Lab 2 Polynomial_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 `LAB_SECTION` below:"
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        "NAME = \"Abdullah Hasan Sajjad Rafi\"\n",
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        "---"
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        "# Part 1: Representing a Polynomial\n",
        "\n",
        "Polynomials are function of the following format\n",
        "\n",
        "$$p(x) = a_0 + a_1 x ^ 1 + a_2 x ^ 2 + ... + a_n x ^ n,$$\n",
        "\n",
        "where, $[a_0, a_1, \\cdots a_n]$ are called coefficients and $n$ (called the degree or order) is a non-negative integer.\n",
        "\n",
        "\n",
        "This can also be written as:\n",
        "\n",
        "$$y = f(x) = a_0 x^0 + a_1 x ^ 1 + a_2 x ^ 2 + ... + a_n x ^ n.$$\n",
        "\n",
        "**Example**\n",
        "$$ y = 1 + 2x^2 + 5x^4 $$ is a polynomial of order 4 ($ = n$) with $n+1$ coeffecients $a_0 = 1, a_1 = 0, a_2 = 2, a_3 = 0, a_4 = 5$"
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        "**Method 1**: Using List\n",
        "\n",
        "---"
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      "source": [
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "\n",
        "\n",
        "# numpy is used for efficient array (vector or matrix) operations\n",
        "# pyplot is used for plotting \n",
        "# Must read: [https://www.tutorialspoint.com/numpy/numpy_matplotlib.htm]"
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        "arr = [1, 0, 2, 0, 5] # coeffecients of the polynomial\n",
        "n = len(arr) - 1 # degree. Remember: number of coeff = degree + 1\n",
        "\n",
        "'''\n",
        "For a single value of x, finding p(x)\n",
        "\n",
        "Note that this is an example of block comment in python. A block comment \n",
        "starts with three ' and ends with three '.\n",
        "'''\n",
        "\n",
        "x = 5.0\n",
        "p_x = 0.0\n",
        "\n",
        "for i in range(n + 1):\n",
        "    '''\n",
        "    p_x = arr[i] * x # WRONG, because no power\n",
        "    p_x = arr[i] * (x ** i) # WRONG, have to add the terms\n",
        "    '''\n",
        "    p_x += arr[i] * (x ** i) # a ** b means pow(a, b) or a^b\n",
        "\n",
        "'''\n",
        "For an array of x, finding p(x) for each element\n",
        "'''\n",
        "\n",
        "x_arr = [1.0, 2.0, 3.0, 4.0, 5.0]\n",
        "p_x_arr = []\n",
        "\n",
        "'''\n",
        "# naive way:\n",
        "for i in range(len(x_arr)):\n",
        "    print(x_arr[i])\n",
        "'''\n",
        "\n",
        "# better way: array traversing\n",
        "for x in x_arr:\n",
        "    temp = 0.0\n",
        "    for i in range(n + 1):\n",
        "        temp += arr[i] * (x ** i)\n",
        "    \n",
        "    p_x_arr.append(temp) # insert the element at the end of the array\n",
        "    \n",
        "\n",
        "print(f\"p({x_arr}) = \", p_x_arr)\n",
        "# note how we formatted the string. A formatted string starts with 'f'."
