{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Pn06U4V2sMUE"
      },
      "source": [
        "# Intuition behind Eigen Values and Eigen Vectors\n",
        "\n",
        "\n",
        "**Eigen vectors** are vectors which are fixed in direction under a given linear transformation. The scaling factor of these Eigen vectors are called **Eigen values**.\n",
        "\n",
        "These **Eigen-pairs**(Eigen value and corresponding vector) are analogous to the roots of a polynomial. For example, consider cubic polynomials in a single variable, having 3 distinct roots. The general structure of these polynomials remain the same, but **rooted** at different points.\n",
        "![Screenshot 2024-08-15 115817.png](data:image/png;base64,[encoded data, 81364 characters])\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "E6kSmViruiBy"
      },
      "source": [
        "### **So, what does this mean?**\n",
        "\n",
        "If you are working with an invariant line/plane/hyperplane, the amount of transformation that can be performed to the surrounding space is limited. So, the eigen vectors, being fixed, act like skewers that restrict the linear transformations and hold it in place.\n",
        "\n",
        "So, verrrryyyy roughly, the eigen values are a measure of the distortion induced by the linear transformation, and the eigen vectors specify the direction of orientation of the distortion."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "En41C_AYqr5Y"
      },
      "outputs": [],
      "source": [
        "import math\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import sympy as sp"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "0hR3Dq_hniNm"
      },
      "source": [
        "### Eigenvalue and Eigenvector Calculation for a 2x2 Matrix\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "7rT6LquAq3Eq"
      },
      "outputs": [],
      "source": [
        "def compute_eigenvalues_2x2(a, b, c, d):\n",
        "    trace = a + d\n",
        "    determinant = a * d - b * c\n",
        "\n",
        "    discriminant = trace**2 - 4 * determinant\n",
        "    sqrt_discriminant = math.sqrt(discriminant)\n",
        "\n",
        "    lambda1 = (trace + sqrt_discriminant) / 2\n",
        "    lambda2 = (trace - sqrt_discriminant) / 2\n",
        "\n",
        "    return lambda1, lambda2\n",
        "\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "WunC_0el7x9c"
      },
      "outputs": [],
      "source": [
        "def compute_eigenvector_2x2(a, b, c, d, eigenvalue):\n",
        "    # Solve (A - lambda I) v = 0\n",
        "    if b != 0:\n",
        "        x = 1\n",
        "        y = (eigenvalue - a) / b\n",
        "    else:\n",
        "        x = 0\n",
        "        y = 1\n",
        "\n",
        "    return x, y"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "GPaxVNCRtVhE"
      },
      "source": [
        "### Eigenvalue and Eigenvector Calculation for a 3x3 Matrix"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "ULso29J7tXR5"
      },
      "outputs": [],
      "source": [
        "def compute_eigenvalues_3x3(A):\n",
        "    a, b, c = 1, -np.trace(A), np.linalg.det(A)\n",
        "    p = np.poly(A)\n",
        "\n",
        "    eigenvalues = np.roots(p)\n",
        "\n",
        "    return eigenvalues\n",
        "\n",
        "def compute_eigenvectors_3x3(A, eigenvalues):\n",
        "    eigenvectors = []\n",
        "\n",
        "    for eigenvalue in eigenvalues:\n",
        "        I = np.eye(3)\n",
        "        M = A - eigenvalue * I\n",
        "        _, v = np.linalg.qr(M)  # Use QR decomposition for simplicity\n",
        "        eigenvectors.append(v[:, 0])  # Take the first vector from QR\n",
        "\n",
        "    return eigenvectors"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "TgGsNGattbz4",
        "outputId": "5214b15f-b1aa-4c2d-9512-82de01b1cc38"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Eigenvalues: [1.00000657+0.00000000e+00j 0.99999671+5.69015387e-06j\n",
            " 0.99999671-5.69015387e-06j]\n",
            "Eigenvector for eigenvalue (1.0000065704250627+0j): [-6.57042506e-06+0.j  0.00000000e+00+0.j  0.00000000e+00+0.j]\n",
