01_MEF_Cuadraturas_Gauss.ipynb

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   "source": [
    "# Integración numérica mediante puntos de Gauss\n",
    "\n",
    "Análisis de Estructuras. Grado en Ingeniería Civil.\n",
    "\n",
    "Autores:\n",
    "\n",
    "- Alejandro E. Martínez-Castro, Esther Puertas García, Rafael Gallego.\n",
    "- Departamento de Mecánica de Estructuras e Ingeniería Hidráulica. Universidad de Granada.\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: Cuadraturas de Gauss-Legendre</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>.\n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "## Cuadraturas de Gauss-Legendre en 1 dimensión\n",
    "\n",
    "En este cuaderno se plantea la integración numérica mediante cuadraturas de [Gauss-Legendre](https://es.wikipedia.org/wiki/Cuadratura_de_Gauss). \n",
    "\n",
    "En una dimensión, una cuadratura de Gauss-Legendre permite realizar la integral mediante un sumatorio\n",
    "\n",
    "\\begin{equation}\n",
    "\\int_{-1}^{1} f(t)\\,dt = \\sum_{n=1}^{N} f(t_i)\\,W_i\n",
    "\\end{equation}\n",
    "\n",
    "Siendo $t_i$ las abscisas de los puntos de Gauss y $W_i$ los \"pesos\" correspondientes a cada punto. \n",
    "\n",
    "La integral anterior es exacta si $f(t)$ es un polinomio, si se elige de forma adecuada el número de puntos de la cuadratura."
   ]
  },
  {
   "cell_type": "markdown",
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     "slide_type": "slide"
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   "source": [
    "Si el intervalo de integración es diferente a $[-1,1]$, se acude a un cambio de variable. \n",
    "\n",
    "Si el intervalo en $x$ es $[x_1,x_2]$, el cambio es: \n",
    "\n",
    "\\begin{equation}\n",
    "x = x_1 + \\dfrac{x_2-x_1}{2}\\,(t + 1)\n",
    "\\end{equation}\n",
    "\n",
    "La integral se transforma como sigue, teniendo en cuenta que: \n",
    "\n",
    "\\begin{equation}\n",
    "d x = \\dfrac{x_2 - x_1}{2} \\,d t\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "\\int_{x_1}^{x_2} f(x)\\, \\mathrm{d} x = \\int_{-1}^{1} f(t) \\,\\dfrac{d x}{dt}\\,dt = \\int_{-1}^{1} f(t) \\,\\dfrac{x_2 - x_1}{2} \\,dt  \n",
    "\\end{equation}\n",
    "\n",
    "\n",
    "Y que esta integral se puede evaluar de forma aproximada como: \n",
    "\n",
    "\\begin{equation}\n",
    "\\sum_{i=1}^{N} f(t_i)\\,W_i \\,\\dfrac{x_2-x_1}{2}\n",
    "\\end{equation}\n",
    "\n",
    "Siendo $t_i$ y $W_i$ las abscisas y los pesos de una cuadratura de Gauss de $N$ puntos.\n",
    "\n",
    "Una propiedad importante de esta integración es que la fórmula es exacta para un polinomio de grado $2N-1$. "
   ]
  },
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   "source": [
    "Si se indexa desde el 0, estas expresiones pueden evaluarse como: \n",
    "\n",
    "\\begin{equation}\n",
    "\\sum_{i=0}^{N-1} f(t_i)\\,W_i \\,\\dfrac{x_2-x_1}{2}\n",
    "\\end{equation}\n",
    "\n",
    "Esta expresión es exacta para un polinomio de grado $2\\,N-1$. \n",
    "\n",
    "Así, \n",
    "\n",
    "- Para $N=1$ la integral es exacta para un polinomio de grado 1. \n",
