Chapter-4-Scipy.ipynb

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   "source": [
    "# Chapter 4: SciPy - Library of scientific algorithms for Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "This notebook is part of the 'Data Science with Python Handbook' by [Alireza Samar](http://alirezasamar.com), read the other chapters at [https://git.io/datascihandbook](https://git.io/datascihandbook)."
   ]
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   "source": [
    "# what is this line all about? Answer in lecture 4\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.display import Image"
   ]
  },
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   "source": [
    "## Introduction"
   ]
  },
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   "source": [
    "The SciPy framework builds on top of the low-level NumPy framework for multidimensional arrays, and provides a large number of higher-level scientific algorithms. Some of the topics that SciPy covers are:\n",
    "\n",
    "* Special functions ([scipy.special](http://docs.scipy.org/doc/scipy/reference/special.html))\n",
    "* Integration ([scipy.integrate](http://docs.scipy.org/doc/scipy/reference/integrate.html))\n",
    "* Optimization ([scipy.optimize](http://docs.scipy.org/doc/scipy/reference/optimize.html))\n",
    "* Interpolation ([scipy.interpolate](http://docs.scipy.org/doc/scipy/reference/interpolate.html))\n",
    "* Fourier Transforms ([scipy.fftpack](http://docs.scipy.org/doc/scipy/reference/fftpack.html))\n",
    "* Signal Processing ([scipy.signal](http://docs.scipy.org/doc/scipy/reference/signal.html))\n",
    "* Linear Algebra ([scipy.linalg](http://docs.scipy.org/doc/scipy/reference/linalg.html))\n",
    "* Sparse Eigenvalue Problems ([scipy.sparse](http://docs.scipy.org/doc/scipy/reference/sparse.html))\n",
    "* Statistics ([scipy.stats](http://docs.scipy.org/doc/scipy/reference/stats.html))\n",
    "* Multi-dimensional image processing ([scipy.ndimage](http://docs.scipy.org/doc/scipy/reference/ndimage.html))\n",
    "* File IO ([scipy.io](http://docs.scipy.org/doc/scipy/reference/io.html))\n",
    "\n",
    "Each of these submodules provides a number of functions and classes that can be used to solve problems in their respective topics.\n",
    "\n",
    "In this lecture we will look at how to use some of these subpackages.\n",
    "\n",
    "To access the SciPy package in a Python program, we start by importing everything from the `scipy` module."
   ]
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  {
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   "execution_count": 2,
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   "source": [
    "from scipy import *"
   ]
  },
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   "source": [
    "If we only need to use part of the SciPy framework we can selectively include only those modules we are interested in. For example, to include the linear algebra package under the name `la`, we can do:"
   ]
  },
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   "execution_count": 3,
   "metadata": {
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   "source": [
    "import scipy.linalg as la"
   ]
  },
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   },
   "source": [
    "## Special functions"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "A large number of mathematical special functions are important for many computional physics problems. SciPy provides implementations of a very extensive set of special functions. For details, see the list of functions in the reference documention at http://docs.scipy.org/doc/scipy/reference/special.html#module-scipy.special. \n",
    "\n",
    "To demonstrate the typical usage of special functions we will look in more detail at the Bessel functions:"
   ]
  },
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   "cell_type": "code",
   "execution_count": 4,
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   "source": [
    "#\n",
    "# The scipy.special module includes a large number of Bessel-functions\n",
    "# Here we will use the functions jn and yn, which are the Bessel functions \n",
    "# of the first and second kind and real-valued order. We also include the \n",
    "# function jn_zeros and yn_zeros that gives the zeroes of the functions jn\n",
    "# and yn.\n",
    "#\n",
    "from scipy.special import jn, yn, jn_zeros, yn_zeros"
   ]
  },
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   "execution_count": 5,
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "J_0(0.000000) = 1.000000\n",
      "Y_0(1.000000) = 0.088257\n"
     ]
    }
   ],
   "source": [
    "n = 0    # order\n",
    "x = 0.0\n",
    "\n",
    "# Bessel function of first kind\n",
    "print \"J_%d(%f) = %f\" % (n, x, jn(n, x))\n",
    "\n",
    "x = 1.0\n",
    "# Bessel function of second kind\n",
    "print \"Y_%d(%f) = %f\" % (n, x, yn(n, x))"
   ]
  },
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   "execution_count": 6,
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rBas9V+Ny3GWdCqmJwa+/sqKUpTZZuX0bePdd4KOPWLiPDtE52vAo/xG6OXXD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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108b03890>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = linspace(0, 10, 100)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "for n in range(4):\n",
    "    ax.plot(x, jn(n, x), label=r\"$J_%d(x)$\" % n)\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  2.40482556,   5.52007811,   8.65372791,  11.79153444])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# zeros of Bessel functions\n",
    "n = 0 # order\n",
    "m = 4 # number of roots to compute\n",
    "jn_zeros(n, m)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Numerical integration: quadrature"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Numerical evaluation of a function of the type\n",
    "\n",
    "$\\displaystyle \\int_a^b f(x) dx$\n",
    "\n",
    "is called *numerical quadrature*, or simply *quadature*. SciPy provides a series of functions for different kind of quadrature, for example the `quad`, `dblquad` and `tplquad` for single, double and triple integrals, respectively.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import quad, dblquad, tplquad"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The `quad` function takes a large number of optional arguments, which can be used to fine-tune the behaviour of the function (try `help(quad)` for details).\n",
    "\n",
    "The basic usage is as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# define a simple function for the integrand\n",
