{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Root Finding"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose that you want to find the solution(s) of a transcendental equation. For example, you might want to know what values of $x$ that solve the equation\n",
    "$$ \\sin x = 2x\\cos x.$$ \n",
    "The graphical approach is to plot both sides of the equation and see where they cross as shown below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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35f2735fd+fOQnHz1sVp27845GNeRI1ZRhoRY7wTKlbO+zz4oV/Zl7gPbFSli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\nqrcVAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f2d83958750>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pylab import * \n",
    "#from scipy import *\n",
    "\n",
    "x = linspace(0,3*pi,1000)\n",
    "yL = sin(x)\n",
    "yR = 2*x*cos(x)\n",
    "\n",
    "figure()\n",
    "plot(x,yL,label=r'$\\sin x$')\n",
    "plot(x,yR,label=r'$2x\\cos x$')\n",
    "legend(loc='upper left')\n",
    "ylim(-15,15)\n",
    "xlabel(r'$x$')\n",
    "axhline(y=0, c=\"k\") # draw horizontal line at y=0\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the graph above, it is possible to find approximate solutions. There is a solution at $x=0$. You can also see that there are solutions between $n\\pi$ and $(n+1)\\pi$ for integer values of $n$.  \n",
    "Another way to find the solution graphically is to plot the difference of the two sides of the original equation. If we define the function\n",
    "$$f(x) = \\sin x - 2x\\cos x,$$ \n",
    "the original equation has a solution where $f(x)=0$. These are known as the roots of $f(x)$. It is slightly easier to find the solutions by looking at a graph of this funciton as shown below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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QHIhICfnb3+DUU+Gaa+Dqq7N7jPqKFWGy/J13YPx4aNYse19bcq8Yd6KLSBZ1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\ntjY0uJhTAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f2d83958ad0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = linspace(0,3*pi,1000)\n",
    "fx = sin(x) - 2*x*cos(x)\n",
    "\n",
    "figure()\n",
    "plot(x,fx)\n",
    "#ylim(-15,15)\n",
    "xlabel(r'$x$')\n",
    "ylabel(r'$f(x)$')\n",
    "axhline(y=0, c=\"k\") # draw horizontal line at y=0\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are several functions in the **`scipy.optimize`** library for finding the roots of a function. For all of the methods, you must define the function whose roots you want to find. In general, you must know the approximate location of the root.  \n",
    "  \n",
    "First, you can use the **`optimize.brentq`** function if you know values that bracket the root. As mentioned above, the roots of the example function $f(x)$ are between $n\\pi$ and $(n+1)\\pi$ for integer values of $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n",
      "4.6042167772\n",
      "7.78988375114\n",
      "10.9499436485\n",
      "14.1017251336\n",
      "17.2497818346\n",
      "20.3958423573\n",
      "23.5407082923\n",
      "26.6848024909\n",
      "29.8283692131\n"
     ]
    }
   ],
   "source": [
    "from scipy import optimize\n",
    "\n",
    "# Define the function\n",
    "def f(x):\n",
    "    return sin(x) - 2*x*cos(x)\n",
    "\n",
    "# The function has roots between 0, pi, 2*pi, etc.\n",
    "for n in arange(10):\n",
    "    print(optimize.brentq(f, n*pi, (n+1)*pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that for the range 0 to $\\pi$, the function found the root $x=0$, not the root near $x=1$. The other root can be found by changing the lower limit of the range slightly as shown below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.16556118521\n"
     ]
    }
   ],
   "source": [
    "print(optimize.brentq(f,0.1,pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The function $f(x)$ must have opposite signs at the two limits given or the the root-finding function will fail as shown below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "f(a) and f(b) must have different signs",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-6-6e832f7132e2>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[1;32mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0moptimize\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mbrentq\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m4\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m8\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32m/projects/sage/sage-7.3/local/lib/python2.7/site-packages/scipy/optimize/zeros.pyc\u001b[0m in \u001b[0;36mbrentq\u001b[1;34m(f, a, b, args, xtol, rtol, maxiter, full_output, disp)\u001b[0m\n\u001b[0;32m    436\u001b[0m     \u001b[1;32mif\u001b[0m \u001b[0mrtol\u001b[0m \u001b[1;33m<\u001b[0m \u001b[0m_rtol\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    437\u001b[0m         \u001b[1;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"rtol too small (%g < %g)\"\u001b[0m \u001b[1;33m%\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mrtol\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0m_rtol\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m--> 438\u001b[1;33m     \u001b[0mr\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0m_zeros\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0m_brentq\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0ma\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mb\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mxtol\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mrtol\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mmaxiter\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0margs\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mfull_output\u001b[0m\u001b[1;33m,\u001b[0m\u001b[0mdisp\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m    439\u001b[0m     \u001b[1;32mreturn\u001b[0m \u001b[0mresults_c\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mfull_output\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mr\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m    440\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mValueError\u001b[0m: f(a) and f(b) must have different signs"
     ]
    }
   ],
   "source": [
    "print(optimize.brentq(f,4,8))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Second, you can use the **`optimize.fsolve`** function if you know approximate values of the roots. In the example below, initial guesses of $x\\approx (n+\\frac{1}{2})\\pi$ for integers $n$ from 0 to 9 are used."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  1.16556119   4.60421678   7.78988375  10.94994365  14.10172513\n",
      "  17.24978183  20.39584236  23.54070829  26.68480249  29.82836921]\n"
     ]
    }
   ],
   "source": [
    "n = arange(10)  # make a list of integers from 0 to 9 \n",
    "guesses = (n+0.5)*pi  # make a list of guesses\n",
    "print(optimize.fsolve(f,guesses))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Additional Documentation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "More information is available at  \n",
    "http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.brentq.html  \n",
    "http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fsolve.html"
   ]
  }
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