doc/source/notebooks/08_02_Quantum_inspired_classical_algorithm_Part1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "量子ソフトウェア勉強会:「量子インスパイアード古典アルゴリズム」ハンズオン資料 2  \n",
    "2023年9月20日(水)\n",
    "高比良宗一（名城大学情報工学部情報工学科）\n",
    "$$\n",
    "\\newcommand{\\norm}[1]{\\lVert#1\\rVert}\n",
    "\\renewcommand{\\top}{\\mathsf{T}}\n",
    "\\newcommand{\\SQ}{\\mathrm{SQ}}\n",
    "$$\n",
    "# $x = A^+b$に関する量子インスパイアード古典アルゴリズム Part 1\n",
    "### (行列$C$の特異値分解まで)\n",
    "\n",
    "このノートブックでは，量子インスパイアード古典アルゴリズムにおける，$A$から構成される行列$C$を得るまでの部分，及びその特異値分解について実装します．そして$C$の左特異ベクトル$w^{(\\ell)}$と特異値$\\tilde{\\sigma}$によって，$A$の左右の特異ベクトルと特異値が近似される様子を確認してみます．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 行列$A$を定義する\n",
    "低ランク行列$A$を生成する関数を定義します．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "import cmath, math\n",
    "import time\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import scipy.linalg\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def low_rank_matrix(m, n, rank, cond=10):\n",
    "\n",
    "    \"\"\"\n",
    "    ランクが`rank`で，条件数が`cond`であるような m x n の実行列を生成する．\n",
    "    ここで条件数とは最大特異値/最小特異値である．したがってrank=1のときは条件数の値は1になる．\n",
    "    \n",
    "    \"\"\"\n",
    "    \n",
    "    U, _  = np.linalg.qr(np.random.rand(m, m))\n",
    "    Vh, _ = np.linalg.qr(np.random.rand(n, n))\n",
    "    \n",
    "    if rank==1:\n",
    "        diag = np.array([cond])\n",
    "        \n",
    "    elif rank==2: \n",
    "        diag = np.array([1, 1/cond])\n",
    "    else:\n",
    "        diag = [random.uniform(1/cond, 1) for i in range(rank)]\n",
    "        diag[1] = 1\n",
    "        diag[2] = 1/cond\n",
    "    \n",
    "    return U[:,:rank] @ np.diag(diag) @ Vh[:rank,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Aのランク = 2\n",
      "Aのフロべニウスノルム = 1.0077822185373184\n",
      "Aの2ノルム = 0.9999999999999997\n",
      "Aの特異値\n",
      " [1.00000000e+00 1.25000000e-01 5.19213071e-16 4.79118019e-16\n",
      " 4.71713714e-16 4.58222417e-16 4.18376202e-16 3.81834822e-16\n",
      " 3.54408030e-16 3.47628970e-16 3.21416227e-16 3.06267277e-16\n",
      " 2.94596201e-16 2.77563960e-16 2.60622934e-16 2.58545313e-16\n",
      " 2.29930845e-16 2.29601626e-16 2.19035211e-16 2.08345819e-16\n",
      " 2.02803805e-16 1.74125391e-16 1.59430441e-16 1.47403420e-16\n",
      " 1.44858870e-16 1.29917459e-16 1.17028191e-16 1.00297735e-16\n",
      " 9.37942832e-17 8.61352464e-17 8.54481002e-17 8.25804231e-17\n",
      " 8.02026027e-17 7.59935124e-17 6.71003385e-17 6.55470690e-17\n",
      " 6.29616202e-17 5.99590794e-17 5.90842981e-17 5.83481073e-17\n",
      " 5.42559429e-17 5.32616988e-17 5.20634581e-17 4.88518527e-17\n",
      " 4.77845279e-17 4.71140163e-17 4.53903152e-17 4.34603235e-17\n",
      " 4.18865006e-17 4.11653036e-17 4.03700291e-17 3.93219745e-17\n",
      " 3.78008155e-17 3.61795232e-17 3.48760210e-17 3.44060987e-17\n",
      " 3.21714941e-17 3.18224187e-17 3.15013830e-17 3.09899747e-17\n",
      " 3.05030105e-17 2.94799811e-17 2.87547844e-17 2.82762851e-17\n",
      " 2.71794738e-17 2.66188204e-17 2.65593766e-17 2.58955778e-17\n",
      " 2.57313346e-17 2.50097543e-17 2.48568056e-17 2.42186036e-17\n",
      " 2.40435255e-17 2.36537034e-17 2.35260358e-17 2.30242899e-17\n",
      " 2.23809209e-17 2.17939005e-17 2.16015771e-17 2.14806951e-17\n",
      " 2.08692603e-17 2.05871356e-17 2.04536573e-17 2.03399627e-17\n",
      " 1.95932932e-17 1.94572013e-17 1.92152275e-17 1.89255407e-17\n",
      " 1.86323067e-17 1.85739870e-17 1.82830250e-17 1.77793150e-17\n",
      " 1.77569421e-17 1.76039948e-17 1.69915772e-17 1.67849530e-17\n",
      " 1.65287465e-17 1.64180495e-17 1.62054369e-17 1.58363436e-17\n",
      " 1.57454327e-17 1.55593677e-17 1.54212713e-17 1.51981088e-17\n",
      " 1.49675088e-17 1.46895809e-17 1.44847141e-17 1.43737005e-17\n",
      " 1.41737181e-17 1.40435267e-17 1.38353451e-17 1.37652725e-17\n",
      " 1.35983264e-17 1.31958979e-17 1.30107535e-17 1.29555106e-17\n",
      " 1.28594865e-17 1.25156029e-17 1.24675914e-17 1.21872844e-17\n",
      " 1.21783613e-17 1.19813660e-17 1.19021017e-17 1.16668045e-17\n",
      " 1.14976212e-17 1.13525816e-17 1.13306028e-17 1.11254143e-17\n",
      " 1.11109004e-17 1.07572074e-17 1.07482652e-17 1.06404959e-17\n",
      " 1.04345157e-17 1.02335277e-17 1.02200967e-17 1.00440568e-17\n",
      " 9.89968573e-18 9.78431182e-18 9.67081664e-18 9.48908699e-18\n",
      " 9.36944023e-18 9.24094514e-18 9.11871447e-18 8.93315308e-18\n",
      " 8.83238984e-18 8.80597694e-18 8.61609837e-18 8.55448812e-18\n",
      " 8.46266622e-18 8.29933641e-18 8.19595843e-18 8.06216710e-18\n",
      " 7.94296967e-18 7.81537940e-18 7.66151453e-18 7.50469568e-18\n",
      " 7.44760795e-18 7.35429286e-18 7.27895009e-18 7.20166496e-18\n",
      " 7.06097760e-18 6.88315576e-18 6.77897885e-18 6.76091588e-18\n",
      " 6.62876084e-18 6.56428001e-18 6.44513690e-18 6.41760489e-18\n",
      " 6.26344653e-18 6.13970093e-18 6.06707466e-18 5.99930842e-18\n",
      " 5.91770843e-18 5.83532236e-18 5.77706755e-18 5.71123285e-18\n",
      " 5.62476419e-18 5.61991301e-18 5.57047856e-18 5.52333805e-18\n",
      " 5.37102942e-18 5.35480686e-18 5.25387834e-18 5.10437091e-18\n",
      " 5.06835214e-18 4.99402028e-18 4.81752983e-18 4.77295842e-18\n",
      " 4.71640294e-18 4.67109667e-18 4.62029149e-18 4.55662552e-18\n",
      " 4.48249051e-18 4.44125242e-18 4.41581291e-18 4.28132619e-18\n",
      " 4.18357302e-18 4.09950584e-18 4.03595753e-18 3.98391045e-18\n",
      " 3.95288911e-18 3.91938640e-18 3.85110454e-18 3.80333119e-18\n",
      " 3.77776324e-18 3.72927218e-18 3.64924670e-18 3.60160992e-18\n",
      " 3.52530987e-18 3.47131489e-18 3.43039738e-18 3.42287556e-18\n",
      " 3.36699887e-18 3.32916140e-18 3.23370117e-18 3.18840443e-18\n",
      " 3.11620071e-18 3.09894801e-18 3.06071617e-18 3.03852313e-18\n",
      " 2.99294541e-18 2.95033138e-18 2.89146352e-18 2.83005899e-18\n",
      " 2.78690702e-18 2.75055694e-18 2.70925601e-18 2.63355929e-18\n",
      " 2.60526269e-18 2.55233348e-18 2.52277879e-18 2.47486980e-18\n",
      " 2.41134561e-18 2.39090624e-18 2.32997985e-18 2.29931649e-18\n",
      " 2.28435486e-18 2.24070395e-18 2.21644341e-18 2.18247479e-18\n",
      " 2.13062578e-18 2.04035556e-18 2.02726658e-18 2.00668714e-18\n",
      " 1.96523986e-18 1.94403667e-18 1.89506717e-18 1.86914043e-18\n",
      " 1.82731690e-18 1.80884057e-18 1.79533469e-18 1.74379845e-18\n",
      " 1.72882613e-18 1.68547459e-18 1.65578022e-18 1.62886682e-18\n",
      " 1.59356446e-18 1.55032771e-18 1.49730237e-18 1.47917764e-18\n",
      " 1.40963443e-18 1.39897517e-18 1.35584901e-18 1.32530291e-18\n",
      " 1.30014849e-18 1.27567866e-18 1.24250612e-18 1.18449954e-18\n",
      " 1.17514589e-18 1.17329050e-18 1.15475950e-18 1.11678209e-18\n",
      " 1.09808184e-18 1.08287195e-18 1.04762457e-18 9.99933617e-19\n",
      " 9.47765773e-19 9.33397107e-19 8.83519416e-19 8.66250827e-19\n",
      " 8.50368417e-19 8.04423706e-19 7.81105493e-19 7.78676724e-19\n",
      " 7.37535138e-19 7.29599221e-19 6.73237315e-19 6.63651314e-19\n",
      " 6.43575106e-19 6.29562939e-19 5.79981955e-19 5.55028008e-19\n",
      " 5.30057187e-19 4.82473887e-19 4.74345389e-19 4.05071912e-19\n",
      " 3.90661947e-19 3.26599040e-19 2.91767554e-19 2.76088986e-19]\n",
      "Aの条件数\n",
      " 7.999999999999996\n"
     ]
    }
   ],
   "source": [
    "m, n = 400, 300\n",
    "A = low_rank_matrix(m, n, 2, 8)\n",
    "\n",
    "rankA = np.linalg.matrix_rank(A)\n",
    "normF = np.linalg.norm(A)\n",
    "norm2 = np.linalg.norm(A, 2) \n",
    "svdvals = sp.linalg.svdvals(A)\n",
    "condnumber = max(svdvals[:rankA])/min(svdvals[:rankA])\n",
    "\n",
    "print(\"Aのランク =\", rankA)\n",
    "print(\"Aのフロべニウスノルム =\", normF)\n",
    "print(\"Aの2ノルム =\", norm2)\n",
    "print(\"Aの特異値\\n\", svdvals)\n",
    "print(\"Aの条件数\\n\", condnumber)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "行列$A$の特異値分解の結果を保存しておきます．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "exact_U, exact_S, exact_Vh = np.linalg.svd(A)\n",
    "exact_V = exact_Vh.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $\\mathrm{SQ}(A)$を定義する．\n",
    "`quantum_inspired.py`に`1_データ構造.ipynb`で実装した`SegmentTree`, `VectorBasedDataStructure`と`MatrixBasedDataStructure`を記述しています．ここから`MatrixBasedDataStructure`をインポートし，それを用いて行列に基づくデータ構造$\\SQ(A)$を用意します．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "from quantum_inspired import MatrixBasedDataStructure\n",