    "- Para $N=2$ la integral es exacta para un polinomio de grado 3. \n",
    "- Para $N=3$ la integral es exacta para un polinomio de grado 5. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "from __future__ import division\n",
    "%pylab inline "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "En Python los puntos y pesos de una cuadratura de Gauss se obtienen mediante una función de numpy. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.57735027,  0.57735027]), array([ 1.,  1.]))"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NPGauss = 2\n",
    "pgauss = np.polynomial.legendre.leggauss(NPGauss)\n",
    "        \n",
    "pgauss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Observe que se obtiene un objeto de tipo \"tupla\". \n",
    "- En la posición [0] se incluye un vector de numpy, con las abscisas $t_i$.\n",
    "- En la posición [1], un vector de numpy con los pesos $W_i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<type 'tuple'>\n",
      "\n",
      "[-0.57735027  0.57735027]\n",
      "<type 'numpy.ndarray'>\n",
      "\n",
      "[ 1.  1.]\n",
      "<type 'numpy.ndarray'>\n"
     ]
    }
   ],
   "source": [
    "print type(pgauss)  # <- Le preguntamos a Python sobre \"pgauss\". ¿De qué tipo es?\n",
    "print \"\"\n",
    "print pgauss[0]\n",
    "print type(pgauss[0])\n",
    "print\"\"\n",
    "print pgauss[1]\n",
    "print type(pgauss[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "A continuación se van a emplear estos puntos para el cálculo de diversas integrales"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Integral de un polinomio cúbico\n",
    "\n",
    "Consideremos el polinomio: \n",
    "\n",
    "\\begin{equation}\n",
    "f(x) = a + b\\,x + c\\,x^2 + d\\,x^3\n",
    "\\end{equation}\n",
    "\n",
    "Supongamos que deseamos evaluar la integral \n",
    "\n",
    "\\begin{equation}\n",
    "\\int_{x_1}^{x_2} f(x) \\,dx\n",
    "\\end{equation}\n",
    "\n",
    "Planteemos esta integral simbólica exacta con SymPy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.0 (Python 2.7.12-32-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": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Integración analítica. \n",
    "\n",
    "Para integrar de forma exacta un polinomio de grado 3, hace falta una cuadratura de Gauss de orden 2. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
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   "outputs": [
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      "text/latex": [
       "$$- a x_{1} + a x_{2} - \\frac{b x_{1}^{2}}{2} + \\frac{b x_{2}^{2}}{2} - \\frac{c x_{1}^{3}}{3} + \\frac{c x_{2}^{3}}{3} - \\frac{d x_{1}^{4}}{4} + \\frac{d x_{2}^{4}}{4}$$"
      ],
      "text/plain": [
       "                   2       2       3       3       4       4\n",
       "               b⋅x₁    b⋅x₂    c⋅x₁    c⋅x₂    d⋅x₁    d⋅x₂ \n",
       "-a⋅x₁ + a⋅x₂ - ───── + ───── - ───── + ───── - ───── + ─────\n",
       "                 2       2       3       3       4       4  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a,b,c,d,x1,x2 = symbols('a b c d x1 x2')\n",
