{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parte 3\n",
    "\n",
    "## Esercizio 1\n",
    "Dire per quali valori di  la soluzione dell’equazione $x^2 - 4x + \\alpha = 0$ risulta essere un problema mal condizionato.\n",
    "\n",
    "\n",
    "### Svolgimento\n",
    "\n",
    "Un indice ci condizionamento elevato indica che ci trova di fronte ad un problema mal condizionato.\n",
    "\n",
    "Per il calcolo dell'indice di condizionamento si ricorre alla seguente formula:\n",
    "\n",
    "\n",
    "$$ K(\\alpha) = \\left| \\frac{f'(\\alpha)}{f(\\alpha)} \\alpha \\right| $$\n",
    "\n",
    "Per valutare il malcondizionamento del problema bisogna considerare le due radici dell'equazione date da:\n",
    "\n",
    "\n",
    "$$ x_{1,2} = 2 \\pm \\sqrt{4 - \\alpha} $$\n",
    "\n",
    "Bisogna studiare per quali valori di  l'indice di condizionamento cresce esponenzialmente. Si ottiene che l'indice di condizionamento tende a infinito per $2 - \\sqrt{4 -\\alpha} \\to 0,\\ \\Rightarrow \\alpha = 4$ .\n",
    "\n",
    "$\\alpha$ assume questo valore se si ha che $|{\\alpha}|$ è più piccolo dello spacing in $[2^2,2^3]$, in quel caso infatti  si approssima a 4. \n",
    "\n",
    "In generale il problema risulta mal condizionato se  è più piccolo dello spacing in $[2^2,2^3]$. \n",
    "\n",
    "Nel secondo caso si ottiene un indice di condizionamento con un denominatore che non si annulla mai. Perciò la valutazione di  risulta ben condizionata per qualunque valor di ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Esercizio 2\n",
    "\n",
    "Mostrare, con degli esempi, che la risoluzione del sistema lineare\n",
    "\n",
    "\n",
    "$$ \\begin{cases} 3x + 5y = 10 \\\\ 3.01x + 5.01y = 1 \\end{cases} $$\n",
    "\n",
    "al variare (per esempio) del coefficiente di  nella seconda equazione risulta essere un problema mal condizionato. Mostrare inoltre cosa si può dire del seguente sistema:\n",
    "\n",
    "\n",
    "$$ \\begin{cases} 5x + 10y = 15 \\\\ 2x + y = 1 \\end{cases} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x =\n",
      "\n",
      "  -2.255000000000211e+03\n",
      "   1.355000000000127e+03\n",
      "\n"
     ]
    }
   ],
   "source": [
    "format long e\n",
    "A = [3 5; 3.01 5.01]; % si memorizzano su ciascuna riga i coefficienti dell'equazione\n",
    "b = [10; 1]; % si memorizzano i termini noti a parte\n",
    "x = A \\ b % risolve un sistema lineare"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si costruisce la matrice perturbata $A+ \\Delta A = A_p$ e si ottiene la relativa soluzione perturbata $x_{p}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_p =\n",
      "\n",
      "  -6.442857142857080e+02\n",
      "   3.885714285714247e+02\n",
      "\n"
     ]
    }
   ],
   "source": [
    "delta_A = [0 0; 0.01 0]; % fattore di perturbazione\n",
    "A_p = A + delta_A; % matrice perturbata\n",
    "x_p = A_p\\b % soluzione perturbata"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si calcola l'errore che si ha quando si sostituisce la matrice $A$ con la matrice $A + \\Delta A$, ovvero l'errore relativo è dato da:\n",
    "\n",
    "\n",
    "$$ \\varepsilon_r = \\frac{\\|(A + \\Delta A) - A \\|}{\\|A\\|} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Errore sui dati: 0.12 %\n",
      "Errore relativo: 71.43 %"
     ]
    }
   ],
   "source": [
    "% si è scelta la norma ad infinito\n",
    "err_dati = norm(delta_A,inf) / norm(A,inf);\n",
    "err_sol = norm(x-x_p,inf) / norm(x,inf);\n",
    "\n",
    "fprintf(\"Errore sui dati: %.2f %%\\n\", err_dati*100)\n",
    "fprintf(\"Errore relativo: %.2f %%\", err_sol*100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La perturbazione dovuta a `err_dati` dello 0.12% provoca un `err_sol` del 70% nel passaggio dal valore esatto $x$ al valore perturbato $x_p$.\n",