    "SQA = MatrixBasedDataStructure(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(209, 138, 0.0017541011869238918, 0.06069818774819866, 1.0077822185373182)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SQA.sample1(), SQA.sample2(0), SQA.query(0, 0), SQA.norm(0), SQA.normF()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 1. 行添字のサンプリング: $R^\\mathsf{T} R \\approx A^\\mathsf{T}A$を満たすような$R$を考える．\n",
    "## 行添字をサンプルする関数の実装\n",
    "Step 1の「確率$\\frac{\\norm{A_{i,\\ast}}^2 }{\\norm{A}_\\mathrm{F}^2}$行添字$i$を$r$回サンプリングする」という部分について実装します．これは，$\\mathrm{SQ}(A)$の$\\mathrm{Sample1}()$を$r$回用いるだけなので，その通りに書きます．ここでの行添字$i_0,\\dots,i_{r-1}$は行列$R$を表します．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Step 1\n",
    "# rはサンプル数\n",
    "def sample_rows(SQA, r): \n",
    "    row_indices = []\n",
    "    for s in range(r):\n",
    "        i_s = SQA.sample1()\n",
    "        row_indices.append(i_s)    \n",
    "    return row_indices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = 10\n",
    "row_indices = sample_rows(SQA, r)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "サンプルされた行添字:\n",
      "[i_0, i_1, ..., i_r-1] =  [238, 48, 264, 8, 265, 55, 114, 131, 238, 289]\n"
     ]
    }
   ],
   "source": [
    "print(\"サンプルされた行添字:\\n[i_0, i_1, ..., i_r-1] = \", row_indices)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## $A^\\top A \\approx R^\\top R$の確認\n",
    "量子インスパイアード古典アルゴリズムでは$r\\times n$の行列$R$を実際には構成しません．しかし，サンプルされた行添字が表す行列$R$について，$A^\\mathsf{T}A \\approx R^\\mathsf{T}R$であるかどうか気になると思います．そこで，行列$R$をサンプルされた行添字から構成する関数`construct_R`を以下のセルで定義し，$A^\\top A$と$R^\\top R$を直接計算して$A^\\top A \\approx R^\\top R$であることを確認してみます．`row_indices`はサンプルされた行添字です．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "# サンプルされた行添字からRを構成する\n",
    "# 確認用\n",
    "def construct_R(SQA, A, row_indices):\n",
    "    r = len(row_indices)\n",
    "    n = A.shape[1]\n",
    "    R = np.zeros((r, n))\n",
    "    for s in range(r):\n",
    "        i_s = row_indices[s]\n",
    "        R[s,:] = (SQA.normF()/math.sqrt(r)) * A[i_s,:]/SQA.norm(i_s)\n",
    "    return R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "この関数`construct_R`を使って行列$R$を作り，$A^\\top A \\approx R^\\top R$であることを確認してみます．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "誤差 0.02244468398310593\n"
     ]
    }
   ],
   "source": [
    "r = 10\n",
    "row_indices = sample_rows(SQA, r)\n",
    "R = construct_R(SQA, A, row_indices)\n",
    "\n",
    "error = np.linalg.norm(A.T @ A - R.T @ R)\n",
    "print(\"誤差\", error)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "さらに，$\\mathbb{E}\\left[ \\lVert R^\\top R - A^\\top A \\rVert_\\mathrm{F}^2  \\right] \\le \\frac{\\lVert A \\rVert_\\mathrm{F}^4}{r}$であることを確認します．具体的には，$r$を増やしてみて，誤差$\\lVert R^\\top R - A^\\top A \\rVert_\\mathrm{F}^2$が減少することを見てみます．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "r=1 0.011036088608588065 1.031494140624999\n",
      "r=2 0.012900277652816452 0.5157470703124994\n",
      "r=4 0.004301352915359831 0.2578735351562497\n",
      "r=8 0.002086925152078077 0.12893676757812486\n",
      "r=16 0.002153392048352806 0.06446838378906243\n",
      "r=32 0.0009459364084887019 0.032234191894531215\n",
      "r=64 0.0002901357148178583 0.016117095947265608\n",
      "r=128 0.00025792208961707787 0.008058547973632804\n",
      "r=256 0.00019129401249491832 0.004029273986816402\n",
      "r=512 8.136658623254179e-05 0.002014636993408201\n",
      "r=1024 4.0273416560124284e-05 0.0010073184967041005\n",
      "r=2048 1.6310662494746377e-05 0.0005036592483520502\n"
     ]
    }
   ],
   "source": [
    "sample_size = 10\n",
    "\n",
    "errors = []\n",
    "\n",
    "rs = [2**k for k in range(12)]\n",
    "for r in rs:\n",
    "    mean = 0\n",
    "    for _ in range(sample_size):\n",
    "        row_indices = sample_rows(SQA, r)\n",
    "        R = construct_R(SQA, A, row_indices)\n",
    "        \n",
    "        mean += np.linalg.norm(R.T @ R - A.T @ A)**2  \n",
    "        \n",
    "    mean = mean/sample_size\n",
    "    errors.append(mean)\n",
    "\n",
    "    print(f\"r={r}\", mean, np.linalg.norm(A)**4/r)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 6))\n",
    "plt.plot(rs, errors, label=\"$(1/10)\\sum_i^{10} || R^T R - A^T A ||_F^2$\")\n",
    "plt.plot(rs, [np.linalg.norm(A)**4/r for r in rs], label=\"upper bound\")\n",
    "plt.xlabel(\"r (sample size of row)\")\n",
    "plt.ylabel(\"E[||R^T R - A^T A||_F]\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "なお，関数`construct_R`はただの確認用なので，実際には使われないことに注意してください．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.022269051271467524,\n",
       " 0.022269051271467524,\n",
       " 0.02226905127146753,\n",
       " 1.0077822185373182,\n",
       " 1.0077822185373184)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.linalg.norm(R[0,:]), np.linalg.norm(R[1,:]), np.linalg.norm(A)/math.sqrt(r), np.linalg.norm(R), np.linalg.norm(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 2. 列のサンプリング：$CC^\\top \\approx RR^\\top$を満たすような$C$を構成する．\n",
    "次に，Step2について実装します．\n",
    "## 列添字をサンプルする関数の実装\n",
    "Step 2の「等確率で${0,1,\\dots,r-1}$から数字をサンプリングし結果を$s$とする．そして確率$\\frac{R_{s,j}^2}{\\norm{R_{s,\\ast}}^2} = \\frac{A_{i_s,j}^2}{\\norm{A_{i_s,\\ast}}^2}$で列添字をサンプリングし，結果を$j_t$とする．この操作を$c$回行い，サンプリングされた列添字を$j_0,\\dots,j_{c-1}$とする」の部分について実装します．\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "def sample_cols(SQA, row_indices, c):\n",
    "    r = len(row_indices)\n",
    "    col_indices = []\n",
    "    for t in range(c):\n",
    "        s   = np.random.randint(r)\n",
    "        i_s = row_indices[s]\n",
    "        j_t = SQA.sample2(i_s)\n",
    "        col_indices.append(j_t)\n",
    "        \n",
    "    return col_indices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "サンプルされた列添字:\n",
      "[j_0, i_1, ..., j_c-1] =  [12, 290, 76, 146, 2, 287, 150, 48, 182, 114, 182, 194, 122, 247, 243, 47, 76, 41, 198, 206]\n"
     ]
    }
   ],
   "source": [
    "r = 10\n",
    "row_indices = sample_rows(SQA, r)\n",
    "\n",
    "c = 20\n",
    "col_indices = sample_cols(SQA, row_indices, c)\n",
    "\n",
    "print(\"サンプルされた列添字:\\n[j_0, i_1, ..., j_c-1] = \", col_indices)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## $r \\times c$の行列$C$を構成する関数の実装\n",
    "Step2の「$r \\times c$の行列$C$を第$t$列が\n",
    "$\\frac{\\norm{R}_\\mathrm{F}}{\\sqrt{c}} \\frac{R_{\\ast,j_t}}{\\norm{R_{\\ast,j_t}}} = \\frac{\\norm{A}_\\mathrm{F}}{\\sqrt{c}} \\frac{R_{\\ast,j_t}}{\\norm{R_{\\ast,j_t}}}$」となるように構成する」という部分について実装します．\n",
    "$R_{\\ast,j_t}$について注目します．行列$R$の$s$行目は$\\frac{\\norm{A}_\\mathrm{F}}{\\sqrt{r}} \\frac{A_{i_s,\\ast}}{\\norm{A_{i_s,\\ast}}}$であるので，\n",
    "行列$R$の第$j_t$列$R_{\\ast,j_t}$は\n",
    "$$\n",
    " R_{\\ast, j_t} = \\frac{\\norm{A}_\\mathrm{F}}{\\sqrt{r}} \n",
    " \\left(\n",
    " \\frac{A_{i_0,\\ast}}{\\norm{A_{i_0,j_t}}}, \n",
    " \\frac{A_{i_1,\\ast}}{\\norm{A_{i_1,j_t}}},\\dots,\n",
    "  \\frac{A_{i_{r-1},\\ast}}{\\norm{A_{i_{r-1},j_t}}}, \n",
    "\\right)^\\top\n",
    "$$\n",
    "となります．このことを踏まえると，行列$C$を構成する関数は次のようになります．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "def construct_C(SQA, row_indices, col_indices):\n",
    "    normAF = SQA.normF()\n",
    "    r, c = len(row_indices), len(col_indices)\n",
    "\n",
    "    # 行列Cを構成する\n",
    "    C = np.zeros((r, c))\n",
    "    for t in range(c):\n",
    "        \n",
    "        # Rのj_t列を構成する\n",
    "        R_jt = np.zeros(r)        \n",
    "        j_t = col_indices[t]\n",
    "        for s in range(r):\n",
    "            i_s = row_indices[s]\n",
    "            R_jt[s] = (normAF/(math.sqrt(r) * SQA.norm(i_s))) * SQA.query(i_s, j_t)\n",
    "            \n",
    "        # Cのt列を構成\n",
    "        C[:, t] = (normAF/math.sqrt(c)) * R_jt/np.linalg.norm(R_jt)\n",
    "        \n",
    "    return C"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##  $CC^\\top \\approx RR^\\top$の確認\n",
    "$CC^\\top \\approx RR^\\top$であることを前述と時と同様にして確認してみます．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, c = 10, 20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "誤差 0.12079629623014908\n"
     ]
    }
   ],
   "source": [
    "row_indices = sample_rows(SQA, r)\n",
    "col_indices = sample_cols(SQA, row_indices, c)\n",
    "\n",
    "R = construct_R(SQA, A, row_indices)\n",
    "C = construct_C(SQA, row_indices, col_indices)\n",
    "\n",
    "error = np.linalg.norm(R @ R.T - C @ C.T)\n",
    "print(\"誤差\", error)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\mathbb{E}\\left[ \\lVert RR^\\top - AA^\\top \\rVert_\\mathrm{F}^2  \\right] \\le \\frac{\\lVert R \\rVert_\\mathrm{F}^4}{c}$であることを確認します．具体的には，$c$を増やしてみて，誤差$\\lVert RR^\\top - AA^\\top  \\rVert_\\mathrm{F}^2$が減少することを見てみます．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "c=1 0.0439018565995124 1.0314941406249971\n",
