Sympy.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "if get_ipython().Completer.use_jedi==True:\n",
    "\tget_ipython().Completer.use_jedi=False\n",
    "else:\n",
    "\tget_ipython().Completer.use_jedi=True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "\n",
       "IPython.keyboard_manager.edit_shortcuts.add_shortcut('ctrl-enter', {\n",
       "\thelp : \"run cell and keep focus\", //This is optional\n",
       "\thandler : function (event) {\n",
       "\t\tIPython.notebook.execute_cell();\n",
       "\t\tIPython.notebook.edit_mode();\n",
       "\t\treturn false;\n",
       "\t}}\n",
       ");\n",
       "\n",
       "IPython.keyboard_manager.edit_shortcuts.add_shortcut('ctrl-x', {\n",
       "\thelp : \"jedi\", //This is optional\n",
       "\thandler : function (event) {\n",
       "\t\tvar index=IPython.notebook.get_edit_index();\n",
       "\t\tIPython.notebook.execute_cells([0]);\n",
       "\t\tIPython.notebook.get_cell(index).focus_cell();\n",
       "\t\tIPython.notebook.edit_mode();\n",
       "\t\treturn false;\n",
       "\t}}\n",
       ");\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%javascript\n",
    "\n",
    "IPython.keyboard_manager.edit_shortcuts.add_shortcut('ctrl-enter', {\n",
    "\thelp : \"run cell and keep focus\", //This is optional\n",
    "\thandler : function (event) {\n",
    "\t\tIPython.notebook.execute_cell();\n",
    "\t\tIPython.notebook.edit_mode();\n",
    "\t\treturn false;\n",
    "\t}}\n",
    ");\n",
    "\n",
    "IPython.keyboard_manager.edit_shortcuts.add_shortcut('ctrl-x', {\n",
    "\thelp : \"jedi\", //This is optional\n",
    "\thandler : function (event) {\n",
    "\t\tvar index=IPython.notebook.get_edit_index();\n",
    "\t\tIPython.notebook.execute_cells([0]);\n",
    "\t\tIPython.notebook.get_cell(index).focus_cell();\n",
    "\t\tIPython.notebook.edit_mode();\n",
    "\t\treturn false;\n",
    "\t}}\n",
    ");\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "%config IPCompleter.greedy=True\n",
    "from IPython.core.interactiveshell import InteractiveShell \n",
    "InteractiveShell.ast_node_interactivity='all'\n",
    "get_ipython().Completer.use_jedi = False"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 方程式求解"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.interactive import printing\n",
    "printing.init_printing(use_latex=True)\n",
    "from sympy import Eq , solve_linear_system,Matrix\n",
    "from sympy.solvers.solveset import linsolve\n",
    "from numpy import linalg\n",
    "import numpy as np\n",
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2 x - y = -4$$"
      ],
      "text/plain": [
       "2⋅x - y = -4"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAAWCAYAAADgreP7AAAABHNCSVQICAgIfAhkiAAAA0hJREFUaIHt2UmIXFUUxvFfYhAxgiZROgZxIYJZBFsk0i6iotGN4MJoFgomQlQURVqjKEExG4egQZzHRZw2igOCGKLi2Cgq3Q7RoNhSGFA7RkMjGqcQF+cW/ayu6XV11aPw/aGox3nn3vvd6dxTtyj537IJb2An9uIXjOEWLCpQV0ljFuESvIhvxLxN4j2sw9yq45yagn9hFF9iF+bjZCzH9+l5Z3e1l+TkcjyEH/AmvsMAVuFQPI/V2F9b8KAGFd6anB/sgtiSzjgD58js4MRiMfH7cV6eCgdToddq7NuSfVWNfQ62pHd35GmoQ9anNtc3eH8c/sQ7PVNULBvEeNyXp9BNqdDmGvsg9omwf0DGvjn5PzpjmTNjKLX7XIP32/APju+ZomK5XozH3TCvgdN1OETE/OVYgc9M36Wf4imsxUViN2/AtXhWnCe9ZFQkKEN13q3GWbhX9KUewzgsR3uf4KU8AnvIPKxJz1ubOf4oVkT186o49OtxlBjgCq5K/ltxYMdyZ8bbScOSjG2+SCwnxAJuRMV/+93qs2VWlc8udwmNr7RbYADn4iuRjZ/YwO92UwMwgoPbrL8i3+A+3Uadt5meR2xKtovb1NVNKma/z7VcncruwMKqsVEYrzIhfr+N4ms8iWV1/H7KPK/D722KGscfbfoSC64VI+l7CC9gKa7B+3giR1vdoht9znIl7hF51EpxV4Lpv7ObMYYTcAR2Z+wX4BmxMBbjYVyRU+BssgA/412chtdxusg9xlqU7fcze1gkY9vFRO+aaUUTIjQsyNjOFhcxn4tFsAN/i91UJF/gN1wo3/1ARf+e2TcITWM4vJXzUrEza5lr6lJlJGNfIcL1tzgy2c5PfkWv9keSjl/FEbOwuXvfc7Po78ea9DUbxodxp7hwGBehcECEwmNEhr5SnAWDIuvdKyZ9PFPPRyJknipCaRGsMXU+X4rHC9LRC9aKCLNPXJ5M1vGpqIlCy/CAOId2i8uHSTF5G02tmGPFxO9R/3LiTLHKPuigA51yStLwoXx5ST+yUevj5q2CtPWEl8VKP6loISXdpZqU3V+0kJLucDRuxGPij47t2r/YKekzLhO7eY+4k1/S3L2kpKSkpKTv+BfrYfiKkC+6zAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$3 x - y = -2$$"
