lectures/lecture5/lecture5.ipynb

{
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  {
   "cell_type": "markdown",
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   "source": [
    "# Лекция 5\n",
    "\n",
    "# Часть 1. Симметричная задача на собственные значения и вычисление сингулярного разложения."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
    }
   },
   "source": [
    "## На прошлой лекции\n",
    "\n",
    "- Собственные векторы и собственные значения\n",
    "- Степенной метод и его сходимость\n",
    "- Разложение Шура и QR алгоритм"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## План на первую часть сегодня\n",
    "\n",
    "- Ускорение QR алгоритма\n",
    "- Концепция сдвигов\n",
    "- Подход \"разделяй и властвуй\" к вычислению сингулярных чисел\n",
    "- Бисекция\n",
    "- Метод Якоби"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сложность вычисления разложения Шура\n",
    "\n",
    "- Напоминаем, что мы хотим избежать сложности $\\mathcal{O}(n^4)$, которую получили при наивной реализации QR алгоритма\n",
    "- Это можно сделать, приведя матрицу к более простой форме\n",
    "- Эта форма призвана удешевить одну итерацию QR алгоритма"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Верхне-гессенбергова форма матрицы\n",
    "\n",
    "Матрица $A$ имеет верхне-гессенбергову форму, если\n",
    "\n",
    "$$a_{ij} = 0, \\; \\mbox{при  } i \\geq j+2.$$\n",
    "\n",
    "$$H = \\begin{bmatrix} * & * & * & * & * \\\\ * & * & * & * & * \\\\ 0 & * & * & * & *\\\\ 0 & 0 & * & * & *\\\\ 0 & 0 & 0 & * & * \\\\ \\end{bmatrix}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Приведение произвольной матрицы к верхне-гессенберговой форме\n",
    "\n",
    "С помощью отражений Хаусхолдера можно привести любую матрицу к верхне-гессенберговой форме:\n",
    "\n",
    "$$U^* A U = H.$$\n",
    "\n",
    "- Единственное отличие от вычисления разложения Шура заключается в занулении последних $n-2, n-3, ...$ элементов в первом, втором и так далее столбцах\n",
    "\n",
    "- Сложность такого приведения $\\mathcal{O}(n^3)$ операций\n",
    "\n",
    "- Если матрица приведена к верхне-гессенберговой форме, то одна итерация QR алгоритма имеет сложность $\\mathcal{O}(n^2)$ операций (например, используя вращения Гивенса) (почему?)\n",
    "\n",
    "- Также верхне-гессенбергова форма матрицы сохраняется после выполнения одной итерации QR алгоритма (проверьте!)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Случай симметричной (эрмитовой) матрицы\n",
    "\n",
    "- Если матрица $A$ симметричная (эрмитова), то $A = A^*$, тогда $H = H^*$ и верхне-гессенбергова форма оказывается трёхдиагональной матрицей\n",
    "\n",
    "- Далее мы будем говорить только о симметричном трёхдиагональном виде верхне-гессенберговой формы\n",
    "\n",
    "- Любая эрмитова матрица может быть приведена к трёхдиагональной форме с помощью отражений Хаусхолдера\n",
    "\n",
    "- **Основная идея:** трёхдиагональная форма сохраняется при выполнении QR алгоритма, и сложность одной итерации может быть сокращена до $\\mathcal{O}(n)$!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Одна итерация QR алгоритма\n",
    "\n",
    "- Одна итерация QR алгоритма имеет следующий вид:\n",
    "\n",
    "$$A_k = Q_k R_k, \\quad A_{k+1} = R_k Q_k.$$\n",
    "\n",
    "- Если $A_0 = A$ <font color='red'> симметричная трёхдиагональная матрица </font>, то эта форма сохраняется.\n",
    "\n",
    "- Давайте это проверим!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
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     "execution_count": 6,
     "metadata": {},
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
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     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAASAAAAEKCAYAAACytIjQAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDMuMC4yLCBodHRwOi8vbWF0cGxvdGxpYi5vcmcvOIA7rQAAHM5JREFUeJzt3Xl0TFkCBvCvQmInCKPFFsZLtyQSgowtdChpM8ceIqE0ia2t0cYSrQnRkbH1tDBomWFogyAhlrE32t5DCwkx3UZCYg2xJC0S8uYPp16nslZJlZsq3++cnJN6eXXvfa8qX91337uvVLIsyyAiEsBKdAOI6P3FACIiYRhARCQMA4iIhGEAEZEwDCAiEsbgADpz5gwcHR3h4eGB7OzsAn+XZRmLFy+Gh4cH3NzcsGnTJgDAo0eP8Ouvv5a+xWZi5syZcHR01FmWnZ2N+/fvC2lPbm4uUlJSlMfnzp2Do6MjoqOjhbSnJKmpqdBoNGjZsiU8PDzw+PHjYte/du0agoODoVar0bJlS3h5eSEoKAiXLl0qdH1HR8dCf9zd3dG3b19s2LABxrpCJTo6Go6Ojjh37pxRytNXSkoKHB0dERERobP89u3bOo8dHR0xc+bMd9k0RXlDn7Bnzx5UrlwZT548wdGjR/HJJ5/o/P3YsWOIjIxE165d0b17d7i7u+P48eP485//jJiYGFSuXNlojS/LfH190b59e+VxamoqAgICMGbMGPTv3/+dtiUjIwPDhw9Hly5dMHHiRABAs2bNsGjRIrRu3fqdtkVff/nLX/Cf//wHEyZMQJ06dVCrVq0i1/3uu+8QFhaGevXqoXfv3rC3t8e9e/cQGxsLX19fTJgwARMmTIBKpdJ5XtOmTTF27FidZXfv3kV0dDS++uorZGVlYfTo0SbZvnehVq1aWLRokc4H4d/+9jfExMTg0KFDAlv2G4MCKDs7GwcPHkSfPn2wZ88exMTEFAig69evAwA+//xzZcP37t2LZ8+eGanJ5qFVq1Zo1aqV8jglJQVJSUlC2vLkyRNcuXIFXbp0UZbZ2dmhT58+Qtqjj+vXr+Ojjz7C+PHji13v+++/R2hoKHr06IElS5agQoUKyt8+++wzfPnll1ixYgVq164Nf39/necWtQ+GDh2KTz75BJGRkRg+fDhsbGyMs1HvWOXKlQts35kzZ/D69WtBLSrIoEOw48eP49mzZ/Dw8ECnTp3www8/4OHDhzrr5OTkAACqVKlivFbSeycnJ6fE91Bubi5CQ0Nhb2+PRYsW6YQPAJQvXx6hoaGQJAlLlixBRkaGXnVXrVoV3bp1w9OnT/G///3vrbeBSmZQAO3evRsqlQpt27aFWq3G69evsWvXLuXvXl5eWLFiBQCgW7du8PLywsyZM3WWaTQaZf1ffvkF48ePR5s2beDq6orBgwfjhx9+0KlTo9EgMDAQX3/9NVq1aoX27dsrvazCXL9+HePGjUObNm3QsmVLDBo0CIcPHy60zBMnTqB///5wcXFB165dERERgdzcXJ119WljYfKOAUVHR2PYsGEAgODgYJ0u8dOnTxEaGorOnTvD2dkZPXv2xD//+U+d8YeIiAi4uLjg0KFD6NixI1q1aoVt27YBABISEjBx4kR06NABTk5OaN++PaZOnYp79+4BeDPW061bNwDAihUr4OjoiJSUlELHgF6/fo3IyEh4e3vD2dkZnTp1wty5c3XGX7TPO3XqFObNm4f27dvD1dUVn376KRITE0vcLyXVoS0/NTUV58+fL3QMQ+vixYtITU3FoEGDUKlSpULXKV++PDQaDTIzM3HkyJES26el71BBcnIyZsyYAU9PTzg7O6Ndu3YYO3Ysfv7552Kfl5GRgXnz5qFTp05wc3PD2LFjceHChVK9JjExMejVqxdcXFwQHBxcYAzIy8sL58+fR2pqaqH7df369ejevTtcXFzQq1cvHDhwQOfvjo6OWLt2Lb799lt07doVrq6u0Gg0SE5Oxs2bNxEYGAg3Nzd4eXlhw4YNeu0/vQ/BMjIycOzYMbi5ucHOzg5dunSBjY0NYmJiMHLkSADArFmzsHPnThw6dAjBwcFo0KABateujYyMDGVZ8+bNAbwJCn9/f9jZ2WHMmDGwtrbGnj17MHr0aCxduhR//OMflbovXryI5ORkTJs2DSkpKfj9739faBsvX76MYcOGoWrVqhgxYgSqVKmCXbt2Yfz48ZgzZw6GDBmirPvf//4XQUFB8PX1ha+vL/bs2YMVK1agVq1aynqGtLE4bdu2xdixY7F69Wr4+vrC3d0dAPDrr79i6NChuHv3Lvz9/VGvXj2cPXsWYWFhSEpKwty5c5UyXr16hdmzZyMwMBDZ2dlwd3dX2te4cWOMHj0alSpVwsWLF7Fr1y48ePAAGzduRLNmzRAcHIyFCxdCrVZDrVajVq1aSE1NLdDOKVOm4MCBA+jRoweGDRuGmzdvYvPmzTh79iy2bduG6tWrK+vOnj0bdevWxbhx4/D06VNERkZi1KhR+P7771G+fNFvq5Lq0I5NLVy4EDVr1sTYsWMLDOZrXbhwAQBKHMfy8PBQ1tfnsDM3NxcnT55E5cqV0aRJkyLXS0tLw6BBg1C1alUMHToUNWvWxLVr1xAVFYUbN27gwIEDsLIq+Bn/+vVrjBw5EpcvX1Zev71792LcuHEF1jXkNZk/fz769OmDgQMHon79+gXKmjVrFpYuXYr09PQCH4T79+/HmTNnMGTIENjY2GD9+vUICgrC9u3b4eTkpKy3ceNGVKpUCQEBAXj06BEiIyMxceJEPHnyBF27doVarca2bdvw1Vdf4cMPP0S7du2K39mynrZv3y5LkiT//e9/V5aNHj1aliRJjouLU5YtX75cliRJvn37drHLhg4dKnfv3l3OzMxUluXk5Mj+/v5yhw4d5JcvXyrrSZIknz17tsQ2Dhw4UHZzc5Pv3r2rLMvKypL79esnt2zZUn706JFOmUeOHNFZr23btrKvr6/BbSzMjBkzZEmSlMdnz56VJUmSd+zYobNfnJyc5MTERJ3nLl26VJYkSb527ZqyniRJ8vLly3XWmzNnjuzq6iqnp6frLJ8yZYosSZKy/Pbt2wWen789x48flyVJkhcsWKBT1r59+2RJkuRFixbpPG/AgAHyq1evlPXWrFkjS5Iknzx5ssh9om8dsizLH3/8sTx06NAiy5JlWZ4/f74sSZJ848aNYtfLzMyUJUmSx4wZoyyTJEn29fWVHz16pPw8fPhQvnz5sjx16lRZkiR55cqVxZar3eZffvlFZ/mSJUtkSZLk+Ph4WZZleceOHTrv4ZiYGFmSJDkqKkp5TnZ2tuzj41Oq1yT//irsdR86dKj88ccf66wnSVKB/5vz58/LkiTJf/3rX3XWc3V1lR8+fKgsmzRpkixJkrx48WJlWVJSkixJkrxs2bJi958sy7Leh2C7d+8GAKjVamWZ9veYmBh9iwEApKen4/z58+jSpQuysrLw+PFjPH78GM+ePYNarUZaWhquXLmirF+xYkW0bdu22DLT0tIQFxeHPn36oF69esryChUqIDAwEFlZWTh9+rSyvFKlSujatavOeg4ODkhLS3urNr6NgwcPQpIk1KlTRyn/8ePH6N69O4A3A6x5derUSedxSEgIjh49CltbW2VZRkaGMhZiyGUPR48eBQCMGTNGZ3nPnj3h4OBQ4DC2R48eKFeunPL4o48+AoACY4KlqaMksp6nybXtzD/4+tNPP6F9+/bKT8eOHeHj44OLFy/iiy++KLRHktfo0aNx+vRpNGvWTFmWlZWl9HqK2v+HDx9GjRo1dM6GWltbY8SIETrrGbq/8r8/DNG6dWud/xsXFxcAUP4ftFq1agU7OzvlsbaHmDcXGjRoAAB48OBBifXqdQj24MEDnD9/Hk2aNIFKpVKuJ/nwww+hUqmwd+9eBAcH6322QHsdwsaNG7Fx48ZC17l7967yu62tbaFd2by0hxQODg4F/qZ9g9y5c6fYMm1sbJQxIEPb+DZu3bqFrKwsndP1xZVfu3ZtnccqlQrp6elYs2YNrl+/jlu3buHOnTvKP2b+8azipKSkoHr16jpvLq1mzZrhxIkTOsvynxbXvvbF1WloHSX53e9+B+BN6DVt2rTI9bT/CPnrzXv9S3p6OjZs2ICff/4Z06ZNQ8+ePfVqQ05ODr7++mskJCTg1q1bSElJUYKuqH2RnJyMBg0a6AQ4gALbUNrXxBD531sVK1YE8NtJpaLW0x5u561bu136fEDoFUD79u3D69evkZSUpAxo5vX06VMcPnxY7zER7Qs0ZMgQ5dM+v7zjPPlfqMIUt7HaN4K1tbWyrKRAM7SNb+P169dwd3fHhAkTCv173bp1dR7nb/OxY8cwbtw41K1bF3/4wx+UgdCTJ09izZo1BrWlpP2Xd98V1hZT1FES7VjahQsXlHGewvz4448AADc3N53lNWrUQIcOHZTHarUaw4YNw+effw6VSlXgEpP84uPjodFoULFiRXTo0AEDBgxAixYtcOvWLcyfP7/I5xV1hi//B7ih+0uf/5Oi6Pt6FjW+l/8aK33pFUDas1/h4eGoWrWqzt8SExMRERGBmJgYvQPI3t4ewJsdlvcNALw565SSklLkWY2SyizstOnNmzcBQKeLKaKNhdWRmZlZoPynT5/izJkzaNy4cbHPDw0NRePGjbFjxw6dszbaw2VD23Ly5EmkpaUV+MS9efMmPvjgA4PLNHUd7u7usLe3x7Zt2zB8+HBlH6SlpSEsLAzDhw+Hk5MTNm7ciMqVK+scJhTGxsYGy5YtQ69evfDFF1/AxcVFeR8UZtGiRbCxscHevXt1egCrV68utp6GDRviypUrkGVZ5x83OTlZZ7138ZqIVmLsJSUlIT4+Hu3atUPfvn3RvXt3nZ8xY8agTp06OHXqVJHTDLTpqk30unXrwtnZGTExMTrPycnJwaxZszBp0iS8evXKoA2pU6cOnJ2dERsbq5yCBt5cPLlu3TrY2NigY8eOepdn7DZqP53ydsu9vLyQmJiIY8eO6ay7atUqTJ48ucRTuU+ePEH9+vV1wufu3bs4ePAggN96cYXVnZ+XlxcAFOg5HT58GDdv3tQZL3tbxq5DpVJh7ty5uHPnDqZNm4aXL18CeDPt58aNG/D19YW/vz+uXr2KsWPH6nWIUr9+fUyfPh0ZGRkICQkpdt0nT56gVq1aOuU+f/5cGRMt6oI/tVqN9PR0/Pvf/1aW5ebmYsuWLTrrmeI1sbKyMujQ3NRK7AFpP019fHwK/bu1tTUGDBiA1atX61wTlJf2BYqMjISnpye6deuG2bNn49NPP8WAAQPg5+cHW1tb7N27F3FxcZg6dSpq1qxp8MZoy/Tx8YGfnx+qVKmC2NhYJCQkYPbs2TqnLA0pzxht1K4bGxsLWZbRr18/jBkzBgcPHsSECRMwePBgNG/eHBcuXMCuXbvg6ekJT0/PYsv09PTEvn37MGfOHLi4uCAlJQVRUVF48eIFACAzMxPAb+NdR48eRf369dGjR48CZXXp0gXdunXDhg0bcP/+fXh4eCApKQmbN29Gw4YNCwyEvg1T1NGlSxeEhIRgwYIF6NmzJ/r06QN7e3uo1WrcunULly5dQoUKFYocZyvMoEGDsHPnTpw4cQK7d+9Gr169Cl3P09MTa9euxeTJk9GpUyc8fPgQ27dvVwZutfs/v379+mHLli2YPn06fvrpJzRp0gQHDhxAXFwcgN8OZ0yxv2rVqoUff/wR69atQ+vWreHq6mpwGcZUYg9oz549qFatWqFvWq1BgwbBysqqyLNhf/rTn9ChQwdER0djyZIlAN6Mpm/evBnOzs5Yt24dFi9ejBcvXiA8PPyt599oy3RycsI//vEPfPPNN6hQoQJWrlypcwGkoeUZo43NmjWDRqNBfHw8wsLCcOfOHdja2mLr1q3o378/9u/fjwULFiAuLg7jxo3D8uXLSzwuDwkJgY+PD44ePYoFCxZg//796Nu3L9avXw8AOHv2LIA3Z/ymTJmCe/fuYcGCBYVeMKhSqfDNN99g8uTJSExMxMKFC3Hw4EH4+vpi+/btBod3YUxVh5+fH3bs2AEPDw/s3r0b8+bNQ1RUFDw8PBAeHg5nZ2cMHjxY57qqktoZGhoKa2trhIWFIT09vdD1Jk6ciICAAFy6dAmhoaGIjo5Ghw4dsHPnTlhZWSn7Pz9ra2tERkaiV69eiI2NxeLFi1GtWjWlx6UdCzLF/ho5ciSaNGmCpUuXYseOHQY/39hUsr7nMonMVG5uLvbs2YOUlJQST62/C0+ePEGVKlUKDCIfOHAAkyZNwvr16w3qsZmzMns/oNzcXMyZMwe+vr7K5d6WpG/fvtBoNNBoNAgODhbdnFKLi4tTepnJycnw8/ODv78/5s6dK3zMwcrKCr17936r8Mm7XQkJCejcubPyuu3bt++t2rNhwwa4ubnpjFUCbyZtly9fHi1atHircs2RwbfjeFcOHz6M7OxsbN26FZcuXUJ4eDhWrVolullGoR0sLer6InOzdu1axMbGKmcFFy5ciKCgIHh4eGDOnDk4cuRIiWegyqL823X16lWMGDECAQEBpSq3Z8+e+PbbbxEQEIBBgwahYsWKOHXqFA4ePIjPPvsMNWrUMEbzzUKZ7QFduHABnTt3BvDm+o34+HjBLTKexMREvHjxAgEBARg2bFiRN80yF40aNdKZ2JiQkKDMAfL09NS5At2c5N+u+Ph4HDt2DEOGDMGsWbP0nl2fX/PmzbFp0ybY29tjzZo1CA8Px+3btxEaGoqgoCBjNd8slNkeUEZGhs41R+XKlcOrV6+KnehoLipWrIjAwEAMHDgQSUlJGDVqFPbv32+22+bt7a1zt8W817dUqVIFz58/F9W0Usm/XS1btsTAgQPh7OyMVatWYeXKlZgxY8Zble3q6oq1a9caq6lmq8z2gKpWrapzGjM3N9ds/0Hzc3BwQO/evaFSqeDg4ABbW9ti51CZm7xn7zIzM41yBq0sUKvVcHZ2Vn6/evWq4BaZvzIbQK1bt1bmuly6dAmSJAlukfFs374d4eHhAID79+8jIyMDderUEdwq42nRooVy/+MTJ06gTZs2gltkHIGBgbh8+TKAN3cWzHubCno7ZbZLoVarcerUKQwePBiyLCMsLEx0k4zGx8cHwcHB8PPzg0qlQlhYmMX07gBgxowZ+PLLL7Fs2TI0bdoU3t7eoptkFCEhIcr1QXZ2dggNDRXdJLPH64CISJgyewhGRJaPAUREwjCAiEgYBhARCcMAIiJhzCKAtm7dKroJJmGp2wVY7rZZ6naJwgASyFK3C7DcbbPU7RLFLAKIiCxTmboQsVmzZoXeZ7lcuXJF3l9XH9pvTyhr0tPT3+rWs+bAUrfNErYrNTVVmSojWpm6/v/Vq1e4deuW0cu1tJuZEZVG3i9EFM2kAZSbm4uQkBBcv34dNjY2WLBgQYlfNUNE7w+TjgHlvavh1KlTlRngRESAiQPIku9qSESlZ9IAKuquhkREgIkDyJLvakhEpWfSALLkuxoSUemZtDtiyXc1JKLSM2kAWVlZYf78+aasgojM2HsxIKP9ihh9lKELw4ksHueCEZEwDCAiEoYBRETCMICISBgGEBEJwwAiImEYQEQkDAOIiIRhABGRMAwgIhKmTAWQu7s7ZFnW68dUVCqV3j9EVDplKoCI6P3CACIiYRhARCQMA4iIhGEAEZEwDCAiEoYBRETCMICISBgGEBEJwwAiImHM9lsxDJmOYappE/qWy2/aICoce0BEJAwDiIiEYQARkTAMICIShgFERMIwgIhIGAYQEQnDACIiYRhARCQMA4iIhDHbqRiGED1tw5AyOW2D3ifsARGRMAwgIhKGAUREwjCAiEgYBhARCcMAIiJhGEBEJAwDiIiEYQARkTDvxZXQhuBV00TvDntARCQMA4iIhDH5IVjfvn1RrVo1AECDBg2wcOFCU1dJRGbCpAH08uVLAMDGjRtNWQ0RmSmTHoIlJibixYsXCAgIwLBhw3Dp0iVTVkdEZsakPaCKFSsiMDAQAwcORFJSEkaNGoX9+/ejfHmefCMiEweQg4MDGjduDJVKBQcHB9ja2uLhw4f44IMPTFktEZkJkx6Cbd++HeHh4QCA+/fvIyMjA3Xq1DFllURkRkzaA/Lx8UFwcDD8/PygUqkQFhbGwy8iUpg0DWxsbLB06VJTVkFEZozdkVLQdyqEKaZsGFoup21QWcQroYlIGAYQEQnDACIiYRhARCQMA4iIhGEAEZEwDCAiEoYBRETCMICISBgGEBEJw6kY74Dob9owtFxO26B3hT0gIhKGAUREwjCAiEgYBhARCcMAIiJhGEBEJAwDiIiEYQARkTAMICIShgFERMJwKkYZw2kb9D5hD4iIhGEAEZEwDCAiEoYBRETCMICISBgGEBEJwwAiImEYQEQkDAOIiIRhABGRMJyKYcY4bYPMHXtARCQMA4iIhGEAEZEwDCAiEoYBRETCMICISBgGEBEJwwAiImEYQEQkDAOIiIThVIz3BKdtUFnEHhARCcMAIiJhjBpAcXFx0Gg0AIDk5GT4+fnB398fc+fORW5urjGrIiILYLQAWrt2LWbPno2XL18CABYuXIigoCD861//gizLOHLkiLGqIiILYbQAatSoESIiIpTHCQkJaNeuHQDA09MTp0+fNlZVRGQhjBZA3t7eKF/+t5NqsiwrZz2qVKmC58+fG6sqIrIQJhuEtrL6rejMzExUr17dVFURkZkyWQC1aNEC586dAwCcOHECbdq0MVVVRGSmTBZAM2bMQEREBHx9fZGTkwNvb29TVUVEZsqoV0I3aNAAUVFRAAAHBwd89913xiyeiCwMp2JQAZy2Qe8Kr4QmImEYQEQkDAOIiIRhABGRMAwgIhKGAUREwjCAiEgYBhARCcMAIiJhGEBEJAynYlCpcNoGlQZ7QEQkDAOIiIRhABGRMAwgIhKGAUREwjCAiEgYBhARCcMAIiJhGEBEJAwDiIiE4VQMemfMadoGp2y8G+wBEZEwDCAiEoYBRETCMICISBgGEBEJwwAiImEYQEQkDAOIiIRhABGRMAwgIhKGUzGoTBI9bYPftPFusAdERMIwgIhIGAYQEQnDACIiYRhARCQMA4iIhGEAEZEwDCAiEoYBRETC8EpoMnu8atp8sQdERMIwgIhIGKMGUFxcHDQaDQAgISEBnTt3hkajgUajwb59+4xZFRFZAKONAa1duxaxsbGoVKkSAODq1asYMWIEAgICjFUFEVkYo/WAGjVqhIiICOVxfHw8jh07hiFDhmDWrFnIyMgwVlVEZCGMFkDe3t4oX/63DlXLli0xffp0bNq0CQ0bNsTKlSuNVRURWQiTDUKr1Wo4Ozsrv1+9etVUVRGRmTJZAAUGBuLy5csAgDNnzsDJyclUVRGRmTLZhYghISEIDQ2FtbU17OzsEBoaaqqqiMhMGTWAGjRogKioKACAk5MTtmzZYsziicjCcCoGvVf0nQphiikbhpb7Pkzb4JXQRCQMA4iIhGEAEZEwDCAiEoYBRETCMICISBgGEBEJwwAiImEYQEQkDAOIiIThVAyiQoj+pg1DyzXXaRvsARGRMAwgIhKGAUREwjCAiEgYBhARCcMAIiJhGEBEJAwDiIiEYQARkTAMICIShgFEVEqyLOv9YyoqlUrvn7KEAUREwjCAiEgYBhARCcMAIiJhGEBEJAwDiIiEYQARkTAMICIShgFERMIwgIhIGH4rBtE7VBa+baMsYQ+IiIRhABGRMAwgIhKGAUREwjCAiEgYBhARCcMAIiJhGEBEJAwDiIiEYQARkTAq2ZS36jeQh4cH7O3tCyxPT09HzZo1BbTItCx1uwDL3TZL2K7U1FScO3dOdDMAlLEAKkr//v0RHR0tuhlGZ6nbBVjutlnqdonCQzAiEoYBRETClAsJCQkR3Qh9ODs7i26CSVjqdgGWu22Wul0imMUYEBFZJh6CEZEwDCAiEoYBRETCMICISBgGEBEJ838DMH2BbFUP7gAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "sns.set_style('white')\n",
    "\n",
    "#Generate a random tridiagonal matrix\n",
    "n = 20\n",
    "d = np.random.randn(n)\n",
    "sub_diag = np.random.randn(n-1)\n",
    "\n",
    "mat = np.diag(d) + np.diag(sub_diag, -1) + np.diag(sub_diag, 1)\n",
    "plt.spy(mat)\n",
    "plt.title(\"Original matrix\", fontsize=18)\n",
    "q, r = np.linalg.qr(mat)\n",
    "plt.figure()\n",
    "b = r.dot(q)\n",
    "b[abs(b) <= 1e-12] = 0\n",
    "plt.spy(b)\n",
    "plt.title(\"After one iteration of QR algorithm\", fontsize=18)\n",
    "#plt.figure()\n",
    "#plt.imshow(np.abs(r.dot(q)))\n",
    "b[0, :]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Трёхдиагональная форма\n",
    "\n",
    "- Работая с трёхдиагональной формой, вам не нужно вычислять матрицу $Q$: нужно лишь вычислить трёхдиагональную часть, которая получается после итерации \n",
    "\n",
    "    $$A_k = Q_k R_k, \\quad A_{k+1}  = R_k Q_k,$$\n",
    "\n",
    "    в случае $A_k = A^*_k$.\n",
    "\n",
    "- Такая матрица определяется $\\mathcal{O}(n)$ параметрами\n",
    "- Вычисление QR разложения более сложное, но возможно вычислить $A_{k+1}$ напрямую без вычисления $Q_k$.\n",
    "- Это называется **неявный QR-шаг**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Теорема о неявном QR алгоритме\n",
    "\n",
    "\n",
    "- Все реализации неявного QR алгоритма основаны на следующей теореме \n",
    "\n",
    "**Теорема.** Пусть \n",
    "\n",
    "$$Q^* A Q = H$$ \n",
    "\n",
    "верхне-гессенбергова форма матрицы.\n",
    "Тогда первый столбец матрицы $Q$ определяет все остальные её столбцы. Он может быть найден из следующего уравнения\n",
    "\n",
    "$$A Q = Q H. $$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сходимость QR алгоритма\n",
    "\n",
    "- Сходимость QR алгоритма – непростой вопрос (см. Е.Е. Тыртышников \"Краткий курс численного анализа\")! \n",
