Richardson_Extrapolation.ipynb

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        "Before you turn this problem in, make sure everything runs as expected. First, **restart the kernel** (in the menubar, select Kernel$\\rightarrow$Restart) and then **run all cells** (in the menubar, select Cell$\\rightarrow$Run All).\n",
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        "NAME = \"Prabal Chowdhury\"\n",
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        "---"
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      "source": [
        "## CSE330 Lab: Richardson Extrapolation\n",
        "---"
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        "## Instructions\n",
        "\n",
        "\n",
        "Today's assignment is to:\n",
        "1. Implement Richardson Extrapolation method using Python\n",
        "\n",
        "## Richardson Extrapolation:\n",
        "We used central difference method to calculate derivatives of functions last task. In this task we will use Richardson extrapolation to get a more accurate result.\n",
        "Let,\n",
        "$$ D_h = \\frac{f(x_1+h) -f(x_1-h)}{2h}\\tag{5.1}$$ \n",
        "\n",
        "\n",
        "General Taylor Series formula:\n",
        "$$ f(x) = f(x_1) + f'(x_1)(x - x_1) + \\frac{f''(x_1)}{2}(x - x_1)^2+... $$\n",
        "Using Taylor's theorem to expand we get,\n",
        "\n",
        "\\begin{align}\n",
        "f(x_1+h) &= f(x_1) + f^{\\prime}(x_1)h + \\frac{f^{\\prime \\prime}(x_1)}{2}h^2 + \\frac{f^{\\prime \\prime \\prime}(x_1)}{3!}h^3 + \\frac{f^{(4)}(x_1)}{4!}h^4 + \\frac{f^{(5)}(x_1)}{5!}h^5 + O(h^6)\\tag{5.2} \\\\\n",
        "f(x_1-h) &= f(x_1) - f^{\\prime}(x_1)h + \\frac{f^{\\prime \\prime}(x_1)}{2}h^2 - \\frac{f^{\\prime \\prime \\prime}(x_1)}{3!}h^3 + \\frac{f^{(4)}(x_1)}{4!}h^4 - \\frac{f^{(5)}(x_1)}{5!}h^5 + O(h^6)\\tag{5.3}\n",
        "\\end{align}\n",
        "Subtracting $5.3$ from $5.2$ we get,\n",
        "$$ f(x_1+h) - f(x_1-h) = 2f^{\\prime}(x_1)h + 2\\frac{f^{\\prime \\prime \\prime}(x_1)}{3!}h^3 + 2\\frac{f^{(5)}(x_1)}{5!}h^5 + O(h^7)\\tag{5.4}$$\n",
        "So,\n",
        "\\begin{align}\n",
        "D_h &= \\frac{f(x_1+h) - f(x_1-h)}{2h} \\\\\n",
        "&= \\frac{1}{2h} \\left( 2f^{\\prime}(x_1)h + 2\\frac{f^{\\prime \\prime \\prime}(x_1)}{3!}h^3 + 2\\frac{f^{(5)}(x_1)}{5!}h^5 + O(h^7) \\right) \\\\\n",
        "&= f^{\\prime}(x_1) + \\frac{f^{\\prime \\prime \\prime}(x_1)}{6}h^2 + \\frac{f^{(5)}(x_1)}{120}h^4 + O(h^6) \\tag{5.5}\n",
        "\\end{align}\n",
        "We get our derivative $f'(x)$ plus some error terms of order $>= 2$ Now, we want to bring our error order down to 4.\n",
        "\n",
        "If we use $h, \\text{and} \\frac{h}{2}$ as step size in $5.5$, we get,\n",
        "\\begin{align}\n",
        "D_h &= f^{\\prime}(x_1) + f^{\\prime \\prime \\prime}(x_1)\\frac{h^2}{6} + f^{(5)}(x_1) \\frac{h^4}{120} + O(h^6) \\tag{5.6} \\\\\n",
        "D_{h/2} &= f^{\\prime}(x_1) + f^{\\prime \\prime \\prime}(x_1)\\frac{h^2}{2^2 . 6} + f^{(5)}(x_1) \\frac{h^4}{2^4 . 120} + O(h^6) \\tag{5.7}\n",
        "\\end{align}\n",
        "Multiplying $5.7$ by $4$ and subtracting from $5.6$ we get,\n",
        "\\begin{align}\n",
        "D_h - 4D_{h/2} &= -3f^{\\prime}(x) + f^{(5)}(x_1) \\frac{h^4}{160} + O(h^6)\\\\ \n",
        "\\Longrightarrow D^{(1)}_h = \\frac{4D_{h/2} - D_h}{3} &= f^{\\prime}(x) - f^{(5)}(x_1) \\frac{h^4}{480} + O(h^6) \\tag{5.8}\n",
        "\\end{align}\n",
        "Let's calculate the derivative using $5.8$\n",
        "\n",
        "\n",
        "### 1. Let's import the necessary headers"
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        "import numpy as np\n",
        "import pandas as pd\n",
        "import matplotlib.pyplot as plt \n",
        "from numpy.polynomial import Polynomial"
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        "### 2. Let's create a function named `dh(f, h, x)`\n",
        "function `dh(f, h, x)` takes three parameters as input: a function `f`, a value `h`, and a set of values `x`. It returns the derivatives of the function at each elements of array `x` using the Central Difference method. This calculates equation $(5.1)$."
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        "def dh(f, h, x):\n",
        "    '''\n",
        "    Input:\n",
        "        f: np.polynomial.Polynonimial type data. \n",
        "        h: floating point data.\n",
        "        x: np.array type data.\n",
        "    Output:\n",
        "        return np.array type data of slope at each point x.\n",
        "    '''\n",
        "    # --------------------------------------------\n",
        "    return (f(x+h) - f(x-h)) / (2*h)\n",
        "    # --------------------------------------------"
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        "### 3. Let's create another funtion `dh1(f, h, x)`.\n",
        "`dh1(f, h, x)` takes the same type of values as `dh(f, h, x)` as input. It calculates the derivative using previously defined `dh(f, h, x)` function and using equation $5.8$ and returns the values."
