aula-18-integracao-newtonCotes.ipynb

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   "source": [
    "# Integração Numérica: Regras de Newton-Cotes\n",
    "\n",
    "As fórmulas de Newton-Cotes são os esquemas mais comuns de integração numérica. Elas são baseadas na estratégia de substituir uma função complicada ou dados tabulados por uma função aproximadora simples que seja fácil de integrar:\n",
    "\n",
    "$$\n",
    "I = \\int _a^b f(x)dx \\cong \\int _a^b f_n (x)dx \\qquad (1)\n",
    "$$\n",
    "\n",
    "em que $f_n (x)$ é um polinômio da forma\n",
    "\n",
    "$$\n",
    "f_n (x) = a_0 + a_1 x + \\dots + a_{n-1} x^{n-1} + a_n x^n\n",
    "$$\n",
    "\n",
    "em que $n$ é o grau do polinômio.\n",
    "\n",
    "## A Regra do Trapézio\n",
    "\n",
    "A **regra do trapézio** corresponde ao caso no qual o polinômio na Equação (1) é de primeiro grau:\n",
    "\n",
    "$$\n",
    "I = \\int _a^b f(x)dx \\cong \\int _a^b f_1 (x)dx\n",
    "$$\n",
    "\n",
    "onde\n",
    "\n",
    "$$\n",
    "f_1 (x) = f(a) + \\dfrac{f(b) - f(a)}{b - a} (x - a)\n",
    "$$\n",
    "\n",
    "O resultado da integração é \n",
    "\n",
    "$$\n",
    "I = (b - a) \\dfrac{f(a) + f(b)}{2} \\qquad (2)\n",
    "$$\n",
    "\n",
    "Geometricamente, a regra dos trapézios é equivalente a aproximar a integral pela área do trapézio sob a reta ligando $f(a)$ e $f(b)$ (conforme figura abaixo). Portanto, a estimativa da integral pode ser representada por\n",
    "\n",
    "$$\n",
    "I \\cong \\text{largura} \\times \\text{altura média} \\\\\n",
    "I \\cong (b - a) \\times \\text{altura média}\n",
    "$$\n",
    "\n",
    "em que, para a regra dos trapézios, a altura média é a média dos valores da função nas extremidades, ou $[f (a) + f (b)]/2$, conforme Equação (2)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import sympy as sy\n",
    "import scipy as sp\n",
    "from scipy import interpolate\n",
    "from scipy import integrate\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Representação geométrica da regra do trapézio\n",
    "\n",
    "def f(x):\n",
    "    return -x**2 + 15*x - 10\n",
    "\n",
    "x = np.linspace(0.5,10,100)\n",
    "y = f(x)\n",
    "\n",
    "plt.plot(x,y, label = 'f(x)')\n",
    "plt.plot([2,8], [f(2),f(8)], 'r', label = 'f1(x)')\n",
    "plt.plot(2,f(2), 'o', label = 'f(a)')\n",
    "plt.plot(8,f(8), 'o', label = 'f(b)')\n",
    "plt.plot(np.mean([2,8]), np.mean([f(2),f(8)]), 'o', label = 'Altura Média')\n",
    "plt.fill([2, 2, 8, 8],[0, f(2), f(8), 0],color='r',alpha=0.4, label = 'Área do Trapézio')\n",
    "plt.legend()\n",
    "plt.axvline(x=0,color='k',linewidth=0.6,linestyle='--')\n",
    "plt.axhline(y=0,color='k',linewidth=0.6,linestyle='--');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Aplicação Múltipla da Regra do Trapézio\n",
    "\n",
    "Uma maneira de melhorar a acurácia da regra do trapézio é dividir o intervalo de integração de $a$ a $b$ em diversos segmentos e aplicar o método a cada segmento (conforme figura abaixo). As áreas correspondentes aos segmentos individuais podem então ser somadas parafornecer a integral para o intervalo inteiro. As equações resultantes são chamadas fórmulas de integração por aplicações múltiplas ou compostas.\n",
