfpYS0u7Ybd79g1ls1npDOCFBn/47ABPrSrg7XunMv40a10HOpc7gjWb5HJPd7mOHGtmra7i04IqNh48SrvTRZTNQl5WMhOyHYwf0BuVkUh2SrxXpwp2ugyKjjaTX1LHVwePsvHgUYqONgMwboCDBSMzuGJcP1ITor36N/pSKP0/nq2oKFuPvwX9VthVVQ0eb9jhiKe2ttnjXO5obHNy5Z83MTIriWevHh00udwVrNkkl3vOlKu1o5PNR2rZVlxPfmkduysaaXN2FYPdaiG7dxzZveNIS4gmLTGa1Pho4qNtRNmsRNutuFwGrU4XrR2dNLQ5qWxop7KxjZK6VvZXN/19W4kxNib0dzAp28G5Q1IZnp0SkuNlFm9ypacn9VjYEX+WyFvbSqhrdfKdGQPNjiLEGcVG2Zg1KJVZg7qm7jo6XeyrauJgTTMHapo5WNPE4WMtbC2uo77VecbtxditZCTFkJkUw8KxWQxJS2BYn0SGpCWExJkqkSaiC7uh1cnizSXMHpzKyEy5S7MIPVE2K3mZSeR18/3b5nRxrLmdVqeLdqeLNqcLq9VCXJSVWLuNhGgbybF2mYcOIV4Vttb6EPCt6ZBQ8fqWYhranHxneo7ZUYTwuRi7Vc6JDjPht5jBWapr6eCNrSWcOzQNlZFodhwhhDijiC3sN7aW0NTeKXvXQoiQEZGF3djm5O1tpcwdksqQ9ASz4wghxFmJyMJe+k0ZDW1O7piabXYUIYQ4axFX2G1OF4u3lDA52yFnhgghQkrEFfbyXeXUNLVzxxRZPlUIEVoiqrCdLoNXvi5mZGYSk+XeckKIEBNRhb1aV1Fa18odUwbIxQJCiJATMYVtGAavbDpCbmo8s4fIinxCiNATMYW98fAxCqubuG1yf6yydy2ECEERU9ivbykhNSGa+cP7mB1FCCE8EhGFfaCmiY2HjnHtuCyivFgvWAghzBQR7fXm1hJi7FYWjulrdhQhhPBY2Bd2bXMH/7e7kgUj+uCIjzI7jhBCeCzsC3tpfhltThc3TOhndhQhhPBKWBd2R6eLt7eVMi2nN4PTZJEnIURoC+vCXl1QRXVTOzdOlL1rIUTo8+iOM0opK/AiMBZoA+7RWhf6MpgvvLm1lIEpcUwb2POd0IUQIlR4uod9JRCrtZ4O/Ah42meJfGR3eQO7yxu4dlxfuVBGCBEWPC3sWcBHAFrrjcAknyXykSU7Som1W7k4L8PsKEII4ROe3oQ3Gag76eNOpZRda+088YnExBjsdptHG7fZrDgc8R5G67pf40pdxRXj+tI/I9nj7fg6lz8FazbJ5R7J5Z5Iy+VpYdcDJ6/+bz25rAEaG9s8DuVwxFNb2+zx89/YWkJrh4vLhvfxaju+zuVPwZpNcrlHcrknHHOlp/d8YxVPp0S+AC4GUEpNA/I93I7PGYbBeztKGZWVJHdDF0KEFU8LeynQqpTaADwDfM93kbyz5Ugdh462sHBsltlRhBDCpzyaEtFau4D7fZzFJ5bsKCU51s75w9LNjiKEED4VVhfOVDe2sbawhktHZhAb5dkBTyGECFZhVdgf7Kqg02WwcKysyieECD9hU9iGYbBsZzkTB/Qiu3ec2XGEEMLnwqawt5XUUVzbyuWjMs2OIoQQfhE2hb0sv5yEaBvzhqaZHUUIIfwiLAq7sc3J6oJqLhyeLgcbhRBhKywKe5Wuos3pkukQIURYC4vC/mBnOYNS4xmZ2fMlnUIIEepCvrAP1DSRX9bA5aMyscgyqkKIMBbyhb0svwKb1cKCvD5mRxFCCL8K6cJ2drpYsaeCcwalkBIfbXYcIYTwq5Au7I2Hj3G0uYNLR8rBRiFE+Avpwl6xu5JesXZm5Mo9G4UQ4S9kC7uxzcmn+2u4QKUTZQvZf4YQQpy1kG26tfuqaXO6WCD3bBRCRIiQLewVeyrp74hldJacey2EiAwhWdiVDW1sLqplwYg+cu61ECJieFXYSqmrlFKv+yrM2fp4byUGcNEImQ4RQkQOT++ajlLqWWA+sN1nac7Sij2VjMpKknWvhRARxZs97A3Ad30V5GwVVjWxr6qJBSPkykYhRGQ54x62Uupuvn1X9Du11m8ppeb29LzExBjsds+WOrXZrDgc8d0+tnbTEWxWCwun5OBICOzVjafLZbZgzSa53CO53BNpuc5Y2Frrl4CX3N1wY2ObR4EAHI54amubv/V5wzBYtr2UaTm9sXU4qa11evwavswVDII1m+Ryj+RyTzjmSk/v+cy3kDpLZGdZA+UNbVyg0s2OIoQQARdShb26oIoom4U5Q1LNjiKEEAHn8VkiAFrrdcA6nyQ5A5dhsFpXMX1gCokxXsUWQoiQFDJ72Pml9VQ2tst0iBAiYoVMYa/SVcTYrZwzOMXsKEIIYYqQKOxOl8Hqgmpm5KaQEC3TIUKIyBQShb29pI6aJpkOEUJEtpAo7FW6ili7lVmDZDpECBG5gr6wnS6DtfuqmTUolbgoz66cFEKIcBD0hb2tuJajzR1cMFymQ4QQkS3oC3tNQTWxdiszBsp9G4UQkS2oC9tlGKwrrGFGbgqxMh0ihIhwQV3Yu8oaqG5qZ+5QuRRdCCGCurDX7qvGbrUwK1cKWwghgrawDcNgXWE1k7IdJMXKxTJCCBG0hb2/ppkjta2cKyvzCSEEEMSFvXZfNRZg9pA0s6MIIURQCNrCXrevmjF9k0kL8G3AhBAiWAVlYR851kxBVRNzh8retRBCnODR0TylVC/gr0AyEA08orX+0lehVu2uAGCuzF8LIcTfebqH/QjwidZ6DnAH8DufJQJW7alkaHoC/R1xvtysEEKENE/Pl3sGOHFbdDvQ6ps4UNPUzpaiY9w7LcdXmxRCiLBwxsJWSt0NfO+UT9+ptf5aKZVJ19TIw6c+LzExBrvd/cvJDzW0E2WzctWkATgc8W4/359sNmvQZTohWLNJLvdILvdEWi6LYRgePVEpNRp4E/iB1nrFqY9XVTV4tmEgKi6ajpZ2T5/uNw5HPLW1zWbH6FawZpNc7pFc7gnHXOnpSZaeHvNoDlsplQe8A9zUXVl7K0Huii6EEN/iaTP+CogFnlVKAdRpra/wWSohhBDf4lFhSzkLIUTgBeWFM0IIIb5NClsIIUKEFLYQQoQIKWwhhAgRUthCCBEiPL5wRgghRGDJHrYQQoQIKWwhhAgRUthCCBEigmLRjrO5IYJS6l7gPsAJPKm1/jCA+a4CrtVa39TNY88BM4GG45+6QmtdFwS5Aj5eSqk4uv4f+9A1HrdrratO+ZqAjZdSygq8CIylaznge7TWhSc9fhnwc7rGaJHW+k/+yOFBrkeAu4ETY3ef1loHItvx158K/FprPfeUz5syXmeRy5TxUkpFAYuAgUAMXT9ny0563OfjFRSFzT9uiPA/qmtxkjeACScePL6M64PAJLrWMFmvlFqltW7rdms+pJR6FpgPbO/hSyYA87XW1f7OcrLT5TJxvL4L5GutH1dK3QD8FHjolK8J5HhdCcRqracrpaYBTwNXwN9/2J4BJgNNwBdKqQ+01uVm5jpuAnCb1npLALL8E6XUvwO30jUmJ3/ezPHqMddxZo3XLUCN1vpWpVQqsA1YBv4br2CZEnkG+MPxv3d3Q4QpwBda67bje2OFwJgAZdtAVxF9y/E9paHAH5VSXyil7gpQptPmwrzxmgV8dPzvK4DzT37QhPH6ex6t9Ua6foGdMAIo1Fof01q3A+uBc/yc52xyAUwEHlVKrVdKPRqgTCfsB67u5vNmjtfpcoF54/UO8LOTPnae9He/jFfA97A9vCFCMnDy2+YGoFeAcr2llJrbw9MSgOeB3wI2YK1SarPW+huTc5k1XhUnvW53r+n38TrFqePQqZSya62d3Tzm8zHyMBd0rTP/O6AeWKqUujRQU4Ba6yVKqYHdPGTmeJ0uF5g0XlrrRgClVBLwLl3vKE/wy3gFvLC11i8BL536+VNuiPDpKQ/XA0knfZwE1AYi1xk0A89qrZsBlFJr6JqX9FkBeZjLlPFSSr130ut295p+H69TnDoO1pNK0e9j5EkupZQF+J8T8/pKqeXAeCBgx2x6YOZ49cjs8VJKDQCWAi9qrV8/6SG/jFdQzGGfdEOE67XWO7r5kk3AfyqlYuma3B8B7AxgxJ4MA95USk2ga3ppFvCKuZEA88brC+Di46+/APj8lMcDPV5fAJcBbx+fK84/6bE9wFClVArQCMwGnvJjlrPNlQzsVEqNoGvucx5dB7bMZuZ4nY5p46WUygBWAg9orT855WG/jFdQFDY93BDh+NHfQq31suNnF3xO1w/6T7TWPrvxr7tOybUY2Ah0AK9qrXcFSS4zxut/gVeUUuuBduCmbnIFcryWAhcopTYAFuBOpdRNQKLW+o/Hc31M1xgt0lqX+DGLO7l+DKyl6wyST7TW/xegXN8SJON1plxmjdePgd7Az5RSJ+ay/wQk+Gu85NJ0IYQIEcFylogQQogzkMIWQogQIYUthBAhQgpbCCFChBS2EEKECClsIYQIEVLYQggRIqSwhRAiRPx/fJoT3Ytgy8YAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-2,2,1000)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ceac9baf-4ce1-4aab-a12d-cafbc2892491",
   "metadata": {},
   "source": [
    "Again, this code is all the same as in the previous section, with the exception of the fact that we start here at -0.5 instead of at -1.5."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "5bc5bfa0",
   "metadata": {},
   "outputs": [],
   "source": [
    "z = -0.5\n",
    "\n",
    "reps = 3\n",
    "\n",
    "arr = np.zeros(reps+1)\n",
    "arr[0] = z\n",
    "\n",
    "for i in range(reps):\n",
    "    arr[i+1] = newt(arr[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c93d0e7b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.5       ,  1.8       ,  1.25181347,  0.71203147])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c46c74e7-c030-46bf-980a-8d3dd320b1cd",
   "metadata": {},
   "source": [
    "As a reality check, we make sure that `newt(arr[1])` really is `arr[2]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "6cca589c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2518134715025906"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newt(1.8)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7771e1ae-069f-45f6-be2f-13d3707833ea",
   "metadata": {},
   "source": [
    "We will get a much better sense of Newton's Method if we can draw tangent lines.  The following function takes as input an $x$-value `x0` and as output returns the function corresponding to the tangent line of `f` at `x0`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "872dfd77",
   "metadata": {},
   "outputs": [],
   "source": [
    "def tang(x0):\n",
    "    return lambda x: df(x0)*(x-x0)+f(x0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6dbc4484-4e39-4875-b235-344d701b4905",
   "metadata": {},
   "source": [
    "For each of our three iterations of Newton's method, we draw the corresponding tangent line using `ax.plot(x, tang(arr[i])(x))` and we draw a bullet point at the corresponding root estimate (i.e., where the tangent line crosses the $x$-axis) using `ax.plot(arr[i+1],0,'ko')`.  The syntax `'ko'`, standing for \"black circle\" marker, might look familiar from Matlab.  Instead of letting Matplotlib automatically choose the ranges for `x` and `y`, we set them explicitly using `ax.set(xlim=(-1,2), ylim=(-2,3.5))`.\n",
    "\n",
    "It's worth staring at the resulting plot and understanding how what it shows relates to the values in `arr`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "cf126c31",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1.0, 2.0), (-2.0, 3.5)]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-2,2,1000)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "z = -0.5\n",
    "\n",
    "reps = 3\n",
    "\n",
    "arr = np.zeros(reps+1)\n",
    "arr[0] = z\n",
    "\n",
    "for i in range(reps):\n",
    "    ax.plot(x, tang(arr[i])(x))\n",
    "    arr[i+1] = newt(arr[i])\n",
    "    ax.plot(arr[i+1],0,'ko')\n",
    "    \n",
    "ax.set(xlim=(-1,2), ylim=(-2,3.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14e906b7-f6eb-4930-9141-cbf4b2cac0b3",
   "metadata": {},
   "source": [
    "Let's use the exact same code but with 10 iterations of Newton's Method.  Notice how we don't seem to be converging on the an actual root, even after 10 iterations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "331fa0fa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-1.0, 2.0), (-2.0, 3.5)]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-2,2,1000)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "z = -0.5\n",
    "\n",
    "reps = 10\n",
    "\n",
    "arr = np.zeros(reps+1)\n",
    "arr[0] = z\n",
    "\n",
    "for i in range(reps):\n",
    "    ax.plot(x, tang(arr[i])(x))\n",
