Mini-project Report.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**BISECTION METHOD**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's start with the exploration of the bisection method for root finding. \n",
    "\n",
    "This method takes the values of two extremes that we are sure contains one and only one root, finds the middle point and compares the value of the function calculated at the lower extreme with the function at the middle: if the two values so obtained are of opposite sign, i.e. if their product is negative, this means that the root is located between the middle point and the lower extreme and the middle point will become our new upper extreme. Otherwise, the root will be located between the middle point and the upper extreme and the middle point will become our new lower extreme. This operation is repeated until our condition for the loop is true, which is usually dependent on the difference between the upper and lower values. This guarantees that the root will be located in an interval of the chosen width. The simplicity of this method makes it a good fit for most functions, given that we know where the root we are interested in is approximately located, but is limited by our knowledge of the function and can also take a while to compute according to the extremes and the stopping point we choose. The number of steps needed can be predicted by calculating how many times the initial interval will have to be split in two until we get to the desired width. This can be calculated by taking: \n",
    "\n",
    "$$ floor \\left (\\log \\dfrac{ |l-u|}{h} \\right) + 1 $$\n",
    "\n",
    "since the logarithm in base two of the value we usually obtain is not a whole number. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The way it is computed could a problem in the unlikely case that our root is exactly in the middle of our interval. In this case the program will keep looking for the root in the upper half of the interval, since the product between the value of the function in the middle and the value in the lower extreme will not be negative. This will cause the method to converge to a value that is not a root. We can see an example of this with the function $f_1(x) = x^2 - 9$, looking for a root between $x = 2$ and $x = 4$.     "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The predicted number of steps is 28\n",
      "\n",
      " number of steps       lower            upper         difference   \n",
      "       0          2.000000000000   4.000000000000   2.000000000000\n",
      "       1          3.000000000000   4.000000000000   1.000000000000\n",
      "       2          3.500000000000   4.000000000000   0.500000000000\n",
      "       3          3.750000000000   4.000000000000   0.250000000000\n",
      "       4          3.875000000000   4.000000000000   0.125000000000\n",
      "       5          3.937500000000   4.000000000000   0.062500000000\n",
      "       6          3.968750000000   4.000000000000   0.031250000000\n",
      "       7          3.984375000000   4.000000000000   0.015625000000\n",
      "       8          3.992187500000   4.000000000000   0.007812500000\n",
      "       9          3.996093750000   4.000000000000   0.003906250000\n",
      "      10          3.998046875000   4.000000000000   0.001953125000\n",
      "      11          3.999023437500   4.000000000000   0.000976562500\n",
      "      12          3.999511718750   4.000000000000   0.000488281250\n",
      "      13          3.999755859375   4.000000000000   0.000244140625\n",
      "      14          3.999877929688   4.000000000000   0.000122070312\n",
      "      15          3.999938964844   4.000000000000   0.000061035156\n",
      "      16          3.999969482422   4.000000000000   0.000030517578\n",
      "      17          3.999984741211   4.000000000000   0.000015258789\n",
      "      18          3.999992370605   4.000000000000   0.000007629395\n",
      "      19          3.999996185303   4.000000000000   0.000003814697\n",
      "      20          3.999998092651   4.000000000000   0.000001907349\n",
      "      21          3.999999046326   4.000000000000   0.000000953674\n",
      "      22          3.999999523163   4.000000000000   0.000000476837\n",
      "      23          3.999999761581   4.000000000000   0.000000238419\n",
      "      24          3.999999880791   4.000000000000   0.000000119209\n",
      "      25          3.999999940395   4.000000000000   0.000000059605\n",
      "      26          3.999999970198   4.000000000000   0.000000029802\n",
      "      27          3.999999985099   4.000000000000   0.000000014901\n",
      "      28          3.999999992549   4.000000000000   0.000000007451\n",
      "\n",
      " The root is located at x = 3.999999996, where the function has value 7.00e+00\n"
     ]
    }
   ],
   "source": [
    "from math import floor, log\n",
    "\n",
    "def bisection_step(f, bounds):\n",
    "    \"\"\"This function executes a step of the bisection method, taking as arguments the function to evaluate and a tuple containing\n",
    "    upper and lower bounds in this order.\"\"\"\n",
    "    lower, upper = bounds  \n",
    "    middle = (lower + upper)/2 \n",
    "    if f(lower)*f(middle) < 0 : \n",
    "        return (lower, middle)\n",
    "    else:\n",
    "        return (middle, upper)\n",
    "\n",
    "\n",
    "def f_1(x):\n",
    "    return x**2 - 9\n",
    "\n",
    "n, l, u = 0, 2, 4\n",
    "\n",
    "p = floor(log(abs(l-u)/1e-8, 2)) + 1 #this is the formula I described above \n",
    "\n",
    "print(\"The predicted number of steps is\", p)     \n",
    "\n",
    "print(\"\\n {:^15}  {:^15}  {:^15}  {:^15}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}  {:15.12f}  {:15.12f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(l - u) > 1e-8 and n < 1000:\n",
    "    \"\"\"This loop iterates the bisection method step as long as the distance between the two bounds is bigger than 10^-8, \n",
    "    displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = bisection_step(f_1, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}  {:15.12f}  {:15.12f}\".format(n, l, u, u-l)) \n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(abs(u+l)/2, f_1(abs(u+l)/2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To solve this problem, we just have to change our bisection formula step so that it takes the left interval if the product of the values of the function at the lower extremum and the value of the function at the middle of the interval is smaller **or equal** to zero.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The predicted number of steps is 28\n",
      "\n",
      " number of steps       lower            upper         difference   \n",
      "       0          2.000000000000   4.000000000000   2.000000000000\n",
      "       1          2.000000000000   3.000000000000   1.000000000000\n",
      "       2          2.500000000000   3.000000000000   0.500000000000\n",
      "       3          2.750000000000   3.000000000000   0.250000000000\n",
      "       4          2.875000000000   3.000000000000   0.125000000000\n",
      "       5          2.937500000000   3.000000000000   0.062500000000\n",
      "       6          2.968750000000   3.000000000000   0.031250000000\n",
      "       7          2.984375000000   3.000000000000   0.015625000000\n",
      "       8          2.992187500000   3.000000000000   0.007812500000\n",
      "       9          2.996093750000   3.000000000000   0.003906250000\n",
      "      10          2.998046875000   3.000000000000   0.001953125000\n",
      "      11          2.999023437500   3.000000000000   0.000976562500\n",
      "      12          2.999511718750   3.000000000000   0.000488281250\n",
      "      13          2.999755859375   3.000000000000   0.000244140625\n",
      "      14          2.999877929688   3.000000000000   0.000122070312\n",
      "      15          2.999938964844   3.000000000000   0.000061035156\n",
      "      16          2.999969482422   3.000000000000   0.000030517578\n",
      "      17          2.999984741211   3.000000000000   0.000015258789\n",
      "      18          2.999992370605   3.000000000000   0.000007629395\n",
      "      19          2.999996185303   3.000000000000   0.000003814697\n",
      "      20          2.999998092651   3.000000000000   0.000001907349\n",
      "      21          2.999999046326   3.000000000000   0.000000953674\n",
      "      22          2.999999523163   3.000000000000   0.000000476837\n",
      "      23          2.999999761581   3.000000000000   0.000000238419\n",
      "      24          2.999999880791   3.000000000000   0.000000119209\n",
      "      25          2.999999940395   3.000000000000   0.000000059605\n",
      "      26          2.999999970198   3.000000000000   0.000000029802\n",
      "      27          2.999999985099   3.000000000000   0.000000014901\n",
      "      28          2.999999992549   3.000000000000   0.000000007451\n",
      "\n",
      " The root is located at x = 2.999999996, where the function has value -2.24e-08\n"
     ]
    }
   ],
   "source": [
    "def bisection_step_corrected(f, bounds):\n",
    "    \"\"\"This function executes a step of the bisection method modified to include the case where the value of the function in the \n",
    "    middle of the interval is zero, taking as arguments the function to evaluate and a tuple containing upper and lower bounds \n",
    "    in this order.\"\"\"\n",
    "    lower, upper = bounds  \n",
    "    middle = (lower + upper)/2 \n",
    "    if f(lower)*f(middle) <= 0 : \n",
    "        return (lower, middle)\n",
    "    else:\n",
    "        return (middle, upper)\n",
    "\n",
    "n, l, u = 0, 2, 4\n",
    "\n",
    "p = floor(log(abs(l-u)/1e-8, 2)) + 1 \n",
    "\n",
    "print(\"The predicted number of steps is\", p)     \n",
    "\n",
    "print(\"\\n {:^15}  {:^15}  {:^15}  {:^15}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}  {:15.12f}  {:15.12f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(l - u) > 1e-8:\n",
    "    \"\"\"This loop iterates the bisection method step as long as the distance between the two bounds is bigger than 10^-8, \n",
    "    displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = bisection_step_corrected(f_1, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}  {:15.12f}  {:15.12f}\".format(n, l, u, u-l)) \n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format((u+l)/2, f_1((u+l)/2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This method works for any kind of function, given that we know an interval that only contains the root that we're interested in, and we can see this by looking for a root of the function $f_2(x) = x^8 + 2x^3 - 4$ between $x = 1$ and $x = 4$.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The predicted number of steps is 29\n",
      "\n",
      " number of steps       lower            upper         difference   \n",
      "       0          1.000000000000   4.000000000000   3.000000000000\n",
      "       1          1.000000000000   2.500000000000   1.500000000000\n",
      "       2          1.000000000000   1.750000000000   0.750000000000\n",
      "       3          1.000000000000   1.375000000000   0.375000000000\n",
      "       4          1.000000000000   1.187500000000   0.187500000000\n",
      "       5          1.000000000000   1.093750000000   0.093750000000\n",
      "       6          1.046875000000   1.093750000000   0.046875000000\n",
      "       7          1.046875000000   1.070312500000   0.023437500000\n",
      "       8          1.058593750000   1.070312500000   0.011718750000\n",
      "       9          1.058593750000   1.064453125000   0.005859375000\n",
      "      10          1.058593750000   1.061523437500   0.002929687500\n",
      "      11          1.060058593750   1.061523437500   0.001464843750\n",
      "      12          1.060791015625   1.061523437500   0.000732421875\n",
      "      13          1.061157226562   1.061523437500   0.000366210938\n",
      "      14          1.061157226562   1.061340332031   0.000183105469\n",
      "      15          1.061248779297   1.061340332031   0.000091552734\n",
      "      16          1.061248779297   1.061294555664   0.000045776367\n",
      "      17          1.061271667480   1.061294555664   0.000022888184\n",
      "      18          1.061271667480   1.061283111572   0.000011444092\n",
      "      19          1.061277389526   1.061283111572   0.000005722046\n",
      "      20          1.061280250549   1.061283111572   0.000002861023\n",
      "      21          1.061280250549   1.061281681061   0.000001430511\n",
      "      22          1.061280250549   1.061280965805   0.000000715256\n",
      "      23          1.061280608177   1.061280965805   0.000000357628\n",
      "      24          1.061280786991   1.061280965805   0.000000178814\n",
      "      25          1.061280876398   1.061280965805   0.000000089407\n",
      "      26          1.061280921102   1.061280965805   0.000000044703\n",
      "      27          1.061280943453   1.061280965805   0.000000022352\n",
      "      28          1.061280943453   1.061280954629   0.000000011176\n",
      "      29          1.061280943453   1.061280949041   0.000000005588\n",
      "\n",
      " The root is located at x = 1.061280946, where the function has value 2.67e-08\n"
     ]
    }
   ],
   "source": [
    "def f_2(x):\n",
    "    return x**8 + 2*x**3 - 4\n",
    "\n",
    "n, l, u = 0, 1, 4\n",
    "\n",
    "p = floor(log(abs(l-u)/1e-8, 2)) + 1 \n",
    "\n",
    "print(\"The predicted number of steps is\", p)     \n",
    "\n",
    "print(\"\\n {:^15}  {:^15}  {:^15}  {:^15}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}  {:15.12f}  {:15.12f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(l - u) > 1e-8:\n",
    "    \"\"\"This loop iterates the bisection method step as long as the distance between the two bounds is bigger than 10^-8, \n",
    "    displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = bisection_step_corrected(f_2, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}  {:15.12f}  {:15.12f}\".format(n, l, u, u-l)) \n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format((u+l)/2, f_2((u+l)/2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**REGULA FALSI METHOD**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now explore a second method for root finding: the regula falsi. \n",
    "Here we approximate the function with a straight line that goes through the values of the function at the lower and at the upper bound, choosing as \"middle point\" the intercept of that straight line with the x axis. In other words, we are substituting the secant of that function through the two points. The values of the function at the lower bound and \"middle point\" are compared in the same way as in the previous method, returning a new couple of bounds at each step and iterating this with a loop. Using the same condition as before for this kind of loop would be unwise, since we are not halving the interval each time and there is no guarantee that the interval will actually be reduced in size when we get close to the desired root. We can see this if we look for again a root of the function $f_1(x) = x^2 - 9$ between $x = 1$ and $x = 4$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "  number of steps         lower              upper           difference    \n",
      "        0          2.000000000000000  4.000000000000000  2.000000000000000\n",
      "        1          2.833333333333333  4.000000000000000  1.166666666666667\n",
      "        2          2.975609756097561  4.000000000000000  1.024390243902439\n",
      "        3          2.996503496503496  4.000000000000000  1.003496503496504\n",
      "        4          2.999500249875062  4.000000000000000  1.000499750124938\n",
      "        5          2.999928602027703  4.000000000000000  1.000071397972297\n",
      "        6          2.999989800185637  4.000000000000000  1.000010199814363\n",
      "        7          2.999998542881539  4.000000000000000  1.000001457118461\n",
      "        8          2.999999791840176  4.000000000000000  1.000000208159824\n",
      "        9          2.999999970262882  4.000000000000000  1.000000029737118\n",
      "       10          2.999999995751840  4.000000000000000  1.000000004248160\n",
      "       11          2.999999999393120  4.000000000000000  1.000000000606880\n",
      "       12          2.999999999913303  4.000000000000000  1.000000000086697\n",
      "       13          2.999999999987615  4.000000000000000  1.000000000012385\n",
      "       14          2.999999999998231  4.000000000000000  1.000000000001769\n",
      "       15          2.999999999999747  4.000000000000000  1.000000000000253\n",
      "       16          2.999999999999964  4.000000000000000  1.000000000000036\n",
      "       17          2.999999999999995  4.000000000000000  1.000000000000005\n",
      "       18          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       19          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       20          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       21          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       22          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       23          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       24          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       25          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       26          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       27          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       28          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       29          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       30          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       31          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       32          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       33          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       34          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       35          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       36          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       37          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       38          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       39          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       40          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       41          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       42          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       43          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       44          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       45          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       46          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       47          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       48          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       49          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       50          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       51          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       52          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       53          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       54          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       55          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       56          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       57          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       58          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       59          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       60          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       61          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       62          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       63          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       64          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       65          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       66          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       67          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       68          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       69          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       70          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       71          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       72          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       73          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       74          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       75          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       76          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       77          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       78          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       79          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       80          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       81          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       82          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       83          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       84          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       85          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       86          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       87          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       88          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       89          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       90          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       91          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       92          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       93          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       94          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       95          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       96          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       97          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       98          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       99          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       100         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       101         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       102         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       103         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       104         