    "\n",
    "Existem $n + 1$ pontos base igualmente espaçados $(x_0, x_1, x_2, \\dots , x_n)$. Consequentemente, existem $n$ segmentos de largura igual:\n",
    "\n",
    "$$\n",
    "h = \\dfrac{b − a}{n} \\qquad (3)\n",
    "$$\n",
    "\n",
    "Se $a$ e $b$ forem designados por $x_0$ e $x_n$, respectivamente, a integral total pode ser representada como\n",
    "\n",
    "$$\n",
    "I = \\int _{x_0}^{x_1} f(x)dx + \\int _{x_1}^{x_2} f(x)dx + \\dots + \\int _{x_{n - 1}}^{x_n} f(x)dx \n",
    "$$\n",
    "\n",
    "Substituindo cada integral pela regra do trapézio, obtém-se\n",
    "\n",
    "$$\n",
    "I = h \\dfrac{f(x_0) + f(x_1)}{2} + h \\dfrac{f(x_1) + f(x_2)}{2} + \\dots + h \\dfrac{f(x_{n - 1}) + f(x_n)}{2}\n",
    "$$\n",
    "\n",
    "ou, agrupando termos,\n",
    "\n",
    "$$\n",
    "I = \\dfrac{h}{2} \\left[ f(x_0) + 2 \\sum _{i = 1} ^{n - 1} f(x_i) + f(x_n) \\right] \\qquad (4)\n",
    "$$"
   ]
  },
  {
   "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": [
    "# Representação geométrica da aplicação múltipla da regra do trapézio\n",
    "# É possível observar, visualmente, que houve uma redução do erro do resultado em comparação com o caso anterior\n",
    "\n",
    "xd = np.array([2,5,8])\n",
    "yd = f(xd)\n",
    "\n",
    "plt.plot(x,y, label = 'f(x)')\n",
    "plt.plot(xd,yd, 'r')\n",
    "plt.plot(2,f(2), 'o', label = 'f(x0)')\n",
    "plt.plot(5,f(5), 'o', label = 'f(x1)')\n",
    "plt.plot(8,f(8), 'o', label = 'f(x2)')\n",
    "plt.fill([2, 2, 5, 5],[0, f(2), f(5), 0], color='r', alpha=0.6, label = 'Área do Trapézio 1')\n",
    "plt.fill([5, 5, 8, 8],[0, f(5), f(8), 0], color='r', alpha=0.3, label = 'Área do Trapézio 2')\n",
    "plt.legend()\n",
    "plt.axvline(x=0,color='k',linewidth=0.6,linestyle='--')\n",
    "plt.axhline(y=0,color='k',linewidth=0.6,linestyle='--'); "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A Regra 1/3 de Simpson\n",
    "\n",
    "A **regra 1/3 de Simpson** é obtida quando um polinômio interpolador de segundo grau é substituído na Equação (1):\n",
    "\n",
    "$$\n",
    "I = \\int _a^b f(x)dx \\cong \\int _a^b f_2 (x)dx\n",
    "$$\n",
    "\n",
    "Se $a$ e $b$ forem designados por $x_0$ e $x_2$ e se $f_2(x)$ for representado por um polinômio de Lagrange de segundo grau, a integral se torna\n",
    "\n",
    "$$\n",
    "I = \\int _{x_0}^{x_2} \\left[ \\dfrac{(x - x_1)(x - x_2)}{(x_0 - x_1)(x_0 - x_2)} f(x_0) + \\dfrac{(x - x_1)(x - x_2)}{(x_0 - x_1)(x_0 - x_2)} f(x_1) + \\dfrac{(x - x_1)(x - x_2)}{(x_0 - x_1)(x_0 - x_2)} f(x_2) \\right] dx\n",
    "$$\n",
    "\n",
    "Depois da integração e de manipulações algébricas, obtém-se a seguinte fórmula:\n",
    "\n",
    "$$\n",
    "I \\cong \\dfrac{h}{3} [f(x_0) + 4 f(x_1) + f(x_2)] \\qquad (5)\n",
    "$$\n",
    "\n",
    "em que, para esse caso\n",
    "\n",
    "$$\n",
    "h = \\dfrac{b − a}{2} \\qquad (6)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "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": [
