aula-19-quadratura-gaussiana.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Integração Numérica: Quadratura Gaussiana\n",
    "\n",
    "Uma característica das fórmulas de Newton-Cotes é estimar a integral com base em valores da função tomados em nós igualmente espaçados. Consequentemente, a posição desses nós é predeterminada, isto é, fixada.\n",
    "\n",
    "Agora, suponhamos que a restrição de ter nós fixados seja removida e que pudéssemos escolher arbitrariamente as posições por eles ocupadas. Com uma escolha adequada, será que conseguiríamos reduzir o erro de integração? \n",
    "\n",
    "De fato, o método da Quadratura Gaussiana permite-nos estabelecer um conjunto mínimo de *nós* e *pesos* que realizam uma espécie de \"balanceamento\" do erro (subestimação vs. superestimação) e determinam aproximações numéricas de integrais com alta ordem.\n",
    "\n",
    "As figuras abaixo mostram, por exemplo, como a integral de uma função $f(x)$ é melhor calculada através da colocação de pontos em posições \"especiais\". A regra do trapézio é posta em contraste com uma outra regra, que aprenderemos, cujo par de pontos, embora defina também um trapézio, ajuda a estimar melhor a integral de $f(x)$ do que o observado no primeiro caso. \n",
    "\n",
    "Em particular, daremos enfoque à chamada *Quadratura de Gauss-Legendre*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy.polynomial.legendre as leg\n",
    "import numpy as np"
   ]
  },
  {
   "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"
    },
    {
     "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": [
    "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",
    "# Representação geométrica da regra do trapézio\n",
    "\n",
    "plt.figure()\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.fill([2, 2, 8, 8],[0, f(2), f(8), 0],color='r',alpha=0.4, label = 'I')\n",
    "plt.title('Regra 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='--')\n",
    "\n",
    "# Representação geométrica da quadratura de Gauss\n",
    "\n",
    "def f1(x):\n",
    "    return 5.0*x + 11.0\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(x,y, label = 'f(x)')\n",
    "plt.plot([2,8], [f1(2),f1(8)], 'r', label = 'f1(x)')\n",
    "plt.plot(3,f(3), 'o', label = 'f(a)')\n",
    "plt.plot(7,f(7), 'o', label = 'f(b)')\n",
    "plt.fill([2, 2, 8, 8],[0, f1(2), f1(8), 0],color='r',alpha=0.4, label = 'I')\n",
    "plt.title('Quadratura de Gauss')\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": [
    "## Dedução da Fórmula de Gauss-Legendre para Dois Pontos\n",
    "\n",
    "O objetivo da quadratura de Gauss-Legendre é determinar os coeficientes de uma equação da forma\n",
    "\n",
    "$$\n",
    "I \\cong c_0 f(x_0) + c_1 f(x_1) \\qquad (1)\n",
    "$$\n",
    "\n",
    "em que os $c$’s são coeficientes desconhecidos. Entretanto, em contraste com a regra do trapézio que usa extremidades fixas $a$ e $b$, os argumentos da função $x_0$ e $x_1$ não são fixados nas extremidades, mas são incógnitas. Assim, agora temos um total de quatro incógnitas que devem ser calculadas e necessitamos de quatro condições para determiná-las exatamente.\n",
    "\n",
    "Podemos obter duas dessas condições, supondo que a Equação (1) calcula a integral de uma função constante e de uma função linear exatamente. Então, para chegar a duas outras condições, simplesmente estendemos esse raciocínio supondo que ele também calculará a integral de uma função parabólica ($y = x^2$) e de uma cúbica ($y = x^3$) exatamente. Fazendo isso, determinamos todas as quatro incógnitas e, de quebra, deduzimos uma fórmula de integração linear de dois pontos que é exata para funções cúbicas. As quatro equações a serem resolvidas são:\n",