    "f = a + b*x + c*x**2 + d*x**3\n",
    "Int_analitica = integrate(f,(x,x1,x2))\n",
    "Int_analitica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Consideremos un conjunto de valores"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$841.958333333333$$"
      ],
      "text/plain": [
       "841.958333333333"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 = 1.1\n",
    "b1 = 2.5\n",
    "c1 = 3.2\n",
    "d1 = 4.3\n",
    "xx1 = 0\n",
    "xx2 = 5\n",
    "Int_exacta = Int_analitica.subs([(a,a1),(b,b1),(c,c1),(d,d1),(x1,xx1),(x2,xx2)])\n",
    "Int_exacta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Integración numérica con 2 puntos de Gauss. \n",
    "\n",
    "Determinacion de las abscisas en x. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.05662433,  3.94337567])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi = xx1 + (xx2 - xx1) / 2 * (pgauss[0] + 1)\n",
    "xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$841.958333333$$"
      ],
      "text/plain": [
       "841.958333333"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Int_numerica = 0\n",
    "\n",
    "def pol3(x):\n",
    "    return a1 + b1*x + c1*x**2 + d1*x**3\n",
    "\n",
    "for i in range(NPGauss):\n",
    "    Int_numerica += pol3(xi[i]) * (xx2-xx1)/2 * pgauss[1][i]    \n",
    "\n",
    "Int_numerica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Comparación de resultados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAU0AAAAUBAMAAAD7KHJ8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3O2KugzMqOPIiSjPd/H/jv8A49z3m02ecgMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( 841.958333333333, \\quad 841.958333333\\right )$$"
      ],
      "text/plain": [
       "(841.958333333333, 841.958333333)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Int_exacta,Int_numerica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Pruebe variar los valores de\n",
    "        a1, b1, c1, d1, xx1, xx2\n",
    "        \n",
    "Observe que el resultado siempre se obtiene de forma correcta. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Integración reducida con 1 punto de Gauss\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 0.]), array([ 2.]))"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NPGauss = 1\n",
    "pgauss = np.polynomial.legendre.leggauss(NPGauss)\n",
    "pgauss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAPBAMAAABTt1/kAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$472.6875$$"
      ],
      "text/plain": [
       "472.6875"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi = xx1 + (xx2 - xx1) / 2 * (pgauss[0] + 1)\n",
    "xi\n",
    "\n",
    "Int_reducida = 0\n",
    "\n",
    "for i in range(NPGauss):\n",
    "    Int_reducida += pol3(xi[i]) * (xx2-xx1)/2 * pgauss[1][i]    \n",
    "\n",
    "Int_reducida"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Comparando los valores, se observar que la integración reducida produce un resultado inferior al que debería obtenerse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAbkAAAAUBAMAAADij16fAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\left ( 841.958333333333, \\quad 841.958333333, \\quad 472.6875\\right )$$"
