04a_scipy.ipynb

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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Parte 4 - SciPy"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "\u00bfQu\u00e9 es SciPy?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* Conjunto de paquetes para computaci\u00f3n cient\u00edfica general\n",
      "* Integraci\u00f3n, optimizaci\u00f3n, interpolaci\u00f3n, procesamiento de se\u00f1ales digitales, estad\u00edsticas...\n",
      "* Normalmente interfaces a programas muy utilizados escritos en Fortran, o C++"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Integraci\u00f3n num\u00e9rica"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para integraci\u00f3n num\u00e9rica e integraci\u00f3n de ecuaciones diferenciales, SciPy proporciona el paquete `integrate`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy import integrate"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Por ejemplo, supongamos que queremos integrar esta funci\u00f3n:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$f(x) = e^{-x^2}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Debemos definir una funci\u00f3n en Python cuyo argumento sea la variable de integraci\u00f3n:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(x):\n",
      "    return np.exp(-x ** 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y ahora simplemente utilizamos la funci\u00f3n `quad`, que recibe como argumentos la funci\u00f3n y los l\u00edmites de integraci\u00f3n."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate.quad(f, 0, 5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "(0.8862269254513955, 2.3183115159980698e-14)"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Se devuelve el resultado de la integraci\u00f3n y una estimaci\u00f3n del error cometido."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Puede darse el caso en que nuestra funci\u00f3n dependa de un cierto n\u00famero de par\u00e1metros:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$f(x) = A e^{-B x^2}$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "En este caso, debemos seguir respetando que el primer argumento es la variable de integraci\u00f3n, pero a partir de ah\u00ed podemos a\u00f1adir todos los argumentos que queramos:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(x, A, B):\n",
      "    return A * np.exp(-B * x ** 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A la hora de integrar esta funci\u00f3n, tendremos que usar el par\u00e1metro `args` para dar valores  los argumentos extra de la funci\u00f3n:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate.quad(f, 0, 5, args=(1.0, 1.0))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "(0.8862269254513955, 2.3183115159980698e-14)"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Ecuaciones diferenciales ordinarias (EDOs)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Dentro del paquete `integrate` tenemos tambi\u00e9n funciones para resolver ecuaciones diferenciales ordinarias (EDOs), como es el caso de la funci\u00f3n `odeint`. Esta funci\u00f3n puede resolver cualquier sistema del tipo:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": []
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Por ejemplo, supongamos que queremos resolver la ecuaci\u00f3n diferencial:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$y' + y = 0$$\n",
      "\n",
      "$$y' = f(y)$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Hemos de definir la funci\u00f3n del sistema, las condiciones iniciales y el vector de tiempos donde realizaremos la integraci\u00f3n."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(y, t):\n",
      "    return -y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y0 = 1.0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t = np.linspace(0, 3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sol = integrate.odeint(f, y0, t)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(t, sol)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 14,
       "text": [
        "[<matplotlib.lines.Line2D at 0x7ff8fee2dcd0>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
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jM/kWsj+3hIdf+ZpXJw2mT6cQrcsRQuiMzORtXMd2frz08CCeWvAfso+UNfwDQghhAdLk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\nmrwQQihMmrwQQihMmrwQQijs/wG93x6I6PxyNgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7ff8ff77a4d0>"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y se obtiene la soluci\u00f3n esperada: un decaimiento exponencial."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para resolver ecuaciones de orden superior, habr\u00e1 que realizar una reducci\u00f3n de orden:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$y'' + y = 0$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\\mathbf{f}(\\mathbf{y}) = \\pmatrix{y \\\\ y'}' = \\pmatrix{y' \\\\ y''} = \\pmatrix{y' \\\\ -y}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def f(y, t):\n",
      "    return np.array([y[1], -y[0]])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t = np.linspace(0, 10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y0 = np.array([1.0, 0.0])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sol = integrate.odeint(f, y0, t)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(t, sol[:, 0], label='$y$')\n",
      "plt.plot(t, sol[:, 1], '--k', label='$\\dot{y}$')\n",
      "plt.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 20,
       "text": [
        "<matplotlib.legend.Legend at 0x7ff8fed267d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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dU7lXWYfishp49+JuzZm1a9eib9++pGUwxuNy6ZfVUC6dqxw/fhy+vr4wNWV/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\narVEvgf6AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x7ff8fedb7750>"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Ecuaciones algebraicas no lineales"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "El paquete `optimize` contiene unas cuantas funciones para optimizaci\u00f3n con y sin restricciones, minimizaci\u00f3n de funciones, m\u00faltiples algoritmos... Ahora nos vamos a concentrar en la b\u00fasqueda de ceros de funciones, utilizando la funci\u00f3n `root`."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy import optimize"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Por ejemplo, esta ser\u00eda la funci\u00f3n necesaria para resolver la ecuaci\u00f3n de Kepler:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$M = E - e \\sin{E}$$\n",
      "\n",
      "$$F(E) = E - e \\sin{E} - M$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def F(E, e, M):\n",
      "    return E - e * np.sin(E) - M"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "La funci\u00f3n `root` recibe como argumento la funci\u00f3n de la ecuaci\u00f3n, la condici\u00f3n inicial y posibles argumentos extra de la funci\u00f3n."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sol = optimize.root(F, 0.1, args=(0.016, 0.1))\n",
      "sol.x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 49,
       "text": [
        "array([ 0.10162317])"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Ajuste e interpolaci\u00f3n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "El paquete `optimize` trae tambi\u00e9n funciones para hacer ajustes. Para ver este tema explicado con detalle puedes leer el art\u00edculo [Ajuste e interpolaci\u00f3n unidimensionales b\u00e1sicos con SciPy](http://pybonacci.wordpress.com/2013/08/15/ajuste-e-interpolacion-unidimensionales-basicos-en-python-con-scipy/) y el notebook incluido en los materiales del curso."
     ]
    }
   ],
   "metadata": {}
  }
 ]
}