      ],
      "text/plain": [
       "3⋅x - y = -2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ x : 2, \\quad y : 8\\right \\}$$"
      ],
      "text/plain": [
       "{x: 2, y: 8}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y=sp.symbols('x y')\n",
    "# just for display\n",
    "eq1=Eq(2*x-y,-4)\n",
    "eq2=Eq(3*x-y,-2)\n",
    "display(eq1)\n",
    "display(eq2)\n",
    "# ----------------\n",
    "row1=(2,-1,-4)\n",
    "row2=(3,-1,-2)\n",
    "system=Matrix((row1,row2))\n",
    "solve_linear_system(system,x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{\\left ( 2, \\quad 8\\right )\\right\\}$$"
      ],
      "text/plain": [
       "{(2, 8)}"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve([2*x-y+4,\n",
    "          3*x-y+2],\n",
    "         x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$2 x - y = -4$$"
      ],
      "text/plain": [
       "2⋅x - y = -4"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$3 x - y = -2$$"
      ],
      "text/plain": [
       "3⋅x - y = -2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ x : 2, \\quad y : 8\\right \\}$$"
      ],
      "text/plain": [
       "{x: 2, y: 8}"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import sympy as sp\n",
    "x, y = sp.symbols('x, y')\n",
    "eq1 = sp.Eq(2*x -y , -4)             \n",
    "eq2 = sp.Eq(3*x -y, -2)         \n",
    "ans = sp.solve((eq1, eq2), (x, y))\n",
    "display(eq1)\n",
    "display(eq2)\n",
    "display(ans)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 2., -8.])"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A=[[2,1],[3,1]]\n",
    "s=[-4,-2]\n",
    "r=np.linalg.solve(A,s)\n",
    "r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Solver\n",
    "来源 http://apmonitor.com/che263/index.php/Main/PythonOptimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial Objective: 16.0\n",
      "Final Objective: 17.01401724563517\n",
      "Solution\n",
      "x1 = 1.0\n",
      "x2 = 4.742996096883977\n",
      "x3 = 3.8211546234095715\n",
      "x4 = 1.379407645075325\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.optimize import minimize\n",
    "\n",
    "def objective(x):\n",
    "    return x[0]*x[3]*(x[0]+x[1]+x[2])+x[2]\n",
    "\n",
    "def constraint1(x):\n",
    "    return x[0]*x[1]*x[2]*x[3]-25.0\n",
    "\n",
    "def constraint2(x):\n",
    "    sum_eq = 40.0\n",
    "    for i in range(4):\n",
    "        sum_eq = sum_eq - x[i]**2\n",
    "    return sum_eq\n",
    "\n",
    "# initial guesses\n",
    "n = 4\n",
    "x0 = np.zeros(n)\n",
    "x0[0] = 1.0\n",
    "x0[1] = 5.0\n",
    "x0[2] = 5.0\n",
    "x0[3] = 1.0\n",
    "\n",
    "# show initial objective\n",
    "print('Initial Objective: ' + str(objective(x0)))\n",
    "\n",
    "# optimize\n",
    "b = (1.0,5.0)\n",
    "bnds = (b, b, b, b)\n",
    "con1 = {'type': 'ineq', 'fun': constraint1} \n",
    "con2 = {'type': 'eq', 'fun': constraint2}\n",
    "cons = ([con1,con2])\n",
    "solution = minimize(objective,x0,method='SLSQP',\\\n",
    "                    bounds=bnds,constraints=cons)\n",
    "x = solution.x\n",
    "\n",
    "# show final objective\n",
    "print('Final Objective: ' + str(objective(x)))\n",
    "\n",
    "# print solution\n",
    "print('Solution')\n",
    "print('x1 = ' + str(x[0]))\n",
    "print('x2 = ' + str(x[1]))\n",
    "print('x3 = ' + str(x[2]))\n",