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        "def dh1(f, h, x):\n",
        "    '''\n",
        "    Input:\n",
        "        f: np.polynomial.Polynonimial type data. \n",
        "        h: floating point data.\n",
        "        x: np.array type data.\n",
        "    Output:\n",
        "        return np.array type data of slope at each point x.\n",
        "    '''\n",
        "    # --------------------------------------------\n",
        "    # YOUR CODE HERE\n",
        "    return (4 * dh(f, h/2, x) - dh(f, h, x)) / 3\n",
        "    # --------------------------------------------"
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        "### 4. Now let's create the `error(f, hs, x_i)` function\n",
        "The `error(f, hs, x_i)` function takes a function `f` as input. It also takes a list of different values of h as `hs` and a specific value as `x_i` as input. It calculates the derivatives as point `x_i` using both functions described in **B** and **C**, i.e. `dh` and `dh1`"
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        "def error(f, hs, x_i):  #Using the functions we wrote dh() my c_diff  and dh1() which is my first order c diff, we find the error through appending their diffrences with Y_actual ny f(x)\n",
        "    '''\n",
        "    Input:\n",
        "        f  : np.polynomial.Polynonimial type data. \n",
        "        hs : np.array type data. list of h.\n",
        "        x_i: floating point data. single value of x.\n",
        "    Output:\n",
        "        return two np.array type data of errors by two methods..\n",
        "    '''\n",
        "    f_prime = f.deriv(1)   #first order derivitive f^1(x)\n",
        "    Y_actual = f_prime(x_i)  \n",
        "\n",
        "    diff_error = []\n",
        "    diff2_error = []\n",
        "\n",
        "    for h in hs: #where h is my loop counter iterating through hs\n",
        "        # for each values of hs calculate the error using both methods\n",
        "        # and append those values into diff_error and diff2_error list.\n",
        "\n",
        "        # --------------------------------------------\n",
        "        # YOUR CODE HERE\n",
        "        e1 = Y_actual - dh(f, hs, x_i)\n",
        "        diff_error.append(e1)\n",
        "        \n",
        "        e2 = Y_actual - dh1(f, hs, x_i)\n",
        "        diff2_error.append(e2)\n",
        "        # --------------------------------------------\n",
        "        \n",
        "    print(pd.DataFrame({\"h\": hs, \"Diff\": diff_error, \"Diff2\": diff2_error}))\n",
        "\n",
        "    return diff_error, diff2_error"
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        "### 5. Finally let's run some tests\n",
        "function to draw the actual function"
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        "def draw_graph(f, ax, domain=[-10, 10], label=None):\n",
        "    data = f.linspace(domain=domain)\n",
        "    ax.plot(data[0], data[1], label='Function')"
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        "### Draw the polynomial and it's actual derivative function"
      ]
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        "fig, ax = plt.subplots()\n",
