MEF_puntos_optimos_tensiones_defo.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "# Puntos óptimos para el cálculo de tensiones y deformaciones en el MEF\n",
    "\n",
    "\n",
    "\n",
    "Análisis de Estructuras. Grado en Ingeniería Civil. \n",
    "\n",
    "Autor: \n",
    "- Alejandro E. Martínez-Castro.\n",
    "- Dpto. Mecánica de Estructuras e Ingeniería Hidráulica. Universidad de Granada.\n",
    "\n",
    "\n",
    "\n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\"><img alt=\"Licencia de Creative Commons\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" property=\"dct:title\">Cuadernos MEF: Viga de Bernoulli-Euler</span> by <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\">Alejandro Martínez Castro, Esther Puertas García, Rafael Gallego Sevilla</span> is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional License</a>."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Motivación\n",
    "\n",
    "Este cuaderno es un apoyo a la clase presencial. Utilizaremos las habilidades de Python para mostrar con ejemplos una interesante propiedad de los puntos de una cuadratura de Gauss. \n",
    "\n",
    "En el intervalo $x \\in [-1,1]$, consideremos un polinomio de grado $N$, y otro de grado $N-1$ obtenido como ajuste por mínimos cuadrados a partir del anterior. \n",
    "\n",
    "Puede demostrarse que ambos polinomios se cortan en las abscisas de los puntos de Gauss de una cuadratura de orden $N$. \n",
    "\n",
    "En este cuaderno se plantean unos ejemplos. Es interesante que pruebe modificar los coeficientes del polinomio original, para ver que la coincidencia no depende de dichos coeficientes. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.0 (Python 2.7.13-32-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/1.0/\n"
     ]
    }
   ],
   "source": [
    "from sympy import init_session\n",
    "init_session()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ejemplo 1. Polinomio de grado $N=2$\n",
    "\n",
    "Consideremos el polinomio\n",
    "\n",
    "$f(x) = 1 + x + x^2$, \n",
    "\n",
    "con $x \\in [-1,1]$. \n",
    "\n",
    "Este polinomio es de grado $N=2$. Un polinomio de grado $N-1$, genérico, es de la forma: \n",
    "\n",
    "$g(x) = a + b x$\n",
    "\n",
    "Ajusteremos las constantes $a$ y $b$ para que $g(x)$ sea el ajuste por mínimos cuadrados del polinomio $f(x)$ en el intervalo $x\\in[-1,1]$. \n",
    "\n",
    "En primer lugar, definiremos los símbolos para $a$ y $b$, y definiremos los polinomios para $f$ y $g$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a,b = symbols('a b')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFwAAAAWBAMAAAC/AuZhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NuzJU\nme9mZ+xlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA90lEQVQ4EWNggIDctQJQFjEUmwPzBGLUQdVw\nOHB/IUV5AMtXEpQzMPB8JEk5hwJJynNJUs1+gbBynodwNWEM0RD2IbgQnAEVEk6Ge497zpmFhJQz\nsMOV8/3//4E45ULKrmoKcIsZYI5BEoYJgUxnTGBv4kBKAFA5ZGFk5WwCbB9ZHTBMRxZGVs7IwLUA\nrjhz5sy+mTMnA/lwYYQQ1Kv8CnDlQAbMKAaEMEwIEjL7N2BVjhBGVs4joM/AJIDQAJVDFkZWHn9g\nPYM7QjXMMcjCMOVMwIAUVhK6XoCpHFkYqpzX7ud1JIUQJswoJAksQnBZQTgLzkASAgDD9UjIFbG9\nPgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{2} + x + 1$$"
