Lectures/Lecture03_solving_nonLinearEquations_part2.ipynb

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      "source": [
        "# **Solving non-linear equations - Part 2**\n",
        "\n",
        "&nbsp;\n",
        "\n",
        "Lecturer: **Hamidreza Norouzi**\n",
        "\n",
        "&nbsp;"
      ],
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      "source": [
        "### **Notes**❕\n",
        "This content forms a part of the instructional presentations for the **`numerical methods in chemical engineering`** course designed for undergraduate chemical engineering students at Amirkabir University of Technology.\n",
        "\n",
        "Feel free to utilize the information and source codes provided in this material, ensuring appropriate acknowledgment of the original document.\n",
        "\n",
        "The visual elements featured in this document are either original or have been obtained from the following sources, unless specified otherwise:\n",
        "\n",
        "* Steven C. Chapra, Applied Numerical Methods with Matlab for Engineers and Scientists, 3rd edition, McGraw-Hill (2012).\n",
        "* Amos Gilat and Vish Subramanian, Numerical Methods for Engineers and Scientists, 3rd edition, Wiley (2014).\n",
        "\n",
        "<div align=\"center\">\n",
        "🟥 🟥 🟥\n",
        "</dive>\n",
        "\n",
        "---"
      ],
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      "source": [
        "# 🟡 1) Fixed-point method\n",
        "* Consider the equation $f(x) = 0$ that we aim to solve. We can reformulate this equation as $x = g(x)$, effectively seeking a point (referred to as the fixed point) for the function g, where $x_{i+1} = g(x_i)$.\n",
        "\n",
        "* When the absolute difference between $x_{i+1}$ and $x_i$ is smaller than the specified tolerance, $x_{i+1}$ is considered the root of the function f(x).\n",
        "\n",
        "* The solution is deemed as converged if the estimated error falls below a specified tolerance:\n",
        "$$\n",
        "ϵ_a = |\\frac{x_{i+1}-x_i}{x_{i+1}}| × 100 \\%\n",
        "$$"
      ],
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      "source": [
        "### ❓ **Example 1**\n",
        "Use fixed-point method to find the root of $f(x)=e^{-x}-x$. Take tol = 1%.\n",
        "\n",
        "<table width=\"100%\">\n",
        "<tr>\n",
        "<td width = \"65%\">\n",
        "\n",
        "💡 *solution*\n",
        "\n",
        "We re-arrage the equation $f(x)=0$  to  $x=e^{-x}$, so we have:\n",
        "$$\n",
        "x_{i+1}=e^{-x_i}\n",
        "$$\n",
        "&nbsp;\n",
        "\n",
        "and we start with the initial guess $x_0 = 0$. The following table is obtained:\n",
        "\n",
        "| Iteration  | $x_i$ | $x_{i+1}$  | $ϵ_a$%|\n",
        "| :---  |    :----:   | :---: | :---: |\n",
        "|0  | 0.0000 | 1.0000 | 100%|\n",
        "|1  | 1.0000 | 0.3679 | 171.8%|\n",
        "|2  | 0.3679 | 0.6922 | 46.85%|\n",
        "|3  | 0.6922 | 0.5005 | 38.31%|\n",
        "|4  | 0.5005 | 0.6062 | 17.44%|\n",
        "|5  | 0.6062 | 0.5454 | 11.15%|\n",
        "|6  | 0.5454 | 0.5796 | 5.90%|\n",
        "|7  | 0.5796 | 0.5601 | 3.48%|\n",
        "|8  | 0.5601 | 0.5711 | 1.93%|\n",
        "|9  | 0.5711 | 0.5649 | 1.11%|\n",
        "|10 | 0.5649 | 0.5684 | 0.62%|\n",
        "\n",
        "The solution is converged after 11 iterations (note that iterations start from 0).\n",
        "</td>\n",
        "\n",
        "<td>\n",
        "<img src=\"https://drive.google.com/uc?id=1l-wDlbIKOI1Y69i1ZeKeBC6ZSxlIfw4A\" width=\"300\">\n",
        "</td>\n",
        "\n",
        "</tr>\n",
        "</table>\n",
        "\n"
