notebooks_completos/014-NumPy-AlgerbraLineal.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"../images/aeropython_logo.png\" alt=\"AeroPython\" style=\"width: 300px;\" />"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Álgebra Lineal con NumPy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Una vez hemos visto el manejo básico de arrays en Python con NumPy, es hora de pasar a operaciones más interesantes como son las propias del Álgebra Lineal._\n",
    "\n",
    "_Los productos escalares y las inversiones de matrices están por todas partes en los programas científicos e ingenieriles, así que vamos a estudiar cómo se realizan en Python._"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Álgebra lineal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como sabemos, las operaciones del álgebra lineal aparecen con mucha frecuencia a la hora de resolver sistemas de ecuaciones en derivadas parciales y en general al linealizar problemas de todo tipo, y suele ser necesario resolver sistemas con un número enorme de ecuaciones e incógnitas. Gracias a los arrays de NumPy podemos abordar este tipo de cálculos en Python, ya que todas las funciones están escritas en C o Fortran y tenemos la opción de usar bibliotecas optimizadas al límite."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El paquete de álgebra lineal en NumPy se llama `linalg`, así que importando NumPy con la convención habitual podemos acceder a él escribiendo `np.linalg`. Si imprimimos la ayuda del paquete vemos que tenemos funciones para:\n",
    "\n",
    "* Funciones básicas (norma de un vector, inversa de una matriz, determinante, traza)\n",
    "* Resolución de sistemas\n",
    "* Autovalores y autovectores\n",
    "* Descomposiciones matriciales (QR, SVD)\n",
    "* Pseudoinversas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Puede que ya sepas que en la biblioteca `SciPy` se pueden encontrar también funciones de Álgebra Lineal. ¿Cuáles usar? Puedes encontrar la respuesta en este enlace: https://docs.scipy.org/doc/scipy-0.18.1/reference/tutorial/linalg.html#scipy-linalg-vs-numpy-linalg\n",
    "\n",
    "Como de momento sólo hemos usado `NumPy` no importaremos las funciones de `SciPy`, aunque como ves, es recomendable hacerlo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on package numpy.linalg in numpy:\n",
      "\n",
      "NAME\n",
      "    numpy.linalg\n",
      "\n",
      "DESCRIPTION\n",
      "    Core Linear Algebra Tools\n",
      "    -------------------------\n",
      "    Linear algebra basics:\n",
      "    \n",
      "    - norm            Vector or matrix norm\n",
      "    - inv             Inverse of a square matrix\n",
      "    - solve           Solve a linear system of equations\n",
      "    - det             Determinant of a square matrix\n",
      "    - lstsq           Solve linear least-squares problem\n",
      "    - pinv            Pseudo-inverse (Moore-Penrose) calculated using a singular\n",
      "                      value decomposition\n",
      "    - matrix_power    Integer power of a square matrix\n",
      "    \n",
      "    Eigenvalues and decompositions:\n",
      "    \n",
      "    - eig             Eigenvalues and vectors of a square matrix\n",
      "    - eigh            Eigenvalues and eigenvectors of a Hermitian matrix\n",
      "    - eigvals         Eigenvalues of a square matrix\n",
      "    - eigvalsh        Eigenvalues of a Hermitian matrix\n",
      "    - qr              QR decomposition of a matrix\n",
      "    - svd             Singular value decomposition of a matrix\n",
      "    - cholesky        Cholesky decomposition of a matrix\n",
      "    \n",
      "    Tensor operations:\n",
      "    \n",
      "    - tensorsolve     Solve a linear tensor equation\n",
      "    - tensorinv       Calculate an inverse of a tensor\n",
      "    \n",
      "    Exceptions:\n",
      "    \n",
      "    - LinAlgError     Indicates a failed linear algebra operation\n",
      "\n",
      "PACKAGE CONTENTS\n",
      "    _umath_linalg\n",
      "    info\n",
      "    lapack_lite\n",
      "    linalg\n",
      "    setup\n",
      "\n",
      "DATA\n",
      "    absolute_import = _Feature((2, 5, 0, 'alpha', 1), (3, 0, 0, 'alpha', 0...\n",
      "    division = _Feature((2, 2, 0, 'alpha', 2), (3, 0, 0, 'alpha', 0), 8192...\n",
      "    print_function = _Feature((2, 6, 0, 'alpha', 2), (3, 0, 0, 'alpha', 0)...\n",
      "\n",
      "FILE\n",
      "    /miniconda/lib/python3.5/site-packages/numpy/linalg/__init__.py\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(np.linalg)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recordemos que si queremos usar una función de un paquete pero no queremos escribir la \"ruta\" completa cada vez, podemos usar la sintaxis `from package import func`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function numpy.linalg.linalg.norm>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numpy.linalg import norm, det\n",
