MMQ-cont.ipynb

{
 "metadata": {
  "name": "",
  "signature": "sha256:952084081dee0b0cd795e77642b0eafd6ba1b4521246f9779ca409aad55a1b07"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Aproxima\u00e7\u00e3o de fun\u00e7\u00f5es pelo  m\u00e9todo dos m\u00ednimos quadrados\n",
      "\n",
      "Vamos substituir a tabela $T$ por uma fun\u00e7\u00e3o $f:[a,b]\\to \\mathbb{R}$ e tomar uma fam\u00edlia de fun\u00e7\u00f5es $\\mathbf{F}=\\{ f_1, \\dots f_n\\}$, todas definidas no intervalo $[a,b]$. E considerar o problema de encontrar os coeficientes $a_1,\\dots,a_n$ que minimize o\n",
      "res\u00edduo quadr\u00e1tico $\\text{Res}(f,a_1,\\dots,a_n) = \\int_a^b(\\sum a_if_i(x) -f(x))^2 dx$.\n",
      "De modo parecido ao caso discreto resolvemos este problema ao solucionar  o  sistema normal associado:\n",
      "$$ Ax=b$$  \n",
      "onde as componentes da matriz de coeficientes s\u00e3o:\n",
      "$$ \\begin{gather} a_{ij} = \\langle f_i, f_j \\rangle = \\int_a^b f_i(x)f_j(x)dx\\\\\n",
      "b_i = \\langle f , f_i \\rangle = \\int_a^b f_i(x)f_j(x)dx \\end{gather} $$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Exemplo\n",
      "Achar uma aproxima\u00e7\u00e3o de $f(x) = \\sin(x)$ por um polin\u00f4mio menor ou igual a tr\u00eas no intervalo $[-\\pi,\\pi]$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Uma boa oportunidade para usar o sympy!\n",
      "import sympy as sp\n",
      "from sympy.interactive import printing\n",
      "printing.init_printing()\n",
      "x, y, z = sp.symbols(\"x y z\")\n",
      "f, g, h = map(sp.Function, 'fgh')\n",
      "f1 = map(sp.Function, 'f1')\n",
      "f2 = map(sp.Function, 'f2')\n",
      "f3 = map(sp.Function, 'f3')\n",
      "f4 = map(sp.Function, 'f4')\n",
      "f = sp.sin(x)\n",
      "f1 = 1\n",
      "f2 = x\n",
      "f3= x**2\n",
      "f4= x**3\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* C\u00e1lculo do sistema normal"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def escalar(f,g):\n",
      "    ''' produto de duas fun\u00e7\u00f5es entre -Pi e Pi '''\n",
      "    return sp.integrate(f*g,(x,-sp.pi,sp.pi))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sp.integrate(f1*f3,(x,0,1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZolU\nRDK1XGk9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAe0lEQVQYGWNgYGBUYAACZn8QxeSaD6IYGOqp\nQYX1T1sAMow88B8EPhCl13q3D1AdcwDD+QUMDOx/GPgnMDDw9jLoC4B1gwSBQBRE8O0MAFEMDDse\ngCmOJqA+AwaWPwwM/J8YWH4xMHA4MHB+ZmDgWsBwvgCo5GqoLwMDAHD/Ibjj/t+AAAAAAElFTkSu\nQmCC\n",
       "prompt_number": 3,
       "text": [
        "1/3"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f1,f1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 \\pi$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAOBAMAAAAyKEr9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAgUlEQVQIHWNgVDJ2YICCMAb2HzB2KgPDTBh7\nFgNDvgOUsz8AwQYKrQ/g+Q8ECkAmx3eGct+DbwRBCrkMWC6wCxwBMRmUgZiZAWgeAwNPApBwY1gA\nJBkuMTAKMOgxfAUyWRQYeARYvzKALJcqL8pi4FnAMB3I3v///y8G5gcM6QwMAFToHkgvNkeDAAAA\nAElFTkSuQmCC\n",
       "prompt_number": 4,
       "text": [
        "2\u22c5\u03c0"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f1,f2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 5,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f1,f3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 \\pi^{3}}{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAvBAMAAACfwL4sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABG0lEQVQoFWNgQALsx1OReCCmHoMymog6w3s0\nEQYGbXQR1ikgESFlE5gE6+kDQCZrAkN9A0wIbDJbAAPnBLiI/wUGBs4FDGxfICJAU+QfMDBwf4WL\nMCxkeBkAluT6ynD+////nxjcjydCVPcXsJfckTsoAOGByMUMvgyFPA4IASYDINsAaCccHAOyWBK4\n4HwGXgUGQQamB/wINU4MDAcZ4gP4G2CKGJeXqy1gOMXAWQAT4Qa6ZAGDHQMbXAQmM1hooANRwAea\nO4yxvPgCqiVCDLz/UEXOX2D4hSrS2cD6G1WEAUMXMEUooKlxX4QmwMDADk6CKMIzA1C4ggwM9qhO\n/B/AYP8ARc1kBob1Aigi7xh4/6AIMHAoKTdARQChy0zr4POTVwAAAABJRU5ErkJggg==\n",
       "prompt_number": 6,
       "text": [
        "   3\n",
        "2\u22c5\u03c0 \n",
        "\u2500\u2500\u2500\u2500\n",
        " 3  "
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f1,f4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 7,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f2,f2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 \\pi^{3}}{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAvBAMAAACfwL4sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABG0lEQVQoFWNgQALsx1OReCCmHoMymog6w3s0\nEQYGbXQR1ikgESFlE5gE6+kDQCZrAkN9A0wIbDJbAAPnBLiI/wUGBs4FDGxfICJAU+QfMDBwf4WL\nMCxkeBkAluT6ynD+////nxjcjydCVPcXsJfckTsoAOGByMUMvgyFPA4IASYDINsAaCccHAOyWBK4\n4HwGXgUGQQamB/wINU4MDAcZ4gP4G2CKGJeXqy1gOMXAWQAT4Qa6ZAGDHQMbXAQmM1hooANRwAea\nO4yxvPgCqiVCDLz/UEXOX2D4hSrS2cD6G1WEAUMXMEUooKlxX4QmwMDADk6CKMIzA1C4ggwM9qhO\n/B/AYP8ARc1kBob1Aigi7xh4/6AIMHAoKTdARQChy0zr4POTVwAAAABJRU5ErkJggg==\n",
       "prompt_number": 8,
       "text": [
        "   3\n",
        "2\u22c5\u03c0 \n",
        "\u2500\u2500\u2500\u2500\n",
        " 3  "
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f2,f3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 9,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f2,f4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 \\pi^{5}}{5}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAvBAMAAACfwL4sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABGElEQVQoFWNgQAJuju+QeCBm//8ENJFeVzQB\nhg50AYamM6ogMSFlE5hUIMNlIJM1gaG+ASbEEC/AwMAWwMA5ASpiyBAPlORcwMD2BSqykEE+gIGB\n+ytCpJShCCLH9ZXh/P///z8xsBcVQET6C9hL7sgdBBoKA4sZfBkKeRxgXAYGJgMg2wBoJxwcA7JY\nErjgfAZeBQZBBqYH/Ag1TgwMBxniA/gbYIoYl5erLWA4xcBZABPhBrpkAYMdAxtcBCYzWGigA1HA\nB5o7jPsoiwaqJZz//x9AFeEuO4gqwMCGxgemEEyRd4ccUAWZAxh/oIoAeVYYIvoCKELZDMipESQ1\nm4FBPwBFjQIDw2oUAQYhBr6JqCKM6kkwTQCI8EhyxTcmGAAAAABJRU5ErkJggg==\n",
       "prompt_number": 10,
       "text": [
        "   5\n",
        "2\u22c5\u03c0 \n",
