{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%load_ext sympyprinting\n",
      "from sympy import *\n",
      "from numpy import *\n",
      "from sympy.abc import x\n",
      "from matplotlib import pyplot as plt\n",
      "%matplotlib inline "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The sympyprinting extension is already loaded. To reload it, use:\n",
        "  %reload_ext sympyprinting\n"
       ]
      }
     ],
     "prompt_number": 257
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "h=symbols('h')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#For A_{i,i}, linear hat functions completely overlap\n",
      "f1 = x/h;\n",
      "plot(f1.subs(h,.1),(f1*f1).subs(h,.1),(x,0,.1))\n",
      "2*integrate(f1*f1,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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JibJ1cGiop6PyHRcuyL76b74JtG4tG44lJgI3cMNsIpdhoreTeaG1Rw92tNZV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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x110db1f90>"
       ]
      },
      {
       "latex": [
        "$$\\frac{2 h}{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAArBAMAAABhvA5FAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNu0SrdlQQ3e8y\niWbzIQYJAAAA9klEQVQoFWNgEFJ2ZWCYtosBBBgTGOobGBiMwBy2CQxcHxgYfoA53DsZ2L4ysP4E\nc5h/MjB9ZeBdAOYACa6fDMx65Y4Q7vkLDPMNGdQgHBMGhv4DDClgDpMDA4MWA8NaMKcMSK5lYPwK\n4nAoMAhyAs3bwAnkHGJgeMj3gYH7wkOgc9bevbSAzYBh/oELQEv+//+/gLmAgenuBJCmAQBAB8AA\nMFyoAc7e9YAbw6jAEF8A4/H+YOBfAONwIHOAgghlQI4ZTBUDA8/lBASHgaE6AInHDYk5oAjjAQaW\nfzCZ+T+ROPwGDFwfYTLsDxjiL8A4DE1KFmA2AKJ9ROdolAMbAAAAAElFTkSuQmCC\n",
       "prompt_number": 76,
       "text": [
        "2\u22c5h\n",
        "\u2500\u2500\u2500\n",
        " 3 "
       ]
      }
     ],
     "prompt_number": 76
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#For A_{i,i+1} and A_{i,i-1}, linear hat functions partially overlap\n",
      "f1 = x/h;\n",
      "f2 = 1-x/h;\n",
      "plot(f1.subs(h,.1),f2.subs(h,.1),(f1*f2).subs(h,.1),(x,0,.1))\n",
      "integrate(f1*f2,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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S1u+5RxK8t+06SOQqZrMU64wM6edPnQrceqv9m69xVa39WOhbqTW9+Opq4JNP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\noZvVAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x11106bc10>"
       ]
      },
      {
       "latex": [
        "$$\\frac{h}{6}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAArBAMAAAC+8nRaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvv3c2ZVESJZna7\nIjIGqvRyAAAAoklEQVQYGWNgEDJhAIMwCPUdTHF8BFOcDmCKNWv3WSBDPpYhCUjVP2DQAVIpDAye\nQMqTgfcHAwPjDwZOA0YGrg8MzBPWMjAFMMg/mMDAuoGBbaYAUA1F4D8YEGeEZOY8oEJePwYVIMUy\ngSEUSK2HuCAYYsLX19sKgM79soDBC0j9Z2CYLwDkAVUBRb8CqQ0MDEcgvHqwHAPPBEYrkNakpAYG\nAKbMKZ+3N+QJAAAAAElFTkSuQmCC\n",
       "prompt_number": 75,
       "text": [
        "h\n",
        "\u2500\n",
        "6"
       ]
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f1 = x/h;\n",
      "integrate(f1,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{h}{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAArBAMAAAC+8nRaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvv3c2ZVESJZna7\nIjIGqvRyAAAApElEQVQYGWNgEDJhAIMwCPUdTHF8BFOcDmCKNWv3WSBDPpYhCUjVP2DQAVIpDAye\nQMqTgfcHAwPjDwZOA0YGrg8MzBPWMjAFMMg/mMDAuoGBbaYAUA1F4D8YEGUEb1pMA1DhJQZukC8U\nGRj8gZSxAIM+UDRegCEfSAGBP9hJfD/BHNYDYCoHTHIrgKkyBt4LDAx8CQxcQOrKzJlAnzL4////\nhQEA5IkmyHWuBKwAAAAASUVORK5CYII=\n",
