text/Chapter00.ipynb

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    "# Chapter 0: 数学編\n",
    "\n",
    "## 紹介するモジュール\n",
    "1. [math](https://docs.python.jp/3/library/math.html)\n",
    "2. [numpy](http://www.numpy.org/)\n",
    "3. [scipy](https://docs.scipy.org/doc/)\n",
    "\n",
    "## モジュールの利用\n",
    "モジュールを利用する場合，以下のように書く．\n",
    "```python\n",
    "import module # moduleの利用\n",
    "module.function() # module内にある関数functionを使う\n",
    "```\n",
    "\n",
    "毎回`module`と書くことはプログラムを長くする可能性があり不便だ．\n",
    "そのため，以下のような書き方もできる．\n",
    "```python\n",
    "from module import function # module内の関数functionのみを読み込む\n",
    "function()\n",
    "```\n",
    "この書き方の場合，`module`内の関数functionのみを読み込んでいるため，`module`内の他の関数を利用することはできない．\n",
    "\n",
    "また，モジュール名や関数名を置き換えるときに，`as`という予約語を用いる．\n",
    "例えば，次のように書くことできる\n",
    "```python\n",
    "import numpy as np # numpyというモジュールをnpと表す\n",
    "print(np.e) \n",
    "```"
   ]
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   "source": [
    "## 0-1. math \n",
    "mathモジュールは，Pythonの標準ライブラリに含まれている．\n",
    "\n",
    "mathモジュールには，例えば次のような関数がある．\n",
    "- `math.sqrt`: 平方根\n",
    "- `math.exp`: 指数関数\n",
    "- `math.sin`: sin関数"
   ]
  },
  {
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    {
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     "text": [
      "1.4142135623730951\n",
      "7.38905609893065\n",
      "0.9092974268256817\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "print(math.sqrt(2))\n",
    "print(math.exp(2))\n",
    "print(math.sin(2))"
   ]
  },
  {
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   "source": [
    "他にも次のような定数も利用できる．\n",
    "- `math.pi`\n",
    "- `math.e`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.141592653589793 2.718281828459045\n"
     ]
    }
   ],
   "source": [
    "print(math.pi, math.e)"
   ]
  },
  {
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   "source": [
    "## 0-2. Numpy\n",
    "\n",
    "### 特徴\n",
    "- Pythonで数学を扱う場合，必ずといっていいほど利用されるモジュール．\n",
    "- 行列演算をサポート\n",
    "- 通常のPythonを使うよりも数倍の速度で演算できる\n",
    "\n",
    "### よく利用されるメソッド\n",
    "\n",
    "#### numpy配列\n",
    "Pythonのデータ構造にあるリストとは別に，numpyを利用した配列がある．\n",
    "\n",
    "1次元のときはベクトル，2次元では行列，3次元以上はテンソルに対応する．"
   ]
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    {
     "name": "stdout",
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     "text": [
      "[[1 2]]\n",
      "[[1 3]\n",
      " [6 4]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "vec = np.array([[1, 2]]) # 1次元の配列をNumpy配列(ベクトル)にする\n",
    "matrix = np.array([[1, 3], [6, 4]]) # 2次元の配列をNumpy配列(行列)にする\n",
    "print(vec)\n",
    "print(matrix)"
   ]
  },
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   "source": [
    "すべての要素が1や0になる特殊なメソッドがある．"
   ]
  },
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   "execution_count": 4,
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     "name": "stdout",
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     "text": [
      "[ 1.  1.  1.]\n",
      "[ 0.  0.  0.]\n",
      "[[ 1.  1.  1.]\n",
      " [ 1.  1.  1.]]\n",
      "[[ 0.  0.  0.]\n",
      " [ 0.  0.  0.]]\n"
     ]
    }
   ],
   "source": [
    "print(np.ones(3)) # 3次元の1ベクトル\n",
    "print(np.zeros(3)) # 3次元の0ベクトル\n",
    "print(np.ones((2, 3))) #2x3次元の1行列\n",
    "print(np.zeros((2, 3))) #2x3次元の0行列"
   ]
  },