      ],
      "text/plain": [
       " 2        \n",
       "x  + x + 1"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = 1 + x + x**2\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAQBAMAAACxV6F2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA60lEQVQYGWNgwABsSzdgiCEEOBQQbDiL1wHC\n5L8AF0IwYJL1CQgxOAsmaQ0XYWBUVgmD8GCSfS66DGwuD0tMGBhKBTg+oEiy/UjIF0hmmO2wg4Et\nkoFpAYok7zcGfgczBvuEQgambwzcDkBJ5o6O7oiOjgcMDEwNDPkGCQyrgIKsCxjyDwBpIIDaCfRm\nPdAzX4Ai+QYM98FScEn+CQz6CQw8f0CSFxhe8AqApaE6+QuA7rjG8oHBjYHLgXMfM0QrVJJvApcC\n+1+uD4wTGNie+Mi5o0iyqVgCRd2cZkJEoSRUJ4oYnMOWAGdCGACwOjDFHy7MYwAAAABJRU5ErkJg\ngg==\n",
      "text/latex": [
       "$$a + b x$$"
      ],
      "text/plain": [
       "a + b⋅x"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = a + b * x\n",
    "g"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación se va a plantear minimizar la siguiente función: \n",
    "\n",
    "$$ M(a,b) = \\int_{-1}^{1} [f(x) - g(x) ]^2 \\,d\\,x$$\n",
    "\n",
    "Plantearemos por tanto que las derivadas parciales respecto de $a$ y $b$ sean 0, y de ahí podremos obtener los valores de $a$ y $b$ del polinomio de ajuste. \n",
    "\n",
    "\\begin{align}\n",
    "\\dfrac{\\partial M}{\\partial a} &= 0 \\\\\n",
    "\\dfrac{\\partial M}{\\partial b} &= 0\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPoAAAAvBAMAAAA4McqSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$2 a^{2} - \\frac{16 a}{3} + \\frac{2 b^{2}}{3} - \\frac{4 b}{3} + \\frac{22}{5}$$"
      ],
      "text/plain": [
       "                 2           \n",
       "   2   16⋅a   2⋅b    4⋅b   22\n",
       "2⋅a  - ──── + ──── - ─── + ──\n",
       "        3      3      3    5 "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimizar = integrate((f - g)**2,(x,-1,1))\n",
    "minimizar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAqBAMAAAAT9FuoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$4 a - \\frac{16}{3}$$"
      ],
      "text/plain": [
       "4⋅a - 16/3"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = diff(minimizar,a)\n",
    "eq1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADwAAAArBAMAAADWAfZiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n534zX8PNWDbXffwAYPBmsnZycgsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{4 b}{3} - \\frac{4}{3}$$"
      ],
      "text/plain": [
       "4⋅b   4\n",
       "─── - ─\n",
       " 3    3"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq2 = diff(minimizar,b)\n",
    "eq2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Debemos resolver ambas ecuaciones a la vez. Para esto, Sympy dispone de un solver: el resultado está en formato de _diccionario_. