5004_hw_2_1.ipynb

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      "name": "python"
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  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/hechubo/DDM-Coursework/blob/master/5004_hw_2_1.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "**1.Find the first two iterations using (1) the Jacobi method and (2) the Gauss-Seidel method for the following linear system, starting from x(0) = (0, −1, 0)**"
      ],
      "metadata": {
        "id": "HjM7i6Jz7IiH"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "\n",
        "\n",
        "可以按照以下步骤使用 Jacobi 和 Gauss-Seidel 方法求解：\n",
        "\n",
        "Jacobi 方法：\n",
        "\n",
        "\\begin{align*}\n",
        "x_1^{(k+1)} &= \\frac{1}{4}(3 - x_2^{(k)} + x_3^{(k)}), \\\\\n",
        "x_2^{(k+1)} &= \\frac{1}{3}(-6 + x_1^{(k)} - x_3^{(k)}), \\\\\n",
        "x_3^{(k+1)} &= \\frac{1}{5}(4 - 2x_1^{(k)} - 2x_2^{(k)}).\n",
        "\\end{align*}\n",
        "\n",
        "从初始值 $x^{(0)} = (0, -1, 0)$ 开始迭代，计算前两次迭代结果：\n",
        "\n",
        "第一次迭代：\n",
        "\\begin{align*}\n",
        "x_1^{(1)} &= \\frac{3 - (-1) + 0}{4} = \\frac{4}{4} = 1\\\\\n",
        "x_2^{(1)} &= \\frac{-6 + 0 - 0}{3} = \\frac{-6}{3} = -2\\\\\n",
        "x_3^{(1)} &= \\frac{4 - 2(0) - 2(-1)}{5} = \\frac{4 + 2}{5} = \\frac{6}{5} = 1.2\n",
        "\\end{align*}\n",
        "\n",
        "第二次迭代：\n",
        "\\begin{align*}\n",
        "x_1^{(2)} &= \\frac{3 - (-2) + 1.2}{4} = \\frac{6.2}{4} = 1.55\\\\\n",
        "x_2^{(2)} &= \\frac{-6 + 1 - 1.2}{3} = \\frac{-6.2}{3} = -2.067\\\\\n",
        "x_3^{(2)} &= \\frac{4 - 2(1) - 2(-2)}{5} = \\frac{4 - 2 + 4}{5} = \\frac{6}{5} = 1.2\n",
        "\\end{align*}\n",
        "Gauss-Seidel 方法：\n",
        "\n",
        "\\begin{align*}\n",
        "x_1^{(k+1)} &= \\frac{1}{4}(3 - x_2^{(k)} + x_3^{(k)}), \\\\\n",
        "x_2^{(k+1)} &= \\frac{1}{3}(-6 + x_1^{(k+1)} - x_3^{(k)}), \\\\\n",
        "x_3^{(k+1)} &= \\frac{1}{5}(4 - 2x_1^{(k+1)} - 2x_2^{(k+1)}).\n",
        "\\end{align*}\n",
        "\n",
        "从初始值 $x^{(0)} = (0, -1, 0)$ 开始迭代，计算前两次迭代结果：\n",
        "\n",
        "第一次迭代：\n",
        "\\begin{align*}\n",
        "x_1^{(1)} &= \\frac{3 - (-1) + 0}{4} = 1\\\\\n",
        "x_2^{(1)} &= \\frac{-6 + 1 - 0}{3} = \\frac{-5}{3} = -1.667\\\\\n",
        "x_3^{(1)} &= \\frac{4 - 2(1) - 2(-1.667)}{5} = \\frac{4 - 2 + 3.334}{5} = \\frac{5.334}{5} = 1.067\n",
        "\\end{align*}\n",
        "\n",
        "第二次迭代：\n",
        "\\begin{align*}\n",
        "x_1^{(2)} &= \\frac{3 - (-1.667) + 1.067}{4} = \\frac{5.734}{4} = 1.4335\\\\\n",
        "x_2^{(2)} &= \\frac{-6 + 1.4335 - 1.067}{3} = \\frac{-5.6335}{3} = -1.8778\\\\\n",
        "x_3^{(2)} &= \\frac{4 - 2(1.4335) - 2(-1.8778)}{5} = \\frac{4 - 2.867 + 3.7556}{5} = 0.9777\n",
        "\\end{align*}"
      ],
      "metadata": {
        "id": "IdHAJWMs7fwn"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "**2. Write codes using MATLAB (or other programming language) to solve the following linear system**"
      ],
      "metadata": {
