{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 符号积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "积分与求导的关系：\n",
    "\n",
    "$$\\frac{d}{dx} F(x) = f(x)\n",
    "\\Rightarrow F(x) = \\int f(x) dx$$\n",
    "\n",
    "符号运算可以用 `sympy` 模块完成。\n",
    "\n",
    "先导入 `init_printing` 模块方便其显示："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import init_printing\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import symbols, integrate\n",
    "import sympy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "产生 x 和 y 两个符号变量，并进行运算："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFYAAAAlCAYAAADY4B6YAAAABHNCSVQICAgIfAhkiAAAA4FJREFU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\ngg==\n",
      "text/latex": [
       "$$\\sqrt{x^{2} + y^{2}}$$"
      ],
      "text/plain": [
       "   _________\n",
       "  ╱  2    2 \n",
       "╲╱  x  + y  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, y = symbols('x y')\n",
    "sympy.sqrt(x ** 2 + y ** 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对于生成的符号变量 `z`，我们将其中的 `x` 利用 `subs` 方法替换为 `3`："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAlCAYAAADlcn/+AAAABHNCSVQICAgIfAhkiAAAA49JREFU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\n3TKcAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{y^{2} + 9}$$"
      ],
      "text/plain": [
       "   ________\n",
       "  ╱  2     \n",
       "╲╱  y  + 9 "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z = sympy.sqrt(x ** 2 + y ** 2)\n",
    "z.subs(x, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "再替换 `y`："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAASCAYAAACNdSR1AAAABHNCSVQICAgIfAhkiAAAAMxJREFU\nKJHN0TFLQmEUBuDHFIKGFhVcmvwFDW35T/wHQXOLo3/AsSVozTlwCwcbGhoaxHLJTVAIAo0cbLjf\nhevHTW5bL5zh/d5zXt7vHP6AcsQn+MAyaOe4xhMW8fA2qm9cpGIlap5hgDrecIPxb7Ee9mU+2CfG\nKEX8EfeoYoMmrvCaNzzFSYa3MUejSKwyVugVjfoeamd1Q8kfWjnutdjhE885zl+Sy+7gTrKFLE4l\nl+zEzWfo4yjwEm4xwmH6kEULl1jjGC/ohij/BT8UwSaNbxctpgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z.subs(x, 3).subs(y, 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "还可以从 `sympy.abc` 中导入现成的符号变量："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAAXCAYAAABefIz9AAAABHNCSVQICAgIfAhkiAAAAwFJREFU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\nsxfhbTwvakRb4T8H94xyrglMYAKnCf8B3eKPw+iETXAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin^{2}{\\left (\\theta \\right )}$$"
      ],
      "text/plain": [
       "   2   \n",
       "sin (θ)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import theta\n",
    "y = sympy.sin(theta) ** 2\n",
    "y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "对 y 进行积分："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAAZCAYAAAArBywYAAAABHNCSVQICAgIfAhkiAAABZBJREFU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\n9yxvAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{\\theta}{2} - \\frac{1}{2} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\theta \\right )}$$"
      ],
      "text/plain": [
       "θ   sin(θ)⋅cos(θ)\n",
       "─ - ─────────────\n",
       "2         2      "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y = integrate(y)\n",
    "Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "计算 $Y(\\pi) - Y(0)$："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAASCAYAAABfCexoAAAABHNCSVQICAgIfAhkiAAABchJREFU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\nlqIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.5707963267949$$"
