Lab 4 Newtons_Divided_Difference_Interpolation.ipynb

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        "Make sure you remove `raise NotImplementedError()` and fill in any place that says `# YOUR CODE HERE`, as well as your `NAME`, `ID`, and `SECTION` below:"
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        "NAME = \"Abdullah Hasan Sajjad Rafi\"\n",
        "ID = \"19301097\"\n",
        "SECTION = \"12\" "
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        "---"
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        "# Part 1: Hermite Interpolation\n",
        "---\n",
        "Hermite Interpolation is an example of a variant of the interpolation problem, where the interpolant matches one or more **derivatives of $f$**  at each of the nodes, in addition to the function values."
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        "## Importing the necessary libraries"
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        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from itertools import combinations\n",
        "from numpy.polynomial import Polynomial"
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        "## Creating the components for Hermite interpolation\n",
        "\n",
        "For the case of Hermite Interpolation, we look for a polynomial that matches both $f'(x_i)$ and $f(x_i)$ at the nodes $x_i = x_0,\\dots,x_n$. Say you have $n+1$ data points, $(x_0, y_0), (x_1, y_1), x_2, y_2), \\dots, (x_n, y_n)$ and you happen to know the first-order derivative at all of these points, namely, $(x_0, y_0 ^\\prime ), (x_1, y_1 ^\\prime ), x_2, y_2 ^\\prime ), \\dots ,(x_n, y_n ^\\prime )$. According to hermite interpolation, since there are $2n + 2$ conditions; $n+1$ for $f(x_i)$ plus $n+1$ for $f'(x_i)$; you can fit a polynomial of order $2n+1$. \n",
        "\n",
        "General form of a $2n+1$ degree Hermite polynomial:\n",
        "\n",
        "$$p_{2n+1} = \\sum_{k=0}^{n} \\left(f(x_k)h_k(x) + f'(x_k)\\hat{h}_k(x)\\right), \\tag{1}$$\n",
        "\n",
        "where $h_k$ and $\\hat{h}_k$ are defined using Lagrange basis functions by the following equations:\n",
        "\n",
        "$$h_k(x) = (1-2(x-x_k)l^\\prime_k(x_k))l^2_k(x_k), \\tag{2}$$\n",
        "\n",
        "and\n",
        "\n",
        "$$\\hat{h}_k(x) = (x-x_k)l^2_k(x_k), \\tag{3}$$\n",
        "\n",
        "where the Lagrange basis function being:\n",
        "\n",
        "$$l_k(x) = \\prod_{j=0, j\\neq k}^{n} \\frac{x-x_j}{x_k-x_j}. \\tag{4}$$\n",
        "\n",
        "**Note** that, we can rewrite Equation $(2)$ in this way,\n",
        "\n",
        "\\begin{align}\n",
        "h_k(x) &= \\left(1-2(x-x_k)l^\\prime_k(x_k) \\right)l^2_k(x_k) \\\\\n",
        "&= \\left(1 - 2xl^\\prime_k(x_k) + 2x_kl^\\prime_k(x_k) \\right)l^2_k(x_k) \\\\\n",
        "&= \\left(1 + 2x_kl^\\prime_k(x_k) - 2l'_k(x_k)x \\right) l^2_k(x_k) \\tag{5}\n",
        "\\end{align}\n",
        "Replacing $l^\\prime_k(x_k)$ with $m$, we get:\n",
        "$$h_k(x) = (1 - 2xm + 2x_km)l^2_k(x_k). \\tag{6}$$\n",
        "\n",
        "# Tasks:\n",
        "\n",
        "* The functions: `l(k, x)`, `h(k, x)` and `h_hat(k, x)` calculate the corresponding $l_k$, $h_k$, and $\\hat{h}_k$, respectively.\n",
        "\n",
        "* Function `l(k, x)` has already been defined for you. Your task is to complete the `h(k, x)`, `h_hat(k, x)`, and `hermit(x, y, y_prime)` functions.\n",
        "\n",
        "* Later we will draw some plots to check if the code is working.\n",
        "\n",
        "---"
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        "### Part 1: Calculate $l_k$\n",
        "This function uses the following equation to calculate $l_k(x)$ and returns a polynomial:\n",
        "\n",
        "$$l_k(x) = \\prod_{j=0, j\\neq k}^{n} \\frac{x-x_j}{x_k-x_j}.$$"
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        "# Already written for you.\n",
        "\n",
        "def l(k, x):\n",
        "    n = len(x)\n",
        "    assert (k < len(x))\n",
        "    \n",
        "    x_k = x[k]\n",
        "    x_copy = np.delete(x, k)\n",
        "    \n",
        "    denominator = np.prod(x_copy - x_k)\n",
        "    \n",
        "    coeff = []\n",
        "    \n",
        "    for i in range(n):\n",
        "        coeff.append(sum([np.prod(x) for x in combinations(x_copy, i)]) * (-1)**(i) / denominator)\n",
        "    \n",
        "    coeff.reverse()\n",
        "    \n",
        "    return Polynomial(coeff)"
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        "### Part 2: Calculate $h_k$\n",
        "This function calculates $h_k(x)$ using the following equation:\n",
        "$$h_k(x) = \\left(1 + 2x_kl^\\prime_k(x_k) - 2l'_k(x_k)x \\right) l^2_k(x_k).$$\n",
        "\n",
        "This equation is basically a multiplication of two polynomials.\n",
        "\n",
        "First polynomial: $1 + 2x_kl^\\prime_k(x_k) - 2l'_k(x_k)x$.\n",
        "\n",
        "Second polynomial: $l^2_k(x_k)$.\n",
        "\n",
        "The `coeff` variable should contain a python list of coefficient values for the **first** polynomial of the equation. These coefficient values are used to create a polynomial `p`."
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        "def h(k, x):\n",
        "    # initialize with None. Replace with appropriate values/function calls\n",
        "    \n",
        "    l_k = l(k,x)\n",
        "    l_k_sqr = l(k,x)**2\n",
        "    l_k_prime = l_k.deriv(1)\n",
        "    coeff = np.array([ 1 + 2*x[k]*l_k_prime(x[k]), -2*l_k_prime(x[k])])\n",
        "    p = Polynomial(coeff)\n",
        "    \n",
        "    return p * l_k_sqr"
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        "# Test case for the h(k, x) function\n",
        "x = [3, 5, 7, 9]\n",
        "k = 2\n",
        "h_test = h(k, [3, 5, 7, 9])\n",
        "h_result = Polynomial([-2.5, 0.5]) * (l(k, x) ** 2)\n",
        "\n",
        "assert Polynomial.has_samecoef(h_result, h_test)\n",
        "assert h_result == h_test"
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        "### Part 3: Calculate $\\hat{h}_k$\n",
        "This function calculates $\\hat{h}_k(x)$ using the following equation:\n",
        "\n",
        "$$\\hat{h}_k(x) = (x-x_k)l^2_k(x_k).$$\n",
        "\n",
        "This equation is also a multiplication of two polynomials.\n",
        "\n",
        "First polynomial: $x-x_k$.\n",
        "\n",
        "Second polynomial:  $l^2_k(x_k)$.\n",
        "\n",
        "The `coeff` variable should contain a python list of coefficient values for the **first** polynomial of the equation. These coefficient values are used to create a polynomial `p`."
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        "def h_hat(k, x):\n",
        "    # Initialize with none\n",
        "\n",
        "    l_k = l(k,x)\n",
        "    l_k_sqr = l(k,x)**2\n",
        "    coeff = [-x[k],1]\n",
        "    p = Polynomial(coeff)\n",
        "    \n",
        "    return p * l_k_sqr"
      ]
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        "x = [3, 5, 7, 9]\n",
        "k = 2\n",
        "h_test = h_hat(k, [3, 5, 7, 9])\n",
        "h_result = Polynomial([-7, 1]) * (l(k, x) ** 2)\n",
        "\n",
        "assert Polynomial.has_samecoef(h_result, h_test)\n",
        "assert h_result == h_test"
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        "### Part 4: The Hermite Polynomial\n",
        "This function uses the following equation:\n",
        "\n",
        "$$p_{2n+1} = \\sum_{k=0}^{n} \\left(f(x_k)h_k(x) + f'(x_k)\\hat{h}_k(x)\\right).$$\n",
        "\n",
        "The polynomial denoted by the equation is calculated by the variable `f`."
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        "def hermit(x, y, y_prime):\n",
        "    assert len(x) == len(y)\n",
        "    assert len(y) == len(y_prime)\n",
        "    \n",
        "    f = Polynomial([0])\n",
        "    \n",
        "    for i in  range(len(x)):\n",
        "      f += y[i]*h(i,x)  + y_prime[i]*h_hat(i,x)\n",
        "    \n",
        "    return f"
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        "## Testing our methods by plotting graphs.\n",
        "\n",
        "**Note:** \n",
        "\n",
        "* For each of the 5 plots, there will be 2 curves plotted: one being the original function, and the other being the interpolated curve. \n",
        "\n",
        "* The original functions are displayed in orange color, while the hermite interpolated curves are in blue.\n",
        "\n",
        "* `x`, `y`, and `y_prime` contain $x_i$, $f(x_i)$, and $f'(x_i)$ of the given nodes of the original function $f$.\n",
        "\n",
        "Upon calling the `hermit()` function, it returns a polynomial `f`. For example, for plot 1, it is called `f3`.\n",
        "\n",
        "In general, a polynomial may look like the following: $f = 1 + 2x + 3x^2$. Next, we pass in a number of $x$ values to the polynomial by calling the `.linspace()` function on the polynomial object using `f.linspace()`. This function outputs a tuple, which is stored in a variable called `data`. First element of `data` contains a 1D numpy array of $x_i$ values generated by `linspace()`, and the second element of `data` contains a 1D numpy array of the corresponding $y_i$ values outputted by our example polynomial:\n",
        "$f = 1 + 2x + 3x^2$. \n",
        "\n",
        "Using `test_x`, we generate a range of $x_i$ values to plot the original function, and `test_y` contains the corresponding $y_i$ values of the original function. For the first plot, our original function is the *sine curve*.\n",
        "\n",
        "For all the plots:\n",
        "\n",
        "`plt.plot(test_x, test_y)` plots the original function.\n",
        "\n",
        "`plt.plot(data[0], data[1])` plots the interpolated polynomial."
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        "pi      = np.pi\n",
        "x       = np.array([0.0, pi/8.0,  pi, 4.0*pi/2.0])\n",
        "y       = np.array([0.0,    1.0, 0.0,       -1.0])\n",
        "y_prime = np.array([1.0,    0.0, 1.0,        0.0])"
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        "**Plot 1:** trying to interpolate a sine curve (`np.sin()`) using first 2 nodes in `x` and `y`, and their corresponding derivative in `y_prime`."