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        "## Convergence of fixed-point method\n",
        "\n",
        "* Depending on the first derivative of $g(x)$, we can judge whether a solution converges or not!\n",
        "$$\n",
        "|g^{'}(x)| < 1\n",
        "$$\n",
        "* It can be shown that the error for any iteration is linearly proportional to the error from the previous iteration multiplied by the first derivative of $g$:\n",
        "$$\n",
        "E_{i+1} = g^{'}(ξ)E_i\n",
        "$$\n",
        "\n"
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      "source": [
        "<div align=\"center\">\n",
        "<br>\n",
        "<img src=\"https://drive.google.com/uc?id=1CmnA6OfXzfvjHlcywfghQ44S37uZ6gZ1\" width=\"600\">\n",
        "</div>"
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      "source": [
        "<div align=\"center\">\n",
        "🟨 🟨 🟨\n",
        "</div>\n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ],
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      "source": [
        "# 🟢 2) Newton-Raphson method\n",
        "* If we assume $x^{*}$ is the root of equation $f(x)=0$ and $x_i$ is the estimate of the root for this equation, we can use Taylor series expansions as follows ($h = x^{*}-x_i$):\n",
        "$$\n",
        "f(x^{*}) = f(x_i) + (x^{*}-x_i)f^{'}(x_i) + \\frac{(x^{*}-x_i)}{2!}^2f^{\"}(x_i) + ... = 0  \\tag{2-1}\n",
        "$$\n",
        "* Neglecting second and higher order terms, we use this equation to estimate the root:\n",
        "$$\n",
        "x^{*} = x_i - \\frac{f(x_i)}{f^{'}(x_i)} \\tag{2-2}\n",
        "$$\n",
        "\n",
        "* Since we used only the linear part, so the estimate may deviate from the actual root $x^{*}$. So, we need to use an iterative procedure to improve it step-by-step.\n",
        "$$\n",
        "x_{i+1} = x_i - \\frac{f(x_i)}{f^{'}(x_i)} \\tag{2-3}\n",
        "$$\n",
        "* The following equation is used for checking the convergence:\n",
        "$$\n",
        "ϵ_a = |\\frac{x_{i+1}-x_i}{x_{i+1}}| × 100 \\% \\tag{2-4}\n",
        "$$\n"
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        "### ❓ **Example 2:**\n",
        "Use Newton-Raphson method to find the root of $f(x)=e^{-x}-x$ with an initial guess of $x_0=0$. Take tol = 0.1%.\n",
        "\n",
        "💡 *Solution*\n",
        "\n",
        "The first derivative of $f(x)$ is:\n",
        "$$\n",
        "f^{'}(x) = -1-e^{-x}\n",
        "$$\n",
        "so the equation for estimating the root becomes:\n",
        "$$\n",
        "x_{i+1}=x_i-\\frac{e^{-x_i}-x_i}{-1-e^{-x_i}}\n",
        "$$\n",
        "\n",
        "| Iteration  | $x_i$ | $x_{i+1}$  | $ϵ_a$%|\n",
        "| :---  |    :----:   | :---: | :---: |\n",
        "|0  | 0.00000 | 0.50000 | 100%|\n",
        "|1  | 0.50000 | 0.56631 | 11.71%|\n",
        "|2  | 0.56631 | 0.56714 | 0.15%|\n",
        "|3  | 0.56714 | 0.56714 | 0.00%|"
      ],
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      "cell_type": "markdown",
      "source": [
        "## 2-1) Code for Newton-Raphson method"
      ],
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        "def newtonRaphson(fx, dfx, x0, tol=1e-6, maxIter=100):\n",
        "  \"\"\"\n",
        "  Find a root of a function using the Newton-Raphson method.\n",
        "\n",
        "  :param fx: The function for which to find a root.\n",
        "  :param dfx: The derivative of the function.\n",
        "  :param x0: Initial guess for the root.\n",
        "  :param tol: Tolerance for convergence (default is 1e-6).\n",
        "  :param maxIter: Maximum number of iterations (default is 100).\n",
        "\n",
        "  :return: A tuple containing the approximation of the root, the\n",