        "id": "GDW9GGVe8edJ"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "BAm_rIJx7D1u",
        "outputId": "872b970f-2535-4b70-ad90-a129272e90e5"
      },
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Gauss-Seidel solution: [-0.7530598   0.04088658  0.69151617  0.71926212] after 8 iterations.\n",
            "SOR solution: [-0.75345564  0.04124306  0.69171407  0.71916649] after 6 iterations.\n"
          ]
        }
      ],
      "source": [
        "import numpy as np\n",
        "\n",
        "def gauss_seidel(A, b, x_0, tol, max_iter=1000):\n",
        "    x = np.copy(x_0)\n",
        "    n = A.shape[0]\n",
        "    x_prev = np.copy(x)\n",
        "    iteration = 0\n",
        "    diag_A = np.diag(A)\n",
        "\n",
        "    while iteration < max_iter:\n",
        "        for i in range(n):\n",
        "            x[i] = (b[i] - np.dot(A[i, :i], x[:i]) - np.dot(A[i, i+1:], x_prev[i+1:])) / diag_A[i]\n",
        "\n",
        "        iteration += 1\n",
        "        # Stopping criterion\n",
        "        if np.linalg.norm(x - x_prev, np.inf) <= tol:\n",
        "            break\n",
        "        x_prev = np.copy(x)\n",
        "\n",
        "    return x, iteration\n",
        "\n",
        "def SOR(A, b, x_0, tol, omega, max_iter=1000):\n",
        "    x = np.copy(x_0)\n",
        "    n = A.shape[0]\n",
        "    x_prev = np.copy(x)\n",
        "    iteration = 0\n",
        "    diag_A = np.diag(A)\n",
        "\n",
        "    while iteration < max_iter:\n",
        "        for i in range(n):\n",
        "            x_new = (b[i] - np.dot(A[i, :i], x[:i]) - np.dot(A[i, i+1:], x[i+1:])) / diag_A[i]\n",
        "            # Over-relaxation\n",
        "            x[i] = x[i] + omega * (x_new - x[i])\n",
        "\n",
        "        iteration += 1\n",
        "        # Stopping criterion\n",
        "        if np.linalg.norm(x - x_prev, np.inf) <= tol:\n",
        "            break\n",
        "        x_prev = np.copy(x)\n",
        "\n",
        "    return x, iteration\n",
        "\n",
        "# Input data\n",
        "A = np.array([[4, 1, 1, -1],\n",
        "              [1, 4, -1, -1],\n",
        "              [1, -1, 3, 1],\n",
        "              [-1, -1, 1, 5]], dtype=float)\n",
        "\n",
        "b = np.array([-3, -2, 2, 5], dtype=float)\n",
        "\n",
        "x_0 = np.array([0, 0, 0, 1], dtype=float)\n",
        "\n",
        "tol = 1e-3\n",
        "omega = 1.2\n",
        "\n",
        "# Gauss-Seidel method\n",
        "gs_solution, gs_iterations = gauss_seidel(A, b, x_0, tol)\n",
        "print(f\"Gauss-Seidel solution: {gs_solution} after {gs_iterations} iterations.\")\n",
        "\n",
        "# SOR method\n",
        "sor_solution, sor_iterations = SOR(A, b, x_0, tol, omega)\n",
        "print(f\"SOR solution: {sor_solution} after {sor_iterations} iterations.\")"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "**3. Write a code using MATLAB (or other programming language) to compute the derivative**"
      ],
      "metadata": {
        "id": "92j5r7wz8kxP"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "\n",
