{
  "cells": [
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "Given the equations:\n\n$$\n\\begin{eqnarray*}\n10x_1 + 2x_2 - x_3 = 27 \\\\\n-3x_1 -6x_2 + 2x_3 = -61.5 \\\\\nx_1 + x_2 + 5x_3 = - 21.5 \\\\\n\\end{eqnarray*}\n$$"
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "(a) Naive Gauss elimination"
    },
    {
      "metadata": {
        "collapsed": false,
        "trusted": true
      },
      "cell_type": "code",
      "source": "import numpy as np\nA = np.array([[10.,2.,-1.],[-3.,-6.,2.],[1.,1.,5.]])\nB = np.array([[27.],[-61.5],[-21.5]])",
      "execution_count": 1,
      "outputs": []
    },
    {
      "metadata": {
        "collapsed": false,
        "trusted": true
      },
      "cell_type": "code",
      "source": "import copy\ndef upm(A_, B_):\n    A = copy.deepcopy(A_)\n    B = copy.deepcopy(B_)\n    for i in range(len(A)):\n        for j in range((i+1),len(A)):\n            c = A[j][i]/A[i][i]\n            for k in range(len(A)):\n                A[j][k] -= c*A[i][k]\n            B[j][0] -= c*B[i][0]\n    return [A,B]\n    \ndef naive_gauss_elimination(A_, B_):\n    A = copy.deepcopy(A_)\n    B = copy.deepcopy(B_)\n    A, B = upm(A, B)\n    ans = np.array([0. for i in range(len(A))])\n    for i in reversed(range(len(A))):\n        a = B[i]\n        for j in reversed(range(i+1,len(A))):\n            a -= A[i][j]*ans[j]\n        a /= A[i][i]\n        ans[i] = a\n    return ans",
      "execution_count": 2,
      "outputs": []
    },
    {
      "metadata": {
        "collapsed": false,
        "trusted": true
      },
      "cell_type": "code",
      "source": "X = naive_gauss_elimination(A,B)\nprint(X)",
      "execution_count": 3,
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": "[ 0.5  8.  -6. ]\n"
        }
      ]
    },
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "(b) Substitute the results into original equations to check the answer"
    },
    {
      "metadata": {
        "collapsed": false,
        "scrolled": true,
        "trusted": true
      },
      "cell_type": "code",
      "source": "A.dot(X)",
      "execution_count": 4,
      "outputs": [
        {
          "data": {
            "text/plain": "array([ 27. , -61.5, -21.5])"
          },
          "execution_count": 4,
          "metadata": {},
          "output_type": "execute_result"
        }
      ]
    },
    {
      "metadata": {
        "collapsed": false,
        "scrolled": true
      },
      "cell_type": "markdown",
      "source": "(c) Compute the determinant of the matrix of the coefficients"
    },
    {
      "metadata": {
        "collapsed": false,
        "trusted": true
      },
      "cell_type": "code",
      "source": "np.linalg.det(A)",
      "execution_count": 5,
      "outputs": [
        {
          "data": {
            "text/plain": "-289.00000000000006"
          },
          "execution_count": 5,
          "metadata": {},
          "output_type": "execute_result"
        }
      ]
    },
    {
      "metadata": {
        "collapsed": false,
        "trusted": true
      },
      "cell_type": "code",
      "source": "A, _ = upm(A, B)\ndet = 1.\nfor i in range(len(A)):\n    det *= A[i][i]\nprint(det)",
      "execution_count": 6,
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": "-289.0\n"
        }
      ]
    }
  ],
  "metadata": {
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3",
      "language": "python"
    },
    "language_info": {
      "name": "python",
      "version": "3.6.0",
      "mimetype": "text/x-python",
      "codemirror_mode": {
        "name": "ipython",
        "version": 3
      },
      "pygments_lexer": "ipython3",
      "nbconvert_exporter": "python",
      "file_extension": ".py"
    },
    "toc": {
      "threshold": 4,
      "number_sections": true,
      "toc_cell": false,
      "toc_window_display": false,
      "toc_section_display": "block",
      "sideBar": true,
      "navigate_menu": true,
      "moveMenuLeft": true,
      "widenNotebook": false,
      "colors": {
        "hover_highlight": "#DAA520",
        "selected_highlight": "#FFD700",
        "running_highlight": "#FF0000"
      },
      "nav_menu": {
        "height": "12px",
        "width": "252px"
      }
    },
    "varInspector": {
      "window_display": true,
      "cols": {
        "lenName": 16,
        "lenType": 16,
        "lenVar": 40
      },
      "kernels_config": {
        "python": {
          "library": "var_list.py",
          "delete_cmd_prefix": "del ",
          "delete_cmd_postfix": "",
          "varRefreshCmd": "print(var_dic_list())"
        },
        "r": {
          "library": "var_list.r",
          "delete_cmd_prefix": "rm(",
          "delete_cmd_postfix": ") ",
          "varRefreshCmd": "cat(var_dic_list()) "
        }
      },
      "types_to_exclude": [
        "module",
        "function",
        "builtin_function_or_method",
        "instance",
        "_Feature"
      ],
      "oldHeight": 122,
      "position": {
        "height": "144px",
        "left": "880px",
        "right": "20px",
        "top": "96px",
        "width": "350px"
      },
      "varInspector_section_display": "block"
    },
    "gist": {
      "id": "",
      "data": {
        "description": "Numerical Method Lecture - naive Gauss elimination",
        "public": true
      }
    }
  },
  "nbformat": 4,
  "nbformat_minor": 2
}