            "Eigenvector for eigenvalue (0.9999967147874662+5.6901538690326425e-06j): [-6.57042407e-06+0.j  0.00000000e+00+0.j  0.00000000e+00+0.j]\n",
            "Eigenvector for eigenvalue (0.9999967147874662-5.6901538690326425e-06j): [-6.57042407e-06+0.j  0.00000000e+00+0.j  0.00000000e+00+0.j]\n"
          ]
        }
      ],
      "source": [
        "A = np.array([[2, 6, 10],\n",
        "              [3, 8, 13],\n",
        "              [-2, -6, -10]])\n",
        "\n",
        "\n",
        "\n",
        "eigenvalues = compute_eigenvalues_3x3(A)\n",
        "\n",
        "eigenvectors = compute_eigenvectors_3x3(A, eigenvalues)\n",
        "\n",
        "print(f\"Eigenvalues: {eigenvalues}\")\n",
        "for i, vector in enumerate(eigenvectors):\n",
        "    print(f\"Eigenvector for eigenvalue {eigenvalues[i]}: {vector}\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "ql7I-uwLlgQh"
      },
      "source": [
        "## Using numpy"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DI5UgI6ZqCZo"
      },
      "source": [
        "### Eigenvalues and Eigenvectors Calculation\n",
        "\n",
        "In this section, we compute the eigenvalues and eigenvectors of the matrix \\( A \\) using NumPy's `linalg.eig` function.\n",
        "\n",
        "The matrix \\( A \\) is defined as:\n",
        "\n",
        " A = \\begin{pmatrix} 4 & -2 \\\\ 1 & 1 \\end{pmatrix} \\\n",
        "\n",
        "The function `np.linalg.eig(A)` returns two outputs:\n",
        "1. **Eigenvalues**: The scalars \\( \\lambda \\) that satisfy the equation \\( A \\mathbf{v} = \\lambda \\mathbf{v} \\).\n",
        "2. **Eigenvectors**: The vectors \\( \\mathbf{v} \\) associated with each eigenvalue.\n",
        "\n",
        "The eigenvalues and eigenvectors are displayed below.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vOXk3WBSkq1-",
        "outputId": "0ebbf989-2ad0-4fa5-e24b-2cd7b16ce822"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Eigenvalues:\n",
            "[3. 2.]\n",
            "\n",
            "Eigenvectors:\n",
            "[[0.89442719 0.70710678]\n",
            " [0.4472136  0.70710678]]\n"
          ]
        }
      ],
      "source": [
        "A = np.array([[4, -2],\n",
        "              [1, 1]])\n",
        "\n",
        "eigenvalues, eigenvectors = np.linalg.eig(A)\n",
        "\n",
        "print(\"Eigenvalues:\")\n",
        "print(eigenvalues)\n",
        "\n",
        "print(\"\\nEigenvectors:\")\n",
        "print(eigenvectors)\n"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "TPKV4dAmqPsQ"
      },
      "source": [
        "### Visualize Matrix Transformation and Eigenvectors"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "DsenLH5Eqa_r"
      },
      "source": [
        "This section visualizes how the matrix \\( A \\) transforms a grid of vectors and plots the eigenvectors.\n",
        "\n",
        "1. **Matrix Transformation**:\n",
        "   - We create a grid of points and apply the matrix transformation to visualize how the matrix \\( A \\) affects these points. The blue arrows represent the transformed grid.\n",
        "\n",
        "2. **Eigenvectors**:\n",
        "   - The red arrows represent the eigenvectors of the matrix \\( A \\). These vectors point in directions where the matrix transformation acts as simple scaling (stretching or compressing).\n",
        "\n",
        "The plot helps to understand how eigenvectors remain in the same direction while being scaled by their corresponding eigenvalues.\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 717
        },
        "id": "7o0VQYKUnzvy",
        "outputId": "7e8772db-3014-4f14-c50f-f31c43a2d6ef"
      },
      "outputs": [
        {
          "data": {
            "image/png": "[encoded data, 106532 characters]",
            "text/plain": [
              "<Figure size 800x800 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "x = np.linspace(-2, 2, 10)\n",
        "y = np.linspace(-2, 2, 10)\n",
        "X, Y = np.meshgrid(x, y)\n",
        "grid = np.vstack([X.ravel(), Y.ravel()])\n",
        "\n",
        "transformed_grid = np.dot(A, grid)\n",
        "\n",
        "fig, ax = plt.subplots()\n",
        "ax.set_xlim(-5, 5)\n",
        "ax.set_ylim(-5, 5)\n",
        "ax.set_aspect('equal')\n",
        "\n",
