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  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": [],
     "source": [
      "> This is one of the 100 recipes of the [IPython Cookbook](http://ipython-books.github.io/), the definitive guide to high-performance scientific computing and data science in Python.\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# 9.1. Finding the root of a mathematical function"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "1. Let's import NumPy, SciPy, scipy.optimize, and matplotlib."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import scipy as sp\n",
      "import scipy.optimize as opt\n",
      "import matplotlib.pyplot as plt\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "2. We define the mathematical function $f(x)=\\cos(x)-x$ in Python. We will try to find a root of this function numerically, which corresponds to a fixed point of the cosine function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = lambda x: np.cos(x) - x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "3. Let's plot this function on the interval $[-5, 5]$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = np.linspace(-5, 5, 1000)\n",
      "y = f(x)\n",
      "plt.figure(figsize=(5,3));\n",
      "plt.plot(x, y);\n",
      "plt.axhline(0, color='k');\n",
      "plt.xlim(-5,5);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "4. We see that this function has a unique root on this interval (this is because the function's sign changes on this interval). The scipy.optimize module contains a few root-finding functions that are adapted here. For example, the `bisect` function implements the **bisection method** (also called the **dichotomy method**). It takes as input the function and the interval to find the root in."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "opt.bisect(f, -5, 5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's visualize the root on the plot."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.figure(figsize=(5,3));\n",
      "plt.plot(x, y);\n",
      "plt.axhline(0, color='k');\n",
      "plt.scatter([_], [0], c='r', s=100);\n",
      "plt.xlim(-5,5);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "5. A faster and more powerful method is `brentq` (Brent's method). This algorithm also requires that $f$ is continuous and that $f(a)$ and $f(b)$ have different signs."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "opt.brentq(f, -5, 5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The `brentq` method is faster than `bisect`. If the conditions are satisfied, it is a good idea to try Brent's method first."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%timeit opt.bisect(f, -5, 5)\n",
      "%timeit opt.brentq(f, -5, 5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "> You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).\n",
      "\n",
      "> [IPython Cookbook](http://ipython-books.github.io/), by [Cyrille Rossant](http://cyrille.rossant.net), Packt Publishing, 2014 (500 pages)."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}