    "\n",
    "\n",
    "**Итог.** Если у нас есть разложение вида\n",
    "\n",
    "$$A = X \\Lambda X^{-1}, \\quad A = \\begin{bmatrix}A_{11} & A_{12} \\\\ A_{21} & A_{22}\\end{bmatrix}$$\n",
    "\n",
    "и\n",
    "\n",
    "$$\n",
    "\\Lambda = \\begin{bmatrix} \\Lambda_1 & 0 \\\\ \n",
    "0 & \\Lambda_2 \\end{bmatrix}, \\quad \\lambda(\\Lambda_1)=\\{\\lambda_1,\\dots,\\lambda_m\\}, \\ \\lambda(\\Lambda_2)=\\{\\lambda_{m+1},\\dots,\\lambda_r\\},\n",
    "$$\n",
    "\n",
    "а также есть **зазор** между собственными значениями в матрице $\\Lambda_1$ и $\\Lambda_2$ ($|\\lambda_1|\\geq \\dots \\geq |\\lambda_m| > |\\lambda_{m+1}| \\geq\\dots \\geq |\\lambda_r| >0$), тогда блок $A^{(k)}_{21}$  матрицы $A_k$\n",
    "сходится к нулевому в процессе работы QR алгоритма со скоростью\n",
    "\n",
    "$$\\Vert A^{(k)}_{21} \\Vert \\leq  C q^k, \\quad q = \\left| \\frac{\\lambda_{m+1}}{\\lambda_{m}}  \\right |,$$\n",
    "\n",
    "где $m$ размер матрицы $\\Lambda_1$.\n",
    "\n",
    "Таким образом, нам нужно увеличить зазор между $\\Lambda_1$ и $\\Lambda_2$. Это можно сделать с помощью **QR алгоритма со сдвигами**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## QR алгоритм со сдвигами\n",
    "\n",
    "$$A_{k} - s_k I = Q_k R_k, \\quad A_{k+1} = R_k Q_k + s_k I$$\n",
    "\n",
    "\n",
    "- Сходимость такого алгоритма линейная с фактором\n",
    "\n",
    "$$\\left| \\frac{\\lambda_{m+1} - s_k}{\\lambda_{m} - s_k}  \\right |,$$\n",
    "\n",
    "где $\\lambda_m$ – $m$-ое большее по модулю собственное значение. Если сдвиг близок к собственному вектору, сходимость более быстрая.\n",
    "\n",
    "- Существуют различные стратегии выбора сдвигов. \n",
    "\n",
    "- Использование сдвигов – это общий подход к ускорению сходимости итерационных методов вычисления собственных значений. Далее покажем, как выбирать сдвиги для более простых алгоритмов"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сдвиги и степенной метод\n",
    "\n",
    "Напоминаем, что такое степенной метод для вычисления собственых значений.\n",
    "\n",
    "$$x_{k+1} := A x_k, \\quad x_{k+1} := \\frac{x_{k+1}}{\\Vert x_{k+1} \\Vert}.$$\n",
    "\n",
    "- Он сходится к собственному вектору, соответствующему максимальному по модулю собственному значению.\n",
    "\n",
    "- Сходимость может быть очень медленной.\n",
    "\n",
    "- Используем сдвиги: если мы преобразуем матрицу\n",
    "\n",
    "$$  A := A - \\lambda_k I,$$\n",
    "\n",
    "то соответствующее собственное значение уменьшится (а нам нужно максимальное по модулю). Это не то, что нам было нужно!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Обратная итерация и итерация Релея\n",
    "\n",
    "- Для того чтобы из малого собственного значения сделать большое, нам нужно **обратить матрицу**, и это приводит нас к **методу обратной итерации**\n",
    "\n",
    "$$x_{k+1} = (A - \\lambda I)^{-1} x_k,$$\n",
    "\n",
    "где $\\lambda$ – сдвиг, который близок к собственному значению, которое мы хотим найти. Аналогично степенному методу сходимость линейная.\n",
    "\n",
    "- Для ускорения сходимости можно использовать **итерацию Релея**, которая задаётся с помощью **адаптивного выбора** параметра сдвига:\n",
    "\n",
    "$$x_{k+1} = (A - \\lambda_k I)^{-1} x_k,$$\n",
    "\n",
    "$$\\lambda_k = \\frac{(Ax_k, x_k)}{(x_k, x_k)}$$\n",
    "\n",
    "В симметричном случае $A = A^*$ сходимость **локально кубическая**, и **локально квадратичная** иначе."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сингулярные значения и собственные значения (1)\n",
    "\n",
    "\n",
    "- Сингулярное разложение имеет вид\n",
    "\n",
    "$$A = U \\Sigma V^*$$\n",
    "\n",
    "и существует для любой матрицы.\n",
    "\n",
    "- Его также можно считать способом приведения данной матрицы к диагональному виду с помощью двух унитарных преобразований:\n",
    "\n",
    "$$\\Sigma = U^* A V.$$\n",
    "\n",
    "- С помощью двусторонних преобразований Хаусхолдера мы можем привести любую матрицу к **бидиагональной форме** $B$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сингулярные значения и собственные значения (2)\n",
    "\n",
    "**Неявный QR алгоритм** (со сдвигами) вычисляет собственные значения (разложение Шура).\n",
    "Но мы не можем применить его напрямую к бидиагональной матрице, поскольку она может быть недиагонализуема в общем случае.\n",
    "\n",
    "Однако задачу вычисления сингулярного разложения можно свести к **симметричной задаче на собственные значения** двумя способами:\n",
    "\n",
    "1. Работать с трёхдиагональной матрицей\n",
    "\n",
    "$$T = B^* B$$ \n",
    "\n",
    "2. Работать с расширенной матрицей \n",
    "\n",
    "$$T = \\begin{bmatrix} 0 & B \\\\ B^* & 0 \\end{bmatrix}$$\n",
    "\n",
    "\n",
    "Случай 1 практически реализуем, если **не формировать матрицу $T$ явно**!\n",
    "\n",
    "Таким образом, задача вычисления сингулярных чисел может быть сведена к задаче вычисления собственных чисел симметричной трёхдиагональной матрицы."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Алгоритмы для симметричной задачи на собственные значения\n",
    "\n",
    "Рассмотрены:\n",
    "\n",
    "- QR алгоритм: \"золотой стандарт\" для вычисления собственных значений\n",
    "- Метод итерации Релея\n",
    "\n",
    "Далее:\n",
    "\n",
    "- Алгоритм на основе стратегии \"разделяй и властвуй\"\n",
    "- Метод бисекции\n",
    "- Метод Якоби"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Метод разделяй и властвуй\n",
    "\n",
    "Пусть у нас есть трёхдиагональная матрица и мы разделили её на блоки:\n",
    "\n",
    "\n",
    "$$T = \\begin{bmatrix} T'_1 & B \\\\ B^{\\top} & T'_2 \\end{bmatrix}$$\n",
    "\n",
    "Можем записать матрицу $T$ в виде\n",
    "\n",
    "$$T = \\begin{bmatrix} T_1 & 0 \\\\ 0 & T_2 \\end{bmatrix} + b_m v v^*$$\n",
    "\n",
    "где $vv^*$ – матрица ранга 1, $v = (0,\\dots,0,1,1,0,\\dots,0)^T$.\n",
    "\n",
    "Пусть мы уже разложили матрицы $T_1$ и $T_2$:\n",
    "\n",
    "$$T_1 = Q_1 \\Lambda_1 Q^*_1, \\quad T_2 = Q_2 \\Lambda_2 Q^*_2$$\n",
    "\n",
    "Тогда (проверьте!),\n",
    "\n",
    "$$\\begin{bmatrix} Q^*_1 & 0 \\\\ 0 & Q^*_2 \\end{bmatrix} T\\begin{bmatrix} Q_1 & 0 \\\\ 0 & Q_2 \\end{bmatrix} = D + \\rho u u^{*}, \\quad D = \\begin{bmatrix} \\Lambda_1 & 0 \\\\ 0 & \\Lambda_2\\end{bmatrix},$$\n",
    "\n",
    "то есть мы свели задачу к задаче вычисления собственных значений у матрицы вида\n",
    "\n",
    "\n",
    "<font color='red'> диагональная матрица плюс матрица малого ранга. </font>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Матрица вида диагональная матрица плюс матрица малого ранга\n",
    "\n",
    "- Собственные значения матрицы  вида\n",
    "\n",
    "$$D + \\rho u u^* $$\n",
    "\n",
    "вычислить не так просто!\n",
    "\n",
    "- Характеристический многочлен имеет вид\n",
    "\n",
    "$$\\det(D + \\rho uu^* - \\lambda I) = \\det(D - \\lambda I)\\det(I + \\rho (D - \\lambda I)^{-1} uu^*) = 0.$$\n",
    "\n",
    "Тогда (докажите!!) \n",
    "\n",
    "$$\\det(I + \\rho (D - \\lambda I)^{-1} uu^*) = 1 + \\rho \\sum_{i=1}^n \\frac{|u_i|^2}{d_i - \\lambda} = 0$$\n",
    "\n",
    "Подсказка: найдите $\\det(I + w u^*)$ используя два факта: \n",
    "- $\\text{det}(C) = \\prod_{i=1}^n\\lambda_i(C)$\n",
    "- $\\text{trace}(C) = \\sum_{i=1}^n \\lambda_i$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Характеристическое уравнение\n",
    "\n",
    "$$1 + \\rho \\sum_{i=1}^n \\frac{|u_i|^2}{d_i - \\lambda} = 0$$\n",
    "\n",
    "Как найти корни?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "lm = [1, 2, 3, 4]\n",
    "M = len(lm)\n",
    "D = np.array(lm)\n",
    "a = np.min(lm)\n",
    "b = np.max(lm)\n",
    "t = np.linspace(-1, 6, 1000)\n",
    "u = 0.5 * np.ones(M)\n",
    "rho = 1\n",