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       105         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       106         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       107         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       108         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       109         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       110         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       111         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       112         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       113         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       114         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       115         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       116         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       117         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       118         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       119         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       120         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       121         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       122         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       123         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       124         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       125         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       126         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       127         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       128         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       129         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       130         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       131         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       132         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       133         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       134         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       135         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       136         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       137         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       138         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       139         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       140         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       141         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       142         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       143         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       144         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       145         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       146         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       147         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       148         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       149         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       150         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       151         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       152         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       153         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       154         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       155         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       156         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       157         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       158         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       159         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       160         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       161         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       162         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       163         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       164         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       165         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       166         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       167         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       168         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       169         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       170         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       171         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       172         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       173         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       174         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       175         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       176         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       177         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       178         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       179         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       180         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       181         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       183         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       184         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       185         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       186         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       187         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       188         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       189         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       190         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       191         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       193         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       196         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       197         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       198         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       199         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       200         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       201         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       202         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       203         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       204         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       205         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       206         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       207         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       208         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       209         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       210         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       211         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       212         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       213         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       214         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       215         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       217         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       218         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       219         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       220         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       226         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       227         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       228         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       230         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       254         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       421         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       422         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       423         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       424         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       428         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       431         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       758         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       759         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       760         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       761         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       762         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       763         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       764         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       765         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       766         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       767         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       768         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       769         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       770         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       771         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       772         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       773         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       774         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       775         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       776         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       777         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       778         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       779         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       780         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       781         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       782         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       783         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       784         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       785         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       786         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       787         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       788         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       789         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       790         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       791         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       792         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       793         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       794         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       795         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       796         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       797         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       798         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       799         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       800         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       801         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       802         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       803         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       804         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       805         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       806         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       807         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       808         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       809         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       810         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       811         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       812         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       813         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       814         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       815         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       816         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       817         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       818         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       819         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       820         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       821         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       822         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       823         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       824         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       825         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       826         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       827         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       828         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       829         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       830         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       831         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       832         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       833         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       834         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       835         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       836         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       837         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       838         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       839         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       840         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       841         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       842         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       843         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       844         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       845         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       846         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       847         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       848         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       849         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       850         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       851         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       852         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       853         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       854         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       855         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       856         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       857         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       858         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       859         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       860         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       861         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       862         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       863         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       864         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       865         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       866         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       867         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       868         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       869         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       870         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       871         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       872         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       873         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       874         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       875         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       876         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       877         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       878         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       879         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       880         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       881         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       882         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       883         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       884         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       885         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       886         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       887         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       888         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       889         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       890         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       891         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       892         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       893         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       894         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       895         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       896         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       897         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       898         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       899         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       900         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       901         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       902         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       903         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       904         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       905         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       906         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       907         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       908         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       909         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       910         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       911         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       912         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       913         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       914         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       915         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       916         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       917         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       918         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       919         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       920         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       921         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       922         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       923         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       924         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       925         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       926         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       927         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       928         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       929         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       930         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       931         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       932         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       933         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       934         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       935         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       936         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       937         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       938         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       939         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       940         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       941         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       942         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       943         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       944         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       945         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       946         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       947         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       948         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       949         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       950         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       951         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       952         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       953         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       954         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       955         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       956         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       957         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       958         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       959         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       960         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       961         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       962         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       963         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       964         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       965         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       966         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       967         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       968         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       969         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       970         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       971         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       972         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       973         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       974         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       975         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       976         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       977         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       978         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       979         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       980         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       981         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       982         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       983         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       984         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       985         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       986         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       987         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       988         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       989         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       990         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       991         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       992         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       993         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       994         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       995         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       996         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       997         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       998         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       999         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "      1000         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "\n",
