codeSession-2-newton.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Code session 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline \n",
    "import numpy as np \n",
    "import matplotlib.pyplot as plt \n",
    "import sympy as sy "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Determinação de raízes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## `newton`\n",
    "\n",
    "A função `newton` localiza a raiz de uma função dentro de um intervalo dado usando o método de Newton. \n",
    "Os argumentos de entrada obrigatórios desta função são: \n",
    "\n",
    "1. a função-alvo `f` (contínua)\n",
    "2. a estimativa inicial `x0`\n",
    "\n",
    "Parâmetros opcionais relevantes são: \n",
    "\n",
    "- `fprime`: a derivada da função, quando disponível. Caso ela não seja especificada, o _método da secante_ é usado. \n",
    "- `fprime2`: a segunda derivada da função, quando disponível. Se ela for especificada, o _método de Halley_ é usado. \n",
    "- `tol`: tolerância (padrão: 1.48e-08)\n",
    "- `maxiter`: número máximo de iterações (padrão: 50)\n",
    "- `disp`: mostra erro se algoritmo não convergir (padrão: True) \n",
    "\n",
    "O argumento de saída é:\n",
    "\n",
    "- `x`: a estimativa para a raiz de `f`\n",
    "\n",
    "Como importá-la? \n",
    "\n",
    "```python \n",
    "from scipy.optimize import newton\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.optimize import newton "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problema 1\n",
    "\n",
    "Encontre a menor raiz positiva (real) de $x^{3} - 3.23x^{2} - 5.54x + 9.84 = 0$ pelo método de Newton."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Resolução"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definimos a função e sua primeira derivada."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# função\n",
    "def f(x): \n",
    "    return x**3 - 3.23*x**2 - 5.54*x + 9.84\n",
    "\n",
    "# 1a. derivada\n",
    "def df(x):\n",
    "    return 3*x**2 - 2*3.23*x - 5.54"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Realizamos a análise gráfica."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# analise gráfica \n",
    "x = np.linspace(-4,5)\n",
    "plt.plot(x,f(x));\n",
    "plt.plot(x,df(x));\n",
    "plt.axhline(y=0,color='k',ls='--');\n",
    "plt.axvline(x=0,color='r',ls='--');\n",
    "plt.legend(['$f(x)$','$f\\'(x)$','$y=0$','x=0']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos realizar um estudo de diferentes estimativas iniciais e ver o que acontece."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Estimativa inicial: $x_0 = -1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Raiz: x = -2.000000\n"
     ]
    }
   ],
   "source": [
    "# resolução com newton \n",
    "x0 = -1.\n",
    "x = newton(f,x0,df) # raiz \n",
    "print('Raiz: x = %f' % x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Estimativa inicial: $x_0 = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Raiz: x = 1.230000\n"
     ]
    }
   ],
   "source": [
    "# resolução com newton \n",
    "x0 = 0.\n",
    "x = newton(f,x0,df) # raiz \n",
    "print('Raiz: x = %f' % x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Estimativa inicial: $x_0 = 3$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Raiz: x = 4.000000\n"
     ]
    }
   ],
   "source": [
    "# resolução com newton \n",
    "x0 = 3.\n",
    "x = newton(f,x0,df) # raiz \n",
    "print('Raiz: x = %f' % x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Problema 2\n",
    "\n",
    "Determine a menor raiz não nula positiva de $\\cosh(x) \\cos(x) - 1 = 0$ dentro do intervalo $(4,5)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Resolução:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Primeiramente, vamos escrever a função `f(x)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "# função\n",
    "f = lambda x: np.cosh(x)*np.cos(x) - 1 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para computar a primeira derivada, vamos utilizar computação simbólica. Veja no início deste notebook que inserimos a instrução\n",
    "\n",
    "```python\n",
    "import sympy as sy\n",
    "```\n",
    "\n",
    "a qual nos permitirá utilizar objetos do módulo `sympy`.\n",
    "\n",
    "Em primeiro lugar, devemos estabelecer uma variável simbólica `xs`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "xs = sy.Symbol('x')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Em seguida, devemos utilizar as funções `cosh` e `cos` **simbólicas** para derivar `f`. Elas serão **chamadas de dentro do módulo sympy**.\n",
    "\n",
    "Escrevemos a expressão simbólica para a derivada."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\sin{\\left(x \\right)} \\cosh{\\left(x \\right)} + \\cos{\\left(x \\right)} \\sinh{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-sin(x)*cosh(x) + cos(x)*sinh(x)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = sy.diff(sy.cosh(xs)*sy.cos(xs) - 1)\n",
    "d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note que `d` é um objeto do módulo `sympy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.add.Add"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos agora aproveitar a expressão de `d` para criar nossa derivada. Se imprimirmos `d`, teremos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-sin(x)*cosh(x) + cos(x)*sinh(x)\n"
     ]
    }
   ],
   "source": [
    "print(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Porém, precisamos indicar que as funções serão objetos numpy. Logo, adicionamos `np`, de modo que:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "df = lambda x: - np.sin(x)*np.cosh(x) + np.cos(x)*np.sinh(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, realizamos a análise gráfica. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "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": [
    "# analise gráfica \n",
    "x = np.linspace(4,5)\n",
    "plt.plot(x,f(x));\n",
    "plt.plot(x,df(x));\n",
    "plt.axhline(y=0,color='r',ls='--');\n",
    "plt.legend(['$f(x)$','$f\\'(x)$','$y=0$']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Agora, vamos resolver optando pela estimativa inicial $x_0 = 4.9$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Raiz: x = 4.730041\n"
     ]
    }
   ],
   "source": [
    "# resolução com newton \n",
    "x0 = 4.9\n",
    "x = newton(f,x0,df) # raiz \n",
    "print('Raiz: x = %f' % x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## _Homework_ \n",
    "\n",
    "1. Reproduza os Problemas de 3 a 8 da _Code Session 1_ resolvendo com o método `newton`.\n",
    "2. Para os casos possíveis, determine a derivada. Caso contrário, utilize como método da Secante. \n",
    "3. Pesquise sobre o método de Halley e aplique-o aos problemas usando também a função `newton`, mas avaliando-a também com a segunda derivada. "
   ]
  }
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