aula-20-diferenciacao-numerica.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Diferenciação Numérica\n",
    "\n",
    "## Fórmula de Derivação de Alta Acurácia\n",
    "\n",
    "A expansão em série de Taylor progressiva pode ser escrita como\n",
    "\n",
    "$$\n",
    "f(x_{i+1}) = f(x_i) + f'(x_i) h + \\dfrac{f '' (x_i)}{2} h^2 + \\dots\n",
    "$$\n",
    "\n",
    "o que pode ser resolvido por\n",
    "\n",
    "$$\n",
    "f'(x_i) = \\dfrac{f(x_{i+1}) - f(x_i)}{h} - \\dfrac{f''(x_i)}{2} h + O(h^2) \\qquad (1)\n",
    "$$\n",
    "\n",
    "Truncamos esse resultado excluindo os termos da segunda derivada e de derivadas de ordem superior e ficamos com o resultado final\n",
    "\n",
    "$$\n",
    "f' (x_i) = \\dfrac{f(x_{i+1}) - f(x_i)}{h} + O(h) \\qquad (2)\n",
    "$$\n",
    "\n",
    "Em contraste a essa abordagem, agora manteremos o termo da segunda derivada substituindo a seguinte aproximação da segunda derivada\n",
    "\n",
    "$$\n",
    "f'' (x_i) = \\dfrac{f(x_{i+2}) - 2 f(x_{i+1}) + f(x_i)}{h^2} + O(h) \\qquad (3)\n",
    "$$\n",
    "\n",
    "na Equação (1) para obter\n",
    "\n",
    "$$\n",
    "f' (x_i) = \\dfrac{f(x_{i+1}) - f(x_i)}{h} - \\dfrac{f(x_{i+2}) - 2 f(x_{i+1}) + f(x_i)}{2 h^2} + O(h^2)\n",
    "$$\n",
    "\n",
    "ou, agrupando os termos,\n",
    "\n",
    "$$\n",
    "f' (x_i) = \\dfrac{- f(x_{i+2}) + 4 f(x_{i+1}) - 3 f(x_i)}{2 h} + O(h^2) \\qquad (4)\n",
    "$$\n",
    "\n",
    "Observe que a inclusão do termo da segunda derivada melhorou a precisão para $O(h^2)$.\n",
    "\n",
    "## Derivada de Dados Não Uniformemente Espaçados\n",
    "\n",
    "Uma forma de tratar dados não uniformemente espaçados é ajustar um polinômio interpolador de Lagrange aos dados. Para uma derivada de primeira ordem mais acurada, por exemplo, podemos deduzir um polinômio de segundo grau para cada conjunto de três pontos adjacentes analiticamente através das expressões das funções de base de Lagrange. Em seguida, derivamos as funções de base de modo a obter\n",
    "\n",
    "$$f'(x) = f(x_{i-1}) \\dfrac{2 x - x_i - x_{i+1}}{(x_{i-1} - x_i)(x_{i-1} - x_{i+1})} + f(x_i) \\dfrac{2 x - x_{i-1} - x_{i+1}}{(x_i - x_{i-1})(x_i - x_{i+1})} + f(x_{i+1}) \\dfrac{2 x - x_{i-1} - x_i}{(x_{i+1} - x_{i-1})(x_{i+1} - x_i)}, \\qquad (5)$$\n",
    "\n",
    "em que $x$ é o valor no qual se quer estimar a derivada."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np \n",
    "import sympy as sy\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motivação\n",
    "\n",
    "Gerando curvas de deslocamento, velocidade e aceleração para uma partícula."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# eixo temporal\n",
    "x = np.linspace(0,1,30,endpoint=True)\n",
    "\n",
    "# lei de movimento\n",
    "y = 0.93*np.cos(2.5*(x-1))*np.sin(6*x*x)\n",
    "\n",
    "# velocidade e aceleração (funções analíticas)\n",
    "# derivação simbólica\n",
    "t = sy.Symbol('t')\n",
    "st = 0.93*sy.cos(2.5*(t-1))*sy.sin(6*t*t)\n",
    "dt = sy.diff(st,t) # v(t)\n",
    "dt2 = sy.diff(dt,t) # a(t)\n",
    "\n",
    "# converte para numérico\n",
    "dy = np.asarray([dt.subs(t,xn) for xn in x])\n",
    "dy2 = np.asarray([dt2.subs(t,xn) for xn in x])\n",
    "\n",
    "# função para plotagem\n",
    "def plotting(x,y,c,lab):\n",
    "    plt.plot(x,y,c,label=lab)\n",
    "    plt.grid()\n",
    "    plt.xlabel('$t$')\n",
    "    plt.tick_params(\n",
    "        axis='both',        \n",
    "        which='both',      \n",
    "        bottom=False,      \n",