      " The root is located at x = 3.000000000, where the function has value -1.78e-15\n"
     ]
    }
   ],
   "source": [
    "def regula_falsi_step(f, bounds):\n",
    "    \"\"\"This function executes a step of the bisection method, taking as arguments the function to evaluate and a tuple containing\n",
    "    upper and lower bounds in this order.\"\"\"\n",
    "    lower, upper = bounds\n",
    "    m = (lower*f(upper) - upper*f(lower))/(f(upper) - f(lower))\n",
    "    if f(lower)*f(m) < 0 :\n",
    "        return (lower, m)\n",
    "    else : \n",
    "        return (m, upper)\n",
    "\n",
    "n, l, u = 0, 2, 4\n",
    "\n",
    "print(\"\\n {:^17}  {:^17}  {:^17}  {:^17}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(l - u) > 1e-8 and n < 1000:\n",
    "    \"\"\"This loop iterates the regula falsi method step as long as the distance between the two bounds is bigger than 10^-8, \n",
    "    displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = regula_falsi_step(f_1, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l)) \n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_1(l)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Even if the lower bound is exactly the value we were looking for, the loop continues to run since the difference between upper and lower bounds is still bigger than the value we have chosen. A better measure of the \"closeness\" to the exact value of the root is the value itself of the function, which should be close enough to zero in the chosen point to be satisfying. Let's see the same function with the new condition for the loop: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "  number of steps         lower              upper           difference    \n",
      "        0          2.000000000000000  4.000000000000000  2.000000000000000\n",
      "        1          2.833333333333333  4.000000000000000  1.166666666666667\n",
      "        2          2.975609756097561  4.000000000000000  1.024390243902439\n",
      "        3          2.996503496503496  4.000000000000000  1.003496503496504\n",
      "        4          2.999500249875062  4.000000000000000  1.000499750124938\n",
      "        5          2.999928602027703  4.000000000000000  1.000071397972297\n",
      "        6          2.999989800185637  4.000000000000000  1.000010199814363\n",
      "        7          2.999998542881539  4.000000000000000  1.000001457118461\n",
      "        8          2.999999791840176  4.000000000000000  1.000000208159824\n",
      "        9          2.999999970262882  4.000000000000000  1.000000029737118\n",
      "       10          2.999999995751840  4.000000000000000  1.000000004248160\n",
      "       11          2.999999999393120  4.000000000000000  1.000000000606880\n",
      "\n",
      " The root is located at x = 2.999999999, where the function has value -3.64e-09\n"
     ]
    }
   ],
   "source": [
    "n, l, u = 0, 2, 4 \n",
    "\n",
    "print(\"\\n {:^17}  {:^17}  {:^17}  {:^17}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(f_1(l)) > 1e-8 and abs(f_1(u)) > 1e-8:\n",
    "    \"\"\"This loop iterates the regula falsi method step as long as the value of the function at both the upper and lower bounds is\n",
    "    bigger than 10^-8, displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = regula_falsi_step(f_1, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l)) \n",
    "    \n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_1(l)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This also shows that the new method is quicker than the bisection one, since it takes a smaller number of steps for the same function. It also solves the problem that brought us to modify the bisection step function, since if one of the extremes is close enough to the actual root it will be identified immediately. This situation will change quite rapidly the less close the function examined is to linear behaviour. Let's take for example $f_2(x) = x^8 + 2x^3 - 4$, which between $x = 1$ and $x = 4$ took 29 steps with the bisection method. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "  number of steps         lower              upper           difference    \n",
      "        0          1.000000000000000  2.000000000000000  1.000000000000000\n",
      "        1          1.003717472118959  2.000000000000000  0.996282527881041\n",
      "        2          1.007227303414260  2.000000000000000  0.992772696585740\n",
      "        3          1.010539116534168  2.000000000000000  0.989460883465832\n",
      "        4          1.013662300100489  2.000000000000000  0.986337699899511\n",
      "        5          1.016605991121609  2.000000000000000  0.983394008878391\n",
      "        6          1.019379060278457  2.000000000000000  0.980620939721543\n",
      "        7          1.021990099973159  2.000000000000000  0.978009900026841\n",
      "        8          1.024447415001483  2.000000000000000  0.975552584998517\n",
      "        9          1.026759015688094  2.000000000000000  0.973240984311906\n",
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      "       252         1.061280942656268  2.000000000000000  0.938719057343732\n",
      "       253         1.061280942800203  2.000000000000000  0.938719057199797\n",
      "       254         1.061280942934616  2.000000000000000  0.938719057065384\n",
      "       255         1.061280943060135  2.000000000000000  0.938719056939865\n",
      "       256         1.061280943177350  2.000000000000000  0.938719056822650\n",
      "       257         1.061280943286810  2.000000000000000  0.938719056713190\n",
      "       258         1.061280943389027  2.000000000000000  0.938719056610973\n",
      "       259         1.061280943484481  2.000000000000000  0.938719056515519\n",
      "       260         1.061280943573620  2.000000000000000  0.938719056426380\n",
      "       261         1.061280943656861  2.000000000000000  0.938719056343139\n",
      "       262         1.061280943734595  2.000000000000000  0.938719056265405\n",
      "       263         1.061280943807186  2.000000000000000  0.938719056192814\n",
      "       264         1.061280943874974  2.000000000000000  0.938719056125026\n",
      "       265         1.061280943938277  2.000000000000000  0.938719056061723\n",
      "       266         1.061280943997392  2.000000000000000  0.938719056002608\n",
      "       267         1.061280944052595  2.000000000000000  0.938719055947405\n",
      "       268         1.061280944104146  2.000000000000000  0.938719055895854\n",
      "       269         1.061280944152287  2.000000000000000  0.938719055847713\n",
      "       270         1.061280944197242  2.000000000000000  0.938719055802758\n",
      "       271         1.061280944239223  2.000000000000000  0.938719055760777\n",
      "       272         1.061280944278426  2.000000000000000  0.938719055721574\n",
      "       273         1.061280944315036  2.000000000000000  0.938719055684964\n",
      "\n",
      " The root is located at x = 1.061280944, where the function has value -9.76e-09\n"
     ]
    }
   ],
   "source": [
    "In this report, we are analyzing data from neutron diffraction data collected at ISIS, the UK’s national neutron source, since it is a kind of radiation sensitive to magnetism. The material used is methylammonium cobalt(II) formate (CH3NH3Co(HCO2)3).n, l, u = 0, 1, 2\n",
    "\n",
    "print(\"\\n {:^17}  {:^17}  {:^17}  {:^17}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(f_2(l)) > 1e-8 and abs(f_2(u)) > 1e-8:\n",
    "    \"\"\"This loop iterates the regula falsi method step as long as the value of the function at both the upper and lower bounds is\n",
    "    bigger than 10^-8, displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = regula_falsi_step(f_2, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l)) \n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_2(l)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Clearly, with 226 steps this method is not very efficient for this kind of function. What about transcendental functions like exponential and sine? As an example, I've taken the function $f_3(x) = exp(x) - 1$ between $x = -1$ and $x = 1$, and the function $f_4(x) = sin(x) - 1/2$ between $ x = \\dfrac {\\pi}{2} $  and $x = \\pi$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "  number of steps         lower              upper           difference    \n",
      "        0          -1.000000000000000  1.000000000000000  2.000000000000000\n",
      "        1          -0.462117157260010  1.000000000000000  1.462117157260010\n",
      "        2          -0.203030831979271  1.000000000000000  1.203030831979271\n",
      "        3          -0.086811129005843  1.000000000000000  1.086811129005843\n",
      "        4          -0.036646538125036  1.000000000000000  1.036646538125036\n",
      "        5          -0.015383022923406  1.000000000000000  1.015383022923406\n",
      "        6          -0.006441740127726  1.000000000000000  1.006441740127726\n",
      "        7          -0.002694776300347  1.000000000000000  1.002694776300347\n",
      "        8          -0.001126825652959  1.000000000000000  1.001126825652959\n",
      "        9          -0.000471099943012  1.000000000000000  1.000471099943012\n",
      "       10          -0.000196941337446  1.000000000000000  1.000196941337446\n",
      "       11          -0.000082327916827  1.000000000000000  1.000082327916827\n",
      "       12          -0.000034415310270  1.000000000000000  1.000034415310270\n",
      "       13          -0.000014386457840  1.000000000000000  1.000014386457840\n",
      "       14          -0.000006013884357  1.000000000000000  1.000006013884357\n",
      "       15          -0.000002513945469  1.000000000000000  1.000002513945469\n",
      "       16          -0.000001050888065  1.000000000000000  1.000001050888065\n",
      "       17          -0.000000439295742  1.000000000000000  1.000000439295742\n",
      "       18          -0.000000183635862  1.000000000000000  1.000000183635862\n",
      "       19          -0.000000076764069  1.000000000000000  1.000000076764069\n",
      "       20          -0.000000032089169  1.000000000000000  1.000000032089169\n",
      "       21          -0.000000013414020  1.000000000000000  1.000000013414020\n",
      "       22          -0.000000005607373  1.000000000000000  1.000000005607373\n",
      "\n",
      " The root is located at x = -0.000000006, where the function has value -5.61e-09\n"
     ]
    }
   ],
   "source": [
    "from math import exp\n",
    "\n",
    "def f_3(x):\n",
    "    return exp(x) - 1\n",
    "\n",
    "n, l, u = 0, -1, 1\n",
    "\n",
    "print(\"\\n {:^17}  {:^17}  {:^17}  {:^17}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(f_3(l)) > 1e-8 and abs(f_3(u)) > 1e-8:\n",
    "    \"\"\"This loop iterates the regula falsi method step as long as the value of the function at both the upper and lower bounds is\n",
    "    bigger than 10^-8, displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = regula_falsi_step(f_3, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l)) \n",
    "    \n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_3(l)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "  number of steps         lower              upper           difference    \n",
      "        0          1.570796326794897  3.141592653589793  1.570796326794897\n",
      "        1          2.356194490192345  3.141592653589793  0.785398163397448\n",
      "        2          2.586232286319997  3.141592653589793  0.555360367269796\n",
      "        3          2.614934664598647  3.141592653589793  0.526657988991146\n",
      "        4          2.617708122598263  3.141592653589793  0.523884530991530\n",
      "        5          2.617967265862889  3.141592653589793  0.523625387726904\n",
      "        6          2.617991400322655  3.141592653589793  0.523601253267138\n",
      "        7          2.617993647319121  3.141592653589793  0.523599006270672\n",
      "        8          2.617993856515818  3.141592653589793  0.523598797073975\n",
      "        9          2.617993875992102  3.141592653589793  0.523598777597691\n",
      "\n",
      " The root is located at x = 2.617993876, where the function has value 1.73e-09\n"
     ]
    }
   ],
   "source": [
    "from math import sin, pi\n",
    "\n",
    "def f_4(x):\n",
    "    return sin(x) - 0.5\n",
    "\n",
    "n, l, u = 0, pi/2, pi\n",
    "\n",
    "print(\"\\n {:^17}  {:^17}  {:^17}  {:^17}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(f_4(l)) > 1e-8 and abs(f_4(u)) > 1e-8:\n",
    "    \"\"\"This loop iterates the regula falsi method step as long as the value of the function at both the upper and lower bounds is\n",
    "    bigger than 10^-8, displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = regula_falsi_step(f_4, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l)) \n",
    "    \n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_4(l)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The method doesn't appear to have particular difficulties with solving equations containing transcendental functions, since when we are in a neighbourhood of such a function is closely approximated by the first couple of terms of its Taylor expansion, which makes it close enough to the behaviour of a linear function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**NEWTON-RAPHSON METHOD**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The third and last method for root finding we are going to explore is the Newton-Raphson. Each steps uses an estimate of the root, the value of the function and the value of the derivative of the function at the estimate to find a more accurate estimate. The way this works is by subtracting the ratio of the two values of the function and of the derivative at the estimate from the given estimate, that is we move of an amount proportional to the value at the estimate in the direction opposite to the rate of change of the function. This means that we are moving in the direction where the function increases if we are at a negative value and in the direction where the function decreases if we are at a positive value, thus we are moving towards the zero! Since we have only one value and not two bounds, I will use the closeness to zero of the absolute value of the function as the stopping point of the loop that iterates a step of the method. To focus on the method alone and not being influenced by the method used to find the derivative, I am using the textbook derivatives for these functions.  \n",
    "\n",
    "This method presents a quicker rate of convergence when compared to the previous ones, as we can see by evaluating once more the functions $f_1(x) = x^2 - 9$ close to $ x = 2$ and $f_2(x) =  x^8 + 2x^3 - 4$ close to $x = 1$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " number of steps     estimate    \n",
      "       0          2.000000000000\n",
      "       1          3.250000000000\n",
      "       2          3.009615384615\n",
      "       3          3.000015360039\n",
      "       4          3.000000000039\n",
      "\n",
      " The root is located at x = 3.000000000, where the function has value 2.36e-10\n"
     ]
    }
   ],
   "source": [
    "def newton_step(f, fp, x0):\n",
    "    \"\"\"\"This function executes one step of the Newton-Rhapson method, taking the estimate, the function and the \n",
    "    derivative of the function as arguments. It returns the difference between the estimate and the ratio of the values\n",
    "    of the function at the estimate and the derivative of the function at the estimate.\"\"\"\n",
    "    return x0 - f(x0)/fp(x0)\n",
    "\n",
    "def fp_1(x):\n",
    "    return 2*x\n",
    "    \n",
    "n, l = 0, 2    \n",
    "    \n",
    "print(\"\\n {:^15}  {:^15}\".format(\"number of steps\", \"estimate\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "while abs(f_1(l)) > 1e-8:\n",
    "    \"\"\"This loop iterates the Newton-Raphson method step as long as the value of the function at the estimate is bigger \n",
    "    than 10^-8, displaying the number of step executed and the value of the estimate for each step.\"\"\" \n",
    "    l = newton_step(f_1, fp_1, l)\n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_1(l)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " number of steps     estimate    \n",
      "       0          1.000000000000\n",
      "       1          1.071428571429\n",
      "       2          1.061529756456\n",
      "       3          1.061281096762\n",
      "       4          1.061280944832\n",
      "\n",
      " The root is located at x = 1.061280945, where the function has value 1.07e-12\n"
     ]
    }
   ],
   "source": [
    "def fp_2(x):\n",
    "    return 8*x**7 + 6*x**2\n",
    "    \n",
    "n, l = 0, 1    \n",
    "    \n",
    "print(\"\\n {:^15}  {:^15}\".format(\"number of steps\", \"estimate\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "while abs(f_2(l)) > 1e-8:\n",
    "    \"\"\"This loop iterates the Newton-Raphson method step as long as the value of the function at the estimate is bigger \n",
    "    than 10^-8, displaying the number of step executed and the value of the estimate for each step.\"\"\" \n",
    "    l = newton_step(f_2, fp_2, l)\n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_2(l)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Efficiency when compared to the previous methods is clear-cut, as we can see from the table below: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Table of comparison of the three methods**\n",
    "\n",
    "$f_1(x) = x^2 - 9$ with lower bound = 2 and upper bound = 4 \n",
    "\n",
    "| Method used| Number of iterations |  Root Found  | Magnitude of the value of the function |\n",
    "|----|----|----|----|----|\n",
    "| Bisection   | 28  | 2.999999992549 | 1e-08 |\n",
    "| Regula Falsi |  11 | 2.999999999393| 1e-09 |\n",
    "| Newton-Raphson  | 4  |  3.000000000039 | 1e-10 |\n",
    "\n",
    "\n",
    "$f_2(x) =  x^8 + 2x^3 - 4$ with lower bound = 1 and upper bound = 2\n",
    "\n",
    "| Method used| Number of iterations |  Root Found  | Magnitude of the value of the function |\n",
    "|----|----|----|----|----|\n",
    "| Bisection   | 29  | 1.061280949041 | 1e-08 |\n",
    "| Regula Falsi |  226 | 1.061280944315 | 1e-09 |\n",
    "| Newton-Raphson  | 4  | 1.061280944832 | 1e-12 |\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What could happen with different functions, though, is that the Newton-Raphson method returns a value that is not accurate.  \n",
    "\n",
    "An example of this is provided by the function $f_5(x) = \\exp (x) - 10^{-10}$, due to the fact that it converges very slowly so that after a certain point even if the function doesn't change significantly enough, the estimate point is still changing quite significantly. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " number of steps     estimate    \n",
      "       0          1.000000000000\n",
      "       1          0.000000000037\n",
      "       2         -0.999999999863\n",
      "       3         -1.999999999591\n",
      "       4         -2.999999998852\n",
      "       5         -3.999999996844\n",
      "       6         -4.999999991384\n",
      "       7         -5.999999976543\n",
      "       8         -6.999999936200\n",
      "       9         -7.999999826537\n",
      "      10         -8.999999528441\n",
      "      11         -9.999998718133\n",
      "      12         -10.999996515489\n",
      "      13         -11.999990528096\n",
      "      14         -12.999974252771\n",
      "      15         -13.999930012571\n",
      "      16         -14.999809760559\n",
      "      17         -15.999482921005\n",
      "      18         -16.998594769316\n",
      "      19         -17.996182665985\n",
      "      20         -18.989641685896\n",
      "\n",
      " The root is located at x = -18.989641686, where the function has value 5.56e-09\n",
      "\n",
      " The correct value of the root is -23.025850929940457\n"
     ]
    }
   ],
   "source": [
    "def f_5(x):\n",