      ],
      "text/plain": [
       "(841.958333333333, 841.958333333, 472.6875)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Int_exacta,Int_numerica,Int_reducida"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Integración de un polinomio de grado 5\n",
    "\n",
    "La integración exacta requiere 3 puntos de Gauss\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAskAAAAwBAMAAAARGfomAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZiXbvRGa7\nVKtFbb1tAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAJzUlEQVR4Ae2bf4wcZRnHv3O3v2739nYCse0Z\n01sPgg2ovWDQahRWE/QftZsYS4DELtjeVantoni9RNNuDV4Pm8ABqYpQ2QSj1mi6SiA1R8KCSRMS\n4i0GCH+IPRTlGgLcVaFNyXV9dn6+8877PrO3cyaNYZJ23vf9PN/3eebZd2Znnp0D1nKbPGky0/F0\nag+n5SkwyPgl1GRw4tWbGwz+8vVzDAXSU5wa2HeYla8epstDRb2Kp6lGqqnX8pR0p7iPCPll/dTI\nti8wFMfbCxzGlclZjvfVb+RwD6y/nHhbL+Npqppb0mt5ChgH2CyvZ7P8PKv93Wf1YRHJLBg1zuBu\nbOBwD6y/njyvl/EUSDX12iiaGWEzdZDNcp3zi9+wFAMtnh/lcdf0A/f7puFTk6e+EhguiT25raRf\n/LFj9iVVlhPPVGyceI7N8q5p7iP65cwtcizUXzddtkcLLynVHl9RyhUzRg191Dfor/ltp8VT3zyz\nyW+HWxq64lhWVVlGwVlnaYPLcrKea4YdeiM34Qmv7TdeO+XMvb2SHfOHvZbLk+/gEW8wVsM9VJpk\nMjwRTwX7VFPohJpKmnWylysrs3zctKd5jM0ywH2b0Axby6FojPPvd8ZGWlBcJT1urKBQD8l7GHAP\nlaS5SkjPU9E8KXwe4rjdVtK0s4w2QJnl79hSo8Zmecg0FHnyIngFI3Wv4zbyS26Lcviu2/b3Pj+P\nQsMf773lHirN8Cv8Qp6Hp771cCmpiNblGtpXsQ1eHN97m2sq7K+w2/nx8XeYIx0oJ7kLyhFsN4U5\n7aZ/VH0N1Vr2+VVrtJb7Fqdb2HD3TYdqiY+8/JQcEE+RnrnT1vaXM/fK2gi6bWqr4xiKTCQn7lyB\nMfPPX08BijQaU/T9ZUWdNtOVkOf1Mzsd8e2Ky2Du9LUksdTZZr4YUlvc9v1b/CWEexkY+aDxtlHN\nbekvDrTb3pnkzsTT5GfwtK1NTO0vuxp3z9PsEcybthi5+Tdckbs/Vh5cxno8WrsaeOJCyx1294+U\nk1VHvG/CHfT22WtwgyNOT1a9Ya9RqNE9unXMeHzS9Ia9RofbvnMTqlsUz7DrxikTZ2k9LOdDWepM\nwdOBKt7Qa3k63MCt0IqNB5Eq4nYcMF9QHUnyKNaVtGLc2DJ26MV06pTo0Vp7zDbX+lbFEzl2K4yz\nBvqaakOe7qV1oNdG0k/oxall9Fdh4sPqsIYenPm9XozTJw6bejGtHZpVH7fNtb7VEUWMXoEs3R10\nzhEkiiFbgSqqRQ9Z9jqtQJVa695AIx5o2vcG58hDenforC0ssJ6dyoZGDNh3LxrXNLPNtXLL9+r+\nexe0brC1REneMxaS+lRRLTLOWvYarUDVWut+SSMeqVoLLvEf8vBd/F2Oq1DlPMOOSycG/syqHa6X\ny9F00T+P4UbevAcZE/lwln2qqhbRzVter/WpQkv3+4PFF7Rielp4Ol9+LrOEg7gm/GBBl3zUtGL8\nhMpA0IqBn1Je9GqbM/IusiqbvI6TON6ax83kOJxln6qqRXtg7NBrfarSbsKjC1WtuK+WuzadXRlc\nSjbwKWyvS1FnilhX1orxAxg79WJYlyq92uKMXIqlq+666QY2zmy4raHMskAVZd70904wWoEqtBsP\nXb9Y1zo2Fu/42o+MxYPTc52DoDsdaZs+zEWd2TUHRmw90GpdAx3OyKVQVttVrGVxCkUtycdxtKqT\nyJ+ZWlsCPakT4VkhTjf9KVRqkSvkvrinlsqjMJGiluTTONqoLCse7nzHqxf/jR6rvU0Rd4Dzvr1p\nVtFQeBTUilqSQONooxL1uOAn3OQ9Iyz+9KXCJAp1gIflgthqviQPRPR9j/la2FRRSxKM4miFLKtC\nzlazJcGT3OQ9K8QndgpTKNQiV8gFsdVUhSzbiP38rNtTZFlVS3LNaR9HK4hVIX/u5b+Gvv269YwI\nseBaccyIlFMcqpCF8ORm8usXas6YwqOqluTPEEcLQawK+eF223cUagliRdTgxaJrlVqWb5u8y7SL\neX4cbsjJiV33ulVEn3ItpUdOILA42tUuDMEtNWN57kZNdcTNbjHPK0e4WT5W7ltyC4HhmkAwUrsX\nJ9442os8y1RH3O0U8/xyhJPlThWx6RQCFTWB97IcubIu+VhnuxxUZdziFvPsr81vj45+cnS08xN0\np4pYcwqBdk3AuKyjurpEsNAObjSUHh297L7R0QU1/R9pl8ibHzKCQdkXZvdYyVLCHXHPUVs/XTBq\n4aeNh2B0yldWMc+7OXHWcqeKWCJ4jv4pagI0GtrinPVxtBf3FYOqjF4d0b/7dLIsVhFVNYFQjuN9\nj/wfZ1lZR3SzLFQRI2oCbsLjZCqO9uJey1DWEZ0si