Scipy-slides.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SciPy - Library of scientific algorithms for Python"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "J.R. Johansson (jrjohansson at gmail.com)\n",
    "\n",
    "The latest version of this [IPython notebook](http://ipython.org/notebook.html) lecture is available at [http://github.com/jrjohansson/scientific-python-lectures](http://github.com/jrjohansson/scientific-python-lectures).\n",
    "\n",
    "The other notebooks in this lecture series are indexed at [http://jrjohansson.github.io](http://jrjohansson.github.io)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy.linalg as la"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Special functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "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)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = linspace(-56, 2.5, 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": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 2.40482556  5.52007811  8.65372791 11.79153444]\n"
     ]
    },
    {
     "ename": "TypeError",
     "evalue": "'numpy.ndarray' object is not callable",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-11-ec337d731271>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0mm\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m4\u001b[0m \u001b[0;31m# number of roots to compute\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      5\u001b[0m \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mjn_zeros\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 6\u001b[0;31m \u001b[0moptimize\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfsolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mjn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m)\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[0m\u001b[1;32m      7\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mjn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      8\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py3/lib/python3.6/site-packages/scipy/optimize/minpack.py\u001b[0m in \u001b[0;36mfsolve\u001b[0;34m(func, x0, args, fprime, full_output, col_deriv, xtol, maxfev, band, epsfcn, factor, diag)\u001b[0m\n\u001b[1;32m    146\u001b[0m                'diag': diag}\n\u001b[1;32m    147\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 148\u001b[0;31m     \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_root_hybr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mjac\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mfprime\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0moptions\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    149\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mfull_output\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    150\u001b[0m         \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mres\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py3/lib/python3.6/site-packages/scipy/optimize/minpack.py\u001b[0m in \u001b[0;36m_root_hybr\u001b[0;34m(func, x0, args, jac, col_deriv, xtol, maxfev, band, eps, factor, diag, **unknown_options)\u001b[0m\n\u001b[1;32m    212\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtuple\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    213\u001b[0m         \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 214\u001b[0;31m     \u001b[0mshape\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdtype\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_check_func\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'fsolve'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'func'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfunc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx0\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn\u001b[0m\u001b[0;34m,\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[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    215\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mepsfcn\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    216\u001b[0m         \u001b[0mepsfcn\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfinfo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdtype\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meps\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/miniconda3/envs/py3/lib/python3.6/site-packages/scipy/optimize/minpack.py\u001b[0m