        "def f(x):\n",
        "    return np.sin(x) / x**2\n",
        "\n",
        "def central_difference_derivative(f, x, h):\n",
        "    return (f(x + h) - f(x - h)) / (2 * h)\n",
        "\n",
        "def true_derivative(x):\n",
        "    return (x**2 * np.cos(x) - 2*x*np.sin(x)) / x**4\n",
        "\n",
        "x = 3.0\n",
        "true_value = true_derivative(x)\n",
        "\n",
        "# Step sizes\n",
        "hs = [10**(-i) for i in range(1, 6)]\n",
        "\n",
        "print(\"Approximations and errors:\")\n",
        "errors = []\n",
        "for h in hs:\n",
        "    derivative_approx = central_difference_derivative(f, x, h)\n",
        "    error = np.abs(derivative_approx - true_value)\n",
        "    errors.append(error)\n",
        "    print(f\"h={h:.0e}, Approximation={derivative_approx:.6f}, Error={error:.6e}\")\n",
        "\n",
        "# Check the order of convergence\n",
        "print(\"\\nChecking the order of convergence:\")\n",
        "for i in range(1, len(hs)):\n",
        "    ratio = hs[i-1] / hs[i]\n",
        "    order = np.log(errors[i-1] / errors[i]) / np.log(ratio)\n",
        "    print(f\"Order of convergence when h is decreased by a factor of {ratio}: {order:.2f}\")"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "2vgUsuqZ8yDl",
        "outputId": "3a40513b-f13a-4851-b33b-7542f33439a1"
      },
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Approximations and errors:\n",
            "h=1e-01, Approximation=-0.120607, Error=1.544207e-04\n",
            "h=1e-02, Approximation=-0.120454, Error=1.542962e-06\n",
            "h=1e-03, Approximation=-0.120453, Error=1.542948e-08\n",
            "h=1e-04, Approximation=-0.120453, Error=1.545546e-10\n",
            "h=1e-05, Approximation=-0.120453, Error=2.176495e-12\n",
            "\n",
            "Checking the order of convergence:\n",
            "Order of convergence when h is decreased by a factor of 10.0: 2.00\n",
            "Order of convergence when h is decreased by a factor of 10.0: 2.00\n",
            "Order of convergence when h is decreased by a factor of 10.0: 2.00\n",
            "Order of convergence when h is decreased by a factor of 10.0: 1.85\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "**4. Write codes using MATLAB (or other programming language) to solve the ODE**"
      ],
      "metadata": {
        "id": "R_yCMiuT9EV0"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "def f(t, y):\n",
        "    return (t * y - 0.5 * y**2) / t**2\n",
        "\n",
        "def forward_euler(f, t0, y0, t_end, h):\n",
        "    t = np.arange(t0, t_end + h, h)\n",
        "    y = np.zeros(t.shape)\n",
        "    y[0] = y0\n",
        "\n",
        "    for i in range(1, len(t)):\n",
        "        y[i] = y[i-1] + h * f(t[i-1], y[i-1])\n",
        "\n",
        "    return t, y\n",
        "\n",
        "def backward_euler(f, t0, y0, t_end, h):\n",
        "    t = np.arange(t0, t_end + h, h)\n",
        "    y = np.zeros(t.shape)\n",
        "    y[0] = y0\n",
        "\n",
        "    for i in range(1, len(t)):\n",