        "ax.quiver(grid[0, :], grid[1, :], transformed_grid[0, :] - grid[0, :], transformed_grid[1, :] - grid[1, :], angles='xy', scale_units='xy', scale=1, color='blue', alpha=0.5)\n",
        "\n",
        "for i in range(len(eigenvalues)):\n",
        "    eigenvector = eigenvectors[:, i]\n",
        "    ax.quiver(0, 0, eigenvector[0], eigenvector[1], angles='xy', scale_units='xy', scale=1, color='red', label=f'Eigenvector {i+1}')\n",
        "\n",
        "ax.set_title('Matrix Transformation and Eigenvectors')\n",
        "ax.set_xlabel('x')\n",
        "ax.set_ylabel('y')\n",
        "ax.legend()\n",
        "ax.grid(True)\n",
        "plt.show()\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "qoKZGmG2bgX1"
      },
      "outputs": [],
      "source": []
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "_IptZs0USr0y"
      },
      "source": [
        "# **GRADIENT DESCENT PROBLEMS**"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "bTChUmCiauFm",
        "outputId": "0f3e089d-1e35-4fc0-bc41-8223379b8c39"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Iteration 0: x = 3.6000, f(x) = 23.0400\n",
            "Iteration 10: x = -3.4359, f(x) = 1.6437\n",
            "Iteration 20: x = -4.4692, f(x) = 0.5859\n",
            "Iteration 30: x = -4.9032, f(x) = 0.3008\n",
            "Iteration 40: x = -5.1438, f(x) = 0.1833\n",
            "Iteration 50: x = -5.2972, f(x) = 0.1235\n",
            "Iteration 60: x = -5.4037, f(x) = 0.0889\n",
            "Iteration 70: x = -5.4820, f(x) = 0.0671\n",
            "Iteration 80: x = -5.5421, f(x) = 0.0524\n",
            "Iteration 90: x = -5.5896, f(x) = 0.0421\n",
            "Final value: x = -5.6246, f(x) = 0.0352\n"
          ]
        }
      ],
      "source": [
        "def f1(x):\n",
        "    return (1/4)*(x**2) + 3*x + 9\n",
        "\n",
        "def derivative_f1(x):\n",
        "    return ((1/2)*x + 3)**2\n",
        "\n",
        "def gradient1(x):\n",
        "    return derivative_f1(x)\n",
        "\n",
        "x = 10\n",
        "x1 = None\n",
        "step_size = 0.1\n",
        "iterations = 100\n",
        "\n",
        "for i in range(iterations):\n",
        "    grad1 = gradient1(x)\n",
        "    x = x - step_size * grad1\n",
        "    if i % 10 == 0:\n",
        "        print(f\"Iteration {i}: x = {x:.4f}, f(x) = {f1(x):.4f}\")\n",
        "\n",
        "print(f\"Final value: x = {x:.4f}, f(x) = {f1(x):.4f}\")\n",
        "# Correction: remove the square operation in the derivative function"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "xMurdw_XRpnB",
        "outputId": "a74c9986-ea5d-43cf-d329-d4f5bba49994"
      },
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "Iteration 0: x = -7.6333, f(x) = 4496.4599\n",
            "Iteration 10: x = -33.4699, f(x) = 0.2846\n",
            "Iteration 20: x = -33.9333, f(x) = 0.1267\n",
            "Iteration 30: x = -34.2503, f(x) = 0.0651\n",
            "Iteration 40: x = -34.4814, f(x) = 0.0369\n",
            "Iteration 50: x = -34.6577, f(x) = 0.0225\n",
            "Iteration 60: x = -34.7968, f(x) = 0.0146\n",
            "Iteration 70: x = -34.9095, f(x) = 0.0098\n",
            "Iteration 80: x = -35.0026, f(x) = 0.0069\n",
            "Iteration 90: x = -35.0809, f(x) = 0.0050\n",
            "Final value: x = -35.1414, f(x) = 0.0038\n"
          ]
        }
      ],
      "source": [
        "def f2(x):\n",
        "    return (1/144)*(x**4) + x**3 + 54*(x**2) + 1296*x + 11664\n",
        "\n",
        "def derivative_f2(x):\n",
        "    return (1/36)*(x**3) + 3*x**2 + 108*x + 1296\n",
        "\n",
        "def dderivative_f2(x):\n",
        "    return (1/12)*(x**2) + 6*x + 108\n",
        "\n",
        "def gradient2(x):\n",
        "    return dderivative_f2(x)\n",
        "\n",
        "x = 10\n",
        "x1 = None\n",
        "step_size = 0.1\n",
        "iterations = 100\n",
        "\n",
        "for i in range(iterations):\n",
        "    if x1 is not None:\n",
        "        grad2 = gradient2(x1)\n",
        "    else:\n",
        "        grad2 = gradient2(x)\n",
        "    x1, x = x, x - step_size * grad2\n",
        "    if i % 10 == 0:\n",
        "        print(f\"Iteration {i}: x = {x:.4f}, f(x) = {f2(x):.4f}\")\n",
        "\n",
        "print(f\"Final value: x = {x:.4f}, f(x) = {f2(x):.4f}\")\n",