    "print('x4 = ' + str(x[3]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 解非线性方程组-nonlinsolve（）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$$\\left\\{\\left ( -1, \\quad 1\\right ), \\left ( -1, \\quad - \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right ), \\left ( -1, \\quad - \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right ), \\left ( 1 - i, \\quad -1 + i\\right ), \\left ( 1 + i, \\quad -1 - i\\right ), \\left ( 1 - \\frac{i \\sqrt{- 6 \\sqrt{3} + 12}}{2} - \\frac{i \\sqrt{- 2 \\sqrt{3} + 4}}{2}, \\quad \\frac{1}{2} + \\frac{\\sqrt{3}}{2} + \\frac{\\sqrt{2} \\sqrt{-2 + \\sqrt{3}}}{2}\\right ), \\left ( 1 - \\frac{\\sqrt{-12 - 6 \\sqrt{3}}}{2} + \\frac{\\sqrt{-4 - 2 \\sqrt{3}}}{2}, \\quad - \\frac{\\sqrt{3}}{2} + \\frac{1}{2} - \\frac{\\sqrt{-8 + \\left(- \\sqrt{3} + 1\\right)^{2}}}{2}\\right ), \\left ( 1 - \\frac{\\sqrt{-4 - 2 \\sqrt{3}}}{2} + \\frac{\\sqrt{-12 - 6 \\sqrt{3}}}{2}, \\quad - \\frac{\\sqrt{3}}{2} + \\frac{1}{2} + \\frac{\\sqrt{-8 + \\left(- \\sqrt{3} + 1\\right)^{2}}}{2}\\right ), \\left ( 1 + \\frac{\\sqrt{-4 + 2 \\sqrt{3}}}{2} + \\frac{\\sqrt{-12 + 6 \\sqrt{3}}}{2}, \\quad \\frac{1}{2} + \\frac{\\sqrt{3}}{2} - \\frac{\\sqrt{2} \\sqrt{-2 + \\sqrt{3}}}{2}\\right )\\right\\}$$"
      ],
      "text/plain": [
       "⎧                                                                             \n",
       "⎪                                                                             \n",
       "⎨         ⎛      1   √3⋅ⅈ⎞  ⎛      1   √3⋅ⅈ⎞                                  \n",
       "⎪(-1, 1), ⎜-1, - ─ - ────⎟, ⎜-1, - ─ + ────⎟, (1 - ⅈ, -1 + ⅈ), (1 + ⅈ, -1 - ⅈ)\n",
       "⎩         ⎝      2    2  ⎠  ⎝      2    2  ⎠                                  \n",
       "\n",
       "                                                                     ⎛        \n",
       "  ⎛        ____________       ___________                _________⎞  ⎜      __\n",
       "  ⎜    ⅈ⋅╲╱ -6⋅√3 + 12    ⅈ⋅╲╱ -2⋅√3 + 4   1   √3   √2⋅╲╱ -2 + √3 ⎟  ⎜    ╲╱ -\n",
       ", ⎜1 - ──────────────── - ───────────────, ─ + ── + ──────────────⎟, ⎜1 - ────\n",
       "  ⎝           2                  2         2   2          2       ⎠  ⎝        \n",
       "\n",
       "                                          _________________⎞  ⎛               \n",
       "__________     ___________               ╱               2 ⎟  ⎜      _________\n",
       "12 - 6⋅√3    ╲╱ -4 - 2⋅√3     √3   1   ╲╱  -8 + (-√3 + 1)  ⎟  ⎜    ╲╱ -4 - 2⋅√\n",
       "────────── + ─────────────, - ── + ─ - ────────────────────⎟, ⎜1 - ───────────\n",
       "  2                2          2    2            2          ⎠  ⎝          2    \n",
       "\n",
       "                                   _________________⎞                         \n",
       "__     ____________               ╱               2 ⎟  ⎛      ___________     \n",
       "3    ╲╱ -12 - 6⋅√3     √3   1   ╲╱  -8 + (-√3 + 1)  ⎟  ⎜    ╲╱ -4 + 2⋅√3    ╲╱\n",
       "── + ──────────────, - ── + ─ + ────────────────────⎟, ⎜1 + ───────────── + ──\n",
       "           2           2    2            2          ⎠  ⎝          2           \n",
       "\n",
       "                                      ⎫\n",
       "____________                _________⎞⎪\n",
       " -12 + 6⋅√3   1   √3   √2⋅╲╱ -2 + √3 ⎟⎬\n",
       "────────────, ─ + ── - ──────────────⎟⎪\n",
       "    2         2   2          2       ⎠⎭"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"\n",
    "    x**2+y**2-2=0\n",
    "    x**3+y**3=0\n",
    "\"\"\"\n",
    "\n",
    "import sympy as sy\n",
    "x, y = sy.symbols(\"x y\")\n",
    "\n",
    "eq = [x**2+y**3-2, x**3+y**3]\n",
    "result = sy.nonlinsolve(eq, [x, y])\n",
    "display(result)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$16$$"
      ],
      "text/plain": [
       "16"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sy\n",
    "x, y = sy.symbols(\"x y z\")\n",
    "\n",
    "eq = [x**2+y**2-z**2, x**3+y**3]\n",
    "result = sy.nonlinsolve(eq, [x, y])\n",
    "display(result)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sin{\\left (x \\right )}}{x}$$"
      ],
      "text/plain": [
       "sin(x)\n",
       "──────\n",