      ],
      "text/plain": [
       "1.57079632679490"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "np.set_printoptions(precision=3)\n",
    "\n",
    "Y.subs(theta, np.pi) - Y.subs(theta, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "计算 $\\int_0^\\pi y d\\theta$ ："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAAZCAYAAAC/zUevAAAABHNCSVQICAgIfAhkiAAAAc5JREFU\nSInt1k2IjlEUB/DfMOWzMQtponzUu5gtZVZYCCVlYTaMmrKwsdDEUlamLJSyYEXKR5RCDRELUT6i\nkFnZDQtJioWJmfKxOM+bd67rfXg9Q03+9dS5/+fcc//3nOee5zJFseVfC+jG5d+dNK1iEX04l3A9\nuIhrGMYJLKp43Ql4jFkN4xW4gc5iPBd38AZLJ0NAD04n3FXUEm45vuL8ZIg4go0J9wEvsSDh3+Ft\n1QKm4xnaE34YY1iW8K8xmgvUIVL0UaQr93zB6szc9Tia4eegK+EWFrFu1Ym68jacwhOcFGf9EUYw\ngGMYxyfcyyzWh+MZftSPO95dbGZf6txvYj0viBTDUCZ4I2biabGRMtTEdzJY5tiJS4U9Gw9K/Htx\n8BcEzMBDHE5f5JrVdtwv7G58Lgm+DWdLfNpEma9jT4kvIrUrC3sTnjfxnSd2V4ZB7E+4/rqRZmIN\nlojOR5yYxaLuOfSKltwMO8SHeCDhV9WN9FzvxU3fS/CqELAOVzILbMXOJgLW4pAow5kGvl0c0yxG\nsKFh3IEX2Jzx7cLtJgKIzviznpNmpiUMYFcVgf4EdzG/ikCt3idqeK+in1CrInKXl7+OIXFB+Y+p\nh284OVxxrb4HkAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\pi}{2}$$"
      ],
      "text/plain": [
       "π\n",
       "─\n",
       "2"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(y, (theta, 0, sympy.pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "显示的是字符表达式，查看具体数值可以使用 `evalf()` 方法，或者传入 `numpy.pi`，而不是 `sympy.pi` ："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAASCAYAAABfCexoAAAABHNCSVQICAgIfAhkiAAABchJREFU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\nlqIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.5707963267949$$"
      ],
      "text/plain": [
       "1.57079632679490"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(y, (theta, 0, sympy.pi)).evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAASCAYAAABfCexoAAAABHNCSVQICAgIfAhkiAAABchJREFU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\nlqIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1.5707963267949$$"
      ],
      "text/plain": [
       "1.57079632679490"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(y, (theta, 0, np.pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "根据牛顿莱布尼兹公式，这两个数值应该相等。\n",
    "\n",
    "产生不定积分对象："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAxCAYAAAC8snXfAAAABHNCSVQICAgIfAhkiAAABXZJREFU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\n",
      "text/latex": [
       "$$\\int \\sin^{2}{\\left (\\theta \\right )}\\, d\\theta$$"
      ],
      "text/plain": [
       "⌠           \n",
       "⎮    2      \n",
       "⎮ sin (θ) dθ\n",
       "⌡           "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y_indef = sympy.Integral(y)\n",
    "Y_indef"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'sympy.integrals.integrals.Integral'>\n"
     ]
    }
   ],
   "source": [
    "print type(Y_indef)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "定积分："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAA4CAYAAAAhHqx3AAAABHNCSVQICAgIfAhkiAAABoBJREFU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\nHA6HIwn+B7jjVvTlLJflAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\int_{0}^{\\pi} \\sin^{2}{\\left (\\theta \\right )}\\, d\\theta$$"
      ],