    "# Descrição gráfica da regra 1/3 de Simpson: ela consiste em tomar a área sob uma parábola ligando três pontos\n",
    "\n",
    "def f(x):\n",
    "    return 0.6*np.sin(x+np.pi/2) + 0.2*x + 2\n",
    "\n",
    "def f2(x):\n",
    "    return -0.127382111059529*x**2 + 1.73647155618593*x - 2.0569711669823 # Obtido pelo MMQ\n",
    "\n",
    "x = np.linspace(2,10,100)\n",
    "y = f(x)\n",
    "\n",
    "xc = np.linspace(2,10,100)\n",
    "yc = f2(xc)\n",
    "\n",
    "xf = np.linspace(3,9,100)\n",
    "yf = f2(xf)\n",
    "\n",
    "xf = np.concatenate((np.array([3]), xf, np.array([9])))\n",
    "yf = np.concatenate((np.array([0]), yf, np.array([0])))\n",
    "\n",
    "plt.plot(x,y, label = 'f(x)')\n",
    "plt.plot(xc,yc, ':r', label = 'f2(x)')\n",
    "plt.plot(3,f(3), 'o', label = 'f(x0)')\n",
    "plt.plot(6,f(6), 'o', label = 'f(x1)')\n",
    "plt.plot(9,f(9), 'o', label = 'f(x2)')\n",
    "plt.fill(xf,yf, color='r', alpha=0.4)\n",
    "plt.legend()\n",
    "plt.axvline(x=0,color='k',linewidth=0.6,linestyle='--')\n",
    "plt.axhline(y=0,color='k',linewidth=0.6,linestyle='--');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Aplicações Múltiplas da Regra 1/3 de Simpson\n",
    "\n",
    "Do mesmo modo como no caso da regra do trapézio, a regra de Simpson pode ser melhorada dividindo-se o intervalo de integração em diversos segmentos de mesmo comprimento\n",
    "\n",
    "$$\n",
    "h = \\dfrac{b − a}{2}\n",
    "$$\n",
    "\n",
    "A integral total pode ser representada como\n",
    "\n",
    "$$\n",
    "I = \\int _{x_0}^{x_2} f(x)dx + \\int _{x_2}^{x_4} f(x)dx + \\dots + \\int _{x_{n - 2}}^{x_n} f(x)dx \n",
    "$$\n",
    "\n",
    "Substituindo cada integral individual pela regra 1/3 de Simpson, obtemos\n",
    "\n",
    "$$\n",
    "I \\cong 2 h \\dfrac{f(x_0) + 4 f(x_1) + f(x_2)}{6} + 2 h \\dfrac{f(x_2) + 4 f(x_3) + f(x_4)}{6} + \\dots + 2 h \\dfrac{f(x_{n - 2}) + 4 f(x_{n - 1}) + f(x_n)}{6}\n",
    "$$\n",
    "\n",
    "ou \n",
    "\n",
    "$$\n",
    "I \\cong (b - a) \\dfrac{ f(x_0) + 4 \\sum _{i = 1, 3, 5}^{n-1} f(x_i) + 2 \\sum _{i = 2, 4, 6}^{n-2} f(x_j) + f(x_n) }{3 n} \\qquad (7)\n",
    "$$\n",
    "\n",
    "## Regra 3/8 de Simpson\n",
    "\n",
    "De uma maneira parecida com a dedução da regra do trapézio e da regra 1/3 de Simpson, um polinômio de Lagrange de ordem três pode ser ajustado a quatro pontos e integrado\n",
    "\n",
    "$$\n",
    "I = \\int _a^b f(x)dx \\cong \\int _a^b f_3 (x)dx\n",
    "$$\n",
    "\n",
    "para fornecer\n",
    "\n",
    "$$\n",
    "I \\cong \\dfrac{3 h}{8} [f(x_0) + 3 f(x_1) + 3 f(x_2) + f(x_3)] \\qquad (8)\n",
    "$$\n",
    "\n",
    "em que\n",
    "\n",
    "$$\n",
    "h = \\dfrac{b − a}{3} \\qquad (9)\n",
    "$$\n",
    "\n",
    "Essa equação é chamada regra 3/8 de Simpson porque $h$ é multiplicada por 3/8."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": [
    "# Descrição gráfica da regra 3/8 de Simpson: ela consiste em tomar a área sob uma equação cúbica ligando quatro pontos\n",
    "\n",
    "def f3(x):\n",
    "    return -0.0166482246490701*x**3 + 0.189467273747325*x**2 - 0.117878777555138*x + 1.10393743650415 # Obtido pelo MMQ\n",
    "\n",
    "xc = np.linspace(2,10,100)\n",