1VEdPdcHidTcbQXeZaVdUQny4EqYvRzeWc9\nx8lUHO1FnmX3VLdSNOb0nCyLTPNcnhx/tiSaBTMlU9GS3pbY9/2WOCJpJSpadtqJSmBEDlnCAVts\nMj/PeO47mPhW0D7YS0/dERgIxk3vhx8qBbjU8UoK75MAdTWP9RuRDrwGZZRFqUxFBmxDnn578reg\nVqa+nd3qbwZG5JAlHLClx1/7Bz53NOiZSgJVlyj2xpX0poK4BdW4pJ4qilhqCw/2EqHuw+qawCvA\nA2Fjd4Sn97Twb9cyvOcpcFczrBFGWPyhxcBiEGSdZt/+H0ojge5gBc8GBqTOSQw2paG43fuBt0ra\nSXh6rG6c0UrBUyT/0NRr6WrE4gVOiiGW4jj3CdGvUsyfJfAT6+lmk8syT+m7MnDFkL2wNK173cae\nhccLsqtAPyLLzwSMQ53B2dDQWgxcZ3KzsPTSKidl6WN8lnn8+okdjOOhuekSg898YbrO4MLlU99k\ncI8oG/wikWZh6Q0f5z4glho1NssR+KnwKwRC3IMm92cyxpkGjgrWcnPkqsi/OZElXfT7FzgjnqZ/\nxmk5mta+oGfNGIHp7wTHOM/4k54abRPfYFbHyDJS4XeF9fN1R8ZZM57iASZcunvR0+f5LEdgKjfy\n31F7y/qjovuiJ+t6XBhDLnBzqzftnmQqnC1LvwrMt7RqlhpVNssReKgo3eUHg6CX6w7Ug0Nij96L\nfrIhDgTb/UXk2C/1oHl3vReRZD53ltKpx2SZpfk333zr5zVthBF4oGi9SKmT34dO1V27nebX8tDs\n2q9lepDN6LPMU3qB9jq9lqfAQFObhw7gcAqgFafdKgCtV+22la7LWkhP/nRdnmV4L+gr+yb/odfx\ndA75c3otT+l3h6ZeS4TDyQr+qP946ck/d4SZe6hqcPcYeA3DLUbeC9rcbjNPyTzNju+u633yFOn5\nCw29OAJv238Lo01O/MtkMCb3lDicHn+Vw++xtc7AfwG+z8GSVjDeJQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- a x_{1} + a x_{2} - \\frac{b x_{1}^{2}}{2} + \\frac{b x_{2}^{2}}{2} - \\frac{c x_{1}^{3}}{3} + \\frac{c x_{2}^{3}}{3} - \\frac{d x_{1}^{4}}{4} + \\frac{d x_{2}^{4}}{4} - \\frac{e x_{1}^{5}}{5} + \\frac{e x_{2}^{5}}{5} - \\frac{f x_{1}^{6}}{6} + \\frac{f x_{2}^{6}}{6}$$"
      ],
      "text/plain": [
       "                   2       2       3       3       4       4       5       5  \n",
       "               b⋅x₁    b⋅x₂    c⋅x₁    c⋅x₂    d⋅x₁    d⋅x₂    e⋅x₁    e⋅x₂   \n",
       "-a⋅x₁ + a⋅x₂ - ───── + ───── - ───── + ───── - ───── + ───── - ───── + ───── -\n",
       "                 2       2       3       3       4       4       5       5    \n",
       "\n",
       "     6       6\n",
       " f⋅x₁    f⋅x₂ \n",
       " ───── + ─────\n",
       "   6       6  "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e,f = symbols('e,f')\n",
    "g = a + b*x + c*x**2 + d*x**3 + e*x**4 + f*x**5\n",
    "\n",
    "Int_analitica = integrate(g,(x,x1,x2))\n",
    "Int_analitica"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK8AAAAPBAMAAABgoIKoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$-16007.3333333333$$"
      ],
      "text/plain": [
       "-16007.3333333333"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1 = 3.2\n",
    "b1 = 4.2\n",
    "c1 = 3.2\n",
    "d1 = 4.3\n",
    "e1 = -2\n",
    "f1 = -6\n",
    "\n",
    "xx1 = 1\n",
    "xx2 = 5\n",
    "Int_exacta = Int_analitica.subs([(a,a1),(b,b1),(c,c1),(d,d1),(e,e1),(f,f1),(x1,xx1),(x2,xx2)])\n",
    "Int_exacta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Integración numérica con 3 puntos de Gauss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.77459667,  0.        ,  0.77459667]),\n",
       " array([ 0.55555556,  0.88888889,  0.55555556]))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NPGauss = 3\n",
    "pgauss = np.polynomial.legendre.leggauss(NPGauss)\n",
    "pgauss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.45080666,  3.        ,  4.54919334])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi = xx1 + (xx2 - xx1) / 2 * (pgauss[0] + 1)\n",