      "text/plain": [
       "π           \n",
       "⌠           \n",
       "⎮    2      \n",
       "⎮ sin (θ) dθ\n",
       "⌡           \n",
       "0           "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y_def = sympy.Integral(y, (theta, 0, sympy.pi))\n",
    "Y_def"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "产生函数 $Y(x) = \\int_0^x sin^2(\\theta) d\\theta$，并将其向量化："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Y_raw = lambda x: integrate(y, (theta, 0, x))\n",
    "Y = np.vectorize(Y_raw)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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jB/zlL2EKvohIKVEiz2L06LA5xKhRcOqpUUcjIrI5XeyswR13wLXXhmVpO3aM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\ngg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb8bd320>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "x = np.linspace(0, 2 * np.pi)\n",
    "p = plt.plot(x, Y(x))\n",
    "t = plt.title(r'$Y(x) = \\int_0^x sin^2(\\theta) d\\theta$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 数值积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "数值积分：\n",
    "\n",
    "$$F(x) = \\lim_{n \\rightarrow \\infty} \\sum_{i=0}^{n-1} f(x_i)(x_{i+1}-x_i) \n",
    "\\Rightarrow F(x) = \\int_{x_0}^{x_n} f(x) dx$$\n",
    "\n",
    "导入贝塞尔函数："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.special import jv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return jv(2.5, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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s6VEF3xna01fKQyUnJ9O6dWv27NnjVU98+qJTp05Rs2ZNli5dSuPGjW3H+R/a\n01fKRwQGBtKoUSPee+8921H83rRp02jcuLHHFXxnaE9fKQ8WHx/P008/TUpKilfeE+4L/vrrL2rW\nrMnnn39OvXr1bMe5gvb0lfIhLVq0oEyZMsTGxtqO4rcmTpxImzZtPLLgO0N7+kp5uEWLFvH222/z\n1Vdf2Y7id/744w9q1arF119/TY0aNWzHyZb29JXyMZ06deLXX3/lyy+/tB3F77z55ps8/PDDHlvw\nnaE9faW8wNSpU1m9ejXLli2zHcVvHD58mMDAQBITE6lUqZLtODnKa09fi75SXuDkyZNUrVqV+Ph4\n6tSpYzuOXxg0aBDGGI+fB8ltRV9EygIfAZWBPUAXY8yxbI6LAcKAX40xOa4Dp0VfqasbNWoU+/fv\n95oFuL3Z3r17adCgAdu2baN8+fK241yVO4v+G8ARY8wbIvISUMYYE5XNcc2BNOA9LfpKOe/IkSPU\nqlWL5ORkjy9E3i48PJzKlSvz6quv2o5yTe4s+tuBlsaYwyJSHnAYY7L93SkiVYAVWvSVck2/fv24\n6aabGD16tO0oPmvLli20a9eO1NRUbrzxRttxrsmdRf8PY0yZzG0Bjl7cz+bYKmjRV8plP/74I02b\nNmX37t2UKlXKdhyfFBYWRvv27YmMjLQdJVfyWvSLXONka4DsfkcOzbpjjDEi4nLFHjly5KXtkJAQ\nQkJCXD2lUj6levXqtGrVitmzZzNw4EDbcXyOw+EgJSWFJUuW2I6SI4fDgcPhcPrzrg7vhBhjDolI\nABCnwztKFbyEhAQeffRRUlNTvW5Rbk9mjKFp06YMHDiQxx57zHacXHPnw1mxQHjmdjigNxAr5QaN\nGzematWqLFy40HYUn7J48WLOnj1Lt27dbEcpUK7esvkxUIkst2yKyO3ATGNMWOZxC4CWwM3Ar8Ar\nxpg52ZxPe/pK5VJcXBz//Oc/SUlJoUiRq47SqlxIT0/nzjvvZPLkydx///224+SJPpyllJ8ICQnh\niSeeIDw8/NoHq6uKjo5m0aJFrFmzhoz7UryHFn2l/ER8fDwRERGkpKTo2L4L0tLSqFWrFrGxsTRq\n1Mh2nDzTCdeU8hMtW7akcuXKvP/++7ajeLUJEybQokULryz4ztCevlJebMOGDfTq1YsdO3Zob98J\nv/32G3Xq1GHjxo1eO5Om9vSV8iP33XcfNWrUYO7cubajeKXXXnuN7t27e23Bd4b29JXycl9//TXd\nu3dn587fphsOAAANG0lEQVSdXHfddbbjeI3k5GRatGhBcnIyt956q+04TtOevlJ+5p577qFu3brE\nxMTYjuI1jDEMGjSIYcOGeXXBd4b29JXyAd9++y2PPPIIqampXH/99bbjeLzly5fz8ssvs3nzZq+/\nFqI9faX8UJMmTQgKCmLWrFm2o3i806dPM3jwYCZOnOj1Bd8Z2tNXykds2rSJhx9+mF27dlGsWDHb\ncTzWmDFjSEhIYOnSpbaj5At9OEspP/bQQw/Rrl07BgwYYDuKR9q/fz933XUXCQkJVKtWzXacfKFF\nXyk/lpiYSFhYGLt27aJEiRK243icHj16ULVqVV577TXbUfKNFn2l/FzXrl0JDAxkxIgRtqN4lA0b\nNtC9e3e2b99OyZIlbcfJN1r0lfJzFxf1/v7776lcubLtOB7h/PnzNG7cmBdeeIHu3bvbjpOv9O4d\npfxc5cqViYyM5IUXXrAdxWPMnj2bkiVL+vxc+bmhPX2lfNCpU6eoW7cuc+fO9ftlR//44w/q1q3L\np59+SnBwsO04+U6Hd5RSACxatIhXX32V77//3q8XWunbty8XLlwgOjradpQC4dbhHREpKyJrRGSn\niKwWkZuyOaaiiMSJyDYR2Soi3rHEvFJernPnzpQrV44ZM2bYjmJNXFwcK1asYOzYsbajeAyXevoi\n8gZwxBjzhoi8BJQxxkRddkx5oLwx5gcRKQV8BzxsjEm57Djt6SuVz5KSkmjTpg0pKSncfPPNtuO4\nVVpaGvXr1+fdd98lLCzMdpwC49bhHRHZDrQ0xhzOLO4OY0yda3xmGfCuMWbtZa9r0VeqAAwYMIDz\n588zdepU21Hcqn///qSlpfn8tNPuLvp/GGPKZG4LcPTifg7HVwHigTuNMWmXvadFX6kCcPToUerW\nrcvq1au56667bMdxC4fDQc+ePUlKSqJMmRxLkk/Ia9G/5tUdEVkDlM/mraFZd4wxRkRyrNqZQzuL\ngIGXF/yLRo4ceWk7JCTE7+86UCo/lC1bllGjRhEZGYnD4fC6hb/zKi0tjYiICKZPn+6TBd/hcOBw\nOJz+fH4M74QYYw6JSAAQl93wjogUBVYCnxpjJuRwLu3pK1VAzp8/T8OGDfnXv/5Fly5dbMcpUAMG\nDOD48ePMmzfPdhS3cPfwzhvA78aYcSISBdyUzYVcAeZlHjf4KufSoq9UAfriiy/o0aMHSUlJlC5d\n2nacAhEfH3/p7+iLvfzsuLvolwU+BioBe4AuxphjInI7MNMYEyYi9wHrgS3AxcZeNsZ8dtm5tOgr\nVcD69u3LsWPH+OCDD3xumOfEiRPUr1+fiRMn8uCDD9qO4zb6cJZSKkcnT56kcePGREVF8fjjj9uO\nk68iIyP5888//WZY5yIt+kqpq9qyZQtt2rThm2++oXr16rbj5It169bRq1cvvxrWuUgnXFNKXVX9\n+vUZNmwYjz32GOnp6bbjuGzPnj306NGDuXPn+l3Bd4b29JXyQ8YYwsLCCA4OZvTo0bbjOO3EiRM0\na9aM3r17M2jQINtxrNDhHaVUrhw+fJjg4GA+/PBDr3wmxhhD165dKVGiBHPmzPG5C9O5pcM7Sqlc\nue2224iJiaFXr14cPXrUdpw8e/3119m7dy/R0dF+W/CdoT19pfzcoEGD2LdvH4sWLfKa4rly5Uqe\neeYZEhISuP32223HsUp7+kqpPBk7diy7du1i1qxZtqPkSkpKCk8++SSLFi3y+4LvDP9dWUEpBUCx\nYsVYsGABrVq1okKFCoSGhtqOlKNjx47RsWNHxo0bxz333GM7jlfS4R2lFADffPMNDz30EAsWLKBN\nmza241zh/PnzPPjgg9SqVYuJEyfajuMx9O4dpZTTvvjiCzp37szixYtp3ry57TiXnD59mp49e5KW\nlsaKFSsoWrSo7UgeQ8f0lVJOa968OQsWLKBz58588803tuMAcPz4cR544AEKFSrE8uXLteC7SIu+\nUup/tGnThnnz5tGxY0e+//57q1kOHz5MSEgIgYGBLFiwgOuvv95qHl+gRV8pdYUHHniAGTNmEBoa\nSlJSkpUMP/30E82aNaNjx45MnjyZwoULW8nha/TuHaVUtjp27MiZM2do3749a9eupW7dum5r+4cf\nfiAsLIzhw4fTp08ft7XrD7Snr5TKUZcuXXjzzTdp3rw5kyZN4vz58wXepsPh4P7772fixIla8AuA\n3r2jlLqmHTt28PTTT5Oens7s2bMJDAzM9zaOHj3KmDFjmDdvHh9//DGtWrXK9zZ8kdvu3hGRsiKy\nRkR2ishqEbkpm2OKichGEflBRJJF5HVn21NK2VO7dm0cDgfh4eG0bNmSUaNGcfbs2Xw595kzZxg/\nfjy1a9fmr7/+YsuWLVrwC5ArwztRwBpjTC1gbeb+/zDGnAZaGWP+H1AfaJW5fKJSyssUKlSIPn36\nkJiYyHfffUeDBg1cuq3zwoULfPjhh9SpU4f4+HjWr1/P9OnTCQgIyMfU6nJOD++IyHagpTHmsIiU\nBxzGmDpXOb4EEA+EG2OSs3lfh3eU8hLGGD7++GMGDRpEw4YNCQkJoUWLFgQHB1/zPvojR46wceNG\nRowYQaFChXjzzTdp2bKlm5L7Hrc9kSsifxhjymRuC3D04v5lxxUCvgeqA9OMMS/mcD4t+kp5mWPH\njrFmzRrWr1/P+vXr2b17N02bNqVFixbcd999nDlzhpSUlP/5k56ezp133smAAQPo0qULhQrp/SSu\nyNeiLyJrgPLZvDUUmJe1yIvIUWNM2aucqzSwCogyxjiyed+MGDHi0n5ISIhXLuyglD87evQoX375\nJevXr2fDhg2UKFGCunXr/s+f8uXLe80Uzp7I4XDgcDgu7Y8aNcptPf3tQIgx5pCIBABxVxveyfzM\ncOCUMeatbN7Tnr5SSuWRO+feiQXCM7fDgWXZhCl38a4eESkOtAMSXWhTKaWUC1zp6ZcFPgYqAXuA\nLsaYYyJyOzDTGBMmIvWBuWT841IIeN8Y82YO59OevlJK5ZFOrayUUn5Ep1ZWSimVIy36SinlR7To\nK6WUH9Gir5RSfkSLvlJK+REt+kop5Ue06CullB/Roq+UUn5Ei75SSvkRLfpKKeVHtOgrpZQf0aKv\nlFJ+RIu+Ukr5ES36SinlR7ToK6WUH3G66ItIWRFZIyI7RWT1xRWycji2sIgkisgKZ9tTSinlOld6\n+lHAGmNMLWBt5n5OBgLJgK6SkgtZFz32d/pd/E2/i7/pd+E8V4r+Q8C8zO15wMPZHSQidwChwCwg\n16u7+DP9D/pv+l38Tb+Lv+l34TxXiv5txpjDmduHgdtyOO4d4AXgggttKaWUygdFrvamiKwBymfz\n1tCsO8YYIyJXDN2IyIPAr8aYRBEJcSWoUkop1zm9MLqIbAdCjDGHRCQAiDPG1LnsmDHA48A5oBhw\nI7DYGNMrm/PpeL9SSjkhLwuju1L03wB+N8aME5Eo4CZjTI4Xc0WkJfC8MeYfTjWolFLKZa6M6Y8F\n2onITqB15j4icruI/DeHz2hvXimlLHK6p6+UUsr7WH8iV0Q6iMh2EUkVkZds57FFRCqKSJyIbBOR\nrSISaTuTbfpQXwYRuUlEFolIiogki0hT25lsEZGXM/8fSRKRD0XketuZ3EVEYkTksIgkZXkt1w/J\nXmS16ItIYWAy0AEIBLqLSF2bmSxKBwYbY+4EmgL9/Pi7uEgf