    "def f(x):\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "integral value = 0.5 , absolute error = 5.55111512313e-15\n"
     ]
    }
   ],
   "source": [
    "x_lower = 0 # the lower limit of x\n",
    "x_upper = 1 # the upper limit of x\n",
    "\n",
    "val, abserr = quad(f, x_lower, x_upper)\n",
    "\n",
    "print \"integral value =\", val, \", absolute error =\", abserr "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "If we need to pass extra arguments to integrand function we can use the `args` keyword argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.736675137081 9.3891268825e-13\n"
     ]
    }
   ],
   "source": [
    "def integrand(x, n):\n",
    "    \"\"\"\n",
    "    Bessel function of first kind and order n. \n",
    "    \"\"\"\n",
    "    return jn(n, x)\n",
    "\n",
    "\n",
    "x_lower = 0  # the lower limit of x\n",
    "x_upper = 10 # the upper limit of x\n",
    "\n",
    "val, abserr = quad(integrand, x_lower, x_upper, args=(3,))\n",
    "\n",
    "print val, abserr "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "For simple functions we can use a lambda function (name-less function) instead of explicitly defining a function for the integrand:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "numerical  = 1.77245385091 1.42026367809e-08\n",
      "analytical = 1.77245385091\n"
     ]
    }
   ],
   "source": [
    "val, abserr = quad(lambda x: exp(-x ** 2), -Inf, Inf)\n",
    "\n",
    "print \"numerical  =\", val, abserr\n",
    "\n",
    "analytical = sqrt(pi)\n",
    "print \"analytical =\", analytical"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "As show in the example above, we can also use 'Inf' or '-Inf' as integral limits.\n",
    "\n",
    "Higher-dimensional integration works in the same way:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.785398163397 1.63822994214e-13\n"
     ]
    }
   ],
   "source": [
    "def integrand(x, y):\n",
    "    return exp(-x**2-y**2)\n",
    "\n",
    "x_lower = 0  \n",
    "x_upper = 10\n",
    "y_lower = 0\n",
    "y_upper = 10\n",
    "\n",
    "val, abserr = dblquad(integrand, x_lower, x_upper, lambda x : y_lower, lambda x: y_upper)\n",
    "\n",
    "print val, abserr "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Note how we had to pass lambda functions for the limits for the y integration, since these in general can be functions of x."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Ordinary differential equations (ODEs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "SciPy provides two different ways to solve ODEs: An API based on the function `odeint`, and object-oriented API based on the class `ode`. Usually `odeint` is easier to get started with, but the `ode` class offers some finer level of control.\n",
    "\n",
    "Here we will use the `odeint` functions. For more information about the class `ode`, try `help(ode)`. It does pretty much the same thing as `odeint`, but in an object-oriented fashion.\n",
    "\n",
    "To use `odeint`, first import it from the `scipy.integrate` module"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import odeint, ode"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "A system of ODEs are usually formulated on standard form before it is attacked numerically. The standard form is:\n",
    "\n",
    "$y' = f(y, t)$\n",
    "\n",
    "where \n",
    "\n",
    "$y = [y_1(t), y_2(t), ..., y_n(t)]$ \n",
    "\n",
    "and $f$ is some function that gives the derivatives of the function $y_i(t)$. To solve an ODE we need to know the function $f$ and an initial condition, $y(0)$.\n",
    "\n",
    "Note that higher-order ODEs can always be written in this form by introducing new variables for the intermediate derivatives.\n",
    "\n",
    "Once we have defined the Python function `f` and array `y_0` (that is $f$ and $y(0)$ in the mathematical formulation), we can use the `odeint` function as:\n",
    "\n",
    "    y_t = odeint(f, y_0, t)\n",
    "\n",
    "where `t` is and array with time-coordinates for which to solve the ODE problem. `y_t` is an array with one row for each point in time in `t`, where each column corresponds to a solution `y_i(t)` at that point in time. \n",
    "\n",
    "We will see how we can implement `f` and `y_0` in Python code in the examples below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### Example: double pendulum"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Let's consider a physical example: The double compound pendulum, described in some detail here: http://en.wikipedia.org/wiki/Double_pendulum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<img src=\"http://upload.wikimedia.org/wikipedia/commons/c/c9/Double-compound-pendulum-dimensioned.svg\"/>"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(url='http://upload.wikimedia.org/wikipedia/commons/c/c9/Double-compound-pendulum-dimensioned.svg')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The equations of motion of the pendulum are given on the wiki page:\n",
    "\n",
    "${\\dot \\theta_1} = \\frac{6}{m\\ell^2} \\frac{ 2 p_{\\theta_1} - 3 \\cos(\\theta_1-\\theta_2) p_{\\theta_2}}{16 - 9 \\cos^2(\\theta_1-\\theta_2)}$\n",
    "\n",
    "${\\dot \\theta_2} = \\frac{6}{m\\ell^2} \\frac{ 8 p_{\\theta_2} - 3 \\cos(\\theta_1-\\theta_2) p_{\\theta_1}}{16 - 9 \\cos^2(\\theta_1-\\theta_2)}.$\n",
    "\n",
    "${\\dot p_{\\theta_1}} = -\\frac{1}{2} m \\ell^2 \\left [ {\\dot \\theta_1} {\\dot \\theta_2} \\sin (\\theta_1-\\theta_2) + 3 \\frac{g}{\\ell} \\sin \\theta_1 \\right ]$\n",
    "\n",
    "${\\dot p_{\\theta_2}} = -\\frac{1}{2} m \\ell^2 \\left [ -{\\dot \\theta_1} {\\dot \\theta_2} \\sin (\\theta_1-\\theta_2) +  \\frac{g}{\\ell} \\sin \\theta_2 \\right]$\n",
    "\n",
    "To make the Python code simpler to follow, let's introduce new variable names and the vector notation: $x = [\\theta_1, \\theta_2, p_{\\theta_1}, p_{\\theta_2}]$\n",
    "\n",
    "${\\dot x_1} = \\frac{6}{m\\ell^2} \\frac{ 2 x_3 - 3 \\cos(x_1-x_2) x_4}{16 - 9 \\cos^2(x_1-x_2)}$\n",
    "\n",
    "${\\dot x_2} = \\frac{6}{m\\ell^2} \\frac{ 8 x_4 - 3 \\cos(x_1-x_2) x_3}{16 - 9 \\cos^2(x_1-x_2)}$\n",
    "\n",
    "${\\dot x_3} = -\\frac{1}{2} m \\ell^2 \\left [ {\\dot x_1} {\\dot x_2} \\sin (x_1-x_2) + 3 \\frac{g}{\\ell} \\sin x_1 \\right ]$\n",
    "\n",
    "${\\dot x_4} = -\\frac{1}{2} m \\ell^2 \\left [ -{\\dot x_1} {\\dot x_2} \\sin (x_1-x_2) +  \\frac{g}{\\ell} \\sin x_2 \\right]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "g = 9.82\n",
    "L = 0.5\n",
    "m = 0.1\n",
    "\n",