    "Il condizionamento di questo problema si studia con l'indice di condizionamento della matrice."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Indice di condizionamento: 1.975302e+10"
     ]
    }
   ],
   "source": [
    "fprintf(\"Indice di condizionamento: %.6e\", cond(A,inf))\n",
    "% oppure norm(A*inv(A), inf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il valore dell'indice di condizionamento amplifica il piccolo inevitabile errore sui dati commesso. Rappresenta un indice piuttosto elevato.\n",
    "\n",
    "Se i coefficienti della matrice nascono da misure sperimentali e la matrice ha un indice di condizionamento elevato allora i risultati saranno poco affidabili (un problema si dice mal condizionato quando ad una piccola perturbazione sui dati corrisponde una grande variazione dei risultati)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rappresentazione grafica\n",
    "\n",
    "Si può ora realizzare una rappresentazione grafica della discrepanza dei risultati."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = linspace(-2255.005, -2254.995,100);\n",
    "yy = (10 - 3 * xx ) / 5;\n",
    "yy1 = (1 - 3.01 * xx) / 5.01;\n",
    "yy2 = (1 - 3.02 * xx) / 5.01;\n",
    "\n",
    "plot(xx,yy,'b-',xx,yy1,'r--')\n",
    "title('Grafico delle due rette nel sistema non perturbato');\n",
    "legend(\"yy\", \"yy1\");\n",
    "box off"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure\n",
    "plot(xx,yy,'b-',xx,yy2,'r--')\n",
    "title('Grafico delle due rette nel sistema perturbato');\n",
    "legend(\"yy\", \"yy2\",'location','eastoutside');\n",
    "box off"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sistema 2\n",
    "Si memorizzano i dati, si calcola l'errore sui dati e sulle soluzioni introducendo le stesse perturbazioni viste in precedenza."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x_2 =\n",
      "\n",
      "   1.00000\n",
      "   1.00000\n",
      "\n",
      "x_p_2 =\n",
      "\n",
      "   1.00658\n",
      "   0.98684\n",
      "\n",
      "Errore sui dati: 0.67 %\n",
      "Errore relativo: 1.32 %\n",
      "Indice di condizionamento: 6.00 \n"
     ]
    }
   ],
   "source": [
    "A_2 = [5 10; 4 2];\n",
    "b_2 = [15; 6];\n",
    "x_2 = A_2 \\ b_2\n",
    "\n",
    "A_p_2 = [5 10.1; 4 2];\n",
    "x_p_2 = A_p_2 \\ b_2\n",
    "\n",
    "% si è scelta la norma ad infinito\n",
    "err_dati_2 = norm(A_2 - A_p_2, inf) / norm(A_2, inf);\n",
    "err_sol_2 = norm(x_2 - x_p_2, inf) / norm(x_2, inf);\n",
    "\n",
    "fprintf(\"Errore sui dati: %.2f %%\\n\", err_dati_2*100)\n",
    "fprintf(\"Errore relativo: %.2f %%\\n\", err_sol_2*100)\n",
    "fprintf(\"Indice di condizionamento: %.2f \\n\", cond(A_2, \"inf\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sia l'errore sui dati che l'errore relativo risultano molto piccoli. Lo stesso vale per l'indice di condizionamento relativo alla risoluzione di questo secondo sistema lineare. Ciò significa che l'errore sui dati subirà un fattore di amplificazione ridotto.\n",
    "\n",
    "Si può concludere che il problema risulta ben condizionato."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Esercizio 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x =\n",
      "\n",
      "   136422.33758\n",
      "   -25999.49957\n",
      "     1237.44420\n",
      "\n",
      "x_p =\n",
      "\n",
      "   67.09867\n",
      "   -9.66740\n",
      "    0.31242\n",
      "\n",
      "Errore sui dati: 0.000012 %\n",
      "Errore relativo: 99.95 %\n",
      "Indice di condizionamento: 1.975e+10 \n"
     ]
    }
   ],
   "source": [
    "clear A b x A_p x_p delta_A err_dati err_sol cond_A\n",
    "A = [6 63 662.2; 63 662.2 6967.8; 662.2 6967.8 73393.5664];\n",
    "b = [1.1; 2.33; 1.7];\n",
    "x = A \\ b\n",
    "\n",
    "delta_A = zeros(3); delta_A(1,1) = 0.01;\n",
