2_Raices.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Busqueda de Raices\n",
    "En este capitulo expondremos la necesidad y la importancia para la busqueda de raices en  funciones o conjunto de datos. Despues definiremos varios metodos para busqueda de raices. Podemos clasificar los metodos como metodos cerrados, metodos abiertos, metodos para raices complejas, y para polinomios.\n",
    "\n",
    "Los metodos cerrados nos enfocaremos en buscar raices en rangos cerrados, por ejemplo buscar la raiz o el numero que hace cero la funcion entre el rango de numero de \"a\" a \"b\".\n",
    "\n",
    "$$ f(x) = 0$$\n",
    "Despues los metodos abiertos solo necesitamos definir una condicion inicial y por caracteristicas propias del metodo convergeremos a la raiz mas cercana a este punto.\n",
    "\n",
    "Los metodos anteriores se enfocan en buscar raices reales, es decir no podemos encontrar raices compleejas como en muchos casos aprecen en polinomios. Presentaremos metodos para buscar raices complejas.\n",
    "\n",
    "Por ultimo presentaremos metodos para buscar raices en polinomios. Y algunas operacione simportante entre poliniomios, como representar polinomios en computadores, multiplicacion de polinomios. Division de polinomios."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introducción\n",
    "La imporatancia de la busqueda de raices es evidente con el siguiente ejemplo\n",
    "si deseamos resolver la ecuación, es decir encontrar el valor de \"x\" que resuelve la ecuación:\n",
    "$$x^2=4$$\n",
    "\n",
    "$$e^x = x^2$$\n",
    "No encontraremos con el problema que la solucion es una funcion trasendental o solo se puede expresar como una sumatoria de terminos infinitos. Por lo tanto si definimos la ecuacion como una funcion implicita.\n",
    "\n",
    "Mientras la funciones simples contiene raices faciles de buscar por medios analiticos como:\n",
    "$$f(x) = e^x - x^2$$\n",
    "y graficamos esta funcion, podemos determinar graficamente que el valos que hace cero esta funcin es la solucion.\n",
    "Esta entre tantas otras aplicaciones es la utilidad de la busqueda de raices. Se espera que el lector este familiarizado con los metodos analiticos para busqueda de raices. Sin embargo un concepto primordial de los metodos numericos es que un metodo numerico bien definido sirve para resolver una amplia gama de problemas.\n",
    "\n",
    "Para presentar por primera vez los metodos de busqueda de raice vamos a defnir dos funciones de prueba:\n",
    "\n",
    "$$f_1(x) = x^2 - 4$$\n",
    "\n",
    "$$f_2(x) = e^x - x^2$$\n",
    "\n",
    "La primera función tiene un a respues analitica facil, y por lo tanto nos servira para validar nuestro metodo numerico. La segunda funcion de prueba es nuestro primer ejemplo. Y demotrara la utilidad de nuestros metodos"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Raices Reales\n",
    "Lo valores que hacen una funcion cero se conocen como raices, precisamnete los valores que pertencesn al conjunto de los numeros reales se conocen como raices relaes. Puede existir el caso el la funcion no tenga raices reales o tenga raice con numeros complejos. Este ultimo caso lo dejaremos para las ultimas secciones de este capitulo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math \n",
    "\n",
    "#Funcion de prueba f1\n",
    "def f1(x):\n",
    "    return x**2 - 4\n",
    "\n",
    "#Funcion de prueba f2\n",
    "def f2(x):\n",
    "    return math.exp(x)-x**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La función la podemos evaluar en varios puntos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f2(1) = 1.718281828459045\n",
      "f2(2) = 3.3890560989306504\n",
      "f2(3) = 11.085536923187668\n"
     ]
    }
   ],
   "source": [
    "print(\"f2(1) = \" + str(f2(1)))\n",
    "print(\"f2(2) = \" + str(f2(2)))\n",
    "print(\"f2(3) = \" + str(f2(3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La pregunta fundamental es: que valor de $x$ hace que la funcion $f_2$ es igual a $0$?\n",
    "La solucion mas simple es probar aletoriamente la funcion hasta que encontremos un valor muy cercano a cero."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SAbemos que valor de $f2(1)$ es el mas cercano a 0y por lo tanto podemos  evaluar la funcion con numero mas pequeños."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f2(0) = 1.0\n",
      "f2(-1) = -0.6321205588285577\n",
      "f2(-2) = -3.864664716763387\n"
     ]
    }
   ],
   "source": [
    "print(\"f2(0) = \" + str(f2(0)))\n",
    "print(\"f2(-1) = \" + str(f2(-1)))\n",
    "print(\"f2(-2) = \" + str(f2(-2)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nuestro mejor resultado fue $x=-1$, sin embargo, no es una raiz valida. Por otro lado con la funcion de prueba $f_1(x)$ podemos calcular la raiyz de forma analiitca. Por ejemplo\n",
    "$$f_1(x) = x^2 - 4 = 0$$\n",
    "$$x^2 - 4 = 0$$\n",
    "$$x^2 = 4$$\n",
    "$$x = 2$$\n",
    "El valor de $x=2$ se puede comprar que es una raiz, evaluando la funcion en este punto:\n",
    "$$f_1(x) = x^2 - 4 $$\n",
    "$$f_1(x=2) = 2^2 - 4 $$\n",
    "$$f_1(2) = 4 - 4 $$\n",
    "$$f_1(2) = 0 $$\n",
    "Y a continuacion podemos evaluar la funciona arbitrariamente por varios puntos alrededor de su raiz:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f1(1) = -3\n",
      "f1(2) = 0\n",
      "f1(3) = 5\n"
     ]
    }
   ],
   "source": [
    "print(\"f1(1) = \" + str(f1(1)))\n",
    "print(\"f1(2) = \" + str(f1(2)))\n",
    "print(\"f1(3) = \" + str(f1(3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos observar que este metodo es como jugar a la loteria, es question de suerte. El siguiente metodo es conocido como el metodo Grafico. Este metodo trata de seleccionar muchos puntos en una region definida de la funcion y graficarla."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11632ed30>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "dominio = [-1,0,1,2,3]\n",
    "y = [f1(-1),f1(0),f1(1),f1(2),f1(3)]\n",
    "#Graficamos los puntos\n",
    "plt.plot(dominio,y,\"o\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1164f62e8>,\n",