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ulRfLalmzYh7jR43o/w4iOUShIDnnB785zKgR+XxOewkil1EoSE6paznPv+0+zur3z2GSPg5b5DIKBckpz2w9Rpc7D902N+pSRNKSQkFyRkdnN8+8dYw7Fk1hnr5SUyQphYLkjBfLTlDf2s6aFfOiLkUkbSkUJGc8s/UYc4tHc3vJlKhLEUlbCgXJCVVNZ9l6pIlP3jKLvDyLuhyRtKVQkJzw87erMYMHls6KuhSRtKZQkKzn7vzr2zXctqCYmRNGRV2OSFpTKEjW23b0FMeazvKftJcg0i+FgmS9X+yqYXRhPiuvnx51KSJpT6EgWa2zq5vNZbXcde1UxhTpO6VE+qNQkKz21tEmGts6uP+GGVGXIpIRFAqS1V7cW8vIEXncsVjvTRAZCIWCZK2ubuelfbXcuXgqows1dCQyEAoFyVo73j1FfWs792noSGTAFAqStbbsr2VEvnHXtVOjLkUkYygUJGv9+sBJbp1fzFiddSQyYAoFyUrHGs9yuL6NO7WXIDIoCgXJSr8+UAegoSORQVIoSFZ67WA98yePYb6+TEdkUBQKknXOdnTy+8pG7lysvQSRwVIoSNZ5s7KRjs5uDR2JXAGFgmSdN8obKSrIo3TexKhLEck4oYWCmX3dzGrMbFdwuT9u3lfMrMLMDprZvXHtK4O2CjN7LKzaJLv97nADpfMmMnJEftSliGScsPcU/s7dbwoumwDMbAmwGrgOWAn8wMzyzSwfeBK4D1gCPBgsKzJgDWfaOVDbyoqFk6MuRSQjRfGunlXAs+7eDhwxswpgWTCvwt0rAczs2WDZ/RHUKBnqd4cbAfjDaxQKIlci7D2FR81sj5mtN7OLA7wzgaq4ZaqDtt7aRQbsdxUNjBtZwA0zx0ddikhGGlIomNkrZlaW5LIKeApYCNwEnAD+NgX1XnzctWa23cy219fXp+rXShb498MNLF9QTH6eRV2KSEYa0vCRu394IMuZ2d8Dvwxu1gCz42bPCtrooz3xcdcB6wBKS0t9ECVLFqtqOktV0zke/sP5UZcikrHCPPso/vOKHwDKgumNwGozKzKz+UAJ8BawDSgxs/lmVkjsYPTGsOqT7LPtaBMAty4ojrgSkcwV5oHmvzazmwAHjgKfB3D3fWb2PLEDyJ3AI+7eBWBmjwKbgXxgvbvvC7E+yTLbjjYxbmQBi6aNi7oUkYwVWii4+3/tY963gG8lad8EbAqrJslu246eonTuRB1PEBkCvaNZskJTWwcVJ89QOm9S1KWIZDSFgmSF7cHxhGXzFQoiQ6FQkKyw7WgThfl5en+CyBApFCQrbDt6ihtnj9fnHYkMkUJBMt75C12U1TRzy1wNHYkMlUJBMl5ZTTOd3c7SOROiLkUk4ykUJOPtqjoNwE0KBZEhUyhIxttVdZqZE0YxddzIqEsRyXgKBcl4u6pOc9Ns7SWIpIJCQTJaw5l2qk+dUyiIpIhCQTLarmOx4wk3KhREUkKhIBltV9Vp8vNMb1oTSRGFgmS0XVWnWTxtHKMK9aY1kVRQKEjGcnf2VJ/W0JFICikUJGNVnzpHy/lODR2JpJBCQTJWWU0zANfPvCriSkSyh0JBMlbZ8Wby80zftCaSQgoFyVhlNS2UTB2rT0YVSSGFgmQkd6esppnrdTxBJKUUCpKR6lraaWzr4PqrdTxBJJUUCpKR3jvIrD0FkVRSKEhGKjvejBm8b4b2FERSSaEgGamspoX5k8cwpqgg6lJEsopCQTLSgdoWlmgvQSTlFAqScVrPX6D61DkNHYmEQKEgGedQXSsAi/WmNZGUUyhIxnnnRCwUrp2hUBBJNYWCZJyDta2MKypg5oRRUZciknUUCpJxDtS2sHj6OMws6lJEso5CQTKKu3OgtlVDRyIhUShIRjnefJ7W851cO11nHomEQaEgGeXAiRYArp2uPQWRMCgUJKMcqI2debRIoSASCoWCZJRDda3MnDCKq0aOiLoUkaykUJCMUl53hpJpY6MuQyRrKRQkY3R1O4frz1AyVaEgEhaFgmSM6lNnae/spmSqjieIhEWhIBmjvO4MAAu1pyASmiGFgpl90sz2mVm3mZUmzPuKmVWY2UEzuzeufWXQVmFmj8W1zzezrUH7c2ZWOJTaJPuUn4yFwjUKBZHQDHVPoQz4OPB6fKOZLQFWA9cBK4EfmFm+meUDTwL3AUuAB4NlAb4D/J27XwOcAh4eYm2SZSpOnmHaVUWMH6Uzj0TCMqRQcPd33P1gklmrgGfdvd3djwAVwLLgUuHule7eATwLrLLYh9jcBfwsuP/TwMeGUptkn4qTrTqeIBKysI4pzASq4m5XB229tRcDp929M6E9KTNba2bbzWx7fX19SguX9OTulJ88o6EjkZD1+wW3ZvYKMD3JrMfd/YXUl9Q/d18HrAMoLS31KGqQ4XW8+TxnO7r0HgWRkPUbCu7+4Sv4vTXA7Ljbs4I2emlvBCaYWUGwtxC/vAjlwbetafhIJFxhDR9tBFabWZGZzQdKgLeAbUBJcKZRIbGD0Rvd3YHXgE8E918DRLIXIumpQmceiQyLoZ6S+oCZVQO3Ab8ys80A7r4PeB7YD7wEPOLuXcFewKPAZuAd4PlgWYAvA//LzCqIHWP40VBqk+xyuL6NCaNHMGmMzlQWCVO/w0d9cfcNwIZe5n0L+FaS9k3ApiTtlcTOThK5zJGGMyyYPCbqMkSynt7RLBmhsr6NBVM0dCQSNoWCpL0z7Z2cbG1nvvYUREKnUJC0d6S+DYCFUxQKImFTKEjaq2yInXk0f7KGj0TCplCQtFdZ34YZzC0eHXUpIllPoSBp70hDG7MmjmLkiPyoSxHJegoFSXuVDWc0dCQyTBQKktbcnSP1bXqPgsgwUShIWjvZ2k5bR5fOPBIZJgoFSWuVwemoGj4SGR4KBUlrI0fkcc+SafrIbJFhMqTPPhIJ281zJrLuodL+FxSRlNCegoiI9FAoiIhID4WCiIj0UCiIiEgPhYKIiPRQKIiISA+FgoiI9FAoiIhID3P3qGsYEjOrB969wrtPBhpSWE6qqK7BUV2Do7oGJ1vrmuvuUxIbMz4UhsLMtrt72r1dVnUNjuoaHNU1OLlWl4aPRESkh0JBRER65HoorIu6gF6orsFRXYOjugYnp+rK6WMKIiJyqVzfUxARkTgKBRER6ZH1oWBmnzSzfWbWbWalCfO+YmYVZnbQzO7t5f7zzWxrsNxzZlYYQo3Pmdmu4HLUzHb1stxRM9sbLLc91XUkebyvm1lNXG3397LcyqAPK8zssWGo6/+a2QEz22NmG8xsQi/LDUt/9ff8zawoeI0rgnVpXli1xD3mbDN7zcz2B+v//0yyzB1m1hz3+n417LqCx+3zdbGYJ4L+2mNmS4ehpsVx/bDLzFrM7E8SlhmW/jKz9WZ20szK4tommdkWMysPrif2ct81wTLlZrbmigpw96y+AO8DFgO/AUrj2pcAu4EiYD5wGMhPcv/ngdXB9A+BL4Zc798CX+1l3lFg8jD23deB/93PMvlB3y0ACoM+XRJyXfcABcH0d4DvRNVfA3n+wH8HfhhMrwaeG4bXbgawNJgeBxxKUtcdwC+Ha30a6OsC3A+8CBiwHNg6zPXlA7XE3tw17P0F3A4sBcri2v4aeCyYfizZOg9MAiqD64nB9MTBPn7W7ym4+zvufjDJrFXAs+7e7u5HgApgWfwCZmbAXcDPgqangY+FVWvweJ8CfhLWY4RgGVDh7pXu3gE8S6xvQ+PuL7t7Z3DzTWBWmI/Xj4E8/1XE1h2IrUt3B691aNz9hLu/HUy3Au8AM8N8zBRaBfzYY94EJpjZjGF8/LuBw+5+pZ+UMCTu/jrQlNAcvw719n/oXmCLuze5+ylgC7BysI+f9aHQh5lAVdztai7/oykGTsf9A0q2TCp9EKhz9/Je5jvwspntMLO1IdYR79FgF359L7usA+nHMH2O2FZlMsPRXwN5/j3LBOtSM7F1a1gEw1U3A1uTzL7NzHab2Ytmdt0wldTf6xL1OrWa3jfMougvgGnufiKYrgWmJVkmJf1WMPja0o+ZvQJMTzLrcXd/YbjrSWaANT5I33sJH3D3GjObCmwxswPBVkUodQFPAd8k9kf8TWJDW58byuOloq6L/WVmjwOdwL/08mtS3l+ZxszGAj8H/sTdWxJmv01siORMcLzoF0DJMJSVtq9LcMzwo8BXksyOqr8u4e5uZqG9lyArQsHdP3wFd6sBZsfdnhW0xWsktutaEGzhJVsmJTWaWQHwceCWPn5HTXB90sw2EBu6GNIf00D7zsz+HvhlklkD6ceU12VmnwH+I3C3BwOqSX5HyvsriYE8/4vLVAev83hi61aozGwEsUD4F3f/18T58SHh7pvM7AdmNtndQ/3wtwG8LqGsUwN0H/C2u9clzoiqvwJ1ZjbD3U8EQ2knkyxTQ+y4x0WziB1LHZRcHj7aCKwOzgyZTyzx34pfIPhn8xrwiaBpDRDWnseHgQPuXp1sppmNMbNxF6eJHWwtS7ZsqiSM4z7Qy+NtA0osdpZWIbFd740h17US+Avgo+5+tpdlhqu/BvL8NxJbdyC2Lv26tyBLleCYxY+Ad9z9u70sM/3isQ0zW0bs/0GoYTXA12Uj8FBwFtJyoDlu6CRsve6tR9FfceLXod7+D20G7jGzicFQ7z1B2+CEfSQ96guxf2bVQDtQB2yOm/c4sTNHDgL3xbVvAq4OphcQC4sK4KdAUUh1/hPwhYS2q4FNcXXsDi77iA2jhN13/wzsBfYEK+WMxLqC2/cTO7vl8DDVVUFs7HRXcPlhYl3D2V/Jnj/wDWKhBTAyWHcqgnVpwTD00QeIDfvtieun+4EvXFzPgEeDvtlN7ID9imGoK+nrklCXAU8G/bmXuLMGQ65tDLF/8uPj2oa9v4iF0gngQvC/62Fix6BeBcqBV4BJwbKlwD/E3fdzwXpWAXz2Sh5fH3MhIiI9cnn4SEREEigURESkh0JBRER6KBRERKSHQkFERHooFEREpIdCQUREevx/ztCfF7cAX9wAAAAASUVORK5CYII=\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "n      = 1\n",
        "f3     = hermit(x[:(n+1)], y[:(n+1)], y_prime[:(n+1)])\n",
        "data   = f3.linspace(n=50, domain=[-3, 3])\n",
        "test_x = np.linspace(-10, 10, 50, endpoint=True)\n",
        "test_y = np.sin(test_x)\n",
        "\n",
        "plt.plot(data[0], data[1])\n",
        "plt.plot(test_x, test_y)\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "W7v9mEcLPlvh",
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "d8f7bf96ae4f35419f880d25e72cff09",
          "grade": false,
          "grade_id": "plot02",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        }
      },
      "source": [
        "**Plot 2:** trying to interpolate a sine curve (`np.sin()`) using first 3 nodes in `x` and `y` and their corresponding derivative in `y_prime`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "qnNNOyC5Plvh",
        "nbgrader": {
          "cell_type": "code",
          "checksum": "25f490548e99af7d5fccb9fd5c653e80",
          "grade": true,
          "grade_id": "plot02_ans",