    "    return exp(x) - 0.0000000001\n",
    "\n",
    "def fp_5(x):\n",
    "    return exp(x)\n",
    "\n",
    "n, l = 0, 1\n",
    "    \n",
    "print(\"\\n {:^15}  {:^15}\".format(\"number of steps\", \"estimate\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "while abs(f_5(l)) > 1e-8:\n",
    "    \"\"\"This loop iterates theNewton-Raphson method step as long as the value of the function at the estimate is bigger \n",
    "    than 10^-8, displaying the number of step executed and the value of the estimate for each step.\"\"\" \n",
    "    l = newton_step(f_5, fp_5, l)\n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_5(l)))\n",
    "\n",
    "print(\"\\n The correct value of the root is\", log(0.0000000001))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To solve this problem, we just have to change the condition for stopping the iteration fo the step from one linked to the value of the function to one linked to how much the function has changed from the previous step. Once we've modified this, the algorithm functions as intended. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " number of steps     estimate    \n",
      "       0          1.000000000000\n",
      "       1          0.000000000037\n",
      "       2         -0.999999999863\n",
      "       3         -1.999999999591\n",
      "       4         -2.999999998852\n",
      "       5         -3.999999996844\n",
      "       6         -4.999999991384\n",
      "       7         -5.999999976543\n",
      "       8         -6.999999936200\n",
      "       9         -7.999999826537\n",
      "      10         -8.999999528441\n",
      "      11         -9.999998718133\n",
      "      12         -10.999996515489\n",
      "      13         -11.999990528096\n",
      "      14         -12.999974252771\n",
      "      15         -13.999930012571\n",
      "      16         -14.999809760559\n",
      "      17         -15.999482921005\n",
      "      18         -16.998594769316\n",
      "      19         -17.996182665985\n",
      "      20         -18.989641685896\n",
      "      21         -19.971977379161\n",
      "      22         -20.924801547130\n",
      "      23         -21.802473555147\n",
      "      24         -22.508238809246\n",
      "      25         -22.912296921644\n",
      "      26         -23.019640937364\n",
      "      27         -23.025831687788\n",
      "      28         -23.025850929755\n",
      "      29         -23.025850929940\n",
      "\n",
      " The root is located at x = -23.025850930, where the function has value -3.88e-26\n",
      "\n",
      " The correct value of the root is -23.025850929940457\n"
     ]
    }
   ],
   "source": [
    "n, l, u = 0, 1, 100    \n",
    "    \n",
    "print(\"\\n {:^15}  {:^15}\".format(\"number of steps\", \"estimate\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "while abs(l - u) > 1e-8:\n",
    "    \"\"\"This loop iterates the Newton-Raphson method step as long as the value of the estimate is changing \n",
    "    of an amount more than 10^-8, displaying the number of step executed and the value of the estimate for \n",
    "    each step.\"\"\" \n",
    "    u = l\n",
    "    l = newton_step(f_5, fp_5, l)\n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_5(l)))\n",
    "\n",
    "print(\"\\n The correct value of the root is\", log(0.0000000001))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This method can also present a considerably longer number of steps if we start our search very close to zero. In this case the ratio between the values of the function and the derivative will be quite small, thus the algorithm will move quite slowly in the direction of the actual root. We can see an example of this with the function $f_1(x) = x^2 - 9$, starting our search around several negative powers of $10$ and comparing the number of steps.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " Starting from 1.00e+00, we find the root after 6 steps\n",
      "\n",
      " Starting from 1.00e-01, we find the root after 10 steps\n",
      "\n",
      " Starting from 1.00e-02, we find the root after 13 steps\n",
      "\n",
      " Starting from 1.00e-03, we find the root after 16 steps\n",
      "\n",
      " Starting from 1.00e-04, we find the root after 20 steps\n",
      "\n",
      " Starting from 1.00e-05, we find the root after 23 steps\n",
      "\n",
      " Starting from 1.00e-06, we find the root after 26 steps\n",
      "\n",
      " Starting from 1.00e-07, we find the root after 30 steps\n",
      "\n",
      " Starting from 1.00e-08, we find the root after 33 steps\n",
      "\n",
      " Starting from 1.00e-09, we find the root after 36 steps\n",
      "\n",
      " Starting from 1.00e-10, we find the root after 40 steps\n",
      "\n",
      " Starting from 1.00e-11, we find the root after 43 steps\n",
      "\n",
      " Starting from 1.00e-12, we find the root after 46 steps\n",
      "\n",
      " Starting from 1.00e-13, we find the root after 50 steps\n",
      "\n",
      " Starting from 1.00e-14, we find the root after 53 steps\n",
      "\n",
      " Starting from 1.00e-15, we find the root after 56 steps\n",
      "\n",
      " Starting from 1.00e-16, we find the root after 60 steps\n"
     ]
    }
   ],
   "source": [
    "for i in range(17):\n",
    "    \"\"\"This loop uses the Newton-Raphson method to find a root for different starting points, each time going\n",
    "    a power of ten closer to zero. It then prints the starting point and the total number of steps executed before \n",
    "    the root was found.\"\"\"\n",
    "    l = 10**(-i)\n",
    "    n = 0\n",
    "    u = 100\n",
    "    while abs(l - u) > 1e-8:\n",
    "        \"\"\"This loop iterates the Newton-Raphson method step as long as the value of the estimate is changing \n",
    "        of an amount more than 10^-8.\"\"\" \n",
    "        u = l\n",
    "        l = newton_step(f_1, fp_1, l)\n",
    "        n += 1  \n",
    "    print(\"\\n Starting from {:5.2e}, we find the root after {:^d} steps\".format(10**(-i), n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the number of steps increases each time we get closer to zero, something that didn't happen with the other methods.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**The black sheep of root finding: the asymptote**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A problem that is common to all root finding methods is to find a root that is close to an asymptote, which can either cause a non convergence, as in the case of the regula falsi method, or a convergence to a value that is not a root, as in the case of the bisection method. \n",
    "\n",
    "We can see this by looking for a root of the function $f_6 (x) = \\dfrac {x - 5}{x - 4}$ between $x = 3$ and $x = 6$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "  number of steps         lower              upper           difference    \n",
      "        0          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "        1          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "        2          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "        3          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "        4          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "        5          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "        6          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "        7          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "        8          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "        9          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       10          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       11          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       12          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       13          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       14          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       15          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       16          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       17          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       18          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       19          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       20          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       21          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       22          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       23          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       24          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       25          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       26          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       27          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       28          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       29          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       30          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       31          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       32          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       33          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       34          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       35          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       36          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       37          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       38          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       39          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       40          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       41          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       42          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       43          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       44          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       45          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       46          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       47          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       48          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       49          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       50          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       51          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       52          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       53          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       54          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       55          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       56          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       57          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       58          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       59          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       60          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       61          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       62          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       63          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       64          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       65          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       66          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       67          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       68          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       69          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       70          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       71          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       72          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       73          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       74          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       75          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       76          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       77          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       78          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       79          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       80          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       81          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       82          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       83          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       84          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       85          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       86          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       87          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       88          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       89          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       90          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       91          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       92          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       93          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       94          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       95          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       96          3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       97          7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       98          3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       99          6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       100         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       101         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       102         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       103         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       104         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       105         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       106         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       107         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       108         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       109         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       110         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       111         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       112         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       113         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       114         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       115         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       116         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       117         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       118         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       119         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       120         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       121         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       122         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       123         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       124         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       125         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       126         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       127         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       128         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       129         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       130         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       131         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       132         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       133         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       134         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       135         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       136         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       137         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       138         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       139         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       140         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       141         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       142         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       143         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       144         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       145         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       146         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       147         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       148         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       149         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       150         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       151         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       152         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       153         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       154         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       155         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       156         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       157         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       158         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       159         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       160         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       161         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       162         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       163         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       164         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       165         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       166         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       167         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       168         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       169         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       170         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       171         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       172         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       173         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       174         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       175         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       176         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       177         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       178         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       179         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       180         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       181         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       182         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       183         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       184         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       185         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       186         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       187         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       188         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       189         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       190         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       191         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       192         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       193         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       194         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       195         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       196         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       197         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       198         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       199         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       200         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       201         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       202         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       203         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       204         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       205         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       206         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       207         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       208         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       209         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       210         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       211         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       212         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       213         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       214         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       215         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       216         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       217         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       218         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       219         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       220         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       221         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       222         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       223         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       224         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       225         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       226         