    "yc = f3(xc)\n",
    "\n",
    "xf = np.linspace(3,9,100)\n",
    "yf = f3(xf)\n",
    "\n",
    "xf = np.concatenate((np.array([3]), xf, np.array([9])))\n",
    "yf = np.concatenate((np.array([0]), yf, np.array([0])))\n",
    "\n",
    "plt.plot(x,y, label = 'f(x)')\n",
    "plt.plot(xc,yc, ':r', label = 'f3(x)')\n",
    "plt.plot(3, f(3), 'o', label = 'f(x0)')\n",
    "plt.plot(5, f(5), 'o', label = 'f(x1)')\n",
    "plt.plot(7, f(7), 'o', label = 'f(x2)')\n",
    "plt.plot(9, f(9), 'o', label = 'f(x3)')\n",
    "plt.fill(xf,yf, color='r', alpha=0.4)\n",
    "plt.legend()\n",
    "plt.axvline(x=0,color='k',linewidth=0.6,linestyle='--')\n",
    "plt.axhline(y=0,color='k',linewidth=0.6,linestyle='--');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integração com Segmentos Desiguais\n",
    "\n",
    "Até esse ponto, todas as fórmulas para integração numérica foram baseadas em dados igualmente espaçados. Na prática, existem muitas situações nas quais essa hipótese não é válida e precisamos lidar com segmentos de tamanhos distintos. Por exemplo, dados obtidos experimentalmente, muitas vezes, são desse tipo. Para tais casos, um método é aplicar a regra do trapézio para cada segmento e somar os resultados:\n",
    "\n",
    "$$\n",
    "I = h_1 \\dfrac{f(x_0) + f(x_1)}{2} + h_2 \\dfrac{f(x_1) + f(x_2)}{2} + \\dots + h_n \\dfrac{f(x_{n - 1}) + f(x_n)}{2}\n",
    "$$\n",
    "\n",
    "em que $h_i$ é a largura do segmento $i$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementações de Newton-Cotes: Regra do Trapézio e 1/3 Simpson Generalizadas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ingegral exata (simbólica): -9700*log(114000) + 9700*log(122649)\n",
      "Integral exata (numérica): 709.3432392521672\n",
      "*** Método de integração: TRAPEZIO ***\n",
      "no. de pontos de integração = 11 \t I = 644.6326388025 \t EREL = 9.1226076276%\n",
      "no. de pontos de integração = 101 \t I = 702.3175905243 \t EREL = 0.9904441657%\n",
      "no. de pontos de integração = 1001 \t I = 708.6345799441 \t EREL = 0.0999035825%\n",
      "no. de pontos de integração = 10001 \t I = 709.2723117737 \t EREL = 0.0099990350%\n",
      "no. de pontos de integração = 100001 \t I = 709.3361458882 \t EREL = 0.0009999903%\n",
      "*** Método de integração: 13SIMPSON ***\n",
      "no. de pontos de integração = 11 \t I = 644.8576901994 \t EREL = 9.0908809000%\n",
      "no. de pontos de integração = 101 \t I = 702.3200388852 \t EREL = 0.9900990068%\n",
      "no. de pontos de integração = 1001 \t I = 708.6346046475 \t EREL = 0.0999000999%\n",
      "no. de pontos de integração = 10001 \t I = 709.2723120210 \t EREL = 0.0099990001%\n",
      "no. de pontos de integração = 100001 \t I = 709.3361458907 \t EREL = 0.0009999900%\n"
     ]
    }
   ],
   "source": [
    "''' Newton-Cotes: Regra do Trapezio\n",
    "    assume Y igualmente espaçado e \n",
    "    com pelo menos 2 pontos \n",
    "'''\n",
    "def integral_trapezio(h,Y):\n",
    "    val = 0.0\n",
    "    for i in range(1,Y.size-1):\n",
    "        val += 2*Y[i]\n",
    "    val = 0.5*h*( val + Y[0] + Y[-1] )\n",
    "    return val\n",