    "        left=False,         \n",
    "        labelbottom=False, \n",
    "        labelleft=False)\n",
    "    plt.legend()\n",
    "    ax = plt.gca()\n",
    "    ax.set_facecolor('#F9F9F9')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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X6e2tOmZ6e1XJ5zvUIjcQugBCWcrlVDl5UmZwUMbzZAYHVTl5kptoa2D1AoDQolSn1hWMdAHAIkIXACwidAHAIkIXACwidAHAIkIXACwidAHAIkIXACwidAHAIkIXgLNcLKLONmAATnK1iLpnjKl7MpPJmEKhEOrCMzMzSqfTYdvlJPqcDPQ5GvYdPqyNU1Orjt/q79eF06ebvn4zfc5msxeNMXtrnWs40u3r69Pw8HCoN52YmAj9WlfR52Sgz9HQPT1d83jP9HRL2tquPjOnC8BNjhZRJ3QBOMnVIuqELgArWr3SwNUi6qxeANB27Vpp4GIRdUa6ANqOx7X/G6ELoP14XPsKQhdA+zm60qAdCF0AbefqSoN2IHQBtJ2rKw3agdAFYMVSLqeFUkkLc3NaKJXqBq6LRWyCYMkYgMhwtYhNEIx0AURGEpaWEboAoiMBS8sIXQDRkYClZYQugMhIwtIyQhdAZCRhaRmrFwBEiotFbIJgpAsAFhG6AGARoQsAFhG6AGARoQsAFnnGmPonPW9a0j/sNQcAYmGHMWZbrRMNQxcA0FpMLwCARYQuAFhE6AKARYQuAFhE6AKARYQunON53n96nvffnW4HEAahCxeNSPpzpxsBhME6XTjF87z9kj6T9Kuk3yQ9ZYy52tlWAf4RunCO53n/K+l/jDF/73RbgKCYXoCLHpb0f51uBBAGoQuneJ73H5L+3xiz0Om2AGEQunDNLkn/7HQjgLAIXbjmkqQHPM/7u+d5/9XpxgBBcSMNACxipAsAFhG6AGARoQsAFhG6AGARoQsAFhG6AGARoQsAFv0LlIZqFyQ7j00AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# curva: deslocamento\n",
    "plotting(x,y,'or','$x(t)$')"
   ]
  },
  {
   "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": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# curva: velocidade\n",
    "plotting(x,dy,'og','$v(t)$')"
   ]
  },
  {
   "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": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# curva: aceleração\n",
    "plotting(x,dy2,'ob','$a(t)$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Derivada numérica progressiva e regressiva (primeira ordem)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "h=0.010\n",
      "f'e(1.2) = 0.36235775\n",
      "f'p(1.2) = 0.35769156\n",
      "f'r(1.2) = 0.36701187\n",
      "f'c(1.2) = 0.36235172\n"
     ]
    },
    {
     "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": [
    "# Exemplo: f(x) = sen(x); \n",
    "# Derivada analítica (verdadeira): f'(x=x0) = cos(x0)\n",
    "x0 = 1.2\n",
    "h = .01\n",
    "n = 10\n",
    "x = np.linspace(x0-2*h,x0+2*h,n)\n",
    "\n",
    "# expressões analíticas de f(x) e f'(x)\n",
    "f = np.sin(x)\n",
    "df = np.cos(x)\n",
    "\n",
    "# valor \"exato\" de f'(x) no ponto x=x0\n",
    "dfe = np.cos(x0)\n",
    "\n",
    "dfp = (np.sin(x0+h) - np.sin(x0))/h       # DF progressiva\n",
    "dfr = (np.sin(x0) - np.sin(x0-h))/h       # DF regressiva\n",
    "dfc = (np.sin(x0+h) - np.sin(x0-h))/(2*h) # DF centrada\n",
    "\n",
    "plt.plot(x,df,'ok') # derivada exata\n",
    "plt.plot(x0,dfe,'sy') # derivada exata no ponto x0\n",
    "plt.plot(x0,dfp,'or') # derivada numérica PROGRESSIVA no ponto x0 \n",
    "plt.plot(x0,dfr,'ob') # derivada numérica REGRESSIVA no ponto x0 \n",
    "plt.plot(x0,dfc,'og') # derivada numérica CENTRADA no ponto x0 \n",
    "plt.grid(True)\n",
    "\n",
    "print('h={0:.3f}'.format(h))\n",
    "print('f\\'e({0:g}) = {1:.8f}'.format(x0,dfe))\n",
    "print('f\\'p({0:g}) = {1:.8f}'.format(x0,dfp))\n",
    "print('f\\'r({0:g}) = {1:.8f}'.format(x0,dfr))\n",
    "print('f\\'c({0:g}) = {1:.8f}'.format(x0,dfc))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparando resultados\n",
    "\n",
    "**Exemplo:** considere a função $f(x) = \\frac{2^x}{x}$. Calcule a segunda derivada em $x=2$ numericamente com a fórmula de diferença centrada a 3 pontos usando:\n",
    "\n",
    "- Os pontos $x = 1.8$, $x = 2$ e $x = 2.2$.\n",