    "def fun(lam):\n",
    "    return 1 + rho * np.sum(u**2/(D - lam))\n",
    "res = [fun(lam) for lam in t]\n",
    "plt.plot(t, res, 'k')\n",
    "plt.plot(t, np.zeros_like(t))\n",
    "plt.ylim([-6, 6])\n",
    "plt.xticks(fontsize=18)\n",
    "plt.yticks(fontsize=18)\n",
    "_ = plt.xlabel(\"$\\lambda$\", fontsize=18)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- Эта функция имеет только один корень на каждом отрезке $[d_i, d_{i+1}]$\n",
    "\n",
    "- Мы показали справедливость теоремы Коши о чередовании (что происходит с собственными числами после внесения возмущения ранга 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Как найти корни\n",
    "\n",
    "- Метод Ньютона не сработает (изобразите касательные к графику).\n",
    "\n",
    "- Заметим, что метод Ньютона – это по сути линейная аппроксимация функции $f(\\lambda)$ на каждой итерации.\n",
    "\n",
    "- Лучше аппроксимировать с помощью **гиперболы** вида:\n",
    "\n",
    "$$f(\\lambda) \\approx c_0 + \\frac{c_1}{d_i - \\lambda} + \\frac{c_2}{d_{i+1} - \\lambda}.$$\n",
    "\n",
    "- Для вычисления коэффициентов нам нужно вычислить $f(\\lambda)$ и $f'(\\lambda)$  в некоторых точках.\n",
    "\n",
    "- После чего получить апроксимацию из решения **квадратного уравнения**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Важный недостаток метода\n",
    "\n",
    "- Устойчивость: этот метод игнорировали долгое время из-за неустойчивого вычисления собственных векторов.\n",
    "\n",
    "- Нам нужно вычислить собственные векторы матрицы $D + \\rho uu^*$.\n",
    "\n",
    "- Точное выражение для собственных векторов $v_i$, для которых найдены собственные значения $\\alpha_i$:\n",
    "\n",
    "$$(D + \\rho uu^*)v_i = \\alpha_i v_i \\qquad (D - \\alpha_i I)v_i + \\rho (u^*v_i) u  = 0$$\n",
    "\n",
    "То есть $v_i \\in \\mathrm{span}(\\{(D - \\alpha_i I)^{-1}u\\})$.\n",
    "\n",
    "- Причины неустойчивости:\n",
    "    - если есть два близких собственных числа $\\alpha_i$ и $\\alpha_{i+1}$, то соответствующие векторы $(D - \\alpha_i I)^{-1}u$ и $(D - \\alpha_{i+1} I)^{-1}u$ будут близки, хотя долждны быть ортогональны\n",
    "    - если $\\alpha_i$ и $\\alpha_{i+1}$ очень близки, то они близки к числу $d_i$ между ними, то есть матрицы $D - \\alpha_i I$ и $D - \\alpha_{i+1} I$ близки к вырожденным"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Теорема Лёвнера (Charles Loewner)\n",
    "\n",
    "- Решение проблемы неустойчивости можно получить с помощью теоремы Лёвнера:\n",
    "\n",
    "Если $\\alpha_i$ и $d_i$ удовлетворяют **теореме о чередовании** \n",
    "\n",
    "\n",
    "$$d_n < \\alpha_n < \\ldots < d_{i+1} < \\alpha_{i+1} \\ldots$$\n",
    "\n",
    "Тогда существует вектор $\\widehat{u}$ такой что $\\alpha_i$ – точное собственное значение матрицы\n",
    "\n",
    "$$\\widehat{D} = D + \\widehat{u} \\widehat{u}^*$$\n",
    "\n",
    "и \n",
    "\n",
    "$$ \\widehat{u}^2_i = \\frac{\\prod_{j=1}^n (\\alpha_j - d_i)}{\\prod_{j=1, j \\neq i}^n (d_j - d_i)}  $$\n",
    "\n",
    "- Использование вектора $\\widehat{u}$ вместо $u$ даёт устойчивое вычисление собственного вектора!\n",
    "- Таким образом, сначала вычисляются собственные значения, затем $\\widehat{u}$ и только потом собственные векторы."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Ещё несколько алгоритмов\n",
    "\n",
    "- Абсолютно другой подход основан на идеи **бисекции**\n",
    "\n",
    "- Дана матрица $A$, инерция которой определяется как триплет $(\\nu, \\zeta, \\pi)$, где $\\nu$ – число отрицательных, $\\zeta$ - число нулевых, и $\\pi$ - число положительных собственных значений. \n",
    "\n",
    "- Если $X$ невырождена, тогда\n",
    "\n",
    "$$Inertia(A) = Inertia(X^* A X)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Бисекция с помощью метода Гаусса\n",
    "\n",
    "- Для данного $z$ мы можем запустить метод Гаусса и получить разложение:\n",
    "\n",
    "$$A - zI = L D L^*,$$\n",
    "\n",
    "а инерция для диагональной матрицы вычисляется просто.\n",
    "\n",
    "- С помощью инерции мы можем легко посчитать число собственных значений в заданном интервале.\n",
    "- Пример: если $Inertia(A)=(5,0,2)$ и после сдвига $Inertia(A-zI)=(4,0,3)$, $z\\in[a,b]$ тогда это значит, что $\\lambda(A)\\in[a,z]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Метод Якоби\n",
    "\n",
    "- Вспомним что такое вращения Гивенса (Якоби):\n",
    "\n",
    "на плоскости им соответствуют ортогональные матрицы $2 \\times 2$ вида\n",
    "\n",
    "$$\\begin{pmatrix} \\cos \\phi & \\sin \\phi \\\\ -\\sin \\phi & \\cos \\phi \\end{pmatrix},$$\n",
    "\n",
    "а в $n$-мерном пространстве мы выбираем два индекса $i$ и $j$ и вращаем относительно соответствующих элементов $n$-мерного вектора."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Метод Якоби: продолжение\n",
    "\n",
    "- Идея метода Якоби состоит в минимизации суммы квадратов недиагональных элементов:\n",
    "\n",
    "$$\\Gamma(A) = \\mathrm{off}( U^* A U), \\quad \\mathrm{off}^2(X) = \\sum_{i \\ne j} \\left|X_{ij}\\right|^2 = \\|X \\|^2_F - \\sum\\limits_{i=1}^n x^2_{ii}$$\n",
    "\n",
    "с помощью последовательных вращений Якоби для их зануления. \n",
    "\n",
    "- Когда элементы выбраны, их легко занулить.\n",
    "\n",
    "- **Главный вопрос:** в каком порядке нужно проводить зануление?\n",
    "\n",
    "- Если мы всегда зануляем максимальный недиагональный элемент, метод имеет глобально линейную сходимость и локально квадратичную.\n",
    "\n",
    "- На практике используется циклический порядок (то есть, $(1, 2), (1, 3), \\ldots, (2, 3), \\ldots$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Главное про метод Якоби\n",
    "\n",
    "Метод Якоби был первым численным методом для вычисления собственных значений, предложен в 1846.\n",
    "\n",
    "- Большая константа в оценке сложности\n",
    "- Очень точный (высокая относительная точность для малых собственных значений по сравнению с другими методами)\n",
    "- Хорошая возможность параллелизации"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Резюме по первой части\n",
    "\n",
    "- Методы решения симметричной задачи на собстенные значения\n",
    "- Использование сдвигов для ускорения сходимости\n",
    "- Способы вычисления сингулярного разложения"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Часть 2. Разреженные матрицы и прямые методы для решения больших разреженных систем"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## План на вторую часть сегодняшней лекции\n",
    "\n",
    "- Плотные неструктурированные матрицы и распределённое хранение\n",
    "- Разреженные матрицы и форматы их представления\n",
    "- Быстрая реализация умножения разреженной матрицы на вектор \n",
    "- Метод Гаусса для разреженных матриц: упорядоченность\n",
    "- Заполнение и графы: сепараторы\n",
    "- Лапласиан графа"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Плотные матрицы большой размерности\n",
    "\n",
    "- Если размер матрицы очень бльшой, то она не помещается в память\n",
    "- Возможные способы работы с такими матрицами\n",
    "    - Если матрица **структурирована**, например блочно Тёплицева с Тёплицевыми блоками (в следующих лекциях), тогда возможно сжатое хранение\n",
    "    - Для неструктурированных матриц помогает **распределённая память**\n",
    "    - MPI для обработки распределённо хранимых матриц"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Распределённая память и MPI\n",
    "\n",
    "- Разбиваем матрицу на блоки и храним их на различных машинах\n",
    "- Каждая машина имеет своё собственное адресное пространство и не может повредить данные на других машинах\n",
    "- В этом случае машины передают друг другу данные для агрегирования результата вычислений\n",
