sistemas_lineares(Jacobi_Seidel).ipynb

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      "source": [
        "import numpy as np\n",
        "import math as m\n",
        "\n",
        "# Gauss-Jacobi\n",
        "def gauss_jacobi(A,b,x0,tol,flag):\n",
        "    n = np.shape(A)[0];\n",
        "    D = np.diag(np.diag(A));\n",
        "    aux_C = np.linalg.solve(D,A);\n",
        "    C = np.eye(n) - aux_C;\n",
        "    g = np.linalg.solve(D,b);\n",
        "    kmax = 10000; k = 0;\n",
        "\n",
        "    error = np.zeros(kmax);\n",
        "    if flag:\n",
        "        statment = (np.linalg.norm(b-A.dot(x0))>tol and k<kmax);\n",
        "    else:\n",
        "        statment = (k<kmax);\n",
        "\n",
        "    while statment:\n",
        "        error[k] = np.linalg.norm(b-A.dot(x0));\n",
        "        k = k+1;\n",
        "        x0 = C.dot(x0)+g;\n",
        "        if flag:\n",
        "            statment = (np.linalg.norm(b-A.dot(x0))>tol and k<kmax);\n",
        "        else:\n",
        "            statment = (k<kmax);\n",
        "    \n",
        "    if (k == kmax and flag):\n",
        "        print('\\nErro: o método não converge.\\n')\n",
        "    \n",
        "    x = x0;\n",
        "    return x,k,error\n",
        "\n",
        "# Gauss-Seidel\n",
        "def gauss_seidel(A,b,x0,tol,flag):\n",
        "    L = np.tril(A); R = np.triu(A,1);\n",
        "    C = -np.linalg.solve(L,R);\n",
        "    g = np.linalg.solve(L,b);\n",
        "    kmax = 10000; k = 0;\n",
        "\n",
        "    error = np.zeros(kmax); \n",
        "\n",
        "    if flag: \n",
        "        statment = (np.linalg.norm(b-A.dot(x0))>tol and k<kmax);\n",
        "    else:\n",
        "        statment = (k<kmax);\n",
        "        \n",
        "    while statment:\n",
        "        error[k] = np.linalg.norm(b-A.dot(x0));\n",
        "        k = k+1;\n",
        "        x0 = C.dot(x0)+g;\n",
        "        if flag:\n",
        "            statment = (np.linalg.norm(b-A.dot(x0))>tol and k<kmax);\n",
        "        else:\n",
        "            statment = (k<kmax);\n",
        "\n",
        "    if (k == kmax and flag):\n",
        "        print('\\nErro: o método não converge.\\n')\n",
        "    \n",
        "    x = x0;\n",
        "    return x,k,error\n",
        "\n",
        "# Exemplo\n",
        "A = np.array([[ -8, 1, 1],  \n",
        "              [ 1, -5, 1],  \n",
        "              [1, 1, -4]], dtype='double')\n",
        "print(A)\n",
        "b = np.array([1,16,7], dtype='double')\n",
        "print(b)\n",
        "\n",
        "# Função Python\n",
        "print('\\nSolução Python:\\n')\n",
        "x = np.linalg.solve(A,b);\n",
        "print(x);\n",
        "\n",
        "input();\n",
        "\n",
        "# Métodos Iterativos\n",
        "n = np.shape(A)[0];\n",
        "x0 = np.zeros(n); tol = 0.00000001;\n",
        "\n",
        "print('\\nSolução Gauss-Jacobi:\\n')\n",
        "(x_jacobi,k_jacobi,erro_jacobi) = gauss_jacobi(A,b,x0,tol,1);\n",
        "print(x_jacobi);\n",
        "print('%d iterações usadas' %(k_jacobi))\n",
        "\n",
        "input();\n",
        "\n",
        "print('\\nSolução Gauss-Siedel:\\n')\n",
        "(x_siedel,k_siedel,erro_siedel) = gauss_seidel(A,b,x0,tol,1);\n",
        "print(x_siedel);\n",
        "print('%d iterações usadas' %(k_siedel))"
      ],
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      "outputs": [
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          "text": [
            "[[-8.  1.  1.]\n",
            " [ 1. -5.  1.]\n",
            " [ 1.  1. -4.]]\n",
            "[ 1. 16.  7.]\n",
            "\n",
            "Solução Python:\n",
            "\n",
            "[-1. -4. -3.]\n",
            "\n",
            "\n",
            "Solução Gauss-Jacobi:\n",
            "\n",
            "[-1. -4. -3.]\n",
            "22 iterações usadas\n",
            "\n",
            "\n",
            "Solução Gauss-Siedel:\n",
            "\n",
            "[-1. -4. -3.]\n",
            "12 iterações usadas\n"
          ]
        }
      ]
    }
  ]
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