    "    arr[i+1] = newt(arr[i])\n",
    "    ax.plot(arr[i+1],0,'ko')\n",
    "    \n",
    "ax.set(xlim=(-1,2), ylim=(-2,3.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "747683db-0de0-4951-9c78-9948a5597981",
   "metadata": {},
   "source": [
    "On the other hand, if we choose a better starting point like -1.5, then we very quickly converge on the actual root.  The following chart includes 10 iterations of Newton's Method, but most of these iterations are so close to each other that we can't distinguish between them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "ca8402c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-2.0, -1.0), (-2.0, 3.5)]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-2,2,1000)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "z = -1.5\n",
    "\n",
    "reps = 10\n",
    "\n",
    "arr = np.zeros(reps+1)\n",
    "arr[0] = z\n",
    "\n",
    "for i in range(reps):\n",
    "    ax.plot(x, tang(arr[i])(x))\n",
    "    arr[i+1] = newt(arr[i])\n",
    "    ax.plot(arr[i+1],0,'ko')\n",
    "    \n",
    "ax.set(xlim=(-2,-1), ylim=(-2,3.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5524cbb6-fd61-4077-bbea-4f0a2afdddaa",
   "metadata": {},
   "source": [
    "If we instead start at 0 for this function $f$, we find another example of what can go wrong when applying Newton's Method.  In this case, Newton's Method oscillates forever between 0 and 1, neither of which is anywhere near the actual root."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "6e64fdab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-2.0, 2.0), (-2.0, 3.5)]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-2,2,1000)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "z = 0\n",
    "\n",
    "reps = 10\n",
    "\n",
    "arr = np.zeros(reps+1)\n",
    "arr[0] = z\n",
    "\n",
    "for i in range(reps):\n",
    "    ax.plot(x, tang(arr[i])(x))\n",
    "    arr[i+1] = newt(arr[i])\n",
    "    ax.plot(arr[i+1],0,'ko')\n",
    "    \n",
    "ax.set(xlim=(-2,2), ylim=(-2,3.5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "8704fd24",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 1., 0., 1., 0., 1., 0., 1., 0., 1., 0.])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "227e2d33",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([2., 1., 2., 1., 2., 1., 2., 1., 2., 1., 2.])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(arr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11bc7bfe",
   "metadata": {},
   "source": [
    "## Analyzing Newton's method\n",
    "\n",
    "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/ZaT6p2ZAC90\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen></iframe>\n",
    "\n",
    "We've seen one starting value where Newton's Method works well, and we've seen two starting values where Newton's Method does not work well.  Here we are going to investigate Newton's Method more systematically.\n",
    "\n",
    "* Let $f(x) = x^3 - 2x + 2$.  For each value of $z$ in `np.linspace(-2,2,100)`, apply Newton's method 7 times, and call the result $z_{7}$.  For how many of these 100 values do we have $-0.001 \\leq f(z_{7}) \\leq 0.001$?\n",
    "\n",
    "The initial setup is the same as in the previous sections."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "99697a84",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "plt.style.use('seaborn-darkgrid')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b747dbc6",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambda x: x**3 - 2*x + 2\n",
    "df = lambda x: 3*x**2 - 2\n",
    "\n",
    "newt = lambda z: z - f(z)/df(z)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12228811-bfa8-4738-b8b4-3b90084b8703",
   "metadata": {},
   "source": [
    "Next we make the array that will hold the 100 starting values we will consider."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "87ef4644",
   "metadata": {},
   "outputs": [],
   "source": [
    "z = np.linspace(-2,2,100)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "41487b5b-6ced-4187-b344-e9b76da4e8ec",
   "metadata": {},
   "source": [
    "We will make a $100 \\times 8$ NumPy array to hold the results.  We think of each row as holding a starting value as well as the first 7 applications of Newton's Method to that starting value.  We initialize all the values to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "dca3bab1",
   "metadata": {},
   "outputs": [],
   "source": [
    "reps = 7\n",
    "arr = np.zeros((len(z), reps+1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a730e0f7-6cf3-4ddd-945b-c8a9a3f45bc1",
   "metadata": {},
   "source": [
    "We fill in the zeroth column with the starting values, and then we one-by-one fill in the next columns using a for loop.  This code is very similar to what we did above, except instead of applying `newt` to a single entry, here we are applying `newt` to an entire column."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "942f9c7d",
   "metadata": {},
   "outputs": [],
   "source": [
    "arr[:, 0] = z\n",
    "\n",
    "for i in range(reps):\n",
    "    arr[:, i+1] = newt(arr[:, i])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25bcc3e0-0681-4557-86e1-91518d4ad9dd",
   "metadata": {},
   "source": [
    "Here is what `arr` looks like.  For example, the top row of `arr` is saying that, if we start at $z_0 = -2$ and apply Newton's Method 7 times, we reach approximately $z_7 = -1.76929$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "a7f166fd",
   "metadata": {
    "tags": [
     "output_scroll"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-2.00000000e+00, -1.80000000e+00, -1.76994819e+00,\n",
       "        -1.76929266e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.95959596e+00, -1.79093206e+00, -1.76962110e+00,\n",
       "        -1.76929243e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.91919192e+00, -1.78321559e+00, -1.76942955e+00,\n",
       "        -1.76929237e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.87878788e+00, -1.77700676e+00, -1.76933475e+00,\n",
       "        -1.76929236e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.83838384e+00, -1.77248656e+00, -1.76929966e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.79797980e+00, -1.76986592e+00, -1.76929259e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.75757576e+00, -1.76939218e+00, -1.76929236e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.71717172e+00, -1.77135720e+00, -1.76929541e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.67676768e+00, -1.77610789e+00, -1.76932547e+00,\n",
       "        -1.76929236e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.63636364e+00, -1.78405978e+00, -1.76944655e+00,\n",
       "        -1.76929237e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.59595960e+00, -1.79571479e+00, -1.76978006e+00,\n",
       "        -1.76929252e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.55555556e+00, -1.81168492e+00, -1.77052745e+00,\n",
       "        -1.76929345e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.51515152e+00, -1.83272408e+00, -1.77199981e+00,\n",
       "        -1.76929760e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.47474747e+00, -1.85977167e+00, -1.77465685e+00,\n",
       "        -1.76931290e+00, -1.76929235e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.43434343e+00, -1.89401326e+00, -1.77915850e+00,\n",
       "        -1.76936154e+00, -1.76929236e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.39393939e+00, -1.93696679e+00, -1.78643431e+00,\n",
       "        -1.76949961e+00, -1.76929239e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.35353535e+00, -1.99060765e+00, -1.79777892e+00,\n",
       "        -1.76985803e+00, -1.76929258e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.31313131e+00, -2.05755477e+00, -1.81498754e+00,\n",
       "        -1.77072260e+00, -1.76929382e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.27272727e+00, -2.14135575e+00, -1.84055561e+00,\n",
       "        -1.77268324e+00, -1.76930058e+00, -1.76929235e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.23232323e+00, -2.24693804e+00, -1.87798643e+00,\n",
       "        -1.77690004e+00, -1.76933359e+00, -1.76929236e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.19191919e+00, -2.38135163e+00, -1.93228936e+00,\n",