    "- Os pontos $x = 1.9$, $x = 2$ e $x = 2.1$.\n",
    "\n",
    "Compare os resultados com a derivada analítica."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solução\n",
    "\n",
    "Vamos primeiro computar a seguda derivada analítica de $f(x)$ por computação simbólica."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2^{x} \\left(\\log{\\left(2 \\right)}^{2} - \\frac{2 \\log{\\left(2 \\right)}}{x} + \\frac{2}{x^{2}}\\right)}{x}$"
      ],
      "text/plain": [
       " x ⎛   2      2⋅log(2)   2 ⎞\n",
       "2 ⋅⎜log (2) - ──────── + ──⎟\n",
       "   ⎜             x        2⎟\n",
       "   ⎝                     x ⎠\n",
       "────────────────────────────\n",
       "             x              "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import x \n",
    "sy.init_printing()\n",
    "\n",
    "# função\n",
    "f = 2**x/x\n",
    "\n",
    "# derivada segunda\n",
    "d2fdx2 = sy.diff(f,x,2)\n",
    "d2fdx2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# derivada em x = 2 \n",
    "d2fdx2_p2 = d2fdx2.subs(x,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.5746116667165122$"
      ],
      "text/plain": [
       "0.5746116667165122"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# valor numérico\n",
    "d2fdx2_p2_num = float(d2fdx2_p2)\n",
    "d2fdx2_p2_num "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, vamos montar uma função para a fórmula da segunda derivada a 3 pontos (supondo que os pontos dados sejam igualmente espaçados:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# função \n",
    "def der_num_2(x1,x2,x3):\n",
    "    d2fdx2_num = lambda x1,x2,x3: (f.subs(x,x1) - 2*f.subs(x,x2) + f.subs(x,x3))/(x2 - x1)**2\n",
    "    return float(d2fdx2_num(x1,x2,x3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Aplicamos a nossa aproximação numérica ao primeiro conjunto de pontos para estimar a derivada numérica no ponto $x=2$ por diferença centrada."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.5774817738923302$"
      ],
      "text/plain": [
       "0.5774817738923302"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2_1 = der_num_2(1.8,2.0,2.2)\n",
    "df2_1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em seguida, aplicamos a nossa aproximação numérica ao segundo conjunto de pontos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.5753244156644123$"
      ],
      "text/plain": [
       "0.5753244156644123"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df2_2 = der_num_2(1.9,2.0,2.1)\n",
    "df2_2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Esperamos que o segundo conjunto de pontos nos dê uma estimativa mais próxima para a derivada analítica já que os pontos estão mais próximos um do outro. Para verificar isto, vamos medir o erro relativo percentual entre as derivadas numéricas e a derivada analítica."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "erro relativo percentual 1: 0.49948640831095764%\n",
      "erro relativo percentual 2: 0.12404011077131777%\n"
     ]
    }
   ],
   "source": [
    "# erro relativo percentual \n",
    "erp = lambda v,va: abs(v-va)/abs(v)*100\n",
    "\n",
    "# erro para o grupo 1\n",
    "print('erro relativo percentual 1: ' + str(erp(d2fdx2_p2_num,df2_1)) + '%' )\n",
    "\n",
    "# erro para o grupo 2\n",
    "print('erro relativo percentual 2: ' + str(erp(d2fdx2_p2_num,df2_2)) + '%' )"
   ]
  },
  {
   "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": 14,
   "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
}