          "locked": true,
          "points": 1,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "3b6386ad-1590-4e66-b920-dfdebe6c2adc"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "n      = 2\n",
        "f5     = hermit(x[:(n+1)], y[:(n+1)], y_prime[:(n+1)])\n",
        "data   = f5.linspace(n=100, domain=[-0.85, 2.5])\n",
        "test_x = np.linspace(-5*pi, 5*pi, 100, endpoint=True)\n",
        "test_y = np.sin(test_x)\n",
        "\n",
        "plt.plot(test_x, test_y) # 25-\n",
        "plt.plot(data[0], data[1]) # 10-33\n",
        "plt.show()\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "InK1vx-bPlvi",
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "aae683a514e99dfecaa8672e1d6a4cb3",
          "grade": false,
          "grade_id": "plot03",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        }
      },
      "source": [
        "**Plot 3:** trying to interpolate a sine curve (`np.sin()`) using first 4 nodes in `x` and `y` and their corresponding derivative in `y_prime`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "eondZ70xPlvi",
        "nbgrader": {
          "cell_type": "code",
          "checksum": "29fc16454184db75748c0381eff20504",
          "grade": true,
          "grade_id": "plot03_ans",
          "locked": true,
          "points": 1,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "40e5b528-c1ef-4b41-c0eb-e546b4552566"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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6pbB1NSw4Aea9C6omfp1bnq+iqI0Ow6t/gJV3w8p7off1Ny5vnB3CfsGJsN+JMG1eado5HiMD8MKd8NTPYO1jb1xm1TBjP5j1Vpi1EN7xP2GP/UvTzvHofBUe+z48/e8QT9NdUT8VmufCcZ+Hd5xTFi/IbYMxNvcOs6Blyo47E3F49pfw0DehZ336X6yqhSP+F5zwRZi61xsWpT4VrOsa5KX2Prb2j5Qu4DtegkevhS0vQfc66N/8l4+Zdyy88zPw1veV7Rsx8Ri8+gisb4MNbbBhKQxs3bH84SuhaU84+ENwyIdh7mGVWeiNQ5n+hyKybT08cBm8dF+osGqnwFveAwd9BQ44Ndz36sPwSvLy/H+G39v7aDjzB9ByUGnbP9bGZ2HpLaGNwz0wY39Y9NWwgw5tg80roOPF8HPzCnjxbnj0O3Dil+Bdn4Oa/OdZiUz7cnj027DsdqiqhkM/Ft6QRgdhoAsGu2CwEwY6Yd0T8Jt/gCdvgtOuhHnvLGnTl7++DYCD95waqsNV98OSy2HzC7DXkXDMZ6CmAaproboueakN+9dTP4Onfw5H/h2c8IXwCYxQyQOs7xpgbWc/8yLsChq3kX545FvhTbd2UvhbDjoNpu4TLtP2gcY5sOo+eOIG+M9zoXlvOPpTcMS5MHlG8ducyevPwJ0XQvvzgIXX8YGnw95Hwl6tMH1fWPUALLst/C2P/wD2eAsc/GE46lPQmOZTWgWzcuoHbG1t9ba2tmhW9vyv4XdfgMQovOOjcNAZoVqvzTDszT2E5OoH4Q/XhJ3+5MvgmM+WtoIc6oG7vxDCvboeFp4JR54L+x6368qjdxPcewksvyNU9e//HuxzVPHanc7rz8B/fxNeuje84bb+HRx7ATTvmfl3Egl4/j9CkPZuhEM+Aosuh6l7F6nRb3TjIy/zjXte5NnzpjP10StgzaPh09PJl8HCD+76f9K1Bv7wLXj61hD8refB8RfRWzOdQy6/n/992lv5xZNrOGLedL57zuHF+YPcQ0Fw7yWwbR0c+nE45Wu7DrpEPPwPn/hRqJZrJsFR58F7vpL59VUMo8Pw8L+GAmJKC5x6BRxwSugGzGSgE1b8Nry+Xns0/N4Hr4cDFhWv3REws6Xu3pp2WaFD3sxOA74LVAM/dvcrMz02kpAf7IK7L4Zlv4Z93gln3QAzFmS3jt52+O3n4aXfhzA987rs1xGFDUvh1+dB9xp49z/DMf8Q+qqzsfL34fno2RCqlJMvC8ciiskdHr8OliyG+ubwdxz1qeyqv+G+8OJ97PtgVXD8RXDcRUUPlc/f/CDv33Ati+KPwuSZcNIlcOTfhtAer85XQtX87C+hdjK87zsce/cMDp83jfuWt/PZk/bni+8twqfIzlfh918Kn0ZmLYS/ugb2fVd262hfDo/9AJ79Bcx5B3zkp6XpIlzfBndeEAq1wz4RAj7b10r7C3DbeeFT2TGfDcVEhRzjKlnIm1k18BJwCrAe+DPwMXd/Id3j8w75Vx+BOz4Dfe1w4iVw/D/l3mfoDs/8IlQ4iTic+i/hY3Yx+u0SifAR8sGvho/IZ/8Y9j029/UN98J/XREqr6a58P7vwIGnRtfeXRnoDC++lfeEPtwzf5D9i2+srjWhC+6F38DsQ+CjtxQtVPpWP862n/81c6q6qT7hC3Dc56C+KfcVblkNd10Iax/nD83v41ObP8wwdVz70UP50BEF/KTiHrq/HvgKVNXASZfCOz+d3RvVzl68J3SrJeJh/zrkw9G1d1dig/Bf/wJ/+mFy3/5uqN7zWd8Di+HJG8L+dfaPwzGuMrerkMfdC3YBjgXuG3P7UuDSTI8/8sgjPSexIfd7/4/74qnu3zvCff3S3NaTTtda95++331xs/vPznLv2RjdutPp3ez+8w+F7f3y4+79W6Nb97o29x++K6z795e6x4ajW3c6a590v/bt7l/dw/3x690TiejWvfJe9yv3db9iL/fnb4tuvekkEr7y9m96bPF0X3vZ/r7q6YejW/foiPv9l7kvbvblXznET7zkJm/vGYxu/Tsb6HL/1SfCPvDvH3HftiG6dXetdf/xKWHdd/6j+3B/dOtOZ9My9+8fFbZ31+fcB7ujW/fKe92v2s/967Pcn7wp2n030/a61+X860CbZ8rhTAuiuAAfJnTRpG7/DfCDnR5zPtAGtM2bNy+3v3DpLeEf/dt/ch/uy20duxKPuz9xo/vXZ7tftcD9xd9Hvw1391UPuF99gPvXWgq3Y8WG3O/+5/B83XCi+9ZXot9GIuH+x++7f3WG+7cPdl/fFv023EOo3HRy+Ft+98Xwt0VtoMs33nC2++JmX3LZe/zWh56Jfhvu3vX0Xd69eE8fuHxO4d601i91//Yh4f/yx+8VZv8aHXF/4PLwP7nuGPf2FdFvI5EIr4+vtYTXy6ol0W/D3b1nUyjsFje73/rRcDtqw33ud30+mV8X5byasg75sZecK/l43H3N47n9bjbaV7j/8LjwD7n7YveRgWjWO9Dl/pvPhvV+/yj3jc9Hs95deeEu92/u4/6Nvd2X3R7dervWhgox9UlkoCu6dacTG05+imt2/9G7o33TSoZibPF0v+6Ki3xwOBbdutMY7Vzj8bFvWlFVwomE+59+FD5RXbPQfe0T0ax3V1Y9sKMSfuSaEP5R6N8a9qvFzeETb+/maNabSTzu/th14Q3lyvnRvlbWtbl/9/DQA3Hfl/MqUnYV8oXukz8WuNzdT03evjTZRfTNdI+PdHRNoYwOw5Kvwp+uCwerzv4JzF6Y+/peug9+exH0bQrjwU+8pHgHE7vWwK8/GcYRt34STv1GGD6Xi9GRcBzhkavD7ZMXh37eYo09fvHu0CfshINuh34s9+Mxg11hhNUTN+BTWvho52c49JhF/N/35fF/Hq94LIwkevwHYYjiKV+Fg8/O/Xkc6AyDCFbcBQeeFkaOFGu4Y+8muOfiMHpl9sFhhNfeR+a+vjWPw22fCq+VRZfDMRcUb+Rbx8pwvO/1p8L/44xv5f48xkfD/vXwVeE4wlk/Cidn5aGUB15rCAdeTwY2EA68ftzdl6d7fEWEfMqqJSFUhraFIWetn8xuLPpAZzio+9z/C28WZ14Hex1RuPZmEo/Bg1+Dx74Xxgofe0E48agui7HarzwcXsxbXgoHV0+7MoyrLrau1+C2v4f1T4bzCE66JLwgq8Z5YlFsKBxw+8M1Yejqoefw3Nu/xAduXsH1nziC0w+ZW9Dmv8Frfwz7x6bnYJ9j4PQrYc8shlX2bgojmtr+LZx/sOhyOPbC0pzws+J3cM8/hyGw7/w0vOf/ZnfAetsGePLGsI9OmwcfvjmM4y+2+GgY4fXwlTB5D/jA97MfwND5Ctx+Pqz/MxzyUTjjapg0Le+mlXoI5RnAdwhDKG929ysyPbaiQh6grwPu/GwYgtYwDRZ+IIzj3ve49MHiDltfhtf+AA99I5zwc8IX4YSLS3+y0qolYTTPpufC33LkuXDU3+86rHs2wv1fDieVTJ8Pp18NB763aE1Oyz2M5HnoG9C+DGYeGMJ+4VmZq75EHJ77jzBKo2c9vGVRONFszsH8eul6Lv7PZ3no4pNYMHNK+t8vlEQcnrk1vAn3bwlDA0++DJpmZ/6dzlfgj98NI8MSo/D2s+D4L8Ccg4vX7nSGesLf8ecfh/MiTr8qPM+ZPjnGY+FT7lO3wOol4IlwwtwZ3yr+EOCdbXwuVPWbl4fzVlLn4GQ636N/azivYOU94W+pqYe/ujbSEUglDflsVFzIQwiV1UvCyRQv3h3Oom2cEyrIhWeGSn9DW/L06qUw1B1+b8474IM/DHOBlAt3WPt4GGq54rfhvre+Dw49JwzD7F4X5jDpXhdOnOleC1gYqnr8Rbl39RRCIhG6KP77m2Hs9KyF4X/iHqZQiI+EIImPwNo/hTeEuYeFT2X7nbh9Ndc9tJqr71vJi18/rXRTDQz1hG6wP10fAmKfo5Pz+yTn+Zk8I5zws3pJOPmtqia8IRz3uXCiVjlZ92ToPtr8AmChOJj1tjD30qy3hbNrV90X3qT62kN3xmGfgMP/ujTnqmQyOhy6W9r+LRRrAHscEPadBSeGtr78UAj2dU+EN6mmPeGg08PZzhGfzKeQL5aRgfCOvey2UN3Hk9/baVXQ8rYdp1Xv3Rpul8FcLBl1r4M/3xSmUki9MQFMmRWq+6n7hNPDjzi3vOfHScRD8D18VehOAsBCWKamHGiaG154aar9r/xmGXc9+zrPLi7xJxQInwIfuRq2rNox9cNgN+FABFDXBEd9MpzIk5wyoSyNjoQgb1++YzqOravDJw8IczAdeGqY6+ctp5Tv/DgQion2ZTumR1nzGMT6dyyfc0io9A86vaDz4yjkS2GwG17+r3Ca9J6HQ31jqVuUm5GBMENk4+wwsVY5VevZcA8nulTXha60cb7Y/v5nbazdOsB9//TuAjcwR4l4+LQ42BVmuczn5KxSGh2BzpdDd9OeR4SJ6SrR6Eg4ONv5Csw/oWjHpnYV8mX8FlnhJk0LM9xVurrJMP+4Urcif2bZHUxOau8ZYs7UEs7HsjtV1aG7ppwmCMtFTV3orpn1tlK3JD81dTDvmHApE2XcXyBSepu2DTEnou9yFSkFhbzILnQPxJg+pYymaRbJkkJeJIOhWJyReIKmBvVqSuVSyItk0DsURns0K+SlginkRTLoHYoB0KiQlwqmkBfJIFXJN9XnMc+6SIkp5EUy2B7yquSlginkRTJIddc0NaiSl8qlkBfJQJW8TAQKeZEMepKVfLMqealgCnmRDFKVvEbXSCVTyItk0D88yuS6aqqrSvBFGyIRUciLZDAYizOpVHPIi0REIS+SwVAsUbovChGJiEJeJIOh0Tj1tXqJSGXTHiySwdBInIYaVfJS2RTyIhkMjcZpUCUvFU57sEgGQ7EEk+pUyUtlU8iLZDAUU3eNVD6FvEgGQ7G4RtdIxStYyJvZ5Wa2wcyeSV7OKNS2RAphKJbQ6BqpeIU+X/vb7v6tAm9DpCBUyctEoDJFJAP1yctEUOiQv9DMnjOzm81seroHmNn5ZtZmZm0dHR0Fbo7I+A2NJphUpzpIKltee7CZLTGzZWkuZwLXA/sDhwEbgWvSrcPdb3T3VndvbWlpyac5IpGJxRPEE65KXipeXn3y7r5oPI8zs5uA3+WzLZFiGorFAdQnLxWvkKNr5o65eRawrFDbEonaUCwBoDNepeIVcnTNv5rZYYADrwGfLuC2RCKVquTrVclLhStYyLv73xRq3SKFNjyq7hqZGPRZVCSN4dHQXVNXrZeIVDbtwSJpxOIOQF2NvvpPKptCXiSNWDxVyau7RiqbQl4kjViyu6a2WpW8VDaFvEgaI8lKvrZGLxGpbNqDRdLY3ievA69S4bQHi6SR6pOvVchLhdMeLJLGjpBXn7xUNoW8SBojo6rkZWLQHiySxo5x8nqJSGXTHiyShvrkZaLQHiyShvrkZaJQyIukMaJKXiYI7cEiacRGQ5+8Ql4qnfZgkTRi8QTVVUZ1lbprpLIp5EXSiMUT6o+XCUEhL5LGSDyhrhqZELQXi6QRiyc0b41MCNqLRdKIjboqeZkQtBeLpBGLJ6jVt0LJBKCQF0lDffIyUWgvFklDffIyUeS1F5vZR8xsuZklzKx1p2WXmtlqM1tpZqfm10yR4orF1ScvE0NNnr+/DPgQcMPYO81sIXAO8HZgT2CJmR3o7vE8tydSFCOjGicvE0NepYq7r3D3lWkWnQn8yt2H3f1VYDVwdD7bEikm9cnLRFGovXgvYN2Y2+uT9/0FMzvfzNrMrK2jo6NAzRHJTiye0FzyMiHstrvGzJYAc9Is+rK735lvA9z9RuBGgNbWVs93fSJROHa/PZhSn29vpkjp7XYvdvdFOax3A7DPmNt7J+8TqQhfOu2tpW6CSCQK9Xn0LuAcM6s3swXAAcCTBdqWiIhkkO8QyrPMbD1wLHC3md0H4O7Lgf8AXgDuBS7QyBoRkeLLq9PR3e8A7siw7ArginzWLyIi+dHwARGRCUwhLyIygSnkRUQmMIW8iMgEppAXEZnAzL18TjI1sw5gTZpFM4EtRW5ONsq9fVD+bSz39kH5t1Hty1+5tzFT+/Z195Z0v1BWIQmyOtIAAARBSURBVJ+JmbW5e+vuH1ka5d4+KP82lnv7oPzbqPblr9zbmEv71F0jIjKBKeRFRCawSgn5G0vdgN0o9/ZB+bex3NsH5d9GtS9/5d7GrNtXEX3yIiKSm0qp5EVEJAcKeRGRCaysQ97MrjazF83sOTO7w8ymjVl2qZmtNrOVZnZqidr3ETNbbmYJM2sdc/98Mxs0s2eSlx+VU/uSy0r+/O3MzC43sw1jnrczSt0mADM7Lfk8rTazS0rdnnTM7DUzez75vLWVQXtuNrPNZrZszH0zzOwBM1uV/Dm9zNpXVvufme1jZg+Z2QvJ1/Hnk/dn9zy6e9legPcCNcnrVwFXJa8vBJ4F6oEFwMtAdQna9zbgIOC/gdYx988HlpXB85epfWXx/KVp7+XAxaVux05tqk4+P/sBdcnnbWGp25Wmna8BM0vdjjHteTdwxNjXAfCvwCXJ65ekXs9l1L6y2v+AucARyetNwEvJ125Wz2NZV/Lufr+7jyZv/onwNYIAZwK/cvdhd38VWA0cXYL2rXD3lcXe7njton1l8fxViKOB1e7+iruPAL8iPH+yC+7+CNC5091nArckr98CfLCojRojQ/vKirtvdPenktd7gRXAXmT5PJZ1yO/kk8Dvk9f3AtaNWbY+eV85WWBmT5vZw2Z2Qqkbs5Nyfv4uTHbP3VzKj/NjlPNzNZYD95vZUjM7v9SNyWC2u29MXt8EzC5lYzIot/0PCF3AwOHAE2T5PJb86+jNbAkwJ82iL7v7ncnHfBkYBW4tZtuS295t+9LYCMxz961mdiTwGzN7u7v3lEn7SmZX7QWuB75OCKyvA9cQ3txl94539w1mNgt4wMxeTFarZcnd3czKbfx2We5/ZtYI3AZc5O49ZrZ92Xiex5KHvLsv2tVyM/tb4H3AyZ7shAI2APuMedjeyfuK3r4MvzMMDCevLzWzl4EDgcgPiOXSPor4/O1svO01s5uA3xW4OeNRsucqG+6+Iflzs5ndQehmKreQbzezue6+0czmAptL3aCx3L09db1c9j8zqyUE/K3ufnvy7qyex7LurjGz04AvAR9w94Exi+4CzjGzejNbABwAPFmKNqZjZi1mVp28vh+hfa+UtlVvUJbPX3KHTTkLWJbpsUX0Z+AAM1tgZnXAOYTnr2yY2RQza0pdJwxYKIfnbmd3Aecmr58LlNUnzXLb/yyU7D8BVrj7tWMWZfc8lvoI8m6OLq8m9Ic+k7z8aMyyLxNGPawETi9R+84i9NEOA+3Afcn7zwaWJ9v8FPD+cmpfuTx/adr7c+B54Lnkjjy31G1KtusMwsiGlwndYCVv007t248w6ufZ5H5X8jYCvyR0W8aS++B5wB7Ag8AqYAkwo8zaV1b7H3A8oevouTEZeEa2z6OmNRARmcDKurtGRETyo5AXEZnAFPIiIhOYQl5EZAJTyIuITGAKeRGRCUwhLyIygf1/5m4fGKuzFIEAAAAASUVORK5CYII=\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "n      = 3\n",
        "f7     = hermit(x[:(n+1)], y[:(n+1)], y_prime[:(n+1)])\n",
        "data   = f7.linspace(n=50, domain=[-0.45, 3.3])\n",
        "test_x = np.linspace(-6*pi, 6*pi, 50, endpoint=True)\n",
        "test_y = np.sin(test_x)\n",
        "\n",
        "plt.plot(data[0], data[1])\n",
        "plt.plot(test_x, test_y)\n",
        "plt.show()\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "qR1mUG58Plvi",
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "d8edc2554aa6f2946081a47711e0f556",
          "grade": false,
          "grade_id": "plot04",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        }
      },
      "source": [
        "**Plot 4:** trying to interpolate an exponential curve (`np.exp()`) using all nodes in `x` and `y` and their corresponding derivatives in `y_prime`."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "tMgsKsC9Plvi",
        "nbgrader": {
          "cell_type": "code",
          "checksum": "4978eef584205c7563997760c20241c6",
          "grade": true,
          "grade_id": "plot04_test",
          "locked": true,
          "points": 1,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "16b03b6a-64cf-433a-9113-a5d0aeb4b690"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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58uIP3XarICnkRaTX2PnEN2j1MLEL/pGy4t5xmZBCXkR6hYPrX2d4zTM8UXoVV328OuhycqZ3/CkTkd7NnV2P/wMt3p+p1/0LReHec3zbe/ZURHqtFS/NZXzjUt4cewuTxo4MupycUsiLSEE71NhE2cvfocZG8LHrvxZ0OTmnkBeRgvbGQ7cz1mto/sS3KCkp7AufOqKQF5GCtXrx81ywYw5vD7yMCR+7LuhyAqGQF5GC1HywgfJnb2VXaAinfPbHQZcTGIW8iBSk9+bcytBELXUX3U1Z/4FBlxMYhbyIFJxNLz/MlLqneXHIjZx53qygywlURs6TN7NNwAEgDsTcvdrMBgK/BsYAm4Br3X1vJrYnInI0e3ZsouKFv2eVncxHbroz6HICl8kj+U+4+1R3b7+U7HZggbtPABakXouIZE1rW4yaX3yOIm8jcs19lPftE3RJgctmd81sYE5qeg5wZRa3JSLCH+7/R85sfYv3pn2dUyZPC7qcvJCpkHfg92b2ppndnGob6u47UtM7gaEdvdHMbjazJWa2pK6uLkPliEhvs/CxH/DJnfexYvClTJ39laDLyRuZunfN+e6+zcyGAM+b2Zr0me7uZuYdvdHd7wXuBaiuru5wGRGRY3n3D3OZsfxfWV5azaRbHgKzoEvKGxk5knf3balxLfBbYAawy8yGA6TGtZnYlohIutpVLzP6xVtZGx7PSX/zGOGiwn8QSFd0O+TNrMzM+rVPAxcDK4B5wE2pxW4CnuzutkRE0jVtX0nJ/3yaWgZSctPj9O9fEXRJeScT3TVDgd9a8t+jCPCwuz9nZm8Ac83s88Bm4NoMbEtEBICWPTU03n8liUSI7Vf8inNPGhN0SXmp2yHv7huAMztorwd6x/O1RCSnWmrXs+feK+gbO8AfznuQy8/qPQ8B6So9NEREepTmjYto+eW1lMTjvH7uz7j84t59RevxKORFpMdoWj6P0ONfYJ/3Z/Wf/IJLPn5B0CXlPYW8iPQIzX/8McXzv8GyxDh2Xj6HWTPOCLqkHkEhLyL5Ld5G8zPfoOTNn/F8ohq/+ufMmjou6Kp6DIW8iOSvundpmfsFSuqW8VB8FiOv/wEzJ48IuqoeRbcaFpH8k0jAonuI33MBjbUb+Zp9lXE3/lgBfwJ0JC8i+WXfVhK//RtCm/7AS/FpPDz0a3znL2cyYkBp0JX1SAp5EckPsVZ48xckFvwbLa2t/GvbF+j/0c/z01kTKQqr0+FEKeRFJFjusOoJmP+vsHcjizmdb3MzX/n0JVw8eVjQ1fV4CnkRCc7mhfD7f4ZtS9haNJZ/bv0H9gz/GD/7i7OoGqgHfmSCQl5EcssdNrwIC38C657nQFEl/zd2C08lLuRLnzyFL1wwluJIOOgqC4ZCXkRyo60Jlv0aFt0DdWtojg7igdCnufvAJ7ls2jjmz5rIsPKSoKssOAp5Ecmu2jWw7FF4cw407eHAgNO4v99X+UndFE4bNZiHPzuZ6aN1i+BsUciLSOY11MCKx2H5Y7BrOW4haio/zve5iCd2jmFY/1LuuOYU/nz6KEIhPcUpmxTyItJ97lC3BtbNhzXPwJbXAGgZOp2Xqv6O7245jU1b+jK1agB3zxrLpacP02mROaKQF5ET09QAm16Btc/DugWwfysAbYMmsvLkW5lzsJonNxcTMuPSM4bz/fPGqFsmAAp5ETk+d9izAWoWQ82i5Lh2NeBQ3J/GUeezZPhn+eXuCTy/rQi2wbjKMm77xHBuOHs0w8t1tWpQFPIi8kHxGNSvhZ3L04Zl0FgPgBf3p2noWaw/dSavtp7Cb+pG8N7KFgBOH9mfr108jFmnD+PkIf2C3AtJUciL9Ebu0NyQPDrfvQ52v5cM9t3roH4dxJOh7eEorQNPZfeQj7O2aCIvNI7jdzv6U/9eDIABfYo4a3Q51549iEsmD9MFTHlIIS9SiNqa4cAOOLATDmxPjhtqoGELNGxOjlv2H17cLUxT2Sj2lp7EjqFTWRkfzWuHRvDK3goaa97/gnTs4DI+cVoF1SdVUD2mgnGD++rsmDynkBfJd4kEtB6Apr3JoXHPB6cP1ZE4VEfiQC1+qI7QoTrCLQ0fWk1LqA97osOoDQ1le/hCNhUPZnXzQFa2DmWLD6WtKRkHZjB6YB9OruzLZyb35eTKvkwY2o/xlWX0KynK9d5LN2U95M1sFvAjIAzc5+53ZnubIkHweIyW5sbk0NRIW8sh2pobaWtuJNbaRLzlIN5yCG89RKK1EVobsbZDhNoOEmo7RKTtEJHYISKxRqLxAxTHD1EaP0hxopEQftTt7vW+7PF+7Kaceq+g3k9il1ewiwp2+kB2eQU7vYKmcF8GhIupKC5iQJ8og8qiDCsv4bryEoaVlzKivIRh5SUM6VdCNKLTGwtFVkPezMLAj4FPAluBN8xsnruvyuZ2u8UdNr8GFjpi4P1pLK3d0trap0lOt7cddcwH2z70Pt5f5vDkB/81diDhRjzhJDw5xN1JJJKlhDHCISNkHB5/YA3uaWs6Spv7h6c7NQY8kdaWmvbEh18fHqfmJeLvv/YEeDy5TCKemk68P51IzT/8Om2ciEMiRiIRJxZrJdbWRizWRqytjXgsOR2PtZGItRKPteHxNjzWSiLeBvFWLN4GiTYs3kYo0UYo0UrYY4S9lYi3EfEYUVqJehtFFqcE6MqF+Qk3DlHCIUpo9BIarTQ1DKIxdBLN4TKaov1oCZfREulPU1E5sWg5seIKEsUDSJRWUFxcTEkkTJ9omL4lEcqLI4wqjlBWHKGsOEz/kiIqyqKURcOYqWult8n2kfwMYJ27bwAws0eB2UB+h/yDlwVdRacZyX+RdDunYwsB0dQQdyNGmDhhYoSIESZGhDbCxD1EGxHiRIhbmLgVEbciPBQhYX1JhKMkQlE8XISHoxCO4uESiBRDUQkWKSYUKYFoCaGiPoSjfQhFSwkXlxIpLiNc0pdIcV+KSvtSVFpGtLiM4miEgeEQQ8OmEJaMy3bIjwRq0l5vBc5OX8DMbgZuBhg9enSWy+kEM7jxyY6PMDs8Am1vSz+KTT967Xh8oCXGrn1N7NzXzK59Teza30ztgSZiccfSjqrDBn2LI/QtDlMcCRGNhChODUXhENGwETbDQqSO0pNBEbLUga87jpPw5FGjuxNPOHGHeCJBLAHxhBNL+BHT77fF4qnl3InFnZhDWzw5L7n3yWByDPdk9Y6RwFLzkm0JQofb2vfSMeKH20NHtIVIeHI9CUKEwmFCoRAWihAKh4mEI0QiYcLhCOGiIiLhCEWRCEVFEaJFUYqKIhQXRYlGi4gWRYlGo5QURymNFlEaDVNalDz6LYkmx6VFYUqjYQZGI5REQkR0RaYUgMC/eHX3e4F7Aaqrq4/e8ZgrZjDuwoyvdu+hVl5dv5tX3tvNK2vr2L6v+fC8QWVRJg7vx6kT+zOxsoxh/UsY2r+EoeXFDCorJpzHZy8kDv8xSCTHcScWT6T+qCS7j/zwdHJscPiINTmd/OMUTv3BCoc+OETDIYp0lCtyQrId8tuAqrTXo1JtBc/deWtLAy+9W8vLa3ezbGsD7tCvJML5Jw/mr86vYOKw/pw6rB+V/YqDLveEhUJGNGRE9Ux4kbyU7ZB/A5hgZmNJhvv1wKezvM1A7T3UyuNvbeWRxVtYX3eIcMiYWjWAL8+cwMdOqWTKyHJ1A4hIzmQ15N09Zma3Af9L8rvBB9x9ZTa3GQR3Z/HGPTy8eAvPLt9JazzB9NED+N41U7h48jDKS3VusYgEI+t98u7+DPBMtrcTBHfnqWU7+NH891hfd4h+JRFumFHFDWePZuKw/kGXJyIS/BevPdWSTXv4zu9W805NAxOH9eN710zh8ikjKI3qZEYRyR8K+S7atPsQ//7cGp5dsZOh/Yv53jVTuHr6qLw+A0ZEei+FfCfta2rjR/PX8stFmygKh/i7i07hrz82lj5R/QhFJH8poTphaU0Dtz38FtsbmrjuI1X83UWnMKS/niovIvlPIX8M7s4vXt3Ed59dzZB+JTz2pY/q8WUi0qMo5I9iX2Mbf//YO/x+1S4uOm0o//mpKQzoEw26LBGRLlHId+CdmgZuffgtdu5r5p//9DQ+f/5YXVIvIj2SQv4Ijyzewr88uYIh/Ur4ny+eyzR1z4hID6aQT/PAHzfy7adXceGplfzwuqnqnhGRHk8hn/LzlzdwxzOrufT0Ydx9wzSKdH8ZESkACnngJy+t4z+ee5c/nTKcH143VQEvIgWj14f8fy1Yy13Pv8fsqSO461Nn6g6RIlJQem3Iuzs/nL+WHy1Yy9XTR/K9a87UrQlEpOD02pD/wfPvcfcL67i2ehTfvXqKAl5EClKvDPnfvr2Vu19Yx3XVVXz36jMIKeBFpED1ug7oldv38fXfLOfssQP5t6tOV8CLSEHrVSHf0NjKF//7TQaURvnxX0zXWTQiUvB6TXdNPOH8n0feZte+FuZ+8VwG9+25D88WEemsXhPyd/3+XV5Zu5s7rz6DqVUDgi5HRCQnekV/xXMrdvCTl9Zzw4zRXD9jdNDliIjkTMGH/NpdB/jq3HeYNnoA37piUtDliIjkVLdC3sy+ZWbbzGxpargsbd7XzWydmb1rZpd0v9Sua2yNccsv36Q0GuGevziL4ogesi0ivUsm+uR/4O7/md5gZpOA64HJwAhgvpmd4u7xDGyv84U9/x4bdh/ikb8+h2HlelyfiPQ+2equmQ086u4t7r4RWAfMyNK2OrRi2z4eeHUTN8wYzbnjB+Vy0yIieSMTIX+bmS0zswfMrP0JGyOBmrRltqbaciKecL7+m+VU9Ily+6yJudqsiEjeOW7Im9l8M1vRwTAbuAcYD0wFdgB3dbUAM7vZzJaY2ZK6urou70BHHnxtE8u37eObfzaJ8j5FGVmniEhPdNw+eXe/qDMrMrOfA0+nXm4DqtJmj0q1dbT+e4F7Aaqrq70z2zqWbQ1N3PX7d/nEqZVcPmV4d1cnItKjdffsmvQUvQpYkZqeB1xvZsVmNhaYACzuzrY6w9355pMrcIdvzz5dD98WkV6vu2fX/IeZTQUc2ATcAuDuK81sLrAKiAG35uLMmudW7GT+6lq+cdlpVA3sk+3NiYjkvW6FvLt/5hjz7gDu6M76u2J/cxvfnLeSySP687nzxuRqsyIiea1g7l3zvefeZffBFu67qVqP8BMRSSmINHxz817++/XN3PTRMUwZpZuPiYi0K4iQj4SMCyZU8tWLTw26FBGRvFIQ3TVnVg3gob/K6QW1IiI9QkEcyYuISMcU8iIiBUwhLyJSwBTyIiIFTCEvIlLAFPIiIgVMIS8iUsAU8iIiBczcu30L94wxszpgc9B1pAwGdgddRDeo/uD19H1Q/cHr7D6c5O6VHc3Iq5DPJ2a2xN2rg67jRKn+4PX0fVD9wcvEPqi7RkSkgCnkRUQKmEL+6O4NuoBuUv3B6+n7oPqD1+19UJ+8iEgB05G8iEgBU8iLiBQwhXyKmQ00s+fNbG1qXHGU5eJmtjQ1zMt1nR3UM8vM3jWzdWZ2ewfzi83s16n5r5vZmNxXeXSdqP+zZlaX9jP/QhB1Ho2ZPWBmtWa24ijzzczuTu3fMjObnusaj6UT9V9oZvvSfv7/kusaj8XMqszsRTNbZWYrzezLHSyTt59BJ+vv3mfg7hqS30v8B3B7avp24N+PstzBoGtNqyUMrAfGAVHgHWDSEcv8DfDT1PT1wK+DrruL9X8W+H9B13qMffgYMB1YcZT5lwHPAgacA7wedM1drP9C4Omg6zxG/cOB6anpfsB7HfwO5e1n0Mn6u/UZ6Ej+fbOBOanpOcCVAdbSWTOAde6+wd1bgUdJ7ke69P16DJhpZpbDGo+lM/XnNXd/GdhzjEVmAw950iJggJkNz011x9eJ+vOau+9w97dS0weA1cDIIxbL28+gk/V3i0L+fUPdfUdqeicw9CjLlZjZEjNbZGZB/yEYCdSkvd7Kh39BDi/j7jFgHzAoJ9UdX2fqB/jz1L/Zj5lZVW5Ky5jO7mM+O9fM3jGzZ81sctDFHE2qK3Ia8PoRs3rEZ3CM+qEbn0FBPMi7s8xsPjCsg1nfSH/h7m5mRzu39CR332Zm44AXzGy5u6/PdK1y2FPAIwLkhd8AAAHxSURBVO7eYma3kPyv5E8Crqk3eYvk7/xBM7sMeAKYEHBNH2JmfYHHga+4+/6g6+mq49Tfrc+gVx3Ju/tF7n56B8OTwK72f+FS49qjrGNbarwBeInkX96gbAPSj2xHpdo6XMbMIkA5UJ+T6o7vuPW7e727t6Re3geclaPaMqUzn1Hecvf97n4wNf0MUGRmgwMu6wPMrIhkQP7K3X/TwSJ5/Rkcr/7ufga9KuSPYx5wU2r6JuDJIxcwswozK05NDwbOA1blrMIPewOYYGZjzSxK8ovVI8/4Sd+va4AXPPVtTh44bv1H9J1eQbLPsieZB9yYOsPjHGBfWrdg3jOzYe3f4ZjZDJKZkS8HCaRqux9Y7e7fP8piefsZdKb+7n4Gvaq75jjuBOaa2edJ3u74WgAzqwa+6O5fAE4DfmZmCZI/6DvdPbCQd/eYmd0G/C/JM1UecPeVZvZtYIm7zyP5C/RLM1tH8gu264Oq90idrP9vzewKIEay/s8GVnAHzOwRkmc/DDazrcA3gSIAd/8p8AzJszvWAY3A54KptGOdqP8a4EtmFgOagOvz6CABkgdanwGWm9nSVNs/AaOhR3wGnam/W5+BbmsgIlLA1F0jIlLAFPIiIgVMIS8iUsAU8iIiBUwhLyJSwBTyIiIFTCEvIlLA/j8Kxk7StdbmZQAAAABJRU5ErkJggg==\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "#defining new set of given node information: x, y and y'\n",
        "x       = np.array([0.0, 1.0,          2.0       ])\n",
        "y       = np.array([1.0, 2.71828183,  54.59815003])\n",
        "y_prime = np.array([0.0, 5.43656366, 218.39260013])\n",
        "\n",
        "\n",
        "f7      = hermit( x, y, y_prime)\n",
        "data    = f7.linspace(n=50, domain=[-0.7, 2.45])\n",
        "test_x  = np.linspace(-0.5, 2.2, 50, endpoint=True)\n",
        "test_y  = np.exp(test_x**2)\n",
        "\n",
        "plt.plot(data[0], data[1])\n",
        "plt.plot(test_x, test_y)\n",
        "plt.show()\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "DYVL-_XcPlvi",
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "5419db887410ebe73b42fa5f21ffc50e",
          "grade": false,
          "grade_id": "plot05",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        }
      },
      "source": [
        "**Plot 5:** trying to interpolate $y = (x-3)^2 + 1$ using all nodes in `x` and `y` and their corresponding derivatives in `y_prime`.\n",
        "\n",