       "prompt_number": 72,
       "text": [
        "h\n",
        "\u2500\n",
        "2"
       ]
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Now for Hermite polynomials.  As above, we consider on a single interval, with varying overlaps.\n",
      "a2f = lambda a: a[0]+a[1]*x+a[2]*x**2+a[3]*x**3\n",
      "M = Matrix([[1,0,0,0],[0,1,0,0],[1,h,h**2,h**3],[0,1,2*h,3*h**2]]).inv()\n",
      "f1 = a2f(M.dot(Matrix([1,0,0,0])));\n",
      "f2 = a2f(M.dot(Matrix([0,1,0,0])));\n",
      "f3 = a2f(M.dot(Matrix([0,0,1,0])));\n",
      "f4 = a2f(M.dot(Matrix([0,0,0,1])));\n",
      "hval = 1.6\n",
      "plot(f1.subs(h,hval),f2.subs(h,hval),f3.subs(h,hval),f4.subs(h,hval),(x,0,hval))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2bduGnJwcdOrUCR07doSXl9cdx0ZERBS/HxgYKPF5O2DAAMDHB/jHP1jwNHs2N2AtBaUY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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x11173c410>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 121,
       "text": [
        "<sympy.plotting.plot.Plot at 0x111132450>"
       ]
      }
     ],
     "prompt_number": 121
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#test\n",
      "f2u = lambda f: [f.subs(x,0),diff(f,x).subs(x,0),f.subs(x,h),diff(f,x).subs(x,h)]\n",
      "[f2u(f) for f in [f1,f2,f3,f4]]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\begin{bmatrix}1, & 0, & 0, & 0\\end{bmatrix}, & \\begin{bmatrix}0, & 1, & 0, & 0\\end{bmatrix}, & \\begin{bmatrix}0, & 0, & 1, & 0\\end{bmatrix}, & \\begin{bmatrix}0, & 0, & 0, & 1\\end{bmatrix}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAo0AAAAZBAMAAACm6wkCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
       "prompt_number": 122,
       "text": [
        "[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]"
       ]
      }
     ],
     "prompt_number": 122
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_ii for i even\n",
      "2*integrate(f1*f1,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{26 h}{35}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAArBAMAAAAEUfw6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNu0SrdlQQ3e8y\niWbzIQYJAAABW0lEQVQoFc3SMUvDQBTA8X+SGtvGpOooCEWhiIPEIuIgooOgg+A3aBWLk+BkBweL\nq4OdXBySL1DpqovBXSx1LrTfoB2CDajxLiWtugu+4bj34+XuhXswPb8NqWbL4+YBGUqRiwoznu5C\nPhK9SrpLDc2GfiTGPbpvBXI/Fq1oAaqv1aWYrlxlpIPMenkBtKXy1kBeGs4d6TbOMrmBrOB0MQOu\nPY4iUTdxXKw+i4g7ZZxBxsZ6E6niS0hmmUrImpSPWU8JeYJOUp5jdTEaHfEXtVbTZY/btm7jeA3R\nTBiGLuphCe0UtVWVB/33EC3/iO4fN/zc2oGN6mwlvkfJUjjlMfyIAbNPxiV/4g0lGUlxmEcb8dUv\nWYXjcmlUNfEqCnZx9kfEuUz0+jcx5HiZ77EoHolP8fRmNAdSnUCIIR62F9eIWUn3VDF1dizjbQoN\nJcvVZCxczq2JqckdRPAFv41v6RP5q9IAAAAASUVORK5CYII=\n",
       "prompt_number": 123,
       "text": [
        "26\u22c5h\n",
        "\u2500\u2500\u2500\u2500\n",
        " 35 "
       ]
      }
     ],
     "prompt_number": 123
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_ii for i odd\n",
      "2*integrate(f2*f2,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 h^{3}}{105}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAwBAMAAACCgmVcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 119,
       "text": [
        "   3\n",
        "2\u22c5h \n",
        "\u2500\u2500\u2500\u2500\n",
        "105 "
       ]
      }
     ],
     "prompt_number": 119
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_{i,i+1} for i even\n",