        "\u2500\u2500\u2500\u2500\n",
        " 5  "
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f3,f3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 \\pi^{5}}{5}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAvBAMAAACfwL4sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABGElEQVQoFWNgQAJuju+QeCBm//8ENJFeVzQB\nhg50AYamM6ogMSFlE5hUIMNlIJM1gaG+ASbEEC/AwMAWwMA5ASpiyBAPlORcwMD2BSqykEE+gIGB\n+ytCpJShCCLH9ZXh/P///z8xsBcVQET6C9hL7sgdBBoKA4sZfBkKeRxgXAYGJgMg2wBoJxwcA7JY\nErjgfAZeBQZBBqYH/Ag1TgwMBxniA/gbYIoYl5erLWA4xcBZABPhBrpkAYMdAxtcBCYzWGigA1HA\nB5o7jPsoiwaqJZz//x9AFeEuO4gqwMCGxgemEEyRd4ccUAWZAxh/oIoAeVYYIvoCKELZDMipESQ1\nm4FBPwBFjQIDw2oUAQYhBr6JqCKM6kkwTQCI8EhyxTcmGAAAAABJRU5ErkJggg==\n",
       "prompt_number": 11,
       "text": [
        "   5\n",
        "2\u22c5\u03c0 \n",
        "\u2500\u2500\u2500\u2500\n",
        " 5  "
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f3,f4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 12,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f4,f4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 \\pi^{7}}{7}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAvBAMAAACfwL4sAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABAklEQVQoFWNgQAK1oaFIPBAz//9/A1ShjI7m\nAFSRCww3UAUYGFgLQCJCyiZwCTaQJtYEhvoGmJAziAEU5pwAEykHMTgXMLB9gYnsATG4vyKJzINK\ncX1lOP////9PDAwroCL9Bewld+QOCkC5QGoxgy9DIY8DQoDJAMg2ADsFKngMSLMkcCGU8CowCDIw\nPeAHORcCnBgYDjLEB/A3wAQYl5erLWA4xcBZABPhBrpkAYMdAxtcBCYzWGigA1HAB5o7zCupvBw1\nON4DXZCAYq8iMAGgCDAAdZSgijAwcBxAF2FGF2CoQxdhxfAhzwZ0NVwG6CLxCegi/groIvcxRCQv\nwNQAAGo1R3o3N6NSAAAAAElFTkSuQmCC\n",
       "prompt_number": 13,
       "text": [
        "   7\n",
        "2\u22c5\u03c0 \n",
        "\u2500\u2500\u2500\u2500\n",
        " 7  "
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f,f1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 14,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f,f2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 \\pi$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAOBAMAAAAyKEr9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAgUlEQVQIHWNgVDJ2YICCMAb2HzB2KgPDTBh7\nFgNDvgOUsz8AwQYKrQ/g+Q8ECkAmx3eGct+DbwRBCrkMWC6wCxwBMRmUgZiZAWgeAwNPApBwY1gA\nJBkuMTAKMOgxfAUyWRQYeARYvzKALJcqL8pi4FnAMB3I3v///y8G5gcM6QwMAFToHkgvNkeDAAAA\nAElFTkSuQmCC\n",
       "prompt_number": 15,
       "text": [
        "2\u22c5\u03c0"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f,f3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n",
       "prompt_number": 16,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "escalar(f,f4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- 12 \\pi + 2 \\pi^{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAWBAMAAAAx2K8dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
       "prompt_number": 17,
       "text": [
        "           3\n",
        "-12\u22c5\u03c0 + 2\u22c5\u03c0 "