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       227         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       228         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       229         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       230         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       231         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       232         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       233         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       234         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       235         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       236         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       237         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       238         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       239         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       240         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       241         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       242         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       243         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       244         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       245         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       246         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       247         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       248         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       249         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       250         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       251         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       252         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       253         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       254         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       255         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       256         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       257         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       258         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       259         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       260         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       261         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       262         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       263         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       264         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       265         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       266         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       267         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       268         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       269         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       270         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       271         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       272         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       273         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       274         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       275         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       276         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       277         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       278         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       279         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       280         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       281         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       282         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       283         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       284         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       285         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       286         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       287         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       288         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       289         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       290         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       291         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       292         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       293         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       294         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       295         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       296         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       297         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       298         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       299         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       300         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       301         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       302         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       303         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       304         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       305         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       306         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       307         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       308         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       309         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       310         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       311         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       312         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       313         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       314         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       315         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       316         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       317         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       318         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       319         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       320         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       321         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       322         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       323         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       324         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       325         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       326         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       327         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       328         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       329         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       330         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       331         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       332         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       333         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       334         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       335         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       336         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       337         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       338         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       339         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       340         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       341         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       342         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       343         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       344         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       345         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       346         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       347         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       348         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       349         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       350         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       351         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       352         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       353         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       354         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       355         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       356         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       357         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       358         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       359         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       360         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       361         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       362         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       363         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       364         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       365         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       366         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       367         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       368         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       369         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       370         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       371         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       372         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       373         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       374         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       375         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       376         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       377         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       378         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       379         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       380         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       381         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       382         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       383         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       384         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       385         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       386         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       387         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       388         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       389         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       390         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       391         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       392         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       393         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       394         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       395         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       396         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       397         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       398         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       399         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       400         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       401         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       402         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       403         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       404         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       405         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       406         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       407         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       408         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       409         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       410         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       411         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       412         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       413         7.000000000000000  6.000000000000000  -1.000000000000000\n",
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      "       415         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       416         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       417         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       418         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       419         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       420         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       421         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       422         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       423         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       424         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       425         7.000000000000000  6.000000000000000  -1.000000000000000\n",
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      "       427         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       428         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       429         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       430         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       431         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       434         3.000000000000001  6.000000000000000  2.999999999999999\n",
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      "       439         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       450         3.000000000000001  6.000000000000000  2.999999999999999\n",
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      "       459         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       461         7.000000000000000  6.000000000000000  -1.000000000000000\n",
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      "       463         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       469         7.000000000000000  6.000000000000000  -1.000000000000000\n",
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      "       471         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       483         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       499         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       515         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       555         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       563         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       565         7.000000000000000  6.000000000000000  -1.000000000000000\n",
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      "       568         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       569         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       570         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       571         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       572         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       573         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       574         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       575         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       576         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       577         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       578         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       579         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       580         3.000000000000000  6.000000000000000  3.000000000000000\n",
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      "       880         3.000000000000000  6.000000000000000  3.000000000000000\n",
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      "       884         3.000000000000000  6.000000000000000  3.000000000000000\n",
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      "       888         3.000000000000000  6.000000000000000  3.000000000000000\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "       889         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       890         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       891         6.999999999999999  6.000000000000000  -0.999999999999999\n",
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      "       894         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       895         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       896         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       897         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       898         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       899         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       900         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       901         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       902         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       903         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       904         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       905         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       906         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       907         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       908         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       909         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       910         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       911         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       912         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       913         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       914         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       915         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       916         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       917         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       918         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       919         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       920         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       921         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       922         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       923         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       924         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       925         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       926         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       927         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       928         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       929         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       930         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       931         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       932         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       933         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       934         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       935         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       936         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       937         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       938         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       939         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       940         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       941         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       942         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       943         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       944         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       945         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       946         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       947         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       948         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       949         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       950         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       951         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       952         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       953         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       954         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       955         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       956         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       957         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       958         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       959         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       960         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       961         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       962         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       963         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       964         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       965         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       966         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       967         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       968         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       969         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       970         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       971         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       972         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       973         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       974         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       975         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       976         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       977         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       978         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       979         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       980         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       981         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       982         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       983         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       984         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       985         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       986         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       987         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       988         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       989         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       990         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       991         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       992         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       993         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       994         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       995         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "       996         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "       997         7.000000000000000  6.000000000000000  -1.000000000000000\n",