      "2*integrate(f1*f2,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{11 h^{2}}{105}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAAwBAMAAAB6Yp6rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 124,
       "text": [
        "    2\n",
        "11\u22c5h \n",
        "\u2500\u2500\u2500\u2500\u2500\n",
        " 105 "
       ]
      }
     ],
     "prompt_number": 124
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_{i,i+1} for i odd\n",
      "integrate(f2*f3,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{13 h^{2}}{420}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAAwBAMAAAB6Yp6rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 126,
       "text": [
        "    2\n",
        "13\u22c5h \n",
        "\u2500\u2500\u2500\u2500\u2500\n",
        " 420 "
       ]
      }
     ],
     "prompt_number": 126
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_{i,i+2} for i even\n",
      "integrate(f1*f3,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{9 h}{70}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAArBAMAAABhvA5FAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKuZiTIi71S7zXZE\nZt2W8+iPAAABE0lEQVQoFb3Sv0vDQBjG8W+apLb5YaUd3EwXBzcRHRyUgNlcsru4OTajkxT8A8zk\nHBBcRHA2i7u7c9UODg6tSimkRd+7A3UXfAmBD/fkuYQLtDdXoX2CmvoxR4dwodHMqOUw1WgVOGOC\nsUaU489wBt8rc7yd8lxoZzQroks2BNY1pxVnKWsqGJYHU3ZhqIBqG1KfGXiFJX2ZBf6E3l04YiF+\nEDxyRa0gSmPJ7W338W6wkyXz0L/fP39m9PfN3eckSeiUW6oqlOo3XllJBY58YNcf4BcKcqWLXYKJ\nQOae1jquORQrpxfjVnrB7mt8aET8wq2BiT2hC+YqZr2DVDd0dSCQTc2pNgS8sKxeB1eFO/vym/AF\nGQVRRdQQ/CwAAAAASUVORK5CYII=\n",
       "prompt_number": 127,
       "text": [
        "9\u22c5h\n",
        "\u2500\u2500\u2500\n",
        " 70"
       ]
      }
     ],
     "prompt_number": 127
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_{i,i+2} for i odd\n",
      "integrate(f2*f4,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{h^{3}}{140}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAwBAMAAABK7o+KAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIs3dRBCr75lUiWZ2\nuzI+JW5bAAABSklEQVQ4EWNgQAesa5YFoItB+BwbOBSwy7A1cH/ALsPAwIVDDwNDuAOGnrBikFDM\nLQwJBgYNsBjHBUypn2AhRgiFLM3yFciTd2D8jSwIZnMZAKlgAe5SDBmm7jMXGfjOLBPAkIlXZGjD\nEAQLzH/AkItdppmBwQq7jBUD4xesMqxfGLgKWLFJAcOYfcFGbDJsCgzxDxZgk2E6wMCxKgCbzLAQ\nEzIGARMG/v+o4MPg8t1VkHP4EhhYe048QHFZLCiFMjBfYGB2YLVDlglLA8scv8BwmIHhELIMAw9I\nhnHTBQYzBgZ/lGQKluFiusDwg4Hh/QRkTWCZjUwXGH8BZQ6gy7AegMosQJfhYsAlMw8mg24a6wKg\nDNgFG9BM40xLyy87AHT1fExXM7CDfboYWQvEpyAZ5gmsdcgyHO2fuoElS/6PDaxdxx4gy6CzAVId\nar4qvbjsAAAAAElFTkSuQmCC\n",
       "prompt_number": 128,
       "text": [
        "  3 \n",
        "-h  \n",
        "\u2500\u2500\u2500\u2500\n",
        "140 "
       ]
      }
     ],
     "prompt_number": 128
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_{i,i+3} for i even\n",
      "integrate(f1*f4,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{13 h^{2}}{420}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD0AAAAwBAMAAABQ0m8EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 129,
       "text": [
        "     2 \n",
        "-13\u22c5h  \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "  420  "
       ]
      }
     ],
     "prompt_number": 129
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#To calculate B, we need shape function derivatives.\n",
      "df1 = diff(f1,x);\n",
      "df2 = diff(f2,x);\n",
      "df3 = diff(f3,x);\n",
      "df4 = diff(f4,x);\n",
      "hval = 1.6\n",