        "    absolute function value at the root, and the number of iterations and None\n",
        "    if no solution is found.\n",
        "  \"\"\"\n",
        "  x = x0\n",
        "  for iter in range(maxIter):\n",
        "    f_x = fx(x0)\n",
        "    f_prime_x = dfx(x0)\n",
        "    if abs(f_prime_x)<1.0e-15:\n",
        "      print(f\"Derivative is zero at {x}. Newton-Raphson method cannot proceed.\")\n",
        "      return None\n",
        "\n",
        "    x = x0 - f_x / f_prime_x\n",
        "\n",
        "    if abs(x) > 1.0e-15:\n",
        "      ea = abs((x - x0) / x)\n",
        "    else:\n",
        "      ea = abs(x)\n",
        "    x0 = x;\n",
        "    if abs(f_x) < tol:\n",
        "      return x, abs(f_x), iter\n",
        "\n",
        "    if ea < tol:\n",
        "      return x, ea, iter\n",
        "\n",
        "  print(\"Newton-raphson mehtod did not find a solution after \"\n",
        "        f\"{maxIter} iterations\")\n",
        "  return None\n"
      ],
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      "cell_type": "markdown",
      "source": [
        "## 2-2) Playing with the code"
      ],
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      "cell_type": "markdown",
      "source": [
        "### Simple case"
      ],
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    {
      "cell_type": "code",
      "source": [
        "import math\n",
        "def simpleCase_Fx(x):\n",
        "  return 8-4.5*(x-math.sin(x))\n",
        "\n",
        "def simpleCase_dFx(x):\n",
        "  return -4.5 + 4.5 * math.cos(x)\n",
        "\n",
        "# test x0 = 2, 4, 0.1 and 0.2\n",
        "print(newtonRaphson(simpleCase_Fx, simpleCase_dFx, 4) )"
      ],
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          "text": [
            "(2.43046574172363, 1.3562377887410548e-09, 4)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "x = np.linspace(0,4, 30)\n",
        "y = [simpleCase_Fx(xi) for xi in x]\n",
        "plt.plot(x,y)\n",
        "plt.axhline(y=0, c='k')\n",
        "plt.show()"
      ],
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              "<Figure size 640x480 with 1 Axes>"
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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Slow convergence case\n",
        "\n"
      ],
      "metadata": {
        "id": "X2mYTmp7zsbu"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def slowConvergence_Fx(x):\n",
        "  return pow(x,10.0)-1\n",
        "\n",
        "def slowConvergence_dFx(x):\n",
        "  return 10*pow(x,9.0)\n",
        "# test x0 = 0.5 and 1.5\n",
        "print(newtonRaphson(slowConvergence_Fx, slowConvergence_dFx,2) )"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "Rj2YaGJXzxlk",
        "outputId": "506c0745-b30c-4ec6-ed02-bfca859267ee"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "(1.0000000000000009, 1.4004243764631497e-07, 10)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "x = np.linspace(0,1.3, 30)\n",
        "y = [slowConvergence_Fx(xi) for xi in x]\n",
        "plt.plot(x,y)\n",
        "plt.axhline(y=0, c='k')\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 430
        },
        "id": "an53F-B40ijQ",
        "outputId": "d5ea010d-6a51-48b8-8c96-712db4474b36"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### Complicated case with multiple roots"
      ],
      "metadata": {
        "id": "Xscy-J2eq8I9"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import math\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "def complicated_Fx(x):\n",
        "  return math.sin(10*x)+math.cos(3*x)\n",
        "\n",
        "def complicated_dFx(x):\n",