       "        -1.78555984e+00, -1.76947918e+00, -1.76929238e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.15151515e+00, -2.55504910e+00, -2.01083181e+00,\n",
       "        -1.80264288e+00, -1.77006384e+00, -1.76929278e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.11111111e+00, -2.78421900e+00, -2.12488859e+00,\n",
       "        -1.83521470e+00, -1.77220938e+00, -1.76929845e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [-1.07070707e+00, -3.09534448e+00, -2.29267033e+00,\n",
       "        -1.89571317e+00, -1.77941316e+00, -1.76936513e+00,\n",
       "        -1.76929236e+00, -1.76929235e+00],\n",
       "       [-1.03030303e+00, -3.53493070e+00, -2.54579403e+00,\n",
       "        -2.00645071e+00, -1.80156357e+00, -1.77001551e+00,\n",
       "        -1.76929273e+00, -1.76929235e+00],\n",
       "       [-9.89898990e-01, -4.19283168e+00, -2.94481740e+00,\n",
       "        -2.20998294e+00, -1.86429691e+00, -1.77518108e+00,\n",
       "        -1.76931710e+00, -1.76929235e+00],\n",
       "       [-9.49494949e-01, -5.26809655e+00, -3.62311891e+00,\n",
       "        -2.59814796e+00, -2.03148836e+00, -1.80791637e+00,\n",
       "        -1.77032155e+00, -1.76929311e+00],\n",
       "       [-9.09090909e-01, -7.30721003e+00, -4.94570843e+00,\n",
       "        -3.41754056e+00, -2.47681584e+00, -1.97445304e+00,\n",
       "        -1.79411694e+00, -1.76972357e+00],\n",
       "       [-8.68686869e-01, -1.25489622e+01, -8.40579361e+00,\n",
       "        -5.66676469e+00, -3.87913658e+00, -2.75233274e+00,\n",
       "        -2.10844708e+00, -1.83003047e+00],\n",
       "       [-8.28282828e-01, -5.39311359e+01, -3.59625627e+01,\n",
       "        -2.39879225e+01, -1.60116576e+01, -1.07048754e+01,\n",
       "        -7.18419619e+00, -4.86522360e+00],\n",
       "       [-7.87878788e-01,  2.16214141e+01,  1.44334331e+01,\n",
       "         9.64996988e+00,  6.47249122e+00,  4.36860038e+00,\n",
       "         2.98162239e+00,  2.06782443e+00],\n",
       "       [-7.47474747e-01,  8.75499163e+00,  5.87909731e+00,\n",
       "         3.97681505e+00,  2.72387826e+00,  1.89646945e+00,\n",
       "         1.32445388e+00,  8.11227019e-01],\n",
       "       [-7.07070707e-01,  5.41234046e+00,  3.66896797e+00,\n",
       "         2.52132147e+00,  1.76065060e+00,  1.22137662e+00,\n",
       "         6.64168077e-01,  2.08979749e+00],\n",
       "       [-6.66666667e-01,  3.88888889e+00,  2.66603410e+00,\n",
       "         1.85781432e+00,  1.29564866e+00,  7.74023344e-01,\n",
       "         5.29225028e+00,  3.58981161e+00],\n",
       "       [-6.26262626e-01,  3.02561426e+00,  2.09696269e+00,\n",
       "         1.46909445e+00,  9.70187358e-01, -2.10728426e-01,\n",
       "         1.08138869e+00,  3.50851023e-01],\n",
       "       [-5.85858586e-01,  2.47567367e+00,  1.72983568e+00,\n",
       "         1.19714636e+00,  6.22490415e-01,  1.81199535e+00,\n",
       "         1.26098987e+00,  7.25624810e-01],\n",
       "       [-5.45454545e-01,  2.09905020e+00,  1.47056765e+00,\n",
       "         9.71633849e-01, -1.98763933e-01,  1.07134095e+00,\n",
       "         3.18232191e-01,  1.14111631e+00],\n",
       "       [-5.05050505e-01,  1.82839634e+00,  1.27346525e+00,\n",
       "         7.43555739e-01,  3.45020712e+00,  2.37727063e+00,\n",
       "         1.66306528e+00,  1.14323583e+00],\n",
       "       [-4.64646465e-01,  1.62731129e+00,  1.11343008e+00,\n",
       "         4.42477131e-01,  1.29313611e+00,  7.70657255e-01,\n",
       "         4.96922487e+00,  3.37699055e+00],\n",
       "       [-4.24242424e-01,  1.47440442e+00,  9.75388643e-01,\n",
       "        -1.68663359e-01,  1.04958485e+00,  2.39488421e-01,\n",
       "         1.07910154e+00,  3.43610662e-01],\n",
       "       [-3.83838384e-01,  1.35628849e+00,  8.49733800e-01,\n",
       "        -4.65204983e+00, -3.23172447e+00, -2.36957020e+00,\n",
       "        -1.92727627e+00, -1.78464424e+00],\n",
       "       [-3.43434343e-01,  1.26416389e+00,  7.30245184e-01,\n",
       "         3.05123116e+00,  2.11391907e+00,  1.48104525e+00,\n",
       "         9.81845963e-01, -1.19900269e-01],\n",
       "       [-3.03030303e-01,  1.19201601e+00,  6.13193205e-01,\n",
       "         1.76479672e+00,  1.22460920e+00,  6.69472450e-01,\n",
       "         2.13587312e+00,  1.49646726e+00],\n",
       "       [-2.62626263e-01,  1.13560207e+00,  4.97077616e-01,\n",
       "         1.39373967e+00,  8.92143634e-01, -1.49537969e+00,\n",
       "        -1.84514226e+00, -1.77311649e+00],\n",
       "       [-2.22222222e-01,  1.09185185e+00,  3.82690537e-01,\n",
       "         1.20969840e+00,  6.44519621e-01,  1.94290031e+00,\n",
       "         1.35859771e+00,  8.52433243e-01],\n",
       "       [-1.81818182e-01,  1.05849802e+00,  2.73219078e-01,\n",
       "         1.10312474e+00,  4.14836015e-01,  1.25172274e+00,\n",
       "         7.11895268e-01,  2.66553223e+00],\n",
       "       [-1.41414141e-01,  1.03384003e+00,  1.74051236e-01,\n",
       "         1.04208023e+00,  2.09298936e-01,  1.06051703e+00,\n",
       "         2.80563528e-01,  1.10884021e+00],\n",
       "       [-1.01010101e-01,  1.01658906e+00,  9.19650586e-02,\n",
       "         1.01206157e+00,  6.82750213e-02,  1.00672095e+00,\n",
       "         3.90185922e-02,  1.00222936e+00],\n",
       "       [-6.06060606e-02,  1.00576401e+00,  3.36178027e-02,\n",
       "         1.00166006e+00,  9.87840470e-03,  1.00014543e+00,\n",
       "         8.71955853e-04,  1.00000114e+00],\n",
       "       [-2.02020202e-02,  1.00062081e+00,  3.71332121e-03,\n",
       "         1.00002063e+00,  1.23781368e-04,  1.00000002e+00,\n",
       "         1.37885051e-07,  1.00000000e+00],\n",
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       "         1.00001955e+00,  1.17316569e-04,  1.00000002e+00,\n",
       "         1.23858900e-07,  1.00000000e+00],\n",
       "       [ 6.06060606e-02,  1.00531632e+00,  3.10739766e-02,\n",
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       "         6.40627796e-04,  1.00000062e+00],\n",
       "       [ 1.01010101e-01,  1.01449580e+00,  8.11339231e-02,\n",
       "         1.00943313e+00,  5.40602026e-02,  1.00424437e+00,\n",
       "         2.49380548e-02,  1.00091821e+00],\n",
       "       [ 1.41414141e-01,  1.02800913e+00,  1.47645749e-01,\n",
       "         1.03047689e+00,  1.58977231e-01,  1.03522821e+00,\n",
       "         1.80153278e-01,  1.04502798e+00],\n",
       "       [ 1.81818182e-01,  1.04584980e+00,  2.24678939e-01,\n",
       "         1.06965321e+00,  3.12539390e-01,  1.13590519e+00,\n",
       "         4.97782800e-01,  1.39524085e+00],\n",
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       "         1.13131327e+00,  4.86987776e-01,  1.37289512e+00,\n",
       "         8.68889872e-01, -2.59722977e+00],\n",
       "       [ 2.62626263e-01,  1.09519343e+00,  3.92441041e-01,\n",
       "         1.22181852e+00,  6.64896525e-01,  2.09594177e+00,\n",
       "         1.46837377e+00,  9.69478766e-01],\n",
       "       [ 3.03030303e-01,  1.12747281e+00,  4.77769474e-01,\n",
       "         1.35483044e+00,  8.48023211e-01, -4.95648203e+00,\n",
       "        -3.42438586e+00, -2.48081406e+00],\n",
       "       [ 3.43434343e-01,  1.16573570e+00,  5.62556971e-01,\n",
       "         1.56477448e+00,  1.05933958e+00,  2.76293132e-01,\n",
       "         1.10549512e+00,  4.21334452e-01],\n",
       "       [ 3.83838384e-01,  1.21109854e+00,  6.46916878e-01,\n",
       "         1.95908288e+00,  1.37039301e+00,  8.66040962e-01,\n",
       "        -2.80266122e+00, -2.13447158e+00],\n",
       "       [ 4.24242424e-01,  1.26521822e+00,  7.31770980e-01,\n",
       "         3.09068759e+00,  2.14002146e+00,  1.49937511e+00,\n",
       "         9.99407793e-01, -3.56379859e-03],\n",
       "       [ 4.64646465e-01,  1.33058821e+00,  8.18845144e-01,\n",
       "        -7.82770391e+01, -5.21904800e+01, -3.48024160e+01,\n",
       "        -2.32149389e+01, -1.54970330e+01],\n",
       "       [ 5.05050505e-01,  1.41106821e+00,  9.10869311e-01,\n",
       "        -9.98948796e-01, -4.01903504e+00, -2.83775191e+00,\n",
       "        -2.15284808e+00, -1.84436813e+00],\n",
       "       [ 5.45454545e-01,  1.51289009e+00,  1.01212504e+00,\n",
