Assignments_1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test 1 python: root finding problems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from numpy import cos, sin, exp\n",
    "import numpy \n",
    "import numpy as np\n",
    "from matplotlib.pyplot import plot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline \n",
    "#display directly graphs on notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 1: bisection method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f= lambda x: (2*x**3 + 3*x -1) * cos(x) - x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1612092234725591"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "X = numpy.arange(-1,1,0.01)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116133668>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1140d4208>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt(X,f(X))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def bisection_method(f, a, b, tol, N):\n",
    "    '''\n",
    "    Solve root-finding problem using bisection method\n",
    "    \n",
    "    paragraph with more detail\n",
    "    Hypothesis ``f(a) * f(b) < 0``\n",
    "    \n",
    "    Parameters\n",
    "    ==========\n",
    "    f : function\n",
    "        continuous function \n",
    "    a : real number\n",
    "        lower bound for search interval\n",
    "    b : real number\n",
    "        upper bound for search interval\n",
    "    tol : real number\n",
    "        tolerance error\n",
    "    N : integer\n",
    "        maximum number of iterations\n",
    "    \n",
    "    Returns\n",
    "    =======\n",
    "    tuple (c, n, err) where c is the root, n the number of iterations, err the absolute error\n",
    "    \n",
    "    '''\n",
    "    err = (b-a)/2\n",
    "    k = 0\n",
    "    while err >= tol and k <= N:\n",
    "        x = (b+a)/2\n",
    "        if f(x) == 0:\n",
    "            print('done !')\n",
    "            return x, err, k\n",
    "        if f(a) * f(x) < 0:\n",
    "            b = x\n",
    "        if f(x) * f(b) < 0:\n",
    "            a = x\n",
    "        err = (b-a)/2\n",
    "        \n",
    "        if err < tol or f(x) == 0:\n",
    "            print('Convergence in', k, 'steps')\n",
    "            return x, err, k\n",
    "        k = k+1\n",
    "        \n",
    "    if k == N+1:\n",
    "        print('No convergence, need more iterations')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function __main__.bisection_method>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bisection_method\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence in 16 steps\n"
     ]
    }
   ],
   "source": [
    "c, err, n = bisection_method(f, -1, 1, 1.e-5, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "No convergence, need more iterations\n"
     ]
    },
    {
     "ename": "TypeError",
     "evalue": "'NoneType' object is not iterable",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-15-fa1f66d219e7>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0merr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mn\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mbisection_method\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1.e-5\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m10\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m: 'NoneType' object is not iterable"
     ]
    }
   ],
   "source": [
    "c, err, n = bisection_method(f, -1, 1, 1.e-5, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "No convergence, need more iterations\n"
     ]
    }
   ],
   "source": [
    "res = bisection_method(f, -1, 1, 1.e-5, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res is None"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Assessments Exercise 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test 1: respect guidelines "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from types import FunctionType"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "assert isinstance(bisection_method, FunctionType), 'No function bisection_method defined'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\n    Solve root-finding problem using bisection method\\n    \\n    paragraph with more detail\\n    Hypothesis ``f(a) * f(b) < 0``\\n    \\n    Parameters\\n    ==========\\n    f : function\\n        continuous function \\n    a : real number\\n        lower bound for search interval\\n    b : real number\\n        upper bound for search interval\\n    tol : real number\\n        tolerance error\\n    N : integer\\n        maximum number of iterations\\n    \\n    Returns\\n    =======\\n    tuple (c, n, err) where c is the root, n the number of iterations, err the absolute error\\n    \\n    '"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bisection_method.__doc__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert len(bisection_method.__doc__) > 0, 'No documentation'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "AssertionError",
     "evalue": "No input",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-22-3c1dc54dcdc1>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32massert\u001b[0m \u001b[0;34m'input'\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mbisection_method\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__doc__\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlower\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'No input'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m: No input"
     ]
    }
   ],
   "source": [
    "assert 'input' in bisection_method.__doc__.lower(), 'No input'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "AssertionError",
     "evalue": "No output",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-23-6ce801921e7f>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32massert\u001b[0m \u001b[0;34m'output'\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mbisection_method\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__doc__\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlower\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m'No output'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m: No output"
     ]
    }
   ],
   "source": [
    "assert 'output' in bisection_method.__doc__.lower(), 'No output'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence in 16 steps\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.4385528564453125, 7.62939453125e-06, 16)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bisection_method(f=f, a=-1, b=1, tol=1.e-5, N=100)\n",
    "# test the students have used appropriate parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence in 16 steps\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.4385528564453125, 7.62939453125e-06, 16)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bisection_method(f, -1, 1, N=100, tol=1.e-5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence in 16 steps\n"
     ]
    }
   ],
   "source": [
    "c, err, n = bisection_method(f, -1, 1, 1.e-5, 100)\n",
