article/library/sympy/base/base_nb.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## sympy\n",
    "\n",
    "これまで数値計算はすべてmathematicaを用いてやってきましたが、webシステム開発者としてpythonに移行しようと思います。\n",
    "\n",
    "まずは最初の基本として、\n",
    "\n",
    "- http://www.turbare.net/transl/scipy-lecture-notes/packages/sympy.html\n",
    "\n",
    "に記載された問題を実際に手を動かしてやって行こうと思います。\n",
    "\n",
    "### github\n",
    "- jupyter notebook形式のファイルは[こちら](https://github.com/hiroshi0530/wa-src/blob/master/article/library/sympy/base/base_nb.ipynb)\n",
    "\n",
    "### google colaboratory\n",
    "- google colaboratory で実行する場合は[こちら](https://colab.research.google.com/github/hiroshi0530/wa-src/blob/master/article/library/sympy/base/base_nb.ipynb)\n",
    "\n",
    "### 筆者の環境\n",
    "筆者のOSはmacOSです。LinuxやUnixのコマンドとはオプションが異なります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ProductName:\tMac OS X\r\n",
      "ProductVersion:\t10.14.6\r\n",
      "BuildVersion:\t18G6020\r\n"
     ]
    }
   ],
   "source": [
    "!sw_vers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python 3.7.3\r\n"
     ]
    }
   ],
   "source": [
    "!python -V"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "基本的なライブラリをインポートしそのバージョンを確認しておきます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "matplotlib version : 3.0.3\n",
      "scipy version : 1.4.1\n",
      "numpy version : 1.16.2\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy\n",
    "import numpy as np\n",
    "\n",
    "print('matplotlib version :', matplotlib.__version__)\n",
    "print('scipy version :', scipy.__version__)\n",
    "print('numpy version :', np.__version__)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## sympyの3つの数値型\n",
    "\n",
    "sympyは\n",
    "\n",
    "- Real\n",
    "- Rational\n",
    "- Integer\n",
    "\n",
    "の三つの型を持つようです。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 分数の扱い"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "a = Rational(1,2)\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 円周率の扱い"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.14159265358979"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*pi"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi * 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "pi**2"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi ** 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 自然対数の扱い"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(2)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(1) ** 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.38905609893065"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(exp(1) ** 2).evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 無限大"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "oo > 999"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "oo"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "oo + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### evalfで桁数の指定\n",
    "\n",
    "引数に桁数を入れるようです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420199"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi.evalf(1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 対数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "log(10)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.250000000000000"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log(2,2**4).evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.71828182845905"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## シンボルの定義\n",
    "\n",
    "代数的な操作を可能にするために、$xや$yなどの変数を宣言します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol('x')\n",
    "y = Symbol('y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*x + y**2"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + y ** 2 + x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x + y)**3"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + y) ** 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 代数操作\n",
    "\n",
    "expandやsimplifyはmathematicaと同じのようで助かります。"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 展開"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x**3 + 3*x**2*y + 3*x*y**2 + y**3"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x + y)**3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 簡易化"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "y + 1"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify((x + x*y) / x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 因数分解"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x + y)**3"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x**3 + 3*x**2*y + 3*x*y**2 + y**3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 微積分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 極限"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sin(x)/ x, x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "oo"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(x, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1 / x, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(x**x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 微分"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*sin(x)*cos(x)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x) ** 2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2*cos(2*x)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(2 * x) , x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tan(x)**2 + 1"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(tan(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 微分が正しいかの確認\n",
    "極限をとり、微分が正しいかどうかチェックできます。微分の定義に従って計算させるだけです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tan(x)**2 + 1"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit((tan(x+y) - tan(x))/y, y, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "高階微分も可能です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x), x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2階微分で元の値にマイナスをかけた値になっている事がわかります。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x), x, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x), x, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(sin(x), x, 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 級数展開\n",
    "Taylor展開も可能です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 - x**2/2 + x**4/24 + O(x**6)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(cos(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x - x**3/6 + x**5/120 + O(x**6)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(sin(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "第三引数で展開する次数を指定出来るかと思いやってみました。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(6) + (x - 6)*exp(6) + (x - 6)**2*exp(6)/2 + (x - 6)**3*exp(6)/6 + (x - 6)**4*exp(6)/24 + (x - 6)**5*exp(6)/120 + O((x - 6)**6, (x, 6))"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x, 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "違うみたいで、第三引数に数値計算する際の中心の数で、第四引数で展開させる次数のようです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x, 0, 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 積分\n",
    "微分が出来れば、積分ももちろん対応しています。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x**4/4"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**3,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(-sin(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "exp(x)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x*log(x) - x"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(log(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(pi)*erf(x)/2"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "積分区間も指定することが出来ます"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2/3"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(x**2, (x, -1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(sin(x), (x, 0, pi/2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "範囲が無限大の広義積分も可能です。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x), (x, 0, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ガウス積分をやってみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(pi)"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2), (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sqrt(3)*sqrt(pi)"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2 / 3), (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 方程式を解く\n",
    "代数方程式を解くことも可能です。ほぼmathematicaと同じインターフェースです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, -1/2 - sqrt(3)*I/2, -1/2 + sqrt(3)*I/2]"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**3 - 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1, 1, -I, I]"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**4 - 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "多変数の連立方程式も対応しています。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: -3, y: 1}"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + 5*y - 2, -3*x + 6*y - 15], [x, y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "オイラーの式も計算してくれるようです。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[I*pi]"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(exp(x) + 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  行列演算\n",
    "\n",
    "行列演算はnumpyでずっとやっているのでおそらく行列に関しては利用していませんが、一応ここでは手を動かして実行してみます。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Matrix\n",
    "\n",
    "a = Matrix([[1,0], [0,1]])\n",
    "b = Matrix([[1,1], [1,1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[1, 1],\n",
       "[1, 1]])"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[2, 1],\n",
       "[1, 2]])"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a + b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[1, 1],\n",
       "[1, 1]])"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 微分方程式\n",
    "\n",
    "常微分方程式も解く事が可能です。dsolveを利用します。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f(x)"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f, g = symbols('f g', cls=Function)\n",
    "f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f(x) + Derivative(f(x), (x, 2))"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x).diff(x,x) + f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Eq(f(x), C1*sin(x) + C2*cos(x))"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(f(x).diff(x,x) + f(x), f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "実際に数値計算に利用されているのがよくわかる高機能ぶりでした。\n",
    "\n",
    "もちろん有料であるmathematicaよりも劣る部分もあると思いますが、積極的にmathematicaからの移行を進めていきます。すごいなぁ〜"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 参考ページ\n",
    "\n",
    "こちらのページを参考にしました。\n",
    "\n",
    "- http://www.turbare.net/transl/scipy-lecture-notes/packages/sympy.html"
   ]
  }
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