    "- [MPI (Message Passing Interface)](https://en.wikipedia.org/wiki/Message_Passing_Interface) – стандарт в параллельных вычислениях с распределённой памятью"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Пример: умножение матрицы на вектор\n",
    "\n",
    "- Предположим вы хотите посчитать произведение $Ax$ и матрица $A$ не помещается в памяти\n",
    "- В этом случае вы можете разбить матрицу на блоки и поместить их на разные машины\n",
    "- Возможные стратегии:\n",
    "    - Одномерное деление на блоки использует только строки\n",
    "    - Двумерное деление на блоки использует и строки и столбцы"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Пример одномерного деления на блоки\n",
    "<img src=\"./1d_block.jpg\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Общее время вычисления произведения матрицы на вектор для одномерного разбиения на блоки\n",
    "\n",
    "- Каждая машина хранит $n / p $ полных строк и $n / p$ элементов вектора $x$\n",
    "- Общее число операций $n^2 / p$\n",
    "- Общее время для отправки и записи данных $t_s \\log p + t_w n$, где $t_s$ – единица времени на отправку и $t_w$ – единица времени на запись"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Пример двумерного деления на блоки\n",
    "\n",
    "<img src=\"./2d_block.png\" width=400>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Общее время вычисления умнождения матрицы на вектор с использованием думарного разбиения на блоки\n",
    "\n",
    "- Каждая машина хранит блок размера $n / \\sqrt{p} $ и $n / \\sqrt{p}$ элементов вектора\n",
    "- Общее число операций $n^2 / p$\n",
    "- Общее время для отправки и записи данных примерно равно $t_s \\log p + t_w (n/\\sqrt{p}) \\log p$, где $t_s$ – единица времени на отправку и $t_w$ – единица времени на запись"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Пакеты с поддержкой распределённого хранения данных\n",
    "\n",
    "- [ScaLAPACK](http://www.netlib.org/scalapack/)\n",
    "- [Trilinos](https://trilinos.org/)\n",
    "\n",
    "В Python вы можете использовать [mpi4py](https://mpi4py.readthedocs.io/en/stable/) для параллельной реализации ваших алгоритмов.\n",
    "\n",
    "- PyTorch поддерживает распределённое обучение и хранение данных, см подробности [тут](https://pytorch.org/tutorials/intermediate/dist_tuto.html) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Резюме про работу с большими плотными неструктурированными матрицами\n",
    "\n",
    "- Распределённое хранение матриц\n",
    "- MPI\n",
    "- Пакеты, которые используют блочные вычисления\n",
    "- Различные подходы к блочным вычислениям"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Разреженные матрицы\n",
    "\n",
    "- Ограничением в решении задач линейной алгебры с плотными матрицами является память, требуемая для хранения плотных матриц, $N^2$ элементов.\n",
    "\n",
    "- Разреженные матриц, где большинство элементов нулевые позволяют по крайней мере хранить их в памяти.\n",
    "\n",
    "- Основные вопросы: можем ли мы решать следующие задачи для разреженных матриц?\n",
    "    - решение линейных систем\n",
    "    - вычисление собственных значений и собственных векторов\n",
    "    - вычисление матричных функций"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Приложения разреженных матриц\n",
    "\n",
    "Разреженные матрицы возникают в следующих областях:\n",
    "\n",
    "- математическое моделирование и решение уравнений в частных производных\n",
    "- обработка графов, например анализ социальных сетей\n",
    "- рекомендательные системы\n",
    "- в целом там, где отношения между объектами \"разрежены\"."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Разреженные матрицы помогают в вычислительной теории графов \n",
    "\n",
    "- Графы представляют в виде матриц смежности, которые чаще всего разрежены\n",
    "- Численное ршеение задач теории графов сводится к операциям с этими разреженными матрицами\n",
    "    - Кластеризация графа и выделение сообществ\n",
    "    - Обучение ранжированию\n",
    "    - Случайные блуждатели\n",
    "    - И другие....\n",
    "- Пример: возможно, самый большой доступный граф гиперссылок содержит 3.5 миллиарда веб-страниц и 128 миллиардов гиперссылок, больше подробностей см. [тут](http://webdatacommons.org/hyperlinkgraph/) \n",
    "- Различные графы среднего размера для тестирования ваших алгоритмов доступны в [Stanford Large Network Dataset Collection](https://snap.stanford.edu/data/)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Florida sparse matrix collection\n",
    "\n",
    "- Большое количество разреженных матриц из различных приложений вы можете найти в [Florida sparse matrix collection](http://www.cise.ufl.edu/research/sparse/matrices/)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"500\"\n",
       "            height=\"500\"\n",
       "            src=\"http://yifanhu.net/GALLERY/GRAPHS/search.html\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.IFrame at 0x1a1b0dc828>"
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     "execution_count": 24,
     "metadata": {},
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   ],
   "source": [
    "from IPython.display import IFrame\n",
    "IFrame('http://yifanhu.net/GALLERY/GRAPHS/search.html', 500, 500)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Разреженные матрицы и глубокое обучение\n",
    "\n",
    "- DNN имеют очень много параметров\n",
    "- Некоторые из них могут быть избыточными\n",
    "- Как уменьшить число параметров без серьёзной потери в точности?\n",
    "- [Sparse variational dropout method](https://github.com/ars-ashuha/variational-dropout-sparsifies-dnn) даёт существенно разреженные фильтры в DNN почти без потери точности!\n",
    "\n",
    "<img height=\"318\" src=\"http://ars-ashuha.ru/pdf/vdsdnn/conv.gif\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Построение разреженных матриц\n",
    "\n",
    "- Мы можем генерировать разреженные матрицы с помощью пакета **scipy.sparse**\n",
    "\n",
    "- Можно задать матрицы очень большого размера\n",
    "\n",
    "Полезные функции при создании разреженных матриц:\n",
    "- для созданий диагональной матрицы с заданными диагоналями ```spdiags```\n",
    "- Кронекерово произведение (определение будет далее) разреженных матриц ```kron```\n",
    "- также арифметические операции для разреженных матриц перегружены"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Кронекерово произведение\n",
    "\n",
    "Для матриц $A\\in\\mathbb{R}^{n\\times m}$ и $B\\in\\mathbb{R}^{l\\times k}$ Кронекерово произведение определяется как блочная матрица следующего вида\n",
    "\n",
    "$$\n",
    "   A\\otimes B = \\begin{bmatrix}a_{11}B & \\dots & a_{1m}B \\\\ \\vdots & \\ddots & \\vdots \\\\ a_{n1}B & \\dots & a_{nm}B\\end{bmatrix}\\in\\mathbb{R}^{nl\\times mk}.\n",
    "$$\n",
    "\n",
    "Основные свойства:\n",
    "- билинейность\n",
    "- $(A\\otimes B) (C\\otimes D) = AC \\otimes BD$\n",
    "- Пусть $\\mathrm{vec}(X)$ оператор векторизации матрицы по столбцам. Тогда \n",
    "$\\mathrm{vec}(AXB) = (B^T \\otimes A) \\mathrm{vec}(X).$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
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     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy as sp\n",
    "import scipy.sparse\n",
    "from scipy.sparse import csc_matrix, csr_matrix\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.linalg\n",
    "import scipy.sparse.linalg\n",
    "%matplotlib inline\n",
    "n = 5\n",
    "ex = np.ones(n);\n",
    "lp1 = sp.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "e = sp.sparse.eye(n)\n",
    "A = sp.sparse.kron(lp1, e) + sp.sparse.kron(e, lp1)\n",
    "A = csc_matrix(A)\n",
    "plt.spy(A, aspect='equal', marker='.', markersize=5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Шаблон разреженности\n",