in \u001b[0;36m_check_func\u001b[0;34m(checker, argname, thefunc, x0, args, numinputs, output_shape)\u001b[0m\n\u001b[1;32m     25\u001b[0m def _check_func(checker, argname, thefunc, x0, args, numinputs,\n\u001b[1;32m     26\u001b[0m                 output_shape=None):\n\u001b[0;32m---> 27\u001b[0;31m     \u001b[0mres\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0matleast_1d\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mthefunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx0\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0mnuminputs\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     28\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0moutput_shape\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mshape\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mres\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0moutput_shape\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     29\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0moutput_shape\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\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[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mTypeError\u001b[0m: 'numpy.ndarray' object is not callable"
     ]
    }
   ],
   "source": [
    "from scipy import optimize\n",
    "# zeros of Bessel functions\n",
    "n = 0 # order\n",
    "m = 4 # number of roots to compute\n",
    "print(jn_zeros(0, m))\n",
    "optimize.fsolve(jn(0,x),1)\n",
    "type(jn)\n",
    "\n",
    "def k2(x):\n",
    "    return jn(0,x)\n",
    "    \n",
    "k = lambda o:jn(0,o)\n",
    "type(k2)\n",
    "type(k)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Numerical integration: quadrature"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import quad, dblquad, tplquad"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# define a simple function for the integrand\n",
    "def f(x):\n",
    "    return x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "integral value = 0.5 , absolute error = 5.551115123125783e-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": {},
   "source": [
    "If we need to pass extra arguments to integrand function we can use the `args` keyword argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7366751370811073 9.389126882496403e-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": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": {},
   "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": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7853981633974476 1.3753098510218528e-08\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": {},
   "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": {},
   "source": [
    "## Ordinary differential equations (ODEs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import odeint, ode"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": {},
   "source": [
    "#### Example: double pendulum"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Image(url='http://upload.wikimedia.org/wikipedia/commons/c/c9/Double-compound-pendulum-dimensioned.svg')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 21,
   "metadata": {},
   "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": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "# choose an initial state\n",
    "x0 = [pi/4, pi/2, 0, 0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "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": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "# solve the ODE problem\n",
    "x = odeint(dx, x0, t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x288 with 2 Axes>"
      ]
     },
     "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": {},
   "source": [
    "Simple annimation of the pendulum motion. We will see how to make better animation in Lecture 4."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import display, clear_output\n",
    "import time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 288x288 with 1 Axes>"
      ]
     },
     "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": {},
   "source": [
    "#### Example: Damped harmonic oscillator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 32,
   "metadata": {},
