aula-07-secante.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Implementação do método da secante "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Estimativas iniciais: xa = 1.0; xb = 2.0 \n",
      "\n",
      "0  4.300000  -6.831000  5.348837e-01\n",
      "1  2.579345  0.897168  6.670898e-01\n",
      "2  2.779097  0.273423  7.187654e-02\n",
      "3  2.866660  -0.022642  3.054518e-02\n",
      "4  2.859963  0.000487  2.341478e-03\n",
      "5  2.860104  0.000001  4.933559e-05\n",
      "Solução obtida: x = 2.8601041641\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": [
    "# Método da Secante\n",
    "\n",
    "from numpy import linspace\n",
    "from matplotlib.pyplot import plot\n",
    "\n",
    "def secante(xa,xb,f,tol,nmax,var,plotar):\n",
    "\n",
    "    f = eval('lambda x:' + f)\n",
    "\n",
    "    # imprime estimativas iniciais\n",
    "    print('Estimativas iniciais: xa = {0}; xb = {1} \\n'.format(xa,xb))  \n",
    "\n",
    "    # Loop \n",
    "    for i in range(0,nmax):\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",
    "        print('{0:d}  {1:f}  {2:f}  {3:e}'.format(i,x,f(x),e))\n",
    "        \n",
    "        if e < tol:\n",
    "            break\n",
    "        xa = xb\n",
    "        xb = x\n",
    "        \n",
    "    if i == nmax:\n",
    "        print('Solução não obtida em {0:d} iterações'.format(nmax))\n",
    "    else:\n",
    "        print('Solução obtida: x = {0:.10f}'.format(x))\n",
    "\n",
    "    # plotagem\n",
    "    if plotar:        \n",
    "        delta = 3*x\n",
    "        dom = linspace(x-delta,x+delta,30)\n",
    "        plot(dom,f(dom),x,f(x),'ro')\n",
    "\n",
    "    return x\n",
    "      \n",
    "    \n",
    "# parametros    \n",
    "xa = 1.0 # estimativa inicial 1\n",
    "xb = 2.0 # estimativa inicial 2\n",
    "tol = 1e-3 # tolerancia\n",
    "nmax = 100 # numero maximo de iteracoes\n",
    "f = '-0.9*x**2 + 1.7*x + 2.5'   # funcao\n",
    "var = 'x'\n",
    "plotar = True\n",
    "\n",
    "# chamada da função\n",
    "xm = secante(xa,xb,f,tol,nmax,var,plotar)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Problema\n",
    "\n",
    "Determinar a raiz positiva da equação: $f(x) = \\sqrt{x} - 5e^{-x}$, pelo método das secantes com erro inferior a $10^{-2}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Resolução"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para obtermos os valores iniciais $x_0$ e $x_1$ necessários para iniciar o processo iterativo, dividimos a equação original $f(x) = 0$ em outras duas $y_1$ e $y_2$, com $y_1 = \\sqrt{x}$ e $y_2(x) = e^{-x}$, que colocadas no mesmo gráfico, produzem uma interseção próximo a $x = 1.5$. Assim, podemos escolher duas estimativas iniciais próximas deste valor. Podemos escolher $x_0 = 1.4$ e $x_1=1.5$."
   ]
  },
  {
   "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": [
    "from numpy import sqrt, exp\n",
    "from matplotlib.pyplot import plot,legend\n",
    "\n",
    "fx = lambda x: sqrt(x) - 5*exp(-x) \n",
    "\n",
    "x = linspace(0,3,100)\n",
    "plot(x,fx(x),label='$f(x) = x^{1/2} - 5e^{-x}$');\n",
    "plot(x,sqrt(x),label='$y_1(x) = x^{1/2}$');\n",
    "plot(x,5*exp(-x),label='$y_2(x) = 5e^{-x}$');\n",
    "plot(x,fx(x)*0,'--');\n",
    "legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vejamos o valor de $f(x=1.5)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.10909407064943988"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fx(1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos montar uma função anônima para computar o valor da interseção da secante com o eixo $x$, a saber:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "xm = lambda a,b: ( a*fx(b) - b*fx(a) ) / (fx(b) - fx(a) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos usar os nosso valores estimados: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.431\n"
     ]
    }
   ],
   "source": [
    "x0 = 1.4\n",
    "x1 = 1.5\n",
    "x2 = round(xm(x0,x1),3)\n",
    "print(x2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, usamos este novo valor e o anterior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.43\n"
     ]
    }
   ],
   "source": [
    "x3 = round(xm(x1,x2),3)\n",
    "print(x3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculemos o erro relativo entre as estimativas $x_1$ e $x_2$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.048\n",
      "4.821803e-02\n"
     ]
    }
   ],
   "source": [
    "err = lambda a,b: abs(a - b)/abs(a);\n",
    "e1 = err(x2,x1)\n",
    "print(round(e1,3))\n",
    "print(\"{0:e}\".format(e1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, calculemos o erro relativo entre as estimativas $x_2$ e $x_3$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.001\n",
      "6.993007e-04\n"
     ]
    }
   ],
   "source": [
    "e2 = err(x3,x2)\n",
    "print(round(e2,3))\n",
    "print(\"{0:e}\".format(e2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "O erro está diminuindo. Além disso, o valor da raiz está se estabilizando em torno de 1.430. Isto significa que as estimativas iniciais foram muito boas. Com efeito, o uso das interseções proporcionou uma boa escolha."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "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();"
   ]
  }
 ],
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