{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Un esempio di algoritmo\n",
    "======================="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Il problema\n",
    "Dato $q$, trovare $r >0$, tale che $r\\, = \\, \\sqrt{q}$, con un errore più piccolo di $\\epsilon$.\n",
    "\n",
    "### Il procedimento\n",
    "\n",
    "Se $r = \\sqrt{q}$ allora $r \\cdot r = q$. \n",
    "\n",
    "- Partiamo da un valore stimato $g$. \n",
    "- Se $g^2 > q$ allora $g > r$. Inoltre $r = q/r > q/g$.\n",
    "Quindi $g > r > q/g$. (Controllate cosa succede se $g^2 < q$.)\n",
    "- Se $\\vert g\\cdot g - q \\vert < \\epsilon$ accettiamo $g$ come risposta.\n",
    "Altrimenti prendiamo come nuova stima per $g$ il punto medio fra $g$ e $q/g$, $g_1 = 1/2 \\cdot (g + q/g)$.\n",
    "- Ripetiamo finchè la stima finale $g_f$ è tale che $\\vert g_f \\cdot g_f - q \\vert < \\epsilon$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style= 'padding:20px 20px 20px 100px;'>\n",
    "    <img src=\"../Images/Algorithm_My_sqrt_fig.png\" width=\"500\"/>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mysqrt(q,g,epsilon):            # I tre argomenti devono essere passati tutti.\n",
    "    while(abs(g*g - q) > epsilon):\n",
    "        g = (g + q/g)/2.\n",
    "    return g\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "mysqrt() missing 1 required positional argument: 'epsilon'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-2-a2450e0411af>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mmysqrt\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m16.\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m: mysqrt() missing 1 required positional argument: 'epsilon'"
     ]
    }
   ],
   "source": [
    "mysqrt(16.,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.000000000000241"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mysqrt(16.,3.,1.e-7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mysqrt2(q,g,epsilon=1.e-5):   # L'argomento epsilon è opzionale. Se non viene passato, assume il valore 1.e-5\n",
    "    while(abs(g*g - q) > epsilon):\n",
    "        g = (g + q/g)/2.\n",
    "    return g\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.000000000052429"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mysqrt2(16.,6.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.000020480052429"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mysqrt2(16.,6.,1.e-3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mysqrt2(16.,6.,1.e-15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il nostro algoritmo ha:\n",
    "\n",
    "- un punto di partenza (fornito dall'utente)\n",
    "- una sequenza definita di operazioni\n",
    "- un criterio per interrompere il procedimento"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "hide_input": false,
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": true,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}