    "def dx(x, t):\n",
    "    \"\"\"\n",
    "    The right-hand side of the pendulum ODE\n",
    "    \"\"\"\n",
    "    x1, x2, x3, x4 = x[0], x[1], x[2], x[3]\n",
    "    \n",
    "    dx1 = 6.0/(m*L**2) * (2 * x3 - 3 * cos(x1-x2) * x4)/(16 - 9 * cos(x1-x2)**2)\n",
    "    dx2 = 6.0/(m*L**2) * (8 * x4 - 3 * cos(x1-x2) * x3)/(16 - 9 * cos(x1-x2)**2)\n",
    "    dx3 = -0.5 * m * L**2 * ( dx1 * dx2 * sin(x1-x2) + 3 * (g/L) * sin(x1))\n",
    "    dx4 = -0.5 * m * L**2 * (-dx1 * dx2 * sin(x1-x2) + (g/L) * sin(x2))\n",
    "    \n",
    "    return [dx1, dx2, dx3, dx4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# choose an initial state\n",
    "x0 = [pi/4, pi/2, 0, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# time coodinate to solve the ODE for: from 0 to 10 seconds\n",
    "t = linspace(0, 10, 250)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# solve the ODE problem\n",
    "x = odeint(dx, x0, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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FFcDTT5OI/vxzW8ZpBLF4op0kohVPfUIMn/TUVOr6d+qU8eOKlWgi0YmJtIqwZ4+5Y7Ia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\n6CNyrXz5pS8mp2b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/kV8wGs4+2zuNTrvHU51j5EgycVVXW1Y0P6iI/vlnmsSlpUV8rF69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\n/B/NCqfN4Rm/ZwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108f90a50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the angles as a function of time\n",
    "\n",
    "fig, axes = plt.subplots(1,2, figsize=(12,4))\n",
    "axes[0].plot(t, x[:, 0], 'r', label=\"theta1\")\n",
    "axes[0].plot(t, x[:, 1], 'b', label=\"theta2\")\n",
    "\n",
    "\n",
    "x1 = + L * sin(x[:, 0])\n",
    "y1 = - L * cos(x[:, 0])\n",
    "\n",
    "x2 = x1 + L * sin(x[:, 1])\n",
    "y2 = y1 - L * cos(x[:, 1])\n",
    "    \n",
    "axes[1].plot(x1, y1, 'r', label=\"pendulum1\")\n",
    "axes[1].plot(x2, y2, 'b', label=\"pendulum2\")\n",
    "axes[1].set_ylim([-1, 0])\n",
    "axes[1].set_xlim([1, -1]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Simple annimation of the pendulum motion. We will see how to make better animation in Lecture 4."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from IPython.display import display, clear_output\n",
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DltdExM0RMT8z/zhNNdaxy/6N0qa1+y8zTxprXfckwb6Z+XpE7Ae8McZnvNb9938i4h6q\noUErwmS8/vWol/39MW0b5qwGzgeIiOOALcMON9vuSeCwiDg4IvYAzqXqz04RsSAioru8hOrUfD8E\nCfTQP/p7/w23Grigu3wB1RHkx0TEnIiY213+6ILMMc9StthYcz697O+Pm+ZZ5H8G/hv4gGo8diFw\nMXDxsDY3Us0iP8MYM81tfQCnAC9067+q+9rO/gGXAc9SzYz/Fjiu6ZpL9q/f99+wPswHHgReBB4A\n5nVf3x+4r7t8SHc/Pt3dp1c1XfcE+ndm9/fvfeB1YM3I/o21v8d7eNGapCLaNsyR1KcME0lFGCaS\nijBMJBVhmEgqwjCRVIRhIqkIw0RSEf8LB+U5EoL/HJ8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108af5f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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DltdExM0RMT8z/zhNNdaxy/6N0qa1+y8zTxprXfckwb6Z+XpE7Ae8McZnvNb9938i4h6q\noUErwmS8/vWol/39MW0b5qwGzgeIiOOALcMON9vuSeCwiDg4IvYAzqXqz04RsSAioru8hOrUfD8E\nCfTQP/p7/w23Grigu3wB1RHkx0TEnIiY213+6ILMMc9StthYcz697O+Pm+ZZ5H8G/hv4gGo8diFw\nMXDxsDY3Us0iP8MYM81tfQCnAC9067+q+9rO/gGXAc9SzYz/Fjiu6ZpL9q/f99+wPswHHgReBB4A\n5nVf3x+4r7t8SHc/Pt3dp1c1XfcE+ndm9/fvfeB1YM3I/o21v8d7eNGapCLaNsyR1KcME0lFGCaS\nijBMJBVhmEgqwjCRVIRhIqkIw0RSEf8LB+U5EoL/HJ8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108af5f90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(4,4))\n",
    "\n",
    "for t_idx, tt in enumerate(t[:200]):\n",
    "\n",
    "    x1 = + L * sin(x[t_idx, 0])\n",
    "    y1 = - L * cos(x[t_idx, 0])\n",
    "\n",
    "    x2 = x1 + L * sin(x[t_idx, 1])\n",
    "    y2 = y1 - L * cos(x[t_idx, 1])\n",
    "    \n",
    "    ax.cla()    \n",
    "    ax.plot([0, x1], [0, y1], 'r.-')\n",
    "    ax.plot([x1, x2], [y1, y2], 'b.-')\n",
    "    ax.set_ylim([-1.5, 0.5])\n",
    "    ax.set_xlim([1, -1])\n",
    "\n",
    "    clear_output() \n",
    "    display(fig)\n",
    "\n",
    "    time.sleep(0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "#### Example: Damped harmonic oscillator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "ODE problems are important in computational physics, so we will look at one more example: the damped harmonic oscillation. This problem is well described on the wiki page: http://en.wikipedia.org/wiki/Damping\n",
    "\n",
    "The equation of motion for the damped oscillator is:\n",
    "\n",
    "$\\displaystyle \\frac{\\mathrm{d}^2x}{\\mathrm{d}t^2} + 2\\zeta\\omega_0\\frac{\\mathrm{d}x}{\\mathrm{d}t} + \\omega^2_0 x = 0$\n",
    "\n",
    "where $x$ is the position of the oscillator, $\\omega_0$ is the frequency, and $\\zeta$ is the damping ratio. To write this second-order ODE on standard form we introduce $p = \\frac{\\mathrm{d}x}{\\mathrm{d}t}$:\n",
    "\n",
    "$\\displaystyle \\frac{\\mathrm{d}p}{\\mathrm{d}t} = - 2\\zeta\\omega_0 p - \\omega^2_0 x$\n",
    "\n",
    "$\\displaystyle \\frac{\\mathrm{d}x}{\\mathrm{d}t} = p$\n",
    "\n",
    "In the implementation of this example we will add extra arguments to the RHS function for the ODE, rather than using global variables as we did in the previous example. As a consequence of the extra arguments to the RHS, we need to pass an keyword argument `args` to the `odeint` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def dy(y, t, zeta, w0):\n",
    "    \"\"\"\n",
    "    The right-hand side of the damped oscillator ODE\n",
    "    \"\"\"\n",
    "    x, p = y[0], y[1]\n",
    "    \n",
    "    dx = p\n",
    "    dp = -2 * zeta * w0 * p - w0**2 * x\n",
    "\n",
    "    return [dx, dp]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# initial state: \n",
    "y0 = [1.0, 0.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# time coodinate to solve the ODE for\n",
    "t = linspace(0, 10, 1000)\n",
    "w0 = 2*pi*1.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# solve the ODE problem for three different values of the damping ratio\n",
    "\n",
    "y1 = odeint(dy, y0, t, args=(0.0, w0)) # undamped\n",
    "y2 = odeint(dy, y0, t, args=(0.2, w0)) # under damped\n",
    "y3 = odeint(dy, y0, t, args=(1.0, w0)) # critial damping\n",
    "y4 = odeint(dy, y0, t, args=(5.0, w0)) # over damped"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
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SJUvon//8Z0optddfUFBgPx4ZGUm3bNli/zxz5kz66quvUkopfe+992hCQoJT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\nduUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1093bf4d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(t, y1[:,0], 'k', label=\"undamped\", linewidth=0.25)\n",