       " <matplotlib.lines.Line2D at 0x1164f6438>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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ZRrLSEnl07iT69OzidEkSQAp2kRji8Vjuf6uAhz/YzqwsF3/73gS6642no46+oyIxoq6xmZuez+P19fv5/pRB3HPeKcSpkVdUUrCLxICymgaufiKHNbvKue2bI5h/uhp5RTMFu0iU21Faw7xFn7P/cB1///4EzhnT1+mSJMgU7CJRbPXOMuY/kYMxhmeunsrEwb2dLklCQMEuEqVezdvHTUvyGNC7K4vmTWJwshp5xQoFu0iUsdbyjw+2c/+/C5icnsQ/fzCR3mrkFVMU7CJRpLHZw50vb+S51Xs4b1w/fn/xWOLj1Mgr1ijYRaJEVV0jP356LR9tK+Wns4dx49cz1cgrRinYRaLAvoojXPHYagpLqrn/orFcMmmg0yWJgxTsIhFu497DXPn4amrrm1k0bxJfG65GXrFOwS4SwZbnl/CTZ9bSq2snnr9uGiP6qJGXKNhFItaTK3dx97KNjOrXg0d+NIm0HmrkJV4KdpEI4/FY7vt3Pgs/LOKMEan89bLxJKiRl7SinwaRCFLX2MyCxbm8ufEAP5w2mLvOHaVGXvIVCnaRCHGoup6rnsghd08Fd3xrJFeeNkSNvOSoFOwiEWC7u5p5i1ZzsLKOf3x/AnNGq5GXHJuCXSTMrSo6xPwn1xDXwfDc/KmMH6RGXnJ8CnaRMLYsdy83P7+eAUldeWzuZAYld3O6JIkACnaRMGSt5aHlhTzw9lamDPE28urVTY28xD8KdpEw09js4faXNrAkp5gLxvfnvovGqJGXtImCXSSMVNY18uOn1vJxYSk/O3M4C84arpUv0mYKdpEwsbfiCPMWfU6Ru4bff2csF2erkZecHAW7SBjYUHyYKx5fTV1jM49fMZkZw1KcLkkimIJdxGHvbj7I/312HUkJnXn6qilkpiU6XZJEuHbdi2yM+b0xJt8Ys94Y85IxplegChOJBU98tpP5T+YwPK07L/1kukJdAqK9TSbeAUZba8cCW4Hb2l+SSPRr9lh+9dpm7lq2iTNGpPHc/KmkJqo7owRGu6ZirLVvt/rrSuA77StHJPodaWjmhsXreGvTQeZOT+fOc0fRUW9hJwEUyDn2K4DFAdyfSNRxV3kbea0vruCuc0dxxWlDnC5JotAJg90Y8y7Q5yifut1au8y3ze1AE/D0cfYzH5gPMGjQoJMqViSSFZZUMXfRakqr6/nn5RM5+5SjPa1E2u+EwW6tPet4nzfGzAXOBc601trj7GchsBAgOzv7mNuJRKPPth/imidz6BzXgcXzpzFuoNYZSPC0ayrGGDMHuAWYaa2tDUxJItHlpXXF3LJ0PYOTE1g0dxIDk9TIS4KrvXPsfwPigXd8tz2vtNZe2+6qRKKAtZa/vFfIH9/dyrSMZB7+wUR6du3kdFkSA9q7KmZYoAoRiSYNTR5ue3EDL6wt5sIJ/bnvwrF0jtNb2Elo6M5TkQA7fKSR655aw6fbD7HgrEx+duYwNfKSkFKwiwTQnrJarnhsNTsP1fDgJeO4cMIAp0uSGKRgFwmQ9cUVXPFYDg1NzTxxxRSmDU12uiSJUQp2kQB4e9MBrn8ul+TunXlu/hSGparnizhHwS7STos+2cEvX9vM2AG9+NcPs3ElxjtdksQ4BbvISWr2WH79+mYWfbKTb5ySxp++O56unfUWduI8BbvISahtaOL653J5Z/NBrjxtCL84Z6QaeUnYULCLtFFJVR1XPZ7Dxr2Huee8U/jR9HSnSxL5Lwp2kTbYdtDbyKuspoGFP8jmrFFpTpck8hUKdhE/fVpYyjVPraFLp44suWYaYwb0dLokkaNSsIv4YemaYm59YT0ZrgQenTuJAb3VyEvCl4Jd5Distfzx3W385b1tnDYshb9fPoEeXdTIS8Kbgl3kGOqbmrnthQ28uG4vF08cwG8vHEOnjmrkJeFPwS5yFIdrG7nmqRxWFpXx87Mz+clsNfKSyKFgF/mSPWW1zF30OXvKjvDnS0/l/FP7O12SSJso2EVaWbe7nKufyKGx2fLklZOZkqFGXhJ5IirYH/5gOx9tczM7K5VZWakMdSXo5bG0W21DE59tP8TyghKWrikmNbELi+ZNYqiru9OliZyUiAr2hPg43FX1/Pr1Lfz69S0MTOrK7KxUZmelMjUjWX06xG87S2tYXlDC8gI3K4sO0dDkoVvnjpw5Mo17zjuFlO5q5CWRy1hrQ37Q7Oxsm5OTc9L/v7i8lhUFblYUlPBJ4SGONDYTH9eBqRnJzM5yMXtEKoOTEwJYsUS6usZmVu0oY3l+CR9sdbOjtAaAoa4EZvkGB5OG9CY+ToMDCV/GmDXW2uwTbheJwd5aXWMzq3eWsTzfG/RFvidsRorvCTvCxeQhSXrCxqA9ZbWs2OpmRX4Jn27/zwBg+tBkZo9IZVZmKoOSdaORRI6YCfYv21lawwrfS+zPfC+xu3bqyIxhKczyjeb79+oalGOLsxqaPOTsKmNFgZvl+SVsK6kGYFBSN2ZnuZg1IpVpGcl06aRf8hKZYjbYWzvS0MxnRaWsKHDzfn4JxeVHAMhM695yATY7vbduOolgBw7X+X6Rl/DxtlJqGprp3LEDUzKSmJWVyqwsFxkpusgu0UHB/iXWWra7a1pC4PMdZTQ2WxLj4zhteAqzs1KZmeUirUeXkNYlbdPU7GHdngqW53tflW3ZXwlAv55dmDXCO1c+fWgyCfERtS5AxC8K9hOorm/ik8JSb9DnuzlQWQfAKf16+EbzLk4d2Is4jeYdV1pdzwcFbpYXlPDhVjeVdU107GDIHtyb2b4wz0zrrlG5RD0FextYa8k/UMXyghJW5LtZs7ucZo+lZ9dOnJ7pYnaWi5mZLpK1BC4kmj2W9cUVLPetfFpffBgAV2K8d648K5XThqeoGZfEHAV7Oxw+0sjH20q9QV/gprS6HmNg7IBe3uWUWamM6d+TDnortIApr2ngw21uVhS4+WCrm7KaBjoYGD+od0uYj+rbQ+dcYpqCPUA8HsumfZW+m1lKyN1TgbWQnNCZmb6QP324i57dNHpsC4/Hsnl/JcvzS1ix1c263eV4LCQldGZmpotZWS5OH+6id0Jnp0sVCRsK9iApq2ngo23ulhtdymsb6WBgwiDvfO+sLBej+vbQfO9RVNb5Xgn5wtxdVQ/AuAE9ffcceF8J6U2hRY5OwR4CzR5LXnEFK3wrNDbs9c4Fp/WIZ1am9+aoGcNSSIzRuWBrLVsPVntf7eSXsGZXOU0eS48ucb5rF6mcnunClahrFyL+ULA7oKSqjg8KvPPEH25zU1XXRFwHw6T0pJabo4anRvfqjZovVhv57vjcd9i72mhk3x4t7R7Ga7WRyElRsDussdnDut0VLaPV/ANVAPTv1dUb8lmpTB+WTLfOkb3e2lpLUWmNd3qlwM3nO8poaPbQPT6O04alMHuEi5mZqfTpqfsDRNpLwR5m9h8+0nKr+yeF/32H5Gzf/PKQlMhoXFbX2MxnRYdapqB2l9UCMDy1e8t1huzBSXSO06hcJJAU7GGsvqmZnJ3lLRcRC309TdKTu7XcBj81zHqa7D5U27Iy6LPth6hv6cGTzMysVGZluhiYpIZaIsGkYI8ge8pqWxqXfbq9lLpGD106dWD60JSWNdyhDs36pmZW7yhvCfMit7dr5pCUhJappMlDksLql49ItFOwR6i6xmZWFh1qaVz2xTTHsNTuLTdHZacHZ5pjX4VvuqjAO11U29BM51Z97mdlRc50kUg0CmmwG2NuAh4AXNba0hNtr2D3j7WWHaU1LbfWryryXphM6OxtQ/zFfHbfnifXhrix2cOaXeUtrRQKDv7nAu8Zvn1PGxr5F3hFooW/wd7uZ6wxZiBwNrC7vfuS/2aMIcPVnQxXd648bQg19f95b84VBW7e3nwQgBF9EluaYU0YdPylhCWVdd6liAUlfLS1lKp675LMyUOSuH3iSGaPcDHUFd1LMkWiXbtH7MaYpcCvgGVAtkbsoWGtZVtJta99bQk5O/9z88/XfDf/zMx0kZTQmdw9FS3tijfu9ba5TesR39KTfsaw5Ji9iUokkoRkxG6MOR/Ya63N0wgvtIwxZKYlkpmWyDUzh1JV18gnhaUsz/fOkb++fj8A3ePjqK73trmdOKg3t8zJYlZmKiP7JmpULhKlThjsxph3gT5H+dTtwC/wTsOckDFmPjAfYNCgQW0oUfyR2KUTc0b3Zc7ovljrbbC1osBNcfkRZgxL5mvD1KhMJFac9FSMMWYM8B5Q6/unAcA+YLK19sDx/q+mYkRE2i7oUzHW2g1AaqsD7sTPOXYREQke3fMtIhJlArZA2VqbHqh9iYjIydOIXUQkyijYRUSijIJdRCTKKNhFRKKMgl1EJMo40rbXGOMGdp3kf08BwnGtvOpqG9XVNqqrbcK1LmhfbYOtta4TbeRIsLeHMSbHnzuvQk11tY3qahvV1TbhWheEpjZNxYiIRBkFu4hIlInEYF/odAHHoLraRnW1jepqm3CtC0JQW8TNsYuIyPFF4ohdRESOI+yD3RhzsTFmkzHGY4w55pVkY8wcY0yBMabQGHNrCOpKMsa8Y4zZ5vuz9zG2azbG5Po+XgliPcf9+o0x8caYxb7PrzLGpAerljbWNdcY4251jq4KUV2PGmNKjDEbj/F5Y4z5i6/u9caYCWFS1yxjzOFW5+uuENQ00Biz3Biz2fdcvP4o24T8fPlZlxPnq4sx5nNjTJ6vrnuOsk1wn4/W2rD+AEYCWcAKvP3ej7ZNR2A7kAF0BvKAUUGu637gVt/jW4HfHWO76hCcoxN+/cCPgYd9jy8FFodJXXOBvznwc3U6MAHYeIzPnwO8CRhgKrAqTOqaBbwW4nPVF5jge5wIbD3K9zHk58vPupw4Xwbo7nvcCVgFTP3SNkF9Pob9iN1au8VaW3CCzSYDhdbaImttA/AccH6QSzsfeNz3+HHg/wT5eMfjz9ffut6lwJkm+G966sT3xS/W2g+BsuNscj7whPVaCfQyxvQNg7pCzlq731q71ve4CtgC9P/SZiE/X37WFXK+c1Dt+2sn38eXL2YG9fkY9sHup/7AnlZ/Lyb43+A0a+1+3+MDQNoxtutijMkxxqw0xgQr/P35+lu2sdY2AYeB5CDV05a6AC7yvXxfaowZGOSa/OXEz5S/pvle5r9pjDkllAf2TRmMxzsKbc3R83WcusCB82WM6WiMyQVKgHestcc8X8F4PgbsjTba43hvmG2tXRbqer5wgjfybmGttcaYYy0vGmyt3WuMyQDeN8ZssNZuD3StEexV4Flrbb0x5hq8o5gzHK4pnK3F+zNVbYw5B3gZGB6KAxtjugMvADdYaytDcUx/nKAuR86XtbYZONUY0wt4yRgz2lp71OsmwRAWwW6tPaudu9gLtB7pDfD9W7scry5jzEFjTF9r7X7fS86SY+xjr+/PImPMCryjikAHuz9f/xfbFBtj4oCewKEA19Hmuqy1rWv4F95rF+EgKD9T7dU6uKy1bxhj/m6MSbFBfq9hY0wnvOH5tLX2xaNs4sj5OlFdTp2vVsesMMYsB+YArYM9qM/HaJmKWQ0MN8YMMcZ0xnsxImgrUHxeAX7ke/wj4CuvLIwxvY0x8b7HKcAMYHMQavHn629d73eA963vyk0QnbCuL83Dnod3njQcvAL80LfaYypwuNXUm2OMMX2+mIs1xkzG+xwO6i9o3/EeAbZYax88xmYhP1/+1OXQ+XL5RuoYY7oCXwfyv7RZcJ+PobxafDIfwAV45+vqgYPAW75/7we80Wq7c/BeFd+Odwon2HUlA+8B24B3gSTfv2cD//I9ng5swLsaZANwZRDr+crXD/wSOM/3uAvwPFAIfA5khOj7d6K67gU2+c7RcmBEiOp6FtgPNPp+vq4ErgWu9X3eAA/56t7AMVZkOVDXT1udr5XA9BDUdBrei3/rgVzfxzlOny8/63LifI0F1vnq2gjcdZSf+6A+H3XnqYhIlImWqRgREfFRsIuIRBkFu4hIlFGwi4hEGQW7iEiUUbCLiEQZBbuISJRRsIuIRJn/Dy3lOhffNRheAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Grafica de la funcion de prueba 1 con pocos puntos e interpolacion lineal (6 puntos)\n",
    "plt.plot(dominio,y,[-1,3],[0,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "graficamos con mas puntos con 100 puntos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116570b70>,\n",
       " <matplotlib.lines.Line2D at 0x116570d68>,\n",
       " <matplotlib.lines.Line2D at 0x116570f98>,\n",
       " <matplotlib.lines.Line2D at 0x11657a320>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "xx = np.linspace(-1,3,100)\n",
    "yy = np.array([f1(xi) for xi in xx])\n",
    "plt.plot(xx,yy,[-1,3],[0,0],[2,2],[0,0],\"ro\",[2,2],[0,-4],\"r\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ahora aplicaremos el método Grafico para encontrar la raiz de la funcionde prueba 2, en el rango de x entre -3 y 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116639da0>,\n",
       " <matplotlib.lines.Line2D at 0x116639ef0>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "xx = np.linspace(-3,3,100)\n",
    "yy = np.array([f2(xi) for xi in xx])\n",
    "plt.plot(xx,yy,[-3,3],[0,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No es tan facil disernir el valor de x que hace que la funciont enga un valor de 0, pero podemos reducir el rango de busqueda de -1 a 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11669df28>,\n",
       " <matplotlib.lines.Line2D at 0x1166a70b8>]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "xx = np.linspace(-1,0,100)\n",
    "yy = np.array([f2(xi) for xi in xx])\n",
    "plt.plot(xx,yy,[-1,0],[0,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1167ce048>,\n",
       " <matplotlib.lines.Line2D at 0x1167ce198>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "# REfinando la busqueda\n",
    "xx = np.linspace(-0.8,-0.6,100)\n",
    "yy = np.array([f2(xi) for xi in xx])\n",
    "plt.plot(xx,yy,[-0.8,-0.6],[0,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El metodo grafico porve una herramienta visual para encontrar raices, pero es poco util, debido a l constante escalamiento manual , la la presicion depende de la resolucion del grafico."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Método de Bisección\n",