        "  return 10*math.cos(10*x)-3*math.sin(3*x)\n",
        "\n",
        "x = np.linspace(0,6, 200)\n",
        "y = [complicated_Fx(xi) for xi in x]\n",
        "\n",
        "plt.plot(x,y)\n",
        "plt.axhline(y=0, c='k')\n",
        "x_ticks = np.arange(0, 6.01, 0.5)  # Adjust the range as needed\n",
        "\n",
        "# Apply the custom tick positions to the x-axis\n",
        "plt.xticks(x_ticks)\n",
        "plt.grid(True, axis='x', linestyle='--', color='gray', linewidth=0.5)\n",
        "plt.show()\n",
        "\n",
        "\n",
        "all_x0 = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]\n",
        "for x0 in all_x0:\n",
        "  print(\n",
        "      f\"Initial guess is {x0} =>\",\n",
        "      newtonRaphson(complicated_Fx,\n",
        "                    complicated_dFx,\n",
        "                    x0) )"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 534
        },
        "id": "VFKVqMdGjSK3",
        "outputId": "e4467011-3e36-4eb3-d377-ea7840f0bea2"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Initial guess is 1.0 => (0.8458134067357137, 5.297555916161656e-08, 4)\n",
            "Initial guess is 2.0 => (1.5707963267948966, 3.858980605295559e-13, 4)\n",
            "Initial guess is 3.0 => (9.062286500739983, 3.590952343576817e-08, 5)\n",
            "Initial guess is 4.0 => (4.263590029871862, 8.918745741937073e-10, 6)\n",
            "Initial guess is 5.0 => (5.16118793089739, 1.0410491591006377e-07, 5)\n",
            "Initial guess is 6.0 => (6.058785831828935, 5.612201906523713e-07, 3)\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 2-3) Using under-relaxation\n",
        "* In each iteration of the Newton-Raphson method, we advance toward the solution (hopefully) by $δ$. The value of $\\delta$ is obtained by re-arranging Eq. (2-3):\n",
        "$$\n",
        "\\delta = x_{i+1} - x_i = - \\frac{f(x_i)}{f^{'}(x_i)} \\tag{2-5}\n",
        "$$\n",
        "\n",
        "* Now, the new estimate of the root, $x_{i+1}$, is obtained by:\n",
        " $$\n",
        " x_{i+1}  = x_i + \\alpha \\delta \\tag{2-6}\n",
        " $$\n",
        "\n",
        " where $\\alpha$ is under-relaxation factor ($0<\\alpha ≤ 1$) and it is usually around 0.5.\n",
        "* The use of under-relaxation makes the solution stable while the rate of convergence becomes slower.\n"
      ],
      "metadata": {
        "id": "NiQ1cRq5wdXZ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def newtonRaphsonRelaxed(fx, dfx, x0, alpha = 0.5, tol=1e-6, maxIter=100):\n",
        "  \"\"\"\n",
        "  Find a root of a function using the Newton-Raphson method with\n",
        "  under-relaxation factor.\n",
        "  \"\"\"\n",
        "\n",
        "  for iter in range(maxIter):\n",
        "    f_x = fx(x0)\n",
        "    f_prime_x = dfx(x0)\n",
        "    if abs(f_prime_x)<1.0e-15:\n",
        "      print(f\"Derivative is zero at {x}. Newton-Raphson method cannot proceed.\")\n",
        "      return None\n",
        "\n",
        "    delta = - f_x / f_prime_x\n",
        "    x = x0 + alpha*delta\n",
        "\n",
        "    if abs(x) > 1.0e-15:\n",
        "      ea = abs((x - x0) / x)\n",
        "    else:\n",
        "      ea = abs(x)\n",
        "\n",
        "    if abs(f_x) < tol:\n",
        "      return x, abs(f_x), iter\n",
        "\n",
        "    if ea < tol:\n",
        "      return x, ea, iter\n",
        "    x0 = x\n",
        "\n",
        "  print(\"Newton-raphson mehtod did not find a solution after \"\n",
        "        f\"{maxIter} iterations\")\n",
        "  return None"
      ],
      "metadata": {
        "id": "xCRpu0On4_SN"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "import math\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "def newtonRaphson_Fx(x):\n",