6sswEfjEGFMXqA+kWM5jhYhUAZ4G\nGhhjgoDCQDebmdxsDhm1Mqu8PCQL2O/pNwF2GWP2GGPSgYVAR8uZrDDGHDLG/JC5nUbG/9i3201l\njz7Ul0FESgPNjTExAMaYc8aYPy3HsuU4GZ2jEiJSBCgBHLAbyX2MMV8Af1z2cq4eks3KdtGvAOzL\nsr8/8zW/ltmjCQY22k1ilT7Ul6Eq8JuIzBGR70VkpoiUsB3KBmPMUWA88DNwEDhmjPncbirrcvuQ\n7CW2i76//2y/goiUAhYBAzN7/H4n60N9+HEvP1MRoAEw1RjTADhBLn7C+yIRqQ4MAqqQ8Su4lIj0\nsBrKg5iMu3KuWVNtF/0DQMUs+xXJ6O37JREpCiwG5htjltnOY9G9wEMishtYALQWkfcsZ7JlP7Df\nGJOQub+IjH8E/FEj4CtjzO/GmHPAEjL+W/Fnh0WkPEDmQ7K/XusDtov+JqCmiFQRkeuArkCs5UxW\niIgAs4FkY8wE23lsMsb8yxhT0RhTlYwLdeuye4rbHxhjDgH7RKRW5kttgW0WI9m0HWgqIsUz/39p\nS8aFfn8WC4RnbocD1+wsXnXunYJmjDknIv2BVWRciZ9tjPHLOxOAZkBPYIuIJGa+9rIx5jOLmTyF\nvw8DDgA+yOwY/Qg8YTmPFcaYzZm/+DaRca3ne2CG3VTuIyILgJZAORHZB7xCxkOxH4tIBLAH6HLN\n8+jDWUop5T9sD+8opZRyIy36SinlR7ToK6WUH9Gir5RSfkSLvlJK+REt+kop5Ue06CullB/Roq+U\nUn7k/wDY8//eOSbGSgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xbfe3a58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0, 10)\n",
    "p = plt.plot(x, f(x), 'k-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### `quad` 函数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quadrature 积分的原理参见：\n",
    "\n",
    "http://en.wikipedia.org/wiki/Numerical_integration#Quadrature_rules_based_on_interpolating_functions\n",
    "\n",
    "quad 返回一个 (积分值，误差) 组成的元组："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import quad\n",
    "interval = [0, 6.5]\n",
    "value, max_err = quad(f, *interval)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "积分值："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.28474297234\n"
     ]
    }
   ],
   "source": [
    "print value"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最大误差："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.34181853668e-09\n"
     ]
    }
   ],
   "source": [
    "print max_err"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "积分区间图示，蓝色为正，红色为负："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "integral = 1.284742972\n",
      "upper bound on error: 2.34e-09\n"
     ]
    },
    {
     "data": {
      "image/png": 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fj/r3n8batV/iec+6jmKO29n4fA3o2vU910FMnHrhhReYP38+Y8aMOe5tQ745S0SaAq8A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\nR6zoG2NMHLGib4wxceT/AZhppPNX9o1QAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc126860>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print \"integral = {:.9f}\".format(value)\n",
    "print \"upper bound on error: {:.2e}\".format(max_err)\n",
    "x = np.linspace(0, 10, 100)\n",
    "p = plt.plot(x, f(x), 'k-')\n",
    "x = np.linspace(0, 6.5, 45)\n",
    "p = plt.fill_between(x, f(x), where=f(x)>0, color=\"blue\")\n",
    "p = plt.fill_between(x, f(x), where=f(x)<0, color=\"red\", interpolate=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 积分到无穷"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from numpy import inf\n",
    "interval = [0., inf]\n",
    "\n",
    "def g(x):\n",
    "    return np.exp(-x ** 1/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "upper bound on error: 7.2e-11\n"
     ]
    },
    {
     "data": {
      "image/png": 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gixQRkThUYg+VmYWAFcAAYAMwH7jUObe8SJtTgWXOuR1mNhgY65zrddB51EMl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\nTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc260dd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "value, max_err = quad(g, *interval)\n",