    "A_p = A + delta_A; % matrice perturbata\n",
    "x_p = A_p \\ b % soluzione perturbata\n",
    "\n",
    "% si è scelta la norma ad infinito\n",
    "err_dati = norm(delta_A,inf) / norm(A,inf);\n",
    "err_sol = norm(x-x_p,inf) / norm(x,inf);\n",
    "\n",
    "fprintf(\"Errore sui dati: %.6f %%\\n\", err_dati * 100)\n",
    "fprintf(\"Errore relativo: %.2f %%\\n\", err_sol * 100)\n",
    "fprintf(\"Indice di condizionamento: %.3e \\n\", cond(A, \"inf\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Esercizio 5\n",
    "\n",
    "In questo esercizio a differenza dei due precedenti si perturba il dato relativo al vettore dei termini noti e si calcola l'errore relativo sui dati forniti nel vettore dei termini noti."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Indice di condizionamento: 2.84e+04\n",
      "\n",
      "x =\n",
      "\n",
      "    -4.0000\n",
      "    60.0000\n",
      "  -180.0000\n",
      "   140.0000\n",
      "\n",
      "x_p =\n",
      "\n",
      "    1.1600\n",
      "    3.0000\n",
      "  -43.8000\n",
      "   51.8000\n",
      "\n",
      "Errore sui dati: 1.000 %\n",
      "Errore relativo: 75.67 %"
     ]
    }
   ],
   "source": [
    "clear A b x A_p x_p delta_A err_dati err_sol cond_A\n",
    "A = hilb(4);\n",
    "b = ones(4,1);\n",
    "\n",
    "fprintf(\"Indice di condizionamento: %.2e\\n\\n\", cond(A,inf))\n",
    "x = A\\b\n",
    "\n",
    "delta_b = 0.01 * [1; -1; 1; -1]; % fattore di perturbazione\n",
    "b_p = b + delta_b;\n",
    "x_p = A \\ b_p\n",
    "\n",
    "err_dati = norm(delta_b, inf) / norm(b, inf); % si è scelta la norma ad infinito\n",
    "err_sol = norm(x - x_p, inf) / norm(x, inf);\n",
    "\n",
    "% disp(err_dati * 100)\n",
    "fprintf(\"Errore sui dati: %.3f %%\\n\", err_dati * 100)\n",
    "fprintf(\"Errore relativo: %.2f %%\", err_sol * 100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si ottiene che il problema risulta mal condizionato dato che l'indice è particolarmente alto. Inoltre l'errore commesso nell'approssimare i risultati delle operazioni è particolarmente alto."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Esercizio 6\n",
    "\n",
    "Dati due algoritmi di calcolo:\n",
    "\n",
    "$fl(fl(a + b)\\ *\\ fl(a - b))$\n",
    "\n",
    "$fl(fl(a^2) - fl((b^2))$\n",
    "\n",
    "Si vuole determinare l'intervallo in cui il primo algoritmo risulta numericamente più stabile del secondo.\n",
    "Si impiega il concetto di **indice algoritmico** e l'**espressione esplicita** di $fl_A(x)$.\n",
    "\n",
    "Nel primo caso si calcola l'indice algoritmico e si ottiene che $I_{ALG} = 3$ dato che $\\varepsilon_{alg} \\le |\\varepsilon_1| + |\\varepsilon_2| + |\\varepsilon_3| \\le 3u$.\n",
    "\n",
    "Nel secondo caso si ottiene che l'indice algoritmico varia in base ai valori assunti da $a$ e $b$. In particolare:\n",
    "\n",
    "$$ I_{ALG} = 1 + \\left| \\frac{a^2}{a^2 - b^2} \\right| + \\left| \\frac{b^2}{a^2 -b^2} \\right| $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Esercizio 7\n",
    "\n",
    "Si implementa un algoritmo di calcolo di un esponenziale che impiega lo sviluppo in serie di Taylor, detto `esp_taylor`.\n",
    "\n",
    "Questo algoritmo va confrontato con l'output della funzione di libreria `exp` in termini di errore relativo\n",
    "\n",
    "Si effettua un campionamento di $x$ su un numero di punti arbitrario e si calcolano i valori dell'esponenziale in quei punti."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "% dati iniziali\n",
    "a = -10; b = 10; num_camp = 1000; % campionamento di x in [a,b]\n",
    "xc = linspace(a,b,num_camp);      % vettore con i valori campionati di x\n",
    "exp_app = zeros(1, num_camp);     % vettore in cui memorizzare i ris.\n",
    "exp_app2 = zeros(1, num_camp);    % vettore in cui memorizzare i ris.\n",
