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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Plotting and coding the algorithms we learnt in the last class."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Plotting"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\newcommand{\\R}{\\mathbb{R}}$\n",
      "### Numpy and Scipy\n",
      "*Look at [my webpage](http://www.iiserpune.ac.in/~vmallick/2014s1/mth103/ \"installing scipy\") for instructions to install these python packages on your laptop.*\n",
      "\n",
      "\n",
      "### Array/Vectors\n",
      "Suppose you want to plot $y = x^2$, $-1 \\leq x \\leq 1$. From our math knowledge, we know this is a curve in $\\R^2$. The way one plots such functions in matlab/mathematica, is to find a lot of points lying on the curve and then joining them. We can use a list to store those points. So a basic python code to generate such lists will be as follows"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def genxy(f, xmin, xmax, N) :\n",
      "    \"\"\"Given a function f, an interval [xmin, xmax] in R and number of points N, This function returns a 2-tuple\n",
      "       of two lists: First an equispaced points on x-axis, and the second one is the value of f on those points\"\"\"\n",
      "    step = float(xmax - xmin)/N\n",
      "    listx = [xmin + i * step for i in range(N+1)]\n",
      "    listy = [f(x) for x in listx]\n",
      "    return (listx, listy)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = genxy(lambda x : x*x, -1, 1, 10)\n",
      "azipped = zip(a[0], a[1])\n",
      "for x, y in azipped :\n",
      "    print \"%6.3f\\t%6.3f\" % (x, y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "-1.000\t 1.000\n",
        "-0.800\t 0.640\n",
        "-0.600\t 0.360\n",
        "-0.400\t 0.160\n",
        "-0.200\t 0.040\n",
        " 0.000\t 0.000\n",
        " 0.200\t 0.040\n",
        " 0.400\t 0.160\n",
        " 0.600\t 0.360\n",
        " 0.800\t 0.640\n",
        " 1.000\t 1.000\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is not too difficult, but is somewhat tedious. To make tasks simpler, one can use arrays. Arrays are **not** in-built in python. To use them, *you have to install numpy*."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# The above code becomes :\n",
      "from numpy import *\n",
      "xs = array(a[0])\n",
      "ys = array(a[1])\n",
      "type(xs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "numpy.ndarray"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "1. In an array all elements must be of the same type. If they are not, they are converted to the same type as is shown below (in the examples).\n",
      "1. The computations are faster if the size of the array doesn't change during computation.\n",
      "1. You can mathematical operation on a whole array and that is faster than doing it by traversing a list."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Array elements are all of the same type\n",
      "v1 = [4, 5, 100, 4, 23.4]\n",
      "w1 = array(v1)\n",