      "       998         3.000000000000001  6.000000000000000  2.999999999999999\n",
      "       999         6.999999999999999  6.000000000000000  -0.999999999999999\n",
      "      1000         3.000000000000000  6.000000000000000  3.000000000000000\n",
      "\n",
      " The root is located at x = 3.000000000, where the function has value 2.00e+00\n"
     ]
    }
   ],
   "source": [
    "def f_6(x):\n",
    "    return (x-5)/(x-4)\n",
    "\n",
    "n, l, u = 0, 3, 6 \n",
    "\n",
    "print(\"\\n {:^17}  {:^17}  {:^17}  {:^17}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(f_6(l)) > 1e-8 and abs(f_6(u)) > 1e-8 and n < 1000:\n",
    "    \"\"\"This loop iterates the regula falsi method step as long as the value of the function at both the upper and lower bounds is\n",
    "    bigger than 10^-8, displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = regula_falsi_step(f_6, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l)) \n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_6(l))) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The regula falsi has trouble with this because any function in the neighbourhood of one of its asymptotes has a behaviour similar to a high order polynomial."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The predicted number of steps is 29\n",
      "\n",
      " number of steps       lower            upper         difference   \n",
      "       0          3.000000000000   6.000000000000   3.000000000000\n",
      "       1          3.000000000000   4.500000000000   1.500000000000\n",
      "       2          3.750000000000   4.500000000000   0.750000000000\n",
      "       3          3.750000000000   4.125000000000   0.375000000000\n",
      "       4          3.937500000000   4.125000000000   0.187500000000\n",
      "       5          3.937500000000   4.031250000000   0.093750000000\n",
      "       6          3.984375000000   4.031250000000   0.046875000000\n",
      "       7          3.984375000000   4.007812500000   0.023437500000\n",
      "       8          3.996093750000   4.007812500000   0.011718750000\n",
      "       9          3.996093750000   4.001953125000   0.005859375000\n",
      "      10          3.999023437500   4.001953125000   0.002929687500\n",
      "      11          3.999023437500   4.000488281250   0.001464843750\n",
      "      12          3.999755859375   4.000488281250   0.000732421875\n",
      "      13          3.999755859375   4.000122070312   0.000366210938\n",
      "      14          3.999938964844   4.000122070312   0.000183105469\n",
      "      15          3.999938964844   4.000030517578   0.000091552734\n",
      "      16          3.999984741211   4.000030517578   0.000045776367\n",
      "      17          3.999984741211   4.000007629395   0.000022888184\n",
      "      18          3.999996185303   4.000007629395   0.000011444092\n",
      "      19          3.999996185303   4.000001907349   0.000005722046\n",
      "      20          3.999999046326   4.000001907349   0.000002861023\n",
      "      21          3.999999046326   4.000000476837   0.000001430511\n",
      "      22          3.999999761581   4.000000476837   0.000000715256\n",
      "      23          3.999999761581   4.000000119209   0.000000357628\n",
      "      24          3.999999940395   4.000000119209   0.000000178814\n",
      "      25          3.999999940395   4.000000029802   0.000000089407\n",
      "      26          3.999999985099   4.000000029802   0.000000044703\n",
      "      27          3.999999985099   4.000000007451   0.000000022352\n",
      "      28          3.999999996275   4.000000007451   0.000000011176\n",
      "      29          3.999999996275   4.000000001863   0.000000005588\n",
      "\n",
      " The root is located at x = 3.999999999, where the function has value 1.07e+09\n"
     ]
    }
   ],
   "source": [
    "n, l, u = 0, 3, 6\n",
    "\n",
    "p = floor(log(abs(l-u)/1e-8, 2)) + 1 \n",
    "\n",
    "print(\"The predicted number of steps is\", p)     \n",
    "\n",
    "print(\"\\n {:^15}  {:^15}  {:^15}  {:^15}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}  {:15.12f}  {:15.12f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(l - u) > 1e-8:\n",
    "    \"\"\"This loop iterates the bisection method step as long as the distance between the two bounds is bigger than 10^-8, \n",
    "    displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = bisection_step_corrected(f_6, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}  {:15.12f}  {:15.12f}\".format(n, l, u, u-l)) \n",
    "    \n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format((l+u)/2, f_6((l+u)/2))) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The bisection method has trouble with this because the function assumes values of opposite sign at the left of the asymptote when compared to the right, exactly like a root.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " number of steps     estimate    \n",
      "       0          3.000000000000\n",
      "       1          1.000000000000\n",
      "       2         -11.000000000000\n",
      "       3         -251.000000000000\n",
      "       4         -65530.999999999993\n",
      "       5         -4294967290.999999046326\n",
      "       6         -18446744073709543424.000000000000\n",
      "       7         -340282366920938161231919703774474534912.000000000000\n",
      "       8         -115792089237315989735501319857932638482122164996827441197474379810430211063808.000000000000\n",
      "       9         -13407807929942549465471389561312667086994292072327229977483347439645365422331853474425488498520600322433332280977453460721137894551709517822781316115988480.000000000000\n",
      "      10         -179769313486230313435132416858223013283261354317548374104623402464906114175265803317097681185405433648022198593022696178492809485617460480132534858739531251307816991132756480472671530906413122895311358559773208870689834174231107149755221833183778348295978703936325493682566845454613065195398408909457870290944.000000000000\n"
     ]
    },
    {
     "ename": "OverflowError",
     "evalue": "(34, 'Result too large')",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mOverflowError\u001b[0m                             Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-16-a036fa38011d>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m     13\u001b[0m     value of the estimate for each step.\"\"\" \n\u001b[0;32m     14\u001b[0m     \u001b[0mu\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0ml\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 15\u001b[1;33m     \u001b[0ml\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnewton_step\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf_6\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mfp_6\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ml\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     16\u001b[0m     \u001b[0mn\u001b[0m \u001b[1;33m+=\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     17\u001b[0m     \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"{:^15d}  {:15.12f}\"\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ml\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m<ipython-input-9-be50279f2f69>\u001b[0m in \u001b[0;36mnewton_step\u001b[1;34m(f, fp, x0)\u001b[0m\n\u001b[0;32m      3\u001b[0m     \u001b[0mderivative\u001b[0m \u001b[0mof\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mfunction\u001b[0m \u001b[1;32mas\u001b[0m \u001b[0marguments\u001b[0m\u001b[1;33m.\u001b[0m \u001b[0mIt\u001b[0m \u001b[0mreturns\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mdifference\u001b[0m \u001b[0mbetween\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mestimate\u001b[0m \u001b[1;32mand\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mratio\u001b[0m \u001b[0mof\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mvalues\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m     of the function at the estimate and the derivative of the function at the estimate.\"\"\"\n\u001b[1;32m----> 5\u001b[1;33m     \u001b[1;32mreturn\u001b[0m \u001b[0mx0\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx0\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m/\u001b[0m\u001b[0mfp\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx0\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      6\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      7\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mfp_1\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m<ipython-input-16-a036fa38011d>\u001b[0m in \u001b[0;36mfp_6\u001b[1;34m(x)\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mfp_6\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m     \u001b[1;32mreturn\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m/\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m4\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m**\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[0mn\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ml\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m3\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m100\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mOverflowError\u001b[0m: (34, 'Result too large')"
     ]
    }
   ],
   "source": [
    "def fp_6(x):\n",
    "    return 1/(x-4)**2\n",
    "    \n",
    "n, l, u = 0, 3, 100    \n",
    "    \n",
    "print(\"\\n {:^15}  {:^15}\".format(\"number of steps\", \"estimate\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "while abs(l-u) > 1e-8 and n < 1000:\n",
    "    \"\"\"This loop iterates the Newton-Raphson method step as long as the value of the estimate is \n",
    "    changing of an amount bigger than 10^-8, displaying the number of step executed and the \n",
    "    value of the estimate for each step.\"\"\" \n",
    "    u = l\n",
    "    l = newton_step(f_6, fp_6, l)\n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_6(l)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " number of steps     estimate    \n",
      "       0          6.000000000000\n",
      "       1          4.000000000000\n"
     ]
    },
    {
     "ename": "ZeroDivisionError",
     "evalue": "float division by zero",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mZeroDivisionError\u001b[0m                         Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-17-b85992c43ca5>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      9\u001b[0m     than 10^-8, displaying the number of step executed and the value of the estimate for each step.\"\"\" \n\u001b[0;32m     10\u001b[0m     \u001b[0mu\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0ml\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m---> 11\u001b[1;33m     \u001b[0ml\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnewton_step\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mf_6\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mfp_6\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ml\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m     12\u001b[0m     \u001b[0mn\u001b[0m \u001b[1;33m+=\u001b[0m \u001b[1;36m1\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m     13\u001b[0m     \u001b[0mprint\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;34m\"{:^15d}  {:15.12f}\"\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0mformat\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mn\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ml\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m<ipython-input-9-be50279f2f69>\u001b[0m in \u001b[0;36mnewton_step\u001b[1;34m(f, fp, x0)\u001b[0m\n\u001b[0;32m      3\u001b[0m     \u001b[0mderivative\u001b[0m \u001b[0mof\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mfunction\u001b[0m \u001b[1;32mas\u001b[0m \u001b[0marguments\u001b[0m\u001b[1;33m.\u001b[0m \u001b[0mIt\u001b[0m \u001b[0mreturns\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mdifference\u001b[0m \u001b[0mbetween\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mestimate\u001b[0m \u001b[1;32mand\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mratio\u001b[0m \u001b[0mof\u001b[0m \u001b[0mthe\u001b[0m \u001b[0mvalues\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m     of the function at the estimate and the derivative of the function at the estimate.\"\"\"\n\u001b[1;32m----> 5\u001b[1;33m     \u001b[1;32mreturn\u001b[0m \u001b[0mx0\u001b[0m \u001b[1;33m-\u001b[0m \u001b[0mf\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx0\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m/\u001b[0m\u001b[0mfp\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx0\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      6\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      7\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mfp_1\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;32m<ipython-input-14-84ee9617e80f>\u001b[0m in \u001b[0;36mf_6\u001b[1;34m(x)\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0mf_6\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m     \u001b[1;32mreturn\u001b[0m \u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m5\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m/\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m-\u001b[0m\u001b[1;36m4\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      3\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      4\u001b[0m \u001b[0mn\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0ml\u001b[0m\u001b[1;33m,\u001b[0m \u001b[0mu\u001b[0m \u001b[1;33m=\u001b[0m \u001b[1;36m0\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m3\u001b[0m\u001b[1;33m,\u001b[0m \u001b[1;36m6\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      5\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mZeroDivisionError\u001b[0m: float division by zero"
     ]
    }
   ],
   "source": [
    "n, l, u = 0, 6, 100    \n",
    "    \n",
    "print(\"\\n {:^15}  {:^15}\".format(\"number of steps\", \"estimate\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "while abs(l-u) > 1e-8 and n < 1000:\n",
    "    \"\"\"This loop iterates the Newton-Raphson method step as long as the value of the function at the estimate is bigger \n",
    "    than 10^-8, displaying the number of step executed and the value of the estimate for each step.\"\"\" \n",
    "    u = l\n",
    "    l = newton_step(f_6, fp_6, l)\n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_6(l)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With this function, the Newton-Raphson method can either incur in an Overflow Error, since the derivative of the function in the vicinity of the asymptote becomes very large in absolute value, or in a division by zero error, since the derivative of the function at the asymptote is zero. This means that we will notice if the method has failed to return any value, let alone the correct value. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Machine precision error** "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we then want our width to be of the order of $10^{-16}$ or smaller, we incur in a different kind of problem due to the fact that we are representing our numbers as floating points. Due to the fact that the maximum number of digits available with this representation is 17, to ask for such a precision would present us with all kinds of problems, including but not limited to the fact that we are setting the condition that two numbers are exactly equal from the point of view of the machine. We can see this with the function $f_1(x) = x^2 - 9$, looking for a root between $x = 2$ and $x = 4$: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The predicted number of steps is 58\n",
      "\n",
      "   number of steps            lower                 upper               difference     \n",
      "         0             1.00000000000000000   3.00000000000000000   2.00000000000000000\n",
      "         1             2.00000000000000000  3.000000000000000000  1.000000000000000000\n",
      "         2             2.50000000000000000  3.000000000000000000  0.500000000000000000\n",
      "         3             2.75000000000000000  3.000000000000000000  0.250000000000000000\n",
      "         4             2.87500000000000000  3.000000000000000000  0.125000000000000000\n",
      "         5             2.93750000000000000  3.000000000000000000  0.062500000000000000\n",
      "         6             2.96875000000000000  3.000000000000000000  0.031250000000000000\n",
      "         7             2.98437500000000000  3.000000000000000000  0.015625000000000000\n",
      "         8             2.99218750000000000  3.000000000000000000  0.007812500000000000\n",
      "         9             2.99609375000000000  3.000000000000000000  0.003906250000000000\n",
      "         10            2.99804687500000000  3.000000000000000000  0.001953125000000000\n",
      "         11            2.99902343750000000  3.000000000000000000  0.000976562500000000\n",
      "         12            2.99951171875000000  3.000000000000000000  0.000488281250000000\n",
      "         13            2.99975585937500000  3.000000000000000000  0.000244140625000000\n",
      "         14            2.99987792968750000  3.000000000000000000  0.000122070312500000\n",
      "         15            2.99993896484375000  3.000000000000000000  0.000061035156250000\n",
      "         16            2.99996948242187500  3.000000000000000000  0.000030517578125000\n",
      "         17            2.99998474121093750  3.000000000000000000  0.000015258789062500\n",
      "         18            2.99999237060546875  3.000000000000000000  0.000007629394531250\n",
      "         19            2.99999618530273438  3.000000000000000000  0.000003814697265625\n",
      "         20            2.99999809265136719  3.000000000000000000  0.000001907348632812\n",
      "         21            2.99999904632568359  3.000000000000000000  0.000000953674316406\n",
      "         22            2.99999952316284180  3.000000000000000000  0.000000476837158203\n",
      "         23            2.99999976158142090  3.000000000000000000  0.000000238418579102\n",
      "         24            2.99999988079071045  3.000000000000000000  0.000000119209289551\n",
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      "         27            2.99999998509883881  3.000000000000000000  0.000000014901161194\n",
      "         28            2.99999999254941940  3.000000000000000000  0.000000007450580597\n",
      "         29            2.99999999627470970  3.000000000000000000  0.000000003725290298\n",
      "         30            2.99999999813735485  3.000000000000000000  0.000000001862645149\n",
      "         31            2.99999999906867743  3.000000000000000000  0.000000000931322575\n",
      "         32            2.99999999953433871  3.000000000000000000  0.000000000465661287\n",