    "ax.plot(t, y2[:,0], 'r', label=\"under damped\")\n",
    "ax.plot(t, y3[:,0], 'b', label=r\"critical damping\")\n",
    "ax.plot(t, y4[:,0], 'g', label=\"over damped\")\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Fourier transform"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Fourier transforms are one of the universal tools in computational physics, which appear over and over again in different contexts. SciPy provides functions for accessing the classic [FFTPACK](http://www.netlib.org/fftpack/) library from NetLib, which is an efficient and well tested FFT library written in FORTRAN. The SciPy API has a few additional convenience functions, but overall the API is closely related to the original FORTRAN library.\n",
    "\n",
    "To use the `fftpack` module in a python program, include it using:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from numpy.fft import fftfreq\n",
    "from scipy.fftpack import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "To demonstrate how to do a fast Fourier transform with SciPy, let's look at the FFT of the solution to the damped oscillator from the previous section:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "N = len(t)\n",
    "dt = t[1]-t[0]\n",
    "\n",
    "# calculate the fast fourier transform\n",
    "# y2 is the solution to the under-damped oscillator from the previous section\n",
    "F = fft(y2[:,0]) \n",
    "\n",
    "# calculate the frequencies for the components in F\n",
    "w = fftfreq(N, dt)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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004qP5FbgVklrgX3Ax7IGr5N0J2nU737gU+FJOup18Hz5fDbsm8A0YGVWHfpl\nRHzK57N+npBvTJwDXAY8IWl1tu064MvAnZIuBzYAH2xN8zpa6f+vz2XjrgL+V/ZLxK+BT5D+r+c+\nn56My8zMzJrC8wCYmZlZUzhkmJmZWVM4ZJiZmVlTOGSYmZlZUzhkmJmZWVM4ZJiZmVlTOGSYmZlZ\nU/x/8QPGQJ9lUrYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108e03b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(9,3))\n",
    "ax.plot(w, abs(F));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Since the signal is real, the spectrum is symmetric. We therefore only need to plot the part that corresponds to the postive frequencies. To extract that part of the `w` and `F` we can use some of the indexing tricks for NumPy arrays that we saw in Lecture 2:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "indices = where(w > 0) # select only indices for elements that corresponds to positive frequencies\n",
    "w_pos = w[indices]\n",
    "F_pos = F[indices]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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RTIZCtHWKiIiIREbBQkRERCKjYCEiIiKRUbAQERGRyChYiIiISGQULERERCQy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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108d77690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(9,3))\n",
    "ax.plot(w_pos, abs(F_pos))\n",
    "ax.set_xlim(0, 5);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "As expected, we now see a peak in the spectrum that is centered around 1, which is the frequency we used in the damped oscillator example."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Linear algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The linear algebra module contains a lot of matrix related functions, including linear equation solving, eigenvalue solvers, matrix functions (for example matrix-exponentiation), a number of different decompositions (SVD, LU, cholesky), etc. \n",
    "\n",
    "Detailed documetation is available at: http://docs.scipy.org/doc/scipy/reference/linalg.html\n",
    "\n",
    "Here we will look at how to use some of these functions:\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Linear equation systems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Linear equation systems on the matrix form\n",
    "\n",
    "$A x = b$\n",
    "\n",
    "where $A$ is a matrix and $x,b$ are vectors can be solved like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from scipy.linalg import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "A = array([[1,2,3], [4,5,6], [7,8,9]])\n",
    "b = array([1,2,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.33333333,  0.66666667,  0.        ])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = solve(A, b)\n",
    "\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ -1.11022302e-16,   0.00000000e+00,   0.00000000e+00])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check\n",
    "dot(A, x) - b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We can also do the same with\n",
    "\n",
    "$A X = B$\n",
    "\n",
    "where $A, B, X$ are matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "A = rand(3,3)\n",
    "B = rand(3,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X = solve(A, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.19168749,  1.34543171,  0.38437594],\n",
       "       [-0.88153715, -3.22735597,  0.66370273],\n",
       "       [ 0.10044006,  1.0465058 ,  0.39801748]])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0014830212433605e-16"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check\n",
    "norm(dot(A, X) - B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Eigenvalues and eigenvectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The eigenvalue problem for a matrix $A$:\n",
    "\n",
    "$\\displaystyle A v_n = \\lambda_n v_n$\n",
    "\n",
    "where $v_n$ is the $n$th eigenvector and $\\lambda_n$ is the $n$th eigenvalue.\n",
    "\n",
    "To calculate eigenvalues of a matrix, use the `eigvals` and for calculating both eigenvalues and eigenvectors, use the function `eig`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "evals = eigvals(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.08466629+0.j,  0.33612878+0.j, -0.28229973+0.j])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "evals, evecs = eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.08466629+0.j,  0.33612878+0.j, -0.28229973+0.j])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.20946865, -0.48428024, -0.14392087],\n",
       "       [-0.79978578,  0.8616452 , -0.79527482],\n",
       "       [-0.56255275,  0.15178997,  0.58891829]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evecs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The eigenvectors corresponding to the $n$th eigenvalue (stored in `evals[n]`) is the $n$th *column* in `evecs`, i.e., `evecs[:,n]`. To verify this, let's try mutiplying eigenvectors with the matrix and compare to the product of the eigenvector and the eigenvalue:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.243515426387745e-16"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 1\n",
