{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 向量化函数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "自定义的 `sinc` 函数："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "def sinc(x):\n",
    "    if x == 0.0:\n",
    "        return 1.0\n",
    "    else:\n",
    "        w = np.pi * x\n",
    "        return np.sin(w) / w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "作用于单个数值："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sinc(0.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.8981718325193755e-17"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sinc(3.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "但这个函数不能作用于数组："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-4-9d4f36f2aa7a>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[0mx\u001b[0m \u001b[1;33m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[1;33m.\u001b[0m\u001b[0marray\u001b[0m\u001b[1;33m(\u001b[0m\u001b[1;33m[\u001b[0m\u001b[1;36m1\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m2\u001b[0m\u001b[1;33m,\u001b[0m\u001b[1;36m3\u001b[0m\u001b[1;33m]\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 2\u001b[1;33m \u001b[0msinc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;32m<ipython-input-1-dffe464e3332>\u001b[0m in \u001b[0;36msinc\u001b[1;34m(x)\u001b[0m\n\u001b[0;32m      2\u001b[0m \u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      3\u001b[0m \u001b[1;32mdef\u001b[0m \u001b[0msinc\u001b[0m\u001b[1;33m(\u001b[0m\u001b[0mx\u001b[0m\u001b[1;33m)\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[1;32m----> 4\u001b[1;33m     \u001b[1;32mif\u001b[0m \u001b[0mx\u001b[0m \u001b[1;33m==\u001b[0m \u001b[1;36m0.0\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m\u001b[0;32m      5\u001b[0m         \u001b[1;32mreturn\u001b[0m \u001b[1;36m1.0\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0;32m      6\u001b[0m     \u001b[1;32melse\u001b[0m\u001b[1;33m:\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n",
      "\u001b[1;31mValueError\u001b[0m: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()"
     ]
    }
   ],
   "source": [
    "x = np.array([1,2,3])\n",
    "sinc(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "可以使用 `numpy` 的 `vectorize` 将函数 `sinc` 向量化，产生一个新的函数："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  3.89817183e-17,  -3.89817183e-17,   3.89817183e-17])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vsinc = np.vectorize(sinc)\n",
    "vsinc(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "其作用是为 `x` 中的每一个值调用 `sinc` 函数："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0xa24e4e0>]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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x8H7o3z/8rf/9758//sEH4VvsfvuF98PEidn6RptlSvQZYRaWXr70UlgaNmxY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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa0b3ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "x = np.linspace(-5,5,101)\n",
    "plt.plot(x, vsinc(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "因为这样的用法涉及大量的函数调用，因此，向量化函数的效率并不高。"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}