    "\n",
    "$$\n",
    "c_0 f(x_0) + c_1 f(x_1) = \\int _{-1}^{1} 1 dx = 2 \\qquad (2) \\\\\n",
    "c_0 f(x_0) + c_1 f(x_1) = \\int _{-1}^{1} x dx = 0 \\qquad (3) \\\\\n",
    "c_0 f(x_0) + c_1 f(x_1) = \\int _{-1}^{1} x^2 dx = \\dfrac{2}{3} \\qquad (4) \\\\\n",
    "c_0 f(x_0) + c_1 f(x_1) = \\int _{-1}^{1} x^3 dx = 0 \\qquad (5)\n",
    "$$\n",
    "\n",
    "As Equações (2) até (5) podem ser resolvidas simultaneamente por\n",
    "\n",
    "$$\n",
    "c_0 = c_1 = 1 \\\\\n",
    "x_0 = -\\dfrac{1}{\\sqrt{3}} \\\\\n",
    "x_1 = \\dfrac{1}{\\sqrt{3}}\n",
    "$$\n",
    "\n",
    "o que pode ser substituído na Equação (1) para fornecer a fórmula de Gauss-Legendre de dois pontos\n",
    "\n",
    "$$\n",
    "I \\cong f \\left( \\dfrac{-1}{\\sqrt{3}} \\right) + f \\left( \\dfrac{1}{\\sqrt{3}} \\right) \\qquad (6)\n",
    "$$\n",
    "\n",
    "Logo, chegamos ao interessante resultado que uma simples adição de valores da função em $x = 1/\\sqrt{3}$ e $−1/\\sqrt{3}$ fornece uma estimativa da integral que tem acurácia de terceira ordem.\n",
    "\n",
    "Observe que os extremos de integração nas Equações (2) até (5) são de $−1$ a $1$. Isso foi feito para simplificar a matemática e tornar a formulação tão geral quanto possível. Uma simples mudança de variável pode ser usada para transformar outros extremos de integração para essa fórmula, o que se faz supondo que uma nova variável $x_d$ está relacionada à variável original $x$ de uma forma linear como em\n",
    "\n",
    "$$\n",
    "x = a_0 + a_1 x_d \\qquad (7)\n",
    "$$\n",
    "\n",
    "Se o extremo inferior, $x = a$, corresponder a $x_d = −1$, esses valores podem ser substituídos na Equação (7) para fornecer\n",
    "\n",
    "$$\n",
    "a = a_0 + a_1 (- 1) \\qquad (8)\n",
    "$$\n",
    "\n",
    "Analogamente, o extremo superior, $x = b$, corresponde a $x_d = 1$, e fornece\n",
    "\n",
    "$$\n",
    "b = a_0 + a_1 (1) \\qquad (9)\n",
    "$$\n",
    "\n",
    "As Equações (8) e (9) podem ser resolvidas simultaneamente por\n",
    "\n",
    "$$\n",
    "a_0 = \\dfrac{b + a}{2}\n",
    "$$\n",
    "\n",
    "e\n",
    "\n",
    "$$\n",
    "a_1 = \\dfrac{b - a}{2}\n",
    "$$\n",
    "\n",
    "o que pode ser substituído na Equação (7) para fornecer\n",
    "\n",
    "$$\n",
    "x = \\dfrac{(b + a) + (b - a) x_d}{2} \\qquad (10)\n",
    "$$\n",
    "\n",
    "Essa equação pode ser derivada para dar\n",
    "\n",
    "$$\n",
    "dx = \\dfrac{b - a}{2} dx_d \\qquad (11)\n",
    "$$\n",
    "\n",
    "Os valores de $x$ e $dx$, nas Equações (10) e (11), respectivamente, podem ser substituídos na equação a ser integrada. Essas substituições transformam efetivamente o intervalo de integração sem mudar o valor da integral.\n",
    "\n",
    "## Fórmulas com Mais Pontos\n",
    "\n",
    "Além da fórmula de dois pontos descrita acima, versões com mais pontos podem ser desenvolvidas na forma geral\n",
    "\n",
    "$$\n",
    "I \\cong c_0 f(x_0) + c_1 f(x_1) + \\dots + c_{n−1} f(x_{n−1}) \\qquad (12)\n",
    "$$\n",
    "\n",
    "em que $n$ é o número de pontos. Os valores para os $c$’s e os $x$’s para até (e incluindo) a fórmula de quatro pontos estão resumidos na tabela abaixo.\n",
    "\n",
    "Pontos | Fatores de Peso | Argumentos da Função\n",
    "-|-|-\n",
    "3 | $c_0$ = 0.5555556 | $x_0$ = - 0.774596669\n",
    " | $c_1$ = 0.8888889 | $x_1$ = 0.0\n",
    " | $c_2$ = 0.5555556 | $x_2$ = 0.774596669\n",
    "4 | $c_0$ = 0.3478548 | $x_0$ = - 0.861136312\n",
    " | $c_1$ = 0.6521452 | $x_1$ = - 0.339981044\n",
    " | $c_2$ = 0.6521452 | $x_2$ = 0.339981044\n",