   "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": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "# initial state: \n",
    "y0 = [1.0, 0.0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "# time coodinate to solve the ODE for\n",
    "t = linspace(0, 10, 1000)\n",
    "w0 = 2*pi*1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "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": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "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": {},
   "source": [
    "## Fourier transform"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy.fft import fftfreq\n",
    "from scipy.fftpack import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 38,
   "metadata": {},
   "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": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 648x216 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(9,3))\n",
    "ax.plot(w, abs(F));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": null,
   "metadata": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": {},
   "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": {},
   "source": [
    "## Linear algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": {},
   "source": [
    "### Linear equation systems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.linalg import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = array([[1,2,3], [4,5,6], [7,8,9]])\n",
    "b = array([1,2,3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/filipe/miniconda3/envs/py3/lib/python3.6/site-packages/ipykernel_launcher.py:1: LinAlgWarning: scipy.linalg.solve\n",
      "Ill-conditioned matrix detected. Result is not guaranteed to be accurate.\n",
      "Reciprocal condition number2.202823e-18\n",
      "  \"\"\"Entry point for launching an IPython kernel.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-0.23333333,  0.46666667,  0.1       ])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = solve(A, b)\n",
    "\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00000000e+00, -2.22044605e-16,  0.00000000e+00])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check\n",
    "dot(A, x) - b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = rand(3,3)\n",
    "B = rand(3,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = solve(A, B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[  2.75023249,  -4.03591566,  13.88091215],\n",
       "       [ -3.99323951,   7.95135045, -20.93900336],\n",
       "       [  2.58179688,  -4.47897837,  11.45277194]])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1873621467522309e-15"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check\n",
    "norm(dot(A, X) - B)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Eigenvalues and eigenvectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "evals = eigvals(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.46778628+0.j, -0.03542668+0.j,  0.12067771+0.j])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "evals, evecs = eig(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.46778628+0.j, -0.03542668+0.j,  0.12067771+0.j])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.60029496,  0.56076062,  0.22721052],\n",
       "       [ 0.34962785, -0.7585436 , -0.72615358],\n",
       "       [ 0.71930962,  0.33190229,  0.64890396]])"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evecs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5305924413163637e-16"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 1\n",
    "\n",
    "norm(dot(A, evecs[:,n]) - evals[n] * evecs[:,n])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are also more specialized eigensolvers, like the `eigh` for Hermitian matrices. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrix operations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# the matrix inverse\n",
    "inv(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# determinant\n",
    "det(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# norms of various orders\n",
    "norm(A, ord=2),norm(A, ord=Inf)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sparse matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.sparse import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "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": 57,