      "c=2 0.01231327307140965 0.5157470703124986\n",
      "c=4 0.0027570746630107973 0.2578735351562493\n",
      "c=8 0.0015392452097097507 0.12893676757812464\n",
      "c=16 0.002100567090502002 0.06446838378906232\n",
      "c=32 0.0005925130682599771 0.03223419189453116\n",
      "c=64 0.00026716441867974096 0.01611709594726558\n",
      "c=128 0.00017349561289635792 0.00805854797363279\n",
      "c=256 0.0001212114849485813 0.004029273986816395\n",
      "c=512 5.384591647598512e-05 0.0020146369934081975\n"
     ]
    }
   ],
   "source": [
    "sample_size = 10\n",
    "\n",
    "# E(R R.T) = C C.Tを確認する\n",
    "\n",
    "# Rを作る\n",
    "r = 10\n",
    "row_indices = sample_rows(SQA, r)\n",
    "R = construct_R(SQA, A, row_indices)\n",
    "\n",
    "cs = [2**i for i in range(10)]\n",
    "\n",
    "errors = []\n",
    "for c in cs:\n",
    "    mean = 0    \n",
    "    for _ in range(sample_size):\n",
    "        # Cを作る\n",
    "        col_indices = sample_cols(SQA, row_indices, c)\n",
    "        C = construct_C(SQA, row_indices, col_indices)\n",
    "    \n",
    "        mean += np.linalg.norm(R @ R.T - C @ C.T)**2\n",
    "\n",
    "    mean = mean/sample_size\n",
    "    errors.append(mean)\n",
    "\n",
    "    print(f\"c={c}\", mean, np.linalg.norm(R)**4/c)    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 6))\n",
    "plt.plot(cs, errors, label=\"$(1/10)\\sum_i^{10} || RR^T - CC^T ||_F^2$\")\n",
    "plt.plot(cs, [1/c for c in cs], label=\"upper bound\")\n",
    "plt.xlabel(\"c (sample size of column)\")\n",
    "plt.ylabel(\"E[||R R^T - C C^T ||_F]\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.title(r\"$RR^T\\approx CC^T$\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Step 3:特異値分解\n",
    "最後に，Step3について考えます．Step3は行列$C$を特異値分解をするだけなのであり，この部分はNumPyの関数を使います  ．\n",
    "<!-- $r,c$は行列$A$の次元には関係しないことに注意してください． -->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{i_0,...i_r-1} = \n",
      " [79, 131, 236, 138, 78, 41, 85, 354, 123, 169, 337, 258, 140, 76, 150, 308, 235, 238, 96, 231, 138, 258, 308, 211, 254, 260, 36, 307, 199, 195, 186, 124, 305, 85, 227, 323, 8, 129, 383, 8, 246, 132, 59, 145, 84, 129, 80, 129, 316, 95, 30, 11, 1, 211, 155, 271, 129, 106, 30, 5, 155, 256, 367, 266, 304, 182, 146, 195, 194, 55, 314, 4, 109, 186, 383, 220, 199, 220, 371, 159, 238, 153, 354, 289, 132, 314, 304, 181, 68, 55, 333, 229, 143, 231, 168, 35, 234, 137, 137, 79]\n",
      "left singular vector of C =\n",
      " [[-0.10078665  0.05567284  0.32115091 ... -0.00234017  0.00356132\n",
      "   0.93119914]\n",
      " [-0.09960004 -0.07489795  0.1081485  ... -0.01584392  0.02411162\n",
      "  -0.04766102]\n",
      " [-0.09953486 -0.07910364 -0.05188662 ...  0.00095203 -0.00144882\n",
      "   0.01363859]\n",
      " ...\n",
      " [-0.09931901 -0.09232038 -0.08066682 ... -0.09563616  0.14554116\n",
      "   0.03532856]\n",
      " [-0.09931901 -0.09232038 -0.08066682 ...  0.08259732 -0.12569838\n",
      "   0.03532856]\n",
      " [-0.10078665  0.05567284  0.05176083 ... -0.02035416  0.03097539\n",
      "  -0.0232938 ]]\n",
      "singular values of C =\n",
      " [1.00242849e+00 1.03740617e-01 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 9.87669694e-17\n",
      " 9.87669694e-17 9.87669694e-17 9.87669694e-17 3.00449313e-17]\n"
     ]
    }
   ],
   "source": [
    "def qi_svd(SQA, r, c):\n",
    "    \n",
    "    # Step 1\n",
    "    row_indices = sample_rows(SQA, r)\n",
    "    \n",
    "    # Step 2\n",
    "    col_indices = sample_cols(SQA, row_indices, c)\n",
    "    C = construct_C(SQA, row_indices, col_indices)\n",
    "    \n",
    "    # Step 3\n",
    "    W, S, _ = np.linalg.svd(C)\n",
    "    \n",
    "    return row_indices, W, S\n",
    "\n",
    "r, c = 100, 100\n",
    "row_indices, W, S = qi_svd(SQA, r, c)\n",
    "print(\"{i_0,...i_r-1} = \\n\", row_indices)\n",
    "print(\"left singular vector of C =\\n\", W)\n",
    "print(\"singular values of C =\\n\", S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $C$の特異値・特異ベクトルと$A$の特異値・特異ベクトルについて\n",
    "\n",
    "## 特異値について\n",
    "上記のアルゴリズムで得られた$C$の特異値$\\tilde{\\sigma}_\\ell$(`S`)が，行列$A$の特異値$\\sigma_\\ell$(`exact_S`)を近似していることを見てみます．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "400x300の行列Aの特異値: \n",
      " [1.00000000e+00 1.25000000e-01 5.19213094e-16 4.79345167e-16\n",
      " 4.71732210e-16 4.58242570e-16 4.18376202e-16 3.81834821e-16\n",
      " 3.51484432e-16 3.47825823e-16 3.15975052e-16 3.06894888e-16\n",
      " 2.92151597e-16 2.76922056e-16 2.60478793e-16 2.58559675e-16\n",
      " 2.32224610e-16 2.30232802e-16 2.14765425e-16 2.08972139e-16\n",
      " 2.02095864e-16 1.59139236e-16 1.54982739e-16 1.45129604e-16\n",
      " 1.28041119e-16 1.23098986e-16 1.21378664e-16 9.86021132e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 9.71529166e-17 9.71529166e-17 9.71529166e-17\n",
      " 9.71529166e-17 8.78031527e-17 6.59778865e-17 5.63605870e-17\n",
      " 4.49028869e-17 2.86026833e-17 1.75002194e-17 3.56813390e-18]\n",
      "\n",
      "20x20の行列Cの特異値: \n",
      " [1.00490103e+00 7.61506021e-02 1.21927212e-16 4.18890547e-17\n",
      " 3.72992896e-17 3.63914528e-17 2.97914887e-17 2.87506791e-17\n",
      " 2.44828993e-17 2.13046125e-17 1.99266041e-17 1.53090350e-17\n",
      " 1.33693534e-17 9.03175877e-18 8.53422448e-18 5.09049230e-18\n",
      " 3.77324832e-18 2.06136031e-18 1.41230911e-33 6.38038974e-34]\n"
     ]
    }
   ],
   "source": [
    "r, c = 20, 20\n",
    "_, _, S = qi_svd(SQA, r, c)\n",
    "print(f\"{m}x{n}の行列Aの特異値: \\n\", exact_S)\n",
    "print(f\"\\n{r}x{c}の行列Cの特異値: \\n\", S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "サンプル数$r,c$を増やしたときの特異値に関する誤差についてみてみます．行列$A$のランク分だけ$C$の特異値に注目します．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(r,c)=(2,2): 0.007605164936364128 0.10077677401493898\n",
      "(r,c)=(4,4): 0.006664163493095154 0.08303654467706957\n",
      "(r,c)=(8,8): 0.011376142261109523 0.07973403408947627\n",
      "(r,c)=(16,16): 0.009745271459953176 0.06004110776570289\n",
      "(r,c)=(32,32): 0.003609230642555905 0.031184591754424763\n",
      "(r,c)=(64,64): 0.004273053488123823 0.03890602246205678\n",
      "(r,c)=(128,128): 0.002837380586722571 0.024082138257608424\n",
      "(r,c)=(256,256): 0.00208143073207121 0.018202510064449937\n",
      "(r,c)=(512,512): 0.0017952579212976772 0.014526341778711576\n"
     ]
    }
   ],
   "source": [
    "sample_size = 10\n",
    "\n",
    "rs = [2**i for i in range(1, 10)]\n",
    "cs = [2**i for i in range(1, 10)]\n",
    "\n",
    "hist0 = []\n",
    "hist1 = []\n",
    "for r, c in zip(rs, cs):\n",
    "    mean0 = 0\n",
    "    mean1 = 0\n",
    "    for _ in range(sample_size):\n",
    "        _, _, S = qi_svd(SQA, r, c)\n",
    "        mean0 += abs(S[0] - exact_S[0])\n",
    "        mean1 += abs(S[1] - exact_S[1])\n",
    "        \n",
    "    mean0 = mean0/sample_size\n",
    "    mean1 = mean1/sample_size\n",
    "    hist0.append(mean0)\n",
    "    hist1.append(mean1)\n",
    "    \n",
    "    print(f\"(r,c)=({r},{c}):\", mean0, mean1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 6))\n",
    "plt.plot(rs, hist0, label=r\"$\\ell=0$\")\n",
    "plt.plot(rs, hist1, label=r\"$\\ell=1$\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.xlabel(\"r, c\")\n",
    "plt.ylabel(r\"$|\\sigma_\\ell - \\tilde{\\sigma}_\\ell|$\")\n",
    "plt.title(r\"$\\tilde{\\sigma}_\\ell \\approx \\sigma_\\ell$\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "なお，サンプル数$r,c$が行列$A$のサイズよりも小さい場合，$C$のサイズは$A$のサイズよりも小さくなります．そのため，$A$の特異値をそのまま計算するよりも，高速に近似特異値を得ることが期待できます．($\\mathrm{SQ}(A)$の構築時間を含めれると，遅くなることに注意)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SciPyの特異値計算の時間\n",
      " 0.017342090606689453\n",
      "qi_svdを用いた時の特異値計算の時間\n",
      " 0.0013911724090576172\n",
      "特異値の誤差 [0.00457841 0.04470537]\n"
     ]
    }
   ],
   "source": [
    "import time\n",
    "\n",
    "start_time = time.time()\n",
    "exact_S = sp.linalg.svdvals(A)\n",
    "print(\"SciPyの特異値計算の時間\\n\", time.time() - start_time)\n",
    "\n",
    "start_time = time.time()\n",
    "# SQA = MatrixBasedDataStructure(A) # SQ(A)の構築も含めると遅くなる\n",
    "_, _, S = qi_svd(SQA, 10, 10)\n",
    "print(\"qi_svdを用いた時の特異値計算の時間\\n\", time.time() - start_time)\n",
    "\n",
    "print(\"特異値の誤差\", np.abs(exact_S[:2] - S[:2]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 右特異ベクトルについて\n",
    "行列$A$の右特異ベクトル$v^{(\\ell)}$は，\n",
