{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import division\n",
      "import numpy as np\n",
      "import scipy.linalg as la\n",
      "\n",
      "import matplotlib.pyplot as pt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's solve $u''=-30x^2$ with $u(0)=1$ and $u(1)=-1$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 50\n",
      "\n",
      "mesh = np.linspace(0, 1, n)\n",
      "h = mesh[1] - mesh[0]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 61
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = ..."
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 136
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<!--\n",
      "(np.eye(n, k=1) + -2*np.eye(n) + np.eye(n, k=-1))/h**2\n",
      "A[0] = 0\n",
      "A[-1] = 0\n",
      "A[0,0] = 1\n",
      "A[-1,-1] = 1\n",
      "-->\n",
      "(edit cell for solution)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b = -30*mesh**2\n",
      "b[0] = 1\n",
      "b[-1] = -1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 145
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_true = la.solve(A, b)\n",
      "pt.plot(mesh, x_true)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 138,
       "text": [
        "[<matplotlib.lines.Line2D at 0xc1d4710>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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01CMiIe3ii2HpUvjXf4WqKruj8V+q+EUk6Pz2t7BtG6xcaZ7mFcxU8YuIAA8/DJWV8NRT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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb1ada90>"
       ]
      }
     ],
     "prompt_number": 138
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### And now with Jacobi"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.zeros(n)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 440
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_new = np.empty(n)\n",
      "\n",
      "for i in xrange(n):\n",
      "    x_new[i] = b[i]\n",
      "    for j in xrange(i):\n",
      "        x_new[i] -= ...\n",
      "    for j in xrange(i+1, n):\n",
      "        x_new[i] -= ...\n",
      "        \n",
      "    x_new[i] = ...\n",
      "\n",
      "x = x_new\n",
      "pt.plot(mesh, x)\n",
      "pt.plot(mesh, x_true, label=\"true\")\n",
      "pt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 444,
       "text": [
        "<matplotlib.legend.Legend at 0x204e4a10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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H+YmkKdwrHCennsTm3zbj9V2vo6K6QuySqAUkHfzs8RNJl6e9J5ImJ+H2g9sY/cNo3C2/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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1fe04950>"