     "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": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<class 'numpy.int64'>'\n",
       "\twith 6 stored elements in Compressed Sparse Row format>"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# convert from dense to sparse\n",
    "A = csr_matrix(M); A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1, 0, 0, 0],\n",
       "        [0, 3, 0, 0],\n",
       "        [0, 1, 1, 0],\n",
       "        [1, 0, 0, 1]], dtype=int64)"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# convert from sparse to dense\n",
    "A.todense()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<class 'numpy.float64'>'\n",
       "\twith 6 stored elements in LInked List format>"
      ]
     },
     "execution_count": 60,
     "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": 61,
   "metadata": {},
   "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": "markdown",
   "metadata": {},
   "source": [
    "Converting between different sparse matrix formats:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<class 'numpy.float64'>'\n",
       "\twith 6 stored elements in LInked List format>"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<class 'numpy.float64'>'\n",
       "\twith 6 stored elements in Compressed Sparse Row format>"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = csr_matrix(A); A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<4x4 sparse matrix of type '<class 'numpy.float64'>'\n",
       "\twith 6 stored elements in Compressed Sparse Column format>"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = csc_matrix(A); A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compute with sparse matrices like with dense matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "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": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.todense()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "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": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A * A).todense()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "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": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.todense()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "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": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.dot(A).todense()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1],\n",
       "       [2],\n",
       "       [3],\n",
       "       [4]])"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = array([1,2,3,4])[:,newaxis]; v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.],\n",
       "       [6.],\n",
       "       [5.],\n",
       "       [5.]])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# sparse matrix - dense vector multiplication\n",
    "A * v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[1.],\n",
       "        [6.],\n",
       "        [5.],\n",
       "        [5.]])"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# same result with dense matrix - dense vector multiplcation\n",
    "A.todense() * v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import optimize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Finding a minima"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's first look at how to find the minima of a simple function of a single variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return (4*x**3 + (x-2)**2 + x**4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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LwdF5DWvmFXDNgiKv48QEFbqIxKUnN52ipXuQP7t5sddRYoYKXUTiTv+Qn3956zhXzy9k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\nEiHIzknn8AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d368b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax  = plt.subplots()\n",
    "x = linspace(-2, 2, 100)\n",
    "ax.plot(x, f(x));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use the `fmin_bfgs` function to find the minima of a function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: -3.506641\n",
      "         Iterations: 2\n",
      "         Function evaluations: 48\n",