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIgAAAAyBAMAAAB1xY5yAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEImZRO/dMlQiu6vN\nZnZmcXX2AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACsElEQVRIDa1WTWvUQBh+s9mvtM3u0qIHFRME\nPehl6+JNS0A8S0EKfkEV7GWhLPgH6sGTh128SumKHrxZD73ZsqVHEffi2eAHHuv6gdVW68w7yWYm\nM5kmsnNI3vd5n+fJJPMVANIm6/T6P816Fqryc60wzHx/+ziQtJUduZvK0FqoIc/6raQ3lagErqwi\nND6QKgSoXFehMmb+RKzak0sAy69UqIyV/iJWZR2K1d2UJsY3FDoqE7uT1mQ32eQ4jMDEHYFJuTUC\nk0kYgcmDRuPHTGy81Kmh+bAAKSeb3uSa+slxVGtyen89zlfmWhOV4sicjGY2gS3epDxLs+wmO5zJ\n0TNs9bPRWXG5kjbMC5uGzTK2G73ztEqumJvnEghM3iC2gFfr7Ay+Y5PnxeKJjQ9LFGKcwKTdJ0ju\nF8XhUmfiC7kZNzBTX5xNg5IDTmAyTqGTHlUYN8Hs0kDX2jVg+xiSAhNY8+DYC0TMART9yGCfNtoz\noZ0H42sEhCblRTg8jXChC04rIqijLchzgxyalBYBzqHWcaGtVnLoHyAdHrbQhH4P8yqFnSV4We4M\nCergOxzqR5XAJIdnBg7xmG83c4Qgjo7TizQkWocnNA84Fdaryz7Ftj1yMTY+P/pIM2GeFG9RaNim\n3rOOIMdc27tAK8+xrJz2D0/geX9vaJAY7GHFqcsEqw44E125FEM0q7iyA9V58pJ+TCKnGpP8FaBH\ntC2L4ojGhFDxdeISOdeaGLdlgQrRmVin2HJQ6QRMZwLwyRPISYnepDCbpBNwjYnVgTFuwQoyMdGY\nVAcjMCl0ocitevHhQqbpid3POE+UP34XG0+FByYmwY9foZfISFEosV0X9/wUdDWlwn6GLXaaqjkH\nok6XUbb9A6mJBONOjdXs3SBIpCYX7m+GtanVMMp6t15TxT/jM7BIVg7UrwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ a : \\frac{4}{3}, \\quad b : 1\\right \\}$$"
      ],
      "text/plain": [
       "{a: 4/3, b: 1}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.solvers import solve\n",
    "solucion = solve([eq1,eq2],[a,b])\n",
    "solucion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a_sol = solucion[a]\n",
    "b_sol = solucion[b]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El polinomio de ajuste, por tanto, es el siguiente:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAAqBAMAAADsVn5KAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NuzJU\nme9mZ+xlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBUlEQVQ4EWNgwA72YhcGiTbjlGJ6jFPKfTJO\nqQScUuwOOKWEGHBKJeCU4tmAU0qMAafU4bt3/13C6XrcXmZ4hVNT1n9znHIkSBzCrZYEKSFlVzUF\niElouhgT2Js4JmCVYhNg+8jqgFWKkYFrAVgic+bMvpkzwfH0HwQ+AEX5FcBSQALNLgaG/RtwSPEI\n6DMwCUAk0XTFH1jP4A7VhiYlrCR0vQC7FMweEA3X5Za2BlkcyBaE8lkuMJyHmYSmhOkPA38DmhiU\ny/qUgX8CdimgKC4DGRgY23BpYsm4iEuKgUEvAKcc30PsUiwODFx/sEvxf8QpxbeAgeMjdl3sBbj9\nFXt3KQMDAO94QHO72EP3AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x + \\frac{4}{3}$$"
      ],
      "text/plain": [
       "x + 4/3"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g_ajuste = g.subs([(a,a_sol),(b,b_sol)])\n",
    "g_ajuste"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación se dibujarán ambas funciones, pero no se va a dar salida en este cuaderno, sino en ventana aparte, para poder localizar con el puntero los puntos de corte. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x9b56a20>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Dibujo ambas funciones. Ver en figura aparte y localizar los puntos de corte\n",
    "from sympy.plotting import plot\n",
    "\n",