       ]
      }
     ],
     "prompt_number": 444
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<!--\n",
      "    for j in xrange(i):\n",
      "        x_new[i] -= A[i,j]*x[j]\n",
      "    for j in xrange(i+1, n):\n",
      "        x_new[i] -= A[i,j]*x[j]\n",
      "        \n",
      "    x_new[i] = x_new[i] / A[i,i]\n",
      "\n",
      "-->\n",
      "\n",
      "(Edit cell for solution)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* Ideas to accelerate this?\n",
      "* Multigrid"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### And now Gauss-Seidel"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.zeros(n)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 148
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_new = np.empty(n)\n",
      "\n",
      "for i in xrange(n):\n",
      "    x_new[i] = b[i]\n",
      "    for j in xrange(i):\n",
      "        x_new[i] -= A[i,j]*x_new[j]\n",
      "    for j in xrange(i+1, n):\n",
      "        x_new[i] -= A[i,j]*x[j]\n",
      "        \n",
      "    x_new[i] = x_new[i] / A[i,i]\n",
      "\n",
      "x = x_new\n",
      "pt.plot(mesh, x)\n",
      "pt.plot(mesh, x_true, label=\"true\")\n",
      "pt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 425,
       "text": [
        "<matplotlib.legend.Legend at 0x1f603ad0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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+mJgY+Pj4wMPDA+bm5pgwYQJ2795daZs9e/bg5ZdfBgB07doVOTk5SE9PV+ew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\nYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1fe26cd0>"
       ]
      }
     ],
     "prompt_number": 425
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### And now Successive Over-Relaxation (\"SOR\")"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.zeros(n)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 426
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_new = np.empty(n)\n",
      "\n",
      "for i in xrange(n):\n",
      "    x_new[i] = b[i]\n",
      "    for j in xrange(i):\n",
      "        x_new[i] -= A[i,j]*x_new[j]\n",
      "    for j in xrange(i+1, n):\n",
      "        x_new[i] -= A[i,j]*x[j]\n",
      "        \n",
      "    x_new[i] = x_new[i] / A[i,i]\n",
      "\n",
      "direction = x_new - x\n",
      "omega = 1.5\n",
      "x = x + omega*direction\n",
      "\n",
      "pt.plot(mesh, x)\n",
      "pt.plot(mesh, x_true, label=\"true\")\n",
      "pt.legend()\n",
      "pt.ylim([-1.3, 1.3])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 439,
       "text": [