      "plot(df1.subs(h,hval),df2.subs(h,hval),df3.subs(h,hval),df4.subs(h,hval),(x,0,hval))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BMRiUWrVKKS8vepZywTUPZ88qNXGiUp6eSi1fTu9d0IbERKVefVWpChWUmjlT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\nBB0i4i4IgqBD/h8JxGCqELY9KAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x111132bd0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 132,
       "text": [
        "<sympy.plotting.plot.Plot at 0x111628d10>"
       ]
      }
     ],
     "prompt_number": 132
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#B_ii for i even\n",
      "2*integrate(df1*df1,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{12}{5 h}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAAqBAMAAACq4N3gAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAuxCrdpnvzWYiRFTd\nMollB7CEAAABCElEQVQoFb2RMUvDYBCGnyYkaTQEcbOLH/gDUtFFBxXs3kIRHRyKs9A6BHRzdXNx\ncBF/QidX8wtKls5mcS2lmxGN930fke6Ct9z73HvcwR3QUPBw2kKHs6kI23THWu8cKlxFNDdWV+Ff\n4JY1BAu8X5BitKgdyW/FElwZjQwAb20Jjq02zmqPR4O6bQp3NYST2Xum4WSY3kZVVRnQhX8IWVeH\nvcLfdl6rpzFbl3bIsPoWkVpID3IRnxbaOq3Ye2Igzqyz29mH4KyzoXGd5Jnkhr713AGvOdsW4i/O\nYSLQyIhLUaF+lp/hfTgl8cCRX8ioUXOOX8ixwx4v9+6IJC+kb9rfIzjCmyl+AHtrSKnR1K/FAAAA\nAElFTkSuQmCC\n",
       "prompt_number": 494,
       "text": [
        " 12\n",
        "\u2500\u2500\u2500\n",
        "5\u22c5h"
       ]
      }
     ],
     "prompt_number": 494
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#B_ii for i even\n",
      "2*integrate(df2*df2,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{4 h}{15}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAAsBAMAAAB8uT79AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMqvdu3bvImbNiRBU\nmUSRiEKYAAAA+UlEQVQoFWNgYGAQYmDYlAykwcAESM6FsllTgIw/UE7FFAYGvt9QzgIgh8sAwuEo\nAHI4b4S6g7hbGYCc/dMZLoE4C0Cc8wIMKkA2jwCIc5mBwQrI2cUA4lgxsP8Dcg4pKX1T5f7HwJXA\nDeQxMKQwcHxgYFvQDuUwTWDYL7AAxFn0/wZnAAPvqg1gGfoT/xHgAzVsZwF6ZMaGqgMgs7j9gZz8\n/7/AbCV9IGeulgCIw8AQD+QogFnYOKqhehA5kDJvhv0PwDwQh4GBKQGJw/URxmExYOACBTzYaDYD\nBt7vMA4vMCInwDjsFxiOFYA4d/PnNjDIXlIHSyARAIOIS/+Sc5HQAAAAAElFTkSuQmCC\n",
       "prompt_number": 134,
       "text": [
        "4\u22c5h\n",
        "\u2500\u2500\u2500\n",
        " 15"
       ]
      }
     ],
     "prompt_number": 134
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#B_{i,i+1} for i even\n",
      "2*integrate(df1*df2,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{5}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAuxCrdpnvzWYiiVQy\nRN3DX0DuAAAAdklEQVQYGWNgYGBUYAACJnkQxWRsD6IYGPypQYXmlzaADCMP/AeBD0TprVCYtQGo\nMP//L5DyUqsDIMoARMApExdbEE+IQf8CiGZgTgASDAxcf4AB8ICB6y8DA9sDBtafDAysDAzsBQwM\nnAEM2yYAFRwNtmRgAADLIh1xkx+eEQAAAABJRU5ErkJggg==\n",
       "prompt_number": 135,
       "text": [
        "1/5"
       ]
      }
     ],
     "prompt_number": 135
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#B_{i,i+1} for i odd\n",
      "integrate(df2*df3,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{10}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACYAAAArBAMAAADmjedDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIs3dRLsQq3aZ72Yy\nVIm0UfRoAAAAkUlEQVQoFWNgQAFhKDwwJ/Ubhlh6OaYYA8fwEOOc9Wk2hocHoYCQMQiYMPD/hwN6\nuxKYWNhmdB5AthaUWJgd2GyQxMCJpZmBoQlJDJxYzBkY/AWQBEEJ6CcDw/kLqGKM/4BiDVjEHhAp\nhqYXbMcGVL0MQLfcR3cL0M2PkZSB3cx8gW0Nkhg4sbDNaz+AJIZgAgCPSUOYJpQjdAAAAABJRU5E\nrkJggg==\n",
       "prompt_number": 136,
       "text": [
        "-1/10"
       ]
      }