        "ax.axhline(y=0, color='k')\n",
        "\n",
        "p = Polynomial([2.0, 1.0, -6.0, -2.0, 2.5, 1.0])\n",
        "p_prime = p.deriv(1)\n",
        "draw_graph(p, ax, [-2.4, 1.5], 'Function')\n",
        "draw_graph(p_prime, ax, [-2.4, 1.5], 'Derivative')\n",
        "\n",
        "ax.legend()"
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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "bde2685f15d0c8559c267a64bc503a83",
          "grade": false,
          "grade_id": "cell-f00acc547d5112db",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "HqvDHKH98p1k"
      },
      "source": [
        "### Draw the actual derivative and richardson derivative using `h=1` and `h=0.1` as step size."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "35baa24f9a45907e2b18d96da1fce24c",
          "grade": true,
          "grade_id": "dh1_test",
          "locked": true,
          "points": 10,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "6GEWjQ9V8p1l",
        "outputId": "cfdeba82-4cea-48f3-e6b1-20f546ff2717"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "ax.axhline(y=0, color='k')\n",
        "\n",
        "draw_graph(p_prime, ax, [-2.4, 1.5], 'actual')\n",
        "\n",
        "h = 1\n",
        "x = np.linspace(-2.4, 1.5, 50, endpoint=True)\n",
        "y = dh1(p, h, x)\n",
        "ax.plot(x, y, label='Richardson; h=1')\n",
        "\n",
        "h = 0.1\n",
        "x = np.linspace(-2.4, 1.5, 50, endpoint=True)\n",
        "y = dh1(p, h, x)\n",
        "ax.plot(x, y, label='Richardson; h=0.1')\n",
        "\n",
        "ax.legend()\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7ffa07778ef0>"
            ]
          },
          "execution_count": 8,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
            "image/png": 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T3pObDE7ScBkx3CZ8g/zb119BRSlFAU6iTS29tk2/K+SduwynhcNBeuE+o6M0SMHBweTl5ckfdCOktSYvL4/g4GCjo/i1radcF9HFhHhn6iH44Rh5k1aJdLLaydIZRkdpkGJjY0lLSyMnJ8foKMIAwcHBxMZ6rwA1RPtPuYZ9O0Z29do2/a6QE9CEdo4g9gWVnveiEHFxAgMDiY+PNzqGEH7rRM6PAPToOMBr26z10IpSqr1Sap1Sar9Saq9S6r6q1yOUUquVUoerPoZ7Lq5L64BWVJo0J4rkqFwI4VtOlx+jld1O61jPdz08oy5j5Hbgd1rr7sAQ4C6lVA/gMWCt1roLsLbqc4/q0DwRgO+Pe6dPtBBC1FauM4cEmwOae2+IqtaFXGudqbXeXvW8BNgPtAMmA2c62S8ErnFzxl/o1n4IAPtOfu/pTQkhRK3ZHXZyzRXEOJuBF9sFX9SWlFJxQD9gC9Baa50JrmIPePwqnYTEYbSy20krcm+LVSGEqI+dp49hM0HbJm28ut06F3KlVCjwCXC/1rrWt+tRSt2hlEpRSqXUd0ZEUFQC8TYHWbbMeq1HCCHcaetx130647xwn85z1amQK6UCcRXxRVrrT6tezlJKxVR9PQbIru57tdava60Haq0HRkdH1yczmMy0cTQly1SOw1lzAyQhhPCmo5muQt47bphXt1uXWSsK+A+wX2v94jlfWg5Mr3o+HVjmvng1ax3QBqsJUotOeWNzQghxQZmlh2ljt9M23jtdD8+oyxH5cOBW4DKl1M6qxwTgr8BYpdRhYGzV5x7XoYXrrcumI5u9sTkhhLigXEcO8VYHqkUHr2631hcEaa2/A2q6+qZunfLdoGfcMDi4ln2nNsOgG729eSGE+AmH00GOuYJLrN6dsQJ+2GvljLjEIbSz2TlVfMjoKEIIwb6c49hM0CbQuzNWwI8LeUB4e+KtDrId0hNECGG8Lce2ANAx3Ps35fDbQo5StNFhZJsrsTlsF15eCCE86HCGa8ZKzw7enbEC/lzIgZigWBwKdmcdNjqKEKKRyyg5TIzdTvsug72+bb8u5AmRrpvHbj4il+oLIYyV7cgmzurA3LK917ft14W8d+cRmLU++5ZGCCGMYHfayTZX0sYZ6rXbu53Lrwt56/iBdLDZySg/anQUIUQjdrTgJHYFrQ2YsQJ+XsgJjaajDXJ0ntFJhBCN2OajrgsT27f0bo+VM/y7kAMxtCDPbKPMWmF0FCFEI3Uw7QcAunUYasj2/b6QxzbtgFaw6cReo6MIIRqptJLDtLPZ6WjAjBVoAIU8sXVfALYd22hsECFEo5Vlz6ajzUlQhHd7rJzh94W8d9eRBGpNas4Oo6MIIRqhMzNWWjubGTJjBerQNMtXNYvpTYLVRpb9hNFRhBCN0MG849gVtAmMMSyD3x+RE9KSDnYTuarA6CRCiEZo82HXjJWOET0My+D/hRxoa4qgMMBBZnG+0VGEEI3MwQzXjJVe8cMNy9AgCnlCWCcAVh/eZnASIURjk15yhHY2O+07efeuQOdqEIV8QEdXt7GdJ9YbG0QI0ehkO3OIszkxtWhrWIYGUcg7Jo6irc3OqcJdRkcRQjQiNqeNHLOFGIN6rJzRIAo5UYl0t9rJ0plGJxFCNCK7Mo7gUBATZNzRODSUQm4y05EICgKsZJTIHYOEEN6xparHSnxVS22j1LqQK6X+q5TKVkrtOee1p5VS6UqpnVWPCZ6JeWGdm3cFYNWhzUZFEEI0MkdOb0VpTVIn42asQN2OyBcA46t5/SWtdd+qxwr3xKq7fp0uQ2nN7uPrjYoghGhkMsuO0c5up3XcQENz1LqQa603AD47Ubtd4kjibHZOFu+58MJCCOEGuc5c4mwawozpQ36GO8bI71ZK7a4aegl3w/ouioqIp6vVSSZZaK2NiiGEaCQsNiu5ARZidJihM1ag/oX8NaAT0BfIBP5e04JKqTuUUilKqZScHA+ckFSKDiqaErODk8Uye0UI4VmbTx3AoaBdSDujo9SvkGuts7TWDq21E3gDqLEZr9b6da31QK31wOjo6PpstkYJLV1njtcckpa2QgjPSjm+BYBOUb0NTlLPQq6UOrfd17WAoQPUfTuNIUBrfjzxjZExhBCNwPGsbZi0pm/nYUZHqX0bW6XU+0AyEKWUSgOeApKVUn0BDaQCs90fsfbadh9Bl102TpTsNzKGEKIROF1+lPbYaRnb3+gotS/kWuubqnn5P27MUm+qeVs6WxVrA3PRWqMMPgEhhGi4TqtcBls0hLY2OkoDubLzHO3MbSg3OTlWKDeaEEJ4xvH80xQF2IlTEYbPWIEGWMgTwpMAWHvwW4OTCCEaqtWHqk50tuhucBKXBlfI+3YdS5DTyW454SmE8JA9p1z1ZUCn0QYncWlwhbxN4jC6WW2cKDtkdBQhRAOVXrKXOKuNNl0uNToK0AALuWoWSYItgAxViN1pNzqOEKIBOq2z6Wp1Qnic0VGABljIAToExmI1aXZlHTQ6ihCigTlVdJriADvtTdE+caITGmgh79ba1Yls7f71xgYRQjQ4qw99D/jOiU5ooIW8b49xhDqdHMjYZHQUIUQD8+OJbzBpTZ+EZKOjnNUgC3lox4H0sFhJtxwzOooQooE5WbKXBJuN2K4jjI5yVoMs5ASFEm8PIctcisVuMTqNEKKB0FqTqXPobAXVsoPRcc5qmIUciA+Jw6Fg/bGdRkcRQjQQ6SWnKTE7iFVRPnOiExpwIe/dbigA3x1eZ3ASIURDse6w60RnfHNjb7b8cw22kPfoMZYIh4NjOVuNjiKEaCB2pq7HrDW9feSKzjMabCEPaNubHhYbmfZTRkcRQjQQJ4r30clqo0P3kUZH+YkGW8gJCCJOh5EbUEF+eYnRaYQQfk5rTZrOpYsNzC3aGh3nJxpuIQc6h3ZBK/ji4A9GRxFC+LmTxemUmR20U6186kQnNPBCPrTTKAA2H/nK4CRCCH/3ddW9gBNa9jI4yS816ELetnMyiRYrx4q3GR1FCOHndqZ+Q4DW9O0yxugov9CgCznR3ehn0Zw2ZVNhkwuDhBAX70TJfrpYbbTtOtzoKL/QsAu5yUS3kATsJs3nB743Oo0Qwk9prclU+XSymVDNY4yO8wu1LuRKqf8qpbKVUnvOeS1CKbVaKXW46mO4Z2JevEsSxqG0Zv2hlUZHEUL4qYN5qZSbnLQ3G3+j5erU5Yh8ATD+Z689BqzVWncB1lZ97lPa95pAN6uN40Uyc0UIcXHWHPgOgMTIPgYnqV6tC7nWegOQ/7OXJwMLq54vBK5xTyw3iu5KXwtkmHKosFUYnUYI4Yf2nvqWQK0Z0NX3TnRC/cfIW2utMwGqPraqfyQ3U4rEkM44FHx5eLPRaYQQfiit/CBdrVbCE4YaHaVaXjvZqZS6QymVopRKycnJ8dZmARiUcAVmrVl3cIVXtyuE8H9O7eS0qYAEWwCERhsdp1r1LeRZSqkYgKqP2TUtqLV+XWs9UGs9MDrauzujQ9IEelqsHCmQBlpCiLrZnnGYSpOmY6DvzVY5o76FfDkwver5dGBZPdfnESqyE72tJjJMeZTbyo2OI4TwI18f/BaA7tF9jQ1yHnWZfvg+8D3QVSmVppSaBfwVGKuUOgyMrfrc9yhF5+AuOBWsPirzyYUQtXcoYyNBTif9u40zOkqNAmq7oNb6phq+5JuncX+mf/x4AtJeY+3+z5nczS8iCyF8QEblEbo6bDTrOMjoKDVq2Fd2niOu3wSSLBYOFUjfFSFE7ZRaKjltLiTRHghNI4yOU6NGU8hNkfH0sgaSYSqg2FJsdBwhhB/4/OAWbCboGdzR6Cjn1WgKOUBi025oBZ8d/M7oKEIIP/Dt4VUorRmacLnRUc6rURXywd0m0cSp2XDgc6OjCCH8QGrhDyRabbTr8fPuJL6lURXymF5X0Mdi4WjJLqOjCCF8nMVuIdOUTR+LhuhuRsc5r0ZVyGnRjl62JmSbi8kpKzA6jRDCh606koLNpEkMigeTb5dK307nAd2b90Ir+GTveqOjCCF82PqDXwIwMM63x8ehERbyYT2uJtjpZPMR6bsihKjZkfwtJFqsJPSZYHSUC2p0hbxFt8vpZ7FwouJHo6MIIXyUzWEjjSz6WDQqurvRcS6o0RVywlrTy96U3IAyThTW2ONLCNGIbTi5HZtJ0zkwzufHx6ExFnKgd7jrLh8f7lpjcBIhhC9avfcLAPp2vMzgJLXTKAv50N6TaeZ0kpIq4+RCiF/an7uZRIuVLn0nGR2lVhplIQ/uNJqR5RUcd+zDZrcbHUcI4UNsDhun9Gn6WDSBrXsYHadWGmUhp1kkQ4ik0mzj031yub4Q4n82nnKNj3cK7OgX4+PQWAs5MCrxGpo4NZ/t/9ToKEIIH7Jyj2vItW8H/xgfh0ZcyCP7/YphFRUcLfserbXRcYQQPmJ/zvckWqx09ZPxcWjEhZyozgx1hFFqrmTDie1GpxFC+ACbw8YpTpNk0QS06Wl0nFprvIUcGNphPGateX/nJ0ZHEUL4gHWp27CZNF2axPnN+Dg08kIeN/gmBlVWsi9/nQyvCCFYVTU+PijOv24H6ZZCrpRKVUr9qJTaqZRKccc6vUG1SWKYpQkF5lL25R42Oo4QwmCH8r6ni9VK5z4TjY5SJ+48Ih+tte6rtR7oxnV6llIMaD0apTXvbJfhFSEaM6vDSrrKoq9Fo1r5x/zxMxr10ApA90t+TR+Lla0ZK42OIoQw0MrDrv7jXf1sfBzcV8g18JVSaptS6g43rdMrAjtewqWVkG3K51jhSaPjCCEMsmafq//44ISxBiepO3cV8uFa6/7AlcBdSqmRP19AKXWHUipFKZWSk5Pjps26gcnEgIjhAPw3RS4OEqKxOlqwmS5WK/G9rjQ6Sp25pZBrrTOqPmYDS4DB1SzzutZ6oNZ6YHR0tDs26zZ9h9xKN4uVH07JTZkbqmJLOYdzT/NjZiZ7M7M5nJ3Pifxiyiw2o6MJH1BqqSTTlEU/iwY/Gx8HCKjvCpRSzQCT1rqk6vk44I/1TuZFAQkjGLnCwRtNskgrPk1s8zZGRxIXwel0svnUQdYf382+nGNklKdTZMvEQjYEFKFU9VNMtTOQQEcMzc3t6RjWiZ5RiYxO6M3A2DhMfjZWKi7Oot1fYzNpegcn+N34OLihkAOtgSVKqTPre09r7V9nDs2BDG45kNfZyxspS5l32RyjE4laSiss4t2d61h36hsyLDsgMO/s1wKcIUSawmivQomzNyVCO8Fswo7CrhR2FFanJg/NUZOV0+xkR9lGdpTBuycARzNaBfQlOXYMtw8cT0zzMON+UOFRaw8tJcTp5LKu/jXt8Ix6F3Kt9TGgjxuyGGrAwFuI2/QwP5xYDkgh92UZRQW8tHkx36VvoETtR5lsKGcAiaZIxlWGM6Q8m/iS0zR3nnMEHhIBIeGgHeB0gNP+v4+WEnC6hljyzQEcDG/PjqBItigTuxwpLD61kQ9Tn6Mlfbi8w1juHjqRqGZS1BsKh9PBCcs2Lq2spHmvq42Oc1HccUTeIAQkXk7yWhtvNz9FZkkeMWGRRkcS59Ba88WB3cxPWcAp+waUyUqIDmOcjuSq0myG5h8nWB+DyC4QlwwRCVWPeNfHkPCaV+6wQf4xyN5HRPZ+hmbvY2j2fn6bdxQbmo0R7VkcGs73ajefpP/Ax4v/RlxQMr+75A5Gd+rmtX0gPGPVka2Um60MdraE8I5Gx7koUsjPCAzhkrBeLFDHeCPlM+aOnmF0IgHYHA5e2ricjw5/QGXAPtBmhjlacnfOMXqXn0SZgyB+JAy+C7qMdRXuujIHQnRX16Pntf97vSyPwIMrSN63jORj67E7bWxu2Y6FzaLYrNZyz7dridhwCfcNvIMpvQZQNbwo/MySHz8lQGsuTfDPYRUAZUSPkYEDB+qUFN+7kt+x6yOuSJmLWSWyauYyo+M0ala7nWe+eY/lqQtwBOQQ5AjhBnsgM0/vIwoz9JoCva6DuBHQpKnnA1UUwqGVsG8ZHFnLaRz8s013Pgsow2myE2rvw32D5nBj0qWezyLcRmvNyLeG0d2aw+vXLoPWvt3xUCm1rbqr5/3v9KwHmbtewYTSSjJNx/jxdKrRcRolrTX//H4llyy4miVpzxOuK5lXDN+fPMgjxTlEjXwMHtgLU16HxCu8U8QBQlpCnxvhpvfhwX20Gf4Az+Sd5JtTqfzaEkql2s+zO+5kxMJfs+H4Ae9kEvW2O/sQheZShlqC/HLa4RlSyM8V3JyrWiQBmr9//5bRaRqdLw7sZOhbt/DvQw/TxJTLvEIba9P2MiW4FYFT3oT790DyoxDW2tigzaLgsj/A/XtoOeZpHi3KZeOpo/ymrAnFzr38dv1N3Lj4T+SVlRmbU1zQoh2uiwAHtUkGPx4ak0L+M12G/ZZR5RXsKfgCq91qdJxG4WjeaSYuup9HN0+nUh3gjrIAvjl1iCkhMZhmfAG/WQtJUyGgidFRfyq4OQy/D+7/kaYT/s69Viur0k4wxBbM3orFjH7/Kl787jNpkezDdmR+RVKlhW6X3GJ0lHqRQv5znS/namswFrOFf29banSaBs3pdPKH1W9zzbLJnLStY6IlhK9PHecei53gKW/C7Wshzg/GnAODYdAsuDuFNiN/z5vZx3g1p5jmqoy3jj5O8tuz2JstfXx8TVpxBqdNuQyvgIAOlxgdp16kkP+cycyIvjPpYLOx9MBCo9M0WDvTT3Lpwmksy3ieDg5YkpHBcwUZRIyZB3dvhd7X+98VdgFBMOph+O1mRsQMZm3aAW4rM1Pg3M6Nn1/Ps+s/lKNzH7Jwx1IA+rUcBCazsWHqyc/+UrwjeNAMri+pIJuTbM/ca3ScBsXhcPLQl2/y61XXU8oe7il2siz9IJ37TId7d8Lwe11HuP4sIh5uXkzQDe/yYKWNz9JP0d6h+eDEM4x/935ySkuMTiiATamfkWC10e+SW42OUm9SyKvTLJLkqFE0cWr+b/MbRqdpMHZlpjLi7VtYlf1/dHIqlqef4g4dRMBtX8LEF6BphNER3Ucp6H4V3LWFjoPvZGn6YW4qhQzn11z+wRSW75cbfhspr6KANJ3GyAobwV1GGx2n3qSQ1yBu9L1MLCtjd+F6iixFRsfxe8998zG//vIGytjPg0U2Pk0/RNzg38KdG6HjUKPjeU5QKIx7hibTlvF4hZVXs/JpYsrn8c2zuPfzV3E6nUYnbJQWbP8cp4L+zXq5hsT8nBTyGqjYgVzpbIXd5OCVLR8YHcdvFVSUMWnRg7ybOo/2Ds1n6ae4zdwC86w1MO5PEBhidETvSBgFd25iROylfJl+jCSbmXV5rzHm3VlklcqBgrdtOLKE1nY7Q/v792yVM6SQn8fAYXeTVGnhy6OL5CTVRVh7ZA+XvXc9J+yrmVoGS9MP02HIvTB7A8QOMDqe9zWLhJs+IGrsn3kn6wR3F9nIdW7jig9vYFOq3PzbW0qt5Zx0HiG5zEJw9/FGx3ELKeTnEdjneqaUOSg2FbD0wHqj4/iVP6xeyH3fTseksng5u4C55RaaTFsGlz/VIN7KXjSlYMidmG9fw2zVjH9nZWNW2cz++lbe3LrG6HSNwts7VmE3aQYGxbuuBWgApJCfT2AIl3W5jpYOB2+kvGl0Gr9QUF7G+HfvYlnGCyTaFV+kpzK6zWCYs9E1vCBcYvrAHd8wrOMYPsk4SbTTxj/2PsT9X/xb3v152NpDn9Lc4WBEnxuNjuI2UsgvIHz4HK4tKSPNsZuDuXJRx/lsPXWEMe9PJd2+geklDj7MOEab0XPhlo8h1Ldu7+cTgkLhhneIG/YgSzOO0s+qWJv7Tya//wgVNrmq2BMq7VZO2H5kVHklzXpNNjqO20ghv5CIBCaF9gI0f/32P0an8VmvbVnBzNW3gMrklZx8HrKZCZy5Ei693/8u7PEmkwkue4Lm177Jf7MzuKHUwXHbSsa8O43M4nyj0zU4/922GovZwSXmNhDayug4biN/YbWQOOJeRlZUsrvgC