    "A continuacion presentaremos el primer metodo numerico del curso. Este método se basa en un proceso iterativo el cual busca encontrar la raiz de una función. Este metodo utiliza la intucion de que en un rango definido de una funcion si este rango en sus extremos conocido conocemos el valor, podemos inferir que si existe un cambio de signo significa que la funcion paso por la linea 0. Entonces podemos dividir el rango en algun punto interno, evalur donde ocurre el cambio de signo. Especificamente dividiremos el rango justo en sus centro.\n",
    "\n",
    "Por ejemplo, si deseamos buscar la raiz de la funcionde prueba 1, debemos seleccionar un rango de busqueda por ejemplo, entre 0 y 6 ( sabemos que la raiz esta en x=2, e inclusive existe tambien la raiz x=-2). \n",
    "Evaluamos la funcion en tres puntos, los extremos y el punto medio $(0+6)/2 = 3$.\n",
    "$$ f_1(x=0) $$\n",
    "$$ f_1(x=3) $$\n",
    "$$ f_1(x=6) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f1(0.000000) = -4\n",
      "f1(3.000000) = 5\n",
      "f1(6.000000) = 32\n"
     ]
    }
   ],
   "source": [
    "def evaluar(donde, funcion, nombre):\n",
    "    print(nombre+\"(%f) = \"%donde + str(funcion(donde)))\n",
    "    \n",
    "evaluar(0,f1,\"f1\")\n",
    "evaluar(3,f1,\"f1\")\n",
    "evaluar(6,f1,\"f1\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si graficamos estos puntos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116898fd0>,\n",
       " <matplotlib.lines.Line2D at 0x1168a3128>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.plot([0,3,6],[-4,5,32],\"o\", [0,6],[0,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos notar que el valor de la primera seccioón del rango cruza el cero, podemos creer que enta seccion se ecuentra la raiz. Seguimos refinando , ya que tomamos la subseccion con cambio de signo como nuevo rango, y biseccionamos en el punto medio y repetimos el proceso hasta encontrar  un valor  cercano a 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f1(0.000000) = -4\n",
      "f1(1.500000) = -1.75\n",
      "f1(3.000000) = 5\n"
     ]
    }
   ],
   "source": [
    "evaluar(0,f1,\"f1\")\n",
    "evaluar(1.5,f1,\"f1\")\n",
    "evaluar(3,f1,\"f1\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116907550>,\n",
       " <matplotlib.lines.Line2D at 0x116907668>]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "plt.plot([0,1.5,3],[-4,-1.75,5],\"o\", [0,3],[0,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si reptimos el proceso, la nueva subseccion que cruza el cero se encuentra en la seccion de la derecha."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f1(1.500000) = -1.75\n",
      "f1(2.250000) = 1.0625\n",
      "f1(3.000000) = 5\n"
     ]
    }
   ],
   "source": [
    "evaluar(1.5,f1,\"f1\")\n",
    "evaluar((1.5+3)/2,f1,\"f1\")\n",
    "evaluar(3,f1,\"f1\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116a1f898>,\n",
       " <matplotlib.lines.Line2D at 0x116a1f9b0>]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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JZ82pGNuflvQWSWttH5L052q9qaGI+HhE7Lf9BUmPS3pR0q3d/NFpuXyS/lLSJ23PqPWu+wer1VK3XC7pBkkz1blDqXXlwYYFGe9V68qYg5L+V61VUze1k/HP1DqX+bHqNMex6O5No9rJ2EvL5uv1sdJORvX2eDlf0u22B9RaPH82Iu6x/ReSpiPibrW+Mf2DWzdSfE6tK2OK4TdPASAZTsUAQDIUOwAkQ7EDQDIUOwAkQ7EDQDIUOwAkQ7EDQDIUOwAk8/9n8RyqfKzePQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot([1.5,2.25,3],[-1.75,1.0625,5],\"o\", [1.5,3],[0,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos a detenernos en esta iteracion, ya que podemos entender el proceso iterativo , va convergiendo lentamente a x=2, ya que los valores de la funcion se van aproximando a 0. A continuacion vamos a definir el algoritmo necesario para implementar esta función. \n",
    "\n",
    "+ Paso 1: definir un rango de busqueda $x_i$ y $x_f$\n",
    "+ Paso 2: Calcular el punto medio $x_m = (x_i + x_f)/2$\n",
    "+ Paso 3: Si el punto medio es cercano a 0 Parar, de lo contrario seguir\n",
    "+ Paso 4: Buscar la seccion de cambio de signo, y redefinir $x_i$ y $x_f$\n",
    "+ Paso 5: Ir al Paso 2\n",
    "\n",
    "A continuación un función en Python que toma como argumentos una función de prueba y un rango de busqueda:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "def biseccion(funcion, x_i, x_f, tol = 0.001):\n",
    "    iteracion = 0\n",
    "    print(\"Metodo de biseccion: tolerancia %f\"%tol)\n",
    "    \n",
    "    while True:\n",
    "    \n",
    "        x_m = (x_i + x_f) / 2\n",
    "        iteracion += 1\n",
    "        print(\"Iteracion#%i -  x_i = %f ; x_m = %f ; x_f = %f   ;; error=%f\"%(iteracion,x_i,x_m,x_f,funcion(x_m)))\n",
    "        \n",
    "        if abs(funcion(x_m)) < tol:\n",
    "            print(\"Raiz encontrada: \"+ str(x_m))\n",
    "            return x_m\n",
    "    \n",
    "        seccion_izq = funcion(x_i) * funcion(x_m)\n",
    "        seccion_der = funcion(x_m) * funcion(x_f)\n",
    "    \n",
    "        if seccion_izq < 0:\n",
    "            x_f = x_m\n",
    "            continue\n",
    "        elif seccion_der < 0:\n",
    "            x_i = x_m\n",
    "            continue  \n",
    "        else:\n",
    "            print(\"Error: no hay cambio de signo en ninguna seccion\")\n",
    "            return None\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Probamos nuesttro metodo de biseccion automatizado con la funcion de prueba:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Metodo de biseccion: tolerancia 0.001000\n",
      "Iteracion#1 -  x_i = 0.000000 ; x_m = 3.000000 ; x_f = 6.000000   ;; error=5.000000\n",
      "Iteracion#2 -  x_i = 0.000000 ; x_m = 1.500000 ; x_f = 3.000000   ;; error=-1.750000\n",
      "Iteracion#3 -  x_i = 1.500000 ; x_m = 2.250000 ; x_f = 3.000000   ;; error=1.062500\n",
      "Iteracion#4 -  x_i = 1.500000 ; x_m = 1.875000 ; x_f = 2.250000   ;; error=-0.484375\n",
      "Iteracion#5 -  x_i = 1.875000 ; x_m = 2.062500 ; x_f = 2.250000   ;; error=0.253906\n",
      "Iteracion#6 -  x_i = 1.875000 ; x_m = 1.968750 ; x_f = 2.062500   ;; error=-0.124023\n",
      "Iteracion#7 -  x_i = 1.968750 ; x_m = 2.015625 ; x_f = 2.062500   ;; error=0.062744\n",
      "Iteracion#8 -  x_i = 1.968750 ; x_m = 1.992188 ; x_f = 2.015625   ;; error=-0.031189\n",
      "Iteracion#9 -  x_i = 1.992188 ; x_m = 2.003906 ; x_f = 2.015625   ;; error=0.015640\n",
      "Iteracion#10 -  x_i = 1.992188 ; x_m = 1.998047 ; x_f = 2.003906   ;; error=-0.007809\n",
      "Iteracion#11 -  x_i = 1.998047 ; x_m = 2.000977 ; x_f = 2.003906   ;; error=0.003907\n",
      "Iteracion#12 -  x_i = 1.998047 ; x_m = 1.999512 ; x_f = 2.000977   ;; error=-0.001953\n",
      "Iteracion#13 -  x_i = 1.999512 ; x_m = 2.000244 ; x_f = 2.000977   ;; error=0.000977\n",
      "Raiz encontrada: 2.000244140625\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2.000244140625"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "biseccion(f1,0,6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Modificamos la funcion biseccion para que devuelva una tabla de para posterior analisis: y graficar la error vs el numero de iteracion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def biseccion_(f, x_i, x_f, tol = 0.001):\n",
