aula-07-secante.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Implementação do método da secante "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Neste capítulo, utilizamos uma implementação própria do método da secante para resolver equações não-lineares unidimensionais. O algoritmo é capaz de lidar com uma quantidade razoável de funções matemáticas.\n",
    "\n",
    "Para ser executado, o método `secante` requer 5 parâmetros: \n",
    "\n",
    "- estimativas iniciais `xa` e `xb`;\n",
    "- a função $f(x)$ propriamente dita, representada por `f`;\n",
    "- o erro relativo desejado $ER$, representado por `tol`;\n",
    "- o número máximo de iterações $N$ para tentativa de solução, representado por `N`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import inspect, re\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from prettytable import PrettyTable as pt\n",
    "\n",
    "def secante(xa,xb,f,tol,N):\n",
    "    \"\"\"Algoritmo para determinação de raízes pelo método da secante.\n",
    "\n",
    "    Parâmetros: \n",
    "        f: string dependendo de uma variável, i.e., a função-alvo\n",
    "            (e.g., 'x**2 - 1', 'x**2*cos(x)', etc.) \n",
    "        xa: 1a. estimativa \n",
    "        xb: 2a. estimativa        \n",
    "        tol: erro relativo desejado (tolerância)\n",
    "        N: número máximo de iterações a repetir\n",
    "\n",
    "    Retorno: \n",
    "        x: aproximação para a raiz da função\n",
    "    \"\"\"\n",
    "\n",
    "    # construtor de tabela\n",
    "    table = pt()\n",
    "    \n",
    "    # substitui expressões da string pelas chamadas das funções do numpy\n",
    "    f = re.sub('(sin|sinh|cos|cosh|tan|tanh|exp|log|sqrt|log10|arcsin|arccos|arctan|arcsinh|arccosh|arctanh)', r'np.\\1', f)    \n",
    "    \n",
    "    # identifica a variável independente em f\n",
    "    var = re.search(r'([a-zA-Z][\\w]*) ?([\\+\\-\\/*]|$|\\))', f).group(1)\n",
    "    \n",
    "    # cria função\n",
    "    f = eval('lambda ' + var + ' :' + f)\n",
    "    \n",
    "    # checa se a função é de uma variável, senão lança erro        \n",
    "    try: \n",
    "        len(inspect.getfullargspec(f).args) - 1 > 0\n",
    "    except:\n",
    "        raise ValueError('O código é válido apenas para uma variável.')\n",
    "    \n",
    "    it = 0 # contador de iteracoes\n",
    "    \n",
    "    # cabeçalho de tabela\n",
    "    table.field_names = ['i','x','f(x)','ER']\n",
    "\n",
    "    # imprime estimativa inicial\n",
    "    print(f'Estimativas iniciais: xa = {xa:.6f}; xb = {xb:.6f}\\n')  \n",
    "\n",
    "    # Loop \n",
    "    for i in range(0,N):\n",
    "        \n",
    "        x = (xa*f(xb) - xb*f(xa))/(f(xb) - f(xa))\n",
    "        \n",
    "        e = abs(x-xb)/abs(x) # erro\n",
    "        \n",
    "        # tabela\n",
    "        # impressão de tabela\n",
    "        table.add_row([i,np.round(x,8),np.round(f(x),8),f'{e:e}'])\n",
    "        table.align = 'c';      \n",
    "        \n",
    "        if e < tol:\n",
    "            break\n",
    "        xa = xb\n",
    "        xb = x                \n",
    "        \n",
    "    print(table)\n",
    "       \n",
    "    if i == N:\n",
    "        print(f'Solução não obtida em {N:d} iterações')\n",
    "    else:\n",
    "        print(f'Solução obtida: x = {x:.6f}')\n",
    "\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problema\n",
    "\n",