    "nt = zeros(1, num_camp);          % indice n della serie\n",
    "nt2 = zeros(1, num_camp);         % indice n della serie"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Implementazione dell'algoritmo per il calcolo dell'esponenziale\n",
    "Si impiega come criterio di arresto il confronto fra l'approssimazione ottenuta ad una certa iterazione e quella successiva, quando i valori calcolati nelle due iterazioni coincidono non ha senso proseguire."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "function [appr, n]=exp_taylor(x)\n",
    "    old_appr = 0.e0;\n",
    "    appr = 1.e0;\n",
    "    tmp = 1.e0;\n",
    "    n = 0;\n",
    "\n",
    "    while old_appr ~= appr \n",
    "        old_appr = appr;\n",
    "        n = n + 1;\n",
    "        tmp = tmp * x/n;\n",
    "        appr = old_appr + tmp;\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Si calcola:\n",
    "    - $n!$ come $(n-1)! \\cdot n$\n",
    "    - $x^n$ come $x^{(n-1)} \\cdot x$\n",
    "- Ad ogni passo si impiega la formula: \n",
    "\n",
    "\n",
    "$$ Exp_n = Exp_{n-1} + \\frac{x^n}{n!}, per\\ n \\ge 1 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Implementazione della versione migliorata dell'algoritmo per il calcolo dell'esponenziale\n",
    "Maggiore accuratezza grazie all'impiego di un'operazione più stabile,\n",
    "\n",
    "\n",
    "$$ e^x = \\frac{1}{e^{-x}} $$\n",
    "\n",
    "per il calcolo dell'esponenziale nel caso di esponente negativo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "function [appr_new, n] = exp_taylor_mk2(x)\n",
    "    [appr_new, n] = exp_taylor(-x);\n",
    "    appr_new = 1.e0 / appr_new; % exp(-x)=1/exp(x)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Applicazione degli algoritmi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "% calcolo con funzione di libreria\n",
    "exp_esatto = exp(xc);\n",
    "\n",
    "% calcolo con exp_taylor sul vettore delle ascisse e memorizzazione del\n",
    "% risultato nel vettore exp_app\n",
    "for i = 1:num_camp\n",
    "   [exp_app(i), nt(i)] = exp_taylor(xc(i));\n",
    "end\n",
    "\n",
    "% calcolo errore relativo per il confronto tra il primo algoritmo e la\n",
    "% funzione di libreria\n",
    "err_rel_1 = abs(exp_app - exp_esatto)./abs(exp_esatto);\n",
    "\n",
    "% calcolo con esp_taylor se l'esponente è positivo altrimenti con\n",
    "% exp_taylor_mk2 se l'esponente è negativo.\n",
    "for i = 1:num_camp\n",
    "    if xc(i) >= 0\n",
    "        [exp_app2(i), nt2(i)] = exp_taylor(xc(i));\n",
    "    else\n",
    "        [exp_app2(i), nt2(i)] = exp_taylor_mk2(xc(i));\n",
    "    end\n",
    "end \n",
    "\n",
    "% calcolo errore relativo per il confronto tra il primo algoritmo e la\n",
    "% versione migliorata\n",
    "err_rel_2 = abs(exp_app2 - exp_esatto)./abs(exp_esatto); "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rappresentazione grafica dei risultati"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(1)\n",
    "plot(xc,exp_esatto,'r',xc,exp_app,'--g')\n",
    "xlabel('x')\n",
    "ylabel('exp')\n",
    "title('Approssimazione esponenziale con serie di Taylor troncata')\n",
    "legend('exp_{es}','exp_{app}')\n",
    "box off"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il calcolo dell'esponenziale che sfrutta la serie di Taylor dà risultati quasi identici rispetto alla funzione di libreria se si considera un grado di definizione modesto.\n",
    "\n",
    "In realtà, come si nota dalle figure 2 e 3, il risultato ottenuto diverge notevolmente da quello della funzione di libreria. In particolare l'approssimazione per gli esponenziali con esponente negativo è pessima."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(2)\n",
    "plot(xc,err_rel_1)\n",
    "xlabel('x')\n",
    "ylabel('err_{rel}')\n",