       "         6.86139803e-02,  1.00678672e+00,  3.93879565e-02,\n",
       "         1.00227130e+00,  1.34749066e-02],\n",
       "       [ 5.85858586e-01,  1.64672381e+00,  1.12970318e+00,\n",
       "         4.83144541e-01,  1.36525448e+00,  8.60149412e-01,\n",
       "        -3.31202551e+00, -2.41559829e+00],\n",
       "       [ 6.26262626e-01,  1.83237831e+00,  1.27648215e+00,\n",
       "         7.47802127e-01,  3.60959356e+00,  2.48223775e+00,\n",
       "         1.73427248e+00,  1.20065819e+00],\n",
       "       [ 6.66666667e-01,  2.11111111e+00,  1.47906865e+00,\n",
       "         9.79929417e-01, -1.33997171e-01,  1.03015086e+00,\n",
       "         1.57493532e-01,  1.03458666e+00],\n",
       "       [ 7.07070707e-01,  2.58521156e+00,  1.80363793e+00,\n",
       "         1.25460126e+00,  7.16198954e-01,  2.74355396e+00,\n",
       "         1.90959817e+00,  1.33415600e+00],\n",
       "       [ 7.47474747e-01,  3.59661518e+00,  2.47369348e+00,\n",
       "         1.72849684e+00,  1.19608506e+00,  6.20581591e-01,\n",
       "         1.80196241e+00,  1.25331781e+00],\n",
       "       [ 7.87878788e-01,  7.41858586e+00,  4.99410616e+00,\n",
       "         3.39337826e+00,  2.33982171e+00,  1.63751143e+00,\n",
       "         1.12200826e+00,  4.64340933e-01],\n",
       "       [ 8.28282828e-01, -1.48478114e+01, -9.93159828e+00,\n",
       "        -6.67292539e+00, -4.53143285e+00, -3.15588288e+00,\n",
       "        -2.32659456e+00, -1.90938038e+00],\n",
       "       [ 8.68686869e-01, -2.61113064e+00, -2.03778989e+00,\n",
       "        -1.80958417e+00, -1.77041046e+00, -1.76929325e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [ 9.09090909e-01, -1.03761755e+00, -3.44266052e+00,\n",
       "        -2.49150308e+00, -1.98116568e+00, -1.79561389e+00,\n",
       "        -1.76977639e+00, -1.76929252e+00],\n",
       "       [ 9.49494949e-01, -4.08706234e-01,  1.42542737e+00,\n",
       "         9.26007184e-01, -7.19546860e-01,  6.14447620e+00,\n",
       "         4.15197471e+00,  2.83910547e+00],\n",
       "       [ 9.89898990e-01, -6.38458417e-02,  1.00641391e+00,\n",
       "         3.72911176e-02,  1.00203833e+00,  1.21067392e-02,\n",
       "         1.00021813e+00,  1.30737305e-03],\n",
       "       [ 1.03030303e+00,  1.58186516e-01,  1.03488558e+00,\n",
       "         1.78653530e-01,  1.04429414e+00,  2.18385786e-01,\n",
       "         1.06583278e+00,  2.99411544e-01],\n",
       "       [ 1.07070707e+00,  3.16101609e-01,  1.13915147e+00,\n",
       "         5.05270632e-01,  1.41155797e+00,  9.11391114e-01,\n",
       "        -9.87872052e-01, -4.23436798e+00],\n",
       "       [ 1.11111111e+00,  4.36392915e-01,  1.28355057e+00,\n",
       "         7.57620974e-01,  4.06524968e+00,  2.78205426e+00,\n",
       "         1.93526117e+00,  1.35301155e+00],\n",
       "       [ 1.15151515e+00,  5.32764976e-01,  1.47808842e+00,\n",
       "         9.78977278e-01, -1.41115928e-01,  1.03368695e+00,\n",
       "         1.73374096e-01,  1.04175933e+00],\n",
       "       [ 1.19191919e+00,  6.13016042e-01,  1.76393659e+00,\n",
       "         1.22393911e+00,  6.68377506e-01,  2.12610270e+00,\n",
       "         1.48961078e+00,  9.90099232e-01],\n",
       "       [ 1.23232323e+00,  6.81908103e-01,  2.25755078e+00,\n",
       "         1.58103819e+00,  1.07367449e+00,  3.25998976e-01,\n",
       "         1.14842883e+00,  5.26047693e-01],\n",
       "       [ 1.27272727e+00,  7.42511823e-01,  3.41379640e+00,\n",
       "         2.35327848e+00,  1.64670393e+00,  1.12968662e+00,\n",
       "         4.83104863e-01,  1.36517643e+00],\n",
       "       [ 1.31313131e+00,  7.96894925e-01,  1.04123609e+01,\n",
       "         6.97833523e+00,  4.70291660e+00,  3.20163989e+00,\n",
       "         2.21333913e+00,  1.55047066e+00],\n",
       "       [ 1.35353535e+00,  8.46499785e-01, -5.25675646e+00,\n",
       "        -3.61586348e+00, -2.59382463e+00, -2.02939786e+00,\n",
       "        -1.80736909e+00, -1.77029315e+00],\n",
       "       [ 1.39393939e+00,  8.92362474e-01, -1.48818753e+00,\n",
       "        -1.85004035e+00, -1.77360559e+00, -1.76930565e+00,\n",
       "        -1.76929235e+00, -1.76929235e+00],\n",
       "       [ 1.43434343e+00,  9.35245831e-01, -5.83137473e-01,\n",
       "         2.44587023e+00,  1.70966541e+00,  1.18107689e+00,\n",
       "         5.92755621e-01,  1.67398490e+00],\n",
       "       [ 1.47474747e+00,  9.75723500e-01, -1.66043710e-01,\n",
       "         1.04791525e+00,  2.32919980e-01,  1.07483082e+00,\n",
       "         3.29803648e-01,  1.15209851e+00],\n",
       "       [ 1.51515152e+00,  1.01423479e+00,  7.97688608e-02,\n",
       "         1.00912412e+00,  5.23658902e-02,  1.00398608e+00,\n",
       "         2.34499722e-02,  1.00081263e+00],\n",
       "       [ 1.55555556e+00,  1.05112154e+00,  2.45462063e-01,\n",
       "         1.08309815e+00,  3.56194707e-01,  1.17922943e+00,\n",
       "         5.89217371e-01,  1.65980849e+00],\n",
       "       [ 1.59595960e+00,  1.08665348e+00,  3.67127699e-01,\n",
       "         1.19138467e+00,  6.12036749e-01,  1.75920727e+00,\n",
       "         1.22024983e+00,  6.62305782e-01],\n",
       "       [ 1.63636364e+00,  1.12104608e+00,  4.61937263e-01,\n",
       "         1.32578507e+00,  8.12888734e-01,  5.24903769e+01,\n",
       "         3.50018118e+01,  2.33467014e+01],\n",
       "       [ 1.67676768e+00,  1.15447341e+00,  5.39116520e-01,\n",
       "         1.49514654e+00,  9.95389365e-01, -2.83180399e-02,\n",
       "         1.00122705e+00,  7.31744098e-03],\n",
       "       [ 1.71717172e+00,  1.18707749e+00,  6.04072885e-01,\n",
       "         1.72226184e+00,  1.19113269e+00,  6.11574414e-01,\n",
       "         1.75698919e+00,  1.21851669e+00],\n",
       "       [ 1.75757576e+00,  1.21897520e+00,  6.60190718e-01,\n",
       "         2.05721771e+00,  1.44093685e+00,  9.42002486e-01,\n",
       "        -4.95680390e-01,  1.77652335e+00],\n",
       "       [ 1.79797980e+00,  1.25026354e+00,  7.09699829e-01,\n",
       "         2.62810251e+00,  1.83241422e+00,  1.27650934e+00,\n",
       "         7.47840244e-01,  3.61111261e+00],\n",
       "       [ 1.83838384e+00,  1.28102360e+00,  7.54131332e-01,\n",
       "         3.88701543e+00,  2.66480155e+00,  1.85698965e+00,\n",
       "         1.29503001e+00,  7.73196448e-01],\n",
       "       [ 1.87878788e+00,  1.31132363e+00,  7.94572930e-01,\n",
       "         9.40622918e+00,  6.31083609e+00,  4.26182442e+00,\n",
       "         2.91137196e+00,  2.02123769e+00],\n",
       "       [ 1.91919192e+00,  1.34122141e+00,  8.31819615e-01,\n",
       "        -1.12032491e+01, -7.51405549e+00, -5.08117412e+00,\n",
       "        -3.50374256e+00, -2.52738502e+00],\n",
       "       [ 1.95959596e+00,  1.37076616e+00,  8.66466648e-01,\n",
       "        -2.77048628e+00, -2.11778748e+00, -1.83295710e+00,\n",
       "        -1.77201911e+00, -1.76929768e+00],\n",
       "       [ 2.00000000e+00,  1.40000000e+00,  8.98969072e-01,\n",
       "        -1.28877933e+00, -2.10576730e+00, -1.82919995e+00,\n",
       "        -1.77171581e+00, -1.76929656e+00]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "arr"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7e7a9cf2-ff40-4aca-a8c7-cd2965d3fe0f",
   "metadata": {},
   "source": [
    "We are interested in the results after the 7 applications of Newton's Method.  This corresponds to the last column of `arr`.  We store this last column with the variable name `results`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b8d5e4c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "results = arr[:, -1]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "04a496fb-352c-4ae3-9975-b1989da51e2a",
   "metadata": {},
   "source": [
    "Let's get ready to plot the results.  First we plot $f$ over a slightly larger domain."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "f0259303",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1133eb8b0>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "x = np.linspace(-2.5,2.5,1000)\n",
    "ax.plot(x, f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9c1e0bef-d23f-441b-83ef-f5aeb3fa4345",
   "metadata": {},
   "source": [