    "\n",
    "- Команда ```spy``` рисует шаблон разреженности данной матрицы: пиксель $(i, j)$ отображается на рисунке, если соответствующий элемент матрицы ненулевой.\n",
    "\n",
    "- Шаблон разреженности действительно очень важен для понимания сложности агоритмов линейной алгебры для разреженных матриц.  \n",
    "\n",
    "- Зачастую шаблона разреженности достаточно для анализа того, насколько \"сложно\" работать с этой матрицей."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Определение разреженных матриц\n",
    "\n",
    "- Разреженные матрицы – это матрицы, такие что количество ненулевых элементов в них существенно меньше общего числа элементов в матрице. \n",
    "\n",
    "- Из-за этого вы можете выполнять базовые операции линейной алгебры (прежде всего решать линейные системы) гораздо быстрее по сравнению с использованием плотных матриц."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Что нам необходимо, чтобы увидеть, как это работает\n",
    "\n",
    "- **Вопрос 1:** Как хранить разреженные матрицы в памяти?\n",
    "\n",
    "- **Вопрос 2:** Как умножить разреженную матрицу на вектор быстро?\n",
    "\n",
    "- **Вопрос 3:** Как быстро решать линейные системы с разреженными матрицами?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Хранение разреженных матриц\n",
    "\n",
    "Существет много форматов хранения разреженных матриц, наиболее важные:\n",
    "\n",
    "- COO (координатный формат)\n",
    "- LIL (список списков)\n",
    "- CSR (compressed sparse row)\n",
    "- CSC (compressed sparse column)\n",
    "- блочные варианты\n",
    "\n",
    "В ```scipy``` представлены конструкторы для каждого из этих форматов, например\n",
    "\n",
    "```scipy.sparse.lil_matrix(A)```."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Координатный формат (COO)\n",
    "\n",
    "- Простейший формат хранения разреженной матрицы – координатный. \n",
    "- В этом формате разреженная матрица – это набор индексов и значений в этих индексах.\n",
    "\n",
    "```python\n",
    "i, j, val\n",
    "```\n",
    "\n",
    "где ```i, j``` массивы индексов, ```val``` массив элементов матрицы. <br>\n",
    "\n",
    "- Таким образом, нам нужно хранить $3\\cdot$**nnz** элементов, где **nnz** обозначает число ненулевых элементов в матрице.\n",
    "\n",
    "**Q:** Что хорошего и что плохого в использовании такого формата?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Основные недостатки\n",
    "\n",
    "- Он неоптимален по памяти (почему?)\n",
    "- Он неоптимален для умножения матрицы на вектор (почему?)\n",
    "- Он неоптимален для удаления элемента (почему?)\n",
    "\n",
    "Первые два недостатка решены в формате CSR.\n",
    "\n",
    "**Q**: какой формат решает третий недостаток? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Compressed sparse row (CSR)\n",
    "\n",
    "В формате CSR матрица хранится также с помощью трёх массивов, но других:\n",
    "\n",
    "```python\n",
    "ia, ja, sa\n",
    "```\n",
    "\n",
    "где:\n",
    "\n",
    "- **ia** (начало строк) массив целых чисел длины $n+1$ \n",
    "- **ja** (индексы столбцов) массив целых чисел длины **nnz** \n",
    "-  **sa** (элементы матрицы) массив действительных чисел длины **nnz**\n",
    "\n",
    "<img src=\"https://www.karlrupp.net/wp-content/uploads/2016/02/csr_storage_sparse_marix.png\" width=60% />\n",
    "\n",
    "Итак, всего необходимо хранить $2\\cdot{\\bf nnz} + n+1$ элементов."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Разреженные матрицы в PyTorch и Tensorflow\n",
    "\n",
    "- PyTorch поддерживает разреженные матрицы в формате COO\n",
    "- Неполная поддержка вычисления градиентов в операциях с такими матрицами, список и обсуждение см. [тут](https://github.com/pytorch/pytorch/issues/9674)\n",
    "- Tensorflow также поддерживает разреженные матрицы в COO формате\n",
    "- Список поддерживаемых операций приведён [здесь](https://www.tensorflow.org/api_docs/python/tf/sparse) и поддержка вычисления градиентов также ограничена"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### CSR формат позволяет быстро умножить разреженную матрицу на вектор (SpMV)\n",
    "\n",
    "```python\n",
    "\n",
    "   for i in range(n):\n",
    "        \n",
    "        for k in range(ia[i]:ia[i+1]):\n",
    "            \n",
    "            y[i] += sa[k] * x[ja[k]]\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "7.91 ms ± 453 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n",
      "10.8 ms ± 536 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy as sp\n",
    "import scipy.sparse\n",
    "import scipy.sparse.linalg\n",
    "from scipy.sparse import csc_matrix, csr_matrix, coo_matrix\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "n = 1000\n",
    "ex = np.ones(n);\n",
    "lp1 = sp.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "e = sp.sparse.eye(n)\n",
    "A = sp.sparse.kron(lp1, e) + sp.sparse.kron(e, lp1)\n",
    "A = csr_matrix(A)\n",
    "rhs = np.ones(n * n)\n",
    "B = coo_matrix(A)\n",
    "%timeit A.dot(rhs)\n",
    "%timeit B.dot(rhs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Видно, что **CSR** быстрее, и чем менее структурирован шаблон разреженности, тем выше выигрыш в скорости."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Разреженные матрицы и эффективность\n",
    "\n",
    "- Использованиеи разреженных матриц приводит к уменьшению сложности\n",
    "- Но они не очень подходят для параллельных/GPU реализаций \n",
    "- Они не показывают максимальную эффективность из-за случайного доступа к данным.  \n",
    "- Обычно, пиковая производительность порядка $10\\%-15\\%$ считается хорошей.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Вспомним как измеряется эффективность операций\n",
    "\n",
    "- Стандартный способ измерения эффективности операций линейной алгебры – это использование **flops** (число опраций с плавающей точкой в секунду)\n",
    "\n",
    "- Измерим эффективность умножения матрицы на вектор в случае плотной и разреженной матрицы"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time:  1.2e-02, Efficiency:  2.7e+00 Gflops\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import time\n",
    "n = 4000\n",
    "a = np.random.randn(n, n)\n",
    "v = np.random.randn(n)\n",
    "t = time.time()\n",
    "np.dot(a, v)\n",
    "t = time.time() - t\n",
    "print('Time: {0: 3.1e}, Efficiency: {1: 3.1e} Gflops'.\\\n",
    "      format(t,  ((2 * n ** 2)/t) / 10 ** 9))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time:  1.3e-04, Efficiency:  9.0e-02 Gflops\n"
     ]
    }
   ],
   "source": [
    "n = 4000\n",
    "ex = np.ones(n);\n",
    "a = sp.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "rhs = np.random.randn(n)\n",
    "t = time.time()\n",
    "a.dot(rhs)\n",
    "t = time.time() - t\n",
    "print('Time: {0: 3.1e}, Efficiency: {1: 3.1e} Gflops'.\\\n",
    "      format(t,  (3 * n) / t / 10 ** 9))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Случайный доступ к данным и промахи в обращении к кешу\n",
    "\n",
    "- Сначала все элементы матрицы и вектора хранятся в оперативной памяти (RAM – Random Access Memory)\n",
    "- Если вы хотите вычислить произведение матрицы на вектор, часть элементов матрицы и вектора перемещаются в кеш (быстрой памяти малого объёма), см. [лекцию об алгоритме Штрассена и умножении матриц](https://nbviewer.jupyter.org/github/amkatrutsa/nla2019_ozon/blob/master/lectures/lecture3/lecture3.ipynb)\n",
    "- После этого CPU берёт данные из кеша, обрабатывает их и возвращает результат снова в кеш\n",
    "- Если CPU требуются данные, которых ещё нет в кеше, это называется промах в обращении к кешу (cache miss)\n",
    "- Если случается промах в обращении к кешу, необходимые данные перемещаются из оперативной памяти в кеш\n",
    "\n",