      "         Gradient evaluations: 16\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-2.67298161])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_min = optimize.fmin_bfgs(f, -0.79664)\n",
    "x_min "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "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": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fmin_bfgs(f, 0.5) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.46961743402759754"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.brent(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "?optimize.brent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Finding a solution to a function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": 40,
   "metadata": {},
   "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": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python3.5/site-packages/ipykernel_launcher.py:4: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  after removing the cwd from sys.path.\n"
     ]
    },
    {
     "data": {
      "image/png": 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EQZUGvcfU8JdkwDp8IQBAKTNVA9bbT1XEfqqBuPO1QyjMdeDGsyp1LyWp3P1m\nNdbtb8Mdn5iNpQzkB+Set2rwi1cO4tNnTMAvPruAm5qSCIMqDXpHKvDqD1inNwyAQdVAhQ0Lm9hP\nNWAfVHdg3f523LZqGjMtA/DGgTb8at0hfGrheFx/ZoXu5SSVe96qwc/+cQCXLRyP//scA6pk49C9\ngHRksacqYZ3eaKZqNBvVB2TjkU54QwbO49Eq/WZaEj96cR8mjszDTcxS9dvuRhdu//N2zCkvwk8+\nM59lq36yLIkfv7wfa987gksXjMcvGVAlJQZVGrD8l7gOBlUJeX1/O3IcGVg5vVT3UpLGU1sacKDV\ng7uvXoTcrEzdy0kKDV1+3PDgZozMz8YD1y9BXjavW3+EDBPfeGonXtzVguvPrMB/XTKHAVWSYlCl\nASeqJ67DG4YQwChOVO83KSXW7W/DWdNL+ZDrpw5vCD/7xwEsrRiFi0/nGXX90e0L47r7NyFiWnhi\nzTL27vWTOxjBrQ9vxfraTnznollYc/ZUZvcGwB2MoCh3+JTm2dWjgcXdfwnr9IVQkpfF0+0H4FCb\nF43dAZw/e6zupSSNH76wD/6Qif/9zDw+4PrBHYzg+gc2obEngPuuq8L0MTzGpz/a3EF8/g/rsflo\nF+68YgFuXTWN99sAvF/dgbN//ibW7WvTvZRezFRp0Fv+Y3p3wNrcIYzlN+ABeXVvKwDg/Nnsp+qP\ndfva8MLOZnz9ozMZHPSDJxjBdfdvwr4WN+65ZjGWVHCnX39Ut3tw3f2b0eMP44EbluAjM8p0Lymp\nPLm5Hnc8uwdTywowq3z4/J4yqNLgWKO65oUkoVZXEOOKGVT1l5QSz+1owtLKUQxG+6HTG8J3nt2N\n08YW4rZV03QvZ9jzhQzc8MBm7Gp04e6rF+GCOcyG9sfmo124+aEtyMrMwJO3rsC8CTyLs78sS+IX\nrx7EPW/V4CMzSnH3NYuGVfmPQZUG8fIf07wD1+IKYt6EIt3LSBr7Wtyocfo4Y6kfpJT4j6d3whWI\n4OEblyLbwRLzyfhCBm58cDO21Xfjt1ctwoXz2HvWH//Y04qvPLEdE0ry8NCNSzFpVL7uJSWNYMTE\nN57eiZd2teDqZZPxw0/OHXbHRjGo0sCSLP0lImxY6PCy/DcQz+9ohiND4OJ5PAT4VB54/yjePOjE\njy6bi9nlDNxPptsXxvUPbsaeJhfuvGIhPjGf91d/PLL+KP77+b1YOKkEa69bglE8GaLfXIEI1jy8\nBRuPdOHbF83CrcO0oZ9BlQaWlCz9JaDdEwQAlLP81y+mJfH8zmasmlmGkfzwPqndjS789O8HcMHs\nsVi9nOf7nUyLK4DVazehvssFQDosAAAgAElEQVSPe65ZhI/NZYbqVKSU+MUrB/H7t2pwwewx+O1V\ni7gTdwBaXUFc/8Am1Di9+M2VC3HZwgm6l3RCDKo0MKUclhH2cNfqigZV44rzNK8kObx1sB0triD+\n+5I5upcyrHV4Q7j1kS0oK8zBzz/LYZUnU+v0YvXaTXAFInjohqVYMY0T+k8lYlr41l934ZltTbhq\n6ST8z2XzuHt5AA63eXDd/ZvgDhp48Ialw37WHoMqDaTk4M9EtLiYqRqIxzbWo6wwh83DJxE2LHzp\n0W3o8ofxl9vOZDnmJPY0uXDd/ZsAAE+sWc7m6n7whQx88bFteOeQE1+7YCa+fP50Bu0DsL2+G9c/\nsBnZjgw8eetyzB0//O85BlUamBbLf4k4lqliUHUqjd1+vHmwHbefO33YNXIOF1JK/PCFvdh0tAu/\nuXIhg4STWF/TiVse3oLivCw8ctNSTC0boXtJw16nN4QbH9yM3U0u/PQzp+PKpZN1LympbD7ahevv\n34Sywhw8ctOypGnoZ1ClQbSnilHVQLW4gijIzkRhDm/bU3lsYz0EgCuWTNK9lGHrD2/X4rGN9bj1\n7KnDukdDt1f3tuL2x7dj8qh8PHLTUpSz/H5Kjd1+XLt2E5p6Avjj6ip8lNniAfmgpgM3PbgF5SW5\nePyW5Um1OYlPJw2kBDKYqhqw5p4AxhXnMn1+Cp5gBI9uqMNF88oxcWRyfLsbas9ub8TP/nEAl8wv\nx7cunKV7OcPWX7Y24lt/3YV5E4rx4PVLuOGhHw62enDt/RvhD5t45KZlWFrJYagD8e5hJ25+aAum\njM7HYzcvR1lhju4lDQiDKg1Y/ktMfZcfU0YX6F7GsPf4pnp4ggZuXTVV91KGpfcOd+A/n96F5VNH\n4ZefX8AvOCdw37u1+H8v7cdZ00vxx9WLUcAM8SltrevCDQ9sRm5WJp6+bQVmjeNojoGID0WdWjYC\nj960FKNHJFdABTCo0sKSknOqBkhKiYYuP5ZUjNS9lGEtZJhY+94RnDltNOZPLNG9nGFnT5MLtz26\nFdPHjMAfV1chx8Ft7cf78Pb/i08fhzuvWMjr1A8baztxw4ObMbYoFw9zqOeA7W9x48YHN2NCSV7S\nBlQAD1TWwpKcpj5QPf4IPCGDH1Sn8Nz2JrS5QzxipQ97mly45r6NKM7LwgM3LEFx3vA52mK4MC2J\n7z23B79/qwZXLZ2M3161iAFVP6yv6cT1D2xGeXEunlyznJ9TA1TX6cO192/CiBwHHrl5WdIGVAAz\nVVpYLP8NWH2XHwAwmR9WJxQyTNz1ejUWTCzGR2YM71kuQy0eUI3IceCJNcvZbN2HiGnh60/txAs7\nm3Hbqmn41oWn8ctfP3xQ3YEbH9qMSSPz8dgtyzCmMHmaqoeDHn8Y196/CYZp4fFbVmBCSXL/bjKo\n0sCSknOqBqg3qBrNoOpEHttQj6aeAH52OQdYftjeZhe+sPZYQMUswr8KhE186bGtePOgE9+6cBa+\neA4znf2x+WgXbnxoM6aMKsBjtyxDaRJnWHQwTAu3/3k7WnqCeHzNMkwfU6h7SbYxqNKA5b+BiwdV\nk7ibrU/ekIHfvVmNldNH4yxmqXrtbY5mqPKzMvH4LQyo+uIORnDzg1uwua4LP/70PFyzjMf09Ee8\nB2h8SR7+fEtyl6x0+fHL+/FedQd+/tn5WDwlNXZJMqjSwJISGexmG5CGLj9KR2RzB9IJ3PtOLbp8\nYfznxzkeIO7DAdUTa1Ywy9mHTm8I196/CQdbPbjryjNw6YLxupeUFBq6/Lju/k0oyHbg4RuTt6la\np2e