        "  return math.sin(10*x)+math.cos(3*x)\n",
        "\n",
        "def newtonRaphson_dFx(x):\n",
        "  return 10*math.cos(10*x)-3*math.sin(3*x)\n",
        "\n",
        "x = np.linspace(0,6, 200)\n",
        "y = [newtonRaphson_Fx(xi) for xi in x]\n",
        "\n",
        "plt.plot(x,y)\n",
        "plt.axhline(y=0, c='k')\n",
        "x_ticks = np.arange(0, 6.01, 0.5)  # Adjust the range as needed\n",
        "\n",
        "# Apply the custom tick positions to the x-axis\n",
        "plt.xticks(x_ticks)\n",
        "plt.grid(True, axis='x', linestyle='--', color='gray', linewidth=0.5)\n",
        "\n",
        "\n",
        "all_x0 = [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]\n",
        "for x0 in all_x0:\n",
        "  print(\n",
        "      f\"Initial guess is {x0} =>\",\n",
        "      newtonRaphsonRelaxed(newtonRaphson_Fx,\n",
        "                    newtonRaphson_dFx,\n",
        "                    x0, 0.1) )"
      ],
      "metadata": {
        "id": "4Ee0nhCN5tw7",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 534
        },
        "outputId": "22c887f7-69b0-499e-d2e6-09798eaf7cc5"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Initial guess is 1.0 => (0.84582027390396, 9.02066219690556e-07, 97)\n",
            "Initial guess is 2.0 => (1.8124727704046386, 9.483263240603725e-07, 88)\n",
            "Initial guess is 3.0 => (3.7457145757644517, 9.050995310625456e-07, 75)\n",
            "Initial guess is 4.0 => (4.229031928249209, 9.211901675860596e-07, 97)\n",
            "Initial guess is 5.0 => (5.1611438646028445, 9.472111875989459e-07, 91)\n",
            "Initial guess is 6.0 => (6.058733372503755, 9.615461453683276e-07, 69)\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 2-4) Some examples that newton may diverge\n",
        "* In these cases, use of under-relaxation may lead to a converged solution.\n",
        "\n",
        "<table width=\"100%\">\n",
        "<tr>\n",
        "<td width = \"50%\">\n",
        "<img src=\"https://drive.google.com/uc?id=1t7hsNCmNdEgOtpffAVFpm3gTe7rrZ1d0\" width=\"94%\" >\n",
        "</td>\n",
        "<td>\n",
        "<img src=\"https://drive.google.com/uc?id=17sB4r7DmUGmU1d0sdjYLcEcQl_PRAsux\" width=\"97%\" >\n",
        "</td>\n",
        "\n",
        "</tr>\n",
        "</table>\n",
        "\n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "th1SWEtn9FvT"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 2-5) Properties of Newton-Raphson method\n",
        "* The convergence is very rapid and error is reduced very quickly.\n",
        "$$\n",
        "E_{i+1} = \\frac{-f^{\"}(x)}{2f^{'}(x)}E^{2}_{i}\n",
        "$$\n",
        "* It may diverge if the initial guess is not close to the root."
      ],
      "metadata": {
        "id": "wKnsdRF2b77O"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "<div align=\"center\">\n",
        "🟩 🟩 🟩\n",
        "</div>\n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ],
      "metadata": {
        "id": "J2fYuJXA_hdu"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# 🔵 3) Secant method\n",
        "* A potential problem in implementing the Newton-Raphson method is the evaluation of the derivative.\n",
        "* One solution is to use finite-difference method to calculate the first derivative of the function:\n",
        "$$\n",
        "f^{'}(x_i) ≃ \\frac{f(x_i+\\Delta x)-f(x_i)}{\\Delta x} \\tag{3-1}\n",
        "$$\n",
        "* Considering Eq. (2-3) and substituting Eq. (3-1) in it, we obtain:\n",
        "$$\n",
        "x_{i+1} = x_i - \\frac{f(x_i) Δx}{f(x_i+\\Delta x)-f(x_i)}  \\tag{3-2}\n",
        "$$\n",
        "\n"