    "plot(f,g_ajuste,(x,-1,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación localizaremos los puntos de corte entre ambas curvas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAA/BAMAAAA/PtpxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARM1UEDIidquZ3Ylm\n77tJMhrgAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACQElEQVRIDe3XP2vbQBgG8MfYkVv/iQ2dOpSI\nkI4Bg6dMFu0Q8KfwlFIypB+gIeQbeMjipfHQpe0Qg7cYgrcuXQJdW1r6AfLPU1ui3HvWWSfd+9pg\nEwjBB1Gce+4XSaeQB2EtDH3MNy7CsIa1er06H8ezekP5ObFm5XvyeyEN8cre6PgSkM5/KFIdTGLB\ne/5UH8eCf1qb6uNY8E14gzP5dzSB3cEfOoXg+3iL7H/xGvrw2mgMRZ/r4qCGv5JXcf4Gla7oi0O8\nqOJa8irOnKPii17dH6Zcv46nXf978s0eHbmh49wXipz9y1TVbFd97X4K1DE9JrF3ekSZ4/d9IP+O\novw5HVMjjrFPZ0p7b/sKKIzRjyCFATsuf2c8crfAVxXsABc1x09ir4XVG87juIWPKggD1pu4MpL8\nXrvUU/4ncBKo7+kRxeVfKIxUlr5/4MmoEKjgm/D8o7g0lJ7/yr8tOmem84H21xlRjOed35S558fm\nhoPsiWTM+EbbXu58TsaMX205xp5Ixoy3F8/8vPS6/16t03ip/uwTw9m+REr19Mj3Lyf+D4+2Ztb9\nfw6cPUxMsD7qNlqYDRLL9Q92NXLedBstLro8UY2cN91GdJvxdjVy3nQb0SHj7WrkvCK6mxkaTcXV\nzvtxt8k+rkbWR90meqsaWY9xt4neqkbB626TPSbVyHnTbZK3q5Hzptskb1cj5023Sd6uRs6bbpO8\nXY2cN90mebsaWS9BZn7pH+b7E/Oo2KnH8PwWfH9+vcj7cz24A43HvdED+JF7AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ - \\frac{\\sqrt{3}}{3}, \\quad \\frac{\\sqrt{3}}{3}\\right ]$$"
      ],
      "text/plain": [
       "⎡-√3   √3⎤\n",
       "⎢────, ──⎥\n",
       "⎣ 3    3 ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "puntos_corte = solve(f-g_ajuste,x)\n",
    "puntos_corte"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Evaluaremos numéricamente estos puntos:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "puntos = []\n",
    "for punto in puntos_corte:\n",
    "    puntos.append(N(punto))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY0AAAAUBAMAAACDl3zdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ -0.577350269189626, \\quad 0.577350269189626\\right ]$$"
      ],
      "text/plain": [
       "[-0.577350269189626, 0.577350269189626]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "puntos\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Estos puntos de corte coinciden con las abscisas de los puntos de Gauss de una cuadratura de orden 2. \n",
    "\n",
    "Veámoslo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.57735027,  0.57735027])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "NPGauss = 2\n",
    "pgauss = np.polynomial.legendre.leggauss(NPGauss)\n",
    "        \n",
    "pgauss[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ejemplo 2. Polinomio de grado $N=3$\n",
    "\n",
    "Consideremos el polinomio\n",
    "\n",
    "$f(x) = 1 + x + x^2 + x^3$, con $x \\in [-1,1]$. \n",
    "\n",
    "Este polinomio es, en esta ocasión, de grado $N=3$. Un polinomio de grado $N-1$, genérico, es de la forma: \n",
    "\n",
    "$g(x) = a + b x + c x^2$\n",
    "\n",
    "Ajusteremos las constantes $a$, $b$, y $c$ para que $g(x)$ sea el ajuste por mínimos cuadrados del polinomio $f(x)$ en el intervalo $x\\in[-1,1]$. \n",
    "\n",
    "Definiremos los símbolos adicionales y realizaremos operaciones análogas al caso anterior. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcMAAAAwBAMAAACf2QzuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJm7MquJRO/dIs12\nVGbfGimAAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHuElEQVRoBd1ab2hbVRQ/L22SJm2S52QyqdDY\nuu6Dm/ujoKJi2Ac/6AfLkLGxuoWOFqZjDSKyb+0YY3MDF4RO0UmzgoPtg8axCRJcyxhzzEKDIgxx\nNvppwpgrjs2trvHce/Peu/fd8241JUV3PvSde36/c849vS95p+8UYBEk0vVo3Vn61tl1+y6i4xuw\npd5skUK4WK/vYvp9BK8W6swXKzTfrdN1Ud0+gM9zdSaM5eK363RdZLfp+j9QiZlG73X9vAl2XsjQ\nnCVdT4H1TJ6BRzVGfGB7GazOJzPw4LsaKABhjk1o8PGBFwBY9GCxuvsLcpRgJsCICbyMh9OeCxVJ\njtUDgzloTSMYYT9UGYPEHCyFCN6Gp1SErQTA7X0aGk/DdEVE1zBmSOwBWFaJ75GjkERhbNsXDFob\nt2GJ66BpDckJ2dAyDKkygr/qhKky3AF8lhwAuKejAmD2ZFpD2+5BKi+iaxgalvbjrb0JYEMtPMWR\nbZ+dkVd+/XcbooFfeC15CN+FIfwtRLPRjN91dw5mAW/RbYUo8WHjAHd5B972u0b3Q6ooovshvk5i\nwFmAQRGepMjG7HwlNg3LdFlvnYHwTdiBpleufI+F+gVv1HEbtmUieT8CAmDm5q+vrNVhvFFFdALC\ng58B6w+AqQIPT1IkY7KAJca7V+LdTQmeYur0zhWBlNgMnJlcBYerVcL74SwzjtqtvefYzeyXURsi\nk+dbqtUbfgTAOsJsGB12TmY0mJ3iITzFHCIYZR55BLDE3YVWIg3zxBI7zkKsHEQZylqzdkeBTHL8\nfZY9egs6vrCImx2B+CfwHOka34GfVYChLJwoNxU1CitxEO+ECg+vwT5DFku03oPQGp+9tmQl3oDQ\nniDKCOBTLTVBO7ftR3usB6Zt+EtnIBDLwus6wC0Xc3gZATgC4bJGYSU2TSQGEcEoiugdbyKDJYZm\ngHg2cU92o+YheTuAEkoDPlA6skoWb3HABugEeBysW57R0RBYjXiAtOAnh0XHYnRhJcLP5wczPLyM\nEx3vQ4AltqyBDmRTgiXGipCcC6CcA/ZYHMLfpi4nAV4sQ1uaPRaj+mODAXt1L26JF6AJPTA65iWE\nl4jf1TYPLxOIjveHzs4/V+IpTMs8SccSw8N4ijQlkYaTqUrAYVRtVuJ3EC/MUYeBwDHibHnu1Awr\nkUWnb5BaifgtwsJL2wW6490HHWXYklCYrheW2DwDoWGa8hLApVQOP6eU4BGN2s1ZaCvcxlbEz+DA\nHDYqfoCt8ezwy5RHrwDo9wgr8aqNf6HwKGoEquPdi5/c5EhEJTqrKfy0bIRlZZISPzKwPB+uNKUd\ntnK9zhq4EwN9m2ErNkh+4cBysB7zA2ydrOBzkUcP9cCSjEZpwxK/tdsrIrwKE98q26tbrd7zn36j\nEmur5YdPVSDS/RqQlNZqtZq3Vu4iXSHa2ZWD8Wr1Diy5UNEoHIjsuqYB3PBW59PAo8PA9QmNEto0\n+wRE+lbgY4OFx6coa/Zr0uco99dV6umT6furNKca7BOw2efidbw/OWg91wU4JybqSSh8gtN6Pb3U\n8QbT/8EWFuDcmBLHWbPPNy51vAvYJcACnBtTIlY3asNY30X8/neF3iXV1bsunkI7e7hBa1SJ2NNH\nD8K4nJncJdnVy141nXQmeISpUSViN47tRZeckdwl2dXLXjWddCZ4hKlRJXbynv6om/HNUunDUgk7\nPAB8eDviNJLtJSanfShnMRvtjPANBC3uW8qgmnLiiitDI6XSV4dKpR4dRQu0c18qL0OltL7ILLHo\n6a2bjOkIdRB0V+94SFfKWYJNaoNOUfT0CfZ7dIXaZUfWhc0K5Wz2cNHGlMi68WP3IFz80U1Ef+/j\nn0VEVy85Oep/rkTe0z8Pv/VknS3ildol3dVLTo5KOTvYPNfGnOI468aXTr7cm5PSk7ukunrJx1VJ\nZxc1Ko0pkUwp7ZLPHkhSzagRJGek8PmDwf+svb7swmqJrb80r3IhQlFxOS1PqkTW3B/wLHz24C11\nTSNIzviClc0fdCdhuWwDvmOV3l9YaPAEO8qst1I1NrdQcSmtSKpEVr3V1RSbPZjETODzB9pdjES+\n7C3QML7a6r8UBPG5RSAukhoiq2H57EE1qSszgc8fVAdvha9+oMdb+rWw3yCt2fuZQFwkNUSWAnEV\n37GYZR5C8I3aqBJxu5j0X5QoZg+GKs0EMX8g3VmJW6+Rr6AYP3z9XIZdKeGnGIyzpIbIvohi9uAz\nykszoTZ/kB1cnZW4Nvj/AJrs4Dk/KzEY50kNkd0d1BQ+e/Ab5fU8BD5/kPmuzkrEneZdg6Y8q1lq\nBlYiSiDOk5oi1+LULmz2YBQzgc0faBElhoi5lMNfXXA037VWYiDOk5oiS/HE7EEy+FUzoTZ/8DuJ\nNZYYLvJxP4nj/yPxGSEFshIDcZ7UFFmNKGYPqk1ZmQli/qA4eAsssaXo/CHqmR3tkGE4xUoMxHlS\nU2Qng7jy2YNqUldmgpg/qB7uCksM4Qyw6BpUJQ3wsWrxVqzEQJwnNUX24jCNzx5Uk7oyE/j8QXXw\nVlM2a/CuFjyLoo1B8qBikBZsbhGIi6GHIbIUCFU+e1BN6moeAps/BAgfiYz148tqWuLdG4K+6Pjc\nIhjnSQ2R6Xz/X+vfXv+50begwpEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2 a^{2} + \\frac{4 a}{3} c - \\frac{16 a}{3} + \\frac{2 b^{2}}{3} - \\frac{32 b}{15} + \\frac{2 c^{2}}{5} - \\frac{32 c}{15} + \\frac{192}{35}$$"
      ],
      "text/plain": [
       "                         2             2             \n",
       "   2   4⋅a⋅c   16⋅a   2⋅b    32⋅b   2⋅c    32⋅c   192\n",
       "2⋅a  + ───── - ──── + ──── - ──── + ──── - ──── + ───\n",
       "         3      3      3      15     5      15     35"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = symbols('c')\n",
    "f = 1 + x + x**2 + x**3\n",