     ],
     "prompt_number": 136
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_{i,i+2} for i even\n",
      "integrate(df1*df3,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{6}{5 h}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAqBAMAAADc2m9rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIs3dRBCJmTKrZu92\nVLsuIg0pAAABAElEQVQoFWNgQAas6WkNyHwIW7iB0wBT1IyBSQFDlOsrhhBQgGkDNlF+jTO1mOLx\nVgzMCzCE4z8wsGEaHW/AwPUXQy2/AgPXHwxRPqxqebGay/CMIRrTDQzs06dgGDskBISMQcCEgeE/\nAnwYMJdrBYhdYIjcjGb//v/fgCLqaKLq0xqAIuhxPAGkigU9PYBF2QxAckhgzhlgpDFVnXmIJMbA\n8JQh3oEhXpGhEkWUgYFzA8P9BoZ5aKJsPxiKGRjMkEWByYntC1CI8QuyKLcBA/tH1i8MbBtYkYTZ\ngQ5Q4PrAwJ2wEEmUsYDhtgCnAkN8QwKSKEN75UQGpgMM7GkBDACC5UGPNOhg9QAAAABJRU5ErkJg\ngg==\n",
       "prompt_number": 137,
       "text": [
        "-6 \n",
        "\u2500\u2500\u2500\n",
        "5\u22c5h"
       ]
      }
     ],
     "prompt_number": 137
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_{i,i+2} for i odd\n",
      "integrate(df2*df4,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{h}{30}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACYAAAArBAMAAADmjedDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIs3dRBCr75lUiWZ2\nuzI+JW5bAAAA/ElEQVQoFWNgQAJhxUgcGFMDxkCifyKxoUyWr5hiXAaYYkzdZy6ii8YrMrShi81/\nwJCLLtbMwGCFLmbFwPgFTYz1CwNXASuqIPcHBvYFG1HF2BQY4h8sQBVjOsDAsSoAVWxQ8oSMQcAE\nq9v4/8MBVnmaCfKtWjuBgbXnxAMkGyQYWP4wMDuw2iGJrZ/A8JnhMAPDISSx+SAxMwYGfwEkQZDe\nHwwM7ycgizElMP4Cih1AEnulxgAWW4AkxsBTginGUB+IpleOgeH+BpAdG+B6Wf8HAMWAbpmP5JZP\nDAz2AkA3L4YrY2A4zcDyi4F5AmsdkhhnRxMwDLqOPWAAALKlS0gioU9bAAAAAElFTkSuQmCC\n",
       "prompt_number": 138,
       "text": [
        "-h \n",
        "\u2500\u2500\u2500\n",
        " 30"
       ]
      }
     ],
     "prompt_number": 138
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#A_{i,i+3} for i even\n",
      "integrate(df1*df4,(x,0,h))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{10}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAArBAMAAABlSd54AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAuxCrdpnvzWYy3UQi\nVIln4VgtAAAAhUlEQVQoFWNgAAFGBSABAUzyCnCmsT2czcDgPzDs0PzSBpiDaEn/h4MPFFrDqMDA\noOoSCTIFHJirGbQmAJmgwGRZwMBSAJIABiaXAQP3HyibP4CB8y+Uff4BA+cXJPY/HGyYGpDeH1A1\nQDN5YWYC7eJbABVnWMugBnQDAzgwVZ9YgoQRAABX9jA0A/Z4GAAAAABJRU5ErkJggg==\n",
       "prompt_number": 139,
       "text": [
        "1/10"
       ]
      }
     ],
     "prompt_number": 139
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#For 9.2, we need second spatial derivatives.\n",
      "ddf1 = diff(f1,x,2);\n",
      "ddf2 = diff(f2,x,2);\n",
      "ddf3 = diff(f3,x,2);\n",
      "ddf4 = diff(f4,x,2);\n",
      "hval = 1.6\n",
      "plot(ddf1.subs(h,hval),ddf2.subs(h,hval),ddf3.subs(h,hval),ddf4.subs(h,hval),(x,0,hval))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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NHz9e+fr63ounVKlSKi0tTSml1J07Ss2YoVTp0kp99ZVSt29rE+u+ffvU4cOH\nlZubm7p7V6nFi5WqWVOpV15RatMmpbZutew9/mBMj8LS97gxMSn1+Htcr5jMeY9rjSYnbWOGbH79\n9Vd4e3sDABo3bowbN27gypUrWlw+3zE1bdoUJUqUuBfThQsXzBaPsTEBwOzZs9G9e3c8999K8/h4\n5qmdnbmAdvduGt9rsXHFmJjWrFmDbt263esMKqOBGcnVq0wfdOrEXuWQEHpdGxvT888/j5s3bwIA\nbt68idKlS6Pgf48ZhQtzICc8nKkKV1cOEpmapWvRogUKFiyJa9ceHk7q0gX4v/+z7D1uiKlkDr62\nlr7HjYkJePge1zsmc9zj5kIT0X7UkM3Fixdz/Rxz3kDGxJSdJUuWoIOZN9Aa+3XasmULPvnkE9y8\nCcyf74R+/bS1Rs1rTKdOncL169fRsmVLNGrUCCtXrsz39dLSWFysUwcoUYIFxAc3yBgT04ABA3D8\n+HG88MILcHd3x8yZMx+6VqVKnDhcsYK58v79gfxmBwzWqJ6ewN27HE560BrV0vd4XrHEPW4M2e9x\ngHUvvdHyHjc3mpSsjP2iqweOOub8ZuXltffs2YOlS5ciKCjIbPEAxsXk4+MDb28/vPOOE7ZtU+jc\nWWHePPM57RkTU1paGg4fPoxdu3YhOTkZTZs2RZMmTVCjRo08XeuPP1gkvHsX2LePJ+D8xjR58mTU\nr18fgYGBOH36NDw9PREREYGnn376oc99/XV2dCxZwgLnu+/Sj7pUqdxjjovjerTVq+nP/fPPLJA+\nbhLTkvd4XrDUPW4MPj4+8PPzu9ek8ODXTA+0usctgSaibcyQzYOfc+HCBVSoUEGLy+c7JgA4evQo\nBgwYgICAgFwf6cwZk6H7YMeOMGza1AMlSgCFCl3D3r3bceBAIXTu3NniMRmoVKkSypQpgyJFiqBI\nkSJ47bXXEBERYfQNHRvLboojR7gBp3PnnEe2jYnpwIEDGDt2LADA2dkZVatWRXR0NBo1avTI1yxY\nkJ4e3buzy8bFhQXcDz4Annji4c+PjGRXTmws0Lw5zbSef54ibmzc5r7HjcWS97gxhIWFoUePHgBY\nUN6+fTsKFTLfPW4Mpt7jFkWLxHhOQzZ4RCEyODjY7AURYwZ/zp49q5ydnVVwcLBZY8kppmPHTqj1\n65Vq1EgpFxelfvpJqZQUfn6fPn3Uhg0bLB7Tg1+nqKgo1apVK5Wenq5u376t3Nzc1PHjx3N97Vu3\nlPruO6Vari9nAAAWNklEQVRKlWIB9c4d7WIaOnSomjBhglJKqcuXL6sKFSqof/75x7gLKKUiIpR6\n7TWl6tfnQJKB/fuzhpOmTGHxNzuxsbGPLWZZ+h43JiZL3+PGxJQdS9zjBnKKKb/3uB5octIuWLAg\n5syZg7Zt294bsnF1dcXChQvvfU6HDh3w22+/oXr16ihWrBiWLVumxaXzHdPHH3+MiRMnIjEx8V5u\nrVChQjh48KBFYkpPz0CDBh/i3XddkZy8EO3aAXPnfmxxpz1jvk4uLi5o164d6tWrhwIFCmDAgAGo\nnUNiXSkW6YYPpxHSkSPML2sZ05gxY9C3b1+4u7sjMzMT/v7+KGVMvuM/6tXjnsVffqGPd7VqHMo5\ndw744gvGX7jw/X+nZ8+e2Lt3L65du4ZKlSrh66+/Rlpa2r2YLH2PGxOTpe9xY2LSg9xiyus9ricy\nXGNhDItZZ8wA6tYFxozh47eVpD5N5uhRelzfuMH89Wuv6R3R4zEMJ02ZwkJj+fLA22/zzeZBwRYE\na8GKHJTtm4QEjlsbFrNu3Upz/xYt7EOwr19ncc/TE+jRg4U/axVswy7Krl05nDRtGnD5MrfDh4Zy\nkfHWraa3CAqCORDRNjNxcew2qFWL7W7BwexAsEUf6EeRkcHdha6uPF1HRQEDBz66uKc3iYnApElA\n1arczj5mDFMk7drxjdPZmf3vc+eyL75DB/Z5C4I1YaMuFdaPofvgt984Vm3oPrAn/vyTqZBnnuHi\nWXd3vSN6NPHx7A3fsIFPNnv25Nzr3rYt0zxz5tDDpE8fYORI/jsFQW/kpK0xQUGc8hs82PTFrNbK\n+fMs3vXqxT2GgYHWKdgxMXzDdHMDUlPZUmnscNKTT7IgGRpK0XdxAX76iT4pgqAnItoaYLBGbdGC\nxkFeXjxhm7qY1dq4e5eGTC+/zHRIVBQHVawtJx8ezoJis2ZAhQoU7x9+ACpXzvtrlS9Pc67Nm4F5\n87jy7NAh7WMWBGOR7hETMCxmnTWLJzlfXw5v2Ko16uNQioW5oUPZKjd9Oguq1kR2a9S4OC7J7d+f\n2921wuBGOGYMUyZDhgDlymn3+oJgDCLa+SD7YtbKlYEvv2Tu09pOnFoQHU1xOnsWmDmTiwasiYwM\nnoL9/IBbt4BRo5i6MdaHOz/cvMknjiVLKOCDBum/eEFwHES080BiInOifn502Bs1ShunPWvk5k22\nKK5cyTelgQOtS5hSUoBVq5iGunABGD2a4/GWHE46eZJugufO8Q3N09Ny1xYcF8lpG0F2a9T4eHYf\naGWNam1kZvKNycWFb1JHj7Koai2CnZSUlZ5Zt46nXIM1qqWnSV1c2Gs/dSrf1N5/nyvSBMGciGjn\nQEwMxdrNjfnrI0eA777T1hrVmjh4kAW8BQv4prRkifXkbA3WqFWrshD4f/9Ha9SWLfVNSzk5sVvo\n+HH6gb/yCuO8fVu/mAT7RkT7EYSGZnUflC/PxQMzZgAvvqh3ZObhyhWgXz+eVrt1Aw4coPhYA4bh\npN69KdwhIdY5nFS4MOM8coRtnq6uHJG3k8ygYEWIaP+HUsCuXcxLvvUWBdtgKWqvm7RTU4Hvv2c/\neenSzNG+957l0wyP4tgx2qY2bMjluEuX8gmgenW9I8uZihWBNWvovz1lCpckWGAto+BAOHwh0tB9\nMHUq8PTTFApzdx9YA7t2MR9cuTJ7mF1c9I6IBAWxM2fbNnatDBxou73uGRk0B5s3jz4sEyfa7wFA\nsBwOK9qG7gN/f4qCHt0HenDmDCf9IiMp1l5e+rcqKsUuED8/FnrHjePQTpEi+salFdevc/HCunXA\nhAnARx9ZpzeLYBs4nGgnJVGsv/2W1qi+vlxHpbdwmZvbt/m4Pn8+rUeHDtXffjQ9nX7Wfn58s7TX\n4SQDBtva9HT2eVurC6Jg3dj5uTKLq1fZb1y1KvcTPmoxqz2iFAt3Li7M0UdE8KlCT8E2WKPWqMG9\nkf7+HD3v0cN+BRvgNOmePXzTfP99oGdP+rgIQl6we9GOjWVutFYt4No1dh+sXQu89JLekZmfiAgu\nsl26lP/m1atZKNMLgzVqtWq0Rl2zhrEZrFEdAScndulERfFNq359fk3u3tU7MsFWsFvRPnaMp5lG\njeiwd+KEbXQfaME//wCffsqR83fe4QBI8+b6xWMYTmrUCPj7b3qE2OtwkrEUK8bCZGgocPgw8Oab\nwJYt0iIo5I7dibbBGrVNGw7FnDnDXKk9WaM+jvR0diq4urLQFRXFDeR6Fb2yW6OmpTE1sGyZ/Q4n\n5YeqVenzPWoUfUzateP3TRAeR75Fe926dahTpw6eeOIJHD58WMuY8kx2a9Rp09gRYRDrEiV0Dc1i\n7N0LdOzIDoVdu4DZs4E87LnVFMNwko/P/dao9jqcpAVvvMHBnA4dWKD84gtuAhKEh8jvGveoqCgV\nHR2tPDw8VFhY2GM/z4RL5EpamlKrVilVt65S7u5KrV3L33Mkzp1T6p13lHrxRaU2blQqM1OfODIz\nldq9W6nWrZWqWFGpGTOUSkrSJxZb58oVpYYM4ddx6VKlMjL0jkiwJvJ90nZxcUHNmjU1fPswnuRk\ntq7Vrs3FrI7SfZCdu3fZttigQdZCgrfesnxBLyODj/evvMLpyl69OMbt46Otl7UjUbYsn0w2beJw\nTpMmwF9/6R2VYC3YlMQlJjJnO3s2b+SVK2nS40goxYLVF19QsENDgSpVLB9H9uGkkiWBsWMdYzjJ\nkjRqxBrNqlV8Q+7Rg7sqy5fXOzJBT3IUbU9PT1y+fPmh3588eTI6depk9EUmTJhw7/89PDzg4eFh\n9N8F2H2wYAF7ezt1YveBIxazoqJoSxoczBNY69aWjyEpidcODGTh88cfmYN1lJY9S1OgAM2yunSh\nZ7ebG/vsBw+2f6sF4TGYml8xZ047Olqp/v2VKllSqYkTlTp7Nt8vZdPcuKHU0KFKlSmj1Ny5SqWm\nWj6Gq1eV+vJLxtCjh1KHD1s+BkGpkyeVatdOqVq1lNq+Xe9oBD3Q5GFWadxcGhb28GLWceMcr/sg\nM5PDJy4uPOEeP87+a0suJIiL43BSw4b3DydZmzWqo1CrFn1apk2j4VfnzqwhCI5DvkV706ZNqFSp\nEkJCQuDl5YX27dubFIjBGrV1a+bvXn+d04wTJgBlypj00jZJSAgfixct4sj9okUsUFmKyMgsa9Qi\nRVgImz+f23sEfXFyYnvn8eMcmnrjDdYUbt3SOzLBEuhuGJWenmWNaqnFrNbMpUvMWe7cya+Jpf2t\ng4Jo4BQayhP2J584Tq+7rXLxIn9uAgNZGO7ZU2oM9oxuop29+6BBA1bGHbn7IDWVhaapU4EPP6S5\n1dNPW+ba2a1Rn3uO06Te3vZjjeooBAWxQFmsGDus6tfXOyLBHFi85c/QfTBjBivhCxc6hjVqTuzc\nyfxk9epc9WWp9vf0dK7EmjrVMaxR7Z1mzbg/c+lSoG1brpAbNswx04v2jMVO2gkJfPdfu5Z50lGj\npJj199/0tb54keZBHTta5rrJyXTYmzSJm2t8fflD7shvnPZGYiLrQWvXAuPH04NG3oztA4skIwYP\nZtX7yhU6zlnjYlZLcusWzYG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       "text": [
        "<matplotlib.figure.Figure at 0x111a222d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 143,
       "text": [
        "<sympy.plotting.plot.Plot at 0x111a0fd90>"
       ]
      }
     ],
     "prompt_number": 143
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def draw_from_a(a,hval,ax,color=None,label=None):\n",
      "    #take a hermite solution and draw it\n",
      "    u = dstack((a[:-2:2],a[1:-2:2],a[2::2],a[3::2]))[0]\n",
      "    f = [a2f(M.dot(Matrix(xi))) for xi in u]\n",
      "    seg_pts = linspace(0,hval,seg_res)\n",
      "    for i,seg_f in enumerate(f):\n",
      "        seg_f = lambdify((x),seg_f.subs(h,hval),'numpy')\n",
      "        ax.plot(seg_pts+i*hval, [seg_f(xi) for xi in seg_pts],color=color if color else 'c')\n",
      "        ax.plot(seg_pts[[0,-1]]+i*hval,[seg_f(xi) for xi in seg_pts[[0,-1]]],marker='.',linestyle='',color='b')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 409
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def symmetrize(a):\n",
      "    #make a matrix symmetry, assuming you haven't set complementary elements\n",
      "    #http://stackoverflow.com/questions/2572916/numpy-smart-symmetric-matrix\n",
      "    return a + a.T - diag(a.diagonal())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 301
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from numpy.linalg import solve\n",
      "def hermite_beam(n,E,I,f):\n",
      "    #n: how many nodes, int\n",
      "    #E: Young's Modulus, float\n",
      "    #I: area moment of inertia, float\n",
      "    #f: load on nodes, sympy expression\n",
      "    seg_res = 20 #how many points to plot on each segment\n",
      "    hval = 1./(n-1)\n",
      "\n",
      "    integ = lambda func: integrate(func,(x,0,h)).subs(h,hval)\n",
      "    an = arange(n); anm1 = arange(n-1)\n",
      "    A = zeros((2*n,2*n))\n",
      "    #A_i,i\n",
      "    A[2*an,2*an] = 2*integ(ddf1*ddf1) \n",
      "    A[2*an+1,2*an+1] = 2*integ(ddf2*ddf2)\n",
      "    A[2*n-2,2*n-2] /= 2\n",
      "    A[2*n-1,2*n-1] /= 2\n",
      "    #A_i,i+1\n",
      "    A[2*an,2*an+1] = integ(ddf1*ddf2) + integ(ddf3*ddf4)\n",
      "    A[2*n-2,2*n-1] = integ(ddf3*ddf4)\n",
      "    A[2*anm1+1,2*anm1+2] = integ(ddf2*ddf3)\n",
      "    #A_i,i+2\n",
      "    A[2*anm1,2*anm1+2] = integ(ddf1*ddf3)\n",