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          "text": [
            "p([1.0, 2.0, 3.0, 4.0, 5.0]) =  [8.0, 89.0, 424.0, 1313.0, 3176.0]\n"
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        "# Using numpy array for vectorization\n",
        "import numpy as np \n",
        "# numpy is used for efficient array (vector or matrix) operations\n",
        "# Must read: [https://www.tutorialspoint.com/numpy/numpy_matplotlib.htm]\n",
        "\n",
        "\n",
        "array = np.array([1, 0, 2, 0, 5])\n",
        "x_arr = np.array([1, 2, 3, 4, 5])\n",
        "p_x_arr = 0.0\n",
        "\n",
        "# vectorized version. requires only one loop\n",
        "for i in range(n + 1):\n",
        "    p_x_arr += array[i] * (x_arr ** i) # a ** b means pow(a, b) or a^b\n",
        "    \n",
        "print(f\"p({x_arr}) = \", p_x_arr)"
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            "p([1 2 3 4 5]) =  [   8.   89.  424. 1313. 3176.]\n"
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        "**Method 2 (Better)**: Using a Class\n",
        "\n",
        "---\n",
        "Complete the implementation of the polynomial class as showed in the lecture"
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        "'''\n",
        "Lab task 1\n",
        "Here we implement a Polynomial class with three methods: the constructor\n",
        "__init__(), the toString method __repr__(), and a method to make the objects\n",
        "of the class callable, __call__() method\n",
        "'''\n",
        "\n",
        "# Polynomial Class\n",
        "\n",
        "class Polynomial:\n",
        "  # Constructor, note that it starts and ends with two underscores\n",
        "  def __init__(self, coeff):\n",
        "    '''\n",
        "    Every internal variable of the object must be saved and initialized\n",
        "    in this method: self.variable = value\n",
        "    '''\n",
        "    self.coeff = coeff\n",
        "    self.degree = len(coeff) - 1\n",
        "\n",
        "  # Constructor to make the object callable \n",
        "  def __call__(self, x_arr):\n",
        "    '''\n",
        "    Here we assumed x_arr is a numpy array. Remember that a numpy array acts \n",
        "    like a vector (1D matrix). So an operation x + 1 would add 1 to each element\n",
        "    of the matrix (unlike python's defaule list). Simlarly, x ** 2 would return\n",
        "    element wise square of the array. \n",
        "\n",
        "    Hence, this method would return an array, where the i'th element is the \n",
        "    (polynomial) interpolated value of x[i], given the coeffecients a[i].\n",
        "    '''\n",
        "    p_x_arr = 0\n",
        "    \n",
        "    for i in  range(self.degree + 1):\n",
        "      p_x_arr +=  self.coeff[i] * ( x_arr**i  )\n",
        "    \n",
        "    return  p_x_arr\n",
        "    \n",
        "    # --------------------------------------------\n",
        "\n",
        "  # String representation method of the object (similar to toString() of java)\n",
        "  def __repr__(self):\n",
        "    # remember: a formatted string must start with f.\n",
        "  \n",
        "    str_ret = f'Polynomial of degree {self.degree}\\np(x) = '\n",
        "    for i in range(self.degree + 1):\n",
        "        a_val = self.coeff[i]\n",
        "        if i != 0:\n",
        "            if a_val >= 0:\n",
        "                str_ret += f'+ {a_val}x^{i} '\n",
        "            else:\n",
        "                str_ret += f'- {-a_val}x^{i} '\n",
        "        else:\n",
        "            str_ret += f'{a_val}x^{i} '\n",
        "            \n",
        "    return str_ret\n",
        "\n",
        "  # custom method 1: to get the degree of the polynomial\n",
        "  def get_degree(self):\n",
        "    # --------------------------------------------\n",
        "    return  self.degree\n",
        "    # --------------------------------------------\n",
        "\n",
        "  # custom method 2: to get the coefficients of the polynomial\n",
        "  def get_coeffs(self):\n",
        "    # --------------------------------------------\n",
        "    return  self.coeff\n",
        "    # --------------------------------------------"
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        "# test cases for your answer\n",
        "x_arr = np.array([1, 2, 3, 4, 5])\n",
        "\n",
        "coeff = np.array([1.0, 0.0, 2.0, 0.0, 5.0])\n",
        "p = Polynomial(coeff)\n",
        "\n",
        "results = [8, 89, 424, 1313, 3176]\n",
        "test = p(x_arr)\n",
        "\n",
        "np.testing.assert_array_equal(results, test)"
      ],
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        "arr = [1,2,3]\n",
        "print(arr * 3)"
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          "text": [
            "[1, 2, 3, 1, 2, 3, 1, 2, 3]\n"
          ]
        }
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        "# an example to see if our implementation works\n",
        "coeff = np.array([1.0, 0.0, 2.0, 0.0, 5.0])\n",
        "p = Polynomial(coeff)\n",
        "print(p)  # check if printable\n",
        "x_arr = np.array([1, 2, 3, 4, 5])\n",
        "print()\n",
        "print(f\"p({x_arr}) =\", p(x_arr)) # check if the object is callable\n",
        "# should print p([1 2 3 4 5]) =  [   8.   89.  424. 1313. 3176.]"
      ],
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          "text": [
            "Polynomial of degree 4\n",
            "p(x) = 1.0x^0 + 0.0x^1 + 2.0x^2 + 0.0x^3 + 5.0x^4 \n",
            "\n",
            "p([1 2 3 4 5]) = [   8.   89.  424. 1313. 3176.]\n"
          ]
        }
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        "# Part 2: Polynomial Interpolation (Matrix Method)\n",
        "\n",
        "If we have  $n+1$ nodes, that is,  $\\{(x_0, y_0), (x_1, y_1), (x_2, y_2), (x_{n}, y_{n})\\}$ that satisfies a polynomial of order $n$, it can be written as:\n",
        "\n",
        "\\begin{align}\n",
        "&a_0 + a_1  x_0 + a_2  x_0^2 + \\cdots a_n  + x_0^n = y_0\\\\\n",
        "&a_0 + a_1  x_1 + a_2  x_1^2 + \\cdots a_n  + x_1^n = y_1\\\\\n",
        "&a_0 + a_1  x_2 + a_2  x_2^2 + \\cdots a_n  + x_2^n = y_2\\\\\n",
        "&\\cdots\\\\\n",
        "&a_0 + a_1  x_{n-1} + a_2  x_{n}^2 + \\cdots + a_n  x_{n}^n = y_{n}\\\\\n",
        "\\end{align}\n",
        "\n",
        "Here, $p(x) = a_0 + a_1x^1 + a_2x^2 + \\cdots a_nx^n$ is called the fitted polynomial of the given data points (nodes). Using this polynomial to find the $y_k$ corresponding to an $x_k$ with the range of the given nodes is called polynomial interpolation.\n",
        "\n",
        "In matrix form, the equations can be written as  $$\\mathbf{Xa = y},$$\n",
        "\n",
        "where $\\mathbf{X} = $\n",
        "\n",
        "\\begin{bmatrix}\n",
        "x_0^0 & x_0^1 & x_0^2 & \\cdots & x_0^n\\\\\n",
        "x_1^0 & x_1^1 & x_1^2 & \\cdots & x_1^n\\\\\n",
        "\\vdots & \\vdots & \\vdots & \\ddots & \\vdots\\\\\n",
        "x_n^0 & x_{n}^1 & x_n^2 & \\cdots & x_n^n\\\\\n",
        "\\end{bmatrix}\n",
        "\n",
        "$\\mathbf{a} = $\n",
        "\\begin{bmatrix}\n",
        "a_0\\\\\n",
        "a_1\\\\\n",
        "\\vdots\\\\\n",
        "a_n\n",
        "\\end{bmatrix}\n",
        "\n",
        "and $\\mathbf{y} = $\n",
        "\\begin{bmatrix}\n",
        "y_0\\\\\n",
        "y_1\\\\\n",
        "\\vdots\\\\\n",
        "y_n\n",
        "\\end{bmatrix}\n",
        "\n",
        "From this, we can solve for $\\mathbf{a}$ using\n",
        "$$\\mathbf{a = X^{-1}y}.$$\n",
        "\n"
      ]
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        "'''\n",
        "Lab task 2\n",
        "Here we implement a function which takes a discrete x and y array, and returns\n",
        "a Polynomial object (the one we just implemented). This polynomial object can \n",
        "be used to calculate y for any other value of x (not in that list) within the\n",
        "range\n",
        "'''\n",
        "def get_poly(data_x, data_y):\n",
        "    n_nodes = len(data_x)\n",
        "    # np.zeors((a, b)) returns a (a x b) matrix, i.e., a rows and b columns \n",
        "    X = np.zeros((n_nodes, n_nodes))\n",
        "  \n",
        "    for i in  range(n_nodes):\n",
        "\n",
        "      for j in  range(n_nodes):\n",
        "\n",
        "        X[i,j]  = ( data_x[i]**j  )\n",
        "\n",
        "    print(X)\n",
        "    # --------------------------------------------\n",
        "    # We could have also used np.linalg.inv to find the inverse\n",
        "    # but pinv is more efficient\n",
        "    X_inv = np.linalg.pinv(X) # pseudo inverse\n",
        "    a = np.dot(X_inv, data_y)\n",
        "    p = Polynomial(a)\n",
        "\n",
        "    return p"
      ],
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        "# test cases for your answer\n",
        "data_x = np.array([-3., -2., -1., 0., 1., 3.])\n",
        "data_y = np.array([-80., -13., 6., 1., 5., 16.])\n",
        "p = get_poly(data_x, data_y)\n",
        "\n",
        "results = np.array([-80, -74.60997689, -69.36169492, -64.26436346, -59.32622134,\n",
        "                   -54.55456417, -49.95577177, -45.53533558, -41.297886, -37.24721982,\n",
        "                   -33.38632762, -29.71742112, -26.24196062, -22.96068235, -19.87362589,\n",