    "xi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAPBAMAAAAWmjAjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$-16007.3333333$$"
      ],
      "text/plain": [
       "-16007.3333333"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def pol5(x):\n",
    "    return a1 + b1*x + c1*x**2 + d1*x**3 + e1*x**4 + f1*x**5\n",
    "\n",
    "Int_numerica = 0\n",
    "\n",
    "for i in range(NPGauss):\n",
    "    Int_numerica += pol5(xi[i]) * (xx2-xx1)/2 * pgauss[1][i]    \n",
    "\n",
    "Int_numerica"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAW0AAAAUBAMAAAC0dXGsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left ( -16007.3333333333, \\quad -16007.3333333\\right )$$"
      ],
      "text/plain": [
       "(-16007.3333333333, -16007.3333333)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Int_exacta,Int_numerica"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Integración  reducida con 2 puntos de Gauss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.57735027,  0.57735027]), array([ 1.,  1.]))"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NPGauss = 2\n",
    "pgauss = np.polynomial.legendre.leggauss(NPGauss)\n",
    "pgauss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAPBAMAAAAWmjAjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$-15483.9555556$$"
      ],
      "text/plain": [
       "-15483.9555556"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi = xx1 + (xx2 - xx1) / 2 * (pgauss[0] + 1)\n",
    "\n",
    "Int_reducida = 0\n",
    "\n",
    "for i in range(NPGauss):\n",
    "    Int_reducida += pol5(xi[i]) * (xx2-xx1)/2 * pgauss[1][i]    \n",
    "\n",
    "Int_reducida"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAh4AAAAUBAMAAAAw6Sk2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRiTXItf/PdloFv8CCLbtg22ET+AAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( -16007.3333333333, \\quad -16007.3333333, \\quad -15483.9555556\\right )$$"
      ],
      "text/plain": [
       "(-16007.3333333333, -16007.3333333, -15483.9555556)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Int_exacta,Int_numerica,Int_reducida"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Integración con un exceso de puntos de Gauss\n",
    "\n",
    "Es interesante observar que si se toma una cuadratura con más puntos de Gauss de los estrictamente necesarios, el resultado es idéntico, pero con un mayor coste computacional. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-0.86113631, -0.33998104,  0.33998104,  0.86113631]),\n",
       " array([ 0.34785485,  0.65214515,  0.65214515,  0.34785485]))"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "NPGauss = 4\n",
    "pgauss = np.polynomial.legendre.leggauss(NPGauss)\n",
    "pgauss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAPBAMAAAAWmjAjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$-16007.3333333$$"
      ],
      "text/plain": [
       "-16007.3333333"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xi = xx1 + (xx2 - xx1) / 2 * (pgauss[0] + 1)\n",
    "\n",
