week_2_3_exercises.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. 请用 python 的基本语句实现：\n",
    "        \n",
    "              计算 1+2+...+9999999\n",
    "              \n",
    "              对  n! 求和，其中 n = 1, ... 20 \n",
    "              \n",
    "              计算  1 + 1/2 + 1/3 + ... + 1/10000 \n",
    "\n",
    "              判断 10000121 是否是质数\n",
    "              \n",
    "              求 100000 之内的质数\n",
    "              \n",
    "              输出所有的水仙花数 （水仙花数指一个 3 位数，各位数字的立方和等于数字本身。例如 153 = 1^3 + 5^3 + 3^3）\n",
    "              \n",
    "              请打印字符串： \n",
    "                  ha\n",
    "                  haha\n",
    "                  hahaha\n",
    "                  hahahaha\n",
    "                  ...\n",
    "                  （最后一行有10个ha)\n",
    "              \n",
    "              对 1-100 按照它们的 sin 值大小排序；\n",
    "              对排好序的数字，绘制 数字 VS sin(数字）的图像\n",
    "        "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. 写一个通用的矩形法积分函数（参数：被积函数，上下限，切割成多少块）\n",
    "\n",
    "    用它计算  sin, cos, exp 在 0-1 区间积分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. 用蒙卡的方法计算 pi\n",
    "\n",
    "    生成 N 个随机 2维 点，分布在中心为0，边长 2 的正方形内 (用 np.random.uniform(low, high, size) 生成)\n",
    "    \n",
    "    统计多少个点落在中心0,半径 1 的圆内\n",
    "    \n",
    "    利用数目的比值求出 pi\n",
    "    \n",
    "    绘制 N, pi估计值的关系曲线"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "4. 用 np.random.randint 生成 n 个长为 10 的list，存放 0-50 的整数 \n",
    "\n",
    "    将他们合并之后，用 set 去重复，用 sort 排序\n",
    "    \n",
    "    将会有 m 个 0-50 整数没有被覆盖在里面，求出 m\n",
    "    \n",
    "    对 n = 1-40, 绘制 n, m 的曲线图"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5. 请写一个函数，对一组输入的数，函数返回这些数的 sum、average、variance。\n",
    "        （提示：python 的 return 语句可以返回多个对象）"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "6. 请自己写一个函数，计算两个随机变量的协方差 (输入为两个 list 或两个 array)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "7. 请用 np.random.normal 函数随机生成 10^6 个 (mu=1, sigma=3) 的随机数，用 hist 函数画 100 bin 的统计直方图，看是不是你预期的结果。\n",
    "\n",
    "       请你自己写的程序求出它们的 average 和 variance，看是不是你预期的。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "8. 已知有随机变量 x, y, z ，且\n",
    "\n",
    "        x 服从 标准正太分布；\n",
    "    \n",
    "        y = x + 另一个标准正太分布；\n",
    "    \n",
    "        z = (x-y)；\n",
    "    \n",
    "    请生成 100000 组 (x,y,z)\n",
    "    \n",
    "    由此计算 3*3 的协方差矩阵 (请分别用你自己写的协方差程序，以及 np.cov 函数计算，看结果是否一致)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "9. 请生成 10 个 2*2 随机对称矩阵，用 numpy 的 eig 函数求特征值、特征向量；\n",
    "\n",
    "    请以特征向量为主轴、特征值为轴长度，画椭圆。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "10. 用 contourf 画一个环的彩色图\n",
    "\n",
    "    用 fig.savefig 将它存储为 rings.png 文件\n",
    "    \n",
    "    使用 plt.imread 将该文件读入到 np.array 数组\n",
    "    \n",
    "    将该数组适当处理，生成一个分辨率低 8 倍，粗糙很多的图片"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "X, Y = np.linspace(-1,1,100), np.linspace(-1,1,100)\n",
    "Z = np.meshgrid(X,Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<Figure size 432x288 with 0 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%pylab inline\n",
    "fig, ax = plt.subplots()\n",
    "X, Y = np.linspace(-1,1,100), np.linspace(-1,1,100)\n",
    "Z = [[x**2 + y**2  for y in Y] for x in X]; Z = np.array(Z)\n",
    "ax.contourf(X, Y,Z)\n",
    "fig.savefig('./rings.png',format='png')\n",
    "\n",
    "data = plt.imread('rings.png')\n",
    "#plt.imshow(data); plt.show()\n",
    "plt.imshow((data[::2,::2,:]+data[0::2,0::2,:])*0.5); plt.show()\n",
    "plt.imshow(data[::8,::8,:]); plt.show()\n",
    "plt.savefig('./rings_lowres.png',format='png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(Little BOSS)**\n",
    "\n",
    "请写一个随机行走搜索最小值的程序，能够自动调整步长（按当前行走的效率，自动调大、调小）。算法如下：\n",
    "    \n",
    "            任意 f(x), 任意步长 dx 开始，不断生成\n",
    "                x2 = x + np.random.uniform(-dx, dx)\n",
    "                    如果 f(x2)< f(x)， 则 x=x2，dx = dx*1.1\n",
    "                    如果 f(x2)> f(x)， 则 dx = dx*0.9 继续行走。\n",
    "            当 dx < 1.0e-6 时停止。\n",
    "        \n",
    "       请用这个算法搜索 (x-3)^2 的最小值。从 x=10, dx = 1 开始，看多少步能够停止。\n",
    "       \n",
    "       请画出 dx, x, 当前最小 f(x) 随步数的演化图。\n",
    "       \n",
    "       注意：对有多个参量的函数，也可以采用类似的方法搜索，此时可以借助 **梯度矢量** 找出当前最快的下降方向。\n",
    "       \n",
    "       相关的算法有 **随机梯度下降**， 在机器学习中有重要应用"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**(Final BOSS)**\n",
    "\n",
    "请在一个 npix*npix 像素的图片上生成 nregion 块不重叠的区域，用最少的颜色上色，验证四色猜想的成立。典型可取 npix=20, nregion=8。\n",
    "     \n",
    "         提示：\n",
    "            \n",
    "            要快速、有效地生成一系列形状不规则、互不重叠区域。\n",
    "            生成完毕后，判断出哪些区域互为邻域。\n",
    "            用递归搜索可能的着色方案。要很高效才能处理 nreigion >15。\n",
    "            用 contourf 对格点阵列画图。\n",
    "            \n",
    "         本栋的程序30秒内实现： npix=300，nreigion<15 找到所有可能着色方案；nregion<35 找到至少一种方案。"
   ]
  }
 ],
 "metadata": {
  "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}