    "**Q**: что если в кеше нет свободного места?\n",
    "\n",
    "\n",
    "- Чем больше промахов в обращении к кешу, тем медленнее выполняются вычисления"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### План кеша и LRU\n",
    "\n",
    "<img src=\"./cache_scheme.png\" width=\"500\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Умножение матрицы в CSR формате на вектор\n",
    "\n",
    "```python\n",
    "\n",
    "   for i in range(n):\n",
    "        \n",
    "        for k in range(ia[i]:ia[i+1]):\n",
    "            \n",
    "            y[i] += sa[k] * x[ja[k]]\n",
    "            \n",
    "```\n",
    "\n",
    "- Какая часть операций приводит к промахам в обращении к кешу?\n",
    "- Как эту проблему можно решить?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Переупорядочивание уменьшает количество промахов в обращении к кешу\n",
    "\n",
    "- Если ```ja``` хранит последовательно элементы, тогда они могут быть перемещены в кеш одноврменно и количество промахов в обращении к кешу уменьшится\n",
    "- Так происходит, когда разреженная матрица является **ленточной** или хотя бы блочно-диагональной\n",
    "- Мы можем превратить данную разреженную матрицу в ленточную или блочно-диагональную с помощью *перестановок* \n",
    "\n",
    "- Пусть $P$ матрица перестановок строк матрицы и $Q$ матрица перестановок столбцов матрицы\n",
    "- $A_1 = PAQ$ – матрица с шириной ленты меньшей, чем у матрицы $A$\n",
    "- $y = Ax \\to \\tilde{y} = A_1 \\tilde{x}$, где $\\tilde{x} = Q^{\\top}x$ и $\\tilde{y} = Py$\n",
    "- [Separated block diagonal form](http://albert-jan.yzelman.net/PDFs/yzelman09-rev.pdf) призван минимизировать количество промахов в обращшении к кешу\n",
    "- Он также может быть расширен на двумерный случай, где разделяются не только строки, но и столбцы"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#### Пример\n",
    "\n",
    "- SBD в одномерном случае\n",
    "<img src=\"./sbd.png\" width=\"400\">"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Методы решения линейных систем с разреженными матрицами\n",
    "\n",
    "- Прмямые методы\n",
    "    - LU разложение\n",
    "    - Различные методы переупорядочивания для минимизации заполнения факторов\n",
    "- Крыловские методы"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5,1,'Reshaped solution')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 10\n",
    "ex = np.ones(n);\n",
    "lp1 = sp.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "e = sp.sparse.eye(n)\n",
    "A = sp.sparse.kron(lp1, e) + sp.sparse.kron(e, lp1)\n",
    "A = csr_matrix(A)\n",
    "rhs = np.ones(n * n)\n",
    "sol = sp.sparse.linalg.spsolve(A, rhs)\n",
    "_, (ax1, ax2) = plt.subplots(1, 2)\n",
    "ax1.plot(sol)\n",
    "ax1.set_title('Not reshaped solution')\n",
    "ax2.contourf(sol.reshape((n, n), order='f'))\n",
    "ax2.set_title('Reshaped solution')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## LU разложение разреженной матрицы\n",
    "\n",
    "- Почему разреженная линейная система может быть решена быстрее, чем плотная? С помощью какого метода? \n",
    "\n",
    "- В LU разложении матрицы $A$ факторы $L$ и $U$ могут быть также разреженными:\n",
    "\n",
    "$$A = L U$$\n",
    "\n",
    "- А решение линейной системы с разреженной треугольной матрицей может быть вычислено очень быстро.  \n",
    "\n",
    "<font color='red'> Заметим, что обратная матрица от разреженной матрицы НЕ разрежена! </font>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-2.  1.  0.  0.  0.  0.  0.]\n",
      " [ 1. -2.  1.  0.  0.  0.  0.]\n",
      " [ 0.  1. -2.  1.  0.  0.  0.]\n",
      " [ 0.  0.  1. -2.  1.  0.  0.]\n",
      " [ 0.  0.  0.  1. -2.  1.  0.]\n",
      " [ 0.  0.  0.  0.  1. -2.  1.]\n",
      " [ 0.  0.  0.  0.  0.  1. -2.]]\n",
      "[[-0.875 -0.75  -0.625 -0.5   -0.375 -0.25  -0.125]\n",
      " [-0.75  -1.5   -1.25  -1.    -0.75  -0.5   -0.25 ]\n",
      " [-0.625 -1.25  -1.875 -1.5   -1.125 -0.75  -0.375]\n",
      " [-0.5   -1.    -1.5   -2.    -1.5   -1.    -0.5  ]\n",
      " [-0.375 -0.75  -1.125 -1.5   -1.875 -1.25  -0.625]\n",
      " [-0.25  -0.5   -0.75  -1.    -1.25  -1.5   -0.75 ]\n",
      " [-0.125 -0.25  -0.375 -0.5   -0.625 -0.75  -0.875]]\n"
     ]
    }
   ],
   "source": [
    "n = 7\n",
    "ex = np.ones(n);\n",
    "a = sp.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "b = np.array(np.linalg.inv(a.toarray()))\n",
    "print(a.toarray())\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## А факторы...\n",
    "\n",
    "- $L$ и $U$ обычно разрежены\n",
    "- В случае трёхдиагональной матрицы они даже бидиагональны!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x1a1b6e4550>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.sparse.linalg import splu\n",
    "T = splu(a.tocsc(), permc_spec=\"NATURAL\")\n",
    "plt.spy(T.L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Отметим, что ```splu``` со значением параметра ```permc_spec``` по умолчанию даёт перестановку, которая не даёт бидиагональные факторы:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3 5 4 6]\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.sparse.linalg import splu\n",
    "T = splu(a.tocsc())\n",
    "plt.spy(T.L)\n",
    "print(T.perm_c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Двумерный случай\n",
    "\n",
    "В двумерном случае всё гораздо хуже:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x1a1b82a2e8>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 20\n",
    "ex = np.ones(n);\n",
    "lp1 = sp.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "e = sp.sparse.eye(n)\n",
    "A = sp.sparse.kron(lp1, e) + sp.sparse.kron(e, lp1)\n",
    "A = csc_matrix(A)\n",
    "T = scipy.sparse.linalg.spilu(A)\n",
    "plt.spy(T.L, marker='.', color='k', markersize=8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Для правильной перестановки в двумерном случае число ненулевых элементов в $L$ растёт как $\\mathcal{O}(N \\log N)$. Однако сложность равна $\\mathcal{O}(N^{3/2})$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Разреженные матрицы и теория графов\n",
    "\n",
    "- Число ненулей в факторах из LU разложения сильно связано с теорией графов.\n",
    "\n",
    "- Пакет ``networkx`` можно использовать для визуализации графов, имея только матрицу смежности. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import networkx as nx\n",
    "n = 10\n",
    "ex = np.ones(n);\n",
    "lp1 = sp.sparse.spdiags(np.vstack((ex,  -2*ex, ex)), [-1, 0, 1], n, n, 'csr'); \n",
    "e = sp.sparse.eye(n)\n",
    "A = sp.sparse.kron(lp1, e) + sp.sparse.kron(e, lp1)\n",
    "A = csc_matrix(A)\n",
    "G = nx.Graph(A)\n",
    "nx.draw(G, pos=nx.spectral_layout(G), node_size=10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Заполнение (fill-in)\n",
    "\n",
    "- Заполнение матрицы – это элементы, который были **нулями**, но стали **ненулями** в процессе выполнения алгоритма.\n",
    "\n",
    "- Заполнение может быть различным для различных перестановок. Итак, до того как делать факторизацию матрицы нам необходимо переупорядочить её элементы так, чтобы заполнение факторов было наименьшим.\n",
    "\n",
    "**Пример**\n",
    "\n",
    "$$A = \\begin{bmatrix} * & * & * & * & *\\\\ * & * & 0 & 0 & 0 \\\\ * & 0  & * & 0 & 0 \\\\ * & 0 & 0& * & 0 \\\\ * & 0 & 0& 0 & * \\end{bmatrix} $$\n",
    "\n",
    "- Если мы исключаем элементы сверху вниз, тогда мы получим плотную матрицу.\n",
    "- Однако мы можем сохранить разреженность, если исключение будет проводиться снизу вверх.\n",
    "- Подробности на следующих слайдах"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "## Метод Гаусса для разреженных матриц\n",
    "\n",
    "- Дана матрица $A$ такая что $A=A^*>0$. \n",
    "- Вычислим её разложение Холецкого $A = LL^*$.\n",
    "\n",