3JKi42O41McDiczlzzH/P2P0cZpZ2nGSZJjhsKcDdD+F/fgFjXpfT2Bt63gyUonT+aVUKr2MfGjm/nx9CmjkzUoS/e/Q7jDwdheU42O4lZSyGuj8+VMtwRjNVt4eM3/GZ3GZ2SVFnHZu7PYWvwuoypNLEs/SodhD8LNi89/Rx5RvXYD4DfruCE0nlezstGm09zyxa9Zf0yuLnaHQ7mnOM1eri0po2mvKUbHcSsp5LVhMjPokjsZX1rGjoJPOJKfZnQiw208sZ/xH15Pvt7GfYWVvFKQR9NfvQeXPSFDKfXRPAZuW8GlcZfzTmY6IaqYe9bPYvHujUYn83t/37QA0EwI7eW3t3SriVv+4pRS45VSB5VSR5RSj7ljnT5n4CzutjTFjIOH1/zZ6DSGenXL58xZOx2zyuWNrGxuN4dj+s3X0G2C0dEahsAQmLqQXkm/ZnHGSSK0hT9uu4eXN31mdDK/ZXPa2JH7GcMrKul66T1Gx3G7ehdypZQZmA9cCfQAblJK+e+tNmoSGEzHK/7E9OIijlR8y6a0xtcrw+l0Mnv5C7y6/3HaORwsyzjBkLixrn7hUV2MjtewmMww8UU6jniUTzJS6WjXvH7oDzy5RjpyXox3d6+gwlzB5MpA171dGxh3HJEPBo5orY9pra3AB0DDmddzrh7XcFNQZ8LtTp785plGNd+3oLyUcYtms6lgISMrFZ9kHKPdyMfhhrchKMzoeA2TUjDqEaImvMQHmSdIssDS9Bf47WcvGZ3M73z44wLa2uwM7z3N71vWVqfeN19WSl0PjNda3171+a3AJVrru2v6nrCwMD1ggH9eot01rIyxyenMi4ok5NvWND1iNzqSx1lbhlA0zoqpaTa/ySvltoJCnt0Xx+b8FkZHazQujSrk0e6pPNQ6hk1hAXA0kfANuZjw39uTeYu9pZOiawu4K7+IVSvaU2ALNDrSRfvmm288dvPl6n6TfvG/g1LqDqVUilIqxWazuWGzxjhY0oymR5vSxWLDMqgEp7lhH5WXxbekeFIBTYLz+L+sHK7JKuOu7V2liHvZd7kteXx3Z/6amcW4Ygt0OkTBmCicv/xTEz9T0SeIAK2JTgvz6yJ+Pu44Ih8KPK21vqLq898DaK3/UtP3DBw4UKekpNRru4YqSmPjv4cwp3U4E9rN4rnL7zc6kds5nU4eXDWfNVlvEm0P4D9ZqSTEXQZT3nDdUV4Y4/QeHO9cw7MhZj4Ka0KseTTLbnyRJgEBRifzSeW2cka8eyljygt5buLbKH+4deB5KKU8dkS+FeiilIpXSjUBbgSWu2G9vqtFLEMHzCG5rJxVp94mrTjb6ERuVVhRxpXv38Xa7NcZZFEsTz9KwiX3wE0fSBE3WptemG/7kj9YzcwsqiDNsY7x791JmdVidDKf9K+UT7CabEywN0d1HG50HI+pdyHXWtuBu4FVwH5gsda6wV/BYLr0fu4pN4Oycc+XNb758Ds7Mo4y5v2pZNi+4/YiK2/mZhF23X/g8qcb5EkivxTVBdPMlTygQ7m3oJQcvZnx791OUWW50cl8zrKDb9PZamXowNmuk8cNlFvmkWutV2itE7XWnbTWz7pjnT4vKJTEMU9xY3EJRyrWsGjXeqMT1dt/UlYxbeXNaDJ5JTuf+3QI5tvXQK/rjI4mfq5lB5i5kt80acPv84ooVDsZ//4M8srlfqBnrD2WQr46zXUlFoIGNIwbSNRELsGrjz43MTuwHe3sTp7b9pjfdke0Oxzcvuw5XtrzMK2ddpZknCS53TC4Yx207ml0PFGTsDYw4wtuDk3gmZwCStV+rvzgVjKk2RYA87e8SYjTybj24xtM3/GaSCGvD5OJ8Akv8Gp2NkGUM/2LOymzlhmdqk6O5mUy6p1b2FL4LiMqTSxPP0rHYQ/CTR9KvxR/0DQCpi1ncmQSL2TnUmk6xlUf38qJgoZ13qauMorzOFb5PRNLy2h16W+NjuNxUsjrq+NQEpKf4sWcLMr1SWZ8/juc2j9uqLtw+yquXXYdpfoAvy8oZ35+Lk1veFf6pfib4OZwy8dcETOEV7KysKs0rllyK4dy041OZpin1r2Jw+Tk6sD2EJNkdByPk79Wdxh6F5cmXsvv8gs4ULKRuev/YXSi86q0W5m25Cme3/0wUU4rH2ekcXNwW0x3rIfuk4yOJy5Gk6Zw0weM6jCGf2dlgjrNDcumsft0qtHJvO5UUTZb8z9keHkF/YbeZXQcr5BC7g5KoSb9g5ubJTKxpIJlJ9/i7V2+2eAoJe0II96eyo7iT5lQrvgi/Qhd+t8Ot6+BqM5GxxP1ERAENyxkSOdJ/Od0JiZTLr/+YjpbTjauGzs/vPoFtLLxQJmCHtcYHccrpJC7S2AwgTe/z5OVgXSvdPDC9qf47uQuo1OdZXfYeWjlP5mx+kbs6iTP5RbzXEkRIb96D678a+O+IXJDYg6EKW/Qv8dU3snMIEgV8Zs1M/j6qO/8LnrS7qxD7CtdydSSErqOfgICg42O5BVSyN0prDXNbn6PV3Lzaem0c/eae9hz2viZLOuO7WTY29eyKuvf9LSa+Dz9BBPCe6DmbJTWsw2RyQxXvULPvjNYlJlGM13Gfd/cwdJ93xudzOMeW/cMIU4ns2gF/W41Oo7XSCF3t7b9aD15Pq9lZWBShdy84ld8tPdrQ6IUVZYybcmT3LNhOg5nBs/mlfN+1lHaXfoITP8MWrQzJJfwApMJrvwbiYPvYnHmSSKcVp7ccjcLtn1ldDKP+erYBk5ZdjCnqJCYCc81qgvYpJB7Qq/r6HnJfSzOyCDKYeOPW+/n0a+fw+70TqdErTWvbF7OyPcmsqN4KWMqAvk6/RhXt+iEafa3kPwYmKU3R4OnFIydR/vkJ/ko8wTt7A7+vvsR/rHpU6OTuZ3D6eCP3z1LW5uDq1sOhE6jjY7kVVLIPWX0E3QecAdfZBzh8lJYcepdxn94M6eKMz22Sad28nrKcoYsnMzrB5+gpbOC/57O4/8Ks2lx5Qtw20po3fDu+SEu4NIHaHXl3/kg8yRdbJr/HHqauWvfMjqVW72y9T2KdAYP5BcSOek5o+N4Xb27H14Mv+9+WBcHVuBceifLAxRPR7ZEE8zv+j/FtD5Xum0TVoeVV7d+wqIDC6hUGTS1NeXOigpuzjtBkx7XwJXPua4CFI3bnk8pWzKbOa1i2BmsGRExk1evesDoVPVWbCkl+b3L6WnN5802kwia/KLRkTympu6HUsi9ofAUfDKLE5nbuL1VPKeDLESpgdzSczK/7jOe4ICLO7N+rCCDlzcvYX3mRzjMebSwhnJfWTHXFqYREN0NLp8HXce7+YcRfu3IGiwf3sqDEZFsaKroGjKR96c8Q6Aft8Gd/fkzbMr7kLeziul313ZoFml0JI+RQm40hw3W/RnLxhf5e1QcHwU3wR5QiUkHkRh2Cdd1ncSU7mNoYm5S4yqKKotZeXQjK458y76CbVSqDABirC14qCSby4uzMLUbACN+B4lXytWZonont2B/byp/Dm3GR2EBRKr+LL1hPi2DQ41OVmf7sk9w0xeTubK8mL/0vR81/F6jI3mUFHJfcWQtLJmNvSyHb8Pa8FZwG3YFl+EMsIAzmFDiCQ4MIMhsRqNwanA4oNhWSKXpBEo5wRlAW2s4ydrJ1SWp9CgrQCUkw6UPQvzIBt2uU7jJ6T3oRVN5x1TJ8+FhBOsOvH/1v+kc6T8zmRxOB6Pfnk6Z3smnRYqOd6c0+OshpJD7EkspHPwS9n4KR9Zgc1j5pnk73g9tw1ET2J0am9OJUgqUqy6HKhODnIrxZZkMKUqnCUBYW9fZ+UGzoJ1/3gNVGKgoHd7/FeuKj/FAdCu0bs4/kl9hdEIfo5PVym8+e4bN+R/ycF4B08a/Cj2uNjqSx0kh91UVhXBwBez5FI6tg/NNUQxqAfEjICHZ9YjsLEffon4sJfDxLPaf+JqZrdtTajJzV88/Mmewb18o9sb2T3n5x6e4qqSCZ0K7Ypr+eaP4W5BC7g/K86HguOvW1doJaNdHrV1NkVr1lPnfwv0cdlj1e05v+w+3tenIqUAHIyKn8c+JD2D2wYtqtmbuZNaq2+hpsfDfUgiZvR5CWxkdyyukkAshzm/zvyhd9Xt+17oDm0KcRKgk3p38Eu1b+E6RzCjJZNInU2lhL2JxVi7RM1c1ija1Z3jy5stCiIZgyBxCf7WI1/JyeKSgggLHHiZ9MoVP9n1jdDIAym3l3LB0NkoX8++sTKKv/XejKuLnI4VcCPE/3SZgmv0Ntwa1ZnFmOi2clTz1wz3M+fyv2OzeaTFRHa01v/r0AYocx/l7djaJIx5tFCc3a6tehVwp9bRSKl0ptbPq4dtnSIQQFxbVBW5fQ7ekaXyZfoRRFSY25i1i5Ds3sSsz1etxnE4nM5b+kdTKTTxQUMCoThNhxENez+HL3HFE/pLWum/VY4Ub1ieEMFpgCEx6kWbXv8UrhXnMzS+jnMP8euUU7vz8r5RZK7wSI7ssl7Hv3cb24o+5qqSCGc06oyb/s1HMUKkLGVoRQtSs57WYZm9garOOfJF+kr7WIL7LW8TwReN5bevHeHKyxKpj67nio8nk2HZyf6GFP1kCMd30ges/GfET7ijkdyuldiul/quUqvG260qpO5RSKUqplJycHDdsVgjhFRHxMHMVsUPv5+3sI7yRVUi43cKr++Zx6dtT+OLgFrdursJWwZ1fPslD395DC6uFjzLSmWVuifnWT6X5Ww0uOP1QKbUGqG7vPQFsBnJxzXz+ExCjtZ55oY3K9EMh/FT+cVg9F8f+5XzQsi0vhjXDGmAhUg3gxh6Tmd5nPCH1OGL+5vguHt7wKBWkc1UxPJV/kqBBd8DYP8qROF6YR66UigM+11r3utCyUsiF8HPHv4WVv6c0ew//aNWFT5qAPaAS5QwmMWwoN/W4mqu6jjxvEzhwzUbZl7efd3evYv3JbyjhGCGOQF7Iy2KkCkZNng+JV3jph/J9HinkSqkYrXVm1fMHgEu01jde6PukkAvRADgdsONd+PpP2Mty+K55GxY0bcv2JsVosxUczWgdmETr0AgiQ1oQ3awlYU3CMOkQyq1WtmVv4WhpCjZVBEALSzgTnFbmZB8kotPlMHl+o7lis7Y8VcjfAfriGlpJBWafKeznI4VciAbEWgYHVsCej+HIWqxOG6vDO/B+02j2myqxKSvaZIGfTTRRjiZ0trRggq2MyUVHiXbYICQckh+Hwb+RmSnVkEv0hRCeV54P+z9zFfXj3+I6xgMnUK4UBU2aUdwkFAcOupfkEAgQ0we6jHM92g1oVDdNrquaCrl0YBJCuE/TCBgw3fUoOQ3p26GyCJOlmNDKIkIri6CyyNXls+Nw6Hw5NI8xOrXfk0IuhPCMsDbQTS729ga5IEgIIfycFHIhhPBzUsiFEMLPSSEXQgg/J4VcCCH8nBRyIYTwc1LIhRDCz0khF0IIP2fIJfpKqRzgRA1fjsLVGtcXSbaLI9kujmS7eL6crz7ZOmqto3/+oiGF/HyUUinV9RLwBZLt4ki2iyPZLp4v5/NENhlaEUIIPyeFXAgh/JwvFvLXjQ5wHpLt4ki2iyPZLp4v53N7Np8bIxdCCFE3vnhELoQQog4ML+RKqeeVUgeUUruVUkuUUi1rWC5VKfWjUmqnUsortxeqQ7bxSqmDSqkjSqnHvJRtqlJqr1LKqZSq8Qy4QfutttmM2G8RSqnVSqnDVR/Da1jOa/vtQvtBubxc9fXdSqn+nsxTx2zJSqmiqv20Uyk114vZ/quUylZK7anh60butwtlc+9+01ob+gDGAQFVz58DnqthuVQgyteyAWbgKJAANAF2AT28kK070BVYDww8z3JG7LcLZjNwv/0NeKzq+WNG/77VZj8AE4Avcd31cgiwxUv/jrXJlgx87s3fr3O2PRLoD+yp4euG7LdaZnPrfjP8iFxr/ZXW2l716WYg1sg856pltsHAEa31Ma21FfgAmOyFbPu11gc9vZ2LUctshuy3qm0srHq+ELjGC9s8n9rsh8nA29plM9BSKeWN+6MZ9W9UK1rrDUD+eRYxar/VJptbGV7If2Ymrv9Bq6OBr5RS25RSd3gx0xk1ZWsHnDrn87Sq13yF0futJkbtt9Za60yAqo+taljOW/utNvvBqH1V2+0OVUrtUkp9qZTq6YVcteXrf5tu229euWenUmoN0KaaLz2htV5WtcwTgB1YVMNqhmutM5RSrYDVSqkDVf/rGZ1NVfOaW6YC1SZbLRi23y60impe8/h+q8NqPLLfqlGb/eCxfXUBtdnudlyXjZcqpSYAS4Eung5WS0btt9pw637zSiHXWl9+vq8rpaYDk4AxumoAqZp1ZFR9zFZKLcH1tq/ef1huyJYGtD/n81ggo765apOtluswZL/VgiH7TSmVpZSK0VpnVr3Nzq5hHR7Zb9WozX7w2L66gAtuV2tdfM7zFUqpV5VSUVprX+hzYtR+uyB37zfDh1aUUuOBR4GrtdblNSzTTCkVduY5rpOQ1Z4N9nY2YCvQRSkVr5RqAtwILPd0ttowar/VklH7bTkwver5dOAX7x68vN9qsx+WA9OqZmEMAYrODA952AWzKaXaKKVU1fPBuGpKnhey1YZR++2C3L7fvHUW9zxnd4/gGsfaWfX4V9XrbYEVVc8TcJ0x3wXsxfX23Sey6f+dHT+E6wy/t7Jdi+uIwwJkAat8aL9dMJuB+y0SWAscrvoYYfR+q24/AHOAOVXPFTC/6us/cp5ZSgZku7tqH+3CNSFgmBezvQ9kAraq37dZPrTfLpTNrftNruwUQgg/Z/jQihBCiPqRQi6EEH5OCrkQQvg5KeRCCOHnpJALIYSfk0IuhBB+Tgq5EEL4OSnkQgjh5/4fWvKvdHbZSmEAAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "needs_background": "light"