      "print w1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[   4.     5.   100.     4.    23.4]\n"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v2 = ['eggs', 4.3, 3]\n",
      "w2 = array(v2)\n",
      "print w2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "['eggs' '4.3' '3']\n"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "v3 = [3, 2, 5]\n",
      "w3 = array(v3)\n",
      "print w3"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[3 2 5]\n"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# doing operations on array\n",
      "def f(x) :\n",
      "    return x + x*x\n",
      "\n",
      "v = array(range(5))\n",
      "print f(v)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0  2  6 12 20]\n"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# One dimensional arrays are like vectors\n",
      "v = array([1, 4, 5])\n",
      "w = array([5, 2, 1])\n",
      "print v + w\n",
      "print 2 * v\n",
      "print v * w"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[6 6 6]\n",
        "[ 2  8 10]\n",
        "[5 8 5]\n"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Back to plotting.\n",
      "We need two things :\n",
      "1. We need to generate an array of equidistant points on $x$-axis. For this we have a numpy function called **linspace**\n",
      "1. We need to generate another array containing their values."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "xlist = linspace(-1, 1, 11)\n",
      "print xlist"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[-1.  -0.8 -0.6 -0.4 -0.2  0.   0.2  0.4  0.6  0.8  1. ]\n"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def square(x) :\n",
      "    return x*x\n",
      "ylist = square(xlist)\n",
      "print ylist"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 1.    0.64  0.36  0.16  0.04  0.    0.04  0.16  0.36  0.64  1.  ]\n"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Good, it works. Let us now write it as a function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def array_pts(f, xmin, xmax, N) :\n",
      "    xlist = linspace(xmin, xmax, N+1)\n",
      "    ylist = f(xlist)\n",
      "    return (xlist, ylist) # pretty simple, isn't it"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Plotting\n",
      "To plot we have a bunch of functions, I like matplotlib. It has a MatLab like syntax."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from matplotlib.pylab import *\n",
      "\n",
      "a = array_pts(lambda x : x*x, -1, 1, 10)\n",
      "plot(a[0], a[1])\n",
      "show()\n",
      "b = array_pts(sin, -4*pi, 4*pi, 50)\n",
      "plot(b[0], b[1])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
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       "text": [