        "# Correction: replace second derivative with first, compute gradient at the current step"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "L_VpxFoiVYFR"
      },
      "source": [
        "#**Parameters**"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "A5-r9E0Ea1T7"
      },
      "outputs": [],
      "source": [
        "# We will use this code for demonstrations of different function and different step sizes\n",
        "# This code snippet in this notebook need not to be sent to students before tutorials, only made for explaning things\n",
        "import time\n",
        "\n",
        "# def f(x):\n",
        "#     return x**2\n",
        "# def derivative_f(x):\n",
        "#     return 2*x\n",
        "\n",
        "def f(x):\n",
        "    return x**4 - 6*x**3 + 25*x + 30\n",
        "def derivative_f(x):\n",
        "    return 4*x**3 - 18*x**2 + 25\n",
        "\n",
        "initial_x = 10\n",
        "iterations = 100\n",
        "learning_rate = 0.001\n",
        "\n",
        "# Store the values for visualization\n",
        "x_values = []\n",
        "f_values = []\n",
        "\n",
        "x = initial_x\n",
        "start_time = time.time()\n",
        "\n",
        "for i in range(iterations):\n",
        "    x_values.append(x)\n",
        "    f_values.append(f(x))\n",
        "\n",
        "    grad = derivative_f(x)\n",
        "    x = x - learning_rate * grad\n",
        "\n",
        "end_time = time.time()\n",
        "total_time = end_time - start_time\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 751
        },
        "id": "dzSolUmRa8lg",
        "outputId": "3caafc0b-4c40-4461-b980-82496b758b72"
      },
      "outputs": [
        {
          "name": "stderr",
          "output_type": "stream",
          "text": [
            "<ipython-input-9-a486294016c7>:26: MatplotlibDeprecationWarning: Setting data with a non sequence type is deprecated since 3.7 and will be remove two minor releases later\n",
            "  line.set_data(x_values[i], f_values[i])\n"
          ]
        },
        {
          "data": {
            "text/html": [
              "\n",
              "<link rel=\"stylesheet\"\n",
              "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css\">\n",
              "<script language=\"javascript\">\n",
              "  function isInternetExplorer() {\n",
              "    ua = navigator.userAgent;\n",
              "    /* MSIE used to detect old browsers and Trident used to newer ones*/\n",
              "    return ua.indexOf(\"MSIE \") > -1 || ua.indexOf(\"Trident/\") > -1;\n",
              "  }\n",
              "\n",
              "  /* Define the Animation class */\n",
              "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
              "    this.img_id = img_id;\n",
              "    this.slider_id = slider_id;\n",
              "    this.loop_select_id = loop_select_id;\n",
              "    this.interval = interval;\n",
              "    this.current_frame = 0;\n",
              "    this.direction = 0;\n",
              "    this.timer = null;\n",
              "    this.frames = new Array(frames.length);\n",
              "\n",
              "    for (var i=0; i<frames.length; i++)\n",
              "    {\n",
              "     this.frames[i] = new Image();\n",
              "     this.frames[i].src = frames[i];\n",
              "    }\n",
              "    var slider = document.getElementById(this.slider_id);\n",
              "    slider.max = this.frames.length - 1;\n",
              "    if (isInternetExplorer()) {\n",
              "        // switch from oninput to onchange because IE <= 11 does not conform\n",
              "        // with W3C specification. It ignores oninput and onchange behaves\n",
              "        // like oninput. In contrast, Microsoft Edge behaves correctly.\n",
              "        slider.setAttribute('onchange', slider.getAttribute('oninput'));\n",
              "        slider.setAttribute('oninput', null);\n",
              "    }\n",
              "    this.set_frame(this.current_frame);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.get_loop_state = function(){\n",
              "    var button_group = document[this.loop_select_id].state;\n",