    "Here is the Boolean array corresponding to our condition $-0.001 \\leq f(z_{7}) \\leq 0.001$.  For example, the zeroth entry being `True` indicates that $z_0 = -2$ is a starting value for which this condition is satisfied."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "cb6eb991",
   "metadata": {},
   "outputs": [],
   "source": [
    "good_bool = np.abs(f(results)) <= 0.001"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "429dea27",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False,  True,\n",
       "        True, False, False, False, False, False, False, False, False,\n",
       "       False, False, False,  True, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False,  True,\n",
       "        True])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "good_bool"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76de0784-d369-4743-91ac-7095ebd1809e",
   "metadata": {},
   "source": [
    "We are getting ready to plot the good starting values.  As a first step, we use Boolean indexing to select the starting values where the condition is satisfied: `pts = z[good_bool]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c50d26c9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "x = np.linspace(-2.5,2.5,1000)\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "pts = z[good_bool]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50d13bec-b06e-4df4-91dc-a1d4a026e1b2",
   "metadata": {},
   "source": [
    "This variable `pts` represents a one-dimensional NumPy array of length 32."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "d8cd7c03",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(32,)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pts.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "571aad86-3266-4b54-96cd-204557f3d205",
   "metadata": {},
   "source": [
    "Here we try to add in black bullet points along the $x$-axis at all of the good points.  The error message tells us that the error is in the last line `ax.plot(pts, 0, 'ko');`, and that the specific error is \"ValueError: x and y must have same first dimension, but have shapes (32,) and (1,)\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "85746ce8",
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "x and y must have same first dimension, but have shapes (32,) and (1,)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "Input \u001b[0;32mIn [13]\u001b[0m, in \u001b[0;36m<cell line: 7>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      3\u001b[0m ax\u001b[38;5;241m.\u001b[39mplot(x, f(x))\n\u001b[1;32m      5\u001b[0m pts \u001b[38;5;241m=\u001b[39m z[good_bool]\n\u001b[0;32m----> 7\u001b[0m \u001b[43max\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mplot\u001b[49m\u001b[43m(\u001b[49m\u001b[43mpts\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m0\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mko\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m~/miniconda3/envs/math9f22/lib/python3.10/site-packages/matplotlib/axes/_axes.py:1632\u001b[0m, in \u001b[0;36mAxes.plot\u001b[0;34m(self, scalex, scaley, data, *args, **kwargs)\u001b[0m\n\u001b[1;32m   1390\u001b[0m \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[1;32m   1391\u001b[0m \u001b[38;5;124;03mPlot y versus x as lines and/or markers.\u001b[39;00m\n\u001b[1;32m   1392\u001b[0m \n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m   1629\u001b[0m \u001b[38;5;124;03m(``'green'``) or hex strings (``'#008000'``).\u001b[39;00m\n\u001b[1;32m   1630\u001b[0m \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[1;32m   1631\u001b[0m kwargs \u001b[38;5;241m=\u001b[39m cbook\u001b[38;5;241m.\u001b[39mnormalize_kwargs(kwargs, mlines\u001b[38;5;241m.\u001b[39mLine2D)\n\u001b[0;32m-> 1632\u001b[0m lines \u001b[38;5;241m=\u001b[39m [\u001b[38;5;241m*\u001b[39m\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_get_lines(\u001b[38;5;241m*\u001b[39margs, data\u001b[38;5;241m=\u001b[39mdata, \u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mkwargs)]\n\u001b[1;32m   1633\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m line \u001b[38;5;129;01min\u001b[39;00m lines:\n\u001b[1;32m   1634\u001b[0m     \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39madd_line(line)\n",
      "File \u001b[0;32m~/miniconda3/envs/math9f22/lib/python3.10/site-packages/matplotlib/axes/_base.py:312\u001b[0m, in \u001b[0;36m_process_plot_var_args.__call__\u001b[0;34m(self, data, *args, **kwargs)\u001b[0m\n\u001b[1;32m    310\u001b[0m     this \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m args[\u001b[38;5;241m0\u001b[39m],\n\u001b[1;32m    311\u001b[0m     args \u001b[38;5;241m=\u001b[39m args[\u001b[38;5;241m1\u001b[39m:]\n\u001b[0;32m--> 312\u001b[0m \u001b[38;5;28;01myield from\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_plot_args\u001b[49m\u001b[43m(\u001b[49m\u001b[43mthis\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n",
      "File \u001b[0;32m~/miniconda3/envs/math9f22/lib/python3.10/site-packages/matplotlib/axes/_base.py:498\u001b[0m, in \u001b[0;36m_process_plot_var_args._plot_args\u001b[0;34m(self, tup, kwargs, return_kwargs)\u001b[0m\n\u001b[1;32m    495\u001b[0m     \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39maxes\u001b[38;5;241m.\u001b[39myaxis\u001b[38;5;241m.\u001b[39mupdate_units(y)\n\u001b[1;32m    497\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m x\u001b[38;5;241m.\u001b[39mshape[\u001b[38;5;241m0\u001b[39m] \u001b[38;5;241m!=\u001b[39m y\u001b[38;5;241m.\u001b[39mshape[\u001b[38;5;241m0\u001b[39m]:\n\u001b[0;32m--> 498\u001b[0m     \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m(\u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mx and y must have same first dimension, but \u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m    499\u001b[0m                      \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mhave shapes \u001b[39m\u001b[38;5;132;01m{\u001b[39;00mx\u001b[38;5;241m.\u001b[39mshape\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m and \u001b[39m\u001b[38;5;132;01m{\u001b[39;00my\u001b[38;5;241m.\u001b[39mshape\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m)\n\u001b[1;32m    500\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m x\u001b[38;5;241m.\u001b[39mndim \u001b[38;5;241m>\u001b[39m \u001b[38;5;241m2\u001b[39m \u001b[38;5;129;01mor\u001b[39;00m y\u001b[38;5;241m.\u001b[39mndim \u001b[38;5;241m>\u001b[39m \u001b[38;5;241m2\u001b[39m:\n\u001b[1;32m    501\u001b[0m     \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mValueError\u001b[39;00m(\u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mx and y can be no greater than 2D, but have \u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m    502\u001b[0m                      \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mshapes \u001b[39m\u001b[38;5;132;01m{\u001b[39;00mx\u001b[38;5;241m.\u001b[39mshape\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m and \u001b[39m\u001b[38;5;132;01m{\u001b[39;00my\u001b[38;5;241m.\u001b[39mshape\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m\"\u001b[39m)\n",
      "\u001b[0;31mValueError\u001b[0m: x and y must have same first dimension, but have shapes (32,) and (1,)"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "x = np.linspace(-2.5,2.5,1000)\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "pts = z[good_bool]\n",
    "\n",
    "ax.plot(pts, 0, 'ko');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e19c777a-742b-4ba6-9fd8-813a6114ba21",
   "metadata": {},
   "source": [
    "We need to replace the single number `0` with a NumPy array of 32 zeros.  (A length 32 list would also work.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "4f5469d4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "x = np.linspace(-2.5,2.5,1000)\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "pts = z[good_bool]\n",
    "\n",
    "ax.plot(pts, np.zeros(len(pts)), 'ko');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "265b1c81-c56d-4586-b818-9090c0a914de",
   "metadata": {},
   "source": [