    "    i = 0\n",
    "    x_m = (x_i + x_f) / 2\n",
    "    out_i = []\n",
    "    out_error = []\n",
    "    out_xm = []\n",
    "    while abs(f(x_m)) > tol:\n",
    "        x_m = (x_i + x_f) / 2\n",
    "        i += 1\n",
    "        out_i.append(i)\n",
    "        out_xm.append(x_m)\n",
    "        out_error.append(f(x_m))\n",
    "        seccion_izq = f(x_i) * f(x_m)\n",
    "        seccion_der = f(x_m) * f(x_f)\n",
    "        if seccion_izq < 0:\n",
    "            x_f = x_m\n",
    "        elif seccion_der < 0:\n",
    "            x_i = x_m\n",
    "        else:\n",
    "            return None\n",
    "        \n",
    "    return [out_i, out_xm, out_error]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116af5e48>]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "out = biseccion_(f1,0,6)\n",
    "#Grafica de Error vs unmero de iteracion\n",
    "plt.plot(out[0],out[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuacion utlizaremos nuestro primer metodo numerico para encontrar el valor de la funcion de prueba numero 2, que hasta entonces era casi imposible encontrar una raiz:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Metodo de biseccion: tolerancia 0.001000\n",
      "Iteracion#1 -  x_i = -2.000000 ; x_m = -1.000000 ; x_f = 0.000000   ;; error=-0.632121\n",
      "Iteracion#2 -  x_i = -1.000000 ; x_m = -0.500000 ; x_f = 0.000000   ;; error=0.356531\n",
      "Iteracion#3 -  x_i = -1.000000 ; x_m = -0.750000 ; x_f = -0.500000   ;; error=-0.090133\n",
      "Iteracion#4 -  x_i = -0.750000 ; x_m = -0.625000 ; x_f = -0.500000   ;; error=0.144636\n",
      "Iteracion#5 -  x_i = -0.750000 ; x_m = -0.687500 ; x_f = -0.625000   ;; error=0.030175\n",
      "Iteracion#6 -  x_i = -0.750000 ; x_m = -0.718750 ; x_f = -0.687500   ;; error=-0.029240\n",
      "Iteracion#7 -  x_i = -0.718750 ; x_m = -0.703125 ; x_f = -0.687500   ;; error=0.000651\n",
      "Raiz encontrada: -0.703125\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-0.703125"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "biseccion(f2,-2,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El resulto es $x=-0.703125$ con un error de $0.000651$, ademas podemos comprobar si este valor es una raiz:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0006511313011985931"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2(-0.703125)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si deseamos 6 sifras decimales de presicion podemos volver a buscar la raiz con este nuevo parametro:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Metodo de biseccion: tolerancia 0.000001\n",
      "Iteracion#1 -  x_i = -2.000000 ; x_m = -1.000000 ; x_f = 0.000000   ;; error=-0.632121\n",
      "Iteracion#2 -  x_i = -1.000000 ; x_m = -0.500000 ; x_f = 0.000000   ;; error=0.356531\n",
      "Iteracion#3 -  x_i = -1.000000 ; x_m = -0.750000 ; x_f = -0.500000   ;; error=-0.090133\n",
      "Iteracion#4 -  x_i = -0.750000 ; x_m = -0.625000 ; x_f = -0.500000   ;; error=0.144636\n",
      "Iteracion#5 -  x_i = -0.750000 ; x_m = -0.687500 ; x_f = -0.625000   ;; error=0.030175\n",
      "Iteracion#6 -  x_i = -0.750000 ; x_m = -0.718750 ; x_f = -0.687500   ;; error=-0.029240\n",
      "Iteracion#7 -  x_i = -0.718750 ; x_m = -0.703125 ; x_f = -0.687500   ;; error=0.000651\n",
      "Iteracion#8 -  x_i = -0.718750 ; x_m = -0.710938 ; x_f = -0.703125   ;; error=-0.014249\n",
      "Iteracion#9 -  x_i = -0.710938 ; x_m = -0.707031 ; x_f = -0.703125   ;; error=-0.006787\n",
      "Iteracion#10 -  x_i = -0.707031 ; x_m = -0.705078 ; x_f = -0.703125   ;; error=-0.003065\n",
      "Iteracion#11 -  x_i = -0.705078 ; x_m = -0.704102 ; x_f = -0.703125   ;; error=-0.001206\n",
      "Iteracion#12 -  x_i = -0.704102 ; x_m = -0.703613 ; x_f = -0.703125   ;; error=-0.000277\n",
      "Iteracion#13 -  x_i = -0.703613 ; x_m = -0.703369 ; x_f = -0.703125   ;; error=0.000187\n",
      "Iteracion#14 -  x_i = -0.703613 ; x_m = -0.703491 ; x_f = -0.703369   ;; error=-0.000045\n",
      "Iteracion#15 -  x_i = -0.703491 ; x_m = -0.703430 ; x_f = -0.703369   ;; error=0.000071\n",
      "Iteracion#16 -  x_i = -0.703491 ; x_m = -0.703461 ; x_f = -0.703430   ;; error=0.000013\n",
      "Iteracion#17 -  x_i = -0.703491 ; x_m = -0.703476 ; x_f = -0.703461   ;; error=-0.000016\n",
      "Iteracion#18 -  x_i = -0.703476 ; x_m = -0.703468 ; x_f = -0.703461   ;; error=-0.000002\n",
      "Iteracion#19 -  x_i = -0.703468 ; x_m = -0.703465 ; x_f = -0.703461   ;; error=0.000006\n",
      "Iteracion#20 -  x_i = -0.703468 ; x_m = -0.703466 ; x_f = -0.703465   ;; error=0.000002\n",
      "Iteracion#21 -  x_i = -0.703468 ; x_m = -0.703467 ; x_f = -0.703466   ;; error=0.000000\n",
      "Raiz encontrada: -0.7034673690795898\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-0.7034673690795898"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "biseccion(f2,-2,0, 1e-6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuacion probamos esta nueva raiz con mayor presicion:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0159194241410319e-07"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2(-0.7034673690795898)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4948664409517023"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "math.exp(-0.7034673690795898)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Por solo 6 cifras decimales estamos lejos del cero, esto comprueba que que este metodo con 21 iteracion, logra alacanzar una raiy de manera exitosa. A contnuacion vamos a graficar la convergencia de este metodo con la segunda funcion de prueba:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116b5a2e8>]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "out = biseccion_(f2,-2,0,1e-6)\n",
    "#Grafica de Error vs núnmero de iteracion\n",
    "plt.plot(out[0],out[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Posicion Falsa\n",
    "\n",
    "El metodo de la poisicon falsa inteligentemente aproxima la seccion entre los dos puntos cerrados como una linea recta buscando una aproximación dependiendo de los valores iniciales traza una linea recta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Newton-Rhapson"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Probablemente el metodo mas utilizado para calculo de raices. El metodo se basa en la intuicion de utilizar la pendiente la funcion para aproximar su acercamiento al cero. En su version de primer orden, tratamos de predecir que la funcion sigue su comportamiento de linea recta. Para demostrar este comportamiento utilizaremos la funcion de prueba numero 1.\n",
    "$$ f_1(x) = x^2-4$$\n",
    "Si derivamos :\n",
    "$$ f_1'(x) = 2x$$\n",
    "\n",
    "En este metodo no necesitamos definir un rango de busque, si no, un punto incial. El metodo utilizara la informacion de la pendiente para buscar los valores de x que hagan la funcion $0$\n",
    "\n",
    "Por ejemplo podemos utilizar el valor de 6 como punto inicial ($x_i$)\n",
    "\n",
    "\n",
    "El esquema de iteracion de Newton -Rahopson proviene de la aproximacion de ***series d Taylor***:\n",