    "g = a + b*x + c*x**2\n",
    "minimizar = integrate((f - g)**2,(x,-1,1))\n",
    "minimizar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZcAAAAyBAMAAABi7ovXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n5Wees301psq4+Pn/C5uOORBrbVJ+AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( 4 a + \\frac{4 c}{3} - \\frac{16}{3}, \\quad \\frac{4 b}{3} - \\frac{32}{15}, \\quad \\frac{4 a}{3} + \\frac{4 c}{5} - \\frac{32}{15}\\right )$$"
      ],
      "text/plain": [
       "⎛      4⋅c   16  4⋅b   32  4⋅a   4⋅c   32⎞\n",
       "⎜4⋅a + ─── - ──, ─── - ──, ─── + ─── - ──⎟\n",
       "⎝       3    3    3    15   3     5    15⎠"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = diff(minimizar,a)\n",
    "eq2 = diff(minimizar,b)\n",
    "eq3 = diff(minimizar,c)\n",
    "eq1, eq2, eq3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Resolviendo las tres ecuaciones a la vez:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMsAAAAyBAMAAAAASDLRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ a : 1, \\quad b : \\frac{8}{5}, \\quad c : 1\\right \\}$$"
      ],
      "text/plain": [
       "{a: 1, b: 8/5, c: 1}"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solucion = solve([eq1,eq2,eq3],[a,b,c])\n",
    "solucion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a_sol = solucion[a]\n",
    "b_sol = solucion[b]\n",
    "c_sol = solucion[c]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAqBAMAAACkSaOPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$x^{2} + \\frac{8 x}{5} + 1$$"
      ],
      "text/plain": [
       " 2   8⋅x    \n",
       "x  + ─── + 1\n",
       "      5     "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g_ajuste = g.subs([(a,a_sol),(b,b_sol),(c,c_sol)])\n",
    "g_ajuste"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x9d2dc50>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Dibujo ambas funciones. Ver en figura aparte y localizar los puntos de corte\n",
    "from sympy.plotting import plot\n",
    "\n",
    "plot(f,g_ajuste,(x,-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAA/BAMAAABKn6SXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\left [ 0, \\quad - \\frac{\\sqrt{15}}{5}, \\quad \\frac{\\sqrt{15}}{5}\\right ]$$"
      ],
      "text/plain": [
       "⎡   -√15   √15⎤\n",
       "⎢0, ─────, ───⎥\n",
       "⎣     5     5 ⎦"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "puntos_corte = solve(f-g_ajuste,x)\n",
    "puntos_corte"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.774596669241483, 0, 0.774596669241483], dtype=object)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "puntos = []\n",
    "for punto in puntos_corte:\n",
    "    puntos.append(N(punto))\n",
    "\n",
    "puntos = np.array(puntos)\n",
    "puntos.sort()\n",
    "puntos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.77459667,  0.        ,  0.77459667])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NPGauss = 3\n",
    "pgauss = np.polynomial.legendre.leggauss(NPGauss)\n",
    "        \n",
    "pgauss[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}