    "We make the bullet points a little smaller by setting a `markersize`.  (I just guessed-and-checked values to see what size looked good.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "60bdb963",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWwAAAD0CAYAAAC/3RwjAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAAAsTAAALEwEAmpwYAAAfPUlEQVR4nO3deXQc5Z3u8W9LrX2XLFmLLduypfJubMBLsMEsYZ3EBC6EZBImQHKSwNyEhIRkCITkZG5yyVwmcDMBJkDCcEmAsCWQ2CwhYGywscE2xlvZlqzV2vfW3t11/5DtcWzL1tLd1dX9fM7ROepWq+r3qkuPXr1V71suy7IQEZHwF2N3ASIiMjoKbBERh1Bgi4g4hAJbRMQhFNgiIg6hwBYRcQh3sDbc3NztyOsFU1MT8HgG7C4jpNTmyBdt7QXntjk3N8010tfUwz6B2x1rdwkhpzZHvmhrL0RmmxXYIiIOocAWEXEIBbaIiEMosEVEHEKBLSLiEApsERGHUGCLiDiEAltEJECGfH5ufGobb5jNQdm+AltEJEA2lLeyt9FDUlxwolWBLSISIC993EBeajwrpmcHZfsKbBGRAKjt6OP9ynbWLMgnNmbE5UAmRIEtIhIAf/y4gRgXXL2gIGj7UGCLiEzQoNfPyx83sGpmDnlpCUHbjwJbRGSC3j7YQnvfENcsCl7vGhTYIiIT9vxH9RRlJLJsWlZQ96PAFhGZgIrWHrbXdnLNwgJiXME52XiUAltEZAJe/KieuFgXn5o/Oej7UmCLiIxT35CPv+xp5KLSSWQlxwd9fwpsEZFxen1fE54BH9cuKgzJ/hTYIiLjYFkWL3xUT0lOMmcVpYdknwpsEZFx2Hm4i72NHq47qxBXkE82HqXAFhEZh2e2HSYtwc1V84J/svEoBbaIyBg1dPXz1oFm1izIJykuNmT7VWCLiIzR8x/VYwHXnRWak41HKbBFRMagf8jHH3fWc/7MHAozEkO6bwW2iMgYrNvbRGe/lxuWFIV83wpsEZFRsiyLZ7bVUZqbwpIpGSHfvwJbRGSUPqjpoKK1lxuWFIXsUr7jKbBFREbp6Q/ryEqK47LZebbs3z3aFxqGsQy4zzTN1YZhzAKeACxgF3CbaZr+4JQoImK/ytZeNlS08eXlxSS47enrjmqvhmHcCTwGHD0l+u/A3aZprgJcwJrglCciEh7+3wc1JLhjuH5xaC/lO95o/0yUA9cc9/hsYP2Rz9cBlwSyKBGRcNLsGWDtniY+NW9ySFblG8mohkRM03zBMIzpxz3lMk3TOvJ5N3DS6dLU1ATc7tDNAAqU2NgYMjOT7S4jpNTmyBdt7YXAtvnX79fgtyy+flGprT/HUY9hn+D48eo0oOPEF3g8A+PctL0yM5Pp6Oi1u4yQUpsjX7S1FwLXZs+Al99vqebislzSYwj6zzE3N23Er4135Hy7YRirj3x+BbBhnNsREQlrL+2sp2fQxxfPnWJ3KePuYd8BPGoYRjywF3g+cCWJiISHQa+f339Yx7nFmcyZPHLPN1RGHdimaVYCy498vh+4IEg1iYiEhVf3NtHSM8i9l5fZXQqgiTMiIqfk81s8ubWG0twUlk3LsrscQIEtInJKb+5vpqq9j1uWF9syDf1UFNgiIifwWxaPba5mRk4yF5ZOsrucYxTYIiIn+Nv+Fg619vLl5cXEhEnvGhTYIiJ/x29ZPL65mmlZSVxclmt3OX9HgS0icpz1B1s52NLDzcuLiY0Jn941KLBFRI6xLIvHNlUxNTORS21aQvV0FNgiIkdsqGhjf3MPNy0rxh1mvWtQYIuIAMNj14++V0VhRiJXzAm/3jUosEVEAHjrQAv7mjx8ZUUx7tjwjMbwrEpEJIS8fouHN1YyIyeZK+ZMtrucESmwRSTqrd3TSFV7H18/b3rYXRlyPAW2iES1Qa+fR9+rYm5+Gqtn5dhdzmkpsEUkqr24s56G7gFuXTk9bNYMGYkCW0SiVu+gj99sruac4sywWZHvdBTYIhK1fvdBLe19Q9x63nS7SxkVBbaIRKVmzwBPbq3hkrJJLChMt7ucUVFgi0hUenhjJT7L4rZVM+wuZdQU2CISdcwmD3/e3chnFxcxJTPJ7nJGTYEtIlHFsiweWF9BeqKbm5cV213OmCiwRSSqbKxo44PqDr6yYhppiaO+D3lYUGCLSNTw+vw8uL6C4qwkrl1UYHc5Y6bAFpGo8fS2Oqra+7j9gpKwXeDpdJxXsYjIODR1D/DopipWlmSzamZ4T0EfiQJbRKLCg+sr8Pkt7rhwpt2ljJsCW0Qi3gfVHbxuNvNPS6c66jK+EymwRSSieX1+fv7mQQozErnx3Kl2lzMhCmwRiWhPb6vjUFsvd1w4k8S4WLvLmRAFtohErNqOPv7zveETjec79ETj8RTYIhKRLMviZ28cwB3j4nsXz7K7nIBQYItIRHppx2G2VHdw26oZ5Kcn2l1OQCiwRSTitPYM8tN1+1hUmO7IGY0jUWCLSMT5P38rp3fQy92XlhET5rf9GgsFtohElLcPtPDX/c3ctnoW03OS7S4noBTYIhIx2noH+ekbByjLTeErK51zY4LRctbagiIiI7Asi//1+gF6Br08fOVC4t0x9NpdVICphy0iEeGVXY28U97KrStnMHNSit3lBIUCW0Qcr66zj/vfKufsqRl87uwiu8sJGgW2iDiaz2/x43UmLhfce7kRUVeFnGhCY9iGYWwHOo88PGSa5k0TL0lEZPQe21TF9roufnS5QUGETJAZybgD2zCMRADTNFcHrBoRkTHYUtXO45uruWreZK6aN9nucoJuIj3sRUCyYRivH9nOXaZpbg5MWSIip9fSM8g9a/cxPTs5YtYKOROXZVnj+kbDMBYAy4HHgFJgHWCYpukF6OsbtNxu5y1lGBsbg8/nt7uMkFKbI1+ktdfnt7j5vz5gW007L3x1BWWT0056jVPbHBcXO+Ig/ER62PuBg6ZpWsB+wzBagQKgBsDjGZjApu2TmZlMR0ekXb15empz5Iu09j76XhXvVbRy96Wl5CXEnrJtTm1zbu7Jf3yOmshVIjcD9wMYhlEIpAP1E9ieiMgZrT/Yyq83VXHl3Dw+PT/f7nJCaiI97MeBJwzD2AhYwM1Hh0NERILhUGsv967bx5zJqfzLJaW4IvgSvlMZd2CbpjkIfD6AtYiIjKi738t3/rSbBHcMP//0XMff7ms8tJaIiIQ9n9/iB3/ZS11nP49ctzBibkgwVprpKCJh74H1FWyqbOe7F83krCkZdpdjGwW2iIS1339YyzPb6rhhSRHXLiq0uxxbKbBFJGy9ub+ZB96u4KLSSdx+QYnd5dhOgS0iYWlHbSc/XLuPBYXp/PgKg9iY6Loi5FQU2CISdg40e7jjT7vJT0/k/qvnReUVIaeiwBaRsFLZ2ss/P/8xie4Y/u+188lMirO7pLChwBaRsFHX2cdtz+8E4FfXLaQoI8nmisKLrsMWkbDQ0NXPrc99TL/XzyPXL2R6dmTd8TwQFNgiYrvajj5ufW4nXf1efnXdQkpzU+0uKSwpsEXEVpWtvdz6/E4GvX4evn4hc06xVKoMU2DLSYZ8ftp6h+gd9NE3NPwx4PXjckEMLlwuiI1xkeiOITXBTVqim9R4N/FunRKRsTnQ7OGfn/8YgEeuX8Ss3Mi823mgKLCjVLNngIrWXqraemnoGeJAQzdNngFaewbp7B/foosJ7hjyUuPJS0sgLzWBvLQECtMTmJadzLSsJHJS4qNudTUZ2dbqdu58eQ/JcbH86jqNWY+GAjsKeH1+dtV3s6Ouk90N3exu6KbZM3js6ynxsRRnJVGclcTiKRnkpMSTkxJPanwsiXGxJMXFkHDk7kGWZeGzLPx+6Pf66B7w0t3vwzPgpbN/iGbPIM2eAT6q66TJM4jXb520n5JJKczOS2V2Xipleakkx+sa22izdk8jP3ltP8VZSTx4zfyoXcxprBTYEaqpe4D15a28X9nOBzUd9Az6ACjOSuLsqZnMy09j1qQUpmUnMasok87OvoDX4LcsmroHqGrro6q9l8q2Pqraetl0qI2/7G4EwHWkptmTU1lYmM6iogxmTUrRrLYIZVkWT2yp4aGNlZw9NYN/+/Q80hIVQ6Oln1QEafEM8Ob+Fv66v5kddV0AFKYncNnsPJZNz2LJlIxTTkII1jBFjMtFfnoi+emJLJuedex5y7Jo6RlkX6OHfU0ezEYP22o7eW1fMzDcE19QmM5ZRemcVZTBvPw0zXSLAH1DPv71tf28bjZz+Zw87rm0TOc9xkiB7XA+v8XmynZe3FnPxopW/BbMmpTC186bxsWluUzLTgq7cWOXy0VuagK5qQmsmpkDDId4Q/cAO+o6+aiuix11nTzybhUwPDa+qDCdZdOyWDotk7K8VGLCrE1yerUdfdz58h7KW3r4n6tm8MVzp4TdcekECmyH8gx4eWlnPc/tOEx91wDZyXF88dypXDV3MjNynHfyxuVyUZCeSEF6IlfMmQxAV/8QH9V1sbW6g/er2vnlhkOwATIS3ZxbnMWyaZmsmJHN5LQEm6uX09lQ3sqPXjUBePCa+Syfnm1zRc6lwHaY9t5Bntl+mOe2H6Z7wMs5UzP4xvklXDArh7jYyPr3Mj0xjlUzc471wls8A2w5Et5bqjr46/7hIZTS3BRWlmSzsiSHeflpGv8OEwNeP798p4Jntx+mLDeF+z49lymZmmo+ES7Lss78qnFobu4OzoaDLDMzmY6OXrvLOIlnwMuTW2t4+sM6Brx+VpdO4ktLpzI3f+KTDMK1zadjWRblrb28V9HGxkNt7KzrxGdBZlIcn5iRxXkzslkxPXvEE1pObPNEhLq95S093P2XfRxs6eHzZxdx28oZIR+vdup7nJubNmKPQz3sMDfk8/PCR/U8tqmKzn4vl83O5Zbl0xw57BFILpeLWZNSmDUphRuXTqWrf4jNle1srGjj3Yo21u5pItYFi4oyWFmSzaqSnLAcz480Qz4/T2yp4bfvV5Ma7+aBa+Zz3gwNgQSKetgnCKe/yu9XtnPfmweo6ejnnOJMvnn+DGYHYdpuOLU5EHx+i131Xbx7qI2NFW0caO4BYEpmIitLclhVks3qeQX0evptrjR0QvEe767v4iev76e8pZdLjVzuuGgm2cnxQd3n6Tj1uD5dD1uBfYJweJNbewb5xdvlvLavmeKsJL594Uw+MT0raL3DcGhzMDV09bOxYji8t1a3M+izSEmIZfm0LFaWZHPejGyybAyWUAjme9zaM8jDGyt5eVcDuanxfO+SUs4/ct7BTk49rhXYY2Dnm2xZFq/sbuSBtyvo9/r40tKp/NPSYhKCPPbn1AN7PPqGfGyt7mBLbSdv7m2ipWcQFzC/IJ1VM7NZWZLNrEkpETd0Eoz3eNDr59ntdTy+uZp+r5/PLi7kKyumkZoQHiOtTj2uFdhjYNeb3NE7xE//eoC3DrSweEoGd11SyvQQjVM79cCeiMzMZNrbezCbPGyoaGNDeSt7Gz0A5KclDF91MjOHc6ZmBv0PZigE8j0e8vl5eVcDv9lcTZNnkJUl2dx+QQnTwmwtEKce1wrsMbDjTd5c2caPX91PR98Qt66czj+eMyWkE0OcemBPxKna3OIZODbuvbmynX6vn0R3DEunZbGqJJvzSrLJTXXmNd+BeI/7h3ys3dPIb9+voaF7gIWF6Xz1E9NYOi3rzN9sA6ce17pKJEz5/BYPv1vJf22pYUZOMg9cMx8jTwu322VSagJrFhSwZkEBA14/H9Z0sPFI7/ud8lYA5kxOZcWMbJYWZzK/ID0iet9n0tg9wPM7DvPSzno6+70sKEjjB5eWsmxa8M6ryKmph32CUP1V7ugd4u61e3m/qoOrF+Rzx4UzbVsvw6k9kYkYS5sty6K8pZcNFa1srGhjV30Xfuu/p8yfU5zJ0uJMjMlpuMN00s5Y3+Mhn3/48si9TbxzsAULOH9mDjcsKWLJlAxHBLVTj2sNiYxBKN7kvY3d3PmnPbT2DnLnRbO4emFBUPd3Jk49sCdiIm32DHjZVtvJ1uoOtla3U94yvJ3UhFgWF2WwsDCdBYXpYbVo1Wja6/X52V7Xyd/2t/CG2Uxnv5fs5DiunDuZ684qpDDDWUugOvW41pBIGHnrQAv3rN1HZlIcj95wFvMCMFNRQis1wc35M3OOXbrW2jPIhzUdbK3uYFttJxsq2gCIdUFZXioLCtKZX5hGWW4q07KTw6oXXt/Vz7aaTjZWtLGpso2eQR8J7hgumJnDlXMns2x6VljVG+0U2CH09LY6fvFWOfMK0rj/6nm2TiqQwMlJiefS2XlcOjsPGB7u+ri+a/jjcBev7G7gDzsOAxAf62JGTgqlucMf07OTmZqZREF6Au4grwXT0TvEwZYeDrb0sKu+ix11XTR2DwCQnRzHJWW5rCzJZum0LN1UIkxpSOQEwfg3ym9ZPPB2BU9vq2P1rBx+cuXssPlXGZz7r+NEhLLNXr9FZWsv+5s9HGju4WBzD/ubPbT1Dh17TawL8tMTmZKZSF5qwrG7/uSkxJOVFEdS/PCdf5LiYklyxxIb48JvWUc+YNDnp7vfS/eAF8+Al7beIRq6B2js6qehe4Cajn6ajoQzDP+RWXxkvfGzpmRQmpsScUvWOvW41pCIjbw+Pz9cZ/KG2cwNS4q4/YISrSYXZdwxLmblppx0g9nWnkGq2/uo6eijrqOP2o5+ajr6qGjtpa13CJ9/Yn2eGBfkpiZQkJ7AeTNzmJqeQGnu8Porur+mMymwg2jQ6+euP+9lfXkr3zh/Bl88d6rdJUkYOdqDXjwl46Sv+S2Lrj4vLb2DdPQO0Tvko3/o6F3s/fgtixiXixjX8J194mJdpCXGkZYQS1qCm8ykOCalJhwbf3Zqb1P+ngI7SPqHfNz58h42Vbbz3Ytmcf3iQrtLEgeJcbnITI4jM/nkW7pJ9FJgB0HfkI9vv7SLD2s6ufvSUtYssPeyPRGJDArsABvw+vnOH3ezrbaTH19pHLvdlYjIREX+vNoQ8vr8/ODPe9lS3cE9l5UprEUkoMbVwzYMIwZ4CFgEDABfNk3zYCALcxq/ZfGjV03Wl7fy3Ytm8g/z8u0uSUQizHh72FcDiaZprgC+D9wfsIocyLIsfv7mQV7b18ytK6dz/eIiu0sSkQg03jHslcCrAKZpbjYM45zAlWSf733v2zz55G8pLS3jwIH93HjjTQBnfO7cy65jV303PR+9ym//VMZdI7xurNsN9HOjff2NN97Efff9e9B/3hIYR49bu97D4/dv93Fzqt9hu2sKpHHNdDQM4zHgBdM01x15XA2UmKbpPfqavr5By+0On9l8o5GUlIDP5zv2ODZ2uP4zPYcrBpfLheU/w+vGuN1APzfa18fGxtLXN3DS90WS2NgYfD6/3WUExNHj9nTvYTDbe/z+7T5uTvU7bHdNYxUXFxvwmY5dwPGrFsUcH9YAHo+zfkgAN95405h6rlOnz6Sy4iAzVq7h/JIcnnoqMD3ccOhhR/oki0iaSHL0uD3dexjM9h6/f7t/pqf6Hba7prHKzR15Qbjx9rCvBT5lmuaXDMNYDtxrmuYVx78m0tcSqWrr5Zand5CZFMfjnzuLjCTnTnCIpPAarWhrc7S1F5zb5mCsJfIS8EnDMN4DXMBN49yOI3kGvNzxx93EuFw8cM18R4e1iDjHuALbNE0/8LUA1+IIfsvi3nUmtZ39PHTdAqZkJtldkohECU2cGaPHN1fzTnkr37qghCVTMu0uR0SiiAJ7DN4pb+XX71Vx1dw8LeYkIiGnwB6l6vY+frh2H7PzUvn+JaVaS1hEQk6BPQqD3uE1QtwxLn6+Zm5Y3S1GRKKHAnsUfrnhEPuaPNxzmUFBurPuHC0ikUOBfQbrD7byzLY6Pru4kAtm5dhdjohEMQX2aTR09fOT10yMvFS+cX6J3eWISJRTYI/A57f44dp9DPksfvoPc4h360clIvZSCo3g9x/Wsr2uizsvnkVxlibHiIj9FNincLClh4ffrWT1rByunJtndzkiIoAC+ySDXj8/WmeSGu/mrk/qemsRCR8K7BM8tL4cs8nDXZ8sJSs53u5yRESOUWAfZ3dDN4+8U8FVc/NYXTrJ7nJERP6OAvuIIZ+fn7xmMik1njsunGV3OSIiJxnvetgR56kPailv6eWRf1xCWqJ+LCISftTDZvjuMY9tquLisklcPFtXhYhIeIr6wLYsi5/99QDx7hi+c+FMu8sRERlR1Af2K7sa+bCmk2+cX8Kk1AS7yxERGVFUB3Zb7yAPvlPB4ikZrFmQb3c5IiKnFdWB/R/vHKJ30Mddl5QSowkyIhLmojawPz7cxSu7G/n82VOYnpNsdzkiImcUlYHt81v8298Okpsazy3Li+0uR0RkVKIysP+0q4G9jR6+eX4JyfG63ZeIOEPUBXZn3xAPbTjE4ikZXDo71+5yRERGLeoC+5F3K/EMePnuRTO1Ep+IOEpUBXZFaw8v7qzn