    "title('Errore relativo - scala cartesiana')\n",
    "box off"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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wx1J9/Pjx5uZme3s7rAno/fv3Dx8+XDhKOGwzWYFwr7+5ufn48WPEEdXLFlgMIYT7I3InTv+TvzwRRY3es+KZieD5gp8+fXLK8OOPP3769EkeNLpgDvUT6y+DyvnByiOQAPyPbAyUA/d7vd7bt2/lGBAgHwQSgN/d3Nz89NNPNzc37nuzgHwQSACAQmCUHQCgEPILJNM0u92uZVlCiPl83u12O52OXAQAIKdA6vf7m5ubx8fH5+fnQojBYNBsNi8uLvr9fj4FAAAUXE5TB1mWdXx8XKvV5OLGxsZ4PL67u1tfX8+nAACAgtNQQ5rP57Ztu9dYljWfz4UQ/X7fNE3btm3bdtJIWl9fX19fv7u7S18AAMAK0FBDGgwGt7e3JycncrHT6RiGMZ1Oj4+P9/b25Mp6vW5ZVqPRkIv9fv/i4kK+SF8AAMAKSBtI7XZ7Op0eHBzIxdFoZBjG6empbdvdbrfZbMr1+/v75+fnGxsb9XpdCLG3t3d0dOQsAgCg4T4k0zSFELKGZJrm5uamrBhtbW1dX187m8mWPaeS5FkEAFSc/kENhmHIFzs7O+71tVrN3Y3kWXTb2tr65ZdfnEVu3QWAKtAfSM4Ah8lkkngnhBAAVI3m+5AajcZsNhNC2LZN/xAAQJ3mGlKr1er1eqZpTiYT99NWYltbE1SSAKBKMplcdTweG4bhdCbFtbW1df3LLwQSAFRKJjM1OKO9AQBQxGzfAIBCKHAgra0tuwQAgPwUOJAAAFVS1EBiRAMAVExRAwkAUDEEEgCgEAocSE6rHaMbAKACihpIMoSIIgCojKIGEgCgYggkAEAhEEgAgEIodiBxNxIAVEaxAwkAUBmFDyRnoB0j7gBgpRU1kCIa60gmAFhFRQ0kAEDFEEgAgEIoVSDRWAcAq6vwgeTvTGIsOACsosIHEgCgGkoYSDTcAcAqKmEgAQBWUeEDifoQAFRD4QMJAFANJQmktTXv4Lq1NSpPALBKShJIAIBVV4ZAcupG3IEEAKurDIHk8LfR0WoHAKuiJIF0f/979sh6ErUlAFgtJQkkAMCqK2EgeapKgoY7AFgFZQgk8gYAKuCr3I5kmubd3d3e3l6j0ZjP5+fn50IIuRj1Mdl7FDicwd2N5L9RCQBQKjnVkPr9/ubm5vHxscyh8/Pzvb294+PjXq+Xar+EEACsipwCybKsZ8+e1Wo1uXh3d2cYRq1Ws21bdRf391+MtROLmvJo6AOAUtHQZDefz2XAOGssy5J50+/3Z7PZ/v6+bdtOGgkhDg8Pj46OarXafD5PXwBvUAEASkhDIA0Gg9vb25OTE7nY6XQMw5hOp8fHx3t7e3JlvV63LMvpLqrVam/evKnVakdHR4sP4LkJaWF/EgCghNIGUrvdnk6nBwcHcnE0GhmGcXp6att2t9ttNpty/f7+/vn5+cbGRr1eF0LYtt3v9zc2NpwNlAROsQoAWAlpA2k4HJqm6Sw61SDDMCaTibPeMIzXr1/bti3fbTabsonP3dDntuZKmntP75EzU4OnP8l9WxIVJgAoG/3Dvp2M2dnZca+v1WrubqSwKJLuSRQAqBj9o+ycgXPuGpI27nuSokOLSAOAUtEcSI1GYzabCSFs25bdRXkgewCg/DQ32bVarV6vZ5rmZDI5PDzUu/Mv+o3ChjPQgQQA5bSWRW/NeDw2DCO6lyjC1tbW9fW1d61MGplD7hfudwWBBABllclcdvEGcytSjxknusI+QmgBQPGUYbbvhbgbCQDKr4SBFFi5CbthlqwCgJIoTyAFRotnpco2AIBCKk8gSU5NKNYcQmQSABRe2QJJBQMWAKCE8ntibFrOqG5n0V9J8g8HF+QTAJTDStSQiBwAKL8SBpJTVQrsGfKEkz+rwsZB0M8EAEtVwkASaiO/435cIpYAYEnKGUhhZJwEPlI2YhEAUAArFEiex5zHTR06ogBgqVYikKKnrYtYBAAURjkDSTFXiB8AKI+yBVLYTA0RW6qsV3wKLQAgM2ULpOxQnQKApSpzILlrM544kW+5q1MReeN+i1gCgCUpcyA5aGcDgPJbiUBy+JPJP3FDdF0KALAkKxRIyWZq4HHmAFAMKxRIwjcxXdg8dWEdRXQgAcDyrFAgJYgTd/XI/zwLAECOViWQVLLEmU/I3ZPk71KingQAy7AqgeQRPVSB+YQAoHhWNJAiJJ7iAQCQpeoFEgCgkFYokCJqNoEDFvwNd0zZAADLs0KBFI37jQCg2FYrkFT6hzzJREoBQDGUM5D0pohnbzTWAcAylDOQ4vJnDBUjACiYagSSw51M/nlXacoDgOWpTCCFTcGg+JwkAEDGcg2kXq/nvDZNs9vtWpaVZwGECKr6EDwAUAA5BZJt2+12++rqSi72+/3Nzc3j4+Pz8/M8Dh8rhBY21hFgAJCBnALJMIzhcOgsWpb17NmzWq2Wz9F/F/HUc8/66AdYAAB00xBI8/nctm33Gsuy5vO5EKLf75um6XlXCGHb9hLSaKEE8UNQAYAmX6XfxWAwuL29PTk5kYudTscwjOl0enx8vLe3F/iRer1uWVaj0Uh/9LQinjch32KKBwDIRdpAarfb0+n04OBALo5GI8MwTk9PbdvudrvNZjPwU/v7++fn5xsbG/V6PWUBlsB/Iy2JBQCppQ2k4XBomqaz6NR7DMOYTCaejS8uLuQLwzBev35t23ZYJWnNVWu5z/pyT5wAQAFoaLLzMAxDvtjZ2YnYrFarRXQj5RFCabp/qBUBgG76A8kZwuCvIRWLP5OcjPG8IH4AIHuah303Go3ZbCaEsG27HP1DKjM1RMy+SlABgCaaa0itVqvX65mmOZlMDg8P9e48K1SAAKAA1rLorRmPx4ZhOJ1JcW1tbV1fX+stkhBfBo98HVYH8m/v/xQZBgBa6e9DEkKEjfYuluiZGpzeI/HlY/08SQYA0KQys32rcMdM2GvJXdMCAOhQ+UCKaHnzD7qL3h4AkELlA0mkzhjPNKwAgEQIpHDRo+9IIADQqqqBpFIr8rfU0XUEAJmpaiCJoExydxoptuP553QAACRS4UBSFDFNQxhiCQDiI5AAAIVQ1UCKmG0hWfcSACCdqgaSCGpYix7AHTY1uPtdwgkAkqpwIKVHXxEA6EMgaUVEAUBSBJKysOY4QggAdCCQkiKHAECrqgZS4tEH0ZMJ8WQKAEiqqoEkImdqSLAHxtcBQDoVDiTtAh9XAQBQQyABAAqhqoGksaeH+hAA6FDVQBLhMzUoSjDpKgAgXIUDKT3FSYYAAAoIJH2Yzg4AUiCQUuMJSQCgQ1UDKfGDJxzqGUMaAYCCqgZShJRtbsQPACRS4UBK39nj7EFOGgQASKFKgZRbZhBOABBflQIpa4o5RJseAAT5atkFKD9PDrkXZVMeFSYAUEANKamFMzVQEwKAOAikFDyR4yyurZFGABBXroHU6/UiFgEAVZZTINm23W63r66uAhfLTaU+5K480aUEAEFyCiTDMIbDYdhiKXlCyD94wX2XkvsFrXkAEERDIM3nc9u23Wssy5rP50KIfr9vmqbn3QrxjLgDAITTMOx7MBjc3t6enJzIxU6nYxjGdDo9Pj7e29tLv//iknlzf0/YAEB6aWtI7Xb7/PzcWRyNRoZhnJ6evnnz5t27dyl3XlZhM7fSewQA4dIG0nA4PDg4cBYty2o0GkIIwzAmk4ln44uLi4hFtzWXlCUsopX8UgCQjv6ZGgzDkC92dnYS7+S+XJUJf2llO57TphfxrsToOwCVp3+UnTOEwV9DWikq+RFYEyJ4ACCI5kBqNBqz2UwIYdt2vV7Xu/PC8Yzn9q8XqadvoHEPQGVobrJrtVq9Xs80zclkcnh4qHfnAIAVtpZFb814PDYMw+lMimtra+v6+lpvkTLhdPwEdgg5lRvPuPDAx/qF9SHRtwSgMjJ5/ESz2cxit8US3ZjmziR/gx4ZAwA+zPadJYIHAJQRSMUQNskQkQagMgikFMLSwr3eeZ1spgZG2QGoDAIpF+5Y8q9ciFgCUAEEkg5hczHIIFk4IwN5AwAEUnJ6Hy0RuAeCCkCVEEgphAVG4EwN2vcMAKuFQMpe3AfFMrIOQCURSMsT3Uwne5sIJwCVQSBlJuwxff4c8my5cA4IAFhFBFJSKsEQdkMSAMCHQEpBMVoi7p91T8Cq+BEAWFEEUlKBjW/OW4Gv1fmftBR3ZAQAlA2BtFSBd9SGPfgcAFYagaRDWHgsrNDEqvf4530AgBVCIKXgfrxeShF74PlJAKqBQEpBJYcUK0np16sXCQAKiUAqD+pJAFYagVQwnioOkzUAqAwCqQCIHAAgkLKVadL4u4sY/gCgzAikZQu864iZGgBUD4GUsZQR4nm0UuIxewBQeARSwYQ94Jy6EYBVRyDpkHimhojNwp5JkfJYAFBUBFIKWmZqSP+8cwBYCQRSChrzI3DInPpBPSsJNgAl9FXYG7Zt27btWdlsNjMuz8rROBQ74oEXAFB+UYF0dXXlWUkg5S1uCDH2AUBphQZSs9mU8dPv92ez2ZMnTwzDyLFgVRKdOoqPR6L+BKDkFvQhmaY5m80mk8n6+vrR0VE+ZVopeqssYUPvAKD8FgTSdDo9OTlZX19vNBo7Ozvj8TifYlWIuwIUnV7+eVed9bTUASi/BYFUr9d7vd7d3d14PJ5Opyn7kHq9nnwxn8+73W632x2NRml2WHT55wR1JgCltSCQZPWoXq9fXV2dnp4mPoxt2+122xklMRgMms3m6elpv99PvE8AwCoJHdQgmaZ5cnKyt7eX8jCGYQyHw06nIxefPXsmh0jc3d2l3HMhpJ+wzr8H5lcFUDELakibm5tHR0ej0Wg8Hod1IM3nc88dS5ZlzedzIUS/3zdN038/k2EY4/G40+m8fv06ReGXTddMDer78UzrEDayTq6n+Q5AqQTUkGStyFnc2NiwLEu+DuxDGgwGt7e3zkc6nY5hGNPp9Pj4OKxqJRPu4uIiZemXTO9oAv9sC4E7D1zvSTVqUQBKKCCQptPpeDyW2bOwsa7dbk+n04ODA7k4Go0Mwzg9PbVtu9vthg2C+Pe//z2fz2ULXuljKSX18IiuDwFAyQUEUqvVOjo6qtfrzpqIzBgOh6ZpOouWZTUaDSGEYRiTycSzsbOfN2/eRBdrzXWFvefv/WjuKhHhBKC0AgKpXq8vDIwIzoQOOzs7iXdSlRBK9jWdBCJ7AKyQgEAajUbuPqS4nCEM/hoSYlDMqsAqkXu8A11KAEoiYJSd7ENKtrtGozGbzYQQtm27G/1Wk/ZpgVQmrFt4dKfyRA4BKJW0fUj+z/Z6PdM0J5PJ4eGhliLif1QyZuEYPAAoqjV/b41lWZ77VePOGDQejw3DSDw7+NbW1vX1dbLPLkc+V3ynIc4zwtvTpeQ01nlW5lZOAEgkoMmu0WjIhyHZtj2fzxO0vDWbzWo9qyLPqzyJAmBFBQSSaZpyztPZbPbf//73xYsXuZeqbHSNdlOcqcE9WiHsziTxZd0IAAovIJAGg8GbN2+ePHkihDg8PKzX6zx1Ig+KqRZreqGUxwKAHAUE0vr6unvRtm3PGixHrOlWqRgBKJuAUXYvX7588eJFvV63bVs+JVZOvoASCGzECwsnxjgAKJKAUXZCCMuyZDfS5uZm+mdPxFXRUXYqXT5O2LiDJ2LQXeCovMC3AGDZggNpuQikBdsI39juiP0QSABKYsHzkJAfXTM1SAxbAFA2BFIFEE4AyoBAKqHoulHEuzTQASgwAkmHws7U4NSN3HfUOjshnwAUCYGkg5Y2sbW12DM1qHCPaACAAiOQykYltFSqPuQTgIIhkFYUzXEAyoZAqoCwe5WoJAEoEgJp1XlSR/ZUUX8CUDwEUpHoygn/rEIAUHgEUmGohIfGmRrIKgAFQyBVAH1FAMqAQCqhNDM1UDECUFQEkg7Fn6khcA+EE4AiIZAKQ/3R48lmalA/CgAsA4G0cuSo7oW1n4hkIrQALAOBVCUkDYACI5AKQ7FHJ0GoRLTaUU8CUBgE0qpTjBz/hA4AkC8CqZwipv+5vw+YqYEBdQAKj0AqlfQBE/hBT32I9AKwDARSVXke3EcIAVg2AqmEtFSPFEeHA0BeCKSyiTUHq9MW5+5zki8W7idiA4Y8AMgAgVQq8mlGIk4keMJJcdAdNScAucs1kHq9nnwxn8+73e7R0ZFlWXkWAL+LyCcqQACWIadAsm273W5fXV3JxXfv3u3t7b1588aJKOQhLGmoDwEogJwCyTCM4XDoLJ6cnBiGMRqNNjY28ikAfqdYAaKeBCBfGgJpPp/btu1eY1nWfD4XQvT7fdM0Pe867u7u7u7u0hcAUZLlSvSnqFEByMBX6XcxGAxub29PTk7kYqfTMQxjOp0eHx/v7e0FfsQ0zZOTk729vdFolL4A1RIrDJxZG9zTNyTYj2diiIh5IgAgqbSB1G63p9PpwcGBXByNRoZhnJ6e2rbd7XabzWbgpxqNxtHR0cbGhmEYKQtQLeljICKZ3C+cIQ8ED4C8pA2k4XBomqazaFlWo9EQQhiGMZlMPBtfXFzIF61Wq9FozOdzuTGWidoPgGLQ0GTn4VR6dnZ2ojeLqB6tuf6Kv+f66JfsnOia4gEAMqA/kJwhDP4akjpCSDP/TA3Oa8aCAygGzcO+G43GbDYTQti2Xa/X9e4cX0wFpLi9+kwN7nflp8IekkRWAciA5hpSq9Xq9XqmaU4mk8PDQ707h36Bz+ijGwnAMqxl0Tg2Ho+ju4iibW1tXV9f6y3SinCiQr5YmBzu7R3+DqGwbfyH8+8BADTR34ckhAgb7Y1CCIuxwGY6AMgLs32vumTP33OmFfejegQgGwTSSkvw0COVvKEWBSADBFKppKmdhD2XTz2TqBsByBKBVHlUdwAUA4FUTsuaqSHiRloASCeTUXYoNMVYooEOQL6oIZVK1jM1OALTaOF9SHGLBwAuBBLSIXsAaEIgVUlgeDDOG0AxEEjVQ7oAKCQCCUFiTQ2uuB4AIhFI5aQ+Ui7sgeVpLIwcz0gKAFBAIJVK+pka4gqbJjxwJSPFAaRAIGGRiNnB/RUgqkQAkiKQqiRsOjuVT/kXI7LHM1k4KQVAAYGEcP5H+al/iuY7ADERSKWSoKqRxax3gUEV+Aglp1mPShKARQikysguEhYOfAAABQRSlSTOpIgPkkAANCGQsEiyNIp4CDoABCGQEEQlS8KGPKi8BgAfAglqFo64kxvIt9wD7WjTA6CGQCqV/GdqSHYIKkMA4iOQoCBlmFFJAqCAQKoSvcGw8AGymR4dwMohkBBJvRUuYlrxsNnwAMCFQCqV3Ppm1GdqcK9xj2gQv+WQM1MDHUsAIhFIlZEsD+J+inENAJIikKokWUgEPmYizwIAqAYCCUE8N7QufDRf4AcBIA4CCbqRSQASIZAQX9hMDU6dyTOuIcHeAFRProHU6/Xci/1+37KsPAtQegWfqUGEDLdz3vKsIYoAuOQUSLZtt9vtq6sr95p3797d3d3lUwCkVfwsBFByOQWSYRjD4dC95t27d8+ePcvn6PifjGZqAAAdNATSfD63bdu9