    "x = np.linspace(0, 10, 50)\n",
    "fig = plt.figure(figsize=(10,3))\n",
    "p = plt.plot(x, g(x), 'k-')\n",
    "p = plt.fill_between(x, g(x))\n",
    "plt.annotate(r\"$\\int_0^{\\infty}e^{-x^1/2}dx = $\" + \"{}\".format(value), (4, 0.6),\n",
    "         fontsize=16)\n",
    "print \"upper bound on error: {:.1e}\".format(max_err)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 双重积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "假设我们要进行如下的积分：\n",
    "\n",
    "$$ I_n = \\int \\limits_0^{\\infty} \\int \\limits_1^{\\infty} \\frac{e^{-xt}}{t^n}dt dx = \\frac{1}{n}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def h(x, t, n):\n",
    "    \"\"\"core function, takes x, t, n\"\"\"\n",
    "    return np.exp(-x * t) / (t ** n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "一种方式是调用两次 `quad` 函数，不过这里 `quad` 的返回值不能向量化，所以使用了修饰符 `vectorize` 将其向量化："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from numpy import vectorize\n",
    "@vectorize\n",
    "def int_h_dx(t, n):\n",
    "    \"\"\"Time integrand of h(x).\"\"\"\n",
    "    return quad(h, 0, np.inf, args=(t, n))[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "@vectorize\n",
    "def I_n(n):\n",
    "    return quad(int_h_dx, 1, np.inf, args=(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 1.97,  1.  ,  0.5 ,  0.2 ]),\n",
       " array([  9.804e-13,   1.110e-14,   5.551e-15,   2.220e-15]))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_n([0.5, 1.0, 2.0, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "或者直接调用 `dblquad` 函数，并将积分参数传入，传入方式有多种，后传入的先进行积分："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import dblquad\n",
    "@vectorize\n",
    "def I(n):\n",
    "    \"\"\"Same as I_n, but using the built-in dblquad\"\"\"\n",
    "    x_lower = 0\n",
    "    x_upper = np.inf\n",
    "    return dblquad(h,\n",
    "                   lambda t_lower: 1, lambda t_upper: np.inf,\n",
    "                   x_lower, x_upper, args=(n,))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 1.97,  1.  ,  0.5 ,  0.2 ]),\n",
       " array([  9.804e-13,   1.110e-14,   5.551e-15,   2.220e-15]))"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I_n([0.5, 1.0, 2.0, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 采样点积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### trapz 方法 和 simps 方法"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import trapz, simps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`sin` 函数， `100` 个采样点和 `5` 个采样点："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x_s = np.linspace(0, np.pi, 5)\n",
    "y_s = np.sin(x_s)\n",
    "x = np.linspace(0, np.pi, 100)\n",
    "y = np.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Rw+lEZbm6uvLwww9zzz330KdPnzIHq3h4ePD//t//q9La/MI5yB2qDurKlSsMHDiQ7Oxs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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc1954e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = plt.plot(x, y, 'k:')\n",
    "p = plt.plot(x_s, y_s, 'k+-')\n",
    "p = plt.fill_between(x_s, y_s, color=\"gray\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "采用 [trapezoidal 方法](https://en.wikipedia.org/wiki/Trapezoidal_rule) 和 [simpson 方法](https://en.wikipedia.org/wiki/Simpson%27s_rule) 对这些采样点进行积分（函数积分为 2）："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Trapezoidal Integration over 5 points : 1.896\n",