    "$$\n",
    "    v^{(\\ell)} \\approx \\tilde{v}^{(\\ell)} = \\frac{1}{\\tilde{\\sigma}_\\ell}R^\\dagger\\vec{w}^{(\\ell)}\n",
    "$$のようにして近似されます．このことを確認してみましょう．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, c = 100, 100\n",
    "row_indices, W, S = qi_svd(SQA, r, c)\n",
    "\n",
    "R = construct_R(SQA, A, row_indices)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0番目の特異値に対応するAの右特異ベクトル\n",
      "近似:\n",
      " [-0.0317202   0.01337399 -0.11415716  0.06198199 -0.05629502 -0.03856402\n",
      "  0.07927241 -0.05296198  0.02053256  0.07143687 -0.06066683  0.06599246\n",
      "  0.08793431 -0.05312667  0.051438   -0.01644564 -0.05498971  0.03757013\n",
      " -0.00098816  0.02539686  0.06564687 -0.00855814 -0.0259037  -0.05998523\n",
      "  0.05305378  0.08943291  0.07003928  0.00999123  0.04687588  0.00265625\n",
      " -0.09704059  0.07938635 -0.01840886  0.06111095 -0.06052255 -0.07395583\n",
      " -0.0521981   0.11650144 -0.09268968  0.07901958  0.02008417  0.08155085\n",
      " -0.04346777  0.01810488  0.03339253  0.05137885 -0.03231164  0.03518399\n",
      " -0.03111885  0.03459961  0.00977295  0.03586271 -0.10363354 -0.07319737\n",
      " -0.0802782   0.03757656  0.02940295 -0.02734621 -0.03605049 -0.0772996\n",
      "  0.05253919 -0.06384619  0.01248809  0.00919471  0.08133145 -0.07976466\n",
      "  0.07494208 -0.08056276 -0.07593681  0.02836961  0.04319938  0.03126609\n",
      "  0.033294    0.1033259  -0.07034601  0.10302226 -0.09485786  0.08655116\n",
      "  0.0194916   0.01596536 -0.0566828   0.08449272  0.02083656  0.08549993\n",
      " -0.00218206  0.00948511  0.02293383 -0.01769414 -0.07367503  0.00033869\n",
      " -0.04419602  0.10288678 -0.06686099  0.04419152  0.01094282 -0.01183871\n",
      "  0.05867622 -0.02852445 -0.10442126  0.00174744 -0.01631248  0.05210681\n",
      " -0.01571766 -0.02589921 -0.08469993 -0.07157025 -0.0230451  -0.06465406\n",
      "  0.00617937 -0.0159394   0.07770179 -0.02575547 -0.07170193 -0.06723837\n",
      " -0.04396394  0.03729801 -0.00641077 -0.07978389 -0.05345854  0.02760726\n",
      " -0.02578446  0.00591991  0.08548548  0.02656586  0.13203495 -0.03676117\n",
      " -0.06378039 -0.01779187 -0.09531531  0.02193487 -0.09872355 -0.00112499\n",
      "  0.01478052  0.03016424 -0.03330164 -0.04999261  0.04666671 -0.0789648\n",
      "  0.04571095 -0.0003321  -0.05986715 -0.00819154 -0.05777269 -0.05001009\n",
      "  0.06926649 -0.02717667 -0.11020916  0.06314345  0.04643599 -0.01661893\n",
      " -0.03868591 -0.08211097  0.05058854 -0.05164187  0.01467088 -0.04339066\n",
      " -0.01783837  0.08601976 -0.1095638   0.0054588  -0.00342149 -0.09726745\n",
      " -0.03630337  0.02420097 -0.05219577 -0.06499896 -0.0011301  -0.01747065\n",
      "  0.04274382 -0.04308753  0.05683675  0.0512726  -0.01191697  0.03852084\n",
      "  0.08130772  0.03795078 -0.01944249  0.03060062  0.0523495   0.02011647\n",
      " -0.0880724  -0.05803288 -0.11355218 -0.00479497  0.0039829  -0.07985159\n",
      " -0.01587512 -0.03129549 -0.03690483 -0.02547485 -0.04708811 -0.017056\n",
      "  0.01738491  0.00883852 -0.1573275  -0.06953934 -0.04309797 -0.08133134\n",
      " -0.06846626 -0.00274993  0.0155446   0.05527531  0.00888762 -0.08007245\n",
      "  0.02708738 -0.00783943  0.09025781  0.02962462  0.15465978  0.05453309\n",
      "  0.05973076 -0.03082224  0.00290697  0.08462897 -0.08427127  0.13412481\n",
      " -0.04548615  0.04383659 -0.01316888 -0.00543999  0.0320396   0.05633077\n",
      " -0.04379285  0.11285006 -0.01666589  0.10728228  0.00355295 -0.02455303\n",
      "  0.06540515 -0.00571469 -0.14216609 -0.02847352  0.0749826   0.01440262\n",
      "  0.00832115 -0.03356875  0.07404006  0.04974037  0.05205633  0.0340721\n",
      "  0.0260405   0.00697726  0.01628316  0.05733054  0.05572234 -0.01441417\n",
      " -0.09585749 -0.0509137  -0.03943181 -0.03281695  0.04084048  0.15735757\n",
      " -0.00024769 -0.1315968  -0.01233281 -0.01006828  0.02565003  0.03327864\n",
      " -0.05773076  0.02379497 -0.03868227 -0.02009777  0.02232155  0.02321261\n",
      "  0.11474818 -0.01715254  0.00643562  0.07770879  0.00178848  0.02780374\n",
      "  0.06755107  0.01103524 -0.0750486  -0.00611766  0.01811955 -0.04082593\n",
      "  0.02820813 -0.04598371 -0.0077759   0.10142964  0.09072496 -0.01452512\n",
      "  0.0361294  -0.00788169  0.04839007  0.04639304 -0.02924079 -0.09581713\n",
      "  0.05989333 -0.00654495  0.09971479  0.06922174 -0.02830517 -0.01349306\n",
      " -0.06510445  0.01003437  0.07008415  0.06489023  0.0167468  -0.01076254]\n",
      "SciPy:\n",
      " [-0.03178579  0.01339787 -0.1143845   0.06210105 -0.05640416 -0.03864014\n",
      "  0.07942287 -0.05306616  0.0205774   0.0715718  -0.06078328  0.06612246\n",
      "  0.08810335 -0.05323437  0.05153945 -0.01648487 -0.05510182  0.03764083\n",
      " -0.00099093  0.02545148  0.06577178 -0.00858007 -0.02595692 -0.0601031\n",
      "  0.0531583   0.08960578  0.0701723   0.01000783  0.04696291  0.0026565\n",
      " -0.09723311  0.07954568 -0.01844061  0.06122542 -0.06063984 -0.07410657\n",
      " -0.05229871  0.11672668 -0.09286865  0.07917701  0.02012488  0.08171221\n",
      " -0.04355533  0.01813954  0.03345575  0.0514816  -0.03237935  0.03525824\n",
      " -0.03117853  0.03466819  0.00978781  0.03593037 -0.10383397 -0.07333392\n",
      " -0.08043726  0.0376514   0.0294628  -0.02740079 -0.03612539 -0.07744662\n",
      "  0.052643   -0.06397362  0.01251978  0.00921097  0.08149118 -0.07992301\n",
      "  0.07509322 -0.08071959 -0.07609187  0.02842743  0.04328069  0.03132377\n",
      "  0.03336238  0.10352992 -0.07048071  0.10322216 -0.09504197  0.08671549\n",
      "  0.01952856  0.01599235 -0.0567889   0.08465883  0.02087572  0.08566471\n",
      " -0.00219     0.00950963  0.02298541 -0.01772501 -0.07381223  0.00033611\n",
      " -0.04427933  0.1030884  -0.0669901   0.04427171  0.01096017 -0.01186036\n",
      "  0.05879242 -0.02858395 -0.10463549  0.00175337 -0.01635007  0.05220833\n",
      " -0.01574671 -0.02594836 -0.08486516 -0.07170336 -0.02309082 -0.0647826\n",
      "  0.00618636 -0.01597017  0.07785602 -0.02580621 -0.07183828 -0.06737655\n",
      " -0.04404622  0.03736789 -0.00642312 -0.07994316 -0.05355942  0.02766029\n",
      " -0.02584077  0.00592402  0.08565684  0.02661693  0.13228927 -0.03683485\n",
      " -0.06389875 -0.01782499 -0.09549502  0.02198091 -0.09891416 -0.00112374\n",
      "  0.01481104  0.0302231  -0.03336965 -0.05009244  0.04676617 -0.07912072\n",
      "  0.04579826 -0.00032691 -0.05997921 -0.00820417 -0.05788193 -0.05011023\n",
      "  0.06939673 -0.02723142 -0.11042606  0.06326595  0.04652422 -0.01664543\n",
      " -0.03876942 -0.0822678   0.05067961 -0.05173948  0.01470153 -0.04348097\n",
      " -0.01787777  0.0861799  -0.10977688  0.00547187 -0.00343178 -0.09745301\n",
      " -0.03637709  0.02425079 -0.05229407 -0.06512647 -0.00113412 -0.01750239\n",
      "  0.04283423 -0.04316742  0.05694969  0.05137446 -0.0119413   0.03859628\n",
      "  0.08146537  0.03802787 -0.01947816  0.03066369  0.0524491   0.02014959\n",
      " -0.08824002 -0.05813942 -0.1137716  -0.0047998   0.0039882  -0.08001098\n",
      " -0.01590691 -0.03135636 -0.0369706  -0.02552279 -0.04717569 -0.01708627\n",
      "  0.01741501  0.00885184 -0.15763194 -0.0696734  -0.04318288 -0.08148692\n",
      " -0.06860291 -0.00275695  0.01557192  0.05538071  0.00890811 -0.08022934\n",
      "  0.02714474 -0.00785599  0.09043169  0.02968092  0.15496412  0.05464191\n",
      "  0.05984439 -0.03088542  0.00291311  0.08479313 -0.08443903  0.13438914\n",
      " -0.04557197  0.04391762 -0.01319775 -0.0054477   0.03210176  0.05644166\n",
      " -0.04387738  0.11307303 -0.01669799  0.10748609  0.00356494 -0.02459806\n",
      "  0.06553224 -0.00573285 -0.14244482 -0.02852883  0.07513035  0.0144233\n",
      "  0.00833655 -0.0336364   0.07418805  0.04983916  0.05215753  0.03414352\n",
      "  0.02609588  0.00699501  0.01631814  0.05744169  0.05582713 -0.01443581\n",
      " -0.09604381 -0.05101221 -0.03951222 -0.03287923  0.04091922  0.15766364\n",
      " -0.00024434 -0.13185333 -0.01236108 -0.01008769  0.02570283  0.03334367\n",
      " -0.05784436  0.02384133 -0.03875717 -0.02013642  0.02236463  0.02325399\n",
      "  0.11496818 -0.01718518  0.00644599  0.07786043  0.00178812  0.02785964\n",
      "  0.06768607  0.01104995 -0.07519349 -0.00612426  0.01815661 -0.04090824\n",
      "  0.02826305 -0.04607949 -0.00779054  0.10163116  0.09090134 -0.01455158\n",