    "norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "El producto matricial usual (no el que se hace elemento a elemento, sino el del álgebra lineal) se calcula con la misma función que el producto matriz-vector y el producto escalar vector-vector: con la función `dot`, que **no** está en el paquete `linalg` sino directamente en `numpy` y no hace falta importarlo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function numpy.core.multiarray.dot>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.dot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Una consideración importante a tener en cuenta es que en NumPy no hace falta ser estricto a la hora de manejar vectores como si fueran matrices columna, siempre que la operación sea consistente. Un vector es una matriz con una sola dimensión: por eso si calculamos su traspuesta no funciona."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = np.array([\n",
    "    [1, 2],\n",
    "    [3, 4]\n",
    "])\n",
    "v = np.array([1, -1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1, -1])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1, -1])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = np.dot(M, v)\n",
    "u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para hacer comparaciones entre arrays de punto flotante se pueden usar las funciones `np.allclose` y `np.isclose`. La primera comprueba si todos los elementos de los arrays son iguales dentro de una tolerancia, y la segunda compara elemento a elemento y devuelve un array de valores `True` y `False`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([-1, -1]), array([ 1, -1]))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u, v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(u, v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.isclose(0.0, 1e-8, atol=1e-10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__En la versión 3.5 de Python se incorporó un nuevo operador `@` para poder calcular hacer multiplicaciones de matrices de una forma más legible__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1, -1])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = M @ v\n",
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([23, 67, 66, 49])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mat = np.array([[1, 5, 8, 5],\n",
    "                [0, 6, 4, 2],\n",
    "                [9, 3, 1, 6]])\n",
    "\n",
    "vec1 = np.array([5, 6, 2])\n",
    "\n",
    "vec1 @ mat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si quieres saber más, puedes leer [este artículo](http://pybonacci.org/2016/02/22/el-producto-de-matrices-y-el-nuevo-operador/) en Pybonacci escrito por _Álex Sáez_."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### Ejercicios"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1- Hallar el producto de estas dos matrices y su determinante:\n",
    "\n",
    "$$\\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 1 & 1 \\\\ -1 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} 2 & 3 & -1 \\\\ 0 & -2 & 1 \\\\ 0 & 0 & 3 \\end{pmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy.linalg import det"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  0  0]\n",
      " [ 2  1  1]\n",
      " [-1  0  1]]\n",
      "[[ 2  3 -1]\n",
      " [ 0 -2  1]\n",
      " [ 0  0  3]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([\n",
    "    [1, 0, 0],\n",
    "    [2, 1, 1],\n",
    "    [-1, 0, 1]\n",
    "])\n",
    "B = np.array([\n",
    "    [2, 3, -1],\n",
    "    [0, -2, 1],\n",
    "    [0, 0, 3]\n",
    "])\n",
    "print(A)\n",
    "print(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2,  3, -1],\n",
       "       [ 4,  4,  2],\n",
       "       [-2, -3,  4]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = A @ B\n",
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-12.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "det(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2- Resolver el siguiente sistema:\n",
    "\n",
    "$$ \\begin{pmatrix} 2 & 0 & 0 \\\\ -1 & 1 & 0 \\\\ 3 & 2 & -1 \\end{pmatrix} \\begin{pmatrix} 1 & 1 & 1 \\\\ 0 & 1 & 2 \\\\ 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} -1 \\\\ 3 \\\\ 0 \\end{pmatrix} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 2,  2,  2],\n",
       "       [-1,  0,  1],\n",
       "       [ 3,  5,  6]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = (np.array([[2, 0, 0],\n",
    "                        [-1, 1, 0],\n",
    "                        [3, 2, -1]])\n",
    "     @\n",
    "        np.array([[1, 1, 1],\n",
    "                        [0, 1, 2],\n",
    "                        [0, 0, 1]]))\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.5, -4.5,  3.5])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.linalg.solve(M, np.array([-1, 3, 0]))\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(M @ x, np.array([-1, 3, 0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3- Hallar la inversa de la matriz $H$ y comprobar que $H H^{-1} = I$ (recuerda la función `np.eye`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[          2           4           7          12          21          38]\n",