       "  x   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFUKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSxoPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA6XoX52TB9a4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABEAAAALCAYAAACZIGYHAAAABHNCSVQICAgIfAhkiAAAANNJREFUKJG10TFLglEUh/GfurU0SZvQGCI1OBcZ+BFc6hMI+jGaXFzEKQjaWtqiLRChpKU9hcBBMGgRRRBteN9BLu8LDXq289x7Hs79X/ZUN/jGAg84iPklPvCLZxTSBCX8oIYqhnhBEXN0cYF2LMwkSe7Q3OqLWGKCx+DuGypJkgFOA9bBCscBb6EB2eAgg03ARsghH/CNlOfco77VH2KMKV6DoXdcJUnO4oEaztFDXxT4XJRZGbf4TNsErkVfPMMTjmJewRfWouxO0gT/qTDH3dQfdTgpAZFH3n4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAALCAYAAAAa5F88AAAABHNCSVQICAgIfAhkiAAAAO5JREFUOI3d069Lw0EcxvHX1GQxDZtglCEalhUn+CesuL9A0D/DZLGIaTBYs9jEJoigYrFvuoGIoGCR6UDcwveEL4cTDYfgAxee53583tx9jn+sGrp4RROTIV/BFZ5xhJmUEPN4QhVraOMYJfSwj2XsBqhCKpA6tnK+hD4ecBCtPUdl1EEdDH4xmtH+SyxE2R7eMRvlO9j8NBPRZBtvoyi/0H3kCwEwrxuMo4jbXD6Q8Gka2Mj5KdzhESdR4QuspgJZDEWrWMIpzmRN3JP1UBnbuJbwRmBd9n1fcIjpkFfQwoesl+ZSQvxEY38N8K2Gk+85FrKxPB4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "\n",
    "# 初始化\n",
    "x = symbols('x')\n",
    "\n",
    "# 表达式\n",
    "expr1 = sin(x)/x\n",
    "expr2 = 1/x\n",
    "display(expr1)\n",
    "\n",
    "# 求趋于某个值的极限\n",
    "r1 = limit(expr1, x, 0)\n",
    "\n",
    "# 正向趋进\n",
    "r2 = limit(expr2, x, 0, '+')\n",
    "\n",
    "# 负向趋进\n",
    "r3 = limit(expr2, x, 0, '-')\n",
    "\n",
    "display(r1)\n",
    "display(r2)\n",
    "display(r3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r1: -sin(x)\n",
      "r2: 2*x*exp(x**2)\n"
     ]
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "# 初始化\n",
    "x = symbols('x')\n",
    "\n",
    "# 表达式\n",
    "expr1 = cos(x)\n",
    "\n",
    "expr2 = exp(x**2)\n",
    "\n",
    "\n",
    "# 求导\n",
    "r1 = diff(expr1, x)\n",
    "r2 = diff(expr2, x)\n",
    "\n",
    "\n",
    "print(\"r1:\", r1)\n",
    "print(\"r2:\", r2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFUKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSxoPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA6XoX52TB9a4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA8AAAAPCAYAAAA71pVKAAAABHNCSVQICAgIfAhkiAAAAMFJREFUKJHV0D1qQkEUxfFfRAgEm1iHdEIgFuo2rKxcgFV6cQXZQzaRJrWtC/CDNHbp04iQJsSXIvNgvPGJhY0Hhgv/OefeO8OZ1Edx4ulAPQtv8IInvOM1u3vAEDNMsTw0fZA6Pwc+SXyUw1owdVKdB96r4Ht6SxNaga/xjetj4Q9scZWxBnZYRXO+9i3u/X1GkfFuavZv5TxcvncRPJXvPRSOpnaqR9eumtxM9RF3sUGpOX5wE/gQn/jCuCp8QfoFaukqtXE/L4oAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\pi$$"
      ],
      "text/plain": [
       "π"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$0.0833333333333333$$"
      ],
      "text/plain": [
       "0.0833333333333333"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "# 初始化\n",
    "x, y = symbols('x y')\n",
    "\n",
    "# 表达式\n",
    "expr1 = cos(x)\n",
    "expr2 = exp(-x)\n",
    "expr3 = exp(-x**2-y**2)\n",
    "expr4=1/3*x**3\n",
    "\n",
    "# 求不定积分\n",
    "r1 = integrate(expr1, x)\n",
    "\n",
    "# 求定积分\n",
    "r2 = integrate(expr2, (x, 0, oo))\n",
    "\n",
    "# 求多重积分\n",
    "r3 = integrate(expr3, (x, -oo, oo), (y, -oo, oo))\n",
    "\n",
    "r4=integrate(expr4,(x,0,1))\n",
    "\n",
    "display(r1)\n",
    "display(r2)\n",
    "display(r3)\n",
    "display(r4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "178.303810051593 fps\n"
     ]
    }
   ],
   "source": [
    "from time import time\n",
    "from numpy import uint8\n",
    "from numpy.random import rand\n",
    "import cv2\n",
    "\n",
    "xy=(512,512)\n",
    "Nf = 500\n",
    "\n",
    "def fpsopencv(dat):\n",
    "    tic = time()\n",
    "    for i in dat:\n",
    "        cv2.imshow('test',i)\n",
    "        cv2.waitKey(1) #integer milliseconds, 0 makes wait forever\n",
    "    cv2.destroyAllWindows()\n",
    "    return Nf / (time()-tic)\n",
    "\n",
    "\n",
    "imgs = (rand(Nf,xy[0],xy[1])*255).astype(uint8)\n",