        "For this plot you might be able to see only one curve due to the two curves overlapping. This means that our polynomial is accurately interpolating the original function.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "tuHX_czoPlvj",
        "nbgrader": {
          "cell_type": "code",
          "checksum": "92cc6293f29ebf42544cedf49e4c1fc5",
          "grade": true,
          "grade_id": "plot05_test",
          "locked": true,
          "points": 3,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 265
        },
        "outputId": "8f380875-2bd3-4938-c352-42b102155eb6"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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ZWUd/hA03PfoPJ7P/bxlcsoiFb97vdRwRaUTqM+fe3jm3xXd7K9Ded7szsLHa4zb5xn7AzKaYWbqZpRcUFNQjRmhLm3wrWbGjGLrsQdYtz/A6jog0En75hao7sMr2Ea207Zx7xjmX6pxLTUxM9EeMkGQREXS5dBr7LI6Kt35FaYkuLiYih1efct/2/XSL73u+bzwP6FrtcV18Y1JHbTt0ZeOx99O7ci2Lp2tpPhE5vPqU+3vAJb7blwCzqo1f7DtqJg0oqjZ9I3U0dOwFLGozkbStM8ia+57XcUQkyNX2UMhXgXlAXzPbZGZXAPcCJ5vZamCc7z7Ah8BaIBeYClzt99RhavDlT5IX2YlOn93AjvzNXscRkSBmB6bLvZWamurS09O9jtEorF0yjy5vncHypiMY8ruPsAidhyYSrswswzmXWtM2NUMj03PwUWT0u4WhJQtIf/2vXscRkSClcm+ERk++lYy4Yxi64hFys+Z6HUdEgpDKvRGKiIyg5xXPs90SiJv1K/bo6pEichCVeyOV0LY9u057mvZVBax89gpdnkBEfkDl3oj1H3Uy6T1/Q+qez5j/zmNexxGRIKJyb+RG/fIucmKHk7LkHnJzFnodR0SChMq9kYuIjKTTZS+wz5oS/fZl7N290+tIIhIEVO4hoHX7JLad8g+6VOWxcuplmn8XEZV7qBhw9Oks6HENI/Z8Tvqb93kdR0Q8pnIPIWkX/YXFcWkMXfYAqzM+8zqOiHhI5R5CIiIj6fmrlyiIaEuL93/FjnxdjFMkXKncQ0yrNu3YN/F5Wrnd5D33Syr27/c6koh4QOUegpJTjiF7yB0MLlvMgum3eh1HRDygcg9RI8+5kYyECRy1aRoZc17zOo6INDCVe6gyY9CUqXwX3ZM+X93I+pVZXicSkQakcg9hTeKa0+zi16iwaHj9Qop36wJjIuGizuVuZn3NLKva124zu9HM7jSzvGrjp/kzsByZ9kl9yBv3FJ0rN5P7zwtxVZVeRxKRBlDncnfOrXTOpTjnUoARwD5gpm/zI99vc8596I+gUneDxpzBgj63MLT4GxbqF6wiYcFf0zJjgTXOufV+ej3xs6MvuJ2FLcczesNUFs9+yes4IhJg/ir384FXq92/1syyzWyamSXU9AQzm2Jm6WaWXlBQ4KcYcigWEcGQq6axKqoPfb+5hdylWrNWJJTVu9zNLAY4C3jTN/QU0AtIAbYAD9X0POfcM865VOdcamJiYn1jSC3ExsXT+rI3KLVYmrz1S7YXbPE6kogEiD8+uU8AFjvntgE457Y55yqdc1XAVGCUH/YhftK2cw92nvkc7asK2DJ1MuVlpV5HEpEA8Ee5X0C1KRkz61ht29lAjh/2IX7Ue8Q4ckbew6DybL596nJdIlgkBNWr3M0sHjgZeKfa8P1mtsTMsoETgZvqsw8JjOFnXMm8zlcwctcHLJrxZ6/jiIif1avcnXPFzrk2zrmiamMXOecGO+eGOOfOcs5pYjdIjb78ARbFn0Dq6sf4ds7LXscRET/SGaphLCIykgG/eZnV0ckkf3Uzq7O+8jqSiPiJyj3MxTdrTpsr3ma3NafluxexZdNaryOJiB+o3IW2HZMonTyDePaxd9rPKNqla9CINHYqdwGg+4DRrD/xSXpUrmP9Uz+nvKzM60giUg8qd/mvAcefS1bKnQwpyyD7HxfpEEmRRkzlLj+QevYNzEuaQmrRbBY9/1uv44hIHanc5UfSLr2P+a3OYNTGaWS89aDXcUSkDlTu8iMWEcHwq58nK3Y0KUvuZvHHOgZepLFRuUuNYmJi6HPtm6yJTmbA1zeyZN7HXkcSkSOgcpdDatqsJe2unMX2yLYkfXQpq7Pnex1JRGpJ5S4/qVViJ6IvnUWZxZLwznlsWJ3tdSQRqQWVuxxWu6S+lF3wNpFUEf3KOeTrLFYRv3j6izXMWbYtIK+tcpda6dp3GIWTXqWZ20vptDMpKtzsdSSRRu3ZL9dy779XMHvp1oC8vspdai05ZQzrTn2edpXbKHjqDHbv2u51JJFG6eX567n7g+WcNrgD954zOCD7ULnLERl89ASWH/cPulWsY+Pfz2Tv3t1eRxJpVN5M38gf3s1hXP92PHreMKIiA1PDKnc5YsPGTmZp2oP0L1/G2sfPYl/xHq8jiTQK7327mVvfzubY5LY8+YvhxEQFroL9sUD2Ot/KS1lmlu4ba21mc8xste97Qv2jSjBJmXA536bew6CyLHIfP4vSfXu9jiQS1D7K2cpNr2eR2r01z1yUSmx0ZED3569/Nk50zqU451J9928DPnXOJQOf+u5LiBl25tUsHnYXg0ozWfX4REpLir2OJBKUPl+Zz3WvLmZIl5ZMu3QkcTGBLXYI3LTMROAF3+0XgEkB2o94LHXSdWQM/TNDStNZ/fhEykv3eR1JJKhkfvkvbnvpc/p2aM70y0bRrElUg+zXH+XugI/NLMPMpvjG2ldbO3Ur0P7gJ5nZFDNLN7P0goICP8QQr4w85wbmD/wTg0sWseKxSZSp4EUAyPzkNQZ8cin3Np3BS5ePpmVcdIPt2x/lPsY5NxyYAFxjZsdV3+iccxz4B4CDxp9xzqU651ITExP9EEO8lHbuzcwb8EeGlCxg+aOTNEUjYW/xRy8y8Mur2RjdjRFXPkNCfEyD7r/e5e6cy/N9zwdmAqOAbWbWEcD3Pb+++5Hgd9TkW1g08I+klC5g1aNnULynyOtIIp5I/+BZhsy7ge9ikml37WxatPnR5EXA1avczSzezJp/fxs4BcgB3gMu8T3sEmBWffYjjcfIc29hccrdDCzNZP3jE3Sik4SdRe8+ybCFt7CqyQA6X/cRLVq19SRHfT+5twe+MrNvgYXAB865j4B7gZPNbDUwzndfwsTwSdeRnfYwyeUr2PbkKRRtD8zp1SLBZtHbDzEi8w8si02h+/Uf0qyFd0eB1+vXts65tcDQGsa3A2Pr89rSuA2bcDnfxjSl39xr2fz3U6i48l+0aZ/kdSyRgJk/46+krbqfrLhR9LvhXWLj4j3NozNUJWCGjj2fVeOm0a5yKyVPn8KW9au8jiTid66qim+m3kjaqvvJjB9D/xtneV7soHKXABt87FmsP30GLVwR9vwE1i5L9zqSiN9U7C9n4RMXc3Te8yxsfSZDbnqXJrFNvY4FqNylAQwYNY4dP59JFBW0feMsls77yOtIIvVWWlLMkkcmMXrn+8zvchkjr32RyKiGO479cFTu0iC6D0qj4rLZFFlLen/0SzJnv3D4J4kEqaKdhax95FSG7fua+X1vJe1Xj2IRwVWnwZVGQlqHbv1ofvVnrIvuxdBvbmDBG/d5HUnkiBVuXs/2J8bSu2wZGakPknbB772OVCOVuzSoVokdSbrpE7Lj0xi97B4WTL0BV1XldSyRWlmzZAEVz5xEh8otrDzpWUac8WuvIx2Syl0aXFx8cwbd9B4LEs5kdN500h+drMsVSND79rPX6fDWWURQxeZz3mHw8ed4HeknqdzFE1HRMYy67kUW9LiakbvnsO6hsRRs3eB1LJEfcc4x/9W/MeiLK9kS1Rl+/Rm9h47xOtZhqdzFMxYRwehL/kZm2mN027+GyqdPZM2SBV7HEvmviv3lzP/7r0hbeS9L4o+i002f0a5zD69j1YrKXTw3bPyl5J39DhFU0fGtM8maM8PrSCLsKdpBzkOnc1ThWyzocAFDbn6fps1aeR2r1lTuEhR6pxxLxJTPyYtOYshXVzP/xT/qF63imfUrs9j52BgGlaSzaOAfGX3V00RENcwiG/6icpeg0bZTd7re/DmZzY8nbe3jZDxyri4bLA1u8eyXaTNjPM2q9rDi1JcZee4tXkeqE5W7BJXYps0ZfvNM5ne/muG7PyX/kTGsX/Wt17EkDFRWVPDN1BsZPu8aNkd3pfyK/zDo6NO9jlVnKncJOhYRQdqlf2PZ2Om0qtpJm1dOJeMjndEqgbNrez45D57K0XnPsyjhDLrd8h86dO3ldax6UblL0Bp03CTKL/+czdFdGTH/euY9fTX795d7HUtCTG7WXIqfHEP/kkwWDfojI69/mSax3l/Vsb5U7hLU2icl0/2WuSxsezZHbX2FVfefROHmdV7HkhDgqqpYOOMvJM2cRJSrYP2ZbzDy57eAmdfR/ELlLkEvJjaOUddOJ2P4vfQoX0XkM2N0uKTUy478zWQ/MJ5Rqx5iadPRxFw7j+TUcV7H8qs6l7uZdTWzz81smZktNbMbfON3mlmemWX5vk7zX1wJZyPO+g0Fv/iY7ZHtSPn6N8x/4lJKivd6HUsamSVff0DFP46h/74MFvS7lZTffUBC24ZfwDrQzDlXtyeadQQ6OucW+xbJzgAmAZOBvc65B2v7WqmpqS49XYs4SO2Ule4jc/otpG19hXURXak4+1l6D07zOpYEuf3lZSx84TbSNj3P5oiO7D/7OXoOOdrrWPViZhnOudSattX5k7tzbotzbrHv9h5gOdC5rq8nUltNYpuSdtU/yDnpBZpV7aHrW2fwzSt3U1VZ6XU0CVIbli9kw31HcUzeNL5tfSptfzu/0Rf74fhlzt3MugPDgO8vDHKtmWWb2TQzq3H5bzObYmbpZpZeUFDgjxgSZgYdN4moa+axIn4ER69+gBX3nUDemhyvY0kQqdhfzvwX/kCH1yaQUFlARtoTDL/hdeKatfQ6WsDVeVrmvy9g1gz4Avirc+4dM2sPFAIOuIsDUzeX/9RraFpG6sNVVbHo3Sfo9+29RFNBVp/rGXXe7UQ2stPFxb++W5FF+VtT6FuxksXxx9H14qdIbN/F61h+FZBpGd8LRwNvA684594BcM5tc85VOueqgKnAqPrsQ+RwLCKCUefcQMmUr1nVdBhHrX6QVfeNYd2KLK+jiQcqysv4+sX/R8dXx9GhIo/FIx9k2G9nhVyxH059jpYx4DlguXPu4WrjHas97GxA/0+WBtG+c0+G/O4jMkbcS6f9G+j46ji+nn4H5WWlXkeTBrJm8edsvG80x6x9jFXNRlJ51TcMP/3XQbe+aUOoz9EyY4AvgSXA95fv+z1wAZDCgWmZdcCVzrktP/VampYRf9u+bQMbX7qGlL1z2RDRhb1j/8aAY87yOpYESNGOAla+cguphbMosAQ2pf2ZEadeFDInJB3KT03L1HvO3R9U7hIomZ+8Rruv/x+d3TYymp9I5/MepkOXnl7HEj+pqqxi4fvPkJx1D63cbha2n8yAX9xLy1atvY7WIFTuEtZK9hWT+eqdDN/wPJVEktXzSkac93tiY2O9jib1sCZrLmX/uo0BFUtZHdWHyLMeC/nDGw+mchcBNn+3nMI3b2TIvvlstE7sSLuNISdfFJbzsY1Z/sZc8t6+jWG75rCdlqwfciPDJl6PRYbf0VEqd5Fqlv7nTZrO/Qs9qjawKrofleP+TP/R472OJYdRtHMHOW/eyYi8GRiwsOMvGHL+nWEzBVMTlbvIQfbv38+iWX+nV85jtGcHmXFpNDv9bpIHjfQ6mhykdN9eFr/zMH1zn6UNRWS0GEfHc+6hU/e+XkfznMpd5BBKiveQ/fa9DFj7HE1dKRktxtLxjDvo2ne419HCXlnJXnLee5xuy/9JW3axvEkKsaf9hR5Dj/c6WtBQuYscxu7tW1n51l0M3PwmsZTzbYvjaHHy7fQacpTX0cJOcfFesmc9Su9Vz5LITpZEDyHixNsZeLQuMHswlbtILRVu28yKd+9j6ObXaW4lLI5LI+r43zF49FgsxI+Z9lrR9q3kvPcYfdbPIJFdLI8ZzP7jbmXwMWfovT8ElbvIESraUciK9x6g77qXacVeVkT1o2T4rxly8sVERsd4HS+k5OVms2X2IwzM/xdxVk5OXCoxx99Mn9GnhfxJSPWlchepo9K9u8j54O90WPESXdwWCkkgN2kyvSZcS2LHJK/jNVoVFRVkzZ1FxKJnGF4ynzIXRWarU2h36k30HKDLUdWWyl2kniorK8n87E2i06cytCydMhdFVvPjaZJ6IYPHTNQVKGupMC+XNbP/SdKGmXSkgB20YFXX8+h12vX6x7IOVO4ifrRp9bds/eRJ+mz7gBYUk08b1nc+nXbHXkq3fiO8jhd09hbvZdnnrxGXM4OBJYuJMEdOk1r6giMAAAarSURBVGFUpFzEoJMuIKpJU68jNloqd5EAKC/dx5LPXseyX2VIySKirIpVkb3J73YG3Y45j669Bngd0TNlJXtYNncmFTkzGbD7G+KtlK20JbfzRLqNnULXnv28jhgSVO4iAVawZQPfff4CbdfOpGfFGgC+i+xOYeexJAw/m56DjyEiMrQvc7C9MJ/cee8Rvep9+u2ZR1PK2EkLctucQMsRP6f3qNOJ0PSVX6ncRRpQ/voVrP3qDZqvm02/8qVEmmMrbVjXcjQRPY+j58jxtO3Uw+uY9Va5v5zczM/ZmT2bhK1f0Xv/KiLNsZ2WrEw4gaYp5zDw6NOI1tFFAaNyF/FIwbY81s97h+jc2fTYu5gWFAOwKaITm1uNoCppDO0HHE1Sr4FERkZ6nPanFe0o4Lvsuexds4D4gkz6lGQTb6VUOiM3pi9FHY+lzZBT6ZFyoj6hNxCVu0gQqKqoYO3SBWzL/oSmeV/TuzSb5pQAsMfFsbFJMsWtB2CdhtKs2zA69ehPixatGjxnZUUFm9evpHBdDvs2Lye2MIf2e5bSpWrzgT+HMzZFdmFr61TodSJ9R59Gy9aJDZ5TPCp3MxsPPAZEAs865+491GNV7hKOqioqyFuVTv7KBVTkZdFy1zK67V9LnJX/9zGFtKIwuhPF8Unsb9EN17ILTVok0rRVO5q3bk/Lth2Jb55Qq8sWu6oqyspKKd1bxK7CPPYU5FGyczOVRVtwe7cRu28zrfeto2PlFprY/v8+r4AENsQNoLx9Cs17p9Fj8BjiW4bvlRiDSYOXu5lFAquAk4FNwCLgAufcspoer3IXOaC8vJyta7PZuS6b0m25ROz8jvjiDbTZv5n27KjxOftdJGXEsN+iqCSSCqKotEgqiSLK7acJZcS6MmIpI9Jq/vte4mIojGhLYWw3ylr2JLJdX1p2HUDHXoNp3rpDIP/IUg8/Ve6BmhgbBeQ659b6ArwGTARqLHcROSAmJoakfqkk9fvx39fykmKKCjaxe8c2indupbSogIo9hUSUbieiohQq90PVfqyqAqvaD64SImKojIqDqFhcdBxExUFMPNEtOxCX0IkWiV1o3aELTZsl0NWMrh78mSUwAlXunYGN1e5vAkZXf4CZTQGmACQl6cw0kcOJiYsnMakviUm6jrkcnmcH3jrnnnHOpTrnUhMT9csYERF/ClS558EP/ofXxTcmIiINIFDlvghINrMeZhYDnA+8F6B9iYjIQQIy5+6cqzCza4HZHDgUcppzbmkg9iUiIj8WsNPInHMfAh8G6vVFROTQQvtKRiIiYUrlLiISglTuIiIhKCguHGZmBcB6D3bdFij0YL/BTu/Loem9OTS9N4cWqPemm3OuxhOFgqLcvWJm6Ye6LkM40/tyaHpvDk3vzaF58d5oWkZEJASp3EVEQlC4l/szXgcIUnpfDk3vzaHpvTm0Bn9vwnrOXUQkVIX7J3cRkZCkchcRCUFhV+5mdq6ZLTWzKjNLPWjb7WaWa2YrzexUrzIGAzO708zyzCzL93Wa15m8ZmbjfT8buWZ2m9d5gomZrTOzJb6flbBeM9PMpplZvpnlVBtrbWZzzGy173tCoHOEXbkDOcA5wNzqg2Y2gAOXJh4IjAf+4VsLNpw94pxL8X2F9UXgfD8LfwcmAAOAC3w/M/I/J/p+VsL9WPfpHOiQ6m4DPnXOJQOf+u4HVNiVu3NuuXNuZQ2bJgKvOefKnHPfAbkcWAtWBKqtC+ycKwe+XxdY5Aecc3PhR6uZTwRe8N1+AZgU6BxhV+4/oaZ1Xzt7lCVYXGtm2b7/Zgb8v5FBTj8fP80BH5tZhm99ZPmh9s65Lb7bW4H2gd5hwK7n7iUz+wToUMOmO5xzsxo6T7D6qfcJeAq4iwN/ae8CHgIub7h00siMcc7lmVk7YI6ZrfB9gpWDOOecmQX8GPSQLHfn3Lg6PC3s1n2t7ftkZlOBfwU4TrALu5+PI+Gcy/N9zzezmRyYxlK5/882M+vonNtiZh2B/EDvUNMy//MecL6ZNTGzHkAysNDjTJ7x/QB+72wO/CI6nGld4EMws3gza/79beAU9PNysPeAS3y3LwECPoMQkp/cf4qZnQ08ASQCH5hZlnPuVOfcUjN7A1gGVADXOOcqvczqsfvNLIUD0zLrgCu9jeMtrQv8k9oDM80MDnTKDOfcR95G8o6ZvQqcALQ1s03An4B7gTfM7AoOXN58csBz6PIDIiKhR9MyIiIhSOUuIhKCVO4iIiFI5S4iEoJU7iIiIUjlLiISglTuIiIh6P8DOFhL2HgzXRwAAAAASUVORK5CYII=\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "#defining new set of given node information: x, y and y'\n",
        "x       = np.array([1.0, 3.0, 5.0])\n",
        "y       = np.array([5.0, 1.0, 5.0])\n",
        "y_prime = np.array([-4.0, 0.0, 4.0])\n",
        "\n",
        "f7      = hermit( x, y, y_prime)\n",
        "data    = f7.linspace(n=50, domain=[-11, 10.5])\n",
        "test_x  = np.linspace(-10, 10, 50, endpoint=True)\n",
        "test_y  = (test_x-3)**2 + 1\n",
        "\n",
        "plt.plot(data[0], data[1])\n",
        "plt.plot(test_x, test_y)\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "UgP_TWwOPlvj",
        "nbgrader": {
          "cell_type": "markdown",
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        }
      },
      "source": [
        "## Part 2: Polynomial Interpolation Using Newton's Divided Difference Form\n",
        "---\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "AgAbn67kPlvj",
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "030540dd5ee10acfca978c35d1e76cb8",
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      "source": [
        "### Newton's Divided Difference Form\n",
        "\n",
        "Newton form of a $n$ degree polynomial:\n",
        "\n",
        "$$p_n(x) = \\sum_{k=0}^{n} a_kn_k(x),$$\n",
        "where the basis is:\n",
        "$$n_k(x) = \\prod_{j=0}^{k-1}(x-x_j),$$\n",
        "$$ n_0(x)=1,$$\n",
        "\n",
        "and the coefficients are: $$a_k = f[x_0, x_1, ..., x_k],$$\n",
        "\n",
        "where the notation $f[x_0, x_1,\\dots,x_k]$ denotes the divided difference.\n",
        "\n",
        "By expanding the Newton form, we get:\n",
        "\n",
        "$$p(x) = f [x_0] + (x-x_0) f[x_0,x_1] + (x-x_0) (x-x_1) f[x_0,x_1,x_2] + \\dots + (x-x_0) (x-x_1) \\dots (x-x_{k-1}) f[x_0, x_1, \\dots, x_k]$$\n",
        "\n",
        "### Tasks: \n",
        "1. Complete the `calc_div_diff(x,y)` function which takes input `x` and `y`, and calculates all the divided differences. You may use the lambda function `difference()` inside the `calc_div_diff(x,y)` function to calculate the divided differences.\n",
        "\n",
        "2. Complete the `__call__()` function which takes an input `x`, and calculates `y` using all the difference coefficients. `x` can be a single value or a numpy. In this case, it is a numpy array.\n",
        "\n",
        "`res` variable must contain all results (corresponding y for x)."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "id": "iLUOVj3APlvj",
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          "cell_type": "code",
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      },
      "outputs": [],
      "source": [
        "class Newtons_Divided_Differences:\n",
        "  \n",
        "    def __init__(self, differences):\n",
        "        self.differences = differences\n",
        "\n",
        "    def __call__(self, x):\n",
        "        '''\n",
        "        this function is for calculating y from given x using all the difference coefficients\n",
        "        x can be a single value or a numpy\n",
        "        the formula being used:\n",
        "        f(x) = f [x0] + (x-x0) f[x0,x1] + (x-x0) (x-x1) f[x0,x1,x2] + . . . + (x-x0) (x-x1) . . . (x-xk-1) f[x0, x1, . . ., xk]\n",