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      "        820            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        825            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        829            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        830            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        831            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        832            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        833            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        834            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        835            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        836            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        837            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        838            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        839            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        840            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        841            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        842            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        843            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        844            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        845            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        846            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        847            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        848            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        849            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        850            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        851            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        852            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        853            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        854            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        855            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        856            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        857            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        858            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        859            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        860            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        861            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        862            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        863            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        864            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        865            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        866            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        867            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        868            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        869            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        870            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        871            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        872            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        873            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        874            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        875            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        876            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        877            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        878            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        879            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        880            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        881            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        882            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        883            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        884            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        886            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        888            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        889            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        890            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        891            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        892            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        893            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        895            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        897            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        899            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        904            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        905            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        906            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        907            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        908            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        909            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        910            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        911            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        912            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        913            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        914            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        915            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        916            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        917            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        918            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        919            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        920            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        921            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        922            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        923            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        924            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        925            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        926            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        927            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
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      "        929            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        930            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        931            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        932            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        933            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        934            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        935            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        936            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        937            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        938            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        939            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        940            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        941            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        942            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        943            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        944            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        945            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        946            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        947            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        948            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        949            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        950            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        951            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        952            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        953            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        954            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        955            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        956            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        957            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        958            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        959            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        960            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        961            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        962            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        963            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        964            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        965            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        966            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        967            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        968            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        969            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        970            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        971            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        972            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        973            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        974            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        975            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        976            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        977            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        978            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        979            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        980            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        981            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        982            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        983            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        984            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        985            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        986            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        987            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        988            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        989            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        990            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        991            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        992            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        993            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        994            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        995            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        996            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        997            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        998            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        999            2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "        1000           2.99999999999999956  3.000000000000000000  0.000000000000000444\n",
      "\n",
      " The root is located at x = 3.000000000, where the function has value 0.00e+00\n"
     ]
    }
   ],
   "source": [
    "n, l, u = 0, 1, 3\n",
    "\n",
    "p = floor(log(abs(l-u)/1e-17, 2)) + 1\n",
    "\n",
    "print(\"The predicted number of steps is\", p)     \n",
    "\n",
    "print(\"\\n {:^20}  {:^20}  {:^20}  {:^20}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^20d}  {:20.17f}  {:20.17f}  {:20.17f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(l - u) > 1e-17 and n < 1000:\n",
    "    \"\"\"This loop iterates the bisection method step as long as the distance between the two bounds is bigger than 10^-17, \n",
    "    displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = bisection_step_corrected(f_1, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^20d}  {:20.17f}  {:20.18f}  {:20.18f}\".format(n, l, u, u-l)) \n",
    "    \n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format((u+l)/2, f_1((u+l)/2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Had I not put a limit to the number of steps, the loop would have tried to exercute an infinite number of steps, incurring in memory overflow. From the $52^{nd}$ step the function in the loop is trying to calculate a number that is between 2.99999999999999956 and 3.000000000000000000, which with machine precision is equal to 3.000000000000000000 and therefore the loop continues running with the same bounds. This is what happens for the Bisection method.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "  number of steps         lower              upper           difference    \n",
      "        0          2.000000000000000  4.000000000000000  2.000000000000000\n",
      "        1          2.833333333333333  4.000000000000000  1.166666666666667\n",
      "        2          2.975609756097561  4.000000000000000  1.024390243902439\n",
      "        3          2.996503496503496  4.000000000000000  1.003496503496504\n",
      "        4          2.999500249875062  4.000000000000000  1.000499750124938\n",
      "        5          2.999928602027703  4.000000000000000  1.000071397972297\n",
      "        6          2.999989800185637  4.000000000000000  1.000010199814363\n",
      "        7          2.999998542881539  4.000000000000000  1.000001457118461\n",
      "        8          2.999999791840176  4.000000000000000  1.000000208159824\n",
      "        9          2.999999970262882  4.000000000000000  1.000000029737118\n",
      "       10          2.999999995751840  4.000000000000000  1.000000004248160\n",
      "       11          2.999999999393120  4.000000000000000  1.000000000606880\n",
      "       12          2.999999999913303  4.000000000000000  1.000000000086697\n",
      "       13          2.999999999987615  4.000000000000000  1.000000000012385\n",
      "       14          2.999999999998231  4.000000000000000  1.000000000001769\n",
      "       15          2.999999999999747  4.000000000000000  1.000000000000253\n",
      "       16          2.999999999999964  4.000000000000000  1.000000000000036\n",
      "       17          2.999999999999995  4.000000000000000  1.000000000000005\n",
      "       18          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       19          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       20          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       21          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       22          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       23          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       24          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       25          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       26          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       27          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       28          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       29          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       30          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       31          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       32          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       33          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       34          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       35          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       36          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       37          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       38          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       39          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       40          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       41          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       42          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       43          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       44          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       45          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       46          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       47          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       48          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       49          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       50          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       51          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       52          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       53          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       54          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       55          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       56          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       57          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       58          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       59          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       60          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       61          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       62          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       63          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       64          3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       95          3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       98          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       99          3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       100         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       101         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       102         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       103         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       104         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       105         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       106         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       107         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       108         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       109         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       110         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       111         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       112         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       113         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       114         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       115         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       116         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       117         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       118         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       119         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       120         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       121         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       122         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       123         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       124         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       125         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       126         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       130         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       133         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       145         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       146         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       147         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       148         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       149         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       150         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       155         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       156         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       157         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       158         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       160         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       161         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       162         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       164         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       165         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       167         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       170         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       172         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       178         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       179         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       180         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       254         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       371         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       372         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       373         3.000000000000000  4.000000000000000  1.000000000000000\n",