      " [         71         128         265         512        1035        2048]\n",
      " [       4109        8206       16399       32784       65553      131090]\n",
      " [     262145      524308     1048577     2097174     4194305     8388632]\n",
      " [   16777271    33554488    67108921   134217786   268435515   536870972]\n",
      " [ 1073741855  2147483680  4294967329  8589934626 17179869219 34359738404]]\n"
     ]
    }
   ],
   "source": [
    "# aeropython: preserve\n",
    "A = np.arange(1, 37).reshape(6,6)\n",
    "A[1, 1::2] = 0\n",
    "A[3, ::2] = 1\n",
    "A[4, :] += 30\n",
    "B = (2 ** np.arange(36)).reshape((6,6))\n",
    "H = A + B\n",
    "print(H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "456269.58138193813"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.det(H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "Hinv = np.linalg.inv(H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[False, False, False, False, False, False],\n",
       "       [False, False, False, False, False, False],\n",
       "       [False, False,  True,  True, False,  True],\n",
       "       [False, False, False, False, False, False],\n",
       "       [ True, False,  True,  True, False,  True],\n",
       "       [False, False,  True,  True, False,  True]], dtype=bool)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.isclose(np.dot(Hinv, H), np.eye(6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.125  0.25   0.5    1.     1.     4.   ]\n",
      " [-0.094  0.875 -0.25  -0.5   -0.5   -3.   ]\n",
      " [-0.125  0.25   1.     0.     4.     0.   ]\n",
      " [ 0.188  0.25   0.5    2.     1.     6.   ]\n",
      " [ 0.    -0.125  0.     0.     0.     0.   ]\n",
      " [-0.031  0.062  0.     0.     1.     1.   ]]\n"
     ]
    }
   ],
   "source": [
    "np.set_printoptions(precision=3)\n",
    "print(np.dot(Hinv, H))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-warning\">¡No funciona! Y no solo eso sino que los resultados varían de un ordenador a otro.</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4- Comprobar el número de condición de la matriz $H$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0682988788334074e+17"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.cond(H)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-warning\">La matriz está mal condicionada.</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Autovalores y autovectores "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La cosa no queda ahí y también se pueden resolver problemas de autovalores y autovectores:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 1.,  1.,  1.]), array([[  0.000e+00,   0.000e+00,   4.930e-32],\n",
       "        [  1.000e+00,  -1.000e+00,  -1.000e+00],\n",
       "        [  0.000e+00,   2.220e-16,   2.220e-16]]))"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.array([\n",
    "    [1, 0, 0],\n",
    "    [2, 1, 1],\n",
    "    [-1, 0, 1]\n",
    "])\n",
    "\n",
    "np.linalg.eig(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Ya hemos aprendido a efectuar algunas operaciones útiles con NumPy. Estamos en condiciones de empezar a escribir programas más interesantes, pero aún queda lo mejor._\n",
    "\n",
    "* [Álgebra lineal en Python con NumPy en Pybonacci](http://pybonacci.org/2012/06/07/algebra-lineal-en-python-con-numpy-i-operaciones-basicas/)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "<br/>\n",
    "#### <h4 align=\"right\">¡Síguenos en Twitter!\n",
    "<br/>\n",
    "###### <a href=\"https://twitter.com/AeroPython\" class=\"twitter-follow-button\" data-show-count=\"false\">Follow @AeroPython</a> <script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script> \n",
    "<br/>\n",
    "###### Este notebook ha sido realizado por: Juan Luis Cano, Mabel Delgado y Álex Sáez \n",
    "<br/>\n",
    "##### <a rel=\"license\" href=\"http://creativecommons.org/licenses/by/4.0/deed.es\"><img alt=\"Licencia Creative Commons\" style=\"border-width:0\" src=\"http://i.creativecommons.org/l/by/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" property=\"dct:title\">Curso AeroPython</span> por <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\">Juan Luis Cano Rodriguez y Alejandro Sáez Mollejo</span> se distribuye bajo una <a rel=\"license\" href=\"http://creativecommons.org/licenses/by/4.0/deed.es\">Licencia Creative Commons Atribución 4.0 Internacional</a>."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "_Las siguientes celdas contienen configuración del Notebook_\n",
    "\n",
    "_Para visualizar y utlizar los enlaces a Twitter el notebook debe ejecutarse como [seguro](http://ipython.org/ipython-doc/dev/notebook/security.html)_\n",
    "\n",
    "    File > Trusted Notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "/* This template is inspired in the one used by Lorena Barba\n",
       "in the numerical-mooc repository: https://github.com/numerical-mooc/numerical-mooc\n",
       "We thank her work and hope you also enjoy the look of the notobooks with this style */\n",
       "\n",