    "fps = fpsopencv(imgs)\n",
    "\n",
    "print(fps,'fps')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 毛春"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "square=100\n",
    "circle=np.pi*5**2\n",
    "circular_sector=np.pi*100/4\n",
    "irregular=square-circular_sector\n",
    "corner=(square-circle)/4\n",
    "fusiform=square-irregular*2\n",
    "crescent=irregular-corner"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$100.0$$"
      ],
      "text/plain": [
       "100.0"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "corner*4+circle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "None"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x,y,m,n=sp.symbols('x y m n')\n",
    "row1=(2,1,0,0,corner)\n",
    "row2=(2,0,1,0,crescent) #pointless\n",
    "row3=(0,2,0,1,fusiform)\n",
    "row4=(0,0,2,1,circle)\n",
    "row5=(8,4,2,1,square) #pointlless\n",
    "row6=(2,2,1,1,circular_sector-corner)\n",
    "system=Matrix((row1,row3,row4,row6))\n",
    "result=solve_linear_system(system,x,y,m,n)\n",
    "display(result)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{\\left ( 0.25 n - 11.5873852123405, \\quad - 0.5 n + 28.5398163397448, \\quad - 0.5 n + 39.2699081698724, \\quad n\\right )\\right\\}$$"
      ],
      "text/plain": [
       "{(0.25⋅n - 11.5873852123405, -0.5⋅n + 28.5398163397448, -0.5⋅n + 39.2699081698\n",
       "724, n)}"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y,m,n=sp.symbols('x y m n')\n",
    "linsolve([2*x+y-corner\n",
    "          ,2*y+n-fusiform\n",
    "          ,2*m+n-circle\n",
    "#           ,2*x+2*y+m+n+corner-circular_sector\n",
    "         ],\n",
    "         x,y,m,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{\\left ( k\\right )\\right\\}$$"
      ],
      "text/plain": [
       "{(k,)}"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k=sp.symbols('k')\n",
    "linsolve([2*(0.25*k-12)+2*(-0.5*k+29)+1*(-0.5*k+39)+k-73],k)\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAASCAYAAADVCrdsAAAABHNCSVQICAgIfAhkiAAAAQtJREFUSInt1L0uREEYxvEfiYawPloSceLjEtTux3VQCZ2KW9BS6NyDpbAaohQfCYXVzMbsseeYOTayhad5MvO+7z9PMpmXEdUijnCHN3Swj7m/YhV4QBcn2MV5OF9iISNAY9ZpaNou3e+F+8OMEI1YK6F4g/FSbRrPeMFUQoAsVtywFfwMH6XBJ1xgEpsJIbJYcYj14FcV4OvgawkhslhxiFbwx4rB3v1sQogsVvm96jQWvJsxk8SKQ/TStQzWTKmvTlmsOEQ7eNWbrwaveudYjVmFn7/Vq7Qv+itW7oIpsIGJIbD6oPGq3fG1atu+r9pOqC0PgdWnJRzjHu+4xQHmB/TWhchl/Ws09AlCUmedySlDMwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$0.0$$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.5-1-0.5+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "ename": "LinAlgError",
     "evalue": "Singular matrix",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mLinAlgError\u001b[0m                               Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-77-3f83815093d6>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m      4\u001b[0m   [2,2,1,1]] \n\u001b[1;32m      5\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mcorner\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mfusiform\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mcircle\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mcircular_sector\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mcorner\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 6\u001b[0;31m \u001b[0mr\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlinalg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mA\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ms\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      7\u001b[0m \u001b[0mr\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/numpy/linalg/linalg.py\u001b[0m in \u001b[0;36msolve\u001b[0;34m(a, b)\u001b[0m\n\u001b[1;32m    401\u001b[0m     \u001b[0msignature\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'DD->D'\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0misComplexType\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32melse\u001b[0m \u001b[0;34m'dd->d'\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    402\u001b[0m     \u001b[0mextobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mget_linalg_error_extobj\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_raise_linalgerror_singular\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 403\u001b[0;31m     \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgufunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msignature\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msignature\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mextobj\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mextobj\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    404\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    405\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mwrap\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mastype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mresult_t\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcopy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.7/site-packages/numpy/linalg/linalg.py\u001b[0m in \u001b[0;36m_raise_linalgerror_singular\u001b[0;34m(err, flag)\u001b[0m\n\u001b[1;32m     95\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     96\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_raise_linalgerror_singular\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0merr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mflag\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 97\u001b[0;31m     \u001b[0;32mraise\u001b[0m \u001b[0mLinAlgError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"Singular matrix\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     98\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     99\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_raise_linalgerror_nonposdef\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0merr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mflag\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mLinAlgError\u001b[0m: Singular matrix"
     ]
    }
   ],
   "source": [
    "A=[[2,1,0,0,],\n",
    "  [0,2,0,1],\n",
    "  [0,0,2,1],\n",
    "  [2,2,1,1]] \n",
    "s=[corner,fusiform,circle,circular_sector-corner]\n",
    "r=np.linalg.solve(A,s)\n",
    "r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ x : 2, \\quad y : 8\\right \\}$$"
      ],
      "text/plain": [
       "{x: 2, y: 8}"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y=sp.symbols('x y')\n",
    "row1=(2,-1,-4)\n",
    "row2=(3,-1,-2)\n",
    "# row3=(1,1,10)\n",
    "system=Matrix((row1,row2))\n",
    "solve_linear_system(system,x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$100.13981633974484$$"
      ],
      "text/plain": [
       "100.13981633974484"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "5.4*4+circle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACEAAAASCAYAAADVCrdsAAAABHNCSVQICAgIfAhkiAAAAQtJREFUSInt1L0uREEYxvEfiYawPloSceLjEtTux3VQCZ2KW9BS6NyDpbAaohQfCYXVzMbsseeYOTayhad5MvO+7z9PMpmXEdUijnCHN3Swj7m/YhV4QBcn2MV5OF9iISNAY9ZpaNou3e+F+8OMEI1YK6F4g/FSbRrPeMFUQoAsVtywFfwMH6XBJ1xgEpsJIbJYcYj14FcV4OvgawkhslhxiFbwx4rB3v1sQogsVvm96jQWvJsxk8SKQ/TStQzWTKmvTlmsOEQ7eNWbrwaveudYjVmFn7/Vq7Qv+itW7oIpsIGJIbD6oPGq3fG1atu+r9pOqC0PgdWnJRzjHu+4xQHmB/TWhchl/Ws09AlCUmedySlDMwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$0.0$$"
      ],
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "crescent"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 优化"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: -1.000000\n",
      "         Iterations: 26\n",
      "         Function evaluations: 52\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([-1.5708125])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "-0.9999999998692137"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import scipy.optimize as opt\n",