    "Expansion de serie de Taylor\n",
    "$$ f(x_0 + \\Delta x) = f(x_0)  + \\frac{1}{1!}\\frac{\\partial f(x)}{\\partial x}\\biggr\\rvert_{x=x_0}\\Delta x + \\frac{1}{2!}\\frac{\\partial^2 f(x)}{\\partial x^2}\\biggr\\rvert_{x=x_0} \\Delta x^2 + ...$$\n",
    "\n",
    "$$ f(x_o + \\Delta x) = f(x_0) + \\sum_{k=1}^\\infty{\\frac{\\Delta x^k}{k!}\\frac{\\partial^k f(x)}{\\partial x^k}}\\biggr\\rvert_{x=x_0} $$\n",
    "\n",
    "\n",
    "Si solo tomamos en consideración el terminno constante $f(x_0)$ y el termino lineal $\\frac{\\partial f(x)}{\\partial x}\\biggr\\rvert_{x=x_0}\\Delta x$ , y reescibimos $\\Delta x$ como:\n",
    "\n",
    "$$ \\Delta x = x_{i+1} - x_i$$\n",
    "\n",
    "Podemos reescribir la paroximacion de la funcion en terminos lineales:\n",
    "\n",
    "$$ f(x_{i+1}) = f(x_i)  + (x_{i+1} - x_i)\\cdot\\frac{\\partial f(x)}{\\partial x}\\biggr\\rvert_{x=x_0} $$\n",
    "\n",
    "podemos aproximar la Raiz al igualar $f(x_{i+1}) = 0$ y despejamos $x_{i+1}$\n",
    "\n",
    "\n",
    "$$ x_{i+1} = x_i - \\frac{f(x_i)}{f´(x_i)} $$\n",
    "\n",
    "En caso que la funcion solucion no tenga curvatura (un alinea recta), el metodo siempre convergera en un paso. De lo contrario se tiene que aplicar el metodo sucesivamente para ir aproxmando el valor de la raiz."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def d_f1(x):\n",
    "    return 2*x\n",
    "# Definicion de la derivada analitica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "si evaluamos la funcion y su derivada en $x=6$ obtenemos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f1(6.000000) = 32\n",
      "d_f1(6.000000) = 12\n"
     ]
    }
   ],
   "source": [
    "evaluar(donde=6,funcion=f1,nombre=\"f1\")\n",
    "evaluar(donde=6,funcion=d_f1,nombre=\"d_f1\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si graficamos estos valores y la pendiente de la funcion en este punto:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116c78208>,\n",
       " <matplotlib.lines.Line2D at 0x116c78358>,\n",
       " <matplotlib.lines.Line2D at 0x116c785c0>,\n",
       " <matplotlib.lines.Line2D at 0x116c78908>]"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "xx = np.linspace(0,10,100)\n",
    "yy = np.array([f1(xi) for xi in xx])\n",
    "plt.plot(xx,yy,[0,10],[0,0],[6,6],[f1(6),f1(6)],\"ro\",[2,6],[32-(4*12),32],\"r\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Al \"seguir\" la información de la pendiente de la graficapodemos aproximar la raiz de la sigiente manera:\n",
    "$$ x_{i+1} = x_i - \\frac{f(x_i)}{f'(x_i)} $$\n",
    "Es importante mensionar que la primera aproximacion no necesariamente alcanza al cero, dependiendo de la curvatura de la funcion, tendremos diferente grados de convergencia.\n",
    "\n",
    "El diseño de el algoritmo es relativamente sencillo, peor por el monento se necesita suministrar como argumento la funcion que computa la derivada de la funcion.\n",
    "\n",
    "+ PASO1 : Definir un punto inicial ($x_i$)\n",
    "+ PASO2 : calcular $ x_{i+1} = x_i - \\frac{f(x_i)}{f'(x_i)} $\n",
    "+ PASO3 : siguiente iteracion $ i = i+1$ \n",
    "+ PASO4 : Si el valor actual de $f(x_i)$ es menor que la toleracia establecida, Parar.\n",
    "+ Paso5 : Repetir PASO 2\n",
    "\n",
    "A continuacion una implementacion modelo del algoritmo de newthon rhapson:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def newton_rhapson(f,df,xi,tol=0.0001):\n",
    "    i = 0\n",
    "    while abs(f(xi)) > tol:\n",
    "        i += 1\n",
    "        print(\"X_%i  =  %f    ; error=%f\"%(i,xi,(f(xi))))\n",
    "        xi = xi - f(xi)/df(xi)\n",
    "        \n",
    "    print(\"Raiz encontrada: \"+str(xi))\n",
    "    return xi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para probar este metodo para la funcion de prueba 1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X_1  =  6.000000    ; error=32.000000\n",
      "X_2  =  3.333333    ; error=7.111111\n",
      "X_3  =  2.266667    ; error=1.137778\n",
      "X_4  =  2.015686    ; error=0.062991\n",
      "X_5  =  2.000061    ; error=0.000244\n",
      "Raiz encontrada: 2.0000000009313226\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2.0000000009313226"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "newton_rhapson(f1,d_f1,6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos notar la importancia de este metodo y su velocidad de convergencia. A continuacion vamos a modificar el metodo para que retorne informacion tabulada sobre cada paso de iteracion para posterior analisis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "def newton_rhapson_(f,df,xi,tol=0.0001):\n",
    "    i = 0\n",
    "    out_i = []\n",
    "    out_x = []\n",
    "    out_error = []\n",
    "    while abs(f(xi)) >= tol:\n",
    "        i += 1\n",
    "        out_i.append(i)\n",
    "        out_x.append(xi)\n",
    "        out_error.append(f(xi))\n",
    "        xi = xi - f(xi)/df(xi)\n",
    "        \n",
    "    return [out_i, out_x, out_error]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Graficamos su numero de iteracion vs el error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116d3f2e8>]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "out = newton_rhapson_(f1,d_f1,6,1e-6)\n",
    "#Grafica de Error vs unmero de iteracion\n",
    "plt.plot(out[0],out[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Se puede observar una rapida convergencia y muy importante una estabilidad. A continuacion repiteremos el proceso para la funcion de prueba 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "X_1  =  0.000000    ; error=1.000000\n",
      "X_2  =  -1.000000    ; error=-0.632121\n",
      "X_3  =  -0.733044    ; error=-0.056908\n",
      "X_4  =  -0.703808    ; error=-0.000647\n",
      "Raiz encontrada: -0.7034674683317975\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-0.7034674683317975"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def d_f2(x):\n",
    "    return math.exp(x) - 2*x\n",
    "\n",
    "newton_rhapson(f2,d_f2,0,1e-6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-8.716603061564854e-08"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2(-0.7034674683317975)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como siguiente paso mostraremos la