aula-04-bissecao.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Implementações do método da bisseção"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"MB Metodo da bissecao para funcoes unidimensionais\n",
    "entrada: \n",
    "    f: uma string dependendo de x, i.e., a funcao\n",
    "           (e.g., 'x^2 + 1', 'x^2*cos(x)', etc.) \n",
    "    a: limite inferior do dominio\n",
    "    b: limite superior do dominio\n",
    "    tol: tolerancia    \n",
    "    N: numero maximo de iteracoes do metodo\n",
    "\n",
    "saida: \n",
    "    xm: raiz da funcao\n",
    "\"\"\"\n",
    "\n",
    "import inspect, re\n",
    "\n",
    "def bissecao(f,a,b,tol,N,var):\n",
    "        \n",
    "    # TODO identificar a variável usada na função \n",
    "    #      Aqui, tentei assumir que apenas uma era usada (e.g. 'x'),\n",
    "    #      mas foi complicado generalizar quando há objeto numpy\n",
    "    #var = re.search('[a-zA-Z]+',f)\n",
    "    #var = var.group()\n",
    "\n",
    "    # cria função anônima\n",
    "    f = eval('lambda ' + var + ' :' + f)\n",
    "\n",
    "    # Se função não for de uma variável, lança erro.\n",
    "    # Mais aplicável se o caso geral fosse implementado.        \n",
    "    if len(inspect.getfullargspec(f).args) - 1 > 0:    \n",
    "        raise ValueError('O código é válido apenas para uma variável.')\n",
    "\n",
    "    # calcula valor da função nos extremos\n",
    "    fa = f(a) \n",
    "    fb = f(b)\n",
    "    \n",
    "    # verifica sinal da função para o intervalo passado     \n",
    "    if fa*fb >= 0:\n",
    "        raise ValueError('A função deve ter sinais opostos em a e b!')\n",
    "    \n",
    "    # flag usada para prevenir a obtenção da raiz \n",
    "    # antes de o intervalo ter sido \n",
    "    # suficientemente reduzido\n",
    "    done = 0;\n",
    "\n",
    "    # loop principal\n",
    "\n",
    "    # bisecta o intervalo\n",
    "    xm = (a+b)/2\n",
    "\n",
    "    i = 1 # contador \n",
    "\n",
    "    while abs(a-b) > tol and ( not done or N != 0 ):\n",
    "    # avalia a função no ponto médio\n",
    "        fxm = f(xm)\n",
    "        print(\"(i = {0:d}) f(xm)={1:f} | f(a)={2:f} | f(b)={3:f}\".format(i,fxm,fa,fb))\n",
    "   \n",
    "        if fa*fxm < 0:       # Raiz esta à esquerda de xm\n",
    "            b = xm\n",
    "            fb = fxm\n",
    "            xm = (a+b)/2\n",
    "        elif fxm*fb < 0:     # Raiz esta à direita de xm\n",
    "            a = xm\n",
    "            fa = fxm\n",
    "            xm = (a+b)/2\n",
    "        else:               # Achamos a raiz\n",
    "            done = 1\n",
    "    \n",
    "        N -= 1              # Atualiza passo\n",
    "        i += 1              # Atualiza contador\n",
    "            \n",
    "    print(\"Solução encontrada: {0}\".format(xm))\n",
    "\n",
    "    return xm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.optimize import bisect, newton\n",
    "\n",
    "# Dados de entrada\n",
    "\n",
    "t = np.arange(0,520,1)   # tempo [s]\n",
    "c = 1.46   # coeficiente de arrasto [kg/s]\n",
    "m = 90     # massa [kg]\n",
    "g = 9.81   # constante de gravidade [m/s2]\n",
    "\n",
    "# Dados de saída\n",
    "\n",
    "## velocidade terminal [m/s]\n",
    "v_ms1 = (g*m/c)*(1 - np.exp((-c/m)*t))\n",