3N+KB94/ihpUV+HxV6szT4xNKA5b/Bq62w8dxCidQ3+nHH96uwSXzy7FwEnf8AcC+ZjcDqlPo\n8oVxzX0bcaTDhz9dW4VzZ43RvaSk0O2L9gCFDAtP37aCs+AScKTDh+89uwdLK0bhjotn616OUrby\nJUKIXwghDgghdgkhnhVC8BO9HywJTlQfoFqnF9N5NEaffvTiXjgyBO74RGp9OCUqGlBtQF5WJh5f\ns5wBVR+6fWFc/acNONLhw9rrljCg6qeIaeFLj21DU08Aa6+rwsyxyd8DNNTChoUvP74djswM/PrK\nhSl3uLTd/zevAZgnpZwP4BCA79hfUuqzLAnGVP3X4w+jwxvGtDHMVB3vjQNtWLe/HV85fwZ3tOFY\nQJWblYkn1ixndrMP3bEMVW2HD/ddV8UevAH48Uv7sb62Ez/59OmoqkiNHqCh9ts3DmN3kws/u3w+\nxif5Tr++2AqqpJSvSimN2D9uADDR/pJSH4d/DkyN0wcAmMZM1T8JRkz88IV9mFZWgBtWVupejnb7\nWxhQnYonGMHq+zei2unFfddW4SMzynQvKWk8taUBD35wFDedVYnLF/NRl4gDrW7c81YNPnPGBFw4\nb5zu5QwKlXm3GwH8XeH7pSweqDwwNU4vAAZVx/vlqwdR1+nH/1w2D9mO1EqhD1Q0oNqIHEd0lx8D\nqn8VMkyseXgrDrR48MfVi3H2TAZU/XW4zYP//tsenDltNL5zETeDJMK0JL79190oysvC9y6Zo3s5\ng+aUjepCiHUA+gop75BS/i32mjsAGAAeO8n7rAGwBgAmT56c0GJThWlxpMJA1Di9yM7MwMSRqZcq\nTtTWui7c994RXLNsMs6cnt7lmwOt0YAqOzMDT6xZjopSBlTHMy2Jrz25A+trO3HnFQtw7mnsoeqv\nYMTEvz++HQXZjpTsARoqf95Ujx0NPbjzigUYlcIHc58yqJJSXnCyPxdCXA/gEgDnSynlSd7nXgD3\nAkBVVdUJX5cOpJTg72X/HW7zoqI0nx9mMcGIif98ehfGF+fhOym2c2agDrS6cfWfGFCdjJQSP3xh\nL17e3Yo7Lp6NT5/B0tVA/PwfB3Gg1YP7r6/itPQEeYIR/Pq1Q1hWOQqfWjhB93IGla2RCkKICwF8\nE8AqKaVfzZJSH8t/A7Ov2Y3lU9kUGvfzfxxEbYcPj928DCPSeG7X4TYPrvnTRmRlCjzOgOqEHt1Q\nh4fX1+Hmsypxy9lTdS8nqWyt68YDHxzBtSum4LxZY3UvJ2nd+04tOn1h3H/x7JSv0tj96v87AIUA\nXhNC7BBC/EHBmlKeyYnq/dbpDaHVHcSc8UW6lzIsvHmgHfe/H/2QX5nGZb/qdi+u+tNGZGYIPH7L\nclQyoOrT+ppO/PCFfThv1pi0z2oOVNiw8J1ndqG8KBffvJB9VIlqdQXxp3drcemC8ViQBnP0bH3N\nlVJOV7WQdCKlRCZjqn7Z3+IBAMwpL9a8Ev3a3UH8x9M7MWtcIb6bxg/IIx0+XP2nDQCAP9+yHFO5\ngaFPjd1+/Nuft2Hy6Hz8+sqF3HE8QH98uwaH2rxYe11VWmeE7frdm4dhWhL/+bHTdC9lSPBO0cC0\nWP7rr30tLgDA7PL0HrJnWRJfe2oHfGEDT169HLlZmbqXpEVdpw9X3bsBpiXx+JrlmD6GAVVfIrGD\naiOGhfuurUJRbpbuJSWVVlcQd79VjU+cXo7zZ7PslyinJ4SntjTi8kUT02YIL4MqDdhT1X/7mt0Y\nV5Sb9mdr3fN2Dd6v7sRPP3N6SpzknoimngCuuncDQoaJP9+ynNOsT+I36w5jR0MPfnf1GczkJeCX\nrx6EZQHf5vgEW+5//wgM08Ktq6bpXsqQ4XYqDSwJHqjcT/ta3GnfT/XGgTb836sH8ckF43HFktQ5\neHQgunxhXLt2IzwhA4/ctAyzy9P7njiZ9TWduPutany+aiIumc8DkgdqX7Mbf9nWiOvOnIJJo9Ij\nuzIYvCEDj66vw0Wnl6dVzyMf7RpYLP/1SzBiosbpw5w0foBWt3vw5cd3YO74Ivzs8vlpucHBFzJw\nw4Ob0dgdwNrrlmDeBPbXnYg7GMHXn9qBytEF+MEn5+peTlL62T8OoCg3C7efO0P3UpLa33Y0wRMy\ncNNZ6XXaA4MqDXhMTf8caPXAtGTaZqp6/GHc/NAW5GZl4N7VVcjLTr8+qrBh4YuPbcOeJhd+d/Ui\nLK3kaI2T+fk/DqDNHcSdVyxEfja7OwZqZ0MP3j7kxK2rpqI4n31oiZJS4tEN9ZhTXoQz0mDH34cx\nqNLA4kiFftlW1w0AOGNyev1SAoARazRu6gngj6sXp+TBo6cipcS3/7oL7xxy4iefPh0fncOG4ZPZ\ncrQLj26ox/VnVqbF1vXB8Pu3qlGU68Dq5VN0LyWp7WjoiR4dtXxy2j3rGFRpEG1U172K4W9rfTfG\nF+eivDj9Aoofv7wf71V34MefOh2Lp6Rndub3b9Xgme1N+MZHZ+LzadpL1l8R08J3n92NCSV5+MbH\nZupeTlI63ObBK3vbcP2ZFSjkbklbntzcgILsTFyW4tPT+8KgSgNLSmSmWfSeiO113Vg0ZaTuZQy5\npzY34IH3j+LGlZVpG0z8Y08rfvHKQXxq4Xjcfh7H4Z3K45vqcajNi+9fOgcFnKmUkIfWH0W2IwPX\nr0yvHiDVwoaFv+9pxcfnjkvL+V4MqjSweKDyKbW4Amh2BbFocnoFVVuOduGO53bjIzNK8d2L03M7\n954mF7725A4snFSCn6Zpc/5AeIIR/GbdYSyfOool0gR5ghE8u60Jl84fn9KH/Q6Fdw874QpEcOmC\n9Nx5mn5h5DDA8t+pbavrAQAsTqNMVVNPALc9uhUTSvLwu6sWpeUB0j3+MG59ZCtK8rNw77WL03bI\n6UD84e0adPrCeCANzlUbLM9tb4IvbGL1CvZS2fXCzmYU52Wl7TFa6fepPQxw+Oepba3rRm5WRtrs\n/AuETax5eAuCEQv3XVeVljuPpJT4j6d3od0TxD1fWIwxhbm6lzTstbuDWPveEXxywXjMn8jm9EQ9\ntrEep08oxoKJHNdhRzBi4rV9bbho3jhkO9IzvEjP/9eamZZEJg//O6nNR7swf2IJstIgWyOlxH/8\nZSf2tbhx11UL03Zi+tr3jmDd/jZ856LZWMjda/2y9r0jCBsWvv5RNqcn6kCrGwdaPfhc1URm+mxa\nX9sJX9jEx+eN070UbVL/iTUMmRYb1U+mxx/GnmYXVk5Lj/Tx3W9W46VdLfjWhbNw3qz07InZ2dCD\nn/79AD4+dyxuWFmhezlJoccfxqMb6nDJ/PGoSKOJ1ar9bUczMjMEPnF6ue6lJL23DzqR48jAiqmj\ndS9FGwZVGpgc/nlSG2o7ISWwcnrq/2K+urcV//fqIXxq4XjcevZU3cvRIhgx8Y2nd6KsMAc/v3wB\nswX99NAHdfCFTXzp3PQ5V001y5J4fkczzppemvbni6rw9iEnVkwbnda9kAyqNLAsMKg6iferO1GQ\nnZnyAwwPtnrwtSd3YMHE4rTe5fbrdYdR3e7FTy+fn5a9ZIkIGxYe3ViHc04rw6xx6dF3OBi21Xej\nqSeAyxam5041leo6fTjS4cM5M8t0L0UrBlUaGJbF8t9JvF/TgaWVo1K6n6rbF8bND29GQY4Df1xd\nlbbf7LbXd+Ped2pwRdUkrErzD+OBeGVvK5yeEK5bUaF7KUnt1X1tyMoUHEWhwFsHnQCAVaeN0bwS\nvVL3qTWMmRaQwUxVn1pdQdQ6fSm9HTdiWvjSY9vQ5g7hj6sXY1xxeu5yi5gWvvXXXRhblIs7Lpmt\nezlJ5ZENdZg0Kg9nMxC15Y0D7VhWOZoT1BV4v7oDk0bloTLN+/sYVGlgSQkHg6o+vXWwHQBw1ozU\nDar+58V9WF/biZ98+nSckWbDTT/s4fV1ONTmxY8um4ciPtT6rbrdg01HuvCFZVPYRmBDQ5cf1e1e\nnHMaA1O7pJTYVt+NJRXpeaTWhzGo0sAwLX4YnsC6/e2YUJKH08am5liBP2+sx8Pr67Dm7Km4fPFE\n3cvRxukJ4devHcKqmWW4YHZ6lwsG6tntTcgQwKcXpd+5aiq9cSD6Be68Wbz/7Krr9KPDG06rYc0n\nwqBKA0uCwz/7EIyYeK/aiQtmj0nJpu3NR7vw/ef3YNXMMnzrwvQ8gibuF68cQNAw8d+XzknJf9eD\nRUqJv+1oxsrppRyOatObB9tRMTofU8tG6F5K