        "<matplotlib.figure.Figure at 0x1d32dd0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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t7XXmfsGgHh0YcFb0u7vVX6Vs4qToG4Zeou+k09fFJJiQ6CuC06tyT57UY6gO\nONuJdXJtTsbL2bPsv5MnO3M/K1B6Z2RI9BXCyU6sk9N3cssBnTqwjHjx+Zy5nxWcnMg9fx7o6wNS\nU525nx2Q6CsEOX0+Tjo3Hco1TShe+DgdL5mZevwxNCHRV4jMTOccrU5O3+n0ji75WYoXPma8OFEU\nodPI0IREXyGyssi58aCcPh+KFz6TJrH9gZwo89UpXkxI9BXCyVykjs7NCXTqxBQvkXEqZnSKFxMS\nfYXIyQE++0z8fUIhti+6LsE6bZpzh8zoVH7n97MSXyeei05OH3BO9HWKFxMSfYXIyQGamsTfp60N\nmD6drerUAbOc1Yn8tU7OzecDbrjBGaOgm9N3ahSkU7yYkOgrRFoac22iF2jp5toAZ5xbfz87bNzv\nF3sfO3HCKAwOMnELBMTex04ovRMZEn2F8PmcSfHo5toAZzpxezuQng4kJoq9j504IfodHWyfn3Hj\nxN7HTii9ExkSfcW44QbxnZicPh+dyjVNKF74OLH/zuAgS5XqFjMk+orhhHMjp89Hx6E6xQsfJ+Kl\nq4vtmaXTCAgg0VcOJzqxjs7Nia0YSPT56BovZBL4kOgrBjk3Pk44Nx3zsxQvfKZNY0UR58+Lu4eO\n8QKQ6CsHOTc+lN7h4/ezlacia/V1jBfzHAaRo0Md4wUg0VcO0aJ/8SIrCU1PF3cPEfj9bK/7/n5x\n99CxEyckMBcusuJLR6cPiDcKOsYLYEH0u7u7UVFRgYKCAixatAg9PT3c1+Xk5OCWW25BaWkp5s+f\nH3dDvUJ6utha/WCQBWqCZn/uExPZubAi95rRdbgu2ijo6PQB8U5f13iJu+tv3rwZFRUVOHbsGO68\n805s3ryZ+zqfz4eamhrU1dWhtrY27oZ6BdGrLHV1bYDYTmwY+jo3kaLf18cWrM2YIeb6IiGnzydu\n0a+qqsLq1asBAKtXr8Yrr7wS8bWGUwe/ugSRnViXI+94iKzIOH0aSEpS/8xgHiLjpbmZuVmdFqyZ\niK7gaWpiBk034hb9jo4O+L9Yr+73+9HR0cF9nc/nw8KFCzFv3jw8++yz8d7OU4jsxMGgnkN1QKxz\nO34cyMvT6zAME4oXPiLjZXAQaGwEZs0Sc32RJI30w4qKCrS3tw/79yeeeOKa730+H3wResuBAweQ\nkZGBU6dOoaKiAoWFhSgrK+O+dsOGDVf+v7y8HOXl5aM0352Idvq6Tq2I7MQNDUBurphri4ZGhnxE\nxktLCzsYmVAjAAAQN0lEQVQiceJEMdcfiZqaGtTU1MT9/hFFf9++fRF/5vf70d7ejhkzZqCtrQ3p\nEcpBMjIyAABpaWlYuXIlamtroxJ9L5OTA7z3nphrB4PAPfeIubZoAgHg4EEx1zadvo6Q0+fjxMhQ\nBkMN8caNG2N6f9zpncrKSmzbtg0AsG3bNqxYsWLYay5cuIDeL8pQzp8/j71796K4uDjeW3oGcm58\nRE7k6uz0Rdbq6xwv6elsElpEma/O8RK36K9fvx779u1DQUEBXn/9daxfvx4A0NraimXLlgEA2tvb\nUVZWhjlz5mDBggVYvnw5Fi1aZE/LXYwo0TcMvZ2byIk5nZ2+yFp9neNFZJmvzvEyYnpnJKZOnYr9\n+/cP+/fMzEz88Y9/BADMmjULhw4dir91HiW8Vj8lxb7rnjmjb4UKwKpI2tvZyV92V5Po7NyAq0ah\nqMje6+rs9IGrKZ6cHHuv29Cgb5pUsyU63kBUrb7Org1gJ31Nncr2d7eT3l729cX0k5aIGh3qHjOi\n8vo6O30SfUUR0Yl1d22AmE58/DgrvdNtlXI4IuLl7FlWmjhlir3XdRIR8WIYeo8MNQ5zdyOiE+vu\n2gBxoq+razMRGS86rl0wEREvXV0sTZqaau91nYJEX1HI6fMRsa++zq7NhOKFj6h40dkkkOgriqhO\n7AanHwzae01y+nwoXvgcP663SSDRVxRRw3XdnVteHlBfb+813eD0RdTquyle7Nz+i5w+IQRybnyK\nioCjR+29phucvohafTfES3o6E/xTp+y7Jjl9Qgh276sfCgFtbXpuBRtOfj77Y3j5sj3X6+tjtf+6\nO1rAfqPgBqfv8wGFhcAnn9h3TXL6hBDsrtVva2Pnho4da8/1ZDF2LHOfDQ32XK+xkQlbUtzLFNXB\nbtF3g9MH7B8dktMnhGFnJ3aDazOx07m5IZ9vYme8DA6yqpdAwJ7rycTOeDl3jh22ruOhMiYk+gpj\nZyd2Q/mdiZ3OzQ35fBM746Wjgy3KGj/enuvJxO54yc3Ve+0Cib7C2O303TBUB+ztxOT0+VC88NE9\nnw+Q6CsNOX0+dg7XyenzcVO85OQAnZ0sLWMV3fP5AIm+0pBz42OKvh21125y+nbW6rspXhITWdXX\np59avxY5fUIoOTn2Ve+4ybmZx9RZXV4/MMCei47nnPIwa/VPnrR+LTfFC2Df6JCcPiGU9HQ2JP38\nc2vXMQz2x8NNndiOPG0wCKSlAePG2dMmFbBrdEjxwoecPiEUu2r129rYtSIcY6wlRUXWnZub8vkm\ndon+hx8CN99s/TqqYEe8XLzIVvbqnvYi0VccOzrxoUNAaaneZWZDKSy07tzclM83sSNezp1jq5QL\nCuxokRrYES+NjcyE2X1qm9OQ6CuOHZ24rg6YM8eO1qiDHcN1cvp8Dh9mLl93cQunoID9vgcG4r+G\nG/L5AIm+8tjl9N0m+nZMzJHT5+PGeJkwga2ibWyM/xpuyOcDJPrKY2d6x00EAmyCu6cn/muQ0+fj\nxngBrI8OyekTjmC1E/f2Aq2t7srPAtZ3TzQM93TicPx+drbtxYvxX8ONTh+wPplLTp9wBKui78b8\nrImVybn2dlbrP2mSvW2SjdV99S9fZs+0uNjedqmA1clct5gEEn3FsVqr71bXBlgbrrsxn29ixSgc\nPcoqVCZMsLNFamAlXsyFfDNn2tsmGZDoK47VWn235mcBa+kdN+bzTayIvhfiJZ7tO06eZBPBup9H\nAZDoa8GNNwJ//3t873VjuaYJOX0+FC98pk8HkpNZai9W3JLPB0j0teD224F33on9fW7OzwKsEwaD\n7MjDWHGz0483XgB3pwOB+Cdz3ZLPB0j0taCsDHj77djf98knLDU0caL9bVKB5GSWyqivj/29burE\nQyktBU6ciL2c1TDcL/rxTua6KV5I9DUg3k7s5qG6SbzOzU3D9aEkJwPz5wMHDsT2vs8+YxO4btqj\naSgULyT6WjBmDHDrrcC778b2Pre7NiA+59bdzaoxpk8X0yYVKCuLPcVD8RIZcvqE48ST4vFCJ45n\nMtfM57tpA7qhULzwiSde3LaQj0RfE2LtxF7IzwLxDdfd1IEj8aUvsd//pUvRv8fN5Zom118PnDnD\nVqpHS1sbkJLCvtwAib4mfOlLLEcfbSc+eZIdDuL3i22XbG68kR2DNzgY/XvclJ+NxMSJwE03AbW1\n0b/HC3NACQlsS5JYjILb4oVEXxOuu4514vfei+71XnD5ANtGITWVlW5Gy+HDLLfrdmLJ63d3Mwfs\nlqMjRyLW0aHb4oVEXyNiSfF4YahuEsvk3MWLwL59wJIlYtukArHEy+HDQEkJc8JuJ9bJ3F27gLvu\nEtcep/HAr9g9xNKJvTBUN4llcu5PfwLmznV3WaLJV74C/OUvQCg0+mspXvh0dgIffAAsXiy2TU5C\noq8RsXRir6R3gNiG6zt3At/8ptj2qEJaGpCRwc67HQ2KFz6vvAJ8/evA+PFi2+QkJPoakZ7ONn0a\nbV+VM2eA06fdX6FiEu1wva8P+OMfgZUrxbdJFaIdHXpJ9PPz2Qlaly+P/tqXXgLuuUd8m5yERF8z\nounEXsrPAtE7t/372dkCGRni26QK0UzmXrrEKlRmz3amTbIZOxbIzmaluyNx+jQbWbtt/scjsuAe\nohF9L+VnATb66etjnXQkvJTaMTHjZaTthD/+mJUkjhvnXLtkE83osKoKqKhglXNugkRfM8wdFEfq\nxF4aqgNsZe0ddwAvvBD5NZcvs058993OtUsFbriBnZo2kqv1WrwAwFe/OnK8ACy140aTQKKvGTNn\nMpE7cSLya7xUrmny+OPAz34W+YSxN95gudzsbGfbJRufb/TRoRfj5cEH2TN5/33+z8+eBd56C1i2\nzNl2OQGJvmaM1on7+oBjx7yTnzWZMwe4807gySf5P/diasdkNNH3WjoQYCuWH3sMePRR/qj51VeB\n8nL3naEMkOhryUid2Iv5WZP//E/gv/8b6Oi49t8HBljpnduqMKJlpMncwUFW0llS4mybVGDNGqC5\nGdi7d/jP3GwSSPQ1ZKSTkbyYnzWZORO47z6W5gnn7bdZWscNh1rHw+zZwKlT/GMCT5xg21hMnep8\nu2STnAxs2sTcfvjeTb29wOuvu2sVbjgk+hpy881spWC4ozUMNjH1058CS5fKa5tsfvITYPv2aycu\n3ezaoiEhgS3sG2oU3n8f+Na33Jm3jpa772YlnNu3X/23PXvY80pNldcukcQt+i+++CJmz56NxMRE\nfPDBBxFfV11djcLCQuTn52PLli3x3o4IIzERuO22q534+HG2anDTJlZxcO+9ctsnk7Q04OGH2R8/\ngK1efvll76Z2TMJTgufOAQ89BCxfDvzgB8Cvfy23bTLx+YBf/ILFi3nWsttNQtyiX1xcjF27duGr\nX/1qxNeEQiGsW7cO1dXVOHLkCLZv346j8Rxb4wJqampsvV5ZGfDnPwNPPAEsWAAsXAj87W/Al79s\n622iwu7PZpVHHgHefJM9j3ffZSuZCwriv55qny8eTNHfuZPt1nrhApv/+c53gDffrJHdPKGM9vu7\n4w72TP7nf9hz2bsX+MY3nGmbDOIW/cLCQhSM0pNqa2uRl5eHnJwcJCcnY9WqVdi9e3e8t9QaEaL/\nm9+wFYPvvw/86EcsRykD1UTRrMxYv96eZfSqfb54mDePLUZ67DGWyvjtb4Fp09jP3PD5RiKaz7d5\nMxsp79jBjiZ181GaSSIv3tLSguywwuhAIIC//vWvIm/pGW67je2t/w//4O5j/+JlzRrgqaeY04/2\nDAI3M2YMcPAg27JizBjZrVGP4mK23cK6dZHLft3CiE6/oqICxcXFw75effXVqC7uIzUShs/H3Bs9\nYj7JycCWLUzkbrpJdmvUoKSEBH8kHn+c1eWvWCG7JYIxLFJeXm787W9/4/7sL3/5i7F48eIr32/a\ntMnYvHkz97W5ubkGAPqiL/qiL/qK4Ss3NzcmzbYlvWNE2Ahm3rx5qK+vR1NTEzIzM7Fjxw5sD6+N\nCqOhocGOphAEQRAjEPdE7q5du5CdnY2DBw9i2bJlWPLF/qOtra1Y9kXhb1JSErZu3YrFixfjpptu\nwre+9S0UFRXZ03KCIAgiZnxGJJtOEARBuA6pK3IjLfBqamrC+PHjUVpaitLSUjz44IMSWxk/Iy1g\n+/nPf478/HwUFhZiL2/zD83YsGEDAoHAld9ZdXW17CbZgtsXF+bk5OCWW25BaWkp5s+fL7s5lliz\nZg38fj+Ki4uv/Ft3dzcqKipQUFCARYsWoaenR2ILrcH7fHH1u5hmAGzm6NGjxqeffjpsMrixsdG4\n+eabJbbMHiJ9vo8//tgoKSkx+vv7jcbGRiM3N9cIhUISW2qdDRs2GE8++aTsZtjKwMCAkZubazQ2\nNhr9/f1GSUmJceTIEdnNspWcnBzj9OnTspthC2+99ZbxwQcfXKMdP/rRj4wtW7YYhmEYmzdvNh59\n9FFZzbMM7/PF0++kOv1oFnjpTKTPt3v3btx7771ITk5GTk4O8vLyUFtbK6GF9mK4LFPolcWFbvm9\nlZWVIXXIhjlVVVVYvXo1AGD16tV45ZVXZDTNFnifD4j996fshmuNjY0oLS1FeXk53hntkE/NaG1t\nRSAQuPJ9IBBAS0uLxBbZw69+9SuUlJTg/vvv13oYbcJbXOiG31M4Pp8PCxcuxLx58/Dss8/Kbo7t\ndHR0wO/3AwD8fj86hu677QJi7XfCRT+eBV6ZmZkIBoOoq6vDU089hW9/+9vo7e0V3dS4sLqAzUSH\nhWyRPmtVVRUeeOABNDY24tChQ8jIyMAPf/hD2c21jA6/E6scOHAAdXV1eO211/DrX/8ab492ALPG\n+Hw+1/1O4+l3QrdhAIB9+/bF/J4xY8ZgzBdLB+fOnYvc3FzU19dj7ty5djfPMvF8vqysLASDwSvf\nNzc3Iysry85mCSHaz/q9730Pd7lgM/Khv6dgMHjNCM0NZGRkAADS0tKwcuVK1NbWoqysTHKr7MPv\n96O9vR0zZsxAW1sb0tPTZTfJVsI/T7T9Tpn0TnheqqurC6FQCABw4sQJ1NfXY9asWbKaZgvhn6+y\nshIvvPAC+vv70djYiPr6eu0rJ9ra2q