convergencia:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x116da1c88>]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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mptdleG75toNc9+xXPDw0ndED07wuR0SCnJktdc5lVNVOV/gGuW7JjRh8bgsmz9vIoeOFXpcjImFC4R8Cxg9J53BBES/P3+h1KSISJhT+IaBL60SuuqAlr3y1mQNHT3pdjoiEAYV/iBg3OJ2jJ4v42zzN/kXEfwr/EJHWogHDL2zNaws2k3ekwsskRESqTeEfQsYMTudkcQnPz93gdSkiEuIU/iGkfbN63NC9Df9YvIWdh457XY6IhDCFf4h5YGAazjmenZPjdSkiEsIU/iEmuUldbr4kmbeXbGPb/mNVf4GISAUU/iFo9OVpmBlPz872uhQRCVEK/xDUsmE8P+2ZwnvLtrNp71GvyxGREKTwD1H3DuhIXHQUT87M8roUEQlBCv8Q1bxBPD/v3Y6PVuwge/cRr8sRkRCj8A9h9/TvSL24GCbN1L5/ETkzCv8Q1rheHHf0SeWTVTtZs+OQ1+WISAhR+Ie4O/t1IDE+hokzNPsXkeoLSPib2TAzW29mOWb2SAXP1zGzt33PLzaz1ED0K9AwIZZR/Tswc91ulm876HU5IhIi/A5/M4sGngWuBLoAt5pZl3LN7gQOOOc6AROBP/nbr/zHyD7taVw3lgkzdOaPiFRPIGb+PYAc59xG59xJ4C1geLk2w4HXfMvvAoPMzALQtwD168Rw74COzMvKY8nm/V6XIyJnoLC4hH35J9iYl8+yrQeYs34PCzbsrfF+YwLwGm2AbWXWc4GelbVxzhWZ2SGgKXDKCM1sFDAKICUlJQClRY6f9UrlxfmbeGL6et4adanX5YhElOISx+HjhRw8Xsihch+Hjxdy8NjJco8Xccj32NGTxT94vQvbNuSj0X1rtOZAhH/AOOcmA5Oh9A3cPS4npCTERXPfgI784d9rWZCzl96dmnldkkhIKSlxHCko4uDx8kFdyMFjpSFe/rHvwv3IiaLTvnZ8bBQNE2JpmBBLo4Q42jRKoEurxP88Vjf2++XEhFiS6tep8fEGIvy3A8ll1tv6HquoTa6ZxQANgX0B6FvKuLVHCpPnbeSv09fzXsemaM+aRJqSEseRE0WVBvUps/GyIX+sNMDdaaaccTFRvvAuDelWDeM5p2UDEisI8O8/fI/ViYmuvW9CNQUi/JcAaWbWntKQvwW4rVybqcAIYCHwE2C2c6f7NsvZiI+NZvTATjz6wWrmZuVxeefmXpckcsacc+SfKKpg18kPd6lUtIul5DTJEhttp4RzUv06dEqq7wvquB+Ed9lAj48NvgD3h9/h79uHPxqYBkQDrzjn1pjZY0Cmc24q8DLwhpnlAPsp/QMhNeDGi5N5fu4GJkzPYkB6kmb/4gnnHMdOFle966RccH+3XHyaBI+Osu9n4IkJsTSuG0dq03qnhHViJQGeEBut3wmfgOzzd859Cnxa7rHfl1kuAG4MRF9yenExUYwZlMav3l3J9LW7ueK8ll6XJCHKOUdBYUmZ8D5Zwa6Timffh44XUlhceYBHGafMsBMTYklunFDh7pNE337y73ah1ItTgAdCUB3wlcD48UVteH7uBibOyGLIuS2IitIvivzQkYJC3ly8lZ2HCio9wHmyuKTSrzeDxPhTg7p1w4TvQ/qU3SdlZ+N1Y6kfF6OfS48p/MNQTHQUYwanMeat5Xy6eifXdG3tdUkSZBZt3MfD76wg98BxGsTHnBLUac3r06juqbtOvjtLpex6g3gFeChT+IepH3VtzbNzcpg4I4srz29FtH5JBSgoLOav09bz8lebSGlSl/fuvZSL2zXxuizxgG7sFqaiooxxg9PZkHeUj5aXP/NWItGq3EP86OkveenLTdzesx2fjemn4I9gmvmHsSvOa8l5rROZNDObH13Ymtho/a2PRIXFJTw3ZwNPz86maf04XrujB5elJ3ldlnhMaRDGoqKM8UPS2br/GO8tzfW6HPHAhrx8fvL8AibOzOKarq2YPvYyBb8AmvmHvYHnNKdbciOenp3Dj7u3CcorDSXwSkocry3czOOffUtCXDTP3tadq7u28rosCSKa+Yc5M+OhoelsP3ict5dsq/oLJORtP3ic219ezB/+vZY+nZoxfWx/Bb/8gGb+EaBvp2b0SG3CM7NzuCkjOewuU5dSzjneW7adP0xdQ4lzPH79Bdx8SbIuiJIKaeYfAb6b/e85coJ/LNridTlSA/bmn+DuN5by8DsrOLdVIp+P7c8tPVIU/FIpzfwjRM8OTenbqRnPz93ArT1SqFdHmz5cTF+zi9+8v4ojBUU8etW53NG3va7rkCpp5h9Bxg9NZ9/Rk7y2cLPXpUgAHC4o5OF3VjDqjaW0bBjPvx/oy139Oyj4pVo0/Ysg3VMaM/Cc5vzti43c3qsdifGxXpckZ2nBhr386p2V7DpcwAMDO/HAwDTiYjSXk+rTT0uEGT8knUPHC3nly01elyJnoaCwmD/8ew23vbiYOjFRvHvPpTw0tLOCX86YZv4R5vw2DRl2Xktenr+Jkb1TaVQ3zuuSpJpWbDvI+CnL2ZB3lJG9U/mvYeeQEKczt+Ts+DVdMLMmZjbDzLJ9nxtX0KabmS00szVmttLMbvanT/HfuCHp5J8sYvK8jV6XItVQWFzChBlZXP/8Ao6dLOafv+zJ/1x7noJf/OLv/4qPALOcc2nALN96eceAnzvnzgOGAZPMrJGf/YofOrdswI+6tubvCzazN/+E1+XIaWTvPsL1zy3gqVnZDO/Wms/H9qdPp2ZelyVhwN/wHw685lt+DbiufAPnXJZzLtu3vAPYA+jmIh4bMziNgsJiXpi7wetSpAIlJY6X5m/k6qe/ZPvB47xwe3cm3NSNhgk6SC+B4e8+/xbOuZ2+5V1Ai9M1NrMeQBygxPFYx6T6/PiitryxaAt39e9Ai8R4r0sSn237j/HwOytYvGk/g89twf+7/gKSGtTxuiwJM1XO/M1sppmtruBjeNl2zjkHVPqmnWbWCngD+IVzrsL3hjOzUWaWaWaZeXl5ZzgUOVNjBqVRXOJ4dk6O16UIpbdnmLJkG1c+OZ81Ow7zl5905cWfX6zglxpR5czfOTe4sufMbLeZtXLO7fSF+55K2iUCnwCPOucWnaavycBkgIyMjMrf/VkCIqVpXW7MSOatr7dx92UdadMoweuSIlbekRP85v2VzFy3h14dmvDXGy+kbeO6XpclYczfff5TgRG+5RHAR+UbmFkc8AHwunPuXT/7kwB7YGAnAJ6Zne1xJZHrs1U7uWLSPOZl7+W/r+nCm7/speCXGudv+D8ODDGzbGCwbx0zyzCzl3xtbgL6AyPNbLnvo5uf/UqAtG6UwG09U5iSmcuWfUe9LieiHDpeyLi3l3PvP5fRplECnzzQlzv7ttebokutsNJd9cEnIyPDZWZmel1GRNhzuIB+f57D1V1bMeEm/V2uDfOz8/j1uyvZc+QEDwzsxP2Xd9LbbEpAmNlS51xGVe300yY0T4xnRO9UPvxmOzl78r0uJ6wdO1nE7z9azc9e/pq6cdF8cF9vxg5OV/BLrdNPnABwd/8OxMdGM2lmltelhK1lWw9w9VNf8vrCLdzZtz2fPNiPrm11vaN4Q+EvADStX4df9Enl45U7WbfzsNflhJWTRSX8ddp6fvL8Ak4WlfDmXT3572u66B3VxFMKf/neqH4daRAfw8QZmv0HyvpdR7ju2a94Zk4ON3Rvy2dj+9G7o27PIN7TXT3lew3rxvLLvh2YODOLVbmHuKBtQ69LClnFvtszPDE9i8SEGCb/7GKGntfS67JEvqeZv5zijr6pNKoby4QZ670uJWRt3XeMWyYv5P999i2Xn5PEtLH9FfwSdBT+cooG8bHc3b8jc9bnsXTLAa/LCSnOOd5cvJVhT87j251HmHDThbxw+8U0ra/bM0jwUfjLD4zo3Y5m9eM0+z8Dew4XcMffl/DbD1ZxUUojpo3rz/Xd22KmC7YkOCn85QfqxsVw74BOfJWzj4Ub9nldTtD7eOUOhk6ax8KN+/jDtefxxh09aa37JEmQU/hLhX7aM4UWiXWYMGM9wXoVuNcOHjvJg//6htFvfkO7pvX45MF+jOidqtszSEhQ+EuF4mOjGT0wjSWbDzA/e6/X5QSdL7LyuGLSPD5dtZOHhqTz3j2X0jGpvtdliVSbwl8qdXNGMm0aJfDEdM3+v3P0RBGPfrCKEa98TcOEWD68vw8PDEojRrdnkBCjn1ipVFxMFA8O6sSK3EPMWlfhWzVElKVb9nPVU/N58+utjOrfgamj+3J+G10LIaFJ4S+ndX33tqQ2rcsTM7IoKYnM2f+JomL+9Pm33PjCQopLHG/d1YvfXnWubs8gIU3hL6cVGx3FmMFprNt5mM/X7PK6nFq3budhhj/zFc/P3cBNGcl8PrY/PTs09bosEb/5Ff5m1sTMZphZtu9z49O0TTSzXDN7xp8+pfZde2EbOjWvz8QZWRRHyOy/uMTx3Nwcrn3mS/bmn+SVkRk8fkNX6tfRHVEkPPg7838EmOWcSwNm+dYr80dgnp/9iQeio4xxg9PJ3pPPv1fs8LqcGrd571Fu+ttC/vz5eoZ0acH0cf0ZeE4Lr8sSCSh/w3848Jpv+TXguooamdnFQAtgup/9iUeuPL8l57RswKSZWRQVl3hdTo1wzvGPRVu48sn5ZO8+wpO3dOPZ27rTpF6c16WJBJy/4d/CObfTt7yL0oA/hZlFAU8AD/vZl3goKsp4aGhnNu87xvvLtntdTsDtOlTAiFeX8LsPV5OR2pjp4y5jeLc2uj2DhK0qd2Ca2UygolsSPlp2xTnnzKyiHcL3AZ8653Kr+kUys1HAKICUlJSqSpNaNvjc5lzYtiFPzsrmuovaEBcT+ucLOOeYumIH//3hagqLHX+87nxu75mi0JewV2X4O+cGV/acme02s1bOuZ1m1gqo6GTwS4F+ZnYfUB+IM7N859wPjg845yYDk6H0DdyrOwipHWbGuCHpjHx1CVMyt3F7r3Zel+SXA0dP8ruPVvPJyp10T2nEEzd1o32zel6XJVIr/D11YSowAnjc9/mj8g2ccz/9btnMRgIZFQW/hIbL0pPIaNeYZ2bn8JOL24bsue6zv93Nf723ioPHTvLrYZ25u39HonVPHokg/v7f/jgwxMyygcG+dcwsw8xe8rc4CT5mxvih6ew6XMCbi7d6Xc4Zyz9RxG/eX8kdf8+kab04Prq/L/cN6KTgl4hjwXrPloyMDJeZmel1GVKJ215cRNbuI8z79eXUjQuNc9+/3rSfh95ZTu6B49zdvyPjhqRRJyY0/3MRqYyZLXXOZVTVLvSP2IknHhqazt78k7y+cIvXpVSpoLCY//10HTdPXohhTLn7Uh658hwFv0S00JiySdC5uF0TLktP4m9fbOCnPVNoEB/rdUkVWr39EOOnLCdrdz4/7ZnCb686l3q6SldEM385ew8NTefAsUJe/Wqz16X8QFFxCU/Pyua6Z7/i4LFCXv3FJfzfH1+g4Bfx0W+CnLWubRsxpEsLXpy/kRGXptKwbnDM/jfm5TN+ygqWbzvItRe25rHh59Gorq7SFSlLM3/xy/gh6RwpKOLF+Ru9LoWSEsdrCzZz1VPz2bT3KE/fehFP3XqRgl+kApr5i1/ObZXI1V1b8epXm7ijb3vP7oOz4+Bxfv3uSr7M2cuAzkn86YautEiM96QWkVCgmb/4bdzgNI4XFvO3LzbUet/OOd5flssVk+axbOsB/vfHF/DqyEsU/CJV0Mxf/NapeQOu69aG1xZu5s5+7WneoHaCd1/+CR79YDWfr9nFJamN+euNF9KuqW7PIFIdmvlLQDw4KI3CYsdzc2pn9j9j7W6umDSP2d/u4TdXnsNboy5V8IucAYW/BERqs3rceHFb3ly8lR0Hj9dYP0cKCvn1uyu46/VMmjeIZ+oDfbj7Mt2XR+RMKfwlYEYP7ITD8cycnBp5/YUb9jFs0nzeXZrL6Ms78eH9fTinZWKN9CUS7hT+EjBtG9fllktSmLJkG9v2HwvY6xYUFvPHj9dy64uLiIuJ4p17evPwFZ3D4v0ERLyi3x4JqNEDS++Q+eSs7IC83srcg1zz9Je8/OUmfn5pOz55sC8Xt2sckNcWiWQKfwmoFonx3N6rHe8vy2VjXv5Zv05hcQmTZmbx4+cWkF9cvwSAAAAHDklEQVRQxBt39uCx4eeHzB1ERYKdwl8C7t4BHakTE33Ws/+cPfnc8PwCJs3M5toLWzNtbH/6pSUFuEqRyOZX+JtZEzObYWbZvs8V/j9uZilmNt3M1pnZWjNL9adfCW7N6tdhZJ9Upq7YwfpdR6r9dSUljle+3MTVT81n2/5jPPfT7ky8uVvQ3DNIJJz4O/N/BJjlnEsDZvnWK/I68Bfn3LlADyp+r18JI6P6daBeXAyTZmZVq33ugWP89KXFPPbxWvp2asa0cf256oJWNVylSOTyN/yHA6/5ll8DrivfwMy6ADHOuRkAzrl851zgTgWRoNS4Xhx39m3PZ6t3sXr7oUrbOed4J3MbwybNZ2XuQf58Q1deGpFRa1cJi0Qqf8O/hXNup295F9CigjbpwEEze9/MvjGzv5iZ3kIpAtzZrz0NE2KZOKPi2f/e/BOMemMpv3p3JV1aJ/L52P7cdEkyZrpgS6SmVXnqhJnNBFpW8NSjZVecc87MKnpD4BigH3ARsBV4GxgJvFxBX6OAUQApKSlVlSZBLjE+llH9O/CXaev5ZusBLkr5zyGhz1fv4tEPVnHkRBG/u/pc7ujTnihdpStSa6qc+TvnBjvnzq/g4yNgt5m1AvB9rmhffi6w3Dm30TlXBHwIdK+kr8nOuQznXEZSks7uCAcje6fSpF4cE3yz/8MFhYyfspx7/rGUVo3i+eSBvvyyXwcFv0gt83e3z1RghG95BPBRBW2WAI3M7Ls0Hwis9bNfCRH16sRw72UdmZ+9l+fnbmDYxHl8tHwHDw5K44P7+pDWooHXJYpEJH/D/3FgiJllA4N965hZhpm9BOCcKwYeBmaZ2SrAgBf97FdCyO292tG8QR3+9Pm3xMdF8969vRk/JJ3YaF1mIuIVc66i3fTey8jIcJmZmV6XIQHyRVYey7ceZFT/DiTE6Xi/SE0xs6XOuYyq2ulaeakVl6UncVm6juOIBAv93y0iEoEU/iIiEUjhLyISgRT+IiIRSOEvIhKBFP4iIhFI4S8iEoEU/iIiEShor/A1szxgix8v0QzYG6ByvBQu4wCNJViFy1jCZRzg31jaOeeqvKIyaMPfX2aWWZ1LnINduIwDNJZgFS5jCZdxQO2MRbt9REQikMJfRCQChXP4T/a6gAAJl3GAxhKswmUs4TIOqIWxhO0+fxERqVw4z/xFRKQSIR3+ZvaKme0xs9WVPG9m9pSZ5ZjZSjOr8L2Dg0E1xjLAzA6Z2XLfx+9ru8bqMLNkM5tjZmvNbI2ZjamgTUhsl2qOJei3i5nFm9nXZrbCN44/VNCmjpm97dsmi80stfYrrVo1xzLSzPLKbJNfelFrdZlZtJl9Y2YfV/BczW0X51zIfgD9KX0z+NWVPH8V8Bmlbx3ZC1jsdc1+jGUA8LHXdVZjHK2A7r7lBkAW0CUUt0s1xxL028X3fa7vW44FFgO9yrW5D3jBt3wL8LbXdfsxlpHAM17XegZjGg+8WdHPUU1ul5Ce+Tvn5gH7T9NkOPC6K7WI0jeSb1U71Z2ZaowlJDjndjrnlvmWjwDrgDblmoXEdqnmWIKe7/uc71uN9X2UP9g3HHjNt/wuMMjMrJZKrLZqjiVkmFlb4GrgpUqa1Nh2Cenwr4Y2wLYy67mE4C9vGZf6/t39zMzO87qYqvj+Rb2I0tlZWSG3XU4zFgiB7eLbtbAc2APMcM5Vuk2cc0XAIaBp7VZZPdUYC8ANvl2K75pZci2XeCYmAb8GSip5vsa2S7iHfzhZRull2xcCTwMfelzPaZlZfeA9YKxz7rDX9fijirGExHZxzhU757oBbYEeZna+1zWdrWqM5d9AqnOuKzCD/8ycg4qZXQPscc4t9aL/cA//7UDZv/ptfY+FHOfc4e/+3XXOfQrEmlkzj8uqkJnFUhqW/3TOvV9Bk5DZLlWNJZS2C4Bz7iAwBxhW7qnvt4mZxQANgX21W92ZqWwszrl9zrkTvtWXgItru7Zq6gNca2abgbeAgWb2j3Jtamy7hHv4TwV+7ju7pBdwyDm30+uizoaZtfxuX5+Z9aB02wXdL6evxpeBdc65CZU0C4ntUp2xhMJ2MbMkM2vkW04AhgDflms2FRjhW/4JMNv5jjIGk+qMpdzxo2spPVYTdJxzv3HOtXXOpVJ6MHe2c+72cs1qbLvEBOJFvGJm/6L0bItmZpYL/B9KDwDhnHsB+JTSM0tygGPAL7yptGrVGMtPgHvNrAg4DtwSjL+clM5mfgas8u2XBfgtkAIht12qM5ZQ2C6tgNfMLJrSP05TnHMfm9ljQKZzbiqlf+TeMLMcSk88uMW7ck+rOmN50MyuBYooHctIz6o9C7W1XXSFr4hIBAr33T4iIlIBhb+ISARS+IuIRCCFv4hIBFL4i4hEIIW/iEgEUviLiEQghb+ISAT6/9wFGiFsBgh3AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "out = newton_rhapson_(f2,d_f2,0,1e-6)\n",