    "\n",
    "norm(dot(A, evecs[:,n]) - evals[n] * evecs[:,n])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "There are also more specialized eigensolvers, like the `eigh` for Hermitian matrices. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Matrix operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2.0031935 , -0.63411453,  0.49891784],\n",
       "       [-4.63643938, -0.2212669 ,  3.35170585],\n",
       "       [ 1.06421936,  1.37366073, -1.42726809]])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# the matrix inverse\n",
    "inv(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.10292296739753022"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# determinant\n",
    "det(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.3060382297688262, 1.591998214728641)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# norms of various orders\n",
    "norm(A, ord=2), norm(A, ord=Inf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Sparse matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Sparse matrices are often useful in numerical simulations dealing with large systems, if the problem can be described in matrix form where the matrices or vectors mostly contains zeros. Scipy has a good support for sparse matrices, with basic linear algebra operations (such as equation solving, eigenvalue calculations, etc).\n",
    "\n",
    "There are many possible strategies for storing sparse matrices in an efficient way. Some of the most common are the so-called coordinate form (COO), list of list (LIL) form,  and compressed-sparse column CSC (and row, CSR). Each format has some advantanges and disadvantages. Most computational algorithms (equation solving, matrix-matrix multiplication, etc) can be efficiently implemented using CSR or CSC formats, but they are not so intuitive and not so easy to initialize. So often a sparse matrix is initially created in COO or LIL format (where we can efficiently add elements to the sparse matrix data), and then converted to CSC or CSR before used in real calcalations.\n",
    "\n",
    "For more information about these sparse formats, see e.g. http://en.wikipedia.org/wiki/Sparse_matrix\n",
    "\n",
    "When we create a sparse matrix we have to choose which format it should be stored in. For example, "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from scipy.sparse import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 0, 0, 0],\n",
       "       [0, 3, 0, 0],\n",
       "       [0, 1, 1, 0],\n",
       "       [1, 0, 0, 1]])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# dense matrix\n",
    "M = array([[1,0,0,0], [0,3,0,0], [0,1,1,0], [1,0,0,1]]); M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<type 'numpy.int64'>'\n",
       "\twith 6 stored elements in Compressed Sparse Row format>"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# convert from dense to sparse\n",
    "A = csr_matrix(M); A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 0, 0, 0],\n",
       "        [0, 3, 0, 0],\n",
       "        [0, 1, 1, 0],\n",
       "        [1, 0, 0, 1]])"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# convert from sparse to dense\n",
    "A.todense()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "More efficient way to create sparse matrices: create an empty matrix and populate with using matrix indexing (avoids creating a potentially large dense matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<type 'numpy.float64'>'\n",
       "\twith 6 stored elements in LInked List format>"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = lil_matrix((4,4)) # empty 4x4 sparse matrix\n",
    "A[0,0] = 1\n",
    "A[1,1] = 3\n",
    "A[2,2] = A[2,1] = 1\n",
    "A[3,3] = A[3,0] = 1\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 1.,  0.,  0.,  0.],\n",
       "        [ 0.,  3.,  0.,  0.],\n",
       "        [ 0.,  1.,  1.,  0.],\n",
       "        [ 1.,  0.,  0.,  1.]])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.todense()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Converting between different sparse matrix formats:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<type 'numpy.float64'>'\n",
       "\twith 6 stored elements in LInked List format>"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<type 'numpy.float64'>'\n",
       "\twith 6 stored elements in Compressed Sparse Row format>"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = csr_matrix(A); A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<type 'numpy.float64'>'\n",
       "\twith 6 stored elements in Compressed Sparse Column format>"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = csc_matrix(A); A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We can compute with sparse matrices like with dense matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 1.,  0.,  0.,  0.],\n",
       "        [ 0.,  3.,  0.,  0.],\n",
       "        [ 0.,  1.,  1.,  0.],\n",
       "        [ 1.,  0.,  0.,  1.]])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.todense()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 1.,  0.,  0.,  0.],\n",
       "        [ 0.,  9.,  0.,  0.],\n",
       "        [ 0.,  4.,  1.,  0.],\n",
       "        [ 2.,  0.,  0.,  1.]])"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A * A).todense()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 1.,  0.,  0.,  0.],\n",
       "        [ 0.,  3.,  0.,  0.],\n",
       "        [ 0.,  1.,  1.,  0.],\n",
       "        [ 1.,  0.,  0.,  1.]])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.todense()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 1.,  0.,  0.,  0.],\n",
       "        [ 0.,  9.,  0.,  0.],\n",
       "        [ 0.,  4.,  1.,  0.],\n",
       "        [ 2.,  0.,  0.,  1.]])"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(A).todense()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1],\n",
       "       [2],\n",
       "       [3],\n",
       "       [4]])"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = array([1,2,3,4])[:,newaxis]; v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.],\n",
       "       [ 6.],\n",
       "       [ 5.],\n",
       "       [ 5.]])"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sparse matrix - dense vector multiplication\n",
    "A * v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[ 1.],\n",
       "        [ 6.],\n",
       "        [ 5.],\n",