      ],
      "metadata": {
        "id": "UOfaxw0W_r32"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### ❓ **Example 3:**\n",
        "Use secant method to find the root of $f(x)=e^{-x}-x$ with an initial guess of $x_0=0$. Take tol = 0.1%.\n",
        "\n",
        "💡 *Solution*\n",
        "\n",
        "We use Eq. (3-2) to find the root and take $\\delta$=0.001.\n",
        "\n",
        "| Iteration  | $x_i$ | $x_{i+1}$  | $ϵ_a$%|\n",
        "| :---  |    :----:   | :---: | :---: |\n",
        "| 0  | 0.00000 | 0.50013| 100 %|\n",
        "| 1  | 0.50013 | 0.56633| 11.69%|\n",
        "| 2  | 0.56633 | 0.56714| 0.14% |\n",
        "| 3  | 0.56714 | 0.56714| 0.00% |\n",
        "\n"
      ],
      "metadata": {
        "id": "ywfd9r7tcyBG"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 3-1) Code for secant method"
      ],
      "metadata": {
        "id": "TWKpq7dmNl7E"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def secantRelaxed(fx, x0, alpha = 0.5, tol=1e-6, maxIter=100):\n",
        "  \"\"\"\n",
        "  Find a root of a function using the secant method with\n",
        "  under-relaxation factor.\n",
        "\n",
        "  :param fx: The function for which to find a root.\n",
        "  :param x0: Initial guess for the root.\n",
        "  :param alpha: Under-relaxation factor (default is 0.5)\n",
        "  :param tol: Tolerance for convergence (default is 1e-6).\n",
        "  :param maxIter: Maximum number of iterations (default is 100).\n",
        "\n",
        "  :return: A tuple containing the approximation of the root, the\n",
        "    absolute function value at the root, and the number of iterations and None\n",
        "    if no solution is found.\n",
        "  \"\"\"\n",
        "  delta_x = 0.001;\n",
        "  for iter in range(maxIter):\n",
        "\n",
        "    f_x = fx(x0)\n",
        "    f_delta_x = fx(x0+delta_x)\n",
        "\n",
        "    if abs(f_x - f_delta_x)<1.0e-15:\n",
        "      print(f\"Derivative is zero at {x}. Secant method cannot proceed.\")\n",
        "      return None\n",
        "\n",
        "    delta = - f_x * delta_x /(f_delta_x - f_x)\n",
        "    x = x0 + alpha*delta\n",
        "\n",
        "    if abs(x) > 1.0e-15:\n",
        "      ea = abs((x - x0) / x)\n",
        "    else:\n",
        "      ea = abs(x)\n",
        "\n",
        "    if abs(f_x) < tol:\n",
        "      return x, abs(f_x), iter\n",
        "\n",
        "    if ea < tol:\n",
        "      return x, ea, iter\n",
        "    x0 = x\n",
        "\n",
        "  print(\"Secant mehtod did not find a solution after \"\n",
        "        f\"{maxIter} iterations\")\n",
        "  return None"
      ],
      "metadata": {
        "id": "kgW6RNKiD3TA"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "import math\n",
        "\n",
        "def secant_Fx(x):\n",
        "  return 8-4.5*(x-math.sin(x))\n",
        "\n",
        "print(secantRelaxed(secant_Fx, 4, 1) )"
      ],
      "metadata": {
        "id": "QuQXE3Y7Zwsr",
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "outputId": "5f33a679-64e8-46d6-f5bc-f8d1643453db"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "(2.430465741724792, 4.952367937960389e-08, 4)\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<div align=\"center\">\n",
        "🟦 🟦 🟦\n",
        "</div>"
      ],
      "metadata": {
        "id": "e_j292TDf4kn"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# 🟢 4) System of non-linear equations"
      ],
      "metadata": {
        "id": "PjOCaWDPgTQ-"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "* Consider a set of two non-linear equations:\n",
        "$$\n",
        "f_1(x_1,x_2)=0 \\\\\n",