    "\n",
    "''' Newton-Cotes: Regra 1/3 de Simpson\n",
    "    assume Y igualmente espaçado e \n",
    "    com pelo menos 3 pontos \n",
    "'''\n",
    "def integral_onethird_simpson(h,Y):\n",
    "    val = 0.0\n",
    "    for i in range(1,Y.size-1,2):\n",
    "        val += 4*Y[i]\n",
    "    \n",
    "    for i in range(2,Y.size-2,2):\n",
    "        val += 2*Y[i]        \n",
    "    \n",
    "    val = h/3.0*( val + Y[0] + Y[-1] )\n",
    "    return val\n",
    "\n",
    "''' HELPER''' \n",
    "def print_metodo(flag):            \n",
    "    print('*** Método de integração: ' + str.upper(flag) + ' ***')\n",
    "\n",
    "# limites de integração\n",
    "a = 0\n",
    "b = 93\n",
    "\n",
    "# integrando \n",
    "f = lambda v: 97000*v/(5*v**2 + 570000)\n",
    "\n",
    "# pontos de integração (ímpares para testar com 1/3 simpson)\n",
    "npi = [11,101,1001,10001,100001]\n",
    "\n",
    "# integral exata\n",
    "x = sy.Symbol('x')\n",
    "vex = sy.integrate(f(x),(x,a,b))\n",
    "print('Ingegral exata (simbólica): ' + str(vex))\n",
    "vex = float(vex)\n",
    "print('Integral exata (numérica): ' + str(vex))\n",
    "\n",
    "metodos = ['trapezio','13simpson']\n",
    "no = False\n",
    "\n",
    "# integração numérica\n",
    "for metodo in metodos:\n",
    "    I = []\n",
    "    EREL = []\n",
    "    for n in npi:\n",
    "        v = np.linspace(a,b,num=n,endpoint=True)\n",
    "        fv = f(v)\n",
    "        h = (b-a)/n    \n",
    "                \n",
    "        if metodo == 'trapezio':\n",
    "            if no == False:\n",
    "                print_metodo(metodo)\n",
    "                no = True            \n",
    "            val = integral_trapezio(h,fv)\n",
    "            erel = abs((vex - val)/vex)*100\n",
    "            I.append(val)\n",
    "            EREL.append(erel)\n",
    "            print(\"no. de pontos de integração = {0:d} \\t I = {1:.10f} \\t EREL = {2:.10f}%\".format(n,val,erel) )\n",
    "\n",
    "        elif metodo == '13simpson':\n",
    "            if no == False:\n",
    "                print_metodo(metodo)\n",
    "                no = True\n",
    "            val = integral_onethird_simpson(h,fv)\n",
    "            erel = abs((vex - val)/vex)*100\n",
    "            I.append(val)\n",
    "            EREL.append(erel)        \n",
    "            print(\"no. de pontos de integração = {0:d} \\t I = {1:.10f} \\t EREL = {2:.10f}%\".format(n,val,erel) )\n",
    "    \n",
    "    no = False\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tarefa\n",
    "Implemente uma função para realizar a integração numérica pela regra 3/8 de Simpson, use-a para calcular o valor de \n",
    "\n",
    "$$\\int_{a=0}^{b=93} \\frac{97000v}{(5v^2 + 570000)} \\, dx$$ \n",
    "\n",
    "e compare o resultado com os obtidos pelas anteriores."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integração simbólica \n",
    "\n",
    "Vamos ver alguns exemplos de integração simbólica. Para termos uma impressão mais elegante de expressões, antes usamos a seguinte instrução:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "sy.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regra quadratura de Simpson\n",