       "        [ 5.]])"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# same result with dense matrix - dense vector multiplcation\n",
    "A.todense() * v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Optimization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Optimization (finding minima or maxima of a function) is a large field in mathematics, and optimization of complicated functions or in many variables can be rather involved. Here we will only look at a few very simple cases. For a more detailed introduction to optimization with SciPy see: http://scipy-lectures.github.com/advanced/mathematical_optimization/index.html\n",
    "\n",
    "To use the optimization module in scipy first include the `optimize` module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from scipy import optimize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Finding a minima"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Let's first look at how to find the minima of a simple function of a single variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return 4*x**3 + (x-2)**2 + x**4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109feb690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax  = plt.subplots()\n",
    "x = linspace(-5, 3, 100)\n",
    "ax.plot(x, f(x));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We can use the `fmin_bfgs` function to find the minima of a function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: -3.506641\n",
      "         Iterations: 6\n",
      "         Function evaluations: 30\n",
      "         Gradient evaluations: 10\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-2.67298164])"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_min = optimize.fmin_bfgs(f, -2)\n",
    "x_min "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: 2.804988\n",
      "         Iterations: 3\n",
      "         Function evaluations: 15\n",
      "         Gradient evaluations: 5\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 0.46961745])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fmin_bfgs(f, 0.5) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We can also use the `brent` or `fminbound` functions. They have a bit different syntax and use different algorithms. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.46961743402759754"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.brent(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.6729822917513886"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fminbound(f, -4, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Finding a solution to a function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "To find the root for a function of the form $f(x) = 0$ we can use the `fsolve` function. It requires an initial guess: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "omega_c = 3.0\n",
    "def f(omega):\n",
    "    # a transcendental equation: resonance frequencies of a low-Q SQUID terminated microwave resonator\n",
    "    return tan(2*pi*omega) - omega_c/omega"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/rob/miniconda/envs/py27-spl/lib/python2.7/site-packages/IPython/kernel/__main__.py:4: RuntimeWarning: divide by zero encountered in divide\n"
     ]
    },
    {
     "data": {
      "image/png": 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oACdaa08BsBjAk8mH5A4rVMEqLJTmYI5MaLhHHgGuuQY4+2zXkaS+JUuAX/5S\nKnkvvcQlqoayVjY+dOsmW/hPPtl1RH6wVnbzFRUBH33E46LCct99MrDzkUdcR5LkLj9r7ei4f50C\n4IbkwnGLFapgzZ8PnHgiP+AaavRoaabOznYdSerLzpaDop99Vg6VpYaJxWTDw8SJ8sUt/MHp2FFG\nI4wbx2GyYfn4YzmYe/r01PicCTJnvhNAvwDfTx0rVMGqTaio/kpKpLGyZ082otdl8mQp9b/6qvT9\nUMPEYsC990pSmpUF7B/pho3U0rev9EBOnszdfGFZskQeBkaNSp3juOpMqIwxowG02sVfPWWtHVrz\nmqcBbLPWfhxwfKpYoQrW/PnASSe5jiJannhCdli1b+86ktQ2erSce/b+++77JqIoPpn68kt+6Adp\n2jQZKDt2LE82CEssJpsmnn4aOO0019FsV2dCZa29/If+3hjzJwBXArh0d6/p2LHj93/OyMhARkZG\nfeNTxQpVsObPB266yXUU0TFyJDBkCDBnjutIUttnn8kxPJ99Jr1T1DBMpsKzdi1w/fXA22+zFy1M\nvXpJj9qDDwbzfllZWcjKykr6fYy1tvH/sTHtAXQFcJG1dv1uXmOT+R6a7r1XKir33us6kuizVo6p\nWLCAfRn1sWYNcPrpsl39ootcR5O63ntPnkqHD0+tJ9OosBb4+9+BKVNk+CGTqeDEYlItPeMM4Pnn\nXUfjr1WrgFNPlWXqsFpKjDGw1ja4KyvZHqrXATQFMNpIR9i31trIpiOsUAUnP1/mADGZqlssBtx2\nm/ROMZnave7dZSfauHHAcce5jiaaOneWsQgTJjCZCtp//yuHwcctyFAIHnwQuOee1OzPTXaXn1cj\nG2MxJlRBmTlTKi5Uty5d5BDaf/7TdSSpyVp54v/gA9nWf+SRriOKpnfflRldkybxXMigzZgBvPyy\nHBXFWVPhycqSz5a+fV1HsmucjBGnuppN6UFhQlU/Y8ZI5WXKFM6p2RVrpVF/xAipqrTa1fYYqtNX\nXwFPPQWMHw8cdpjraPxSWQnceSfQtSvQpo3raPwVi8msqRdeSN2zJZk+xGGFKjizZjGhqsvy5TJF\nuW9fVl12JRaToX1jx8qTKZOpxsnNlfvsk0+4VBqGV16R4cW33OI6Er998okUPH7/e9eR7B6fieOw\nQhWcmTOBHj1cR5G6ysrk5PlHH5XjUmhHVVXAXXcBS5dKFY8zuRqnuFhmdT33HPvzwpCTI5WpVBks\n6avqaplTlGR0AAATfklEQVTm36NHan9GM6GKwwpVMNaskQNBWXXZNWtlJ2nbtrLjinZUXS1N+hs2\nyLb+vfd2HVE0WSuzei64QJp4KXgPPCBnbnKpL1yffQa0aCGHn6cyJlRxWKEKxqRJwC9+wSe23enc\nWZZEJ07kNUpUXQ3ccQewbp3M5NprL9cRRdebb8pu2/79eZ+FITMTyMuTpIrCU7sp5d//Tv37mAlV\nHFaogjFpEnD++a6jSE19+wJvvSXXKFWOS0gVsZiMjsjPlzlTTKYaLztbtu9PmiTjSyhY1dWyXN+5\nM69v2EaNkt8NV1/tOpK6sR4ThxWqYEyaBJx3nusoUs/YscDDD8uOtcMPdx1NaqldBl28GBg6lMt8\nySgvB/7wBxnHcaxXg21SR9++cvbhdde5jsR/b74ps6dSvToFsEK1A1aokldWBsydC5x1lutIUkt2\nthzg+8knqTmQzrXHHpMjd1LpoNOoev55oF076Z+i4FVXyzV+661ofMhH2fLl0hrxcUROCWZCFYcV\nquTNmCEJAysM261cCVx1lcybuvhi19Gknu7dgWHDgG++4fTuZM2bJ8M758zhh31YBg6UY7X4sxy+\nXr2AW28F9tnHdST1w4QqDitUyfvmGy73xSsqkmTqvvuAm292HU3qGTBA5vh88w1w4IGuo4m22h60\nf/+bwzvDUtsg3aULE9awVVUBvXvLDLqoYD0mDitUyWNCtd22bTJr6oILZEmLdpSVBdx/vzSgH3WU\n62ii77335EP+7rtdR+KvsWPln7/+tds40sG4cTJ65/jjXUdSf0wf4rBClZyqKjlrjQME5Un2z3+W\nUvVrr/FpNlFenkw87tcPOOUU19FEX0mJnAX56qt8KAxTjx7yEMCf5/D16yebK6KEP3pxWKFKzvTp\n8kTRsqXrSNz75z9lx1q/fkzSE5WUyO6oZ55J/UF9UfHyyzJx/8wzXUfirxUr5DzJP/7RdST+q6gA\nBg8Gfvc715E0DHuo4lRX84DaZIweDVx2meso3OvVS4Ypfvstm/MTWSuDO888U570KXlr1wJvvCHD\nYik8PXtKgzR3oYZv5Ejg5z+P3ngZpg9xqqpYTUjGV1/JMQzpbPhwOTft66+Bgw92HU3qeeUVedIf\nP57LJkHp2lWWRnjUU3hiMeCDD+SDnsI3aFBqH4K8O0yo4lRVsULVWCUlciDyL3/pOhJ3pk8H/vQn\nGUzZtq3raFLP1KmyO2raNGDPPV1H44f162Un1OzZriPx24QJsgv1pJNcR+K/WEzO8Hz+edeRNBw7\nhuJwya/xJkyQZZyozAsJWl4ecM01wNtvA+ee6zqa1LNli1RR3nyTO/qC1L07cOONQOvWriPxW9++\n7J3SMn06cMgh0ay4Mn2IwyW/xhsxIn23Em/YAFxxBfD00zyKYnceeED662680XUk/igrk76eb791\nHYnfKiqAzz5jFVDLyJHy+zSKmFDFYYWqcawFhgyRMm26KSuTytS118rwTtrZiBHSU5ad7ToSv/Tr\nB5x9NpeXw5aZKUt9rALqGDkymst9AJf8dsAKVePMmSMnrp9wgutIdFVXA7fcIqXpl15yHU1q2rIF\n+NvfZCk0XZeDw2At8PrrUvmjcA0bJg9MFL7164EFC2QYchQxoYrDClXjDBkiVZp027X1yCOy3Nen\nD+eX7c6TTwK/+hXnTQVt0iRg61a5thQea6XCetVVriNJD+PGycamZs1cR9I4TB/icJdf4wwZItvh\n08nrr8sS56RJ0f3hD9uMGdJ7snCh60j806cPcNddTOTDNmeO7Eht1851JOlh4kTgwgtdR9F4TB/i\ncMmv4VasAJYuBc4/33UkeoYMAV58Uc4tPOAA19GkJmuBhx+Wg3pbtHAdjV/Ky2VOz9y5riPx3/Dh\nUp1Kt+q7KxMnysNqVPH5Jg6X/BpuwADgt78FmjRxHYmO6dOlMjB4MHD00a6jSV2DBkn/1B13uI7E\nP8OHA6edBhxxhOtI/MflPj1btgDffQeccYbrSBqPCVUcVqgarn//6B1g2VjLl0tz6ttvy+4q2rXy\ncuDRR4Fu3fjzFIYPP5TNEBSuoiKpAvKwdx2TJ0syFeUWCiZUcVihapicHGDlSiAjw3Uk4du8Gbjy\nSkkUOGvqh73zDnDiicDFF7uOxD9FRdK4e8MNriPx38SJ8uAU5Q/4KJkyJfpDkZk+xGGFqmH69wdu\nusn/a1ZVJedKXXIJ8NBDrqNJbeXlMkLiiy9cR+KnL7+ULeX77ec6Ev+NHx/tBumomTEj+tPoWaGK\nw11+9WetDBbs0MF1JOF74gk5X6pbNzan1uWdd4DTT492H0QqGzYM+M1vXEeRHiZM4HKfphkzov97\nw1hrw/0Gxtiwv0dQ9tkHWLcO2Htv15Gkvrlz5Rf70qV+b93+8EOgUyc52PfAA11Hk9oqK4FjjgE+\n/1zOdaRgVVUBrVoBs2ZxanfYSkrkWq9bB+y1l+to/Ld2rQyG3rAhNR5ajTGw1jY4EtZj4nDJr/4+\n+AC4+Wa/k6lp04C//116VphM1W3QIOCnP2UyFZbJkyWRYjIVvhkz5LgZJlM6ZsyQynYqJFPJYEIV\nh03p9bNtm1RuJk50HUl4CgqA668HevWSX6xUt+7dZXmUwjF0KHD11a6jSA/TpgFnneU6ivQxc6Yk\nVFHncX2hYayVhMrniktQhg0Djj8eOPZY15GEo6pKesPuvFNmbFHdJk+W5RH294Rn1CjgiitcR5Ee\nmFDpmj8fOPlk11Ekj+lDjVhMlvuiXnLU0Lu3JBu+6tRJBpU++6zrSKLjzTeB++7jknlYNm4Elizh\nh7wWJlS65s+XUStRx6b0GhUVwP77y7Zv2r1Vq+RJIj9fmvh9M2qUTPeeORNo2dJ1NNGwZQtw5JEy\nl+zgg11H46fBg4H//U/GJlC41q+XXsBNm7hioaGqCmjeXBrSU2VDGJvSk8SG9Prp00dmT/mYTK1e\nDdx+O/Dxx0ymGmLAAJnRxWQqPFlZ6TFANxXMni1H+zCZ0rFkCXDYYamTTCWDt0wNNqTXraoK6NkT\n+OtfXUcSvFhMhsrdey8nfDfUe+/xzL6wjR/PhEpLdrYf/TxRsWAB8LOfuY4iGEknVMaYfxhjYsaY\nSG8sZ4WqbkOGyJZtH3ZjJOreXZLqp55yHUm05OTIE2b79q4j8VdJCbB4sVRNKHzz5nFnryZf+qeA\nJBMqY0xrAJcDWB5MOO6wQlW3Hj2ABx5wHUXw5s8HXnxRljOZVDfMoEFyrlyTJq4j8deMGVIx4Zly\nOphQ6Vq0SHaN+yDZCtV/ATwWRCCusUL1w+bNkxv/+utdRxKsbduAW28FXnhBpnxTw3z2mX/3RKqZ\nMgU45xzXUaSHWEwesJhQ6cnLA9q2dR1FMBqdUBljrgWw0lo7N8B4nOE5fj/sjTeAu+8GmjZ1HUmw\nnn8eOPRQ4M9/dh1J9KxYIb8MeYBsuJhQ6Vm2TE5F2H9/15Gkj7w8fx5mfzCFMMaMBtBqF3/1NIAn\nAfwq/uW7e5+OHTt+/+eMjAxkpGB3JZf8dm/zZqB/f2ke9MnChZIozpnD+WONMXiwDPLkcl+4pk4F\nXn7ZdRTpYd48f/p5oqCkRMautNpVlqEoKysLWVlZSb9Po+ZQGWNOAjAGQGnN/3QEgFUAzrbWFia8\nNhJzqHJzpbE2N9d1JKmna1dg+nSgXz/XkQQnFpNdUzfd5GdfmIaMDDnr8JprXEfir9qZSJs3M+nX\n8MorMmuvWzfXkaSH7Gw5lWL+fNeR7Eh1DpW1dh6A7yf1GGOWAjjDWruxMe+XClih2rXKStkBN3iw\n60iC1acPUFYmYxKo4YqLJcm+7DLXkfgtOxv4+c+ZTGnJzeXIBE0+LfcBwc2hSv0SVB3YlL5r/fsD\n7doBZ5zhOpLgrFsHPPmkzNTi/+eNM2GCHM3hwzC+VDZ3riRUpCM3158G6ShYsoQJ1U6stcdEuToF\nsEK1K9YCXboAjz7qOpJgPfOMlJl9nKelZcwY4NJLXUfhPyZUunJy/D30PRXl5cmSti84Kb0Gd/nt\nLDNT/vnrX7uNI0jz5gGffw7E7ZOgRhgzhst9GubMYUKlpbwcKCiQcylJR14ecPTRrqMIDhOqGlzy\n21ltdcqn/o1HHgGefho44ADXkURXYSGwfDlw5pmuI/FbdbXsROVMJB1LlwJHHcUHa00rV8rpG75g\nQlWDS347mjlTjrvo0MF1JMHJzJQ1+3vucR1JtI0dK7On+PMSrmXLgIMOApo3dx1JemD/lL5Vq4DD\nD3cdRXCYUNVghWpHL78M/N//+TNjyFppRH/pJf+Gk2obP54HSGtYvBg47jjXUaQPJlS6yspkDtVB\nB7mOJDhMqGqwQrXdd99JFeLuu11HEpwhQySp4jEpyZs8GTjvPNdR+G/xYtlhSzpWrGD/lKba6pRP\nLSVMqGqwQrXdCy8ADz7oz1KDtUCnTsCzz/r1w+vC1q3yQX/qqa4j8d/ixdxxpsm3fp5U59tyH9DI\nwZ4+4i4/kZcHDB8uvUa+GDpUJqNfe63rSKJv+nTZddasmetI/JeTA1x9teso0kd+PhMqTT4mVKxQ\n1aiuZoUKAF58UaaH+3I4qLXAv/4l1akf8W5P2uTJwLnnuo4iPXDJT1d+PnDEEa6jSB8+JlSsydSo\nrPSnAbuxVqwAPvtMnox98fXXckzKdde5jsQPU6cCN97oOgr/lZXJTKSjjnIdSXqorJQTFA47zHUk\n6WPVKv961vjMXoMJFdC5M/CXvwAHHug6kuC8+irw0EOsTgVl9mzgtNNcR+G/vDygTRu2IWhZswY4\n5BBeb00rV7JC5a10T6hWrwb69QMWLXIdSXCWLpUt/u+/7zoSP2zZIlUTNkqHb/lySahIB/un9Pm4\n5Mfn9hqVlen9dNKlC/CnP8lTmi969ADuvBPYd1/Xkfhh7lzg5JPZa6hh+XL/lkNSGfun9BUUAIce\n6jqKYKVxCrGjqqr0rVAVFkoVZ/5815EEp6QE6NNHJr5TMGbPBk45xXUU6WHFCvZPaeLIBH3r1gEH\nH+w6imCxQlUjnZf8unYFbr7Zr6eFgQOB88/nh1KQ5szh/CktrFDpWrmSFSpNZWXymevLrMNaTKhq\n+JRQZWVl1fu1GzYA77wDPP54ePG48N57stynpSHXPKrmzpUZVKnC52ueqhUqX6/52rVAy5auo9g1\nH6/5+vVy5Ixvg5aZUNXwqYeqIT+A3bsDN9zgV7k7J0eOz7nqKr3v6eMvvXjWyoaF4493Hcl2Pl/z\nVK1Q+XrNCwuZUGlav96/5T6APVTf86lCVV+bNwNvvSWzhXzSpw9wyy3p9/9nmNaulev5k5+4jsR/\nlZVyvX3bAZXKCgv92pCT6tat8+tQ5FpMqGqkY1N6jx5ytMUxx7iOJDjWAn37Al984ToSv3z3HXDc\nca6jSA+rV0u1JN1+H7nEhEqXrxUqY60N9xsYE+43ICIiIgqQtbbBHV6hJ1REREREvmNTOhEREVGS\nmFARERERJSmwhMoY094Ys8gYk2OM2eVUI2PMazV/P8cYwyNWk1TXNTfGZBhjiowxs2q+nnERpy+M\nMe8aY9YaY7J/4DW8xwNU1zXnPR48Y0xrY8w4Y8x8Y8w8Y8yDu3kd7/WA1Oea814PljFmT2PMFGPM\nbGPMAmPMi7t5Xf3vc2tt0l8A9gCQC6ANgCYAZgM4IeE1VwIYUfPncwBMDuJ7p+tXPa95BoAhrmP1\n5QvALwGcBiB7N3/Pe1z/mvMeD/6atwJwas2f9wXwHX+fp8Q1570e/HXfu+afPwYwGcAFCX/foPs8\nqArV2QByrbXLrLWVAPoDuDbhNdcAeB8ArLVTALQwxqToKLVIqM81BwDPZtG6Y639GsCmH3gJ7/GA\n1eOaA7zHA2WtLbDWzq75cwmAhQAOS3gZ7/UA1fOaA7zXA2WtLa35Y1NIkWJjwksadJ8HlVAdDiA/\n7t9X1vxvdb2Gpyc1Xn2uuQVwXk2pcoQx5mdq0aUn3uP6eI+HyBjTBlIhnJLwV7zXQ/ID15z3esCM\nMT8yxswGsBbAOGvtgoSXNOg+D2qwZ31nLyRm15zZ0Hj1uXYzAbS21pYaY64AMBhAu3DDSnu8x3Xx\nHg+JMWZfAJ8CeKimarLTSxL+nfd6kuq45rzXA2atjQE41RizP4BMY0yGtTYr4WX1vs+DqlCtAhB/\nGlxrSCb3Q685ouZ/o8ap85pba4trS5rW2pEAmhhjDtQLMe3wHlfGezwcxpgmAAYB+MhaO3gXL+G9\nHrC6rjnv9fBYa4sADAdwZsJfNeg+Dyqhmg7gWGNMG2NMUwC/BzAk4TVDANwGAMaYcwFsttauDej7\np6M6r7kxpqUxcp63MeZsyCDXxDViCg7vcWW8x4NXcz17A1hgre2+m5fxXg9Qfa457/VgGWMOMsa0\nqPnzXgAuBzAr4WUNus8DWfKz1lYZY+4HkAlp7OptrV1ojPlrzd/3tNaOMMZcaYzJBbAVwB1BfO90\nVZ9rDuBGAPcYY6oAlALo4CxgDxhj+gG4CMBBxph8AM9BdljyHg9JXdccvMfDcD6AWwDMNcbUfsA8\nBeBIgPd6SOq85uC9HrRDAbxvjPkRpLj0obV2TDJ5C4+eISIiIkoSJ6UTERERJYkJFREREVGSmFAR\nERERJYkJFREREVGSmFARERERJYkJFREREVGSmFARERERJYkJFREREVGS/h/Ek2+h/tsImAAAAABJ\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a14e610>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax  = plt.subplots(figsize=(10,4))\n",
    "x = linspace(0, 3, 1000)\n",
    "y = f(x)\n",
    "mask = where(abs(y) > 50)\n",
    "x[mask] = y[mask] = NaN # get rid of vertical line when the function flip sign\n",
    "ax.plot(x, y)\n",
    "ax.plot([0, 3], [0, 0], 'k')\n",
    "ax.set_ylim(-5,5);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.23743014])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fsolve(f, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.71286972])"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fsolve(f, 0.6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.18990285])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fsolve(f, 1.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Interpolation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Interpolation is simple and convenient in scipy: The `interp1d` function, when given arrays describing X and Y data, returns and object that behaves like a function that can be called for an arbitrary value of x (in the range covered by X), and it returns the corresponding interpolated y value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from scipy.interpolate import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "n = arange(0, 10)  \n",
    "x = linspace(0, 9, 100)\n",
    "\n",
    "y_meas = f(n) + 0.1 * randn(len(n)) # simulate measurement with noise\n",
    "y_real = f(x)\n",
    "\n",
    "linear_interpolation = interp1d(n, y_meas)\n",
    "y_interp1 = linear_interpolation(x)\n",
    "\n",
    "cubic_interpolation = interp1d(n, y_meas, kind='cubic')\n",