xrKs+XwuhOj3+6Zpet4VQpimube3l/7QyFz6mRoCm+8AwOer9LsYDAa3t7cnJydysdPpGIYxnU6Pj4/DUmdzc3M0Gk2n07u7u2azmb4MVZHbHDxxZ2qQPMXzzNQQ9nGyCoAQIn0gtdvt6XR6cHAgF0ejkWEYp6entm13u92wsJFBZZrmkydPUhYAqvIMM89EQdG1K2a6AyCESB9Iw+HQNE1n0bKsRqMhhDAMYzKZeDa+uLhwLzqVKr811yXsnqvVcnH+AeRCQ5Odh2EY8sXOzk7inRBCK0uxuQ9A9egfZecMYfDXkLBS4uaH/yYkEgiAi+ZAajQas9lMCGHbdr1e17tzeC3rgq44U0OyWVkZTQ5UleYmu1ar1ev1TNOcTCaHh4d6d47SVCn8Y+oCB3+rDCUvy1cGkNpaFr014/HYMAynMymura2t6+trvUWCzov7wl058eNs6X4hfCnlWRlrKjwAq0L/oAYhBLcWFZH2aRpi7TDsARZOJYngASqP2b6hRu+jY/3TOviTic4koGIIpFIpyDU6Yui2Z9GTLnHnHwJQJQQSsrFwpgZPB5JnPckEVA+BBH3S9yoBqLBMBjVgBenKD3IIQAhqSEgkZa4oNtORXkCVEEiIb2EHT9hMDQsDxn1/EoCKIZBKpSw1Bs9MDe5Zhfxb+t9ytmeAA1AlBBLy5XlUkhsP7gOqjUBCItprLZ6JWQFUD4GEbATmitNA53/Xn3C01AEVQyAhPsVKjH+mhugdkkBAtRFIyEbgTA3qSQageggkAEAhEEjIncqzlABUD4GERNKMhQsbvxB4QxKAyiCQkIjKIIWUSCOgYggkZC9ipoaw7SWenwRUCYGEApOz4VFVAqqBQELuAgOGmRqAyiOQkA0tueK0yNE0B1QAgYTMqGeSZ8uISb4XPhkdQGkRSMie+gzf0RsoJhBBBZQTgYTy4EYlYKURSCgGle4iZ8QddSBgFRFISERv1UTxmeiKGyfYEkABEEhIJIdrved2WtIFWHUEErKXeKYGEZlD8rbZ6I8DKA8CCQBQCAQSikFlpoawKpFH3OcBAigGAgkF5h56R7oAq45AQgGoz9Tg3oZhDsBqyTWQer2efGFZVqfT6XQ6pmnmWQAsR5rkSHYzLHEFlNBX+RzGtu2jo6P19fXDw0MhxN3dXavV2tvby+foWFkLn4ZOQx9QHjnVkAzDGA6HzuJ0Oh2Px0dHR+PxOJ8CYJkU6zSeF2EbxD0cVSWgJDTUkObz+d3dnWEYzhrLsgzDqNVq/X5/Npvt7++73xVC1Ov1nZ0dwzBevHjhDiqURj4zNbiH3pErwKrTEEiDweD29vbk5EQudjodwzCm0+nx8XFYo5xt2/Kt9fX19AXAEuTQGuZMW0e7HFANaQOp3W5Pp9ODgwO5OBqNDMM4PT21bbvb7TabzcBPra+vHx0dbWxs1Ov1lAXAClqYQM5m0VUrz7tEF1BsaQNpOBy6R8pZltVoNIQQhmFMJhPPxhcXF/JFq9VqNpu2bcuN/dZcl5J7LiJQR+oApaV/lJ3TXbSzsxOxWa1Wq9VqYe8SQpWjMlND4n0CKAP9o+xs25Yv/DUkrI58rvUqMzWQOsCq0BxIjUZjNpsJIWzbpn8IqhbO1BDYXcS4OyBCbreH6zuK5ia7VqvV6/VM05xMJvIeWCCVwJkaZIUp+tmygZ8FkG6zlZgAAAoZSURBVEBeXbNrWfTWjMdjwzA89x6p29raur6+1lsklIk7b5zfp7PS89+IzwrxxWaMd0DRZPqb1Ljz6F3pO1AmUweFjfYGgmn8l0PkIDvl+psmTVGX9E2Z7RsFpmXOIQAlQSChqBRvj62OCn7lglj4JJSK01eXIpBQNooPhC1R04qKIjcWFfmKHLdsRf4u+VjqGSCQsEI8/5YKewVHnpZ1hdU+6nqJT/nK658SgYTiIUiKIIdrXz6X18CJDdPsSuPGijtMVvgS1vYIJBSY597YQGHr3Z/lptocJGgcy+0vj5T/u933HlREslvRU9fhCCQUQIJH6uX5hKSCXIZyLsaKXYUVwy9BRgbeDOdXltOY4AwodusqIJBQNsl+9AundSi4lWzGzPrM++egEtl07YSJ9X8t07OR4Fu7mzrz+jdCIGEVaew2wErK4Q8Ulf3EOlbOMxonk66QBBIKQNclIP0fpGRYPnLrQNIySXzc6kXgztWP6D9WFudK5eaq3P85EEgoIXcHUnSDScQsq4n/kZcrtBKUdkkXowVUetRzOEp2FH/VsXaoV0THrabzRiChwPT+i3LvbeGei3ZDZdHiwS1sotviSDDrx1LibeE5XDiWR+N3Uf+UvvEvBBLKRt+QngB661J65d9uo3jcxGmU0YmN/lL396rj4hIcouB/N/jpGpGhCYGEclL815XFk9EjDlcuWTSCFeQjEXewLfyse/tkc3+oV1Oc0Tfubip/hSN9I2rgc8XCuEsiN0vTARYHgYTi0d6eFv3ZwGaQuAG2lFpUPlf/LCS4jcz/kbA9xJokfuHe1Pfp2Zt6PSyw0pYFLRXiLBFIKDBd/34W/sWabEoY7f90s0uLBIPEElyC02+QIBVyiNj87xH219I8CRdr+EOyG889IsYH+dM36YkikFAAmn7NMcTqA3CvXGINI9mh01zE1W/n0tualOwjiZuVAke7hD2MOOIj6Ud2ONmTz88srMaZ+FswUwMqJ9YVVqXlpCCtWNGip+xL/BW0VIOKMIA+/Y2uiUMu4jqe/gZt/w9Y11NXnBpPgmF7KntOhEACfuP+x1/G7hmVVI419j1/yaYo1NsF6Mn46MVYpVLZJk3HmLr0+Z0NAgkFoOt2jbjNU8kG7wY232cxYm2hpQyb1rLP9K1bCw+kd5BCLIGDFBaOXIhouM7/T4clxRKBhApYWC1YSpO9X0Zj02MN+XV/UMT/Ez6siqMykEy9cTWigz2liObcHFIhVuucuujB64rDFxVvLk42IuY3BBIKLLtLQNjfsCofTHPQxO1RS/mLNeecThYzca+AYbGXZuiKYlU71miFsO8VnS4RhVlYhgS/bc//stQ/GAIJJaTSZ6AxzBT/Nsxa3BrGwhGDYdtE7HbhaGD1EQQq401SitW066fYCZTgcpz+axaktTblcEcfAgkrLetKj3DVe2LtR9eVV/s9Q4W9VsYS0dSm2HCa7Na0wEO4x9p50ivw1qJAsf6/eGp76rHhbznQfjtaJAIJxRPxbyBBZcV/sXAuELoOpOtOmoU87S2KTf+xakUqnT1aaDlpicfXJW46C+sqiztqI6KzzdmD4gCN9FRCKHEixkEgocBybg2LuKHEv7FnM8XrYPQlVeWP99zCL3GPgkaKgy+0J2hgJSnwr5DAH0Oy/fs3CHsrVrfiwupR4N6S9eSlRiChANL/+nX1Qi/cOM01N8HgJZUNkn0wVmeP4m611K7idmuFfSTiRxLxjVIOWnF2nmwAi4j/9ePWEdPXfuLuPA4CCcWT84gy5y/ErJuqIvafZkS4lo5oxS0TN5EtPIr2kdxaiprsI55qk64/aMIOl13yqW/mSFweIQSBhMpJ9w9Gg2T1pED38Z8x6u8LiXWU7Cy970qxQpl4FH6sqnBG40gTfPeUkRwTgYQVkrjbY2E/v/jykpR46NHCA2VX3UnzqWSZlD7GElQcVWpaSx+1kfX/5WTjKQqAQEL1BP4Fl6AvJ4t/1Wn+7FXpWUksWcFi3QqaTOKBHrEK5j+K+leLdaz0I2USdMIlOEo2cg2kXq/nvDZN8+joaDQa5VkAlEOCkUsJrko5jz1L/O85cRtRFhYOd/Zvv5TxWooXZf/Q6iKc59wqcOrNtnnJKZBs226321dXV3Kx3+8/ePDgzZs3lmXlUwAgWA7/GpfeQBQtn0bCBPLpuErwkaX0qBUhKbOXUyAZhjEcDp3F2Wy2vr5umub+/n4+BUCEtWr81kMl+xM+boOP63WME55Dz01hexT0JV/UCfcMJVAc5ZG42pdPnC/7f2jiS4qGQJrP57Ztu9dYljWfz4UQ/X7fNE3Pu0KI6XQ6m8329/e73W76AgBpJRtNUMzaVZ7dA1nnZcrsz5TG3j69cm6L1uqr9LsYDAa3t7cnJydysdPpGIYxnU6Pj4/39vYCP1Kv1/f39w3DSH90QI/CNqwVqYnfK59hgbl9pLCtl3EV9se8SNpAarfb0+n04OBALo5GI8MwTk9PbdvudrvNZjPwU/v7++fn5xsbG2QSoKSwSVbYi3ICRf4uRS6bPmv3qb+naZpCCFlDMk1zc3NTVoy2traur6/DPiUb+hqNhv+tra2tlEUCACxXxPU/jIYmOw+n0rOzsxOxWa1Wq9VqgW/98ssv7sX0kYloa2sa/i6BOk54zjjhOUt8wvUHkjOEYTKZJNsDP52cccJzxgnPGSc8Z4lPuOZh341GYzabCSFs267X63p3DgBYYZprSK1Wq9frmaY5mUwODw/17hwAsMIyaVodj8eGYTCCDgCgLpOZGprNZrI0irjHFlgZ/KqxktJfwP/v73//u+ZCpfDPf/7zw4cPT548kYudTse27V6vR30rU3/5y1/G4/G//vWv//znP87JR0b4VeeMn3du0l/A9Y+ySyzZPbZISQ4/ubi4WHZBKoFfdc74eedGywW8QM9DGg6HzpcRQliWJW+bNQwj8QhyLGTbdq1W63a7pmnSjpQ1ftU54+edGy0X8AIFkp/iPbZIYz6fb2xstFqtBw8eHB0dLbs4q49fdZ74eS9Rgp/6kpvsxuPx1dWVM9uQR/p7bBHGfeZbrZYQotls8rzEHPCrzlOr1eLnvSwJfupLDqRmsxnWtthoNOTj+7jHNgvOme/1evV6nc6MfPCrzhk/72VJ9lMv0KAGD+6xzcfOzs7R0dGzZ8+m02nY40KgC7/qnPHzXpZkP/WizznIPbb54DznibOdM074ssQ980UPJABARRR6lB0AoDoIJABAIRBIAIBCIJAAAIVAIAEACoFAAgAUAoEEACgEAgkAUAgEEpCf0WjU6/WEELZtM/k04EEgAflptVqj0ciyrG63y9RqgAeBBOTq9evXL168YApqwI9AAnJVq9WWXQSgoAgkIFfdbvfNmzeTyUQ+LQaAg0AC8mOapmyse/369d/+9rdlFwcoFh4/AQAoBGpIAIBCIJAAAIXw/wFPVCvVQ7ZGdAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(3)\n",
    "semilogy(xc, err_rel_1,'--r')\n",
    "xlabel('x')\n",
    "ylabel('err_{rel}')\n",
    "title('Errore relativo - scala semilogaritmica')\n",
    "box off\n",
    "warning (\"off\", \"Octave:negative-data-log-axis\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dal grafico in figura 4 si nota che il numero di iterazioni richieste per il calcolo di un esponenziale con esponente negativo è tendenzialmente superiore rispetto al caso con esponente positivo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(4)\n",
    "plot(xc,nt)\n",
    "xlabel('x')\n",
    "ylabel('iterazioni')\n",
    "title('Numero di iterazioni')\n",
    "box off"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In figura 5 si nota che l'errore relativo che si ottiene nel calcolo di un esponenziale nell'intervallo $[a, b]$ con l'algoritmo più stabile determina ad un errore relativo totale minore, distribuito in modo quasi equo tra gli esponenziali con esponente negativo e positivo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(5)\n",
    "semilogy(xc,err_rel_2)\n",
    "xlabel('x')\n",
    "ylabel('err_{rel}')\n",
    "title('Errore relativo - scala semilogaritmica')\n",
    "box off"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Esercizio 8\n",
    "Calcolo della derivata di $f(x) = sin(x)$ in $x = 1$ mediante il rapporto incrementale."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "k = (0:-1:-20)';\n",
    "h = 10.^k;\n",
    " \n",
    "der_es = cos(1); % derivata esatta\n",
    "x = 1; % punto in cui calcolare la derivata\n",
    "rap_inc = (sin(x + h) - sin(x))./h; % rapporto incrementale\n",
    "err_rel = abs(rap_inc - der_es)./abs(der_es); % errore relativo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dal grafico si nota che per valori di $h < 1 \\times 10^{-8}$ l'accuratezza dell'approssimazione fornita dal rapporto incrementale inizia a calare bruscamente da quella fornita dalla funzione di libreria."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "loglog(h, err_rel, h, h)\n",
    "\n",
    "xlabel('h')\n",
    "ylabel('err_{rel}')\n",
    "title('Approssimazione derivata con rapporto incrementale')\n",
    "legend('Err_{rel} rapporto incrementale','h','location','southoutside');\n",
    "box off"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://metakernel.readthedocs.io/en/latest/source/README.html"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "5.2.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}