      "Simpson Integration over 5 points : 2.005\n",
      "Trapezoidal Integration over 100 points : 2.000\n"
     ]
    }
   ],
   "source": [
    "result_s = trapz(y_s, x_s)\n",
    "result_s_s = simps(y_s, x_s)\n",
    "result = trapz(y, x)\n",
    "print \"Trapezoidal Integration over 5 points : {:.3f}\".format(result_s)\n",
    "print \"Simpson Integration over 5 points : {:.3f}\".format(result_s_s)\n",
    "print \"Trapezoidal Integration over 100 points : {:.3f}\".format(result)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 使用 ufunc 进行积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Numpy` 中有很多 `ufunc` 对象："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ufunc"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(np.add)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "accumulate(array, axis=0, dtype=None, out=None)\n",
      "\n",
      "Accumulate the result of applying the operator to all elements.\n",
      "\n",
      "For a one-dimensional array, accumulate produces results equivalent to::\n",
      "\n",
      "  r = np.empty(len(A))\n",
      "  t = op.identity        # op = the ufunc being applied to A's  elements\n",
      "  for i in range(len(A)):\n",
      "      t = op(t, A[i])\n",
      "      r[i] = t\n",
      "  return r\n",
      "\n",
      "For example, add.accumulate() is equivalent to np.cumsum().\n",
      "\n",
      "For a multi-dimensional array, accumulate is applied along only one\n",
      "axis (axis zero by default; see Examples below) so repeated use is\n",
      "necessary if one wants to accumulate over multiple axes.\n",
      "\n",
      "Parameters\n",
      "----------\n",
      "array : array_like\n",
      "    The array to act on.\n",
      "axis : int, optional\n",
      "    The axis along which to apply the accumulation; default is zero.\n",
      "dtype : data-type code, optional\n",
      "    The data-type used to represent the intermediate results. Defaults\n",
      "    to the data-type of the output array if such is provided, or the\n",
      "    the data-type of the input array if no output array is provided.\n",
      "out : ndarray, optional\n",
      "    A location into which the result is stored. If not provided a\n",
      "    freshly-allocated array is returned.\n",
      "\n",
      "Returns\n",
      "-------\n",
      "r : ndarray\n",
      "    The accumulated values. If `out` was supplied, `r` is a reference to\n",
      "    `out`.\n",
      "\n",
      "Examples\n",
      "--------\n",
      "1-D array examples:\n",
      "\n",
      ">>> np.add.accumulate([2, 3, 5])\n",
      "array([ 2,  5, 10])\n",
      ">>> np.multiply.accumulate([2, 3, 5])\n",
      "array([ 2,  6, 30])\n",
      "\n",
      "2-D array examples:\n",
      "\n",
      ">>> I = np.eye(2)\n",
      ">>> I\n",
      "array([[ 1.,  0.],\n",
      "       [ 0.,  1.]])\n",
      "\n",
      "Accumulate along axis 0 (rows), down columns:\n",
      "\n",
      ">>> np.add.accumulate(I, 0)\n",
      "array([[ 1.,  0.],\n",
      "       [ 1.,  1.]])\n",
      ">>> np.add.accumulate(I) # no axis specified = axis zero\n",
      "array([[ 1.,  0.],\n",