2kWFlOam2l2OiMiYRFVg//KdQyTHx/KVFdPsLkVEZMyiJrC3VLWzsaKNm5cVk5kcZ3c5IiJjFhWB7bcsHlxfQUF6AtcvLrK7HBGRcYmKwF63p4n9zT3ctnIGCbr7uYg4VMSnV/+Qj4c2HmJufhqf1GV8IuJgER/Yz2yro8kzyDcvmKH1QkTE0SI6sLv7vTy5tZaVJdksmZJpdzkiIhMS0YH91Ie1dA94+dp50+0uRURkwiI2sNt7B3nmwzouKcvFyNMkGRFxvogN7Ce21NDv9fHVT2iSjIhEhogM7KbuAZ7fcZgr507WWtciEjEiMrB/8341fgtNQReRiBJxgV3b0ccfP25gzYJ8CjMS7S5HRCRg3OP5JsMwXEAtcODIU5tM0/yXgFU1AU+8X4M7xqU7yYhIxBlXYAMzgW2maX4qkMVMVH1XP3/e08j/WFRAbmqC3eWIiATUeAP7bKDIMIy3gD7gW6ZpmoEra3ye3FKDC/jCOVPsLkVEJODOGNiGYdwCfOuEp28Dfmaa5nOGYawEngLOPf4FqakJuN2hu19iY1c/L+9u5NolRcwuzh73dmJjY8jMjK4rS9TmyBdt7YXIbLPLsqwxf5NhGMmA1zTNwSOPDwNFpmke21hzc/fYNzwBv3i7nGe31fH8zecyJTNp3NvJzEymo6M3gJWFP7U58kVbe8G5bc7NTRtx0aPxXiVyL3A7gGEYi4Dq48M61Np7B3nxo3oum5M3obAWEQln4x3D/t/AU4ZhXAV4gS8FrKJxeHpbHQNePzct1ZUhIhK5xhXYpmm2A1cFuJZx6eof4g/bD3NxWa5mNYpIRHP8xJnndhymZ9DHTcum2l2KiEhQOTqwB7x+/rD9MJ+YkUWZVuQTkQjn6MBeu6eRtt4hXXctIlHBsYHttyx+90Ets/NSOWdqpt3liIgEnWMDe0N5G1XtfXzhnCm4dK9GEYkCjg3s331QQ35aAheXTbK7FBGRkHBkYO+q72J7XRefO7sId6wjmyAiMmaOTLunPqglNSGWNQvy7S5FRCRkHBfYtR19vHWghWsXFZISP96JmiIizuO4wH52+2FiXC4+u7jQ7lJERELKUYHdM+jllV0NXFw2STcoEJGo46jA/svuJnoGfdywpMjuUkREQs4xge23LJ7bUcfc/DTmF6TbXY6ISMg5JrC3VLVT2dansWsRiVqOCexntx8mOzmOS8py7S5FRMQWjgjs2o4+3q1o4zMLC4h3O6JkEZGAc0T6PbfjMDExLq5dVGB3KSIitgn7wO4d9PHyrgYuLtWlfCIS3cI+sF/d24hnwMf1OtkoIlEurAPbsixe3NlAaW4KCwt1KZ+IRLewDuw9jR7MJg/XLCzQmtciEvXCOrBf+qiepLgYLp+TZ3cpIiK2C9vA9gx4eW1fE5fOziM1QavyiYiEbWCv29tEv9fPNQt1KZ+ICIRpYFuWxUs765mdl8rc/DS7yxERCQthGdi76rs50NzDZzRRRkTkmLAM7Bd31pMcF8tls7VuiIjIUWEX2N39Xt4wm7l8Tp5uASYicpywC+zqjj4sy9K6ISIiJwi7Luy8/DRev3WFetciIicIux42oLAWETmFsAxsERE5mQJbRMQhFNgiIg6hwBYRcQgFtoiIQyiwRUQcQoEtIuIQLsuy7K5BRERGQT1sERGHUGCLiDiEAltExCG0aMcJDMPIAJ4C0oF44NumaW6yt6rQMAzjM8B1pml+3u5agsEwjBjgIWARMAB82TTNg/ZWFRqGYSwD7jNNc7XdtQSbYRhxwG+A6UAC8K+mab5sa1EBoh72yb4NvGma5gXAl4Bf2VtOaBiG8SDwMyL7mLgaSDRNcwXwfeB+e8sJDcMw7gQeAxLtriVEvgC0mqa5CrgC+A+b6wmYSP7lHK9fAP955HM30G9jLaH0HvB1u4sIspXAqwCmaW4GzrG3nJApB66xu4gQeg6457jHXrsKCbSoHhIxDOMW4FsnPH2TaZpbDcPIZ3ho5PaQFxZEp2nzs4ZhrLahpFBKBzqPe+wzDMNtmmbE/EKfimmaLxiGMd3uOkLFNE0PgGEYacDzwN32VhQ4UR3Ypmk+Djx+4vOGYSwAngG+Y5rm+pAXFkQjtTlKdAFpxz2OifSwjlaGYUwFXgIeMk3z93bXEygaEjmBYRhzGf6X6vOmaa6zux4JqHeBKwEMw1gOfGxvORIMhmFMBl4Hvmea5m/srieQorqHPYKfMXxy5kHDMAA6TdNcY29JEiAvAZ80DOM9wAXcZHM9Ehx3AVnAPYZhHB3LvsI0zT4bawoITU0XEXEIDYmIiDiEAltExCEU2CIiDqHAFhFxCAW2iIhDKLBFRBxCgS0i4hAKbBERh/j/FDtCsiQoLSYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "x = np.linspace(-2.5,2.5,1000)\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "pts = z[good_bool]\n",
    "\n",
    "ax.plot(pts, np.zeros(len(pts)), 'ko', markersize=2);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08d53807-fc71-4c70-a426-078cf9ce2efa",
   "metadata": {},
   "source": [
    "Let's make a similar plot for the bad points.  Here is a reminder of how `good_bool` looked."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "559b07a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False,  True,\n",
       "        True, False, False, False, False, False, False, False, False,\n",
       "       False, False, False,  True, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False,  True,\n",
       "        True])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "good_bool"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35e3c73c-efe4-4ce6-ac91-2d9b979dffee",
   "metadata": {},
   "source": [
    "To take the elementwise negation of this Boolean array in Numpy, we use the tilde symbol `~`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "29b98c71",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "       False, False, False, False, False, False, False, False, False,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True, False,\n",
       "       False,  True,  True,  True,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True, False,  True,  True,  True,  True,  True,\n",
       "        True,  True,  True,  True,  True,  True,  True,  True, False,\n",
       "       False])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "~good_bool"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a30878b-abd6-496e-bd95-16d87ccbcb99",
   "metadata": {},
   "source": [
    "Here we use Boolean indexing with `~good_bool` to keep only the bad values from `z`.  We also change the color from black to red, by changing `k` to `r`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "42731661",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "x = np.linspace(-2.5,2.5,1000)\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "pts = z[~good_bool]\n",
    "\n",
    "ax.plot(pts, np.zeros(len(pts)), 'ro', markersize=2);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc9e19ab-8776-4cd6-8abd-132470bafd79",
   "metadata": {},
   "source": [
    "Here is the same plot from above with the good values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "4f8815fe",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAWwAAAD0CAYAAAC/3RwjAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAAAsTAAALEwEAmpwYAAAfPUlEQVR4nO3deXQc5Z3u8W9LrX2XLFmLLduypfJubMBLsMEsYZ3EBC6EZBImQHKSwNyEhIRkCITkZG5yyVwmcDMBJkDCcEmAsCWQ2CwhYGywscE2xlvZlqzV2vfW3t11/5DtcWzL1tLd1dX9fM7ROepWq+r3qkuPXr1V71suy7IQEZHwF2N3ASIiMjoKbBERh1Bgi4g4hAJbRMQhFNgiIg6hwBYRcQh3sDbc3NztyOsFU1MT8HgG7C4jpNTmyBdt7QXntjk3N8010tfUwz6B2x1rdwkhpzZHvmhrL0RmmxXYIiIOocAWEXEIBbaIiEMosEVEHEKBLSLiEApsERGHUGCLiDiEAltEJECGfH5ufGobb5jNQdm+AltEJEA2lLeyt9FDUlxwolWBLSISIC993EBeajwrpmcHZfsKbBGRAKjt6OP9ynbWLMgnNmbE5UAmRIEtIhIAf/y4gRgXXL2gIGj7UGCLiEzQoNfPyx83sGpmDnlpCUHbjwJbRGSC3j7YQnvfENcsCl7vGhTYIiIT9vxH9RRlJLJsWlZQ96PAFhGZgIrWHrbXdnLNwgJiXME52XiUAltEZAJe/KieuFgXn5o/Oej7UmCLiIxT35CPv+xp5KLSSWQlxwd9fwpsEZFxen1fE54BH9cuKgzJ/hTYIiLjYFkWL3xUT0lOMmcVpYdknwpsEZFx2Hm4i72NHq47qxBXkE82HqXAFhEZh2e2HSYtwc1V84J/svEoBbaIyBg1dPXz1oFm1izIJykuNmT7VWCLiIzR8x/VYwHXnRWak41HKbBFRMagf8jHH3fWc/7MHAozEkO6bwW2iMgYrNvbRGe/lxuWFIV83wpsEZFRsiyLZ7bVUZqbwpIpGSHfvwJbRGSUPqjpoKK1lxuWFIXsUr7jKbBFREbp6Q/ryEqK47LZebbs3z3aFxqGsQy4zzTN1YZhzAKeACxgF3CbaZr+4JQoImK/ytZeNlS08eXlxSS47enrjmqvhmHcCTwGHD0l+u/A3aZprgJcwJrglCciEh7+3wc1JLhjuH5xaC/lO95o/0yUA9cc9/hsYP2Rz9cBlwSyKBGRcNLsGWDtniY+NW9ySFblG8mohkRM03zBMIzpxz3lMk3TOvJ5N3DS6dLU1ATc7tDNAAqU2NgYMjOT7S4jpNTmyBdt7YXAtvnX79fgtyy+flGprT/HUY9hn+D48eo0oOPEF3g8A+PctL0yM5Pp6Oi1u4yQUpsjX7S1FwLXZs+Al99vqebislzSYwj6zzE3N23Er4135Hy7YRirj3x+BbBhnNsREQlrL+2sp2fQxxfPnWJ3KePuYd8BPGoYRjywF3g+cCWJiISHQa+f339Yx7nFmcyZPHLPN1RGHdimaVYCy498vh+4IEg1iYiEhVf3NtHSM8i9l5fZXQqgiTMiIqfk81s8ubWG0twUlk3LsrscQIEtInJKb+5vpqq9j1uWF9syDf1UFNgiIifwWxaPba5mRk4yF5ZOsrucYxTYIiIn+Nv+Fg619vLl5cXEhEnvGhTYIiJ/x29ZPL65mmlZSVxclmt3OX9HgS0icpz1B1s52NLDzcuLiY0Jn941KLBFRI6xLIvHNlUxNTORS21aQvV0FNgiIkdsqGhjf3MPNy0rxh1mvWtQYIuIAMNj14++V0VhRiJXzAm/3jUosEVEAHjrQAv7mjx8ZUUx7tjwjMbwrEpEJIS8fouHN1YyIyeZK+ZMtrucESmwRSTqrd3TSFV7H18/b3rYXRlyPAW2iES1Qa+fR9+rYm5+Gqtn5dhdzmkpsEUkqr24s56G7gFuXTk9bNYMGYkCW0SiVu+gj99sruac4sywWZHvdBTYIhK1fvdBLe19Q9x63nS7SxkVBbaIRKVmzwBPbq3hkrJJLChMt7ucUVFgi0hUenhjJT7L4rZVM+wuZdQU2CISdcwmD3/e3chnFxcxJTPJ7nJGTYEtIlHFsiweWF9BeqKbm5cV213OmCiwRSSqbKxo44PqDr6yYhppiaO+D3lYUGCLSNTw+vw8uL6C4qwkrl1UYHc5Y6bAFpGo8fS2Oqra+7j9gpKwXeDpdJxXsYjIODR1D/DopipWlmSzamZ4T0EfiQJbRKLCg+sr8Pkt7rhwpt2ljJsCW0Qi3gfVHbxuNvNPS6c66jK+EymwRSSieX1+fv7mQQozErnx3Kl2lzMhCmwRiWhPb6vjUFsvd1w4k8S4WLvLmRAFtohErNqOPv7zveETjec79ETj8RTYIhKRLMviZ28cwB3j4nsXz7K7nIBQYItIRHppx2G2VHdw26oZ5Kcn2l1OQCiwRSTitPYM8tN1+1hUmO7IGY0jUWCLSMT5P38rp3fQy92XlhET5rf9GgsFtohElLcPtPDX/c3ctnoW03OS7S4noBTYIhIx2noH+ekbByjLTeErK51zY4LRctbagiIiI7Asi//1+gF6Br08fOVC4t0x9NpdVICphy0iEeGVXY28U97KrStnMHNSit3lBIUCW0Qcr66zj/vfKufsqRl87uwiu8sJGgW2iDiaz2/x43UmLhfce7kRUVeFnGhCY9iGYWwHOo88PGSa5k0TL0lEZPQe21TF9roufnS5QUGETJAZybgD2zCMRADTNFcHrBoRkTHYUtXO45uruWreZK6aN9nucoJuIj3sRUCyYRivH9nOXaZpbg5MWSIip9fSM8g9a/cxPTs5YtYKOROXZVnj+kbDMBYAy4HHgFJgHWCYpukF6OsbtNxu5y1lGBsbg8/nt7uMkFKbI1+ktdfnt7j5vz5gW007L3x1BWWT0056jVPbHBcXO+Ig/ER62PuBg6ZpWsB+wzBagQKgBsDjGZjApu2TmZlMR0ekXb15empz5Iu09j76XhXvVbRy96Wl5CXEnrJtTm1zbu7Jf3yOmshVIjcD9wMYhlEIpAP1E9ieiMgZrT/Yyq83VXHl3Dw+PT/f7nJCaiI97MeBJwzD2AhYwM1Hh0NERILhUGsv967bx5zJqfzLJaW4IvgSvlMZd2CbpjkIfD6AtYiIjKi738t3/rSbBHcMP//0XMff7ms8tJaIiIQ9n9/iB3/ZS11nP49ctzBibkgwVprpKCJh74H1FWyqbOe7F83krCkZdpdjGwW2iIS1339YyzPb6rhhSRHXLiq0uxxbKbBFJGy9ub+ZB96u4KLSSdx+QYnd5dhOgS0iYWlHbSc/XLuPBYXp/PgKg9iY6Loi5FQU2CISdg40e7jjT7vJT0/k/qvnReUVIaeiwBaRsFLZ2ss/P/8xie4Y/u+188lMirO7pLChwBaRsFHX2cdtz+8E4FfXLaQoI8nmisKLrsMWkbDQ0NXPrc99TL/XzyPXL2R6dmTd8TwQFNgiYrvajj5ufW4nXf1efnXdQkpzU+0uKSwpsEXEVpWtvdz6/E4GvX4evn4hc06xVKoMU2DLSYZ8ftp6h+gd9NE3NPwx4PXjckEMLlwuiI1xkeiOITXBTVqim9R4N/FunRKRsTnQ7OGfn/8YgEeuX8Ss3Mi823mgKLCjVLNngIrWXqraemnoGeJAQzdNngFaewbp7B/foosJ7hjyUuPJS0sgLzWBvLQECtMTmJadzLSsJHJS4qNudTUZ2dbqdu58eQ/JcbH86jqNWY+GAjsKeH1+dtV3s6Ouk90N3exu6KbZM3js6ynxsRRnJVGclcTiKRnkpMSTkxJPanwsiXGxJMXFkHDk7kGWZeGzLPx+6Pf66B7w0t3vwzPgpbN/iGbPIM2eAT6q66TJM4jXb520n5JJKczOS2V2Xipleakkx+sa22izdk8jP3ltP8VZSTx4zfyoXcxprBTYEaqpe4D15a28X9nOBzUd9Az6ACjOSuLsqZnMy09j1qQUpmUnMasok87OvoDX4LcsmroHqGrro6q9l8q2Pqraetl0qI2/7G4EwHWkptmTU1lYmM6iogxmTUrRrLYIZVkWT2yp4aGNlZw9NYN/+/Q80hIVQ6Oln1QEafEM8Ob+Fv66v5kddV0AFKYncNnsPJZNz2LJlIxTTkII1jBFjMtFfnoi+emJLJuedex5y7Jo6RlkX6OHfU0ezEYP22o7eW1fMzDcE19QmM5ZRemcVZTBvPw0zXSLAH1DPv71tf28bjZz+Zw87rm0TOc9xkiB7XA+v8XmynZe3FnPxopW/BbMmpTC186bxsWluUzLTgq7cWOXy0VuagK5qQmsmpkDDId4Q/cAO+o6+aiuix11nTzybhUwPDa+qDCdZdOyWDotk7K8VGLCrE1yerUdfdz58h7KW3r4n6tm8MVzp4TdcekECmyH8gx4eWlnPc/tOEx91wDZyXF88dypXDV3MjNynHfyxuVyUZCeSEF6IlfMmQxAV/8QH9V1sbW6g/er2vnlhkOwATIS3ZxbnMWyaZmsmJHN5LQEm6uX09lQ3sqPXjUBePCa+Syfnm1zRc6lwHaY9t5Bntl+mOe2H6Z7wMs5UzP4xvklXDArh7jYyPr3Mj0xjlUzc471wls8A2w5Et5bqjr46/7hIZTS3BRWlmSzsiSHeflpGv8OEwNeP798p4Jntx+mLDeF+z49lymZmmo+ES7Lss78qnFobu4OzoaDLDMzmY6OXrvLOIlnwMuTW2t4+sM6Brx+VpdO4ktLpzI3f+KTDMK1zadjWRblrb28V9HGxkNt7KzrxGdBZlIcn5iRxXkzslkxPXvEE1pObPNEhLq95S093P2XfRxs6eHzZxdx28oZIR+vdup7nJubNmKPQz3sMDfk8/PCR/U8tqmKzn4vl83O5Zbl0xw57BFILpeLWZNSmDUphRuXTqWrf4jNle1srGjj3Yo21u5pItYFi4oyWFmSzaqSnLAcz480Qz4/T2yp4bfvV5Ma7+aBa+Zz3gwNgQSKetgnCKe/yu9XtnPfmweo6ejnnOJMvnn+DGYHYdpuOLU5EHx+i131Xbx7qI2NFW0caO4BYEpmIitLclhVks3qeQX0evptrjR0QvEe767v4iev76e8pZdLjVzuuGgm2cnxQd3n6Tj1uD5dD1uBfYJweJNbewb5xdvlvLavmeKsJL594Uw+MT0raL3DcGhzMDV09bOxYji8t1a3M+izSEmIZfm0LFaWZHPejGyybAyWUAjme9zaM8jDGyt5eVcDuanxfO+SUs4/ct7BTk49rhXYY2Dnm2xZFq/sbuSBtyvo9/r40tKp/NPSYhKCPPbn1AN7PPqGfGyt7mBLbSdv7m2ipWcQFzC/IJ1VM7NZWZLNrEkpETd0Eoz3eNDr59ntdTy+uZp+r5/PLi7kKyumkZoQHiOtTj2uFdhjYNeb3NE7xE//eoC3DrSweEoGd11SyvQQjVM79cCeiMzMZNrbezCbPGyoaGNDeSt7Gz0A5KclDF91MjOHc6ZmBv0PZigE8j0e8vl5eVcDv9lcTZNnkJUl2dx+QQnTwmwtEKce1wrsMbDjTd5c2caPX91PR98Qt66czj+eMyWkE0OcemBPxKna3OIZODbuvbmynX6vn0R3DEunZbGqJJvzSrLJTXXmNd+BeI/7h3ys3dPIb9+voaF7gIWF6Xz1E9NYOi3rzN9sA6ce17pKJEz5/BYPv1vJf22pYUZOMg9cMx8jTwu322VSagJrFhSwZkEBA14/H9Z0sPFI7/ud8lYA5kxOZcWMbJYWZzK/ID0iet9n0tg9wPM7DvPSzno6+70sKEjjB5eWsmxa8M6ryKmph32CUP1V7ugd4u61e3m/qoOrF+Rzx4UzbVsvw6k9kYkYS5sty6K8pZcNFa1srGhjV30Xfuu/p8yfU5zJ0uJMjMlpuMN00s5Y3+Mhn3/48si9TbxzsAULOH9mDjcsKWLJlAxHBLVTj2sNiYxBKN7kvY3d3PmnPbT2DnLnRbO4emFBUPd3Jk49sCdiIm32DHjZVtvJ1uoOtla3U94yvJ3UhFgWF2WwsDCdBYXpYbVo1Wja6/X52V7Xyd/2t/CG2Uxnv5fs5DiunDuZ684qpDDDWUugOvW41pBIGHnrQAv3rN1HZlIcj95wFvMCMFNRQis1wc35M3OOXbrW2jPIhzUdbK3uYFttJxsq2gCIdUFZXioLCtKZX5hGWW4q07KTw6oXXt/Vz7aaTjZWtLGpso2eQR8J7hgumJnDlXMns2x6VljVG+0U2CH09LY6fvFWOfMK0rj/6nm2TiqQwMlJiefS2XlcOjsPGB7u+ri+a/jjcBev7G7gDzsOAxAf62JGTgqlucMf07OTmZqZREF6Au4grwXT0TvEwZYeDrb0sKu+ix11XTR2DwCQnRzHJWW5rCzJZum0LN1UIkxpSOQEwfg3ym9ZPPB2BU9vq2P1rBx+cuXssPlXGZz7r+NEhLLNXr9FZWsv+5s9HGju4WBzD/ubPbT1Dh17TawL8tMTmZKZSF5qwrG7/uSkxJOVFEdS/PCdf5LiYklyxxIb48JvWUc+YNDnp7vfS/eAF8+Al7beIRq6B2js6qehe4Cajn6ajoQzDP+RWXxkvfGzpmRQmpsScUvWOvW41pCIjbw+Pz9cZ/KG2cwNS4q4/YISrSYXZdwxLmblppx0g9nWnkGq2/uo6eijrqOP2o5+ajr6qGjtpa13CJ9/Yn2eGBfkpiZQkJ7AeTNzmJqeQGnu8Porur+mMymwg2jQ6+euP+9lfXkr3zh/Bl88d6rdJUkYOdqDXjwl46Sv+S2Lrj4vLb2DdPQO0Tvko3/o6F3s/fgtixiXixjX8J194mJdpCXGkZYQS1qCm8ykOCalJhwbf3Zqb1P+ngI7SPqHfNz58h42Vbbz3Ytmcf3iQrtLEgeJcbnITI4jM/nkW7pJ9FJgB0HfkI9vv7SLD2s6ufvSUtYssPeyPRGJDArsABvw+vnOH3ezrbaTH19pHLvdlYjIREX+vNoQ8vr8/ODPe9lS3cE9l5UprEUkoMbVwzYMIwZ4CFgEDABfNk3zYCALcxq/ZfGjV03Wl7fy3Ytm8g/z8u0uSUQizHh72FcDiaZprgC+D9wfsIocyLIsfv7mQV7b18ytK6dz/eIiu0sSkQg03jHslcCrAKZpbjYM45zAlWSf733v2zz55G8pLS3jwIH93HjjTQBnfO7cy65jV303PR+9ym//VMZdI7xurNsN9HOjff2NN97Efff9e9B/3hIYR49bu97D4/dv93Fzqt9hu2sKpHHNdDQM4zHgBdM01x15XA2UmKbpPfqavr5By+0On9l8o5GUlIDP5zv2ODZ2uP4zPYcrBpfLheU/w+vGuN1APzfa18fGxtLXN3DS90WS2NgYfD6/3WUExNHj9nTvYTDbe/z+7T5uTvU7bHdNYxUXFxvwmY5dwPGrFsUcH9YAHo+zfkgAN95405h6rlOnz6Sy4iAzVq7h/JIcnnoqMD3ccOhhR/oki0iaSHL0uD3dexjM9h6/f7t/pqf6Hba7prHKzR15Qbjx9rCvBT5lmuaXDMNYDtxrmuYVx78m0tcSqWrr5Zand5CZFMfjnzuLjCTnTnCIpPAarWhrc7S1F5zb5mCsJfIS8EnDMN4DXMBN49yOI3kGvNzxx93EuFw8cM18R4e1iDjHuALbNE0/8LUA1+IIfsvi3nUmtZ39PHTdAqZkJtldkohECU2cGaPHN1fzTnkr37qghCVTMu0uR0SiiAJ7DN4pb+XX71Vx1dw8LeYkIiGnwB6l6vY+frh2H7PzUvn+JaVaS1hEQk6BPQqD3uE1QtwxLn6+Zm5Y3S1GRKKHAnsUfrnhEPuaPNxzmUFBurPuHC0ikUOBfQbrD7byzLY6Pru4kAtm5dhdjohEMQX2aTR09fOT10yMvFS+cX6J3eWISJRTYI/A57f44dp9DPksfvoPc4h360clIvZSCo3g9x/Wsr2uizsvnkVxlibHiIj9FNincLClh4ffrWT1rByunJtndzkiIoAC+ySDXj8/WmeSGu/mrk/qemsRCR8K7BM8tL4cs8nDXZ8sJSs53u5yRESOUWAfZ3dDN4+8U8FVc/NYXTrJ7nJERP6OAvuIIZ+fn7xmMik1njsunGV3OSIiJxnvetgR56kPailv6eWRf1xCWqJ+LCISftTDZvjuMY9tquLisklcPFtXhYhIeIr6wLYsi5/99QDx7hi+c+FMu8sRERlR1Af2K7sa+bCmk2+cX8Kk1AS7yxERGVFUB3Zb7yAPvlPB4ikZrFmQb3c5IiKnFdWB/R/vHKJ30Mddl5QSowkyIhLmojawPz7cxSu7G/n82VOYnpNsdzkiImcUlYHt81v8298Okpsazy3Li+0uR0RkVKIysP+0q4G9jR6+eX4JyfG63ZeIOEPUBXZn3xAPbTjE4ikZXDo71+5yRERGLeoC+5F3K/EMePnuRTO1Ep+IOEpUBXZFaw8v7qzn2kWFlOam2l2OiMiYRFVg//KdQyTHx/KVFdPsLkVEZMyiJrC3VLWzsaKNm5cVk5kcZ3c5IiJjFhWB7bcsHlxfQUF6AtcvLrK7HBGRcYmKwF63p4n9zT3ctnIGCbr7uYg4VMSnV/+Qj4c2HmJufhqf1GV8IuJgER/Yz2yro8kzyDcvmKH1QkTE0SI6sLv7vTy5tZaVJdksmZJpdzkiIhMS0YH91Ie1dA94+dp50+0uRURkwiI2sNt7B3nmwzouKcvFyNMkGRFxvogN7Ce21NDv9fHVT2iSjIhEhogM7KbuAZ7fcZgr507WWtciEjEiMrB/8341fgtNQReRiBJxgV3b0ccfP25gzYJ8CjMS7S5HRCRg3OP5JsMwXEAtcODIU5tM0/yXgFU1AU+8X4M7xqU7yYhIxBlXYAMzgW2maX4qkMVMVH1XP3/e08j/WFRAbmqC3eWIiATUeAP7bKDIMIy3gD7gW6ZpmoEra3ye3FKDC/jCOVPsLkVEJODOGNiGYdwCfOuEp28Dfmaa5nOGYawEngLOPf4FqakJuN2hu19iY1c/L+9u5NolRcwuzh73dmJjY8jMjK4rS9TmyBdt7YXIbLPLsqwxf5NhGMmA1zTNwSOPDwNFpmke21hzc/fYNzwBv3i7nGe31fH8zecyJTNp3NvJzEymo6M3gJWFP7U58kVbe8G5bc7NTRtx0aPxXiVyL3A7gGEYi4Dq48M61Np7B3nxo3oum5M3obAWEQln4x3D/t/AU4ZhXAV4gS8FrKJxeHpbHQNePzct1ZUhIhK5xhXYpmm2A1cFuJZx6eof4g/bD3NxWa5mNYpIRHP8xJnndhymZ9DHTcum2l2KiEhQOTqwB7x+/rD9MJ+YkUWZVuQTkQjn6MBeu6eRtt4hXXctIlHBsYHttyx+90Ets/NSOWdqpt3liIgEnWMDe0N5G1XtfXzhnCm4dK9GEYkCjg3s331QQ35aAheXTbK7FBGRkHBkYO+q72J7XRefO7sId6wjmyAiMmaOTLunPqglNSGWNQvy7S5FRCRkHBfYtR19vHWghWsXFZISP96JmiIizuO4wH52+2FiXC4+u7jQ7lJERELKUYHdM+jllV0NXFw2STcoEJGo46jA/svuJnoGfdywpMjuUkREQs4xge23LJ7bUcfc/DTmF6TbXY6ISMg5JrC3VLVT2dansWsRiVqOCexntx8mOzmOS8py7S5FRMQWjgjs2o4+3q1o4zMLC4h3O6JkEZGAc0T6PbfjMDExLq5dVGB3KSIitgn7wO4d9PHyrgYuLtWlfCIS3cI+sF/d24hnwMf1OtkoIlEurAPbsixe3NlAaW4KCwt1KZ+IRLewDuw9jR7MJg/XLCzQmtciEvXCOrBf+qiepLgYLp+TZ3cpIiK2C9vA9gx4eW1fE5fOziM1QavyiYiEbWCv29tEv9fPNQt1KZ+ICIRpYFuWxUs765mdl8rc/DS7yxERCQthGdi76rs50NzDZzRRRkTkmLAM7Bd31pMcF8tls7VuiIjIUWEX2N39Xt4wm7l8Tp5uASYicpywC+zqjj4sy9K6ISIiJwi7Luy8/DRev3WFetciIicIux42oLAWETmFsAxsERE5mQJbRMQhFNgiIg6hwBYRcQgFtoiIQyiwRUQcQoEtIuIQLsuy7K5BRERGQT1sERGHUGCLiDiEAltExCG0aMcJDMPIAJ4C0oF44NumaW6yt6rQMAzjM8B1pml+3u5agsEwjBjgIWARMAB82TTNg/ZWFRqGYSwD7jNNc7XdtQSbYRhxwG+A6UAC8K+mab5sa1EBoh72yb4NvGma5gXAl4Bf2VtOaBiG8SDwMyL7mLgaSDRNcwXwfeB+e8sJDcMw7gQeAxLtriVEvgC0mqa5CrgC+A+b6wmYSP7lHK9fAP955HM30G9jLaH0HvB1u4sIspXAqwCmaW4GzrG3nJApB66xu4gQeg6457jHXrsKCbSoHhIxDOMW4FsnPH2TaZpbDcPIZ3ho5PaQFxZEp2nzs4ZhrLahpFBKBzqPe+wzDMNtmmbE/EKfimmaLxiGMd3uOkLFNE0PgGEYacDzwN32VhQ4UR3Ypmk+Djx+4vOGYSwAngG+Y5rm+pAXFkQjtTlKdAFpxz2OifSwjlaGYUwFXgIeMk3z93bXEygaEjmBYRhzGf6X6vOmaa6zux4JqHeBKwEMw1gOfGxvORIMhmFMBl4Hvmea5m/srieQorqHPYKfMXxy5kHDMAA6TdNcY29JEiAvAZ80DOM9wAXcZHM9Ehx3AVnAPYZhHB3LvsI0zT4bawoITU0XEXEIDYmIiDiEAltExCEU2CIiDqHAFhFxCAW2iIhDKLBFRBxCgS0i4hAKbBERh/j/FDtCsiQoLSYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "x = np.linspace(-2.5,2.5,1000)\n",
    "ax.plot(x, f(x))\n",
    "\n",
    "pts = z[good_bool]\n",
    "\n",
    "ax.plot(pts, np.zeros(len(pts)), 'ko', markersize=2);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a56324c-7275-403a-bbde-965bdcf9b5cc",
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   "source": [
    "The plots say that, initially, all the starting values near -2 work well.  Once we pass the local maximum of $f$, after which Newton's Method starts moving us away from the root, we have mostly bad values.  There are a few outliers (good points surrounded by bad points).  It's worth looking at these outliers and thinking about the corresponding tangent lines.  Can you see why Newton's Method could work well for these particular values?  If we used more than 7 repetitions, we would see many more \"good points\" scattered throughout the bad points."
   ]
  }
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