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "F = [f1,f2,f3,f4]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A1 = [escalar(f1,g) for g in F]\n",
      "A2 = [escalar(f2,g) for g in F]\n",
      "A3 = [escalar(f3,g) for g in F]\n",
      "A4 = [escalar(f4,g) for g in F]\n",
      "A = sp.Matrix([A1,A2,A3,A4])\n",
      "A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}2 \\pi & 0 & \\frac{2 \\pi^{3}}{3} & 0\\\\0 & \\frac{2 \\pi^{3}}{3} & 0 & \\frac{2 \\pi^{5}}{5}\\\\\\frac{2 \\pi^{3}}{3} & 0 & \\frac{2 \\pi^{5}}{5} & 0\\\\0 & \\frac{2 \\pi^{5}}{5} & 0 & \\frac{2 \\pi^{7}}{7}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMwAAABzCAMAAADjYB1gAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n8bvr7PKJ46Yx/wFhCmInpoM04wAAAABJRU5ErkJggg==\n",
       "prompt_number": 19,
       "text": [
        "\u23a1               3      \u23a4\n",
        "\u23a2            2\u22c5\u03c0       \u23a5\n",
        "\u23a22\u22c5\u03c0    0    \u2500\u2500\u2500\u2500   0  \u23a5\n",
        "\u23a2             3        \u23a5\n",
        "\u23a2                      \u23a5\n",
        "\u23a2         3           5\u23a5\n",
        "\u23a2      2\u22c5\u03c0         2\u22c5\u03c0 \u23a5\n",
        "\u23a2 0    \u2500\u2500\u2500\u2500   0    \u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2       3           5  \u23a5\n",
        "\u23a2                      \u23a5\n",
        "\u23a2   3           5      \u23a5\n",
        "\u23a22\u22c5\u03c0         2\u22c5\u03c0       \u23a5\n",
        "\u23a2\u2500\u2500\u2500\u2500   0    \u2500\u2500\u2500\u2500   0  \u23a5\n",
        "\u23a2 3           5        \u23a5\n",
        "\u23a2                      \u23a5\n",
        "\u23a2         5           7\u23a5\n",
        "\u23a2      2\u22c5\u03c0         2\u22c5\u03c0 \u23a5\n",
        "\u23a2 0    \u2500\u2500\u2500\u2500   0    \u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a3       5           7  \u23a6"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b=sp.Matrix([escalar(f,g) for g in F])\n",
      "b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\2 \\pi\\\\0\\\\- 12 \\pi + 2 \\pi^{3}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAABkCAMAAACRmb8DAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 20,
       "text": [
        "\u23a1     0      \u23a4\n",
        "\u23a2            \u23a5\n",
        "\u23a2    2\u22c5\u03c0     \u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a2     0      \u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a2           3\u23a5\n",
        "\u23a3-12\u22c5\u03c0 + 2\u22c5\u03c0 \u23a6"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = A**(-1)*b\n",
      "a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0\\\\- \\frac{1}{8 \\pi^{5}} \\left(- 1260 \\pi + 210 \\pi^{3}\\right) + \\frac{75}{4 \\pi^{2}}\\\\0\\\\- \\frac{105}{4 \\pi^{4}} + \\frac{1}{8 \\pi^{7}} \\left(- 2100 \\pi + 350 \\pi^{3}\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAASQAAABlCAMAAADTVG6jAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nsc9+C+WS/lNgE1pStPX+q6/+B8rhgVVgmFqKAAAAAElFTkSuQmCC\n",