        "(-1.3, 1.3)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SPqjtVsePq78IS+vaHB4O/fqpy0Gda2ObHMbYzWP4a9xf2Nvcp1XYBFSqrHPg\nwAF8fHzw8vLC2tqa0NBQNm7ceNc6mzZtYsSIEQB06NCB1NRUkpKSKrNbIYQZufcq3q1b1Ybh0tSu\nrf5KOH269HUiIqBvX3W5a1e4uKM3Pbx78Na2Uq78MjGVSvwJCQl4enoWPPbw8CAhIeG+61y8eLEy\nuxVCmJEnn4TISLWaEEq+cOteAQGlX/WbmKhe75LfPtCundr/f1bnT/jh5A/subCn6oI3UJWq6tGV\nswvCvfVPpb1v5syZBctBQUEEBQVVNDQhhIlwdVW7HG/frt5bNzNTHaakLPlDND/3XPHXIiPVnkL5\n1UA1a6rrnz3qzPy+8xn982hixsZQ06pm1X+YKhAVFUVUSf1YH0ClEr+7uzvx8fEFj+Pj4/G4507i\n965z8eJF3N3dS9xe0cQvhBD5Bg5Uq3vatbt//39QE/lHH5X8Wnh4YTVPvq5d1Wtd3nlnMKuPreb9\nne/zfo/3qyb4KnZvoXjWrFkPvI1KVfW0a9eOs2fPcv78ebKysli3bh0hISF3rRMSEsLKlSsBiI6O\nxtHREVdX18rsVghhZgYNUock+e23+1fzQOEonfd2dsnJUauKHr+n806XLrB7t1obsaDfAsIOhpn0\n8M2VSvxWVlYsWLCAPn360KpVK4YOHUrLli0JCwsjLCwMgH79+tGkSRN8fHwYO3YsCxcurJLAhRDm\no2lTtcrnp5/KbtjN5+amdvO9tzlx3z71wsQGDe5+vnNn2L9fPTE0tG/InJ5zGLVpFDl5OVX2GQyJ\nXMAlhDAKM2bAunVw6lT51u/bV+2rX7QS4s031WtC3i+hFqdVK1i1Sm0YVhSF7iu6M6TVECY8MqFq\nPkA1qUjulEtXhBBGYezY4uP2l6Wkm7IU7cZ5r/zqHiis8pm1YxZXbl+pWMAGTBK/EMIoNGxYvG6+\nLPcm/oQEdQiSDh1KXj+/gTdf6/qtGf7wcKZtLT4agbGTxC+EMEn39uWPjITevUu/4rdrV3XU0KK1\nJjOCZvDbud/YG7+3eoPVM0n8QgiT1LSpOqZUaqr6uKRunEU1aaKOIxUXV/hcHZs6fBT8EePDx5tU\nQ68kfiGESbK0VMftOXJEHcF127ayq4p0uuLVPQDDWg/DwcaBxX8urt6A9UgSvxDCZOXX8+/dC82a\nFb+hy726di1s4M2n0+n4st+XJtXQK4lfCGGy8hN/0dE4y1K0Z09RD9V/yKQaeiXxCyFMVkCAOmZP\nRET5Er+pg/FfAAAPMklEQVS/v3ojopSU4q+ZUkOvJH4hhMlq3Vq9zWhiojrOz/1YW6v37N23r/hr\ndWzq8HHwxybR0CuJXwhhsmxtwccH+vRRG3vLo7TqHoDQ1qEm0dAriV8IYdL694fQ0PKvX1LPnnw6\nnY4J3l8yLWIW19KvVU2AGpCxeoQQooi0NPUq4WvXwMam8HlFgbAweOcdSOv2CqFP12DFsAcYQ6Ka\nyFg9QghRSXXqqF0/Dx0qfO7mTXj2WVi4UK0GGuUzk+9Pf8vZa2c1i7MyJPELIcQ9ilb3HD2qNgzX\nrq0O3dyiBUwaWR/L/VN54zfj7N4piV8IIe7RpYs6bs+SJer4/++8A199pTYWA/j6wkO3JrM39iA7\n43ZqG2wFSB2/EELc4+JFaNRIHaP/++9Lvsfv0qWwcNcqLLp8zv7R+7HQaVOOljp+IYSoAh4esGYN\nHDhQ+o3dn3kGzm0cRnY2rP1rrX4DrCQp8QshRAWNGgU1fHYRbvscp8afwtbaVu8xSIlfCCH0aNQo\n2L68G20btGXe/nlah1NukviFEKKCOnUCCwt42mkuH+/92GhG75SqHiGEqISPPoITJ8Bh6BSycrNY\n2H+hXvdfkdwpiV8IISohKUnt23/kzHXarWjBzhd20rJeKS3C1UASvxBCaGDQIHVMoLSHPiXqfBSb\nhm3S276lcVcIITQwapTar/+V9q9wOPEw0RejtQ6pTFLiF0KISsrJUS/42roV9tz5inXH17F1+Fa9\n7FtK/EIIoQErKxgxApYtgxf8X+B86nl+j/1d67BKJSV+IYSoAmfPqoO7xcfD96dW8eUfX7Jn5B50\nOl217ldK/EIIoZFmzdTB2zZvVu/UlZaZRvjZcK3DKpEkfiGEqCJjxsCcOZCbY8l73d/jn7//kzwl\nT+uwipHEL4QQVeTZZ6FBA3j1VRjoOxArCyt+OPGD1mEVI3X8QghRhW7cgEcegTffhAZdf2XKr1P4\na9xfWFqU827vD0jq+IUQQmMODrBhA7zxBrik9qaeXT2+Pfqt1mHdRRK/EEJUsVatYNEiGDJExxuB\nHzBzx0yycrO0DquAJH4hhKgGQ4ZAaCjMe70bzZ1bsPTQUq1DKlDhOv7r168zdOhQ4uLi8PLy4rvv\nvsPR0bHYel5eXtSpUwdLS0usra05cOBAyYFIHb8QwsTk5EDfvtCg7Z9srz+QsxPPVvnNWvRaxz9n\nzhyCg4M5c+YMPXv2ZM6cOaUGFRUVRUxMTKlJXwghTJGVFaxdC7vWtaMhj7Doz0VahwRUosTv6+vL\njh07cHV1JTExkaCgIE6dOlVsPW9vb/78809cXFzKDkRK/EIIE3XoEPR89gjWL/Yl7rVzVVrq12uJ\nPykpCVdXVwBcXV1JSkoqNahevXrRrl07vvrqq4ruTgghjFZgIHwwyQ8utWdZzDKtw8GqrBeDg4NJ\nTEws9vwHH3xw12OdTlfqeBR79uyhQYMGXL16leDgYHx9fenWrVuJ686cObNgOSgoiKCgoPuEL4QQ\nxmH0aJjV8W3eaziEMW3HUMOyRoW2ExUVRVRUVKViqVRVT1RUFG5ubly+fJnu3buXWNVT1KxZs6hd\nuzavv/568UCkqkcIYeLmzYPZ8X344NmnGR04ukq2qdeqnpCQEFasWAHAihUrGDhwYLF10tPTuXnz\nJgC3b9/mt99+o02bNhXdpRBCGLXRoyFr6z95d/u/ycnL0SyOCif+6dOns2XLFpo3b8727duZPn06\nAJcuXaJ///4AJCYm0q1bN/z9/enQoQMDBgygd+/eVRO5EEIYmVq1YOrT3chK9mDtX2s1i0PG6hFC\nCD1KTYVGQVuo/8Ikzkw+joWuctfRylg9Qghh4Bwd4ZW+vbiZXIcfT/6oSQxS4hdCCD1LSoKm/X7G\n68V3ODY+plJ36ZISvxBCGAFXVxjRaQBXr8IvZ3/R+/6lxC+EEBqIi4PWz3xPixc/4Y+x+ypc6pcS\nvxBCGInGjWFQy6eIS7rBtthtet23JH4hhNDIm9MsydzyNrN+f1+v+5XEL4QQGmnZEnq5hXLyUjy7\n4nbpbb+S+IUQQkNvv2lF7s43mLv7I73tUxK/EEJoqG1baGs1nJ3/jeZUctnjnVUVSfxCCKGxGW/Z\nYRnzCp/s/Uwv+5PunEIIYQA6B1/hyKMtiH3tNPVr1S/3+6Q7pxBCGKl3p9XH+vRQvtj/ZbXvS0r8\nQghhABQFAoJPExvUjcvTzmNnbVeu90mJXwghjJROBx+82gLiO/N1zIrq3ZeU+IUQwjAoCvj23s3N\nHi8SP+0UlhaW932PlPiFEMKI6XTw73FdSEtyYePpTdW3HynxCyGE4cjLA69+66kd/BknXt9z3/Wl\nxC+EEEbOwgLmjBjEf69cZs+FvdWzj2rZqhBCiAob+owldU68xpubPqmW7UviF0IIA2NpCe8/9SLR\nl3fy9/W/q3z7kviFEMIAvfhcLexOjWXahs+rfNuS+IUQwgBZW8PbvSbw8/nVXEu/VqXblsQvhBAG\natJIN6z/+ySzNi+p0u1K4hdCCANlYwOjH57IsqMLycnLqbLtSuIXQggDNmtsIBlX3VkZvbnKtimJ\nXwghDJijI/SsPZF3f/2iyrYpiV8IIQzcZy8NJv7OCQ5eOFEl25PEL4QQBq5Vixr4pI3ltbULqmR7\nkviFEMIIfDDoJXanriUl/UaltyWJXwghjMDg3g2pc7UP09ctr/S2JPELIYQR0OlgfPsJfHN6AXlK\nXqW2JYlfCCGMxDsjOpN9y57/bPutUtuRxC+EEEbCxkbHgPoT+GBr5bp2SuIXQggjMv+lYSQoB9h/\ntuKjdlY48X///fc89NBDWFpacujQoVLXi4yMxNfXl2bNmjF37tyK7k4IIQTg6WZLm9yRvLp6YYW3\nUeHE36ZNGzZs2MCjjz5a6jq5ublMmDCByMhITpw4wZo1azh58mRFdymEEAL4eOg4ojNWcO3mrQq9\nv8KJ39fXl+bNm5e5zoEDB/Dx8cHLywtra2tCQ0PZuHFjRXcphBACCG7vRb30bry2fFWF3l+tdfwJ\nCQl4enoWPPbw8CAhIaE6dymEEGbh1S4TWRdbsUZeq7JeDA4OJjExsdjzs2fP5oknnrjvxnU63QMF\nM3PmzILloKAggoKCHuj9Qghh6qKiooiKikJRFOofdCa+AtsoM/Fv2bKlgqGp3N3diY8vDCs+Ph4P\nD49S1y+a+IUQQhRXtFA8i1kPXMCGKqrqURSlxOfbtWvH2bNnOX/+PFlZWaxbt46QkJCq2KUQQogK\nqnDi37BhA56enkRHR9O/f3/69u0LwKVLl+jfvz8AVlZWLFiwgD59+tCqVSuGDh1Ky5YtqyZyIYQQ\nFaJTSiuu65lOpyv1l4MQQoiSVSR3ypW7QghhZiTxCyGEmZHEL4QQZkYSvxBCmBlJ/EIIYWYk8Qsh\nhJmRxC+EEGZGEr8QQpgZSfxCCGFmJPELIYSZkcQvhBBmRhK/EEKYGUn8QghhZiTxCyGEmZHEL4QQ\nZkYSvxBCmBlJ/EIIYWYk8RugqKgorUMwGHIsCsmxKCTHonIk8Rsg+VIXkmNRSI5FITkWlSOJXwgh\nzIwkfiGEMDM65UFvz15N/P3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       "text": [
        "<matplotlib.figure.Figure at 0x1f008dd0>"
       ]
      }
     ],
     "prompt_number": 439
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}