    "Determinar a raiz positiva da equação: $f(x) = \\sqrt{x} - 2e^{-2x}$, pelo método das secantes com erro inferior a $10^{-5}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolução"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para obter os valores iniciais $x_0$ e $x_1$ necessários ao processo iterativo do método das secantes, fazemos a análise gráfica."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 600x300 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def f(x):\n",
    "    return np.sqrt(x) - 2*np.exp(-2*x) \n",
    "  \n",
    "x = np.linspace(0,3,100)\n",
    "plt.figure(figsize=(6,3))\n",
    "plt.plot(x,f(x),label='$f(x) = \\\\sqrt{x} - 2e^{-2x}$');\n",
    "plt.axhline(y=0,color='r',linestyle=':')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verificamos que a raiz encontra-se próxima a $0.5$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.028652101156337095"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assim, podemos escolher duas estimativas iniciais próximas deste valor, digamos $x_0 = 0.5 - \\delta x$ e $x_1 = 0.5 + \\delta x$, com $\\delta x = 10^{-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimativas iniciais: xa = 0.400000; xb = 0.600000\n",
      "\n",
      "+---+------------+-------------+--------------+\n",
      "| i |     x      |     f(x)    |      ER      |\n",
      "+---+------------+-------------+--------------+\n",
      "| 0 | 0.52143975 |  0.01723111 | 1.506603e-01 |\n",
      "| 1 | 0.51270504 | -0.00126447 | 1.703652e-02 |\n",
      "| 2 | 0.5133022  |   8.57e-06  | 1.163365e-03 |\n",
      "| 3 | 0.51329818 |     0.0     | 7.829741e-06 |\n",
      "+---+------------+-------------+--------------+\n",
      "Solução obtida: x = 0.513298\n"
     ]
    }
   ],
   "source": [
    "x0, dx = 0.5, 1e-1\n",
    "x = secante(x0 - dx,x0 + dx,'sqrt(x) - 2*exp(-2*x)',1e-5,100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exemplo\n",
    "\n",
    "Resolva o problema $f(x) = 0$, para $f(x) = -\\text{arccos}(x) + 4\\text{sen}(x) + 1.7$, no intervalo $-0.2 \\le x \\le 1.0$ e $\\epsilon = 10^{-3}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimativas iniciais: xa = -0.900000; xb = -0.700000\n",
      "\n",
      "+---+-------------+-------------+--------------+\n",
      "| i |      x      |     f(x)    |      ER      |\n",
      "+---+-------------+-------------+--------------+\n",
      "| 0 |  0.01559341 |  0.20716881 | 4.589077e+01 |\n",
      "| 1 | -0.02762483 | -0.00890993 | 1.564471e+00 |\n",
      "| 2 | -0.02584274 |   -1.4e-06  | 6.895899e-02 |\n",
      "| 3 | -0.02584246 |     0.0     | 1.082134e-05 |\n",
      "+---+-------------+-------------+--------------+\n",
      "Solução obtida: x = -0.025842\n"
     ]
    }
   ],
   "source": [
    "x0, dx = -0.8, 1e-1\n",
    "x = secante(x0 - dx,x0 + dx,'-arccos(x) + 4*sin(x) + 1.7',1e-3,20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Resolvemos usando outras estimativas iniciais."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimativas iniciais: xa = -0.901000; xb = -0.899000\n",
      "\n",
      "+---+-------------+-------------+--------------+\n",
      "| i |      x      |     f(x)    |      ER      |\n",
      "+---+-------------+-------------+--------------+\n",
      "| 0 | -0.03737686 | -0.05765454 | 2.305231e+01 |\n",
      "| 1 | -0.02514562 |  0.00348352 | 4.864164e-01 |\n",
      "| 2 | -0.02584253 |   -3.5e-07  | 2.696761e-02 |\n",
      "| 3 | -0.02584246 |     -0.0    | 2.747395e-06 |\n",
      "+---+-------------+-------------+--------------+\n",
      "Solução obtida: x = -0.025842\n"
     ]
    }
   ],
   "source": [
    "x0, dx = -0.9, 1e-3\n",
    "x = secantes(x0 - dx,x0 + dx,'-arccos(x) + 4*sin(x) + 1.7',1e-3,20)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.16"
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 },
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}