Converge on root of function for a given accuracy and initial guess.

NewtonsMethod.ipynb

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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import cos, sin\n",
      "def f(x):\n",
      "    \"\"\"To find the root of x = cos(x),\n",
      "       the function is cos(x) - x.\"\"\"\n",
      "    return cos(x) - x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f_prime(x):\n",
      "    \"\"\"The derivative of f(x) = cos(x) - x\"\"\"\n",
      "    return -sin(x) - 1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def guess(f, f_prime, x0):\n",
      "    \"\"\"A generator function to yield\n",
      "       succesive guesses for Newton's Method.\"\"\"\n",
      "    x = x0\n",
      "    while True:\n",
      "        yield x\n",
      "        x = x - (f(x)/f_prime(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Converge on the answer.\n",
      "x0 = 0\n",
      "iterations = 0\n",
      "places = 6\n",
      "accuracy = 10 ** (-places)\n",
      "for x in guess(f, f_prime, x0):\n",
      "    if abs(f(x)) < accuracy:\n",
      "        break\n",
      "    iterations += 1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print \"The root of x = cos(x) is %.*f radians.\" % (places + 1, x)\n",
      "print \"Calculated to within %f accuracy in %u iteration(s).\" % (accuracy, iterations)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The root of x = cos(x) is 0.7390851 radians.\n",
        "Calculated to within 0.000001 accuracy in 4 iteration(s).\n"
       ]
      }
     ],
     "prompt_number": 39
    }
   ],
   "metadata": {}
  }
 ]
}