    " | $c_3$ = 0.3478548 | $x_3$ = 0.861136312"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementação\n",
    "\n",
    "Vamos implementar abaixo o código para gerar a tabela de pontos (nós) e pesos para integração numérica consoante as fórmulas de quadratura de **Gauss-Legendre**. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "# número de pontos de quadratura\n",
    "n = 20\n",
    "\n",
    "# pontos e pesos\n",
    "(pontos,pesos) = leg.leggauss(n)        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "QG1 = 30.642493444253617\n",
      "QG2 = 291.9238277183161\n",
      "QG3 = 456.0422383934014\n",
      "QG4 = 488.96142750746253\n",
      "QG5 = 492.48121331764094\n",
      "QG6 = 492.71920635748165\n",
      "QG7 = 492.7303338909227\n",
      "QG8 = 492.73071525454685\n",
      "QG9 = 492.73072524440545\n",
      "QG10 = 492.7307254508673\n",
      "QG11 = 492.7307254543184\n",
      "QG12 = 492.73072545436776\n",
      "QG13 = 492.73072545436753\n",
      "QG14 = 492.73072545436855\n",
      "QG15 = 492.7307254543701\n",
      "QG16 = 492.73072545436673\n",
      "QG17 = 492.7307254543697\n",
      "QG18 = 492.7307254543684\n",
      "QG19 = 492.73072545437\n"
     ]
    }
   ],
   "source": [
    "def F(t):\n",
    "    return 5.1*np.exp(5.1*t + 1.1)\n",
    "\n",
    "def QG2(F):\n",
    "    (pontos,pesos) = leg.leggauss(2)  \n",
    "    print(pontos,pesos)\n",
    "    return pesos[0]*F(pontos[0]) + pesos[1]*F(pontos[1])\n",
    "\n",
    "def QG3(F):\n",
    "    (pontos,pesos) = leg.leggauss(3)  \n",
    "    print(pontos,pesos)\n",
    "    return pesos[0]*F(pontos[0]) + pesos[1]*F(pontos[1]) + pesos[2]*F(pontos[2])\n",
    "\n",
    "def QG4(F):\n",
    "    (pontos,pesos) = leg.leggauss(4)  \n",
    "    print(pontos,pesos)\n",
    "    return pesos[0]*F(pontos[0]) + pesos[1]*F(pontos[1]) \\\n",
    "+ pesos[2]*F(pontos[2]) + pesos[3]*F(pontos[3])\n",
    "\n",
    "\n",
    "def QG(F,n):\n",
    "    (pontos,pesos) = leg.leggauss(n)  \n",
    "    \n",
    "    s = 0\n",
    "    for i in range(n):\n",
    "        s += pesos[i]*F(pontos[i])\n",
    "\n",
    "    return s\n",
    "\n",
    "for i in range(1,20):\n",
    "    print(f'QG{i} = {QG(F,i)}')"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {},
   "source": [
    "Isto é, para a regra de 2 pontos, os nós de Gauss-Legendre são"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.9931286  -0.96397193 -0.91223443 -0.83911697 -0.74633191 -0.63605368\n",
      " -0.510867   -0.37370609 -0.22778585 -0.07652652  0.07652652  0.22778585\n",
      "  0.37370609  0.510867    0.63605368  0.74633191  0.83911697  0.91223443\n",
      "  0.96397193  0.9931286 ]\n"
     ]
    }
   ],
   "source": [
    "print(pontos)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "e os pesos correspondentes são:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.01761401 0.04060143 0.06267205 0.08327674 0.10193012 0.11819453\n",
      " 0.13168864 0.14209611 0.14917299 0.15275339 0.15275339 0.14917299\n",
      " 0.14209611 0.13168864 0.11819453 0.10193012 0.08327674 0.06267205\n",
      " 0.04060143 0.01761401]\n"
     ]
    }
   ],
   "source": [
    "print(pesos)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "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": [
    "plt.plot(pontos,pesos)\n",
    "plt.autoscale;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tabela de pesos/pontos - Quadratura de Gauss-Legendre"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para gerarmos uma tabela de pontos e pesos, basta fazer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "REGRA DE 1 PONTO(S):\n",
      "-> Pontos:\n",
      "[0.]\n",
      "-> Pesos:\n",
      "[2.]\n",
      "\n",