    "Int_num_4pga = 0\n",
    "\n",
    "for i in range(NPGauss):\n",
    "    Int_num_4pga += pol5(xi[i]) * (xx2-xx1)/2 * pgauss[1][i]    \n",
    "\n",
    "Int_num_4pga"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAs4AAAAUBAMAAABv+KZ/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQ\nMqvFy5UvAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGXUlEQVRYCe1YUYhUVRj+ZmZndnbmzuxUGIQP\nXiQQ6cEtKSqipoQKU5vIhXJNNzIJCttA6cHK6SXoxV2xBxPDeckIAgfNJIbcG2yKCO282EvQbkuL\nSKStbrnq0PT//7kz59w7Zxwfdt48sN+cc/7/fP93v7lzzr0LRFzcad11IJoj/nvCNSq7q/jp2A8Q\nyOwYqlQ4I1I5XjQASyuPwgAdNibDzIFx+uQPcIYeqIbqaF4rpYQDPOHBc975Is2dB87s3uRLpLEz\ndCqHng8yryiApGkAy7l1C/iCX2YrlTItaK/SiPxOiU8b7J96wEQpnsfzOJ0TiNbr9f84YwmSVwyI\nFDDqarBGDOJgN3mIrnstNqKn6LyOQJ0OlBIOkpkj1n+4fpOn3gAKyBabC866kX3oq9fLClSa5AqI\nHJMq0G/1BavImGlK0r5o6aqojmACiBaajM7sZQ94GLGB6BjSeYE43fJMhz1F/GtA7wL6hzVYI7zM\n1n7eUgNiBazGSuBbBOp0oJSwjZPnlP4XLpBk9O5H0kUk31xwEZhFz6n3oACSpkHktCG2+YKXgCzn\na1+0dFVUR5AeQYJVNdoeD6k5GsSmkawJpIF0lcNfl5x5A1IHMZnXYI00WFs+e8nno1z3SWDUC9Tp\nQCnhFr7mBOlXdwXuXoP0AJzh5oJ5YFw5I/aotGleKSByeGRtrb6gDMhGo33R0lVRHYEzhntNYuKL\njdFE/zRSCwI0+MrPoH0DvHn4QPsGdTVYI5RhaezzYzxPO9LRYrhOB0oJW0h5SvtcXoPUVS/t0qQs\ncK5StJrlJIGwzyKHo9Zm9SU1onKtFy5FdWQtdpnExNe//uQmTBaQuiZA0QE/4zR9hdDgvE1DDdaI\nvzL8wT7P/7qr5PxDPo+E6nSglHCYsDFmn2dWvEa7YW4NcPnKMQr4Cz6j+7mUXb6rCgGVJrkCLKfB\n0vpp9SXm592OJSfwkMlKfJPPoK+4rIzMTQHaNlyVsfRNj07vJkS+20yHbROsEZPa7JPPzryLAxE6\nsabKwTodKCVscgX67PMTWFbCGZDP6fo6XyIljQMvujEvcgMCKk1yGUROgCowsPkC/hKp3Y4luB9v\nqWyFzFdD/NCyAvvMAEw2EtIHqacB35dorMEaaawNfrLPdQ+v/kj3M/kcqtOBUsJBvuaIfabTZZh2\nT/J575HrJRqqBT0jyfEijR6hPwWxYer5IHI8GtubzRfnbz/XeuFSVEc+AZnnPLud2raq7G/9w4je\nMPeNnc3Sn3vU1dB3iIYarJHmWu5McJnt6+lhoAZ6eMGqb8x9o1mnA6WEA7wB/RSJzyWr5HN0Gnua\nEoEVH47TFWJrrgFxPvEbwHJKNDZbgLfFl958I/c2LJlin3Wj7y2RR/QKHU1JPgeTC/Igygl3AU8V\nNURyiC1oMMLGpCYO9tjnbXRhLp2Do3wO6jrG6vbFgmzGiPRn80jfOAfyuS+HjJLoZ1z2/uBNWkDS\nNCg5BlOwa/MlMSA57VXqCP1qW/aN7Bjdz7EBRGsCcOQtBah75LOG/hr5rMEIG5NBsXrEPtPz7KrS\nagIvUMdY3b6YZgr1yI++POK1vZcuXb/AG96M8PlZG0CH4VZPQNI0KDkhNj0k3rAvmJyWeHuVOgIc\naTkHM7Q/j9GrQ5zfU+J5ZOi3zW0/8G5OQ98AEnMajLAxqVa2Ivs8Rfszv6ds5PcUXcdY3b5YK6M/\nQ37EwT9J1kv3Mx4UPo6uzGXmUODfkYCkaVByOM/aiDfsC6aIh1p7lToCOgdPmMTEhz9xtkh3+dmq\ngiT7PJnHciSvGRB1MTqiwRqhFxt7Y5+zZecAEkXnC+g6tKADpYRZjr2RfnrnXUkG4ypS6xBVEnnB\nTm/CpZfk6D4FkqZBybkVb9gXjJLPAV+0dKVSOwYM0R2t25bDG1ykB/8Czr3fgMgBiifovWXw5ZIB\n+HLwcRigw8bkdk9zG734xfkZ4MSWKpyTH1X9YlJnR0dKKZt4x2AzuqJ/yanNNHW8PoPfhjb5fKQ/\nPUiDyNAsfRMMkDQNIueWvCFfcF8x5Itx4aLSsATrg+/dhubF6Cb5xupG+7gbpMTZLd4IPcTR7d+t\nlu4WcblLxN3iTY/QUdElzUR7pkvUDsnuRusWL/9ftPX//It3Be7iUQWYooHR4g26xSv3csRdPKF3\nmKwORHP4H5EA0wr1FPeIAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( -16007.3333333333, \\quad -16007.3333333, \\quad -15483.9555556, \\quad -16007.3333333\\right )$$"
      ],
      "text/plain": [
       "(-16007.3333333333, -16007.3333333, -15483.9555556, -16007.3333333)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Int_exacta,Int_numerica,Int_reducida,Int_num_4pga"
   ]
  }
 ],
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  "anaconda-cloud": {},
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