      "  0.0361944  -0.007898    0.04848326  0.04648396 -0.02929724 -0.09599888\n",
      "  0.06001497 -0.00655647  0.09990745  0.06935388 -0.02836367 -0.01351696\n",
      " -0.06524075  0.01004781  0.07021831  0.06501476  0.01677325 -0.01077529]\n"
     ]
    }
   ],
   "source": [
    "v0 = R.T @ W[:,0] / S[0]\n",
    "\n",
    "print(\"0番目の特異値に対応するAの右特異ベクトル\")\n",
    "print(\"近似:\\n\",   v0)\n",
    "print(\"SciPy:\\n\", exact_V[:,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "サンプル数を増やしていくにつれて，$\\norm{v^{(\\ell)} - \\tilde{v}^{(\\ell)}}$が小さくなることを確認します．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(r,c)=(2,2): 0.02829504908539988 11.953080497719569 [0.05143816 0.21537544]\n",
      "(r,c)=(4,4): 0.037495686271958616 2.864633876497967 [0.00751327 0.10171908]\n",
      "(r,c)=(8,8): 0.018633988145829013 1.7889104770224038 [0.00145007 0.01220476]\n",
      "(r,c)=(16,16): 0.02246828079147237 1.2119560370589524 [0.00659683 0.07613456]\n",
      "(r,c)=(32,32): 0.015194689881244009 1.1144072028039513 [0.00340926 0.03121909]\n",
      "(r,c)=(64,64): 0.011350414319032538 1.0925412277833169 [0.00151848 0.01160038]\n",
      "(r,c)=(128,128): 0.008898171229586123 0.8549066952643022 [0.00144013 0.01211654]\n",
      "(r,c)=(256,256): 0.0041891501005805205 0.28546587539012197 [3.09644045e-05 2.47466442e-04]\n"
     ]
    }
   ],
   "source": [
    "errors = []\n",
    "sample_size = 10\n",
    "k = 0 # 確認する特異ベクトルの番号．\n",
    "rs = [2**i for i in range(1, 9)]\n",
    "cs = [2**i for i in range(1, 9)]\n",
    "\n",
    "hist0 = []\n",
    "hist1 = []\n",
    "\n",
    "for r, c in zip(rs, cs):\n",
    "\n",
    "    mean0 = 0\n",
    "    mean1 = 0\n",
    "    for _ in range(sample_size):\n",
    "        row_indices, W, S = qi_svd(SQA, r, c)\n",
    "\n",
    "        # 確認\n",
    "        R = construct_R(SQA, A, row_indices)\n",
    "        v0 = R.T @ W[:, 0]/S[0]\n",
    "        v1 = R.T @ W[:, 1]/S[1]\n",
    "        \n",
    "        mean0 += np.linalg.norm(v0 - exact_V[:, 0])\n",
    "        mean1 += np.linalg.norm(v1 - exact_V[:, 1])\n",
    "        \n",
    "    mean0 = mean0/sample_size\n",
    "    mean1 = mean1/sample_size   \n",
    "    hist0.append(mean0)\n",
    "    hist1.append(mean1)\n",
    "    \n",
    "    print(f\"(r,c)=({r},{c}):\", mean0, mean1, np.abs(S[:rankA] - exact_S[:rankA]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 6))\n",
    "plt.plot(rs, hist0, label=r\"$\\ell=0$\")\n",
    "plt.plot(rs, hist1, label=r\"$\\ell=1$\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.xlabel(\"r, c\")\n",
    "plt.ylabel(\"error\")\n",
    "plt.title(r\"\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 左特異ベクトルについて\n",
    "行列$A$の左特異ベクトル$u^{(\\ell)}$は，\n",
    "$$\n",
    "    u^{(\\ell)} \\approx \\tilde{u}^{(\\ell)} = \\frac{1}{\\tilde{\\sigma}_\\ell^2}A R^\\top w^{(\\ell)}\n",
    "$$\n",
    "のようにして近似されます．このことを確認してみます．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, c = 1000, 1000\n",
    "row_indices, W, S = qi_svd(SQA, r, c)\n",
    "R = construct_R(SQA, A, row_indices)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0番目の特異値に対応するAの左特異ベクトル\n",
      "近似:\n",
      " [-0.0605458  -0.0704849  -0.07149567 -0.04855803 -0.06494327 -0.0608823\n",
      " -0.00690812 -0.04015194 -0.07384671 -0.0734254  -0.00876735 -0.05075763\n",
      " -0.05635925 -0.03699295 -0.03174713 -0.07583245 -0.00139154 -0.02593728\n",
      " -0.07389579 -0.04645488 -0.05951348 -0.03605792 -0.05254252 -0.01226802\n",
      " -0.00362969 -0.0339557  -0.04472962 -0.0125205  -0.07404588 -0.03484445\n",
      " -0.07841977 -0.08207029 -0.05855517 -0.06039629 -0.06840638 -0.06452022\n",
      " -0.06882283 -0.02948692 -0.06494297 -0.06989275 -0.00083706 -0.07390116\n",
      " -0.06036129 -0.05834352 -0.0173526  -0.05614958 -0.04937652 -0.03994919\n",
      " -0.01978992 -0.03851737 -0.03806637 -0.08059453 -0.02387455 -0.07240957\n",
      " -0.07010345 -0.08319091 -0.00675038 -0.01602705 -0.03474378 -0.02315195\n",
      " -0.05877738 -0.01669903 -0.04517126 -0.03589944 -0.00946414 -0.08191783\n",
      " -0.03327773 -0.07096757 -0.07956595 -0.07393915 -0.03677444 -0.05498509\n",
      " -0.00635478 -0.02331074 -0.0667158  -0.0048446  -0.05927138 -0.04358965\n",
      " -0.07173062 -0.07904759 -0.0728023  -0.04476247 -0.06216394 -0.05068522\n",
      " -0.07986796 -0.08303011 -0.03791738 -0.01840737 -0.03995369 -0.00265248\n",
      " -0.02539848 -0.02249231 -0.08164556 -0.04743769 -0.05679071 -0.05941819\n",
      " -0.0577563  -0.05288016 -0.00592677 -0.03649032 -0.00146731 -0.03588471\n",
      " -0.05995013 -0.01976901 -0.01166355 -0.04299707 -0.05190724 -0.029776\n",
      " -0.04378774 -0.0463587  -0.03521626 -0.05078906 -0.01696562 -0.01960544\n",
      " -0.07342023 -0.06143066 -0.05866729 -0.04179887 -0.08295138 -0.01211161\n",
      " -0.03385545 -0.08145    -0.0477917  -0.08235871 -0.070211   -0.03934347\n",
      " -0.01423589 -0.06718754 -0.02678061 -0.0756882  -0.07849846 -0.04628303\n",
      " -0.06796465 -0.01840771 -0.06584961 -0.03190238 -0.00906975 -0.06983423\n",
      " -0.05323478 -0.03375346 -0.04189315 -0.01298106 -0.03455251 -0.03931664\n",
      " -0.05917124 -0.07962125 -0.03689864 -0.01044661 -0.01134955 -0.0107187\n",
      " -0.06491777 -0.00575106 -0.00521567 -0.07257526 -0.01290219 -0.08276778\n",
      " -0.06561617 -0.07916155 -0.0674428  -0.03894268 -0.03202196 -0.02759434\n",
      " -0.0706788  -0.00872396 -0.03247791 -0.02014104 -0.00996913 -0.01419721\n",
      " -0.07869264 -0.06606762 -0.0274082  -0.00420174 -0.0122052  -0.05808773\n",
      " -0.05176395 -0.0566248  -0.00416218 -0.07744653 -0.06151638 -0.07639261\n",
      " -0.05347447 -0.06627271 -0.07669143 -0.03477293 -0.04141373 -0.03128679\n",
      " -0.03788901 -0.05323336 -0.08258669 -0.02300419 -0.002807   -0.00712448\n",
      " -0.00251053 -0.07134748 -0.05720149 -0.06478203 -0.05903397 -0.00977324\n",
      " -0.02724393 -0.05174461 -0.0596949  -0.06747449 -0.0706302  -0.05346121\n",
      " -0.02997381 -0.04722246 -0.04539428 -0.01989842 -0.05704522 -0.08253875\n",
      " -0.076835   -0.07251446 -0.07125862 -0.05001557 -0.00465223 -0.0815906\n",
      " -0.02634413 -0.08109756 -0.0491986  -0.03975387 -0.06204669 -0.02409887\n",
      " -0.01237883 -0.05061375 -0.07087539 -0.01549627 -0.04723927 -0.0798324\n",
      " -0.01603082 -0.04837712 -0.00276387 -0.05772191 -0.03471284 -0.00823026\n",
      " -0.01678352 -0.07274266 -0.03495152 -0.08061009 -0.07631995 -0.06716318\n",
      " -0.00873076 -0.06160927 -0.08070924 -0.04558438 -0.06143948 -0.07142257\n",
      " -0.056468   -0.03793588 -0.03122556 -0.05919834 -0.00885933 -0.0029276\n",
      " -0.0568097  -0.01650095 -0.06685359 -0.046717   -0.03655642 -0.03812575\n",
      " -0.04561245 -0.00911834 -0.08015103 -0.0662699  -0.04707998 -0.00862236\n",
      " -0.06604481 -0.0664349  -0.0580816  -0.05609323 -0.02730883 -0.02042405\n",
      " -0.01226267 -0.05435141 -0.06087473 -0.03697347 -0.00023972 -0.05346648\n",
      " -0.04716755 -0.02469469 -0.00302745 -0.03113723 -0.02623074 -0.02681332\n",
      " -0.05223701 -0.02821278 -0.0493041  -0.02786236 -0.02784349 -0.03131616\n",
      " -0.0546355  -0.06380391 -0.00023605 -0.05146422 -0.03066548 -0.05251106\n",
      " -0.07201202 -0.04256657 -0.02744108 -0.00875707 -0.02055682 -0.03793859\n",
      " -0.05096003 -0.02404277 -0.02215396 -0.04740809 -0.08059723 -0.08117378\n",
      " -0.04382467 -0.07713837 -0.08115339 -0.05283448 -0.04931161 -0.07286654\n",
      " -0.04332738 -0.03636768 -0.06870996 -0.02358293 -0.0416052  -0.08031554\n",
      " -0.0288785  -0.03039519 -0.00443046 -0.00799646 -0.03985718 -0.02861012\n",
      " -0.0356367  -0.01622956 -0.05600859 -0.00584186 -0.04529722 -0.0414683\n",
      " -0.00295963 -0.05200601 -0.03670453 -0.02870568 -0.03131672 -0.01109076\n",
      " -0.06158986 -0.05923834 -0.07795516 -0.07206882 -0.04505654 -0.08203382\n",
      " -0.04190346 -0.02162796 -0.03967774 -0.04925219 -0.04258829 -0.01581113\n",
      " -0.06189415 -0.02329364 -0.06518351 -0.0329322  -0.06291574 -0.0444857\n",
      " -0.07947668 -0.04571527 -0.06775138 -0.03454912 -0.01213915 -0.02988561\n",
      " -0.00573499 -0.04412681 -0.02397704 -0.01587559 -0.03690571 -0.01068382\n",
      " -0.01793798 -0.03995668 -0.00986041 -0.0808141  -0.00030202 -0.07198661\n",
      " -0.03742878 -0.03056998 -0.04473364 -0.02759723 -0.0140094  -0.01773207\n",
      " -0.04575469 -0.05951234 -0.05822442 -0.08182881 -0.06823165 -0.05164249\n",