    "y_interp2 = cubic_interpolation(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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BDw/Aywupdp1xYFBzeD4PxOvU15hhOQOjWoxCWe2yhYq8cOFCuLq6onTp0jh58iQ6duxY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\n0zQtBegiS9M0TdM0LQX8H1MW7wSfvwdUAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a5c8cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(10,4))\n",
    "ax.plot(n, y_meas, 'bs', label='noisy data')\n",
    "ax.plot(x, y_real, 'k', lw=2, label='true function')\n",
    "ax.plot(x, y_interp1, 'r', label='linear interp')\n",
    "ax.plot(x, y_interp2, 'g', label='cubic interp')\n",
    "ax.legend(loc=3);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The `scipy.stats` module contains a large number of statistical distributions, statistical functions and tests. For a complete documentation of its features, see http://docs.scipy.org/doc/scipy/reference/stats.html.\n",
    "\n",
    "There is also a very powerful python package for statistical modelling called statsmodels. See http://statsmodels.sourceforge.net for more details."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# create a (discreet) random variable with poissionian distribution\n",
    "\n",
    "X = stats.poisson(3.5) # photon distribution for a coherent state with n=3.5 photons"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10aa71fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = arange(0,15)\n",
    "\n",
    "fig, axes = plt.subplots(3,1, sharex=True)\n",
    "\n",
    "# plot the probability mass function (PMF)\n",
    "axes[0].step(n, X.pmf(n))\n",
    "\n",
    "# plot the commulative distribution function (CDF)\n",
    "axes[1].step(n, X.cdf(n))\n",
    "\n",
    "# plot histogram of 1000 random realizations of the stochastic variable X\n",
    "axes[2].hist(X.rvs(size=1000));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# create a (continous) random variable with normal distribution\n",
    "Y = stats.norm()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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y98xuw4cPT70Nuj5dX2u7ttZwfYWIO8APZeXwDIRiY0dHj48G7o/5fCIi0ojY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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ad17150>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = linspace(-5,5,100)\n",
    "\n",
    "fig, axes = plt.subplots(3,1, sharex=True)\n",
    "\n",
    "# plot the probability distribution function (PDF)\n",
    "axes[0].plot(x, Y.pdf(x))\n",
    "\n",
    "# plot the commulative distributin function (CDF)\n",
    "axes[1].plot(x, Y.cdf(x));\n",
    "\n",
    "# plot histogram of 1000 random realizations of the stochastic variable Y\n",
    "axes[2].hist(Y.rvs(size=1000), bins=50);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Statistics:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3.5, 1.8708286933869707, 3.5)"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X.mean(), X.std(), X.var() # poission distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.0, 1.0, 1.0)"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y.mean(), Y.std(), Y.var() # normal distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Statistical tests"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Test if two sets of (independent) random data comes from the same distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t-statistic = -0.901953297251\n",
      "p-value = 0.367190391714\n"
     ]
    }
   ],
   "source": [
    "t_statistic, p_value = stats.ttest_ind(X.rvs(size=1000), X.rvs(size=1000))\n",
    "\n",
    "print \"t-statistic =\", t_statistic\n",
    "print \"p-value =\", p_value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Since the p value is very large we cannot reject the hypothesis that the two sets of random data have *different* means."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "To test if the mean of a single sample of data has mean 0.1 (the true mean is 0.0):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ttest_1sampResult(statistic=-3.1644288210071765, pvalue=0.0016008455559249511)"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.ttest_1samp(Y.rvs(size=1000), 0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Low p-value means that we can reject the hypothesis that the mean of Y is 0.1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ttest_1sampResult(statistic=2.2098772438652992, pvalue=0.027339807364469011)"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats.ttest_1samp(Y.rvs(size=1000), Y.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Further reading"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "* http://www.scipy.org - The official web page for the SciPy project.\n",
    "* http://docs.scipy.org/doc/scipy/reference/tutorial/index.html - A tutorial on how to get started using SciPy. \n",
    "* https://github.com/scipy/scipy/ - The SciPy source code. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Versions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]"
        },
        {
         "module": "IPython",
         "version": "3.2.1"
        },
        {
         "module": "OS",
         "version": "Darwin 14.1.0 x86_64 i386 64bit"
        },
        {
         "module": "numpy",
         "version": "1.9.2"
        },
        {
         "module": "matplotlib",
         "version": "1.4.3"
        },
        {
         "module": "scipy",
         "version": "0.16.0"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]</td></tr><tr><td>IPython</td><td>3.2.1</td></tr><tr><td>OS</td><td>Darwin 14.1.0 x86_64 i386 64bit</td></tr><tr><td>numpy</td><td>1.9.2</td></tr><tr><td>matplotlib</td><td>1.4.3</td></tr><tr><td>scipy</td><td>0.16.0</td></tr><tr><td colspan='2'>Sat Aug 15 11:13:18 2015 JST</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)] \\\\ \\hline\n",
       "IPython & 3.2.1 \\\\ \\hline\n",
       "OS & Darwin 14.1.0 x86\\_64 i386 64bit \\\\ \\hline\n",
       "numpy & 1.9.2 \\\\ \\hline\n",
       "matplotlib & 1.4.3 \\\\ \\hline\n",
       "scipy & 0.16.0 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Sat Aug 15 11:13:18 2015 JST} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 2.7.10 64bit [GCC 4.2.1 (Apple Inc. build 5577)]\n",
       "IPython 3.2.1\n",
       "OS Darwin 14.1.0 x86_64 i386 64bit\n",
       "numpy 1.9.2\n",
       "matplotlib 1.4.3\n",
       "scipy 0.16.0\n",
       "Sat Aug 15 11:13:18 2015 JST"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%reload_ext version_information\n",
    "\n",
    "%version_information numpy, matplotlib, scipy"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
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