    "\n",
    "Vamos usar símbolos para chegar à expressão da regra de quadratura de Simpson."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "a,b,X = sy.symbols(\"a, b, x\")\n",
    "f = sy.Function(\"f\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definimos tuplas para os pontos de amostra e pesos."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = a, (a + b)/2, b # ponto médio \n",
    "w = [sy.symbols(\"w_%d\" % i) for i in range(len(x))] # pesos "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle w_{0} f{\\left(a \\right)} + w_{1} f{\\left(\\frac{a}{2} + \\frac{b}{2} \\right)} + w_{2} f{\\left(b \\right)}$"
      ],
      "text/plain": [
       "              ⎛a   b⎞          \n",
       "w₀⋅f(a) + w₁⋅f⎜─ + ─⎟ + w₂⋅f(b)\n",
       "              ⎝2   2⎠          "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = sum([w[i] * f(x[i]) for i in range(len(x))])\n",
    "q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para calcular valores aproximados dos pesos $w_i$, escolhemos a base polinomial \n",
    "\n",
    "$$\\{ \\phi_n(x) = x^n \\}_{n=0}^2$$\n",
    "\n",
    "para a interpolação de $f(x)$ e um objeto simbólico para representar cada uma dessas funções."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( x \\mapsto 1 \\right), \\  \\left( x \\mapsto x \\right), \\  \\left( x \\mapsto x^{2} \\right)\\right]$"
      ],
      "text/plain": [
       "⎡                   2⎤\n",
       "⎣x ↦ 1, x ↦ x, x ↦ x ⎦"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phi = [sy.Lambda(X, X**n) for n in range(len(x))]\n",
    "phi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora temos que descobrir os valores dos pesos. A integral $\\int_a^b \\phi_n(x) \\, dx$ pode ser calculada analiticamente para cada função de base. Isto nos ajuda a resolver o seguinte sistema:\n",
    "\n",
    "$$\\sum\\limits_{i=0}^2 w_i \\phi_n(x_i) = \\int_a^b \\phi_n(x) \\, dx$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O sistema pode ser construído no `sympy` da seguinte forma: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ a - b + w_{0} + w_{1} + w_{2}, \\  \\frac{a^{2}}{2} + a w_{0} - \\frac{b^{2}}{2} + b w_{2} + w_{1} \\left(\\frac{a}{2} + \\frac{b}{2}\\right), \\  \\frac{a^{3}}{3} + a^{2} w_{0} - \\frac{b^{3}}{3} + b^{2} w_{2} + w_{1} \\left(\\frac{a}{2} + \\frac{b}{2}\\right)^{2}\\right]$"
      ],
      "text/plain": [
       "⎡                       2           2                       3            3    \n",
       "⎢                      a           b              ⎛a   b⎞  a     2      b     \n",
       "⎢a - b + w₀ + w₁ + w₂, ── + a⋅w₀ - ── + b⋅w₂ + w₁⋅⎜─ + ─⎟, ── + a ⋅w₀ - ── + b\n",
       "⎣                      2           2              ⎝2   2⎠  3            3     \n",
       "\n",
       "                 2⎤\n",
       "2         ⎛a   b⎞ ⎥\n",
       " ⋅w₂ + w₁⋅⎜─ + ─⎟ ⎥\n",
       "          ⎝2   2⎠ ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqs = [q.subs(f, phi[n]) - sy.integrate(phi[n](X), (X, a, b)) for n in range(len(phi))]\n",