    "\n",
    "t=opt.fmin(np.sin,  # Objective function to be minimized.\n",
    "              x0=0,  # Initial guess.\n",
    "              maxiter=2000,  # Maximum number of iterations to perform.\n",
    "              full_output=True)  # Return fopt, func_calls, grad_calls, and warnflag in addition to xopt.\n",
    "\n",
    "#display(t)\n",
    "xopt=t[0]\n",
    "fopt=t[1]\n",
    "display(xopt)\n",
    "display(fopt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 符号运算"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$x^{2} + 2 x + 1$$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 2⋅x + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAASCAYAAAADr20JAAAABHNCSVQICAgIfAhkiAAAAeRJREFUWIXt1s+LTWEYB/DPMFlYya9GWVpY4Q+g5EdKSWFjNafGQpFsREpNUmMxC7KYhdK1sFCKBSUrhbKYoqbYSDcLlPJjQxmMxftejjP3nl/3DAv3W6dz7/M+3+d8v89536fDAFbgEG7iBb7gEx5iDIv+nbTSqOXhMObwGtcwgSv4GOM3MLTAwvtFLQ/bsMf8rozgVSTuXzDJv5HEZ22twW3cw+lIupSJ34vxfZn4EFpx7XyVB+nPeB7+8FD23M7G+7dM/AR+4BwWp+KTGMVlnKqrtGH08tATw5gRurWry3orriXxf6ez19UbiInm33iRh66YjIQ7PdbXCtOzjaMx9y6W1BSZaN54kYd5OBYJz7E8J28i5s3hEZaWrN9O8cpcrbLCU+jqYTiHcAQX8Qzb8T4n913q9xg+lxR1AcsysU3Yi6tCY9J4WrJuB1U8gONCl2awuiD3oDDg3kTOVEVxWSSa2epVPICTkfAEKwtyd+NrLL5K2E6zWF9TLM0Yr+IBnImEaflnGjYLW/ol1sTYgci/VUNsB4n+jJfykD7joziL73ggDIUs2sKA2YjbwnfwTmGbEz4Jp4UzuiXW+Zuo4uEXxhVP1ftYh7f4gA1dCu+IuY9rik/Uf+PjynkYYID/BT8BVGirZ/G5L6sAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2 x + 2$$"
      ],
      "text/plain": [
       "2⋅x + 2"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$x^{2} + 2 x$$"
      ],
      "text/plain": [
       " 2      \n",
       "x  + 2⋅x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x,y,z,t=sp.symbols('x,y,z,t')\n",
    "f=x**2+2*x+1\n",
    "display(f)\n",
    "g=sp.diff(f,x)\n",
    "display(g)\n",
    "g1=sp.integrate(g,x)\n",
    "display(g1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{y - 1}{3}$$"
      ],
      "text/plain": [
       "y - 1\n",
       "─────\n",
       "  3  "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r=sp.solve(3*x+1-y,[x])\n",
    "f=r[0]\n",
    "f.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{3^{n} - 1}{2}$$"
      ],
      "text/plain": [
       " n    \n",
       "3  - 1\n",
       "──────\n",
       "  2   "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i,n=sp.symbols('i,n')\n",
    "sum=sp.summation(3**i,(i,0,n-1))\n",
    "sp.factor(sum)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$f{\\left (x \\right )} = C_{1} + x^{3}$$"
      ],
      "text/plain": [
       "             3\n",
       "f(x) = C₁ + x "
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f=sp.Function('f')(x)\n",
    "f1=sp.Derivative(f,x)\n",
    "sp.dsolve(f1-3*x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{g t^{2}}{2}$$"
      ],
      "text/plain": [
       "   2\n",
       "g⋅t \n",
       "────\n",
       " 2  "
      ]
     },
     "execution_count": 147,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g,t=sp.symbols('g,t')\n",
    "v=sp.Function('v')(t)\n",
    "#g=sp.Dummy('g')\n",
    "v=g*t\n",
    "v.integrate(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.subs?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# covid19"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.core.debugger import set_trace\n",
    "from IPython.core.debugger import Pdb as CorePdb\n",
    "import sys\n",
    "\n",
    "def breakpoint(condition=True):\n",
    "    \"\"\"\n",
    "    Set a breakpoint at the location the function is called if `condition == True`.\n",
    "    \"\"\"\n",
    "    if condition:\n",
    "        debugger = CorePdb()\n",
    "        frame = sys._getframe()\n",
    "        debugger.set_trace(frame.f_back)\n",