      " -0.01971849 -0.03670303 -0.05498087 -0.07597282 -0.04423853 -0.01947815\n",
      " -0.06791941 -0.0294482  -0.01051621 -0.04196672 -0.02150881 -0.04872591\n",
      " -0.02877166 -0.06208905 -0.05471735 -0.05511436]\n",
      "SciPy:\n",
      " [-0.06063178 -0.07056625 -0.07158246 -0.04862612 -0.06503195 -0.06095108\n",
      " -0.00693953 -0.04021985 -0.07394289 -0.07351338 -0.00878353 -0.05081582\n",
      " -0.05642742 -0.03704946 -0.0318087  -0.07593304 -0.00141568 -0.02599341\n",
      " -0.07399201 -0.04650953 -0.05959117 -0.03611226 -0.05260113 -0.01228628\n",
      " -0.00363698 -0.03401735 -0.04479451 -0.01254705 -0.0741346  -0.03491016\n",
      " -0.07853127 -0.08218143 -0.05863315 -0.06048798 -0.0684982  -0.06460986\n",
      " -0.06891866 -0.02954067 -0.06503005 -0.06997553 -0.00084306 -0.0740002\n",
      " -0.06045485 -0.05842809 -0.0173783  -0.05623333 -0.04943806 -0.04001221\n",
      " -0.01984121 -0.03857801 -0.03813133 -0.08068746 -0.02390982 -0.07250399\n",
      " -0.0702003  -0.08329509 -0.00678477 -0.01606309 -0.03480238 -0.02319961\n",
      " -0.058862   -0.01672736 -0.04524962 -0.03596486 -0.00950268 -0.08201297\n",
      " -0.03332387 -0.0710696  -0.07965349 -0.07402582 -0.03682144 -0.05505557\n",
      " -0.00636821 -0.02335395 -0.06680174 -0.00485834 -0.05936046 -0.04365236\n",
      " -0.0718282  -0.07914442 -0.07288395 -0.04483074 -0.06224089 -0.05074273\n",
      " -0.07996829 -0.08314655 -0.03797211 -0.01842989 -0.03999778 -0.00266692\n",
      " -0.02543578 -0.02253468 -0.08174462 -0.04750384 -0.05687514 -0.0594967\n",
      " -0.05783011 -0.0529424  -0.00593584 -0.03653522 -0.00148681 -0.03593123\n",
      " -0.06001967 -0.0198167  -0.01170331 -0.04304919 -0.05196441 -0.02982581\n",
      " -0.04385963 -0.04641885 -0.03527912 -0.05086332 -0.01700349 -0.01964498\n",
      " -0.07352595 -0.06151298 -0.05875747 -0.04185704 -0.08305182 -0.01212945\n",
      " -0.03391248 -0.08154055 -0.04785471 -0.0824541  -0.07031419 -0.03940017\n",
      " -0.01425221 -0.06726387 -0.02683388 -0.07579793 -0.07861089 -0.04634977\n",
      " -0.06804526 -0.01844644 -0.06593022 -0.03196295 -0.00909236 -0.06993697\n",
      " -0.05331092 -0.03380142 -0.0419535  -0.01300585 -0.03460886 -0.03936059\n",
      " -0.05925991 -0.07971551 -0.03694779 -0.01046141 -0.01136678 -0.01075114\n",
      " -0.0650068  -0.00576516 -0.00523565 -0.07265599 -0.01292018 -0.08285846\n",
      " -0.06568797 -0.07925332 -0.06752771 -0.03899369 -0.03206217 -0.0276371\n",
      " -0.07076888 -0.00874673 -0.03254125 -0.02016686 -0.00999661 -0.01424195\n",
      " -0.07878899 -0.06616008 -0.02745157 -0.00421571 -0.01222621 -0.0581739\n",
      " -0.05182444 -0.0567016  -0.00417239 -0.07753641 -0.0616052  -0.07648652\n",
      " -0.05355695 -0.06635803 -0.07679893 -0.0348233  -0.0414831  -0.03133529\n",
      " -0.0379397  -0.05329744 -0.08270502 -0.02303596 -0.00281382 -0.00715434\n",
      " -0.00251455 -0.07143812 -0.05728264 -0.06485961 -0.05912167 -0.00980228\n",
      " -0.02728896 -0.05180133 -0.05978358 -0.06756594 -0.07071727 -0.05354538\n",
      " -0.03002496 -0.04729218 -0.04545434 -0.01992118 -0.05712637 -0.08264747\n",
      " -0.07692423 -0.07261944 -0.07133902 -0.05009336 -0.0046716  -0.08170006\n",
      " -0.02638369 -0.08120615 -0.04926476 -0.03980993 -0.06211485 -0.02414838\n",
      " -0.01239285 -0.0506823  -0.07095273 -0.0155263  -0.0473167  -0.07994415\n",
      " -0.01606506 -0.04844704 -0.00277822 -0.05778958 -0.034761   -0.00826202\n",
      " -0.0168026  -0.07282904 -0.03500217 -0.08070908 -0.07642476 -0.0672386\n",
      " -0.008764   -0.06170246 -0.0808191  -0.04564182 -0.06153348 -0.0715151\n",
      " -0.0565438  -0.0379882  -0.03128069 -0.05928644 -0.0088728  -0.00293301\n",
      " -0.05688345 -0.01654686 -0.0669474  -0.04678061 -0.03661127 -0.03818637\n",
      " -0.04568726 -0.00915323 -0.08024214 -0.06635014 -0.04713686 -0.0086446\n",
      " -0.06612905 -0.06651602 -0.05814475 -0.05617944 -0.0273549  -0.02047405\n",
      " -0.01229146 -0.05442679 -0.06096082 -0.0370327  -0.00024619 -0.05353806\n",
      " -0.04724044 -0.02474614 -0.00303486 -0.03118458 -0.0262789  -0.02686085\n",
      " -0.05230675 -0.02825987 -0.04937317 -0.02789618 -0.02789151 -0.03135598\n",
      " -0.05471743 -0.0638749  -0.00026383 -0.05152938 -0.03071228 -0.05257198\n",
      " -0.07211434 -0.04264211 -0.0274817  -0.00878321 -0.0205986  -0.03799543\n",
      " -0.05102354 -0.02409885 -0.0221824  -0.04746076 -0.08069505 -0.08127258\n",
      " -0.04389224 -0.07722425 -0.08125058 -0.05289844 -0.04939296 -0.07295638\n",
      " -0.04337727 -0.03643595 -0.06880348 -0.02362328 -0.04167796 -0.08040976\n",
      " -0.02892012 -0.03043761 -0.0044412  -0.00802596 -0.03990368 -0.02864774\n",
      " -0.03569326 -0.01625035 -0.05607518 -0.00586505 -0.04537514 -0.04152883\n",
      " -0.00298864 -0.05208027 -0.03677207 -0.02873733 -0.03135558 -0.01111267\n",
      " -0.0616602  -0.0593312  -0.07805245 -0.07215897 -0.04512449 -0.0821318\n",
      " -0.04195338 -0.02165334 -0.03974893 -0.0493141  -0.04264729 -0.015829\n",
      " -0.06198071 -0.02332793 -0.06528109 -0.03297081 -0.06300878 -0.04453606\n",
      " -0.07958374 -0.04576957 -0.0678319  -0.03459215 -0.01216611 -0.02993966\n",
      " -0.00575253 -0.04419967 -0.0240268  -0.01591216 -0.03696948 -0.01070977\n",
      " -0.01797275 -0.04002642 -0.00989548 -0.0809272  -0.00032691 -0.07207141\n",
      " -0.03747797 -0.03060903 -0.04479431 -0.02763489 -0.01403073 -0.01775245\n",
      " -0.04581232 -0.05959357 -0.05831362 -0.08193596 -0.06832077 -0.05172321\n",
      " -0.01974377 -0.03675004 -0.05504588 -0.076076   -0.04428945 -0.01952937\n",
      " -0.06800251 -0.02949921 -0.01054801 -0.04201861 -0.02154635 -0.04880282\n",
      " -0.02883157 -0.06216436 -0.05477746 -0.05518749]\n"
     ]
    }
   ],
   "source": [
    "u0 = A @ R.T @ W[:,0] / (S[0]**2)\n",
    "\n",
    "print(\"0番目の特異値に対応するAの左特異ベクトル\")\n",
    "print(\"近似:\\n\",   u0)\n",
    "print(\"SciPy:\\n\", exact_U[:,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "サンプル数を増やしていくにつれて，$\\norm{u^{(\\ell)} - \\tilde{u}^{(\\ell)}}$が小さくなることを確認します．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(r,c)=(2,2): 0.013603883719106055 2130.2608988385095\n",
      "(r,c)=(4,4): 0.014645442873687764 15.121519302163557\n",
      "(r,c)=(8,8): 0.02509946525614708 10.636265086191234\n",
      "(r,c)=(16,16): 0.012625600458247736 3.8287808017213267\n",
      "(r,c)=(32,32): 0.007096984858864107 2.3173803313471515\n",
      "(r,c)=(64,64): 0.005862771726432317 1.7957413769009574\n",
      "(r,c)=(128,128): 0.004717295673775422 1.279604646308625\n",
      "(r,c)=(256,256): 0.004818623948890067 1.378776675271584\n",
      "(r,c)=(512,512): 0.0029659244507325017 0.9435018957612348\n"
     ]
    }
   ],
   "source": [
    "sample_size = 20\n",
    "\n",
    "hist0 = []\n",
    "hist1 = []\n",
    "\n",
    "rs = [2**i for i in range(1, 10)]\n",
    "cs = [2**i for i in range(1, 10)]\n",
    "for r, c in zip(rs, cs):\n",
    "\n",
    "    mean0 = 0\n",
    "    mean1 = 0\n",
    "    for _ in range(sample_size):\n",
    "        row_indices, W, S = qi_svd(SQA, r, c)\n",
    "\n",
    "        # 確認\n",
    "        R = construct_R(SQA, A, row_indices)\n",
    "        u0 = A @ R.T @ W[:, 0]/S[0]**2 \n",
    "        u1 = A @ R.T @ W[:, 1]/S[1]**2\n",
    "        mean0 += np.linalg.norm(u0 - exact_U[:, 0])\n",
    "        mean1 += np.linalg.norm(u1 - exact_U[:, 1])\n",
    "        \n",
    "    mean0 = mean0/sample_size\n",
    "    mean1 = mean1/sample_size\n",
    "    hist0.append(mean0)\n",
    "    hist1.append(mean1)\n",
    "    \n",