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   "cell_type": "code",
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   "metadata": {},
   "outputs": [
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     "text": [
      "dtype int64\n",
      "shape (2, 2)\n"
     ]
    }
   ],
   "source": [
    "print(\"dtype\", matrix.dtype) # 型を表示する\n",
    "print(\"shape\", matrix.shape) # サイズ(次元)"
   ]
  },
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   "attachments": {},
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   "metadata": {},
   "source": [
    "#### 行列の要素積や行列積の計算\n",
    "注意点\n",
    "- np.arrayとnp.matrixでは挙動が異なる\n",
    "    - np.arrayでは`*` で要素積になるが，np.matrixでは行列積になる"
   ]
  },
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     "name": "stdout",
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     "text": [
      "*演算:\n",
      " [[ 1  9]\n",
      " [36 16]]\n",
      "np.dot:\n",
      " [[19 15]\n",
      " [30 34]]\n",
      "*演算:\n",
      " [[19 15]\n",
      " [30 34]]\n",
      "np.dot:\n",
      " [[19 15]\n",
      " [30 34]]\n"
     ]
    }
   ],
   "source": [
    "A = np.array([[1, 3], [6, 4]])\n",
    "print(\"*演算:\\n\", A*A) # 行列Aどうしの要素積\n",
    "print(\"np.dot:\\n\", np.dot(A, A)) # 行列Aどうしの行列積\n",
    "\n",
    "B = np.matrix([[1, 3], [6, 4]])\n",
    "print(\"*演算:\\n\", B*B) # 行列Bどうしの行列積\n",
    "print(\"np.dot:\\n\", np.dot(B, B)) # 行列Bどうしの行列積"
   ]
  },
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   "source": [
    "### よく使うサブモジュール，関数\n",
    "\n",
    "- `np.linalg`: 線形代数モジュール  \n",
    "    - `np.linalg.inv`: 逆行列を求める関数\n",
    "    - `np.linalg.eig`: 固有値，固有ベクトルを求める関数\n",
    "    \n",
    "- `np.random`: 乱数モジュール"
   ]
  },
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   "metadata": {},
   "source": [
    "## 0-3. Scipy\n",
    "ScipyモジュールはNumpyの拡張モジュールであり，よく利用されるモジュール．  \n",
    "科学技術に関わる計算アルゴリズムの多くが実装されているので，自作するよりもここを調べるといい．\n",
    "\n",
    "### 特徴\n",
    "- Numpyよりも高度な数学関数が用意されている．\n",
    "- 最適化や高速フーリエ変換なども扱う\n",
    "\n",
    "### よく使うサブモジュール\n",
    "- `scipy.linalg`: 線形代数\n",
    "- `scipy.optimize`: 最適化\n",
    "- `scipy.sparse`: 疎行列に対する処理\n",
    "- `scipy.stats`: 統計処理\n",
    "\n",
    "### 例題 2-3-1. 連続最適化"
   ]
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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105b20668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.optimize import fmin_bfgs\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "def f(x):\n",
    "    return x ** 2 + 17 * np.sin(x)\n",
    "\n",
    "x = np.arange(-10, 10, 0.1)\n",
    "plt.plot(x, f(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimization terminated successfully.\n",
      "         Current function value: -14.792859\n",
      "         Iterations: 6\n",
      "         Function evaluations: 21\n",
      "         Gradient evaluations: 7\n"
     ]
    }
   ],
   "source": [
    "x_star = fmin_bfgs(f, 0) # 関数f(x)の最適化(最小値をとるxを求める)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10312ca58>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.xlim(-10, 10)\n",
    "plt.plot(x, f(x))\n",
    "plt.scatter(x_star, f(x_star), c='red');"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## [練習問題](../exercise/questions.md#chapter-0)"
   ]
  }
 ],
 "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": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}