    "#Grafica de Error vs unmero de iteracion\n",
    "plt.plot(out[0],out[2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Método Newton-Rhapson Multidimensional\n",
    "El esquema de iteración de Newton -Raphson  multilineal proviene de la aproximacion de ***series d Taylor*** para funciones de multiples \n",
    "Expansion de serie de Taylor multidimensional generalizado para un funcion $T(...)$ con un lista de variable $x_1, x_2, x_3 ...$:\n",
    "\n",
    "$${\\displaystyle {\\begin{aligned}T(x_{1},\\ldots ,x_{d})&=\\sum _{n_{1}=0}^{\\infty }\\cdots \\sum _{n_{d}=0}^{\\infty }{\\frac {(x_{1}-a_{1})^{n_{1}}\\cdots (x_{d}-a_{d})^{n_{d}}}{n_{1}!\\cdots n_{d}!}}\\,\\left({\\frac {\\partial ^{n_{1}+\\cdots +n_{d}}f}{\\partial x_{1}^{n_{1}}\\cdots \\partial x_{d}^{n_{d}}}}\\right)(a_{1},\\ldots ,a_{d})\\\\&=f(a_{1},\\ldots ,a_{d})+\\sum _{j=1}^{d}{\\frac {\\partial f(a_{1},\\ldots ,a_{d})}{\\partial x_{j}}}(x_{j}-a_{j})+{\\frac {1}{2!}}\\sum _{j=1}^{d}\\sum _{k=1}^{d}{\\frac {\\partial ^{2}f(a_{1},\\ldots ,a_{d})}{\\partial x_{j}\\partial x_{k}}}(x_{j}-a_{j})(x_{k}-a_{k})+\\\\&\\qquad \\qquad +{\\frac {1}{3!}}\\sum _{j=1}^{d}\\sum _{k=1}^{d}\\sum _{l=1}^{d}{\\frac {\\partial ^{3}f(a_{1},\\ldots ,a_{d})}{\\partial x_{j}\\partial x_{k}\\partial x_{l}}}(x_{j}-a_{j})(x_{k}-a_{k})(x_{l}-a_{l})+\\cdots \\end{aligned}}}$$\n",
    "\n",
    "$$ f(x_0 + \\Delta x) = f(x_0)  + \\frac{1}{1!}\\frac{\\partial f(x)}{\\partial x}\\biggr\\rvert_{x=x_0}\\Delta x + \\frac{1}{2!}\\frac{\\partial^2 f(x)}{\\partial x^2}\\biggr\\rvert_{x=x_0} \\Delta x^2 + ...$$\n",
    "\n",
    "$$ f(x_o + \\Delta x) = f(x_0) + \\sum_{k=1}^\\infty{\\frac{\\Delta x^k}{k!}\\frac{\\partial^k f(x)}{\\partial x^k}}\\biggr\\rvert_{x=x_0} $$\n",
    "\n",
    "\n",
    "Si solo tomamos en consideración el terminno constante $f(x_0)$ y el termino lineal $\\frac{\\partial f(x)}{\\partial x}\\biggr\\rvert_{x=x_0}\\Delta x$ , y reescibimos $\\Delta x$ como:\n",
    "\n",
    "$$ \\Delta x = x_{i+1} - x_i$$\n",
    "\n",
    "Podemos reescribir la paroximacion de la funcion en terminos lineales:\n",
    "\n",
    "$$ f(x_{i+1}) = f(x_i)  + (x_{i+1} - x_i)\\cdot\\frac{\\partial f(x)}{\\partial x}\\biggr\\rvert_{x=x_0} $$\n",
    "\n",
    "podemos aproximar la Raiz al igualar $f(x_{i+1}) = 0$ y despejamos $x_{i+1}$\n",
    "\n",
    "\n",
    "$$ x_{i+1} = x_i - \\frac{f(x_i)}{f´(x_i)} $$\n",
    "\n",
    "En caso que la funcion solucion no tenga curvatura (un alinea recta), el metodo siempre convergera en un paso. De lo contrario se tiene que aplicar el metodo sucesivamente para ir aproxmando el valor de la raiz.\n",
    "$$  f_i(\\vec x_i + \\Delta x) =  $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Funcion de prueba multidimensconal\n",
    "Para seguir el procedimiento de derivar un algoritmo, necesitamos aplicar una funcion los suficientemente simple para evaluarla mentalmente, como el caso de \" Que numeros sumados da 5 y multiplicados dan 6\"\n",
    "Esto es un sistema no lineal de dos ecuaciones dos incognitas, podemos definir dos variable incognitas $x_1$ y $x_2$\n",
    "\n",
    "\n",
    "$$ x_1 + x_2 = 5$$\n",
    "$$ x_1*x_2 = 6 $$\n",
    "\n",
    "La solucion a este sistema de ecuaciones es $x_1 = 2$ y $x_2 = 3$. A continuacion reescribimos las ecuaciones como funciones implicitas, que su solucion seria la raiz de tal funcion.\n",
    "\n",
    "$$ f_1(x_1,x_2) = x_1 + x_2 - 5 = 0$$\n",
    "$$ f_2(x_1,x_2) = x_1*x_2 - 6 =0$$\n",
    "\n",
    "En notacion vectorial escribirmo el vector de incognitas:\n",
    "\n",
    "$$\n",
    " \\mathbf x= \n",
    " \\begin{bmatrix}\n",
    "  x_1 \\\\\n",
    "  x_2 \n",
    " \\end{bmatrix}\n",
    " $$\n",
    "\n",
    "y La funcion multivariables:\n",
    "$$\n",
    " \\mathbf f(\\mathbf x) = \n",
    " \\begin{bmatrix}\n",
    "  f_1 \\\\\n",
    "  f_2\n",
    " \\end{bmatrix}\n",
    " $$\n",
    "\n",
    "Expandiendo los simbolos quedaria:\n",
    "$$\\mathbf f(\\mathbf x) = \n",
    " \\begin{bmatrix}\n",
    "  x_1 + x_2 - 5 \\\\\n",
    "  x_1*x_2 - 6\n",
    " \\end{bmatrix}$$\n",
    " \n",
    " \n",
    " Utilizando la primera y segundo terminos de la series de Taylor multidimensiona podemo encontrar la aproximación\n",
    "\n",
    "\n",
    "\n",
    "La solucion al sistema se formula como que valor de $\\mathbf x$ hace que $\\mathbf f(\\mathbf x) = \\mathbf 0$\n",
    "$$\n",
    " \\mathbf Jac(f)= \n",
    " \\begin{bmatrix}\n",
    "  1 & 1 \\\\\n",
    "  x_2 & x_1\n",
    " \\end{bmatrix}\n",
    " $$\n",
    " \n",
    " La defnicion del Jacobiano\n",
    " $${\\displaystyle \\mathbf {J} _{ij}={\\frac {\\partial f_{i}}{\\partial x_{j}}}}$$\n",
    " $${\\displaystyle \\mathbf {J} ={\\begin{bmatrix}{\\dfrac {\\partial \\mathbf {f} }{\\partial x_{1}}}&\\cdots &{\\dfrac {\\partial \\mathbf {f} }{\\partial x_{n}}}\\end{bmatrix}}={\\begin{bmatrix}{\\dfrac {\\partial f_{1}}{\\partial x_{1}}}&\\cdots &{\\dfrac {\\partial f_{1}}{\\partial x_{n}}}\\\\\\vdots &\\ddots &\\vdots \\\\{\\dfrac {\\partial f_{m}}{\\partial x_{1}}}&\\cdots &{\\dfrac {\\partial f_{m}}{\\partial x_{n}}}\\end{bmatrix}}}$$\n",
    " \n",
    "\n",
    "\n",
    "Y Por ultimo el esquema de iteracion para newton-Raphson multidimensional:\n",
    "$$ \\vec x_i^{n+1} = x_i^n -  \\left[ \\frac{\\partial^i f(x)}{\\partial x^j} \\right]^{-1} f_j(\\vec x^n)$$\n",
    "\n",
    "$$\n",
    "\\frac{\\partial^i f(x)}{\\partial x^j} = \n",
    " \\begin{bmatrix}\n",
    "  1 & 1 \\\\\n",
    "  x_2 & x_1\n",
    " \\end{bmatrix}\n",
    " $$\n",
    " \n",
    " "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def jacobiano(x):\n",
    "    return  np.array([[1, 1], [x[1],x[0]] ] )\n",
    "\n",
    "def f_p(x):\n",
    "    return np.array([x[0]+x[1] -5, x[0]*x[1]-6])\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on function diff in module __main__:\n",
      "\n",
      "diff(f, x, i, h=1e-06)\n",
      "    i=numero de 0,...,N-1(numero de columnas -1)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(diff)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "def nr_multi(f,xi,jac, tol=1e-3, iter_max=1000):\n",
    "    i = 0\n",
    "    while abs(f(xi).dot(f(xi))) >= tol:\n",
    "        i += 1 \n",
    "        \n",
    "        xi = xi - np.linalg.solve(jac(xi),f(xi))\n",
    "        print(\"x(i+1)=\" +str(xi))\n",
    "        \n",
    "    return 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x(i+1)=[0.94444444 4.05555556]\n",
      "x(i+1)=[1.64186508 3.35813492]\n",