       "prompt_number": 21,
       "text": [
        "\u23a1             0             \u23a4\n",
        "\u23a2                           \u23a5\n",
        "\u23a2      \u239b           3\u239e       \u23a5\n",
        "\u23a2  105\u22c5\u239d-12\u22c5\u03c0 + 2\u22c5\u03c0 \u23a0    75 \u23a5\n",
        "\u23a2- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2            5             2\u23a5\n",
        "\u23a2         8\u22c5\u03c0           4\u22c5\u03c0 \u23a5\n",
        "\u23a2                           \u23a5\n",
        "\u23a2             0             \u23a5\n",
        "\u23a2                           \u23a5\n",
        "\u23a2             \u239b           3\u239e\u23a5\n",
        "\u23a2  105    175\u22c5\u239d-12\u22c5\u03c0 + 2\u22c5\u03c0 \u23a0\u23a5\n",
        "\u23a2- \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2     4             7       \u23a5\n",
        "\u23a3  4\u22c5\u03c0           8\u22c5\u03c0        \u23a6"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sp.N(a[1])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0.85698332779525$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAOBAMAAABa9wqAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1TGX2LkAAAAASUVORK5CYII=\n",
       "prompt_number": 22,
       "text": [
        "0.856983327795250"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sp.N(a[3])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-0.0933876972831854$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAAPBAMAAACre2ZWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBmQ7Ndck328zAAAAAElFTkSuQmCC\n",
       "prompt_number": 23,
       "text": [
        "-0.0933876972831854"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "g = a[1]*x + a[3]*x**3\n",
      "g"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} \\left(- \\frac{105}{4 \\pi^{4}} + \\frac{1}{8 \\pi^{7}} \\left(- 2100 \\pi + 350 \\pi^{3}\\right)\\right) + x \\left(- \\frac{1}{8 \\pi^{5}} \\left(- 1260 \\pi + 210 \\pi^{3}\\right) + \\frac{75}{4 \\pi^{2}}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAArkAAAAyBAMAAACqmN61AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAMLUlEQVR4Ac1cfYhcVxU/b753vvaZlNIiZp9t\nCAomO6Fi/WjME5QoUnf8o0pA3CHRDaVZMlRpKWJm0aKpJe1GEbpFydh/hNJ0pyhaCHEHikEjuENj\nE0Ibs2IrtpG6aTUfxnY89/vjfewkM2PmEuaee8655/zeb967777zZgMQbF8Pqm6oxqmsmX7UIMP6\nKMiZTpTlRul3rpV49CCn5yMw747Q3zh1NgqqgDR6kOEhgc3s838xxocAnFOb2zD2/uJLhkEbjLna\nYBjir+KDWpCp89Ax2ZB+t21mpqZYKtRsBzreUNXVn78AMNZ0HoZyt+vpBk1OLQ2b3Q1NLV1QNCFT\n+/Ax2TAa3W63olhy3rAd6PiYrn3vSWR3P8C9MLZ7o27QZOfY9mGzm1/R8gVFAzI1/x8w2TD+BJDU\nWXrWdiDjPNKptQQOfwGw6CY1pS1ODptd+J6dUh9bkJlp+Jh0CCjXAPYiv7LlUBFouY6hIuy+DTDd\n1OYZDmQw/CNZ9ANJlcKCzAzDx6QAcCnt6ezm/xVwAGj4hhLZdd5