0tta1w0AqErTw98/jEGVBqYl4eDwz3/xQU0HghEL\n589OvabRFlcAX3x0GyaOzMddV52R1pnKw20ePL21EdetqMA0PtAGZGtdNxq7A/jUQmap7AgbFjbU\nduKcNG+qVmVLXTcKcx2YMYa/zzz7TwPT4jE1fVm3vx0F2ZlYNjW1vu0EIyZue3QbAmEDj9+yDMV5\n6d0/9Ot1h5GflYkvnTtd91KSzt92NCM3KyOtJ1arsKfZhWDEwtLK1Pqs0WVrXRcWTR7JDVhgpkqL\n6PBP3asYXqSUeGN/O86eWYYcR+qMF5BS4r+e24OdDT341RULMSNFe8X6a1+zGy/tbsENKysxqiBb\n93KSipQSr+1rwzkzx2BEDr8P27H5SBcAoKqCPUB2uQIRHGrzoor9VAAYVA05KWX0mJoMXvoP29Pk\nRqs7mHKlv0c21OHprY348vkz8PG5zC7cue4QCnMduOUj6Tk93o59LfHfEZas7Np8tBsVo/PZl6bA\n3mYXAGB+ig9r7i8+2YeYJaP/yeGf/+yl3S1wZAicn0I7cTYd6cKPXtiHC2aPwVfPn6F7OdodbvPg\ntX1tuHFlJSenJ+D1/e0QAuwDssmyJLbUdXH7vyL7mt0AgDnlnOwPMKgacmYsqmL57xgpJV7c1Yyz\nZpRiZIqUhDq9Ifz749swaVQ+fnXFQvYaAFj73hHkODJw7YopupeSlF4/0I4FE0tQVsgz6uyocXrR\n449gCfuplNjX4saYwhzelzF8tA8xS8aDKl76uB0NPWjsDuCS+alx/pZlSXztqZ3o9kdw99WLONgS\nQIc3hGe2N+Eziyby4NoEOD0h7GzoSalMri6bj0a3/zNTpca+ZjfmjGeWKo5P9iFmMFP1L17Y2YLs\nzAx8bG5q9FPd83YN3jnkxA8uncsPm5gnNtUjbFi46axK3UtJSutrOwEAqzj927bdTS4U52WhYnS+\n7qUkvZBhorrdi9ks/fXio32Ixct/HKkQZVkSL+1uxqrTylIio7PpSBd++epBXLpgPK5aOkn3coYF\ny5J4cksDVkwdjemcY5OQTUc6MSLHwb4VBfY2uzB3fBGHzipQ6/TBsCRmjUvvXc0fxqBqiFmxoIpn\n/0VtOtqFNncIly5I/tJfjz+MLz++HVNGF+B/Pz2PH9ox62s70dAVwJUMMhO26UgXFk8ZCQdT3LZE\nTAsHWj2YN6FY91JSQo3TCwCYMYZBVRx/Q4fYsfIfH7gA8JetjRiR40iJ89++99wedHhD+O1VZ/DU\n+w95YnMDivOyOFIiQV2+MA61eTmoUoHqdi/ChoW5LMsrUdPuAwBUlhZoXsnwwaBqiMUb1bkbDPCG\nDLy8uwWXzC9HfnZyDzP8244mvLirBV+9YAa/BX+IJxjBK3tb8amF45GblTpDXYfSptigymUMqmzb\nG9v+P3c8f0dVqO3wYkJJHvKy+bsdx6BqiJks//V6eVcL/GETn6uaqHsptrS4Aviv5/bgjMkluG3V\nNN3LGVbW7W9D2LDwSZ5Vl7BNR7qQ48jA6RMZCNh1sNWNbEcGMyuK1Di9mMY+yX/CoGqIsVH9mKe3\nNmBqWQEWTU7e4w2klPjmX3YhYkr86vML2fNynJd2taK8OBdncNpywrbWd2PBpJKUOr5Jl+p2L6aW\nFrD9QgEpJWqdPkxlgPpPlDwBhBDfEEJIIUSpivdLZSZ7qgAAtU4vNh/txucWT0rqhu6ntzbi3cMd\n+O7Fs/jt9zjuYATvHHLi4tPLWe5OUMS0sL/FjQXMUilR7fSm/fmbqrS6g/CHTWaqjmM7qBJCTALw\nMQD19peT+tioHvWXrY3IzBC4fFHyloU6vSH878v7UTVlJK5Zxinhx3t9fxvCpoWLTy/XvZSkdbgt\n2ljNPj37AmETjd0BTC9jEKBCXacfADBlFOd9fZiKTNWdAL4JQCp4r5R3bKJ6+gZVEdPCX7Y2YtXM\nMowpSt4DTf/fS/vhCxn4yWdOZyamD6/vb0dZYQ5LfzbsbuoBAMyfyGtoV43TCymBGWMZVKnQ2B0A\nAExiUPVPbAVVQojLADRJKXcqWk/K6y3/JXHJy65X97ah3RPCNcsm615Kwt473IFntzfhtlXTWE7o\ng2lJvFfdgbNnlDHgtGF3kwuFOQ5mAxSobo/OVOIAWjUau/0QAhhfkrxfjAfDKfexCyHWAehrwMwd\nAL6LaOnvlIQQawCsAYDJk5P3YWoXe6qAh9cfxcSReTjntOScTRUxLXz/+T2oGJ2Pfzt3uu7lDEu7\nGnvQ44/wWBWb9sbOVWNgal9thw8ZApjC42mUaOwOYGxhLjdQHOeUmSop5QVSynnH/wWgFkAlgJ1C\niKMAJgLYJoToc8KflPJeKWWVlLKqrCx9P2jTPag62OrBxiNd+MLyKUl7Df68sR41Th/u+MQczl46\ngbcPOSEE8JHp3LuSKCklqtu8OI1HgCjR0OVHeXEegwBFGrv9mDgyT/cyhp2EJy5KKXcD6E01xAKr\nKillh4J1pSwzzXuqHt1Qh2xHBj5flZxHlrj8Edy57hBWTh+dElPgB8s7h5yYP7EEIwuydS8labW4\ngvCEDJaXFanv8mPSKAYBqjR2B1A1JXnH4QwWDtUZYumcqfIEI3hmWyMumV+OUUn6sP3N64fhDkTw\nvU/MSepREIPJE4xgR0MPzp7BLJUdh9o8AICZ7AFSor7Lj8nsTVPCMC20uIKYOJLX83jKzgaRUlao\neq9Uls6N6s9ub4IvbOLaFRW6l5KQuk4fHl5/FFcsmYzZ5Tw77ER2NPTAkuBZdTYdbos2Vs9kpsq2\nQNiE0xNiUKVIqzsI05Is//WBmaohZqVppkpKiYfX12H+xGIsTNIt9ne9Xo3MDIGvXTBD91KGta11\n3cgQSNp/z8PFoTYPygpzWEJVoLE7OlOJ2//VaHEFAQDjSxhUHY9B1RBL1+Gfbx10orrdi+vPrNC9\nlITUOL14dnsjrl0xJalnaw2FrXXdOG1cEQpzs3QvJalVO72YwdKfEvVdDKpUaneHAABjinI0r2T4\nYVA1xOKN6um2Rfred2pRXpyLSxeM172UhNz1+mHkODJxKw9MPinTkthe34PFU5ilsquu048po3n0\nkQrxoIrlPzXa3NFM1ZhCfsE8HoOqIRYv/znSKKja3ejC+tpO3LiyEllJeOBwjdOL53c247ozK1A6\ngt/MTuZgqwfekIGqKeynssMdjKDLF0YFZyop0eIKItuRgdEspSrR7gkhK1NgZD6z0cdLvidckouX\n/zLSqFH93ndrUZjjwJVLk3OMwn3vHkFWZgZu/kil7qUMe1vruwEAi7nV2pb6+LlqDKqUaHUFMa4o\nlzt2FWn3BDGmkNezLwyqhli6Nao3dPnx8u4WXLVsclL22HR4Q3hmWyMuXzSRWap+2NXQg9EF2dwV\nZFP8sNrJo1j+U6HVHQ2qSI12dwhlhfw87AuDqiGWbsM/73//CASAG1ZW6F5KQh5ZX4eQYeGms5il\n6o99LdFjVfgN1p6jnT4AzFSp0uYOYmwxgypVopkqBlV9YVA1xAwzfXqqXP4IntzcgE8uGI/y4uTL\nXAQjJh7ZUIcLZo/hIaz9EDYsHG7zYs54zvCyq77Tj9IROSjIUTZKMG1JKWPlPwYBqrR7QhjLzF+f\nGFQNMaO3UT31L/1jm+rgD5u45eypupeSkJd2taDLF8aNzFL1S43Ti7BpYe74Yt1LSXp1XT5mqRRx\nBSIIGRaDAEWCERM9/ggzVSeQ+k/2Yca0LACAIzO1M1Uhw8QD7x/FR2aUJu308Sc3N6CytAArpo7W\nvZSksLfZDQCYk6T/voeT5p4gJnCwohJtsZlKDKrUcHo4o+pkGFQNsUialP/+tqMZTk8It56dnHOd\napxebDrahSuWTGJ/UD8daHEjNysDlaVsrrYjXq4qZw+QEq2xmUrjeD2VcHqjQRUb1fvGoGqIxc/+\ncyThvKb+siyJP71Ti9nlRVg5PTmzPE9uboAjQ+DyRRN1LyVpVDu9mFo6Im02YQyWLl8YYdNiUKVI\nW+xIFe7+U6PHHwYAjMznzK++pO6TfZiKmNHyXyo/eN4+5MThdi/WnF2ZlFmesGHhr1sbccHssfw2\nNgA1Ti8b+hWIn6s2Lgk3dwxHvdO/Wa5SotsXAcCg6kQYVA2xeKYqK4V7quJH0lwyPzmPpHn7kBOd\nvjCuWJKcw0p1CEZMNHYHMK2MQZVdxw6rZWZFhU5fGIW5DuQ4MnUvJSV0xzNVnE7fJwZVQyzVD1RO\n9iNpAOCFnc0YmZ+Fs2aU6l5K0qh1+iAlMG0M+6nsanEFALAHSJUuX5jH0yjU7Q8jM0OgKJfjPvqS\nnE+9JBafU5WVoiMVkv1ImkDYxLr9bbhw3rikDQp1qHZ6AYDlPwVaXEFkZQqUFrBcpUKXL8ysikLd\n/ghK8rKSsrVjKPCpMcRMy4IQQEYKZqriR9JcnaRH0gDAWwfb4Q+bSVu61KWm3QshgIrRzFTZ1eoK\nYmxRbkp+RujATJVaPf4wSniQ8gkJGTs2ZUh/qBBOAHWD/GNKAXQM8s9IN7ymavF6qsdrqhavp3q8\npuoNxTWdIqUsO9WLtARVQ0EIsUVKWaV7HamE11QtXk/1eE3V4vVUj9dUveF0TVn+IyIiIlKAQRUR\nERGRAqkcVN2rewEpiNdULV5P9XhN1eL1VI/XVL1hc01TtqeKiIiIaCilcqaKiIiIaMgkfVAlhLhQ\nCHFQCFEthPh2H3+eI4R4MvbnG4UQFUO/yuTSj2t6vRDCKYTYEfvrZh3rTBZCiPuFEO1CiD0n+HMh\nhLgrdr13CSEWDfUak0k/ruc5QgjXh+7P/x7qNSYTIcQkIcSbQoh9Qoi9Qoiv9PEa3qMD0M9ryvu0\nn4QQuUKITUKInbHr+cM+XjMsnvVJHVQJITIB3A3gIgBzAFwlhJhz3MtuAtAtpZwO4E4APxvaVSaX\nfl5TAHhSSrkw9td9QxrZomMAAAM7SURBVLrI5PMggAtP8ucXAZgR+2sNgHuGYE3J7EGc/HoCwLsf\nuj9/NARrSmYGgG9IKecAWA7g3/r4nec9OjD9uaYA79P+CgE4T0q5AMBCABcKIZYf95ph8axP6qAK\nwFIA1VLKWillGMATAC477jWXAXgo9vd/AXC+4Hz9k+nPNaUBkFK+A6DrJC+5DMDDMmoDgBIhRPnQ\nrC759ON60gBIKVuklNtif+8BsB/AhONexnt0APp5TamfYvedN/aPWbG/jm8IHxbP+mQPqiYAaPjQ\nPzfiX2/c3tdIKQ0ALgCjh2R1yak/1xQALo+VAf4ihEjOg/6Gj/5ec+q/FbFSwd+FEHN1LyZZxEom\nZwDYeNwf8R5N0EmuKcD7tN+EEJlCiB0A2gG8JqU84T2q81mf7EEV6fECgAop5XwAr+HYtwOi4WAb\nokdKLADwWwDPaV5PUhBCjADwVwBflVK6da8nFZzimvI+HQAppSmlXAhgIoClQoh5utfUl2QPqpoA\nfDhLMjH23/X5GiGEA0AxgM4hWV1yOuU1lVJ2SilDsX+8D8DiIVpbqurPfUz9JKV0x0sFUsqXAWQJ\nIUo1L2tYE0JkIfrwf0xK+UwfL+E9OkCnuqa8TxMjpewB8Cb+ta9yWDzrkz2o2gxghhCiUgiRDeBK\nAM8f95rnAVwX+/vPAnhDcjjXyZzymh7XS/FJRPsFKHHPA7g2tsNqOQCXlLJF96KSlRBiXLyXQgix\nFNHPOX6ROoHYtVoLYL+U8lcneBnv0QHozzXlfdp/QogyIURJ7O/zAHwUwIHjXjYsnvWOof6BKkkp\nDSHE7QBeAZAJ4H4p5V4hxI8AbJFSPo/ojf2IEKIa0ebWK/WtePjr5zX9shDik4jucOkCcL22BScB\nIcTjAM4BUCqEaATwfUQbLSGl/AOAlwFcDKAagB/ADXpWmhz6cT0/C+CLQggDQADAlfwidVIrAawG\nsDvWswIA3wUwGeA9mqD+XFPep/1XDuCh2O70DABPSSlfHI7Pek5UJyIiIlIg2ct/RERERMMCgyoi\nIiIiBRhUERERESnAoIqIiIhIAQZVRERERAowqCIiIiJSgEEVERERkQIMqoiIiIgU+P/CHZAM1jDh\nlgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d459a58>"
      ]
     },
     "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": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.23743014])"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fsolve(f, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.71286972])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fsolve(f, 0.6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.18990285])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize.fsolve(f, 1.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interpolation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.interpolate import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": {},
   "source": [
    "## Statistics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create a (continous) random variable with normal distribution\n",
    "Y = stats.norm()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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": {},
   "source": [
    "Statistics:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X.mean(), X.std(), X.var() # poission distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Y.mean(), Y.std(), Y.var() # normal distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Statistical tests"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test if two sets of (independent) random data comes from the same distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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": {},
   "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": {},
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stats.ttest_1samp(Y.rvs(size=1000), 0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Low p-value means that we can reject the hypothesis that the mean of Y is 0.1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Y.mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "stats.ttest_1samp(Y.rvs(size=1000), Y.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Further reading"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "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": {},
   "source": [
    "## Versions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%reload_ext version_information\n",
    "\n",
    "%version_information numpy, matplotlib, scipy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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