      "    A[2*anm1+1,2*anm1+3] = integ(ddf2*ddf4)\n",
      "    #A_i,i+3\n",
      "    A[2*anm1,2*anm1+3] = integ(ddf1*ddf4)\n",
      "\n",
      "    A = symmetrize(A)\n",
      "    A *= E*I\n",
      "\n",
      "    #set up RHS\n",
      "    B = zeros(2*n)\n",
      "    B[2*n-2] = integ(f3*(f.subs(x,x+h*(n-1))))\n",
      "    B[2*n-1] = integ(f4*(f.subs(x,x+h*(n-1))))\n",
      "    for i in range(1,n-1):\n",
      "        B[2*i] =   integ(f3*(f.subs(x,x+h*(i-1)))) + integ(f1*(f.subs(x,x+h*i)))\n",
      "        B[2*i+1] = integ(f4*(f.subs(x,x+h*(i-1)))) + integ(f2*(f.subs(x,x+h*i)))\n",
      "\n",
      "    # before solving we want to enforce boundary conditions a0 = a1 = 0\n",
      "    def e(i): r = zeros(2*n); r[i]=1; return r\n",
      "    A[0] = e(0); A[:,0] = e(0); B[0] = 0\n",
      "    A[1] = e(1); B[1] = 0; A[:,1] = e(1)\n",
      "    return solve(A,B)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 487
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = -.8 + 0*x     #gravity load on nodes, use x to create Sympy expression\n",
      "fig,ax = plt.subplots()\n",
      "n=10\n",
      "draw_from_a(hermite_beam(n,1.,1.,f),1./(n-1),ax)\n",
      "plt.savefig('converge.png',dpi=200)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BG/iw+4iIyEmd6u/HIydPItHbG/8nKQkBDuhOYvcREZGTSvL2xuE5cxAmk2FO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\nRGTx/wFS+MUA1Ra+sQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x14019df50>"
       ]
      }
     ],
     "prompt_number": 498
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#f = x*((x-x**3/6+x**5/120-x**7/5040).subs(x,5*x))    \n",
      "f = (1 - x**2 + x**4/2 - x**6/6 + x**8/24 - x**10/120 + x**12/720 - x**14/5040).subs(x,3*(x-.5))-.5\n",
      "fig,ax = plt.subplots()\n",
      "for n in [7,9,11,13]:\n",
      "    draw_from_a(hermite_beam(n,.5,.5,f),1./(n-1),ax)\n",
      "t = linspace(0,1,4*n)\n",
      "f = lambdify((x),f,'numpy')\n",
      "ax.plot(t,.1*f(t),color='r',label='load')\n",
      "ax.legend(loc='upper left')\n",
      "plt.savefig('converge.png',dpi=200)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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bueeee3j00UdPOGbGjBmsWrUKf39/3nrrLQYPHtzg53aEpKCUotT5Ycx363GT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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x141876c50>"
       ]
      }
     ],
     "prompt_number": 497
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "exp(-x**2).series(x, 0, 20)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 - x^{2} + \\frac{x^{4}}{2} - \\frac{x^{6}}{6} + \\frac{x^{8}}{24} - \\frac{x^{10}}{120} + \\frac{x^{12}}{720} - \\frac{x^{14}}{5040} + \\frac{x^{16}}{40320} - \\frac{x^{18}}{362880} + \\mathcal{O}\\left(x^{20}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAqAAAAAwBAMAAAA869WcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 474,
       "text": [
        "          4    6    8    10    12    14      16      18           \n",
        "     2   x    x    x    x     x     x       x       x        \u239b 20\u239e\n",
        "1 - x  + \u2500\u2500 - \u2500\u2500 + \u2500\u2500 - \u2500\u2500\u2500 + \u2500\u2500\u2500 - \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 + O\u239dx  \u23a0\n",
        "         2    6    24   120   720   5040   40320   362880         "
       ]
      }
     ],
     "prompt_number": 474
    }
   ],
   "metadata": {}
  }
 ]
}