              "    for (var i = 0; i < button_group.length; i++) {\n",
              "        var button = button_group[i];\n",
              "        if (button.checked) {\n",
              "            return button.value;\n",
              "        }\n",
              "    }\n",
              "    return undefined;\n",
              "  }\n",
              "\n",
              "  Animation.prototype.set_frame = function(frame){\n",
              "    this.current_frame = frame;\n",
              "    document.getElementById(this.img_id).src =\n",
              "            this.frames[this.current_frame].src;\n",
              "    document.getElementById(this.slider_id).value = this.current_frame;\n",
              "  }\n",
              "\n",
              "  Animation.prototype.next_frame = function()\n",
              "  {\n",
              "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
              "  }\n",
              "\n",
              "  Animation.prototype.previous_frame = function()\n",
              "  {\n",
              "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
              "  }\n",
              "\n",
              "  Animation.prototype.first_frame = function()\n",
              "  {\n",
              "    this.set_frame(0);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.last_frame = function()\n",
              "  {\n",
              "    this.set_frame(this.frames.length - 1);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.slower = function()\n",
              "  {\n",
              "    this.interval /= 0.7;\n",
              "    if(this.direction > 0){this.play_animation();}\n",
              "    else if(this.direction < 0){this.reverse_animation();}\n",
              "  }\n",
              "\n",
              "  Animation.prototype.faster = function()\n",
              "  {\n",
              "    this.interval *= 0.7;\n",
              "    if(this.direction > 0){this.play_animation();}\n",
              "    else if(this.direction < 0){this.reverse_animation();}\n",
              "  }\n",
              "\n",
              "  Animation.prototype.anim_step_forward = function()\n",
              "  {\n",
              "    this.current_frame += 1;\n",
              "    if(this.current_frame < this.frames.length){\n",
              "      this.set_frame(this.current_frame);\n",
              "    }else{\n",
              "      var loop_state = this.get_loop_state();\n",
              "      if(loop_state == \"loop\"){\n",
              "        this.first_frame();\n",
              "      }else if(loop_state == \"reflect\"){\n",
              "        this.last_frame();\n",
              "        this.reverse_animation();\n",
              "      }else{\n",
              "        this.pause_animation();\n",
              "        this.last_frame();\n",
              "      }\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.anim_step_reverse = function()\n",
              "  {\n",
              "    this.current_frame -= 1;\n",
              "    if(this.current_frame >= 0){\n",
              "      this.set_frame(this.current_frame);\n",
              "    }else{\n",
              "      var loop_state = this.get_loop_state();\n",
              "      if(loop_state == \"loop\"){\n",
              "        this.last_frame();\n",
              "      }else if(loop_state == \"reflect\"){\n",
              "        this.first_frame();\n",
              "        this.play_animation();\n",
              "      }else{\n",
              "        this.pause_animation();\n",
              "        this.first_frame();\n",
              "      }\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.pause_animation = function()\n",
              "  {\n",
              "    this.direction = 0;\n",
              "    if (this.timer){\n",
              "      clearInterval(this.timer);\n",
              "      this.timer = null;\n",
              "    }\n",
              "  }\n",
              "\n",
              "  Animation.prototype.play_animation = function()\n",
              "  {\n",
              "    this.pause_animation();\n",
              "    this.direction = 1;\n",
              "    var t = this;\n",
              "    if (!this.timer) this.timer = setInterval(function() {\n",