78/65du66pMNCV8MWF/f392LFjByorK2U3yzYuXLhwZQR9\n/vx57N271xW/t3AqKyuxbds2AMC2bduwwmV7LMTV72ycXI6Zl19+2QgEAsa4ceMMv99vfP3rXzcM\nwzB27txpzJ4925gzZ44xd+5c4w9/+IPMZsZNpM9nGIbxxBNPGLm5ucaNN95oVFdXS2ylPdx3331G\ncXGxccsttxjf+MY3jPb2dtlNsoU9e/YYBQUFRm5urrFp0ybZzbGVEydOGCUlJUZJSYkxe/Zs7T/f\nqlWrjIyMDCM5OdkIBALG73//e+P06dPGnXfeaeTn5xsVFRXGmTNnZDczboZ+vt/97ndx9TtanEUQ\nBOEhlEnvEARBEOIh0ScIgvAQJPoEQRAegkSfIAjCQ5DoEwRBeAgSfYIgCA9Bok8QBOEhSPQJgiA8\nxP8DFvIZAk/GteYAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x2df8d50>"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Some comments on trigonometric functions\n",
      "Note that we have not imported math anywhere. The sin we used here is imported from numpy. Actually if we override with the sin from math, we'll get errors :"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from matplotlib.pylab import *\n",
      "import math as m\n",
      "\n",
      "b = array_pts(m.sin, -4*pi, 4*pi, 50)\n",
      "plot(b[0], b[1])\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "TypeError",
       "evalue": "only length-1 arrays can be converted to Python scalars",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
        "\u001b[0;32m<ipython-input-48-66a6bff9e21c>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mmath\u001b[0m \u001b[0;32mas\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0marray_pts\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msin\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m4\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mpi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m4\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mpi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m50\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      5\u001b[0m \u001b[0mplot\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      6\u001b[0m \u001b[0mshow\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
        "\u001b[0;32m<ipython-input-37-925050db2e56>\u001b[0m in \u001b[0;36marray_pts\u001b[0;34m(f, xmin, xmax, N)\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0marray_pts\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxmin\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxmax\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      2\u001b[0m     \u001b[0mxlist\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlinspace\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxmin\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxmax\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m     \u001b[0mylist\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxlist\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      4\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mxlist\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mylist\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;31m# pretty simple, isn't it\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
        "\u001b[0;31mTypeError\u001b[0m: only length-1 arrays can be converted to Python scalars"