      "\n",
      "REGRA DE 2 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.57735027  0.57735027]\n",
      "-> Pesos:\n",
      "[1. 1.]\n",
      "\n",
      "\n",
      "REGRA DE 3 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.77459667  0.          0.77459667]\n",
      "-> Pesos:\n",
      "[0.55555556 0.88888889 0.55555556]\n",
      "\n",
      "\n",
      "REGRA DE 4 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.86113631 -0.33998104  0.33998104  0.86113631]\n",
      "-> Pesos:\n",
      "[0.34785485 0.65214515 0.65214515 0.34785485]\n",
      "\n",
      "\n",
      "REGRA DE 5 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.90617985 -0.53846931  0.          0.53846931  0.90617985]\n",
      "-> Pesos:\n",
      "[0.23692689 0.47862867 0.56888889 0.47862867 0.23692689]\n",
      "\n",
      "\n",
      "REGRA DE 6 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.93246951 -0.66120939 -0.23861919  0.23861919  0.66120939  0.93246951]\n",
      "-> Pesos:\n",
      "[0.17132449 0.36076157 0.46791393 0.46791393 0.36076157 0.17132449]\n",
      "\n",
      "\n",
      "REGRA DE 7 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.94910791 -0.74153119 -0.40584515  0.          0.40584515  0.74153119\n",
      "  0.94910791]\n",
      "-> Pesos:\n",
      "[0.12948497 0.27970539 0.38183005 0.41795918 0.38183005 0.27970539\n",
      " 0.12948497]\n",
      "\n",
      "\n",
      "REGRA DE 8 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.96028986 -0.79666648 -0.52553241 -0.18343464  0.18343464  0.52553241\n",
      "  0.79666648  0.96028986]\n",
      "-> Pesos:\n",
      "[0.10122854 0.22238103 0.31370665 0.36268378 0.36268378 0.31370665\n",
      " 0.22238103 0.10122854]\n",
      "\n",
      "\n",
      "REGRA DE 9 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.96816024 -0.83603111 -0.61337143 -0.32425342  0.          0.32425342\n",
      "  0.61337143  0.83603111  0.96816024]\n",
      "-> Pesos:\n",
      "[0.08127439 0.18064816 0.2606107  0.31234708 0.33023936 0.31234708\n",
      " 0.2606107  0.18064816 0.08127439]\n",
      "\n",
      "\n",
      "REGRA DE 10 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.97390653 -0.86506337 -0.67940957 -0.43339539 -0.14887434  0.14887434\n",
      "  0.43339539  0.67940957  0.86506337  0.97390653]\n",
      "-> Pesos:\n",
      "[0.06667134 0.14945135 0.21908636 0.26926672 0.29552422 0.29552422\n",
      " 0.26926672 0.21908636 0.14945135 0.06667134]\n",
      "\n",
      "\n",
      "REGRA DE 11 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.97822866 -0.8870626  -0.73015201 -0.51909613 -0.26954316  0.\n",
      "  0.26954316  0.51909613  0.73015201  0.8870626   0.97822866]\n",
      "-> Pesos:\n",
      "[0.05566857 0.12558037 0.18629021 0.23319376 0.26280454 0.27292509\n",
      " 0.26280454 0.23319376 0.18629021 0.12558037 0.05566857]\n",
      "\n",
      "\n",
      "REGRA DE 12 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.98156063 -0.90411726 -0.76990267 -0.58731795 -0.3678315  -0.12523341\n",
      "  0.12523341  0.3678315   0.58731795  0.76990267  0.90411726  0.98156063]\n",
      "-> Pesos:\n",
      "[0.04717534 0.10693933 0.16007833 0.20316743 0.23349254 0.24914705\n",
      " 0.24914705 0.23349254 0.20316743 0.16007833 0.10693933 0.04717534]\n",
      "\n",
      "\n",
      "REGRA DE 13 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.98418305 -0.9175984  -0.80157809 -0.64234934 -0.44849275 -0.23045832\n",
      "  0.          0.23045832  0.44849275  0.64234934  0.80157809  0.9175984\n",
      "  0.98418305]\n",
      "-> Pesos:\n",
      "[0.040484   0.0921215  0.13887351 0.17814598 0.20781605 0.22628318\n",