              "        t.anim_step_forward();\n",
              "    }, this.interval);\n",
              "  }\n",
              "\n",
              "  Animation.prototype.reverse_animation = function()\n",
              "  {\n",
              "    this.pause_animation();\n",
              "    this.direction = -1;\n",
              "    var t = this;\n",
              "    if (!this.timer) this.timer = setInterval(function() {\n",
              "        t.anim_step_reverse();\n",
              "    }, this.interval);\n",
              "  }\n",
              "</script>\n",
              "\n",
              "<style>\n",
              ".animation {\n",
              "    display: inline-block;\n",
              "    text-align: center;\n",
              "}\n",
              "input[type=range].anim-slider {\n",
              "    width: 374px;\n",
              "    margin-left: auto;\n",
              "    margin-right: auto;\n",
              "}\n",
              ".anim-buttons {\n",
              "    margin: 8px 0px;\n",
              "}\n",
              ".anim-buttons button {\n",
              "    padding: 0;\n",
              "    width: 36px;\n",
              "}\n",
              ".anim-state label {\n",
              "    margin-right: 8px;\n",
              "}\n",
              ".anim-state input {\n",
              "    margin: 0;\n",
              "    vertical-align: middle;\n",
              "}\n",
              "</style>\n",
              "\n",
              "<div class=\"animation\">\n",
              "  <img id=\"_anim_img98d1312204a34f709bf40dba425d073d\">\n",
              "  <div class=\"anim-controls\">\n",
              "    <input id=\"_anim_slider98d1312204a34f709bf40dba425d073d\" type=\"range\" class=\"anim-slider\"\n",
              "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
              "           oninput=\"anim98d1312204a34f709bf40dba425d073d.set_frame(parseInt(this.value));\">\n",
              "    <div class=\"anim-buttons\">\n",
              "      <button title=\"Decrease speed\" aria-label=\"Decrease speed\" onclick=\"anim98d1312204a34f709bf40dba425d073d.slower()\">\n",
              "          <i class=\"fa fa-minus\"></i></button>\n",
              "      <button title=\"First frame\" aria-label=\"First frame\" onclick=\"anim98d1312204a34f709bf40dba425d073d.first_frame()\">\n",
              "        <i class=\"fa fa-fast-backward\"></i></button>\n",
              "      <button title=\"Previous frame\" aria-label=\"Previous frame\" onclick=\"anim98d1312204a34f709bf40dba425d073d.previous_frame()\">\n",
              "          <i class=\"fa fa-step-backward\"></i></button>\n",
              "      <button title=\"Play backwards\" aria-label=\"Play backwards\" onclick=\"anim98d1312204a34f709bf40dba425d073d.reverse_animation()\">\n",
              "          <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
              "      <button title=\"Pause\" aria-label=\"Pause\" onclick=\"anim98d1312204a34f709bf40dba425d073d.pause_animation()\">\n",
              "          <i class=\"fa fa-pause\"></i></button>\n",
              "      <button title=\"Play\" aria-label=\"Play\" onclick=\"anim98d1312204a34f709bf40dba425d073d.play_animation()\">\n",
              "          <i class=\"fa fa-play\"></i></button>\n",
              "      <button title=\"Next frame\" aria-label=\"Next frame\" onclick=\"anim98d1312204a34f709bf40dba425d073d.next_frame()\">\n",
              "          <i class=\"fa fa-step-forward\"></i></button>\n",
              "      <button title=\"Last frame\" aria-label=\"Last frame\" onclick=\"anim98d1312204a34f709bf40dba425d073d.last_frame()\">\n",
              "          <i class=\"fa fa-fast-forward\"></i></button>\n",
              "      <button title=\"Increase speed\" aria-label=\"Increase speed\" onclick=\"anim98d1312204a34f709bf40dba425d073d.faster()\">\n",
              "          <i class=\"fa fa-plus\"></i></button>\n",
              "    </div>\n",
              "    <form title=\"Repetition mode\" aria-label=\"Repetition mode\" action=\"#n\" name=\"_anim_loop_select98d1312204a34f709bf40dba425d073d\"\n",
              "          class=\"anim-state\">\n",