      "       [ 1.,  1.]])\n",
      "\n",
      "Accumulate along axis 1 (columns), through rows:\n",
      "\n",
      ">>> np.add.accumulate(I, 1)\n",
      "array([[ 1.,  1.],\n",
      "       [ 0.,  1.]])\n"
     ]
    }
   ],
   "source": [
    "np.info(np.add.accumulate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`np.add.accumulate` 相当于 `cumsum` ："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "result_np = np.add.accumulate(y) * (x[1] - x[0]) - (x[1] - x[0]) / 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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/uQfabHfnV93vKvGra9fyDzep0Ui3ff4Kb2xamnOJid7Pzp2913HO+5mY6ERE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\nRoFfRCTGKPCLiMSY/weoVZxsAST89wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc612a20>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = plt.plot(x, - np.cos(x) + np.cos(0), 'rx')\n",
    "p = plt.plot(x, result_np)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 速度比较"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "计算积分：$$\\int_0^x sin \\theta d\\theta$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy.abc import x, theta\n",
    "sympy_x = x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = np.linspace(0, 20 * np.pi, 1e+4)\n",
    "y = np.sin(x)\n",
    "sympy_y = vectorize(lambda x: sympy.integrate(sympy.sin(theta), (theta, 0, x)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`numpy` 方法："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The slowest run took 4.32 times longer than the fastest. This could mean that an intermediate result is being cached \n",
      "10000 loops, best of 3: 56.2 µs per loop\n",
      "-2.34138044756e-17\n"
     ]
    }
   ],
   "source": [
    "%timeit np.add.accumulate(y) * (x[1] - x[0])\n",
    "y0 = np.add.accumulate(y) * (x[1] - x[0])\n",
    "print y0[-1] "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`quad` 方法："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10000 loops, best of 3: 40.5 µs per loop\n",
      "result =  3.43781337153e-15\n",
      "number of evaluations 21\n"
     ]
    }
   ],
   "source": [
    "%timeit quad(np.sin, 0, 20 * np.pi)\n",
    "y2 = quad(np.sin, 0, 20 * np.pi, full_output=True)\n",
    "print \"result = \", y2[0]\n",
    "print \"number of evaluations\", y2[-1]['neval']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`trapz` 方法："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10000 loops, best of 3: 105 µs per loop\n",
      "-4.4408920985e-16\n"
     ]
    }
   ],
   "source": [
    "%timeit trapz(y, x)\n",
    "y1 = trapz(y, x)\n",
    "print y1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`simps` 方法："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000 loops, best of 3: 801 µs per loop\n",
      "3.28428554968e-16\n"
     ]
    }
   ],
   "source": [
    "%timeit simps(y, x)\n",
    "y3 = simps(y, x)\n",
    "print y3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`sympy` 积分方法："
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100 loops, best of 3: 6.86 ms per loop\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "%timeit sympy_y(20 * np.pi)\n",
    "y4 = sympy_y(20 * np.pi)\n",
    "print y4"
   ]
  }
 ],
 "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": "2.7.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}