       "<link href='http://fonts.googleapis.com/css?family=Source+Sans+Pro|Josefin+Sans:400,700,400italic|Ubuntu+Condensed' rel='stylesheet' type='text/css'>\n",
       "\n",
       "El estilo se ha aplicado =)\n",
       "\n",
       "<style>\n",
       "\n",
       "\n",
       "\n",
       "#notebook_panel { /* main background */\n",
       "    background: #f7f7f7;\n",
       "}\n",
       "\n",
       "div.cell { /* set cell width */\n",
       "    width: 900px;\n",
       "}\n",
       "\n",
       "div #notebook { /* centre the content */\n",
       "    background: #fff; /* white background for content */\n",
       "    width: 950px;\n",
       "    margin: auto;\n",
       "    padding-left: 0em;\n",
       "}\n",
       "\n",
       "#notebook li { /* More space between bullet points */\n",
       "    margin-top:0.7em;\n",
       "}\n",
       "\n",
       "/* draw border around running cells */\n",
       "div.cell.border-box-sizing.code_cell.running { \n",
       "    border: 1px solid #111;\n",
       "}\n",
       "\n",
       "/* Put a solid color box around each cell and its output, visually linking them*/\n",
       "div.cell.code_cell {\n",
       "    font-family: 'Source Sans Pro', sans-serif;\n",
       "    background-color: rgb(256,256,256);\n",
       "    font-size: 110%;\n",
       "    border-radius: 0px; \n",
       "    padding: 0.5em;\n",
       "    margin-left:1em;\n",
       "    margin-top: 1em;\n",
       "}\n",
       "\n",
       "div.text_cell_render{\n",
       "    font-family: 'Josefin Sans', serif;\n",
       "    line-height: 145%;\n",
       "    font-size: 125%;\n",
       "    font-weight: 500;\n",
       "    width:750px;\n",
       "    margin-left:auto;\n",
       "    margin-right:auto;\n",
       "}\n",
       "\n",
       "\n",
       "/* Formatting for header cells */\n",
       ".text_cell_render h1, .text_cell_render h2, .text_cell_render h3,\n",
       ".text_cell_render h4, .text_cell_render h5 {\n",
       "    font-family: 'Ubuntu Condensed', sans-serif;\n",
       "}\n",
       "/*\n",
       ".text_cell_render h1 {\n",
       "    font-family: Flux, 'Ubuntu Condensed', serif;\n",
       "    font-style:regular;\n",
       "    font-weight: 400;    \n",
       "    font-size: 30pt;\n",
       "    text-align: center;\n",
       "    line-height: 100%;\n",
       "    color: #335082;\n",
       "    margin-bottom: 0.5em;\n",
       "    margin-top: 0.5em;\n",
       "    display: block;\n",
       "}\n",
       "*/\n",
       ".text_cell_render h1 {\n",
       "    font-weight: 600;\n",
       "    font-size: 35pt;\n",
       "    line-height: 100%;\n",
       "    color: #000000;\n",
       "    margin-bottom: 0.1em;\n",
       "    margin-top: 0.3em;\n",
       "    display: block;\n",
       "}\n",
       "\n",
       ".text_cell_render h2 {\n",
       "    margin-top:16px;\n",
       "    font-size: 27pt;\n",
       "    font-weight: 550;\n",
       "    margin-bottom: 0.1em;\n",
       "    margin-top: 0.3em;\n",
       "    font-style: regular;\n",
       "    color: #2c6391;\n",
       "}\t\n",
       "\n",
       ".text_cell_render h3 {\n",
       "    font-size: 20pt;\n",
       "    font-weight: 550\n",
       "    text-align: left;\n",
       "    margin-bottom: 0.1em;\n",
       "    margin-top: 0.3em;\n",
       "    font-style: regular;\n",
       "    color:  #387eb8;\n",
       "}\n",
       "\n",
       ".text_cell_render h4 {    /*Use this for captions*/\n",
       "    font-size: 18pt;\n",
       "    font-weight: 450\n",
       "    text-align: left;\n",
       "    margin-bottom: 0.1em;\n",
       "    margin-top: 0.3em;\n",
       "    font-style: regular;\n",
       "    color:  #5797cc;\n",
       "}\n",
       "\n",
       ".text_cell_render h5 {  /*Use this for small titles*/\n",
       "    font-size: 18pt;\n",
       "    font-weight: 550;\n",
       "    color: rgb(163,0,0);\n",
       "    font-style: italic;\n",
       "    margin-bottom: .1em;\n",
       "    margin-top: 0.8em;\n",
       "    display: block;\n",
       "    color:  #b21c0d;\n",
       "}\n",
       "\n",
       ".text_cell_render h6 { /*use this for copyright note*/\n",
       "    font-family: 'Ubuntu Condensed', sans-serif;\n",
       "    font-weight: 300;\n",
       "    font-size: 14pt;\n",
       "    line-height: 100%;\n",
       "    color: #252525;\n",
       "    text-align: right;\n",
       "    margin-bottom: 1px;\n",
       "    margin-top: 1px;\n",
       "}\n",
       "\n",
       ".CodeMirror{\n",
       "        font-family: 'Duru Sans', sans-serif;\n",
       "        font-size: 100%;\n",
       "}\n",
       "\n",
       "</style>\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\"],\n",
       "                           equationNumbers: { autoNumber: \"AMS\", useLabelIds: true}\n",
       "                           },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Esta celda da el estilo al notebook\n",
    "from IPython.core.display import HTML\n",
    "css_file = '../styles/aeropython.css'\n",
    "HTML(open(css_file, \"r\").read())"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}