              "      <input type=\"radio\" name=\"state\" value=\"once\" id=\"_anim_radio1_98d1312204a34f709bf40dba425d073d\"\n",
              "             checked>\n",
              "      <label for=\"_anim_radio1_98d1312204a34f709bf40dba425d073d\">Once</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"loop\" id=\"_anim_radio2_98d1312204a34f709bf40dba425d073d\"\n",
              "             >\n",
              "      <label for=\"_anim_radio2_98d1312204a34f709bf40dba425d073d\">Loop</label>\n",
              "      <input type=\"radio\" name=\"state\" value=\"reflect\" id=\"_anim_radio3_98d1312204a34f709bf40dba425d073d\"\n",
              "             >\n",
              "      <label for=\"_anim_radio3_98d1312204a34f709bf40dba425d073d\">Reflect</label>\n",
              "    </form>\n",
              "  </div>\n",
              "</div>\n",
              "\n",
              "\n",
              "<script language=\"javascript\">\n",
              "  /* Instantiate the Animation class. */\n",
              "  /* The IDs given should match those used in the template above. */\n",
              "  (function() {\n",
              "    var img_id = \"_anim_img98d1312204a34f709bf40dba425d073d\";\n",
              "    var slider_id = \"_anim_slider98d1312204a34f709bf40dba425d073d\";\n",
              "    var loop_select_id = \"_anim_loop_select98d1312204a34f709bf40dba425d073d\";\n",
              "    var frames = new Array(100);\n",
              "    \n",
              "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA+gAAAJYCAYAAADxHswlAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\\\n",
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              "AABOgIIOAAAAAIAToKADAAAAAOAEKOgAAAAAADgBCjoAAAAAAE6Agg4AAAAAgBOgoAMAAAAA4AQo\\\n",
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              "RK5CYII=\\\n",
              "\"\n",
              "\n",
              "\n",
              "    /* set a timeout to make sure all the above elements are created before\n",
              "       the object is initialized. */\n",
              "    setTimeout(function() {\n",
              "        anim98d1312204a34f709bf40dba425d073d = new Animation(frames, img_id, slider_id, 200.0,\n",
              "                                 loop_select_id);\n",
              "    }, 0);\n",
              "  })()\n",
              "</script>\n"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "execution_count": 9,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "import matplotlib.animation as animation\n",
        "import time\n",
        "\n",
        "# Set up the plot\n",
        "fig, ax = plt.subplots(figsize=(10, 6))\n",
        "x_range = np.linspace(-10, 10, 400)\n",
        "y_range = f(x_range)\n",
        "\n",
        "# Plot the function\n",
        "ax.plot(x_range, y_range, label=r'$f(x) = x^2$', color='black', linestyle='--')\n",
        "line, = ax.plot([], [], 'ro', label='Gradient Descent Path', markersize=8)  # Larger marker size for visibility\n",
        "text = ax.text(0.05, 0.9, '', transform=ax.transAxes)\n",
        "\n",
        "# Set plot limits and labels\n",
        "ax.set_xlim(-10, 10)\n",
        "ax.set_ylim(0, 110)\n",
        "ax.set_xlabel('x')\n",
        "ax.set_ylabel('f(x)')\n",
        "ax.legend()\n",
        "ax.grid(True)\n",
        "\n",
        "# Animation function\n",
        "def animate(i):\n",
        "    line.set_data(x_values[i], f_values[i])\n",
        "    text.set_text(f'Iteration: {i+1}\\nTime Elapsed: {total_time:.2f} seconds')\n",
        "    return line, text\n",
        "\n",
        "# Create the animation\n",
        "ani = animation.FuncAnimation(fig, animate, frames=len(x_values), interval=200, repeat=False)\n",
        "\n",
        "plt.close(fig) # Extra static plot was coming\n",
        "\n",
        "# Display the animation in Notebook\n",
        "from IPython.display import HTML\n",
        "HTML(ani.to_jshtml())\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "background_save": true
        },
        "id": "U1M_VqodbBkP"
      },
      "outputs": [],
      "source": []
    }
  ],
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "display_name": "Python 3",
      "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": "3.11.9"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 0
}