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      "       709         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       710         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       711         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       712         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       713         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       714         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       715         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       716         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       717         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       718         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       719         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       720         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       721         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       722         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       723         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       724         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       725         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       726         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       727         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       728         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       729         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       730         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       731         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       732         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       733         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       734         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       735         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       736         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       737         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       738         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       739         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       740         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       741         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       742         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       743         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       744         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       745         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       746         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       747         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       748         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       749         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       750         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       751         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       752         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       753         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       754         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       755         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       756         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       757         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       758         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       759         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       760         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       761         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       762         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       763         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       764         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       765         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       766         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       767         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       768         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       769         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       770         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       771         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       772         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       773         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       774         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       775         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       776         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       777         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       778         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       779         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       780         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       781         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       782         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       783         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       784         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       785         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       786         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       787         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       788         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       789         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       790         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       791         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       792         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       793         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       794         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       795         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       796         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       797         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       798         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       799         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       800         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       801         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       802         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       803         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       804         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       805         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       806         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       807         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       808         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       809         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       810         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       811         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       812         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       813         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       814         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       815         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       816         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       817         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       818         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       819         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       820         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       821         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       822         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       823         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       824         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       825         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       826         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       827         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       828         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       829         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       830         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       831         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       832         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       833         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       834         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       835         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       836         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       837         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       838         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       839         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       840         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       841         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       842         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       843         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       844         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       845         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       846         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       847         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       848         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       849         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       850         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       851         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       852         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       853         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       854         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       855         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       856         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       857         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       858         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       859         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       860         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       861         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       862         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       863         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       864         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       865         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       866         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       867         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       868         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       869         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       870         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       871         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       872         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       873         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       874         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       875         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       876         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       877         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       878         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       879         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       880         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       881         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       882         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       883         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       884         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       885         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       886         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       887         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       888         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       889         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       890         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       891         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       892         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       893         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       894         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       895         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       896         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       897         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       898         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       899         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       900         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       901         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       902         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       903         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       904         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       905         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       906         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       907         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       908         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       909         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       910         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       911         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       912         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       913         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       914         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       915         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       916         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       917         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       918         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       919         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       920         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       921         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       922         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       923         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       924         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       925         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       926         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       927         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       928         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       929         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       930         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       931         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       932         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       933         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       934         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       935         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       936         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       937         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       938         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       939         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       940         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       941         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       942         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       943         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       944         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       945         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       946         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       947         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       948         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       949         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       950         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       951         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       952         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       953         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       954         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       955         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       956         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       957         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       958         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       959         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       960         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       961         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       962         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       963         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       964         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       965         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       966         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       967         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       968         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       969         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       970         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       971         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       972         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       973         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       974         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       975         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       976         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       977         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       978         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       979         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       980         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       981         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       982         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       983         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       984         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       985         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       986         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       987         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       988         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       989         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       990         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       991         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       992         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       993         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       994         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       995         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       996         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       997         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       998         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "       999         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "      1000         3.000000000000000  4.000000000000000  1.000000000000000\n",
      "\n",
      " The root is located at x = 3.000000000, where the function has value -1.78e-15\n"
     ]
    }
   ],
   "source": [
    "n, l, u = 0, 2, 4 \n",
    "\n",
    "print(\"\\n {:^17}  {:^17}  {:^17}  {:^17}\".format(\"number of steps\",\"lower\", \"upper\", \"difference\"))\n",
    "\n",
    "print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l))\n",
    "\n",
    "while abs(f_1(l)) > 1e-17 and abs(f_1(u)) > 1e-17 and n < 1000:\n",
    "    \"\"\"This loop iterates the regula falsi method step as long as the value of the function at both the upper and lower bounds is\n",
    "    bigger than 10^-17, displaying the number of step executed, the upper and lower bounds and the difference for each step.\"\"\"\n",
    "    l, u = regula_falsi_step(f_1, (l, u)) \n",
    "    n += 1\n",
    "    print(\"{:^17d}  {:17.15f}  {:17.15f}  {:17.15f}\".format(n, l, u, u-l)) \n",
    "    \n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_1(l)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The value of the function after 18 steps is reasonably close to zero since it is of the order of $10^{-15}$, but since we have asked that the difference between zero and the value of the function at that point be of the order of $10^{-17}$ (impossible because of machine precision) the algorithm continues to run endlessly. This is what happens for the Regula Falsi method. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " number of steps     estimate    \n",
      "       0          2.000000000000\n",
      "       1          3.250000000000\n",
      "       2          3.009615384615\n",
      "       3          3.000015360039\n",
      "       4          3.000000000039\n",
      "       5          3.000000000000\n",
      "       6          3.000000000000\n",
      "\n",
      " The root is located at x = 3.000000000, where the function has value 0.00e+00\n"
     ]
    }
   ],
   "source": [
    "n, l = 0, 2\n",
    "\n",
    "print(\"\\n {:^15}  {:^15}\".format(\"number of steps\", \"estimate\"))\n",
    "\n",
    "print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "while abs(l-u) > 1e-17 and n < 1000:\n",
    "    \"\"\"This loop iterates the Newton-Raphson method step as long as the value of the estimate is changing \n",
    "    of an amount larger than 10^-17, displaying the number of step executed and the value of the estimate \n",
    "    for each step.\"\"\" \n",
    "    u = l \n",
    "    l = newton_step(f_1, fp_1, l)\n",
    "    n += 1\n",
    "    print(\"{:^15d}  {:15.12f}\".format(n, l))\n",
    "\n",
    "print(\"\\n The root is located at x = {:10.9f}, where the function has value {:5.2e}\".format(l, f_1(l)))"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using the Newton-Raphson method, we can have confirmation of the fact that the only way the loop will stop running is if the two values the loop condition taking the difference of are equal. With this method, it happens when the function at the estimate has value exactly zero, i.e. when the estimate coincides with the root with \"infinite\" precision. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**CONCLUSION**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each method has its strengths and weaknesses and, with the functions we are used to evaluate in the field of undegraduate physics, they don't usually present problems. If we wanted to combine these to find a new method that surpasses the three of them, we would have to: \n",
    "\n",
    "1) Plot the function we're interested in to visually estimate the approximate position of the root we want to find (if there is more than one root, we can iterate the checklist for each one)\n",
    "\n",
    "2.1) If the root is not close to zero and not close enough to an asymptote that we can't discern between the two, use the Newton-Raphson method to find it\n",
    "\n",
    "2.2) If the root is not close to zero but too close to an asymptote to discern between the two, use the Bisection method, find a point, print the value of the function at that point\n",
    "\n",
    "- If the value of the function is very small, you have found the root\n",
    "\n",
    "- If the value of the function is very large, you have found the asymptote and can now repeat step 2.2 with an interval that doesn't include it\n",
    "\n",
    "- If the value of the function is very small but not satisfyingly accurate, we use the Newton-Raphson method to refine it"
   ]
  }
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