02-Zeros-de-funcoes-2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Convergência do Método de Newton\n",
    "====\n",
    "\n",
    "Seja $f:[a,b]\\to \\mathbb{R}$ uma função \n",
    "\n",
    "*  de classe $\\mathcal{C}^2$\n",
    "*  com um zero em $r \\in (a,b)$\n",
    "*  tal que $f^\\prime(r)\\neq 0$\n",
    "\n",
    "Considere a sequência $\\{x_n\\}_{n\\in \\mathbb{N}}$ com $x_0\\in (a,b)$ e\n",
    "\n",
    "$$x_{n+1}=x_n - \\frac{f(x_n)}{f^\\prime(x_n)}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ordem de convergência\n",
    "----\n",
    "\n",
    "Antes de mostrar as condições de convergência, vamos assumir que a sequẽncia converge a $r$, e vamos ver com que rapidez.\n",
    "\n",
    "Seja $e_n=x_n-r$ o erro de $n$-ésima iteração do método de Newton\n",
    "\n",
    "$$ r = x_n - e_n \\implies f(r)= f(x_n-e_n) = f(x_n)- f^\\prime(x_n)e_n + \\frac{1}{2}f^{\\prime\\prime}(\\xi_n)e_n^2 = 0$$\n",
    "\n",
    "$$ \\frac{f(x_n)}{f^\\prime(x_n)} = e_n - \\frac{1}{2}\\frac{ f^{\\prime\\prime}(\\xi_n)}{f^\\prime(x_n)}e_n^2 = x_n - x_{n+1}= e_n - e_{n+1}$$\n",
    "\n",
    "daí\n",
    "\n",
    "$$ e_{n+1} = \\frac{1}{2}\\frac{ f^{\\prime\\prime}(\\xi_n)}{f^\\prime(x_n)}e_n^2 $$\n",
    "\n",
    "$$ \\lim_{n \\to \\infty} \\frac{e_{n+1}}{e_n^2} = \\frac{1}{2}\\frac{f^{\\prime\\prime}(r)}{f^\\prime(r)} = C $$\n",
    "\n",
    "Esta é a definição de convergência de ordem 2 (ou convergência quadrática)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quando converge\n",
    "----\n",
    "\n",
    "A convergência depende da escolha de um $x_0$ bem próximo de $r$. Em primeiro lugar, tomemos um número $\\delta > 0$ tal que a\n",
    "derivada $f^\\prime(x) \\neq 0$ no intervalo $ I= |x-r|\\lt \\delta$ ($I=(r-\\delta, r+\\delta)$).\n",
    "Definimos:\n",
    "$$ c(\\delta) = \\frac{1}{2} \\frac{\\max_{x\\in I}f^{\\prime\\prime}(x)}{\\min_{x\\in I}f^\\prime{x}} $$\n",
    "Note que $\\lim_{\\delta \\to 0}\\delta c(\\delta)=0$ então podemos tomar um $\\delta$ talque $\\delta c(\\delta)=\\alpha < 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Deste modo temos que:\n",
    "\n",
    "$$|x_1-r|=|e_1|=\\frac{1}{2}\\frac{ |f^{\\prime\\prime}(\\xi_0)|}{|f^\\prime(x_0)|}e_0^2 \\leq c(\\delta)|e_0||e_0| \\lt \\alpha|e_0|$$\n",
    "e por recorrência\n",
    "$$ |e_n| \\lt \\alpha^n|e_0| $$\n",
    "O que mostra a convergência $\\square$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Método das secantes\n",
    "====\n",
    "\n",
    "Esta é uma variação do método de Newton. \n",
    "Fazemos a aproximação da derivada por\n",
    "$$ f^\\prime(x_n) = \\frac{f(x_n) - f(x_{n-1})}{x_n-x_{n-1}}$$\n",
    "o que nos dá a iteração de segunda ordem:\n",
    "\n",
    "$$ x_{n+1} = \\frac{x_{n-1}f(x_n)-x_nf(x_{n-1})}{f(x_n) - f(x_{n-1})}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def secante(f,x0,x1,epsilon):\n",
    "    \"\"\" Método da secante para a encontrar a raiz de f(x)=0 \"\"\"\n",
    "    x= (x0*f(x1) - x1*f(x0))/(f(x1) - f(x0))\n",
    "    while abs(x1-x0) > epsilon:\n",
    "        x0,x1 = x1, x\n",
    "        x= (x0*f(x1) - x1*f(x0))/(f(x1) - f(x0))\n",
    "    return x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para testar vamos usar a função $f(x) =x^6 -x -1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib.pyplot import plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x= arange(-1,1.3,0.01)\n",
    "f = lambda x: x**6-x-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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isdR8pat+oV95xQqTCejVywajEbV+z23/2q6mlXhkbufn52sqHpnb+fn5Hnn+Fi1EffnP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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7ff2c1c09f50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(x,f(x))\n",
    "grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1347241385131241"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "secante(f,1,1.3,0.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.7780895986591118"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "secante(f,-1.0,-0.5,0.001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'https://gist.github.com/9490591'"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%pastebin '02-Zeros-de-funcoes-2.ipynb'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}