        "\n",
        "        work on this after implementing 'calc_div_diff'. Then you should have\n",
        "        f[x0], f[x0,x1]. . . . . ., f[x0, x1, . . ., xk] stored in self.differences\n",
        "\n",
        "        'res' variable must return all the results (corresponding y for x)\n",
        "        '''\n",
        "\n",
        "        res = np.zeros(len(x)) #Initialization to avoid runtime error. You can change this line if you wish\n",
        "\n",
        "        #----------------------------------------------\n",
        "        \n",
        "        for i in  range(len(x)):\n",
        "\n",
        "          for c in  range(len(data_x)):\n",
        "\n",
        "            coefff=1\n",
        "\n",
        "            for j in  range(c):\n",
        "              coefff  = coefff*(x[i]-data_x[j])\n",
        "          \n",
        "            res[i] +=  coefff*self.differences[c]\n",
        "            \n",
        "        #----------------------------------------------\n",
        "        return res"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        ""
      ],
      "metadata": {
        "id": "EPdRXfPdskLp"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "id": "HZhqt-6JPlvj",
        "nbgrader": {
          "cell_type": "code",
          "checksum": "abd6bbf6334a986a608c38206f702989",
          "grade": false,
          "grade_id": "cell-fbe4c16afb6b3b32",
          "locked": false,
          "schema_version": 3,
          "solution": true,
          "task": false
        }
      },
      "outputs": [],
      "source": [
        "# basic rule for calculating the difference, implanted in the lambda function. \n",
        "# You may use it if you wish\n",
        "difference = lambda y2, y1, x2, x1: (y2-y1)/(x2-x1)\n",
        "\n",
        "def calc_div_diff(x,y):\n",
        "    assert(len(x)==len(y))\n",
        "    #write this function to calculate all the divided differences in the list 'b'\n",
        "    b = [y[0]]  #initializing # F[x0], \n",
        "\n",
        "    #----------------------------------------------\n",
        "    \n",
        "    for i in  range(1,len(x)):\n",
        "\n",
        "      for j in  range(len(x)-i):\n",
        "\n",
        "        y[j] = difference(y[j+1],y[j],x[i+j],x[j])\n",
        "      \n",
        "      b.append(y[0])\n",
        "\n",
        "    #----------------------------------------------\n",
        "    return b"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "FancUC_xPlvk",
        "nbgrader": {
          "cell_type": "code",
          "checksum": "a23dd58fcab86885a166b1e138b9ea08",
          "grade": true,
          "grade_id": "cell-72b9329c6c32517e",
          "locked": true,
          "points": 5,
          "schema_version": 3,
          "solution": false,
          "task": false
        }
      },
      "outputs": [],
      "source": [
        "data_x = [-3.,-2.,-1.,0.,1.,3.,4.]\n",
        "data_y = [-60.,-80.,6.,1.,45.,30.,16.]\n",
        "\n",
        "test = calc_div_diff(data_x, data_y)\n",
        "\n",
        "assert len(test) == len(data_x)\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "affgKuuNPlvk",
        "nbgrader": {
          "cell_type": "markdown",
          "checksum": "e2b6e9e71eaf31b857393044c2768ac0",
          "grade": false,
          "grade_id": "cell-f6a9f264b02b2642",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        }
      },
      "source": [
        "### Plotting the polynomial\n",
        "* `data_x` and `data_y` are the coordinates of the given nodes.\n",
        "\n",
        "* `differences` is a list which contains the divided differences as each of its elements: $f[x_0], f[x_0,x_1], f[x_0,x_1,x_2], \\dots$\n",
        "\n",
        "* `obj` is an object of type `Newtons_Divided_Differences`. Creating the object runs the constructor of the class where the `difference` are stored in `self.differences`.\n",
        "\n",
        "* `X` contains $x_i$ values through which we want to plot our polynomial.\n",
        "\n",
        "* Calling the object using `obj(X)` executes the `__call__()` function of the class, which returns a numpy array containing the corresponding $y_i$ values, and storing them in variable `F`.\n",
        "\n",
        "* Using `plt.plot(X,F)`, we plot the $(x_i, y_i)$ pairs of the polynomial."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "deletable": false,
        "editable": false,
        "id": "akcjxODFPlvk",
        "nbgrader": {
          "cell_type": "code",
          "checksum": "3f9ba22ae7061e1d74101e40a2d5a7a4",
          "grade": true,
          "grade_id": "cell-a16b9abefbcdc519",
          "locked": true,
          "points": 5,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 268
        },
        "outputId": "beeef9f8-0d92-4313-af60-634753175753"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "data_x = np.array([-5.,-1,-0.5,0.5,3.5,7.,9])\n",
        "data_y = np.array([-30., -50., 36., 31., 75., 60., 46.])\n",
        "differences = calc_div_diff(list(data_x), list(data_y))\n",
        "p = Newtons_Divided_Differences(list(differences))\n",
        "test_x = np.linspace(-7, 10, 50, endpoint=True)\n",
        "test_y = p(test_x)\n",
        "\n",
        "#generating 50 points from -3 to 4 in order to create a smooth line\n",
        "plt.plot(test_x, test_y)\n",
        "plt.plot(data_x, data_y, 'ro')\n",
        "plt.show()\n"
      ]
    }
  ],
  "metadata": {
    "colab": {
      "name": "19301097_Abdullah_Hasan_Sajjad_Rafi_12_Newtons_Divided_Difference_Interpolation.ipynb",
      "provenance": [],
      "collapsed_sections": []
    },
    "kernelspec": {
      "display_name": "Python 3",
      "language": "python",
      "name": "python3"
    },
    "language_info": {
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "file_extension": ".py",
      "mimetype": "text/x-python",
      "name": "python",
      "nbconvert_exporter": "python",
      "pygments_lexer": "ipython3",
      "version": "3.9.1"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}