        "        y_prev = y[i-1]\n",
        "        y_curr = y_prev\n",
        "\n",
        "        for _ in range(10):  # 使用简单迭代法求解隐式方程\n",
        "            y_curr = y_prev + h * f(t[i], y_curr)\n",
        "\n",
        "        y[i] = y_curr\n",
        "\n",
        "    return t, y\n",
        "\n",
        "t0 = 1\n",
        "y0 = 4\n",
        "t_end = 3\n",
        "h = 1/128\n",
        "\n",
        "t_fe, y_fe = forward_euler(f, t0, y0, t_end, h)\n",
        "t_be, y_be = backward_euler(f, t0, y0, t_end, h)\n",
        "\n",
        "plt.figure(figsize=(8, 6))\n",
        "plt.plot(t_fe, y_fe, label='Forward Euler')\n",
        "plt.plot(t_be, y_be, label='Backward Euler')\n",
        "plt.xlabel('t')\n",
        "plt.ylabel('y')\n",
        "plt.legend()\n",
        "plt.grid(True)\n",
        "plt.show()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 542
        },
        "id": "CVfyeZ5A9Eo4",
        "outputId": "bf20c01a-9871-4c46-f571-daf3989acc2c"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 800x600 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "**5.**"
      ],
      "metadata": {
        "id": "W9hqOsW59LJz"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "要证明给定的数值格式在时间和空间上都是二阶精度的。可以使用泰勒级数展开来证明。\n",
        "\n",
        "对于时间方向：\n",
        "\\begin{align*}\n",
        "U_j^{n+1} &= U_j^n + \\frac{\\partial U_j^n}{\\partial t}\\Delta t + \\frac{1}{2}\\frac{\\partial^2 U_j^n}{\\partial t^2}\\Delta t^2 + O(\\Delta t^3)\\\\\n",
        "U_j^{n-1} &= U_j^n - \\frac{\\partial U_j^n}{\\partial t}\\Delta t + \\frac{1}{2}\\frac{\\partial^2 U_j^n}{\\partial t^2}\\Delta t^2 - O(\\Delta t^3)\n",
        "\\end{align*}\n",
        "\n",
        "将以上两式相减，得到：\n",
        "\\begin{equation*}\n",
        "U_j^{n+1} - U_j^{n-1} = 2\\frac{\\partial U_j^n}{\\partial t}\\Delta t + O(\\Delta t^3)\n",
        "\\end{equation*}\n",
        "\n",
        "对于空间方向：\n",
        "\\begin{align*}\n",
        "U_{j+1}^n &= U_j^n + \\frac{\\partial U_j^n}{\\partial x}\\Delta x + \\frac{1}{2}\\frac{\\partial^2 U_j^n}{\\partial x^2}\\Delta x^2 + O(\\Delta x^3)\\\\\n",
        "U_{j-1}^n &= U_j^n - \\frac{\\partial U_j^n}{\\partial x}\\Delta x + \\frac{1}{2}\\frac{\\partial^2 U_j^n}{\\partial x^2}\\Delta x^2 - O(\\Delta x^3)\n",
        "\\end{align*}\n",
        "\n",
        "将以上两式相加，得到：\n",
        "\\begin{equation*}\n",
        "U_{j+1}^n + U_{j-1}^n = 2U_j^n + \\frac{\\partial^2 U_j^n}{\\partial x^2}\\Delta x^2 + O(\\Delta x^4)\n",
        "\\end{equation*}\n",
        "\n",
        "将时间和空间的结果代入原始的数值格式：\n",
        "\\begin{align*}\n",
        "\\frac{U_j^{n+1} - U_j^{n-1}}{2\\Delta t} &= \\frac{\\partial U_j^n}{\\partial t} + O(\\Delta t^2)\\\\\n",
        "&= \\frac{\\partial^2 U_j^n}{\\partial x^2} + O(\\Delta x^2) + O(\\Delta t^2)\\\\\n",
        "&= \\frac{U_{j+1}^n - 2U_j^n + U_{j-1}^n}{\\Delta x^2} + O(\\Delta x^2) + O(\\Delta t^2)\n",
        "\\end{align*}\n",
        "\n",
        "综上所述，原始的数值格式在时间和空间上都具有二阶精度，即截断误差为 $O(\\Delta t^2)$ 和 $O(\\Delta x^2)$。"
      ],
      "metadata": {
        "id": "SJ04eI679O7T"
      }
    }
  ]
}