        "                   -16.98016156, -14.2790178, -11.76830857,  -9.44556072,  -7.30774144,\n",
        "                   -5.35128559, -3.57212312, -1.96570645, -0.52703788, 0.74930302,\n",
        "                   1.86913203, 2.83863291, 3.66432993, 4.35306058, 4.9119481,\n",
        "                   5.34837412, 5.66995126, 5.88449574, 6, 6.02460529,\n",
        "                   5.96657428, 5.83426368, 5.63609684, 5.38053634, 5.07605665,\n",
        "                   4.73111668, 4.35413242, 3.95344953, 3.53731597, 3.1138546,\n",
        "                   2.69103576, 2.27664993, 1.87828029, 1.50327536, 1.15872159,\n",
        "                   0.85141596, 0.58783863, 0.3741255, 0.21604084, 0.1189499,\n",
        "                   0.0877915, 0.12705066, 0.2407312, 0.43232834, 0.70480134,\n",
        "                   1.06054604, 1.50136754, 2.02845277, 2.64234311, 3.34290699,\n",
        "                   4.1293125, 5, 5.95265474, 6.98417944, 8.09066693,\n",
        "                   9.26737272, 10.50868766, 11.80811047, 13.15822045, 14.55065,\n",
        "                   15.97605727, 17.42409876, 18.88340192, 20.34153777, 21.78499351,\n",
        "                   23.19914511, 24.56822994, 25.87531935, 27.1022913, 28.22980298,\n",
        "                   29.23726338, 30.10280593, 30.80326108, 31.31412894, 31.60955188,\n",
        "                   31.6622871, 31.4436793, 30.92363323, 30.07058634, 28.85148136,\n",
        "                   27.23173894, 25.17523021, 22.64424943, 19.59948659, 16])\n",
        "\n",
        "x_arr = np.linspace(-3, 3, 100)\n",
        "test = p(x_arr)\n",
        "\n",
        "np.testing.assert_array_almost_equal(test, results)"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[[   1.   -3.    9.  -27.   81. -243.]\n",
            " [   1.   -2.    4.   -8.   16.  -32.]\n",
            " [   1.   -1.    1.   -1.    1.   -1.]\n",
            " [   1.    0.    0.    0.    0.    0.]\n",
            " [   1.    1.    1.    1.    1.    1.]\n",
            " [   1.    3.    9.   27.   81.  243.]]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "deletable": false,
        "editable": false,
        "nbgrader": {
          "cell_type": "code",
          "checksum": "277c42ed406022472573b16fb698714a",
          "grade": false,
          "grade_id": "plot",
          "locked": true,
          "schema_version": 3,
          "solution": false,
          "task": false
        },
        "id": "iCGAl18agxwk",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 383
        },
        "outputId": "67e37e62-1eaa-4ae1-961a-563bcf54b211"
      },
      "source": [
        "data_x = np.array([-3., -2., -1., 0., 1., 3.])\n",
        "data_y = np.array([-80., -13., 6., 1., 5., 16.])\n",
        "p = get_poly(data_x, data_y)\n",
        "'''\n",
        "np.linspace(a, b, n) returns a numpy array of n points equally \n",
        "spaced from a to b\n",
        "'''\n",
        "x_arr = np.linspace(-3, 3, 100)\n",
        "# interpolated values\n",
        "y_interp = p(x_arr)\n",
        "\n",
        "# pyplot is used for plotting \n",
        "# Must read: [https://www.tutorialspoint.com/numpy/numpy_matplotlib.htm]\n",
        "\n",
        "plt.plot(x_arr, y_interp, 'y')\n",
        "plt.plot(data_x, data_y, 'go')\n",
        "plt.xlabel('x - axis')\n",
        "plt.ylabel('y - axis')\n",
        "    \n",
        "\n",
        "plt.show()"
      ],
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[[   1.   -3.    9.  -27.   81. -243.]\n",
            " [   1.   -2.    4.   -8.   16.  -32.]\n",
            " [   1.   -1.    1.   -1.    1.   -1.]\n",
            " [   1.    0.    0.    0.    0.    0.]\n",
            " [   1.    1.    1.    1.    1.    1.]\n",
            " [   1.    3.    9.   27.   81.  243.]]\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {
            "needs_background": "light"
          }
        }
      ]
    }
  ]
}