Bdr3kg3hri2jDP5JQAgUaCzJT\nDx+TSC/7Aq4NGksHpUEJjyqRSMhu6iLAcq3gpv5tmtRo+EeSNdcrlZpIFmRmHD4mEwSO9gDoLDWC\n62XRolCyi7OfD8TjiuEfifNWVG4AGzLzHD4mG5GzSjWSpQlPeuzazZgurUgVFcjKgOfutIejcz5V\nBT+GdCSZTVtlrgUpBQQbMnMYEibBUwAFXubzVClZGmNjVJbqpRa1lSvmPGQXcN1dbH4MYF/VtMnR\nkI7kNLysUvhStAUbMrMPB5PkyQaB4xwyqLOUkLe1Ui27Sics1815hF1c1xru63juuqZNjqwjKVal\npS/hFDQgwSNMeZGhbMjM0cL028jp12SgPAlM1syJCoDOUkpbzUrz1HnSN+cQdrcAnIEOwCOmSY2s\nI/mDsvQpnYE0Qiat3GJ9yKcNmbmYmIrR00MixqmQJ4HJcltGivCfYum8ctjQpvJvlIZKhN1c0/kH\n3A3ZVy2bHJpHAoekoU8B05IbMWmFFdqFfdiQmY+JacwNm3k9OsITx2RNX0RqDZYUsHU/ZL4a30RR\n+uTlO8HZdXsbUqe2RSE8+c/v1rVMJU8b9CM6H8RIOZY1IxexQEQLMrNbmF4MzLpOBeWJY7JCfKEJ\nJkv7fOnB7mrkyaHfNqW+hdTZo20MhyczrbPB+3a9Eho9NXNfk1XhmJ9wwrtaokMHWWunKDwwcg+Q\nU3gFqoZwGLDMrr0cmDJKycbODYQnjkm6RgiNqjQU/0PEtLYSS9M1CrPKfz1kkBVZZ4Mn4RZfWZV0\nN+QvsSocrcdJw3ITUqt0VIwE1gtko7pO4FBgzo/xfmImlJm5i4admwhPHJPmHCoue1y9oZ26RMQ8\n5ZgrIzvHjzShQSu9bQV4E1SdLbsCmXl9qgg02YTLrL60n9TjWMNVd6oO8H06UmfoTXoAlHuBXG6q\nSRQOBTZWg6OsAKiseC34bGRhZ0rOE8OkfPX5Sp7ocDnnZ2nRPPFfZYyW8l60DYp4aoj2d4DtbfI0\nzepshYpVghOBvlp1rrAKCK3H8ekPw34XxJ4XS0msPSUE3vcCeRnjyEbgUGANH3V6QuoiMFnY2XTO\nk9iHC19mtD/HK1xT3LXbJ2Ip+ubBPUkXGzS7QlxYO+wKdmmdDdOljYtDBcKVgVbhqB+ffXzTh1HC\n/SBpV1kHYLPbC2S8IlQj7FJg9J6uJ6Q+ApOFnc3nPHFM8UTAeAtg3e2f2eSJ3L1AXYNd3BDq7XGX\nnLvkwp728FLBVXISi/j8TiSOBN9/dGh9ifkl0KPb4VGmXSrgPp01wa7A3QvkM2Iy6Qm72B53rxzf\nXGUJwzGhi8IehmkNdsvz4NSy38lJQpIsM00f/aFICfEptHRlGosUeDiszjZVg+LFzN6ndmz0mY8M\ndPwgqy8xv5kTG+//tIiy3KYSXqiscXYl7l4gHxWTSc/YTV90rtThDZYwFBMY2MMwrcXuHGTczIW8\nL7InV4UU10tSwpzGKrqWPHyb7J6AryXa3EUFyuCDA1bh6MHubWb8D3EPgKkqFR8TCs6uxN0L5B+J\nyaRn7OZaTteFP99Ey6vhmHTsoZjWYDe3Ag6MIQGiMajOEwewPdKGcXKJ6o35SVJ0E5FX0Z7rANxK\n5h/4CY7I+o+Hw+psEzW65WslXRZHR/cm0Z37FKvHFWAr90B2m1S02ZW4ObsiZShkvDIUJsYuAruM\n9ZMv8gJgGCYDO4RgkvhDkwKyCzDu4QdvpbVXhszCwhOvLywgh+HNOHcTFXQih4ML7WJzvEP2T8VK\njs7UAuEysMSqcNQPXoA5GZyfuweo4t6FhccWFn5KZY67B8gQPHcJMFzKG3WWMAQTmNiDmDT8Eqwh\n5OZwiAsbOW9oi3ngFC7Yy3NX00nRYPcIpHzK7qOkzoaXGr5kKNXHRT4RCK/RpToe7DmX+sFpUN/y\nsk9D2+uuxN0L5CC7BBjuGRpVljAEE5jYwzDFEwHlOci7k5AQRwuxm8eCYFCQIsakz/zsPSxKsqXU\nxQ4kGLtbSJ0tsQK4o5hwx6vcRQQ6j7dwv0PqS9TPuQDqwTdizyBxx0Au3rWjTjN9guejHbmUKLBF\nXHdZwjBMFvYwTOHszh5zWTrcM0w3l2CHTB79OI8uiIY1QYqchkKy271Kx0ihbJ+dmf0oWxlonQ1+\nAFhi+gCUa9xDBHoQ8m+x+hL1S8zB32QMvldlwVHL72oSdwzkdBd3s6Thd6YaYZcCK9TwJx0MWAgm\nC3sYplB2M35yniUbb8H629btYV8wUeXVOaPgcMnZF8vuPfc8Rx3T6qKGw93uZa3OBjffdhfAryFZ\no55qiUmffbnK6ku0Hleow0e4h3hWU0/CnF2JOwZy+hmOmJ//NCYt+1FgMHuyzQqAYZgs7CGYFH4J\nFoWcX+SPZPJJWJiDJRG8QHhLTATZXbfpJW7NQ77KxNfEhLV7ce7GebLSqKricHblFBuyc9ssBwJp\nIZSFIGdFC71jMtiVPJWr4uX5smdlUaeIMJBNH2snJLvputDhE1ZTyDdz4ZRQrN2rQJG+uN6Rpmo1\nX7ZcbcglgI9zl/SLm10qlmrWpJhh75iwpCiJoG/VeFTx26DFpp0Gb9xm2yLZrUl2pUe+Dql5MdrK\nhT8KxUD6bIWGidkZWJB/DvA5ntnx8wx9cWUgWEQQjkkMaa94gpzHLPt81qvPJSVSKfWQYDfrB9nN\ntMCZ4zOKwlGdzVas6xrm2nQaf2YIC2FBvg/gFunm8CVQnM3S0JfAMekxFE8As9ywXXeg8qRrqjLy\nSW4dbltMG14ZV91MnSszNS4487ZbP+MH2OS4t5YmrOnn5Huvgu/wu/St/UAIzOWYdL3iCbIdbjiv\nO1