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Accessing array elements"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Array elements can be accessed in the same way as list elements\n",
      "x = array(range(1, 10))\n",
      "print x[5]"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = array(range(1, 10))\n",
      "print x[5]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "6\n"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "However, you must be careful about assignment. Assignment just reflects the same array, and does not create a new array, just like lists:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = array(range(1, 10))\n",
      "l = range(1, 10)\n",
      "x = a\n",
      "m = l\n",
      "print a[5], l[5]\n",
      "x[5] = -1\n",
      "m[5] = -1\n",
      "print a[5], l[5]\n",
      "print type(a), type(l)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "6 6\n",
        "-1 -1\n",
        "<type 'numpy.ndarray'> <type 'list'>\n"
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "a before : [ 1  2  3  4  5 -1  7  8  9]\n",
        "y before : [ 1  2  3  4  5 -1  7  8  9]\n",
        " a after : [ 1  2  3  4  5 -1  7  8  9]\n",
        " y after : [ 1  2 -2  4  5 -1  7  8  9]\n"
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "####Note :\n",
      "For vectors\n",
      "\n",
      "            a += b\n",
      "            \n",
      "is faster than\n",
      "\n",
      "            a = a + b\n",
      "\n",
      "since the later creates a new array to store `a + b` before copying it back to `a`. `a +=  b` just modifies a directly. Same holds for -=, \\*= etc."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Putting two graphs in he same picture\n",
      "One uses the hold funciton. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(x, s):\n",
      "    return 1 / (sqrt(2 * pi) * s) * exp(- x * x / (2 * s * s))\n",
      "\n",
      "x = linspace(-5, 5, 1000)\n",
      "y = f(x, 1)\n",
      "plot(x, y)\n",
      "hold('on')\n",
      "z = f(x, 2)\n",
      "plot(x, z)\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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bpKRwFwhbPZ39REvP6oevZtHEAtHjoxF/OR4bT9Gd5OaIAp7wpqCkAGN3jcX6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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x2df2110>"
       ]
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Some decorations"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = linspace(-5, 5, 1000)\n",
      "y = f(x, 1)\n",
      "plot(x, y)\n",
      "hold('on')\n",
      "z = f(x, 2)\n",
      "plot(x, z)\n",
      "xlabel('x')\n",
      "ylabel('y')\n",
      "legend(['s=1', 's=2'])\n",
      "title('normal or Gaussian distribution')\n",
      "show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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A11foSJQvJwewtASyswF9faGjIZpAnmsnzZAmWk8qBc6f176RSmWMjIA2bWjb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\nJiVl8fLimuiqY9zYGNEjovF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       "text": [
        "<matplotlib.figure.Figure at 0x2dbbd10>"
       ]
      }
     ],
     "prompt_number": 68
    }
   ],
   "metadata": {}
  }
 ]
}