          },
          "output_type": "display_data"
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "742a727d9a6f5c75fb1ed0d0709bb75b",
          "grade": false,
          "grade_id": "dvh_intro",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "HXuImbIG8p1l"
      },
      "source": [
        "### Draw error-vs-h cuve"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "22fcce203a5738cd4551043f5cb6311d",
          "grade": true,
          "grade_id": "error_test",
          "locked": true,
          "points": 10,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "taOUekIO8p1l",
        "outputId": "58964cf6-ba48-4464-e3bd-6dbfb9727edf"
      },
      "source": [
        "fig, ax = plt.subplots()\n",
        "ax.axhline(y=0, color='k')\n",
        "hs = np.array([1., 0.55, 0.3, .17, 0.1, 0.055, 0.03, 0.017, 0.01])\n",
        "e1, e2 = error(p, hs, 2.0)\n",
        "ax.plot(hs, e1, label='e1')\n",
        "ax.plot(hs, e2, label='e2')\n",
        "\n",
        "ax.legend()\n"
      ],
      "execution_count": null,
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "       h                                               Diff  \\\n",
            "0  1.000  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "1  0.550  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "2  0.300  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "3  0.170  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "4  0.100  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "5  0.055  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "6  0.030  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "7  0.017  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "8  0.010  [-59.0, -17.636506249999968, -5.22809999999996...   \n",
            "\n",
            "                                               Diff2  \n",
            "0  [0.25, 0.02287656250000225, 0.0020249999999606...  \n",
            "1  [0.25, 0.02287656250000225, 0.0020249999999606...  \n",
            "2  [0.25, 0.02287656250000225, 0.0020249999999606...  \n",
            "3  [0.25, 0.02287656250000225, 0.0020249999999606...  \n",
            "4  [0.25, 0.02287656250000225, 0.0020249999999606...  \n",
            "5  [0.25, 0.02287656250000225, 0.0020249999999606...  \n",
            "6  [0.25, 0.02287656250000225, 0.0020249999999606...  \n",
            "7  [0.25, 0.02287656250000225, 0.0020249999999606...  \n",
            "8  [0.25, 0.02287656250000225, 0.0020249999999606...  \n"
          ]
        },
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7ffa07718588>"
            ]
          },
          "execution_count": 9,
          "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"
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "5bDWA4me8p1m"
      },
      "source": [
        ""
      ],
      "execution_count": null,
      "outputs": []
    }
  ]
}