    "eqs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em seguida, resolvemos o sistema para obter as expressões analíticas para os pesos:  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ w_{0} : - \\frac{a}{6} + \\frac{b}{6}, \\  w_{1} : - \\frac{2 a}{3} + \\frac{2 b}{3}, \\  w_{2} : - \\frac{a}{6} + \\frac{b}{6}\\right\\}$"
      ],
      "text/plain": [
       "⎧      a   b        2⋅a   2⋅b        a   b⎫\n",
       "⎨w₀: - ─ + ─, w₁: - ─── + ───, w₂: - ─ + ─⎬\n",
       "⎩      6   6         3     3         6   6⎭"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w_sol = sy.solve(eqs,w)\n",
    "w_sol"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substituindo a solução na expressão simbólica para a regra de quadratura:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{\\left(a - b\\right) \\left(f{\\left(a \\right)} + f{\\left(b \\right)} + 4 f{\\left(\\frac{a}{2} + \\frac{b}{2} \\right)}\\right)}{6}$"
      ],
      "text/plain": [
       "         ⎛                 ⎛a   b⎞⎞ \n",
       "-(a - b)⋅⎜f(a) + f(b) + 4⋅f⎜─ + ─⎟⎟ \n",
       "         ⎝                 ⎝2   2⎠⎠ \n",
       "────────────────────────────────────\n",
       "                 6                  "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.subs(w_sol).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos verificar no material que esta é, de fato, a expressão para a regra de quadratura de Simpson."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integração múltipla\n",
    "\n",
    "A integração em 2 ou mais variáveis pode ser feita usando as funções `dblquad`, `tplquad` e `nquad`, onde o número de funções e de limites de integração deve se adequar ao tipo de integral.\n",
    "\n",
    "Abaixo, temos alguns exemplos: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Integração dupla\n",
    "\n",
    "Neste exemplo, integramos \n",
    "$$\\int_0^1\\int_0^1 e^{-x^2 - y^2} \\, dx dy$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 0.5577462853510337, \\  8.291374381535408e-15\\right)$"
      ],
      "text/plain": [
       "(0.5577462853510337, 8.291374381535408e-15)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate.dblquad(lambda x, y: np.exp(-x**2-y**2), 0, 1, lambda x: 0, lambda x: 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Integração tripla\n",
    "\n",
    "Neste exemplo, integramos \n",
    "$$\\int_0^1\\int_0^1\\int_0^1 e^{-x^2 - y^2 - z^2} \\, dx dy dz$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 0.4165383858866382, \\  8.291335287314424e-15\\right)$"
      ],
      "text/plain": [
       "(0.4165383858866382, 8.291335287314424e-15)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def f(x, y, z): \n",
    "    return np.exp(-x**2-y**2-z**2) \n",
    "\n",
    "a, b = 0, 1\n",
    "g, h = lambda x: 0, lambda x: 1\n",
    "q, r = lambda x, y: 0, lambda x, y: 1\n",
    "integrate.tplquad(f, 0, 1, g, h, q, r) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Nota\n",
    "\n",
    "O presente texto (conteúdo teórico) é um resumo baseado no livro Métodos Numéricos para Engenharia (Chapra e Canale)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "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
}