      " 0.23255155 0.22628318 0.20781605 0.17814598 0.13887351 0.0921215\n",
      " 0.040484  ]\n",
      "\n",
      "\n",
      "REGRA DE 14 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.98628381 -0.92843488 -0.82720132 -0.6872929  -0.51524864 -0.31911237\n",
      " -0.10805495  0.10805495  0.31911237  0.51524864  0.6872929   0.82720132\n",
      "  0.92843488  0.98628381]\n",
      "-> Pesos:\n",
      "[0.03511946 0.08015809 0.12151857 0.15720317 0.1855384  0.20519846\n",
      " 0.21526385 0.21526385 0.20519846 0.1855384  0.15720317 0.12151857\n",
      " 0.08015809 0.03511946]\n",
      "\n",
      "\n",
      "REGRA DE 15 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.98799252 -0.93727339 -0.84820658 -0.72441773 -0.57097217 -0.39415135\n",
      " -0.20119409  0.          0.20119409  0.39415135  0.57097217  0.72441773\n",
      "  0.84820658  0.93727339  0.98799252]\n",
      "-> Pesos:\n",
      "[0.03075324 0.07036605 0.10715922 0.13957068 0.16626921 0.186161\n",
      " 0.19843149 0.20257824 0.19843149 0.186161   0.16626921 0.13957068\n",
      " 0.10715922 0.07036605 0.03075324]\n",
      "\n",
      "\n",
      "REGRA DE 16 PONTO(S):\n",
      "-> Pontos:\n",
      "[-0.98940093 -0.94457502 -0.8656312  -0.75540441 -0.61787624 -0.45801678\n",
      " -0.28160355 -0.09501251  0.09501251  0.28160355  0.45801678  0.61787624\n",
      "  0.75540441  0.8656312   0.94457502  0.98940093]\n",
      "-> Pesos:\n",
      "[0.02715246 0.06225352 0.09515851 0.12462897 0.14959599 0.16915652\n",
      " 0.18260342 0.18945061 0.18945061 0.18260342 0.16915652 0.14959599\n",
      " 0.12462897 0.09515851 0.06225352 0.02715246]\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# número máximo de pontos\n",
    "N = 16\n",
    "\n",
    "for i in range(1,N+1):\n",
    "    (pontos,pesos) = leg.leggauss(i)       \n",
    "    print('REGRA DE {0} PONTO(S):\\n-> Pontos:'.format(i))\n",
    "    print(pontos)\n",
    "    print('-> Pesos:')\n",
    "    print(pesos)\n",
    "    print('\\n')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A partir daí, podemos organizar uma tabela para a regra de até 8 pontos/pesos como segue:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "| Regra | nó(s) | peso(s) |\n",
      "|---|---|---|\n",
      "|1|0.0|2.0|\n",
      "|2|-0.5773502691896257, 0.5773502691896257|1.0, 1.0|\n",
      "|3|-0.7745966692414834, 0.0, 0.7745966692414834|0.5555555555555557, 0.8888888888888888, 0.5555555555555557|\n",
      "|4|-0.8611363115940526, -0.33998104358485626, 0.33998104358485626, 0.8611363115940526|0.3478548451374537, 0.6521451548625462, 0.6521451548625462, 0.3478548451374537|\n",
      "|5|-0.906179845938664, -0.5384693101056831, 0.0, 0.5384693101056831, 0.906179845938664|0.23692688505618942, 0.4786286704993662, 0.568888888888889, 0.4786286704993662, 0.23692688505618942|\n",
      "|6|-0.932469514203152, -0.6612093864662645, -0.23861918608319693, 0.23861918608319693, 0.6612093864662645, 0.932469514203152|0.17132449237916975, 0.36076157304813894, 0.46791393457269137, 0.46791393457269137, 0.36076157304813894, 0.17132449237916975|\n",
      "|7|-0.9491079123427585, -0.7415311855993945, -0.4058451513773972, 0.0, 0.4058451513773972, 0.7415311855993945, 0.9491079123427585|0.12948496616887065, 0.2797053914892766, 0.3818300505051183, 0.41795918367346896, 0.3818300505051183, 0.2797053914892766, 0.12948496616887065|\n",
      "|8|-0.9602898564975362, -0.7966664774136267, -0.525532409916329, -0.18343464249564978, 0.18343464249564978, 0.525532409916329, 0.7966664774136267, 0.9602898564975362|0.10122853629037669, 0.22238103445337434, 0.31370664587788705, 0.36268378337836177, 0.36268378337836177, 0.31370664587788705, 0.22238103445337434, 0.10122853629037669|\n"