    "assert c == 0.4385528564453125 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert err < 1.e-5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert 15 <= n <=17"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import inspect "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "AssertionError",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-30-088f627e3705>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32massert\u001b[0m \u001b[0;34m\"break\"\u001b[0m \u001b[0;32min\u001b[0m \u001b[0minspect\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgetsource\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mbisection_method\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m: "
     ]
    }
   ],
   "source": [
    "assert \"break\" in inspect.getsource(bisection_method)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 2: Fixed-point iteration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def fixed_point_iteration(g, x_0, tol, N):\n",
    "    '''\n",
    "    solve the fixed point problem g(x) = x iteratively\n",
    "    \n",
    "    Further details\n",
    "    \n",
    "    Parameters\n",
    "    ==========\n",
    "    g : function\n",
    "        continuous function\n",
    "    x_0 : real number\n",
    "        initial guess\n",
    "    tol : real number\n",
    "        error tolerance\n",
    "    N : integer\n",
    "        maximum number of iterations\n",
    "        \n",
    "    Returns\n",
    "    =======\n",
    "    tuple (c, n, err) where c is the root, n the number of iterations and err the absolute error\n",
    "    \n",
    "    '''\n",
    "    err = 1\n",
    "    k = 0\n",
    "    x = x_0\n",
    "    \n",
    "    while err >= tol and k <= N:\n",
    "        c = g(x)\n",
    "        err = abs(c-x)\n",
    "        if err < tol:\n",
    "            c = x\n",
    "            n = k\n",
    "            print('The fixed-point iteration converged in', n, 'steps')\n",
    "            return c, n, err\n",
    "        k = k+1\n",
    "        x = c\n",
    "    if k == N:\n",
    "        n = k\n",
    "        print('The fixed-point iteration did not converge !')\n",
    "\n",
    "        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10ee75278>]"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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n/j/+cXjjkzoncianFIkmzsH8+X6WrEOTMq1dC+3awZgxsHUrfPwxXHNNeOOU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\nP2q/g/d66TyaAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ee75240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g = lambda x: exp(-x)\n",
    "xx  = np.arange(-1, 1, 0.1)\n",
    "plt(xx, g(xx))\n",
    "plt(xx, xx, 'r--')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The fixed-point iteration converged in 40 steps\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.567143290462223, 40, 8.2179596461173787e-11)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fixed_point_iteration(g, 1, 1.e-10, 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Assessments Exercise 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assert len( fixed_point_iteration.__doc__ ) > 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy.solvers import solve\n",
    "from sympy import Symbol\n",
    "from sympy import exp as expo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = Symbol('x')\n",
    "solution = solve(expo(-x)-x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lw = solution[0]\n",
    "exact_solution = lw.evalf(16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The fixed-point iteration converged in 40 steps\n"
     ]
    }
   ],
   "source": [
    "result = fixed_point_iteration(g, 1, 1.e-10, 100)\n",
    "assert abs( exact_solution - result[0] ) < 1.e-10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercise 3: Newton's method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def newtons_method(f, fp, x_0, tol, N=100):\n",
    "    '''\n",
    "    '''\n",
    "    k=0\n",
    "    err = 1\n",
    "    x = x_0\n",
    "    while k<N and err > tol:\n",
    "        c = x - f(x)/fp(x)\n",
    "        err = abs(x-c)\n",
    "        n = k\n",
    "        if err < tol:\n",
    "            print(\"Newton's method converged in \", k, \" steps\")\n",
    "            return c, n, err\n",
    "        k = k+1\n",
    "        x = c\n",
    "    if k == N:\n",
    "        print(\"Newton's method did not converge !\")\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from numpy import arctan, arctan2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "h = lambda x: arctan(x) -1\n",
    "hp = lambda x: 1/(1 + x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Newton's method converged in  4  steps\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(1.5574077246549023, 4, 2.1656232362943229e-11)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newtons_method(h, hp, 2, 1.e-8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Assessments Exercise 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "AssertionError",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-73-4e05a18a951a>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0;32massert\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnewtons_method\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__doc__\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mstrip\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m>\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m: "
     ]
    }
   ],
   "source": [
    "assert len(newtons_method.__doc__.strip()) > 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Newton's method converged in  4  steps\n"
     ]
    },
    {
     "ename": "AssertionError",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m                            Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-83-5c6dca643235>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      2\u001b[0m \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0matan\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0msol\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0matan\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[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mevalf\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m16\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 4\u001b[0;31m \u001b[0;32massert\u001b[0m \u001b[0mabs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msol\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m1.e-20\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mAssertionError\u001b[0m: "
     ]
    }
   ],
   "source": [
    "c, n, err = newtons_method(h, hp, 2, 1.e-8)\n",
    "from sympy import atan\n",
    "sol = solve(atan(x)-1, x)[0].evalf(16)\n",
    "assert abs(sol - c) < 1.e-20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Newton's method did not converge !\n"
     ]
    }
   ],
   "source": [
    "result = newtons_method(h, hp, 2, 1.e-80, 2)\n",
    "assert result is None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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