ストラング線形代数イントロダクション 1.2 長さと内積 練習問題32 単位ベクトル同士の内積平均

Strang-01-02-32-random-unit-vector.ipynb

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   "source": [
    "# ストラング線形代数イントロダクション 1.2 長さと内積 問題32\n",
    "\n",
    "- 単位ベクトルを1個生成（$\\mathbf{u}$)\n",
    "- 単位ベクトルをランダムに30個生成（$\\mathbf{U}$）\n",
    "- $ |\\mathbf{u} \\cdot \\mathbf{U}| $の平均を求める"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 単位ベクトルを1個生成"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "単位ベクトル u =\n",
      "[[ 0.87148942]\n",
      " [ 0.47712807]\n",
      " [-0.11337984]]\n",
      "uの長さ =\n",
      "0.9999999999999999\n"
     ]
    }
   ],
   "source": [
    "v = np.random.randn(3, 1)\n",
    "u = v / np.linalg.norm(v)\n",
    "print('単位ベクトル u =')\n",
    "print(u)\n",
    "print('uの長さ =')\n",
    "print(np.linalg.norm(u))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 単位ベクトルをランダムにN個生成"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Vの最初の列ベクトル =\n",
      "[-2.15380297 -0.15116081 -0.37320251]\n",
      "Uの最初の列ベクトル =\n",
      "[-0.98296999 -0.06898799 -0.17032517]\n"
     ]
    }
   ],
   "source": [
    "# 個数\n",
    "N = 30\n",
    "\n",
    "# 行列の各列が1つのベクトル\n",
    "V = np.random.randn(3, N) # 3xN行列として\n",
    "print('Vの最初の列ベクトル =')\n",
    "print(V[:,0])\n",
    "V_lengths = np.apply_along_axis(np.linalg.norm, 0, V) # 各列ベクトルの長さ\n",
    "U = V / V_lengths\n",
    "print('Uの最初の列ベクトル =')\n",
    "print(U[:,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 内積の絶対値の平均を求める"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "各列ベクトルとの内積 =\n",
      "[[-0.87025261 -0.93521217 -0.92616723 -0.20324186 -0.47228492 -0.18479246\n",
      "  -0.28899362  0.30278933  0.44196373 -0.01457608  0.94871511  0.16772202\n",
      "   0.04550446  0.18094622  0.33200627 -0.25600423 -0.15252904 -0.44812875\n",
      "  -0.5997203  -0.14956317 -0.12887674 -0.69469828 -0.57043962  0.66246135\n",
      "  -0.78135679 -0.01099612  0.44186798  0.37187519 -0.253802   -0.97761432]]\n",
      "内積の絶対値の平均 = 0.427170065350003\n"
     ]
    }
   ],
   "source": [
    "# uとUの各列ベクトルとの内積\n",
    "#uU = np.apply_along_axis(lambda x: u.dot(), 0, U)\n",
    "uU = u.T.dot(U)\n",
    "print('各列ベクトルとの内積 =')\n",
    "print(uU)\n",
    "print('内積の絶対値の平均 =', np.abs(uU).mean())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.6366197723675814"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 本によれば、内積の絶対値の平均は以下の値になるはず\n",
    "2 / np.pi"
   ]
  }
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