Algoritmos de Busca de Raízes III.ipynb

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   "source": [
    "import numpy as np"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "Os métodos listados a seguir são para retornar a raíz da função sem utilizar um intervalo um intervalo pré-definido"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "### Método do Ponto fixo"
   ]
  },
  {
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   "source": [
    "#### Método de Newton - Rapson \n",
    "$f'(x) = \\frac{f(x) - 0}{x_{i} - x_{i+1}}$ <br />\n",
    "<br />\n",
    "$ x_{i+1} = x{i} - \\frac{f(x_{i})}{f'(x{i})}$ <br />\n",
    "<br />\n",
    "Observações: $f'(x{i})$ existe e não é zero "
   ]
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   "source": [
    "## Método da Secante\n",
    "\n",
    "$x_{i+1} = x{i}-\\frac{f(x_{i})(x_{i-1}-x_{i})}{f(x_{i}-1) - f(x_{i})}$ <br />\n",
    "\n",
    "OBS: precisa de $x_{0}$ e $x_{1}$"
   ]
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   "source": [
    "import numpy as np\n",
    "import scipy.optimize as opt\n",
    "\n",
    "def g(x): return np.exp(-x)\n",
    "\n",
    "opt.fixed_point(g, 0)"
   ]
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    "opt.fixed_point(g, 0, xtol=1e-1)"
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   "source": [
    "opt.fixed_point(g, 0, maxiter=7)"
   ]
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     "execution_count": 9,
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    "opt.fixed_point(g, 0, xtol=1e-1, maxiter=5)"
   ]
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   "execution_count": 13,
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   "source": [
    "#opt.fixed_point(g, 0, xtol=1e-1, method=\"iteration\", maxiter=5)"
   ]
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   "execution_count": 11,
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      "text/plain": [
       "      converged: True\n",
       "           flag: 'converged'\n",
       " function_calls: 10\n",
       "     iterations: 5\n",
       "           root: 0.567143290409784"
      ]
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     "execution_count": 11,
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   ],
   "source": [
    "def f1(x): return np.exp(-x) - x #função\n",
    "def f2(x): return -np.exp(-x) - 1 #1ª derivada\n",
    "\n",
    "opt.root_scalar(f1, fprime=f2, x0=0, method='newton')"
   ]
  },
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   "execution_count": null,
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