      "x(i+1)=[1.9252678 3.0747322]\n",
      "x(i+1)=[1.9951413 3.0048587]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nr_multi(f_p,[40,4],jacobiano)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Ejemplo de sistema de ecuaciones no lineales, Sistema de balance de caudal de tuberias\n",
    "# Ecuaciones de lazo cerrado de mecanismos\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "def nr_multi_(f,xi,jac, tol=1e-3, iter_max=1000):\n",
    "    i = 0\n",
    "    \n",
    "    out_i = [i]\n",
    "    out_x = [xi]\n",
    "    out_error = [f(xi)]\n",
    "    \n",
    "    while abs(f(xi).dot(f(xi))) >= tol:\n",
    "        i += 1\n",
    "        out_i.append(i)\n",
    "        out_x.append(xi)\n",
    "        out_error.append(f(xi))\n",
    "        \n",
    "        print(\"%i ------------\\n xi=\"%(i) +str(xi))\n",
    "        print(\"Jac =\")\n",
    "        print(str(jac(xi)))\n",
    "        print(\"Jac^-1 =\")\n",
    "        print(str(np.linalg.inv(jac(xi))))\n",
    "        \n",
    "        xi = xi - np.linalg.solve(jac(xi),f(xi))\n",
    "        \n",
    "        print(\"x(i+1)=\" +str(xi))\n",
    "        \n",
    "    return [out_i, out_x, out_error]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'r' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-42-d17ca1acc36c>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'r' is not defined"
     ]
    }
   ],
   "source": [
    "r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para graficar los componetes de $\\vec f$ podemos generar una superficie para $f_1(x_1,x_2)$ y otra para$f_2(x_1,x_2)$ y visualizar donde se cuzan los ceros:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib qt \n",
    " # wx, gtk, osx, tk, empty uses default, inline lo pone dentor de la celda output, qt, abre la ventana de gnuplot(creoo!!!)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline \n",
    " # wx, gtk, osx, tk, empty uses default, inline lo pone dentor de la celda output, qt, abre la ventana de gnuplot(creoo!!!)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from mpl_toolkits.mplot3d import axes3d, Axes3D\n",
    "\n",
    "x = np.linspace(-6, 6, 30)\n",
    "y = np.linspace(-6, 6, 30)\n",
    "\n",
    "X, Y = np.meshgrid(x, y)\n",
    "Z = f_p([X, Y])\n",
    "fig = plt.figure()\n",
    "ax = plt.axes(projection='3d')\n",
    "ax.plot_surface(X, Y, Z[0,:], rstride=1, cstride=1,\n",
    "                cmap='viridis', edgecolor='none')\n",
    "ax.plot_surface(X, Y, Z[1,:], rstride=1, cstride=1,\n",
    "                cmap='viridis', edgecolor='none')\n",
    "ax.set_title('surface');\n",
    "#ax.contour(X, Y, Z, 16, extend3d=True)\n",
    "ax.set_xlabel('x')\n",
    "ax.set_ylabel('y')\n",
    "ax.set_zlabel('z');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Método de la Secante\n",
    "\n",
    "Anteriormente visitamos probablemente uno de los metodos mas importante del analisis numerico para determinar raices, pero en ocaciones no tenemos una expresion analitica o algebraica de la funcion a determinar la raiz, en este caso podemos aproximar la derivada por medio de una pendiente aproximada utilizando la secante de la curva de una función"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Raices Imaginarias"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Raices de Polinomios"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Los polinomios son un ejemplo especial de funciones. Antes de describir los metodos utilizados para la busqueda de raices, vamos a describir la utilidad y algunas operaciones sobre los poliniomios"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Evaluacions y differenciacion de polinomios\n",
    "un polinomio de tercer orden se puede escribir como:\n",
    "$$ f_3(x) = a_3 x^3 + a_2 x^2 + a_1 x^1 +a_0 $$\n",
    "\n",
    "Notece que el poliniomio puede minimizar la cantidad de operaciones, y a su vez la cantidad de error de redeondeo si se escribe de manera factorizada:\n",
    "$$ f_3(x) = a_0 + x(a_1 + x(a_2 + a_3 x))$$\n",
    "\n",
    "Podemos escribir una funcion que evalue un polinomio si le suministramos como argumento el arreglo de coeficientes \n",
    "$$ \\mathbf a = [a_0, a_1, a_2, a_3, ..., a_n]$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def eval_pol(x, a):\n",
    "    orden = len(a)\n",
    "    resultado = 0\n",
    "    print(\"Evaluacion de polinomio de %i orden.\"%orden)\n",
    "    for a_i in a:\n",
    "        resultado = resultado*x + a_i\n",
    "    return resultado\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuacion un ejemplo para evalua el polinomio:\n",
    "$$p(x) = x^2 - 4$$\n",
    "Podemos probar los puntos $x=0$  y $x=2$:\n",
    "$$p(x=2) = 0$$\n",
    "$$p(x=0) = -4$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "eval_pol(2,[1, 0, -4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    " eval_pol(0,[1, 0, -4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "En caso de calcular la primera derivada de un polinomio podemos utiliza el algoritmo para evulua:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def eval_d_pol(x, a):\n",
    "    resultado = 0\n",
    "    derivada = 0\n",
    "    for a_i in a:\n",
    "        df = df *x + p\n",
    "        p = p *x + a_i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Deflación del polinomio.\n",
    "Es importante recordar que los polinomios de n orden possen n raices, por lo tanto una vez encontrada una raiz de un polinomio es importante factorizar esta raiz del polinomio. Por ejemplo un polinomio de quinto orden como:\n",
    "$$f_5(x) = -120 -46x +79x^2 -3x^3 -7x^4 + x^5$$\n",
    "La version factorizada se puede escribir:\n",
    "$$f_5(x) = (x+1)(x-4)(x-5)(x+3)(x-2)$$\n",
    "Es aparente o hasta bastante obvio que los terminos que hacen cero en cada parentecis es una raiz exacta del polinomio.\n",
    "Si dividimos todo el poliniomo de 5 orden por un factor como $(x+3)$ y los pasamos a un polinomio de cuarto orden como:\n",
    "$$f_4(x) = (x+1)(x-4)(x-5)(x-2) = -40 -2x +27x^2 -10x^3 + x^4$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Raices de polinomios\n",
    "A continuacion se presentara dos metodos muy comunes para la determinacion de raices en polinomios: elemetodo de müller y el metodo de Bairstow\n",
    "\n",
    "### Método de Müller\n",
    "Podemos recordar que el metodo de la secante utiliza dos puntos de la funcion para proyectar una linea recta y poder estimar el valor de la raiz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Practicas\n",
    "1) Encontrar la raiz utilizando el metodo de la secante para la funcion de prueba 2\n",
    "2) Graficar la convergencia Residue vs numero de iteracion.\n",
    "3) Implementar la solucion de dos ecuaciones no lineales utilizando Newthon rapson para dos dimensiones\n",
    "\n",
    "\n",
    "### Casos practicos\n",
    "1) Solucionar la posiscion de un mecanismos de 4 barras ustilizando el metodo Punto fijo para la ecuaciones de posicion\n",
    "\n",
    "2) Solucionar la ecuacion de  Coolbrook-White para  determinar le coeficiente de friccion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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