    "\n",
    "# velocidade terminal [km/h]\n",
    "v_kh1 = (1/3.6)*v_ms1;\n",
    "\n",
    "# gráfico tempo x velocidade\n",
    "plt.figure\n",
    "plt.plot(t,v_ms1)\n",
    "plt.xlabel('t [s]')\n",
    "plt.ylabel('v [m/s]')\n",
    "plt.title('Velocidade terminal - paraquedista');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(c) = -42 + 686.7*(1 - exp(-6*c/35))/c\n",
      "\n",
      "(i = 1) f(xm)=2.600284 | f(a)=7.910578 | f(b)=-1.844622\n",
      "(i = 2) f(xm)=0.281205 | f(a)=2.600284 | f(b)=-1.844622\n",
      "(i = 3) f(xm)=-0.804573 | f(a)=0.281205 | f(b)=-1.844622\n",
      "(i = 4) f(xm)=-0.267556 | f(a)=0.281205 | f(b)=-0.804573\n",
      "(i = 5) f(xm)=0.005337 | f(a)=0.281205 | f(b)=-0.267556\n",
      "(i = 6) f(xm)=-0.131479 | f(a)=0.005337 | f(b)=-0.267556\n",
      "(i = 7) f(xm)=-0.063164 | f(a)=0.005337 | f(b)=-0.131479\n",
      "(i = 8) f(xm)=-0.028937 | f(a)=0.005337 | f(b)=-0.063164\n",
      "(i = 9) f(xm)=-0.011806 | f(a)=0.005337 | f(b)=-0.028937\n",
      "(i = 10) f(xm)=-0.003236 | f(a)=0.005337 | f(b)=-0.011806\n",
      "(i = 11) f(xm)=0.001050 | f(a)=0.005337 | f(b)=-0.003236\n",
      "(i = 12) f(xm)=-0.001093 | f(a)=0.001050 | f(b)=-0.003236\n",
      "(i = 13) f(xm)=-0.000022 | f(a)=0.001050 | f(b)=-0.001093\n",
      "(i = 14) f(xm)=0.000514 | f(a)=0.001050 | f(b)=-0.000022\n",
      "(i = 15) f(xm)=0.000246 | f(a)=0.000514 | f(b)=-0.000022\n",
      "(i = 16) f(xm)=0.000112 | f(a)=0.000246 | f(b)=-0.000022\n",
      "(i = 17) f(xm)=0.000045 | f(a)=0.000112 | f(b)=-0.000022\n",
      "(i = 18) f(xm)=0.000012 | f(a)=0.000045 | f(b)=-0.000022\n",
      "(i = 19) f(xm)=-0.000005 | f(a)=0.000012 | f(b)=-0.000022\n",
      "Solução encontrada: 15.127429962158203\n"
     ]
    }
   ],
   "source": [
    "import sympy as sp\n",
    "\n",
    "time = 12     # tempo [s]\n",
    "mass = 70     # massa [kg]\n",
    "vel = 42     # velocidade [m/s]\n",
    "grav = 9.81   # constante de gravidade [m/s2]\n",
    "\n",
    "# defino variáveis simbólicas\n",
    "g,m,t,v,c = sp.symbols('g,m,t,v,c')\n",
    "\n",
    "# expressão geral\n",
    "f_g = (g*m/c)*(1 - sp.exp((-c/m)*t)) - v\n",
    "\n",
    "# expressão particular com valores substituídos \n",
    "# convertida para string\n",
    "f_s = str(f_g.subs({'g':grav,'m':mass,'v':vel,'t':time}))\n",
    "\n",
    "# TODO\n",
    "# para esta função, teremos que substituir 'exp' por 'np.exp')\n",
    "print('f(c) = ' + f_s + '\\n')\n",
    "f_s = '-42 + 686.7*(1 - np.exp(-6*c/35))/c'\n",
    "\n",
    "# resolve bisseção\n",
    "xm = bissecao(f_s,12,16,1e-5,100,'c')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(i = 1) f(xm)=-1.375000 | f(a)=3.000000 | f(b)=-5.000000\n",
      "(i = 2) f(xm)=0.765625 | f(a)=3.000000 | f(b)=-1.375000\n",
      "(i = 3) f(xm)=-0.322266 | f(a)=0.765625 | f(b)=-1.375000\n",
      "(i = 4) f(xm)=0.218018 | f(a)=0.765625 | f(b)=-0.322266\n",
      "(i = 5) f(xm)=-0.053131 | f(a)=0.218018 | f(b)=-0.322266\n",
      "(i = 6) f(xm)=0.082203 | f(a)=0.218018 | f(b)=-0.053131\n",
      "(i = 7) f(xm)=0.014474 | f(a)=0.082203 | f(b)=-0.053131\n",
      "(i = 8) f(xm)=-0.019344 | f(a)=0.014474 | f(b)=-0.053131\n",
      "(i = 9) f(xm)=-0.002439 | f(a)=0.014474 | f(b)=-0.019344\n",
      "(i = 10) f(xm)=0.006017 | f(a)=0.014474 | f(b)=-0.002439\n",