    "        return debugger"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the infected number is 2.9997\n",
      "the immunity number is 1\n",
      "the suscepteble number is 9996.0003\n",
      "----------------------------\n",
      "the infected number is 8.995500629973\n",
      "the immunity number is 3.9997\n",
      "the suscepteble number is 9987.004799370026\n",
      "----------------------------\n",
      "the infected number is 26.95143238928294\n",
      "the immunity number is 12.995200629972999\n",
      "the suscepteble number is 9960.053366980743\n",
      "----------------------------\n",
      "the infected number is 80.53131147414942\n",
      "the immunity number is 39.94663301925594\n",
      "the suscepteble number is 9879.522055506595\n",
      "----------------------------\n",
      "the infected number is 238.68326036031914\n",
      "the immunity number is 120.47794449340536\n",
      "the suscepteble number is 9640.838795146276\n",
      "----------------------------\n",
      "the infected number is 690.3320508701291\n",
      "the immunity number is 359.1612048537245\n",
      "the suscepteble number is 8950.506744276146\n",
      "----------------------------\n",
      "the infected number is 1853.6465031309226\n",
      "the immunity number is 1049.4932557238535\n",
      "the suscepteble number is 7096.860241145224\n",
      "----------------------------\n",
      "the infected number is 3946.5210507623165\n",
      "the immunity number is 2903.139758854776\n",
      "the suscepteble number is 3150.339190382908\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 3150.339190382907\n",
      "the immunity number is 6849.660809617093\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n",
      "herd immunity is reached\n",
      "the infected number is 0.0\n",
      "the immunity number is 10000.0\n",
      "the suscepteble number is 0.0\n",
      "----------------------------\n"
     ]
    }
   ],
   "source": [
    "class Model:\n",
    "    def __init__(self,r0,total):\n",
    "        self.r0=r0\n",
    "        self.total=total\n",
    "        self.infected=1\n",
    "        self.immunity=0\n",
    "    def printMe(self):\n",
    "        print(f'the infected number is {self.infected}')\n",
    "        print(f'the immunity number is {self.immunity}')\n",
    "        print(f'the suscepteble number is {self.total-self.infected-self.immunity}')\n",
    "        print('----------------------------')\n",
    "    def move_a_period(self):\n",
    "        #set_trace()\n",
    "        immunity_rate=(self.immunity+self.infected)/self.total\n",
    "        r1=self.r0*(1-immunity_rate)\n",
    "        if r1<=1:\n",
    "            print('herd immunity is reached')\n",
    "        self.immunity=self.immunity+self.infected\n",
    "        self.infected=self.infected*r1\n",
    "        if self.infected>self.total-self.immunity:\n",
    "            self.infected=self.total-self.immunity\n",
    "\n",
    "model=Model(3,10000)\n",
    "for i in range(1,20):\n",
    "    model.move_a_period()\n",
    "    #breakpoint(i==4)\n",
    "    model.printMe()"
   ]
  },
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   "source": [
    "# pandas"
   ]
  },
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    "import pandas as pd\n",
    "\n",
    "class A:\n",
    "    def __init__(self,a,b,c):\n",
    "        self.a=a\n",
    "        self.b=b\n",
    "        self.c=c\n",
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    "\n",
    "a=A(1,2,3)\n",
    "b=A(4,5,6)\n",
    "df=pd.DataFrame()\n",
    "df=df.append([a.__dict__])\n",
    "df=df.append([b.__dict__])\n",
    "df"
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   "execution_count": 71,
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   "cell_type": "code",
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