        "f_2(x_1,x_2)=0   \\tag{5-1}\n",
        "$$\n",
        "\n",
        "&nbsp;\n",
        "\n",
        "* Taylor expansions series of these functions around $(x_{1,i},x_{2,i})$ are:\n",
        "\n",
        "&nbsp;\n",
        "\n",
        "$$\n",
        "f_1(x_{1,i+1},x_{2,i+1}) = f_1(x_{1,i},x_{2,i}) + \\frac{\\partial f_1}{\\partial x_1}\\Bigg|_{x_{1,i},x_{2,i}}(\\delta_1) + \\frac{\\partial f_1}{\\partial x_2}\\Bigg|_{x_{1,i},x_{2,i}}(\\delta_2) + . . . = 0 \\\\\n",
        "f_2(x_{1,i+1},x_{2,i+1}) = f_2(x_{1,i},x_{2,i}) + \\frac{\\partial f_2}{\\partial x_1}\\Bigg|_{x_{1,i},x_{2,i}}(\\delta_1) + \\frac{\\partial f_2}{\\partial x_2}\\Bigg|_{x_{1,i},x_{2,i}}(\\delta_2) + . . . =0  \\tag{5-2}\n",
        "$$"
      ],
      "metadata": {
        "id": "U0WazFS7_3gx"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "* Keeping the linear part and neglecting the higher order terms results:\n",
        "\n",
        "$$\n",
        " \\frac{\\partial f_1}{\\partial x_1}\\Bigg|_{x_{1,i},x_{2,i}}(\\delta_1) + \\frac{\\partial f_1}{\\partial y_1}\\Bigg|_{x_{1,i},x_{2,i}}(\\delta_2) = -f_1(x_{1,i},x_{2,i}) \\\\\n",
        " \\frac{\\partial f_2}{\\partial x_1}\\Bigg|_{x_{1,i},x_{2,i}}(\\delta_1) + \\frac{\\partial f_2}{\\partial y_1}\\Bigg|_{x_{1,i},x_{2,i}}(\\delta_2) = -f_2(x_{1,i},x_{2,i})  \\tag{5-3}\n",
        "$$\n",
        "$$\n",
        "\\delta_1 = x_{1,i+1} - x_{1,i} \\\\\n",
        "\\delta_2 = x_{2,i+1} - x_{2,i}  \\tag{5-4}\n",
        "$$"
      ],
      "metadata": {
        "id": "wlQcfz9EDFjH"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "* The above equations form a system of linear equations:\n",
        "\n",
        "$$ \\begin{bmatrix}\n",
        "\\frac{\\partial f_1}{\\partial x_1}\\Bigg|_{x_{1,i},x_{2,i}} & \\frac{\\partial f_1}{\\partial x_2}\\Bigg|_{x_{1,i},x_{2,i}}  \\\\\n",
        "\\frac{\\partial f_2}{\\partial x_1}\\Bigg|_{x_{1,i},x_{2,i}} & \\frac{\\partial f_2}{\\partial x_2}\\Bigg|_{x_{1,i},x_{2,i}}  \\\\\n",
        "\\end{bmatrix}\n",
        "\\begin{bmatrix} \\delta_1 \\\\ \\delta_2 \\end{bmatrix} = -\n",
        "\\begin{bmatrix} f_1(x_{1,i},x_{2,i}) \\\\ f_2(x_{1,i},x_{2,i}) \\end{bmatrix} \\tag{5-5}\n",
        "$$\n",
        "\n",
        "&nbsp;\n",
        "* In compact form we have:\n",
        "$$\n",
        "[J]\\{\\delta\\} = - \\{f\\}  \\tag{5-6}\n",
        "$$\n",
        " where $[J]$ is called **Jacobina matrix**.\n",
        "\n"
      ],
      "metadata": {
        "id": "TiIXH84m-7yT"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "* In each iteration the solution of the above system gives $(\\delta_1, \\delta_2)$.\n",
        "* We can use under-relaxation factor ($\\alpha$) to stabilize the solution:\n",
        "$$\n",
        "x_{1,i+1} =  x_{1,i} + \\alpha \\delta_1 \\\\\n",
        "x_{2,i+1} =  x_{2,i} + \\alpha \\delta_2  \\tag{5-7}\n",
        "$$\n",
        " where $0 < \\alpha ≤ 0$, with typical value of 0.5."
      ],
      "metadata": {
        "id": "zRhY0s8rFoWS"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "### ❓ **Example 4**\n",
        "Use Newton's method to solve the system of non-linear equations:\n",
        "\n",
        "\n",
        "💡 *Solution*\n",
        "\n"
      ],
      "metadata": {
        "id": "UYEFZpf7I4xb"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## 4-1) Code for Newton's method"
      ],
      "metadata": {
        "id": "53OwopqHNWDg"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "* Instead of providing the Jacobian matrix, the code calculates it using a technique called perturbation (similar to what we did in secant method).\n",
        "* In this way, we need to provide the system of equations only (not the Jacobina matrix)."