B5uWmqvonssvLQkbPnTps2HG2/tAd/pIfPwO/AmCesR4QwgD7VYkGmvMhgFuRk94E6h5zzxSuf\nvDjiyCjXYBCY9CmKJ/gSfIVa9Dfu3LVc0eeA4yG7ojw05Ro2Mih1DwEvMJXrwppvCqn/vsRzTvqR\nsSzIcO6KyyFjpafDpz0dOf3aDQKTNlPjqfjk02xrqn4tIh1Lq1IkQgZrPEVessouBVevb//+3Sov\nMOWqxswBD56PjmdBzr5y+DUBeeYa9i/RCXqwaDzhz88Zh4W5wMS0+TjxDK2g6eUhYwbePBv4xkgV\nmAzrIAeRLy1x8TchfwOy77oQCXmQoFSsMJ7GPWUX0kEhkN6pUXb1kpVuhrIL+I5UFJgM02AHsb8w\nNSADftu4EkdCHiwuHi2Up4YbzGXo8nfcsf2vnlEeMmZM4OhOvehlWAc4MApvdlwDsoOXZbIeDdme\nPJBxKE9GYY6nKXfMfLgVMspDhrXsA/xSLzAZ1gEOpjFRZDMhv4rvkPxoyJFR+jQEeEqHvQAWj3Ei\n2TjuyPSSldDTPn0Isp5e9DKsAxwci4tlQt7pwv0xkOMC9WML8FSwTlMWnBX+RaLM0tv1gl6yEgbW\nv3AKqzha0cu0DmwU9ud+WnADcnHbbBViIGvzBigGeXo2ZNnF91jtASYdUKiQP/fTI48iZCe8VJif\n04GPhhy7MGABbQQhhy8MACdHg1ENRbalDcLE0YMM3wrDibqkF2G4Yeqda2UePciR/4MA4I1qpJoj\nKsfRqEYNMqyn97T/AWiPw2ZrKc23AAAAAElFTkSuQmCC\n",
       "prompt_number": 24,
       "text": [
        "   \u239b             \u239b           3\u239e\u239e     \u239b      \u239b           3\u239e       \u239e\n",
        " 3 \u239c  105    175\u22c5\u239d-12\u22c5\u03c0 + 2\u22c5\u03c0 \u23a0\u239f     \u239c  105\u22c5\u239d-12\u22c5\u03c0 + 2\u22c5\u03c0 \u23a0    75 \u239f\n",
        "x \u22c5\u239c- \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + x\u22c5\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u239f\n",
        "   \u239c     4             7       \u239f     \u239c            5             2\u239f\n",
        "   \u239d  4\u22c5\u03c0           8\u22c5\u03c0        \u23a0     \u239d         8\u22c5\u03c0           4\u22c5\u03c0 \u23a0"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "h=f.series(x, 0,4).removeO()\n",
      "h"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{x^{3}}{6} + x$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAvBAMAAAC/LOHRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnlzaPZFvOqhOPIbRg8YAAAAASUVORK5CYII=\n",
       "prompt_number": 31,
       "text": [
        "   3    \n",
        "  x     \n",
        "- \u2500\u2500 + x\n",
        "  6     "
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy\n",
      "t=numpy.linspace(-3,3,100)\n",
      "fl=sp.lambdify(x,f,\"numpy\")\n",
      "gl = sp.lambdify(x,g,\"numpy\")\n",
      "hl = sp.lambdify(x,h, \"numpy\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from matplotlib.pyplot import plot,grid"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "plot(t,fl(t),t,gl(t))\n",
      "grid()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAEACAYAAAC9Gb03AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xaff0614c>"
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(t,fl(t),t,hl(t))\n",
      "grid()\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXoAAAEACAYAAAC9Gb03AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIYQQQgghhLCR/wfyEUWInX+MTQAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0xaff0006c>"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}