     ]
    }
   ],
   "source": [
    "# número máximo de pontos\n",
    "N = 8\n",
    "\n",
    "header='| Regra | nó(s) | peso(s) |\\n|---|---|---|'\n",
    "print(header)\n",
    "for i in range(1,N+1):\n",
    "    (pontos,pesos) = leg.leggauss(i)       \n",
    "    p = ', '.join([str(p) for p in pontos])\n",
    "    w = ', '.join([str(p) for p in pesos])    \n",
    "    row = '|' + str(i) + '|' + p + '|' + w + '|'\n",
    "    print(row)                   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Tabela de quadratura de Gauss-Legendre \n",
    "| Regra | nó(s) | peso(s) |\n",
    "|---|---|---|\n",
    "|1|0.0|2.0|\n",
    "|2|-0.57735026919, 0.57735026919|1.0, 1.0|\n",
    "|3|-0.774596669241, 0.0, 0.774596669241|0.555555555556, 0.888888888889, 0.555555555556|\n",
    "|4|-0.861136311594, -0.339981043585, 0.339981043585, 0.861136311594|0.347854845137, 0.652145154863, 0.652145154863, 0.347854845137|\n",
    "|5|-0.906179845939, -0.538469310106, 0.0, 0.538469310106, 0.906179845939|0.236926885056, 0.478628670499, 0.568888888889, 0.478628670499, 0.236926885056|\n",
    "|6|-0.932469514203, -0.661209386466, -0.238619186083, 0.238619186083, 0.661209386466, 0.932469514203|0.171324492379, 0.360761573048, 0.467913934573, 0.467913934573, 0.360761573048, 0.171324492379|\n",
    "|7|-0.949107912343, -0.741531185599, -0.405845151377, 0.0, 0.405845151377, 0.741531185599, 0.949107912343|0.129484966169, 0.279705391489, 0.381830050505, 0.417959183673, 0.381830050505, 0.279705391489, 0.129484966169|\n",
    "|8|-0.960289856498, -0.796666477414, -0.525532409916, -0.183434642496, 0.183434642496, 0.525532409916, 0.796666477414, 0.960289856498|0.10122853629, 0.222381034453, 0.313706645878, 0.362683783378, 0.362683783378, 0.313706645878, 0.222381034453, 0.10122853629|"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transformação de variáveis\n",
    "\n",
    "Uma integral $\\int_a^b f(x) \\, dx$ sobre $[a,b]$ arbitrário ser transformada em uma integral em $[-1,1]$ utilizando a mudança de variáveis:\n",
    "\n",
    "$$t = \\dfrac{2x - a - b}{b - a} \\Rightarrow x = \\dfrac{1}{2}[(b-a)t + a + b]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(2,4)\n",
    "y = x - 3\n",
    "plt.plot(x,y);\n",
    "plt.plot(x,x*0);\n",
    "plt.xticks([2,4],['a','b']);\n",
    "plt.yticks([-1,1],['-1','1']);\n",
    "plt.annotate('$t = \\dfrac{2x - a - b}{b - a}$',(2.8,0.5));\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tarefa \n",
    "\n",
    "Defina uma função como a seguinte que retorne `output`, tal que `type(output)` seja `str`. \n",
    "```python\n",
    "def print_gauss_legendre_table(N):\n",
    "    header='| Regra | nó(s) | peso(s) |\\n|---|---|---|'\n",
    "    print(header)\n",
    "    for i in range(1,N+1):\n",
    "        (pontos,pesos) = leg.leggauss(i)       \n",
    "        p = ', '.join([str(p) for p in pontos])\n",
    "        w = ', '.join([str(p) for p in pesos])    \n",
    "        row = '|' + str(i) + '|' + p + '|' + w + '|'\n",
    "        print(row)             \n",
    "```\n",
    "\n",
    "Então, reimprima a tabela para 8 pesos/pontos anterior com o código. \n",
    "\n",
    "```\n",
    "output = print_gauss_legendre_table(8)\n",
    "```\n",
    "\n",
    "Em seguida, use o código abaixo para converter a saída da célula de código do Jupyter diretamente para Markdown. \n",
    "```python\n",
    "from IPython.display import display, Markdown\n",
    "display(Markdown(output))\n",
    "```\n",
    "\n",
    "Por último, incorpore esta funcionalidade em `print_gauss_legendre_table(N)`, para `N` dado."