      ],
      "metadata": {
        "id": "p2HKm6uONrNz"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "\n",
        "# Define a function to approximate the Jacobian matrix using finite differences\n",
        "def approximateJacobian(func, x, epsilon=1e-5):\n",
        "  n = len(x)\n",
        "  J = np.zeros((n, n))\n",
        "\n",
        "  for i in range(n):\n",
        "    x_perturbed = x.copy()\n",
        "    x_perturbed[i] += epsilon\n",
        "    delta_f = func(x_perturbed) - func(x)\n",
        "    J[:, i] = delta_f / epsilon\n",
        "\n",
        "  return J\n",
        "\n",
        "# Define the main function\n",
        "def newtonsSet(equations, initial_guess, alpha=0.5, tolerance=1e-6, maxIterations=100):\n",
        "  \"\"\"\n",
        "  Solves a system of nonlinear equations using the Newton-Raphson method.\n",
        "\n",
        "  Parameters:\n",
        "    - equations : A function that represents a system of equations to be solved.\n",
        "    - initial_guess (list or ndarray): The initial guess for the solution.\n",
        "    - alpha (float, optional): under-relaxation factor (default is 0.5).\n",
        "    - tolerance (float, optional): The convergence tolerance (default is 1e-6).\n",
        "    - maxIterations (int, optional): (default is 100)\n",
        "\n",
        "  Returns:\n",
        "    - x0 (ndarray): The approximate solution to the system of equations.\n",
        "    - None: if no solution is obtained.\n",
        "  \"\"\"\n",
        "\n",
        "  x0 = np.array(initial_guess)\n",
        "\n",
        "  for iteration in range(maxIterations):\n",
        "    # Calculate the system values and approximate Jacobian at the current point\n",
        "    f_values = equations(x0)\n",
        "    J = approximateJacobian(equations, x0)\n",
        "\n",
        "    # Solve the linear system J * delta_x = -f_values for delta_x\n",
        "    delta_x = np.linalg.solve(J, -f_values)\n",
        "\n",
        "    # Update the solution\n",
        "    x0 += alpha * delta_x\n",
        "\n",
        "    # Check for convergence\n",
        "    if np.linalg.norm(delta_x) < tolerance:\n",
        "      return x0\n",
        "\n",
        "  print(\"Newton-Raphson did not converge\")\n",
        "  return None"
      ],
      "metadata": {
        "id": "R-i_ab7zKyTL"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "def equations(xs):\n",
        "  x , y = xs;\n",
        "  f1 = y - 0.5*(np.exp(x/2)+np.exp(-x/2))\n",
        "  f2 =  9*x*x + 25*y*y -225\n",
        "  return np.array([f1, f2])\n",
        "\n",
        "\n",
        "initial_guess = [1.0, 1.0]\n",
        "result = newtonsSet(equations, initial_guess)\n",
        "print(result)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "cp59weRgRUsn",
        "outputId": "3af9881b-66e1-477d-a284-d44e95864889"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "[3.03115576 2.38586552]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# 🟣 5) Using libraries for solving non-linear equations\n"
      ],
      "metadata": {
        "id": "p1WeZASTjZYk"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "from scipy.optimize import fsolve\n",
        "\n",
        "def func(xs):\n",
        "  x , y = xs;\n",
        "  return [y - 0.5*(np.exp(x/2)+np.exp(-x/2)),\n",
        "        9*x*x + 25*y*y -225]\n",
        "\n",
        "root = fsolve(func, [2.5, 2.0])\n",
        "print (\"The solution of the set is \", root )\n",
        "np.isclose(func(root), [0.0, 0.0])  # func(root) should be almost 0.0.\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "XaUNKTBPjo8Q",
        "outputId": "0bfe07bf-20e1-4fec-acbc-5638464447bb"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The solution of the set is  [3.03115539 2.38586565]\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "array([ True,  True])"
            ]
          },
          "metadata": {},
          "execution_count": 9
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "def calculate_lnO_sf(Ta):\n",
        "  global oxygenConentration\n",
        "  term1 = -139.34411\n",
        "  term2 = (1.575701e5) / Ta\n",
        "  term3 = (6.642308e7) / (Ta**2)\n",
        "  term4 = (1.243800e10) / (Ta**3)\n",
        "  term5 = (8.621949e11) / (Ta**4)\n",
        "\n",
        "  return term1 + term2 - term3 + term4 - term5 - math.log(oxygenConentration)\n",
        "\n",
        "oxygenConentration = 10\n",
        "\n",
        "secantRelaxed(calculate_lnO_sf, 300 )"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "hwKo0u81WtFB",
        "outputId": "042f213c-2b20-4f4b-f354-61560210e7bc"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(288.5383569509035, 5.058195901766657e-07, 15)"
            ]
          },
          "metadata": {},
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<div align=\"center\">\n",
        "🟪 🟪 🟪\n",
        "</div>\n",
        "\n",
        "\n",
        "---\n",
        "\n"
      ],
      "metadata": {
        "id": "6cNPEKAG1Z98"
      }
    }
  ]
}