      "Solução encontrada: 0.33740234375\n",
      "O número de iterações é k = 9.96578 ≈ 10.\n"
     ]
    }
   ],
   "source": [
    "# Exemplo 5.1\n",
    "f = 'x**3 - 9*x + 3'\n",
    "a = 0\n",
    "b = 1\n",
    "tol = 1e-3\n",
    "N = 100\n",
    "var = 'x'\n",
    "bissecao(f,a,b,tol,N,var)\n",
    "\n",
    "# relação erro desejado x número de iterações\n",
    "k = np.log2((b-a)/tol)\n",
    "print(\"O número de iterações é k = {0:g} ≈ {1:g}.\".format(k,np.round(k)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tarefas\n",
    "\n",
    "* Melhore o código Python tratando os TODOs: \n",
    "\n",
    "Tente generalizar o código da bisseção para que identifique automaticamente a variável de entrada utilizada pelo usuário (use expressões regulares e remova o argumento `var` da definição da função).\n",
    "\n",
    "Note que o trecho simbólico abaixo foi necessário para substituir a função da chamada `exp`, não interpretada por `eval` por uma nova string que usasse `np.exp`.\n",
    "```python\n",
    "# TODO\n",
    "# para esta função, teremos que substituir 'exp' por 'np.exp')\n",
    "print('f(c) = ' + f_s + '\\n')\n",
    "f_s = '-42 + 686.7*(1 - np.exp(-6*c/35))/c'\n",
    "```\n",
    "Tente fazer as correções necessárias no código. **Sugestão:** verifique a função `sympy.core.evalf` do módulo `sympy`)\n",
    "\n",
    "* Adicione mecanismos de plotagem no código Python \n",
    "\n",
    "* Crie um código em Javascript para adicionarmos na página do projeto Numbiosis com o máximo possível de GUI (labels + input data).\n",
    "\n",
    "* Teste a implementação com um problema realista."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Problema sugerido:**\n",
    "Uma reação química reversível\n",
    "\n",
    "$$2A+B \\iff C$$\n",
    "\n",
    "pode ser caracterizada pela relação de equilíbrio\n",
    "\n",
    "$$K = \\dfrac{c_c}{c_a^2c_b},$$\n",
    "\n",
    "onde a nomenclatura $c_i$ representa a concentração do constituinte $i$. Suponha que definamos uma variável $x$ como o número de moles de $C$ que são produzidos. A conservação da massa pode ser usada para reformular a relação de equilíbrio como\n",
    "\n",
    "$$K = \\dfrac{(c_{c,0} + x)}{(c_{a,0} - 2x)^2 (c_{b,0} - x),}$$\n",
    "\n",
    "onde o subscrito $0$ designa a concentração inicial de cada constituinte. Se $K = 0,016$, $c_{a,0} = 42$, $c_{b,0} = 28$ e $c_{c,0} = 4$, determine o valor de $x$. \n",
    "\n",
    "(a) Obtenha a solução graficamente. \n",
    "\n",
    "(b) Com base em (a), resolva a raiz com suposições iniciais de $x_l = 0$ e $x_u = 20$, com critério de erro de $\\epsilon_s = 0,5\\%$. (Vide clipping _Definições de erro_ para entender $\\epsilon_s$.)\n",
    "\n",
    "(c) Use o método da bisseção."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tarefa: Falsa Posição\n",
    "Programe uma nova função para executar o método da falsa posição ou estenda o código anterior para uma nova função que contemple os dois casos (sugestão: use `switch... case...`).\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.core.display import HTML\n",
    "\n",
    "def css_styling():\n",
    "    styles = open(\"styles/custom.css\", \"r\").read()\n",
    "    return HTML(styles)\n",
    "css_styling();"
   ]
  }
 ],
 "metadata": {
  "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}