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    "| Regra | nó(s) | peso(s) |\n",
    "|---|---|---|\n",
    "|1|0.0|2.0|\n",
    "|2|-0.57735026919, 0.57735026919|1.0, 1.0|\n",
    "|3|-0.774596669241, 0.0, 0.774596669241|0.555555555556, 0.888888888889, 0.555555555556|\n",
    "|4|-0.861136311594, -0.339981043585, 0.339981043585, 0.861136311594|0.347854845137, 0.652145154863, 0.652145154863, 0.347854845137|\n",
    "|5|-0.906179845939, -0.538469310106, 0.0, 0.538469310106, 0.906179845939|0.236926885056, 0.478628670499, 0.568888888889, 0.478628670499, 0.236926885056|\n",
    "|6|-0.932469514203, -0.661209386466, -0.238619186083, 0.238619186083, 0.661209386466, 0.932469514203|0.171324492379, 0.360761573048, 0.467913934573, 0.467913934573, 0.360761573048, 0.171324492379|\n",
    "|7|-0.949107912343, -0.741531185599, -0.405845151377, 0.0, 0.405845151377, 0.741531185599, 0.949107912343|0.129484966169, 0.279705391489, 0.381830050505, 0.417959183673, 0.381830050505, 0.279705391489, 0.129484966169|\n",
    "|8|-0.960289856498, -0.796666477414, -0.525532409916, -0.183434642496, 0.183434642496, 0.525532409916, 0.796666477414, 0.960289856498|0.10122853629, 0.222381034453, 0.313706645878, 0.362683783378, 0.362683783378, 0.313706645878, 0.222381034453, 0.10122853629|\n",
    "|9|-0.968160239508, -0.836031107327, -0.613371432701, -0.324253423404, 0.0, 0.324253423404, 0.613371432701, 0.836031107327, 0.968160239508|0.0812743883616, 0.180648160695, 0.260610696403, 0.31234707704, 0.330239355001, 0.31234707704, 0.260610696403, 0.180648160695, 0.0812743883616|\n",
    "|10|-0.973906528517, -0.865063366689, -0.679409568299, -0.433395394129, -0.148874338982, 0.148874338982, 0.433395394129, 0.679409568299, 0.865063366689, 0.973906528517|0.0666713443087, 0.149451349151, 0.219086362516, 0.26926671931, 0.295524224715, 0.295524224715, 0.26926671931, 0.219086362516, 0.149451349151, 0.0666713443087|\n",
    "|11|-0.978228658146, -0.887062599768, -0.730152005574, -0.519096129207, -0.269543155952, 0.0, 0.269543155952, 0.519096129207, 0.730152005574, 0.887062599768, 0.978228658146|0.0556685671162, 0.125580369465, 0.186290210928, 0.233193764592, 0.26280454451, 0.272925086778, 0.26280454451, 0.233193764592, 0.186290210928, 0.125580369465, 0.0556685671162|\n",
    "|12|-0.981560634247, -0.90411725637, -0.769902674194, -0.587317954287, -0.367831498998, -0.125233408511, 0.125233408511, 0.367831498998, 0.587317954287, 0.769902674194, 0.90411725637, 0.981560634247|0.0471753363865, 0.106939325995, 0.160078328543, 0.203167426723, 0.233492536538, 0.249147045813, 0.249147045813, 0.233492536538, 0.203167426723, 0.160078328543, 0.106939325995, 0.0471753363865|"
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