    "    print(f\"(r,c)=({r},{c}):\", mean0, mean1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAfoAAAF3CAYAAABNO4lPAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAAAsTAAALEwEAmpwYAAAw20lEQVR4nO3dfXRV9Z3v8ff3nDw/P5CAJIEAQR5EQQ2KD3WcUlvbKbXT2lbbeWjryKWVTnGtu9Zt75275t5Za26ddXs71mrr0NFxnGm1jm1HbaudFuvUB0AiIvIcRJCAkJCEkATy/Lt/7JPknBAgISdnn+zzea21V3L22Xvne8qpn71/v9/eP3POISIiIsEU8rsAERERmTwKehERkQBT0IuIiASYgl5ERCTAFPQiIiIBpqAXEREJsDS/C5gM06ZNc9XV1X6XISIikhBvvPHGCedc2WjvBSrozWwVsKqmpoa6ujq/yxEREUkIMzt0rvcC1XTvnHvOObe6sLDQ71JERESSQqCCXkRERGIp6EVERAIsUH30IiKSenp7e2loaKCrq8vvUiZdVlYWlZWVpKenj3kfBb2IiExpDQ0N5OfnU11djZn5Xc6kcc7R3NxMQ0MDc+bMGfN+gWq6N7NVZra+ra3N71JERCRBurq6KC0tDXTIA5gZpaWl4265CFTQa9S9iEhqCnrID7qYzxmooBcREZFYCnoREZEAU9CLiIjEQWdnJ2vXrmXTpk0Xtf8LL7zAggULqKmp4b777otbXQp6ERGROHj44Yfp6urilVdeGfe+/f393HPPPTz//PPs2rWLJ554gl27dsWlLgX9hfSegbYGv6sQEZEkN3hFvmzZsnHv+/rrr1NTU8PcuXPJyMjgjjvu4JlnnolLXbqP/nycg0dugZxS+LP4/A8uIiKT538/t5NdR0/F9ZiLZxbw16suO+82XV1d9Pf3s3XrVtatWxfz3gc+8AHa29vP2ufb3/42H/rQhwA4cuQIVVVVQ+9VVlayefPmiRePgv78zGDJ7fDbv4ajb8LMK/2uSEREklB9fT39/f0sXLjwrKfWvfzyyz5V5QlU0EdPUxs3tV+Gl78Dr9wPn/3n+B1XRETi7kJX3pOlqamJffv2jdrcPpYr+oqKCg4fPjz0XkNDAxUVFXGpLVBB75x7Dniutrb27rgdNKsAln8ZXv0uNL8DpfPidmgREQmGo0eP8ulPf5qBgQFaW1spLi4eem8sV/TLly+nvr6ed999l4qKCp588kl+/OMfx6U2DcYbi2u/AqF0eO0BvysREZEk09fXx9atWzl27Bhr1qwhHA6P+xhpaWk8+OCDfOQjH2HRokV89rOf5bLL4tM6Eagr+kmTPx2WfR62/Rhu/u/eaxEREbyQ/s53vjPh43zsYx/jYx/7WBwqiqUr+rG6/msw0Aubf+B3JSIiImOmoB+r0nmw+DbY8gh0aXY8ERGZGhT043HDOug+BXX/5HclIiIiY6KgH4+Zy2DuzbDp+9A7vvmARURE/KCgH68b74WO47D9Sb8rERERuSAF/XjN+QO4ZBm8+gAM9PtdjYiIyHkp6MfLDG5cBy3vwJ5f+F2NiIjIeSnoL8aiT0DJXHjl772Jb0RERJJU0ge9mS0ys4fN7Gkz+4rf9QAQCsP1f+lNdPPu7/2uRkREkkBnZydr165l06ZNF7X/l7/8ZcrLy1myZElc6/Il6M3sUTNrNLMdI9bfamZ7zWy/mX0DwDm32zm3BvgscIMf9Y5q6Z2QW+5d1YuISMp7+OGH6erq4pVXXrmo/b/4xS/ywgsvxLkq/67oHwNujV5hZmHgIeCjwGLgTjNbHHnvE8AvgV8ltszzSM+C674KB34HR7f5XY2IiPjshRdeYMGCBSxbtuyi9r/pppsoKSmJb1H49Kx759zvzax6xOprgP3OuQMAZvYkcBuwyzn3LPCsmf0SGHU6HzNbDawGmDVr1mSVHmtwCttXvwuf0UN0RER89/w34Njb8T3mjMvho/edd5Ouri76+/vZunUr69ati3lvLNPUTqZkmtSmAjgc9boBuNbMbgY+BWRynit659x6YD1AbW1tYkbIZRV6Yf/aA9DyP70BeiIiknLq6+vp7+9n4cKFpKenx7w3lmlqJ1MyBf2onHMvAS/5XMa5rfiK96S8174HH1d/vYiIry5w5T1Zmpqa2LdvH88888xZ7+mKftgRoCrqdWVk3ZiZ2SpgVU1NTTzrOr/8Gd7AvDd/BDd/E/LKE/e3RUQkKRw9epRPf/rTDAwM0NraSnFx8dB7fl/RJ9PtdVuA+WY2x8wygDuAZ8dzAOfcc8651YWFhZNS4Dnd8HXo74FNmsJWRCTV9PX1sXXrVo4dO8aaNWsIh8MXdZw777yT6667jr1791JZWckjjzwSl/p8uaI3syeAm4FpZtYA/LVz7hEzWwv8GggDjzrndo7zuIm/oofIFLaf8KawvfFeyCpI7N8XERHfpKWl8Z3vfGfCx3niiSfiUM3ZfLmid87d6Zy7xDmX7pyrdM49Eln/K+fcpc65ec65v72I4/pzRQ+RKWzb4I3HEv+3RUREziGZmu6ntoqrvAlvNj4Efd1+VyMiIgIELOjNbJWZrW9ra/OngBvXQccx2P4Tf/6+iIjICIEKel+b7gHm/iHMuMJ7gI6msBURSRiXIhOMXcznDFTQ+87MG4zXvB/2/NLvakREUkJWVhbNzc2BD3vnHM3NzWRlZY1rv2S6jz4YFt8GxXPg1fth0Sov/EVEZNJUVlbS0NBAU1OT36VMuqysLCorK8e1T6CC3rfb66KFwnDDX8Iv7oWDL8Ocm/yrRUQkBaSnpzNnzhy/y0hagWq6972PftDSz0emsL3f3zpERCTlBSrok0Z6FqxYA+9sgPe3+12NiIikMAX9ZKm9CzLyvb56ERERnwQq6H2/jz5adhHUfgl2/hxa3vW7GhERSVGBCvqk6aMftOKrEEqDjQ/6XYmIiKSoQAV90im4BJbeAW/+K3QE/7YPERFJPgr6yXb9171n329+2O9KREQkBQUq6JOqj37QtBpY9HHY8kPobve7GhERSTGBCvqk66MfdMO90KUpbEVEJPECFfRJq/JqqP6AprAVEZGEU9Anyo33Qvv7sP0pvysREZEUoqBPlHkfjJrCdsDvakREJEUo6BPFDG74OjTXw95f+V2NiIikCAV9Ii3+JBRXwyt/DwGfN1lERJJDoII+KW+vixZOg+u/Bkfq4NCrflcjIiIpIFBBn7S310Vb9gXILdMUtiIikhCBCvopIT0brl0D+38Dx972uxoREQk4Bb0flt8FGXneCHwREZFJpKD3Q3YxXP1F2PFTaD3odzUiIhJgCnq/XHcPWBhe0xS2IiIyeRT0fimYCUs/B2/+i6awFRGRSaOg99PgFLav/4PflYiISEAp6P1Udiks/CN4/YfQ3eF3NSIiEkCBCvqkf2DOaG68F7pOwtZ/9rsSEREJoEAF/ZR4YM5IlbXeFLavPQh9PX5XIyIiAROooJ+yblgH7Ufh7X/zuxIREQkYBX0yqFkJ0y/XFLYiIhJ3CvpkYAY3roMTe2Hf835XIyIiAaKgTxaLPwlFszWFrYiIxJWCPlkMTmHbsAUOveZ3NSIiEhAK+mSy7AuQMw1evd/vSkREJCAU9MkkI8ebwrb+P+DYDr+rERGRAFDQJ5tr/kJT2IqISNwo6JNNzBS2h/yuRkREprikD3oz+6SZ/dDMfmJmH/a7noRY8VWwEGx8yO9KRERkivMl6M3sUTNrNLMdI9bfamZ7zWy/mX0DwDn37865u4E1wOf8qDfhCivgis/B1seh84Tf1YiIyBTm1xX9Y8Ct0SvMLAw8BHwUWAzcaWaLozb5q8j7qeGGv4S+M/D6er8rERGRKcyXoHfO/R5oGbH6GmC/c+6Ac64HeBK4zTx/BzzvnNua6Fp9U7YAFvyRF/SawlZERC5SMvXRVwCHo143RNZ9DfgQcLuZrTnXzma22szqzKyuqalpcitNlBvvhTOtXhO+iIjIRUimoB+Vc+4B59zVzrk1zrmHz7PdeudcrXOutqysLJElTp6q5TD7Bm9QXn+v39WIiMgUlExBfwSoinpdGVk3Zma2yszWt7W1xbUwX92wDk41wNtP+12JiIhMQckU9FuA+WY2x8wygDuAZ8dzAOfcc8651YWFhZNSoC/m3wLll3mPxdUUtiIiMk5+3V73BLARWGBmDWZ2l3OuD1gL/BrYDTzlnNvpR31JZXAK26Y9UP9rv6sREZEpxlyApkQ1s1XAqpqamrvr6+v9Lid++vvge1dC/iVw13/4XY2IiCQZM3vDOVc72nvJ1HQ/YYFsugdvCtvrvgaHN8OhjX5XIyIiU0iggj7QrvwTyCnVFLYiIjIugQr6QI66HzQ4he2+F+D4Lr+rERGRKSJQQR/YpvtBy/8C0nM1ha2IiIxZoII+8HJK4Oo/hx1Pw8n3/K5GRESmgEAFfaCb7gddd4/3U1PYiojIGAQq6APfdA9QWAmXf9Z7/v3pkfMCiYiIxApU0KeMG74Ovac1ha2IiFyQgn4qKl8ICz4Gmx+Gnk6/qxERkSQWqKBPiT76QTesi0xh+y9+VyIiIkksUEGfEn30g2ZdC7Ouh40PagpbERE5p0AFfcq5cR20HYYdP/W7EhERSVIK+qls/oehfDG8cr+msBURkVEp6KcyM6+vvmk31GtWOxEROVuggj6lBuMNWvIpKKzSZDciIjKqQAV9Sg3GGxROh+vWwnsb4b1NflcjIiJJJlBBn7Ku+lPILvH66kVERKIo6IMgIxeu/S+w73lo3O13NSIikkQU9EFxzWpIz4FXH/C7EhERSSIK+qDIKYGr/hzefgpOHva7GhERSRKBCvqUHHUfbXAK203f97cOERFJGoEK+pQcdR+tqAou/wy88ZimsBURESBgQS9ETWH7Q78rERGRJKCgD5ryRXDprZrCVkREAAV9MN14L5xpgTf/1e9KRETEZwr6IJq1AqpWwGuawlZEJNUp6IPqxnXQ9h7s/LnflYiIiI8U9EE1/yNQtsh7LK5zflcjIiI+UdAHVSjkjcBv3An1v/G7GhER8Umggj7lH5gz0uW3Q0GlprAVEUlhgQr6lH9gzkjhdLh+LRx6FQ6/7nc1IiLig0AFvYziqj+D7GJNYSsikqIU9EGXkQvX/BfY+0to3ON3NSIikmAK+lRwzWpIy4bXNIWtiEiqUdCngtxSrwl/+0/gvc3Qe8bvikREJEHS/C5AEuT6tbD1cXj0w4BBYRVMq4HS+TBtPpTWeD8LKsDM72pFRCROFPSpomgW3LMZjtTBif3QXA8n6uHwj6CnY3i79BwonQfTLo09CSitgcw8/+oXEZGLoqBPJcWzvSWac9D+vhf6zfXDJwENdbDjZ0DUU/XyZ45oBZjvvS6sglA4oR9FRETGRkGf6sygYKa3zP2D2Pd6u6DlwPDVf/N+7+eOp6Er6qFE4UyvFWCw+T+6JSC7KKEfR0REYino5dzSs2D6Ym+J5hx0nog6AYi0BDTugj2/BNc/vG1u2fCVf3RLQHE1hPX1ExGZbPovrYyfGeSVecvs62Pf6++F1oNwYl/sScCeX8Lp5uHtQmlQPMcbCzDyJCC3NKEfR0QkyJI+6M1sLvA/gELn3O1+1yMXEE73Anva/LPfO90y3Pwf3R2w/zfQ3zO8XXbx2XcDlM6HkjmQlpm4zyIiEgC+BL2ZPQp8HGh0zi2JWn8r8F0gDPyjc+4+59wB4C4ze9qPWiWOckog5xqouiZ2fX8ftL0XezdA837YvwG2/Wh4OwtB0Wwv+MsXwbI/gbJLE/sZRESmGL+u6B8DHgQeH1xhZmHgIeAWoAHYYmbPOud2+VKhJE44DUrmegsfjn2v65QX+s37o7oD9sM7v4NXvwvzPwwrvgpzb9b9/yIio/Al6J1zvzez6hGrrwH2R67gMbMngdsABX0qyyqAiqu8JVpHE9Q9Clt+CP/ySSi/DK77Kiy53RtEKCIiQHI9ArcCOBz1ugGoMLNSM3sYuNLMvnmunc1stZnVmVldU1PTZNcqfssrg5v/G9y7E277vrfumXvg/iXw0n3eiYCIiCRV0I/KOdfsnFvjnJvnnPvWebZb75yrdc7VlpWVJbJE8VNaJlz5BfjKq/Bnz8DMK+Glb8HfXwbPrIXG3X5XKCLiq2QadX8EqIp6XRlZN2ZmtgpYVVNTE8+6ZCow8/rp594MTftg8w9g2xPw5r/AvA/CinugZqX68UUk5Zhz7sJbTcYf9vrofzE46t7M0oB9wEq8gN8CfN45t3O8x66trXV1dXVxrFampNMtXj/+6z+EjmNQthBWfAWu+BykZ/tdnYhI3JjZG8652tHe86Xp3syeADYCC8yswczucs71AWuBXwO7gafGG/JmtsrM1re1tV14Ywm+nBK46b/Curfhj/8Bwhnw3Ne9Zv0X/xbaj/tdoYjIpPPtin4y6YpeRuUcHHwFNn0f9j7vPdzn8s94t+fNWHLh/UVEktT5ruiTqY9eZHKZwZwPeEvzO7DpB94Debb9CObcBNethZpbIJT0Y1RFRMYsUFf0UYPx7q6vr/e7HJkKzrTCG4/B5vXQftR71O6Kr8DSOyEjx+/qRETG5HxX9IEK+kFqupdx6++FXc/Axgfh6Jve8/av/hJcsxoKLvG7OhGR80q6wXgiSSecDpffDnf/Dr70AlTfCK/8Pdx/OfxsNRzd5neFIiIXRX30ItHMYPZ13tLyLmz+B+9e/O0/gdk3eo/ZvfRWCIX9rlREZEwC1XSvPnqZFF1tsPVxL/TbDnuT71z7FVj2ecjM87s6ERH10YvERX8f7H7Wuz2vYQtkFcLVX/T68Qsr/a5ORFKY+uhF4iGcBks+BX/xW7jrNzD3D+G178H9V8DTd8GRN/yuUETkLOqjF7kYVdd4S+sheH2917S/42moWuH14y/8uPrxRSQpBKrpXn304puuU96Ddzb9AE4egqLZcO0auPJPIKvA7+pEJODURy+SKAP9sOeXsPEhOLwJMgvgqj/z+vGLZ/tdnYgElProRRIlFIbFn4C7fg13vwjzP+xd5T+wDJ76czj8ut8VikiKUdCLTJaKq+H2R2Dddrj+a3Dgd/DILfDDlbDjZ94ofhGRSXbBoDdPVSKKEQmkwkq45W/g3l3w0f8LZ1rg6S95V/mvPgBnTvpdoYgE2Jj66M3sbefc5QmoZ0I0GE+mhIF+2PcCbPw+HHoFLATll0FlbWRZ7k2uo1n0RGSMJjwYz8z+GXjQObcl3sVNBg3Gkynj6DZv8N6ROmh4A7rbvPWZhVBxlRf6lcu9E4CcEl9LFZHkFY/56K8FvmBmh4BOwADnnLsiTjWKpKaZy7wFYGAAmuuhoc578l5DHbz8bXAD3vslc4eDv+JqmHG5NxmPiMh5jDXoPzKpVYiI11RftsBbrvyCt667A97fNhz8B17yJtgBSMuCS5bFNvkXVHgT84iIRIz5PnozWwp8IPLyZefcW5NW1QSp6V4CyzloaxgO/iN1XvN/f7f3fv4lXuhXRIJ/5jLIyPWzYhFJgAk33ZvZ14G7gZ9FVv2rma13zn0vTjWKyFiYQVGVtyz5lLeurweOv+318Tds8Zbdz0W2D8P0xVFN/rVQWqOBfiIpZKyD8bYD1znnOiOvc4GNydZHr1H3IhGdJ7xJdgaD/8hW6D7lvZdVOHzFX1nr9fdroJ/IlBaPUfdvA8udc12R11nAlmS95U5N9yIjDAzAiX1Rwf8GNO4aHuhXWhMJ/8gJwPTLNNBPZAqJx6j7fwI2m9nPI68/CTwSh9pEJBFCIShf6C1X/am3rrsdjr4ZCf834J0NsP1J7720bK9/fzD4K5dDwUzfyheRi3fBK3ozCwErgC7gxsjql51zb05ybRdNV/QiF8E5OPne8BV/wxZ4/y3o7/Hez58ZFfy13oj/jBxfSxYRz4Su6J1zA2b2kHPuSmBr3KsTkeRg5s2wVzwbLr/dW9fXDcfejrq3fwvsfjayfRhmLInq71/u3euvgX4iSWWsffTfBjYCP3NTYF5bXdGLTKKOpsiT/KIG+vV0eO+F0qHgEu9+/oKZkaUy8rMCCisgt8yb5U9E4iYeg/HagVygD68Jf/DJeAXxLDReFPQiCTTQD017vdBvfRfajsCpo3Aq8nPwHv9BoTTvfv+Yk4HIScDgurzpOhkQGYcJNd1H+uhvdc69GvfKRGTqC0Xu1Z+++Oz3nIPTzcOh39YQOQmInAi8vw32/gr6umL3s3DkZGBm7AlAdAtB3nQIj3U8sUjqGmsf/YPAlQmoR0SCxAxyp3nLJUtH38Y5ONMadRJwZPjE4NQReH877H0B+s6MOHYY8meM0kUw05sauGAm5M3QyYCkvLH+P2CDmX2aJO+jj3pgjt+liMhYmXkP7MkpgUvO8QyuwZOBkScCbZHfj++C+t9A7+kRxw55V/5DLQKDXQSR3wsqvJMFPTNAAmw8ffQ5QD/qoxeRZOQcdJ0c7hoY2U1w6oh3YtDbOWJHi5wMjOgmKJnrPUioeA6kZ/nxiUTGLB4PzCkEvgDMcc79jZnNAi6JV4EiIhNmBtnF3jL9stG3cc57FHDMgMGoFoIT9fDOS9DTHn1gKKyC0kjwl9ZAyTwonQdFs9U1IElvrN/Qh4AB4IPA3wDtwE+B5ZNUl4hI/Jl5z/rPKhx98OCgMye9Owia34Hm/cM/t/8bdLcNbxdKg+Lq2PAvnee9zp+pZwpIUhhr0F/rnLvKzN4EcM61mlnGJNYlIuKf7CLIvhJmjhiDPHgXQXT4N++HlgNw4D9jBwymZUea/6PCf/CEIHead9IhkgBjDfpeMwsDDsDMyvCu8EVEUkf0XQSzVsS+NzAA7e/Hhn/zfmjc7d1CONA3vG1mYewJQHRrQFZhYj+TBN5Yg/4B4OdAuZn9LXA78FeTVpWIyFQTCnmD+QorYO4fxL7X3wcnDw2H/2BrwHub4e2niVxDeXLLIlf/8yInAIO/z4X07IR+JAmGMY26BzCzhcBKvBH3G5xzuyezsInQqHsRmTJ6u0aMB4hqDeg4HrttYdXw3QDR3QFFs5LzFkHnvEmR+rq8eRN6z3g/+7qilsjr3q4R60d5LyMXpl0K5YugbKF3S6YA8Rl1j3NuD7AnblWJiIh36175Im8Zqbt9+AQgujVgx9PQFTUo0MKRQYHzzm4NKKjwug2GgvPMKOE6MnxHvhcd0GMN7MhrJvLoFfNaMdKyvKWrLfb2yNwyL/DLFsT+zC3TGIgoui9ERCRZZebDzGXeEs05ON0CLSPuCmh+Bw6+cvaDgybEvJBNj4RtWqY30DAtcziAs4tjXw9td5790kduO8p7obTYwHbOez5C015o2hNZ9sL2p7zbJgdlF0eCf2HsCUD+jJQ8AVDQi4hMNWaQW+otVdfEvudcZFBgJPzb34dwxnCgDl0hjxLM6dlnrw+nJ084mkFRlbfM/9Dweueg/VhU+EdOAHb+3HuI0qDMwkjoLxg+CShf6LV6JMtnnARj7qP3i5nlAt8HeoCXnHM/utA+6qMXERGcg86m4eBv2gONkROB0yeGt8vIO7v5v2wBFM6aMs9CiEsffTyZ2aPAx4FG59ySqPW3At8FwsA/OufuAz4FPO2ce87MfgJcMOhFREQwg7xyb5lzU+x7nSeiugAiP/dvgG1REZOeA9PmQ9mi2BOA4uopNY2yX033jwEPAo8Projcp/8QcAvQAGwxs2eBSuDtyGb9iS1TREQCafB5CNU3xK4/0wpN+6Bp9/AJwMGXYfuTw9uEM73R/yNbAUrmJOXdD74EvXPu92ZWPWL1NcB+59wBADN7ErgNL/QrgW3A1GhDERGRqSm7GGZd6y3Ruk7BiX2xYwAaXvfugBgUSo+0AIw8AZgHaf49TDaZBuNVAIejXjcA1+I9rOdBM/sj4Llz7Wxmq4HVALNmzZrEMkVEJOVkFUBlrbdE6+6InABEdQMc3QY7/52hWwst7N3yGD0IsOIq75kICZBMQT8q51wn8KUxbLceWA/eYLzJrktERITMPC+0K66KXd97xpsNsWnvcDdA427Y80twA7DiHrj1/ySkxGQK+iNAVdTrysg6ERGRqSU9Gy65wlui9XV7tz2m5ySslGTq894CzDezOZGZ8e4Anh3PAcxslZmtb2tru/DGIiIiiZaWCdMv8wbuJYgvQW9mTwAbgQVm1mBmdznn+oC1wK+B3cBTzrmd4zmuc+4559zqwkLN/iQiIgL+jbq/8xzrfwX86mKPa2argFU1NTUXewgREZFASaam+wnTFb2IiEisQAW9iIiIxApU0GswnoiISKxABb2a7kVERGIFKuhFREQkVqCCXk33IiIisQIV9Gq6FxERiRWooBcREZFYCnoREZEAC1TQq49eREQkVqCCXn30IiIisQIV9CIiIhJLQS8iIhJgCnoREZEAC1TQazCeiIhIrEAFvQbjiYiIxApU0IuIiEgsBb2IiEiAKehFREQCTEEvIiISYAp6ERGRAAtU0Ov2OhERkViBCnrdXiciIhIrUEEvIiIisRT0IiIiAaagFxERCTAFvYiISIAp6EVERAJMQS8iIhJgCnoREZEAC1TQ64E5IiIisQIV9HpgjoiISKxABb2IiIjEUtCLiIgEmIJeREQkwBT0IiIiAaagFxERCTAFvYiISIAp6EVERAJMQS8iIhJgCnoREZEAS/qgN7O5ZvaImT3tdy0iIiJTzaQGvZk9amaNZrZjxPpbzWyvme03s2+c7xjOuQPOubsms04REZGgSpvk4z8GPAg8PrjCzMLAQ8AtQAOwxcyeBcLAt0bs/2XnXOMk1ygiIhJYkxr0zrnfm1n1iNXXAPudcwcAzOxJ4Dbn3LeAj09mPSIiIqnGjz76CuBw1OuGyLpRmVmpmT0MXGlm3zzPdqvNrM7M6pqamuJXrYiIyBQ22U33E+acawbWjGG79cB6gNraWjfZdYmIiEwFflzRHwGqol5XRtZNmJmtMrP1bW1t8TiciIjIlOdH0G8B5pvZHDPLAO4Ano3HgZ1zzznnVhcWFsbjcCIiIlPeZN9e9wSwEVhgZg1mdpdzrg9YC/wa2A085ZzbOZl1iIiIpKrJHnV/5znW/wr4Vbz/npmtAlbV1NTE+9AiIiJTUtI/GW881HQvIiISK1BBLyIiIrECFfQadS8iIhIrUEGvpnsREZFYgQp6ERERiRWooFfTvYiISKxABb2a7kVERGIFKuhFREQkloJeREQkwAIV9OqjFxERiRWooFcfvYiISKxABb2IiIjEUtCLiIgEmIJeREQkwAIV9BqMJyIiEitQQa/BeCIiIrECFfQiIiISS0EvIiISYAp6ERGRAFPQi4iIBFiggl6j7kVERGIFKug16l5ERCRWoIJeREREYqX5XYDIeBxr62LLwRYOnujk6upilleXkB7W+aqIyLko6CVpDQw46hs72HKwhbqDLdQdaqWh9UzMNvlZafzBpWWsXFTOzZeWU5yb4VO1IiLJSUEvSaOrt5/tDW1sOdjCG4daqTvYwqmuPgDK8jNZXl3Ml2+Yw/LqEmaV5rDpQDMv7m7kxb2N/GL7+4QMrppVzMpF01m5qJz55XmYmc+fSkTEX+ac87uGuKutrXV1dXV+lyEX0NLZMxTodYdaebuhjZ7+AQBqyvNYXl1M7ewSlleXUFWSfc7QHhhw7Djaxm93N/LinuPsOHIKgMribFYuLOeDi6azYm4JmWnhhH02EZFEMrM3nHO1o76noJdEcM7xXstpthwcDvb9jR0AZIRDXF5ZSG11Mctnl3D17OIJNcEfa+vid3sb2bD7OK/sP0FX7wA5GWE+MH8aKxdO5+aFZZTnZ8Xro4mI+E5BLwnX1z/ArvdPseVgK28camHLwVaa2rsBKMhKo7a6xAv26hIurygkK31yrra7evvZ+E4zG/YcZ8PuRt5v6wJgaVWRd7W/sJzLZhaoiV9EprSUCXozWwWsqqmpubu+vt7vclJKR3cfb77XSt3BVuoOtfDmeyc53dMPeE3oyyPBXju7hPnleYRCiQ9W5xy732/nxT3H+e3uRt5qOIlzMKMgiw8uKmflwnKunzeN7Aw18YvI1JIyQT9IV/ST7/iprshoeC/Ydx09xYCDkMGiSwpign1GYXI2kze1d/PS3kZe3NPI7/c10dnTT2ZaiBtqprFykXe1f0lhtt9liohckIJeJmRgwLG/qcML9YMtbDnUwuEW7za37PQwV84qora6hOXVxSyrKiI/K93nisevu6+f199tYcPuRjbsOT70+RZfUsDKReWsXDSdKyoKfWmJEBG5EAW9jEtXbz87jrTFDJxrO9MLwLQ87za32uoSamcXs3hmQeAeWOOcY39jBxv2NPLi7kbqDrUw4GBaXgZ/uKCclYvKuXF+GXmZujtVRJKDgl7O6+Rp7za3wWDfHnWb27yy3EgzvBfss0tzUm7g2snTPby0t4kNexp5aW8j7V19ZIRDXDu3hJULvav9qpIcv8sUkRSmoJchzjkOt5yhLjISvu5gC/WR29zSw8blFYUsr/Zucbt6djGleZk+V5xcevsHeONQKxt2H2fDnkYONHUCML88b+hBPVdWFZEWsFYOEUluCvopqK9/gO6+waWf7t4BuiI/z7uub4CuXu+nt75/6Did3X3sONJGY+Q2t/ysNGpnDzfDL60qmrTb3ILq3ROdvLjHu2f/9Xdb6BtwFOWkc/OlZaxcNJ2bLi2jMHvqjVkQkalFQT8Bnd19nOmNCtAR4dnd209X5Gd0MHf1DodxdDAP7ds7YrsRx+sfmNi/S0Y4RGZaiMz0EJlp4cjvYRZMz+PqyMC5S8vzNbgsjk519fLyvhNs2HOc3+1ppPV0L+GQsby6mJULvav9uWV5fpcpIgGkoJ+AT33/Vba+d/Ki9s1IC5EVCdjMtEjwpoXJGgzf9FHWRYXzBddFjpE14lgZaSHCCnBf9Q84th1u9Ubx725k7/F2AOZMy+WDC7179pfP0cx7IhIfCvoJ+MX2o7R29pwVzNFXyrHB7a3LCId0tSxDDrecjjyWt5GN7zTT0z9AfmYaNy0o4+ZLy7huXimVxRrQJyIXR0EvkkQ6u/t4Zf8JXtzdyIY9jZzo8MZMVBZns2JuaWQpUfCLyJgp6EWS1MCAY+/xdjYdaGbTgWY2v9vCydPeMwsU/CIyVlM66M3sk8AfAQXAI865/7jQPgp6maoGBhz7GtvZ9E4zmw60sPndZlqjgv/aOV7or5hbqnv3RWSIb0FvZo8CHwcanXNLotbfCnwXCAP/6Jy7bwzHKga+7Zy760LbKuglKM4X/BVF2UNX+wp+kdTmZ9DfBHQAjw8GvZmFgX3ALUADsAW4Ey/0vzXiEF92zjVG9vt/wI+cc1sv9HcV9BJUCn4RGY2vTfdmVg38IirorwP+l3PuI5HX3wRwzo0M+cH9DbgP+I1z7rfn+TurgdUAs2bNuvrQoUPx/BgiSUnBLyJw/qD3Y1aOCuBw1OsG4NrzbP814ENAoZnVOOceHm0j59x6YD14V/RxqlUkqYVCxsIZBSycUcAXb5jDwICjvrFjaHDfi3uO89OtDYAX/NdGQv+6uaVUFmen3LwFIqko6affcs49ADzgdx0iU0EoZCyYkc+CGfn8+fXVZwX/S3ub+NnWI4CCXyRV+BH0R4CqqNeVkXUTZmargFU1NTXxOJzIlKfgFxE/+ujT8AbjrcQL+C3A551zO+P1NzUYT2RsBgYc+5uGg3/TgRZaOnsAmFmYFXUffylVJQp+kWTl56j7J4CbgWnAceCvnXOPmNnHgPvxRto/6pz72zj9vcEr+rvr6+vjcUiRlKLgF5mapvQDcy6GruhF4sO52KZ+Bb9IclLQi0hcRAf/5gMtbDrQTHNU8NdMz2dabgYluRmU5GVQmptBaW7m0O8luRnkZabphEAkzlIm6NV0L5JYzjn2D13xt3C49TTNHT00d3bT1Tsw6j4ZaaGh0C/JHTwByKQ06mTA+907QcjXiYHIBaVM0A/SFb2I/0739NHc0UNLp7ec6Oge+r158GdH99Dvp3v6Rz1ORjhEcW46JbmZTMu7wAlCbiYF2ToxkNSTbA/MEZEUkJORRk5J2pifyNfV2++FfkcPJzq7aemIPinojrQU9HCo+TQtnT10dPeNepz0sFGcM9wyUJKbGelCiOpOyMscOlkoyEonFNKJgQSXgl5EkkJWepiKomwqirLHtH1Xb/+IFoLhk4GWjuF1b7eepLmjh/ZznBiEQ96JwcjWgrL8zOElL4vyAu/kID0ciufHFpl0gQp6PTBHJHVkpYeZWZTNzDGeGHT39dPa2UtzZ3dUd4J3MuB1I3jrdh49RXNHN6e6zj4xMIOSnKiTgLzM2BOC/EzKIycG6kKQZKE+ehGRUXT19nOio5um9sgS/XvkdeMp72dP39kDDzPCIcryM5k24oSgPKalwPuZlR724RNKkKiPXkRknLLSw1QW51BZfP4xBs45TnX1nXVC0NjeNbSuofU02w630tzZw2jXVvlZaWe1EJTnZ511QlCSm0FY4wlknBT0IiITYGYUZqdTmJ1OTXneebft6x+gpbOHxvZztxTsPHqKpvbuUQcbhgxK8y7cQlCWn5mw5xU45+gfcAw4GHAuskR+Hxj99/4Bh3PQH9neO0bU/tG/Dx5vwFGUk0FNeZ5OdsYpUEGvPnoRSWZp4RDlBVmUF2RdcNvO7r5zdh0MnijsPdbOiY5u+gbObibISve6DopzMgCGwng4mGPDNjZciQTzcCgPOBfZNjbcEy0nI8ySikKWVRWxtLKIpVWFVBTpqYznoz56EZEpbGDAcfJMb9QJQZc3diBycnDydC8hg5AZoZCN+N17HTbDzAiHvPeifx9eiNln5PHCIcMGfzfv93DoPPuaERrxN8Ihr4XkrN8j+x1v7+Ktw2281XCSnUdPDY2NKM3NYGlU8C+tLKI4N8Pnf5nEUh+9iEhAhUI2dFvgghn5fpcz6f74ykoAevoG2HusnW0NJ3nrsLf8bm/j0BiI2aU5XFFZxNJK7+r/spmFZGek5qBHXdGLiEggtHf1suPIKd6KCv+jbV2A17qwYHo+S6uKWFZVyBWVRcwvzyMtIM9F0CNwRUQkJTWe6uKthja2N5xkWyT8B5+RkJ0e5vKKQq+5P9L0X1k8Nfv7UyboNamNiIicj3OOg82neetwJPhH9PeX5GawtDIS/JHwL5kC/f0pE/SDdEUvIiJj1dM3wL7j7UNX/G81nKS+sWOov39WSQ5XRPr6l1YVsSQJ+/sV9CIiIuPQ0d3H25Emf6/Pv40jJ88AXn//pdPzWRYZ4X9FZRGXTve3v19BLyIiMkGN7V1sj9ze91ZDG28dPknbmV7Ae27B5RWFkVv8ilhWldj+fgW9iIhInDnnONR8mreiBvrtiOrvL85JH+rnX1ZVxBWVhZTmZU5KLbqPXkREJM7MjOppuVRPy+W2ZRUA9PZ79/cP3+LXxn/uqx/q768qyeaKyiJWXTGTW5fMSEidgQp6PQJXRET8lB4OsaSikCUVhXzh2tmA19+/40jb0EC/be+dZH55XsKCXk33IiIiCdY/4OI6Oc/5mu6D8UggERGRKSSRM/Ap6EVERAJMQS8iIhJgCnoREZEAU9CLiIgEmIJeREQkwBT0IiIiARaooDezVWa2vq2tze9SREREkkKggt4595xzbnVhYaHfpYiIiCSFQAW9iIiIxFLQi4iIBJiCXkREJMAU9CIiIgEWyNnrzKwJOBTHQxYCfgzln4y/O9FjXsz+491nLNvHY5tpwIlx1JWs9P2c+DHGs89Yt53odzQo30/Qd3Si+49ln9nOubJR33HOabnAAqwPyt+d6DEvZv/x7jOW7eOxDVDnx79rsv2bJtPfjccxJ/s7OtZtJ/odDcr3M17/rsnyd6fCf0NHLmq6H5vnAvR3J3rMi9l/vPuMZft4bRME+n5O/Bjj2Wes2+o7Okzf0YntP6G/Gcime5GxMLM651yt33WIjEbfT4kXXdFLKlvvdwEi56Hvp8SFruhFREQCTFf0IiIiAaagFxERCTAFvYiISIAp6EUAM5trZo+Y2dN+1yIyGjP7pJn90Mx+YmYf9rsemToU9BJYZvaomTWa2Y4R6281s71mtt/MvgHgnDvgnLvLn0olVY3zO/rvzrm7gTXA5/yoV6YmBb0E2WPArdErzCwMPAR8FFgM3GlmixNfmghwcd/Rv4q8LzImCnoJLOfc74GWEauvAfZHruB7gCeB2xJenAjj+46a5++A551zWxNdq0xdCnpJNRXA4ajXDUCFmZWa2cPAlWb2TX9KEwHO8R0FvgZ8CLjdzNb4UZhMTWl+FyCSDJxzzXh9nyJJyTn3APCA33XI1KMrekk1R4CqqNeVkXUiyULfUYkrBb2kmi3AfDObY2YZwB3Asz7XJBJN31GJKwW9BJaZPQFsBBaYWYOZ3eWc6wPWAr8GdgNPOed2+lmnpC59RyURNKmNiIhIgOmKXkREJMAU9CIiIgGmoBcREQkwBb2IiEiAKehFREQCTEEvIiISYAp6ERGRAFPQi8hFicympv+GiCQ5/Z9URMbMzKrNbK+ZPQ7sIPaZ7CO3rTGz35rZW2a21czmJa5SERmkJ+OJyJiZWTVwALjeObfpAttuBu5zzv3czLKAkHPudALKFJEoCnoRGbNI0P/OOTfnAtvlA7udc5UJKUxEzklN9yIyXp1+FyAiY6egF5EJMbPHzeya6HXOuXagwcw+Gdkm08xy/KhPJNUp6EVkoq4Ajo6y/k+BvzSz7cBrwIyEViUigProRWQCzKwAeMQ59xm/axGR0SnoRUREAkxN9yIiIgGmoBcREQkwBb2IiEiAKehFREQCTEEvIiISYAp6ERGRAFPQi4iIBNj/B02iafzwEC/WAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure(figsize=(8, 6))\n",
    "plt.plot(rs, hist0, label=r\"$\\ell=0$\")\n",
    "plt.plot(rs, hist1, label=r\"$\\ell=1$\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.xlabel(\"r, c\")\n",
    "plt.ylabel(\"error\")\n",
    "plt.title(r\"\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# まとめ\n",
    "量子インスパイアード古典アルゴリズムにおける，$C$の特異値分解までをみてきました．このノートブックでは，特異ベクトルの復元に，行列$A$や$R$と$C$の左特異ベクトルに関する行列-ベクトル積を行なっているため，元々の行列$A$の次元に関して多項式の計算量を必要とします．`3_SQ(x).ipynb`では，このような計算量をかけずにSample, Query, Normそれぞれを実行することについて説明します．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 参考文献\n",
    "- András Gilyén, Seth Lloyd, and Ewin Tang, ''Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension,'' arXiv:1811.04909, (2018). https://arxiv.org/abs/1811.04909\n",
    "\n",
    "- Ewin Tang. 2019. A quantum-inspired classical algorithm for recommendation systems. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019). Association for Computing Machinery, New York, NY, USA, 217–228. https://doi.org/10.1145/3313276.3316310"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.10.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}