    "Фактор $L$ может быть плотным даже если $A$ разреженная:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix} * & * & * & * \\\\ * & * &  &  \\\\ * &  & * &  \\\\ * &  &  & * \\end{bmatrix} = \n",
    "\\begin{bmatrix} * &  &  &  \\\\ * & * &  &  \\\\ * & * & * &  \\\\ * & * & * & * \\end{bmatrix}\n",
    "\\begin{bmatrix} * & * & * & * \\\\  & * & * & * \\\\  &  & * & * \\\\  &  &  & * \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "**Q**: как сделать факторы разреженными, то есть минимизировать заполнение?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Метод Гаусса и перестановка\n",
    "\n",
    "- Нам нужно найти перестановку индексов такую что факторы будут разреженными, то есть мы будем вычислять разложение Холецкого для матрицы $PAP^\\top$, где $P$ – матрица перестановки.\n",
    "\n",
    "- Для примера с предыдущего слайда\n",
    "\n",
    "$$\n",
    "P \\begin{bmatrix} * & * & * & * \\\\ * & * &  &  \\\\ * &  & * &  \\\\ * &  &  & * \\end{bmatrix} P^\\top = \n",
    "\\begin{bmatrix} * &  &  & *  \\\\  & * &  & * \\\\  &  & * & * \\\\ * & * & * & * \\end{bmatrix} = \n",
    "\\begin{bmatrix} * &  &  &  \\\\  & * &  &  \\\\  &  & * &  \\\\ * & * & * & * \\end{bmatrix}\n",
    "\\begin{bmatrix} * &  &  & *  \\\\  & * &  & * \\\\  &  & * & * \\\\  & &  & * \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "где\n",
    "\n",
    "$$\n",
    "P = \\begin{bmatrix}  &  &  & 1 \\\\  &  & 1 &  \\\\  & 1 &  &  \\\\ 1 &  &  &  \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "- Такая форма матрицы даёт разреженные факторы в LU разложении"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Original matrix\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "L factor\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U factor\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Column permutation: [0 1 2 3]\n",
      "Row permutation: [2 1 0 3]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy.sparse as spsp\n",
    "import scipy.sparse.linalg as spsplin\n",
    "import scipy.linalg as splin\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "A = spsp.coo_matrix((np.random.randn(10), ([0, 0, 0, 0, 1, 1, 2, 2, 3, 3], \n",
    "                                           [0, 1, 2, 3, 0, 1, 0, 2, 0, 3])))\n",
    "print(\"Original matrix\")\n",
    "plt.spy(A)\n",
    "plt.show()\n",
    "lu = spsplin.splu(A.tocsc(), permc_spec=\"NATURAL\")\n",
    "print(\"L factor\")\n",
    "plt.spy(lu.L)\n",
    "plt.show()\n",
    "print(\"U factor\")\n",
    "plt.spy(lu.U)\n",
    "plt.show()\n",
    "print(\"Column permutation:\", lu.perm_c)\n",
    "print(\"Row permutation:\", lu.perm_r)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Блочный случай\n",
    "\n",
    "$$\n",
    "PAP^\\top = \\begin{bmatrix} A_{11} &  & A_{13} \\\\  & A_{22} & A_{23} \\\\ A_{31} & A_{32} & A_{33}\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "тогда\n",
    "\n",
    "$$\n",
    "PAP^\\top = \\begin{bmatrix} A_{11} & 0 & 0 \\\\ 0 & A_{22} & 0 \\\\ A_{31} & A_{32} & A_{33} - A_{31}A_{11}^{-1} A_{13} - A_{32}A_{22}^{-1}A_{23} \\end{bmatrix} \\begin{bmatrix}  I & 0 & A_{11}^{-1}A_{13} \\\\ 0 & I & A_{22}^{-1}A_{23} \\\\ 0 & 0 & I\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "- Блок $ A_{33} - A_{31}A_{11}^{-1} A_{13} - A_{32}A_{22}^{-1}A_{23}$ является дополнением по Шуру для блочно-диагональной матрицы $\\begin{bmatrix} A_{11} & 0 \\\\ 0 & A_{22} \\end{bmatrix}$\n",
    "- Мы свели задачу к решению меньших линейных систем с матрицами $A_{11}$ и $A_{22}$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Как найти перестановку?\n",
    "\n",
    "- Основная идея взята из теории графов\n",
    "- Разреженную матрицы можно рассматривать как **матрицу смежности** некоторого графа: вершины $(i, j)$ связаны ребром, если соответствующий элемент матрицы не ноль.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Пример\n",
    "\n",
    "Графы для матрицы $\\begin{bmatrix} * & * & * & * \\\\ * & * &  &  \\\\ * &  & * &  \\\\ * &  &  & * \\end{bmatrix}$ и для матрицы $\\begin{bmatrix} * &  &  & *  \\\\  & * &  & * \\\\  &  & * & * \\\\ * & * & * & * \\end{bmatrix}$ имеют следующий вид:\n",
    "\n",
    "<img src=\"./graph_dense.png\" width=300 align=\"center\"> и <img src=\"./graph_sparse.png\" width=300 align=\"center\">\n",
    "\n",
    "* Почему вторая упорядоченность лучше, чем первая?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Сепаратор графа\n",
    "\n",
    "**Определение.** Сепаратором графа $G$ называется множество вершин $S$, таких что их удаление оставляет как минимум две связные компоненты.\n",
    "\n",
    "Сепаратор $S$ даёт следующий метод нумерации вершин графа $G$:\n",
    "- Найти сепаратор $S$, удаление которого оставляет связные компоненты $T_1$, $T_2$, $\\ldots$, $T_k$\n",
    "- Номера вершин в $S$ от $N − |S| + 1$ до $N$\n",
    "- Рекурсивно, номера вершин в каждой компоненте: \n",
    "    - в $T_1$ от $1$ до $|T_1|$\n",
    "    - в $T_2$ от $|T_1| + 1$ до $|T_1| + |T_2|$\n",
    "    - и так далее\n",
    "- Если компонента достаточно мала, то нумерация внутри этой компоненты произвольная"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Сепаратор и структура матрицы: пример\n",
    "\n",
    "Сепаратор для матрицы двумерного лапласиана\n",
    "\n",
    "$$\n",
    " A_{2D} = I \\otimes A_{1D} + A_{1D} \\otimes I, \\quad A_{1D} = \\mathrm{tridiag}(-1, 2, -1),\n",
    "$$\n",
    "\n",
    "имеет следующий вид\n",
    "\n",
    "<img src='./separator.png' width=300> </img>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Если мы пронумеруем сначала индексы в $\\alpha$, затем в $\\beta$, и наконец индексы в сепараторе $\\sigma$ получим следующую матрицу\n",
    "\n",
    "$$\n",
    "PAP^\\top = \\begin{bmatrix} A_{\\alpha\\alpha} &  & A_{\\alpha\\sigma} \\\\  & A_{\\beta\\beta} & A_{\\beta\\sigma} \\\\ A_{\\sigma\\alpha} & A_{\\sigma\\beta} & A_{\\sigma\\sigma}\\end{bmatrix},\n",
    "$$\n",
    "\n",
    "которая имеет подходящую структуру.\n",
    "\n",
    "- Таким образом, задача поиска перестановки была сведена к задаче поиска сепаратора графа!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "### Nested dissection\n",
    "\n",
    "- Для блоков $A_{\\alpha\\alpha}$, $A_{\\beta\\beta}$ можем продолжить разбиение рекурсивно\n",
    "\n",
    "- После завершения рекурсии нужно исключить блоки $A_{\\sigma\\alpha}$ и $A_{\\sigma\\beta}$. \n",
    "\n",
    "- Это делает блок в положении $A_{\\sigma\\sigma}\\in\\mathbb{R}^{n\\times n}$ **плотным**.\n",
    "\n",
    "- Вычисление разложения Холецкого этого блока стоит $\\mathcal{O}(n^3) = \\mathcal{O}(N^{3/2})$, где $N = n^2$ – общее число вершин.\n",
    "\n",
    "- В итоге сложность $\\mathcal{O}(N^{3/2})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Пакеты для nested dissection\n",
    "\n",
    "- MUltifrontal Massively Parallel sparse direct Solver ([MUMPS](http://mumps.enseeiht.fr/))\n",
    "- [Pardiso](https://www.pardiso-project.org/)\n",
    "- [Umfpack как часть пакета SuiteSparse](http://faculty.cse.tamu.edu/davis/suitesparse.html)\n",
    "\n",
    "У них есть интефейс для C/C++, Fortran и Matlab "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Резюме про nested dissection\n",
    "\n",
    "- Нумерация свелась к поиску сепаратора\n",
    "- Подход разделяй и властвуй\n",
    "- Рекурсивно продолжется на два (или более) подмножества вершин после разделения\n",
    "- В теории nested dissection даёт оптимальную сложность (почему?)\n",
    "- На практике этот метод лучше других только на очень больших задачах"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сепараторы на практике\n",
    "\n",
    "- Вычисление сепаратора – это **нетривиальная задача!**\n",
    "\n",
    "- Построение методов разбиения графа было активной сферой научных исследований долгие годы\n",
    "\n",
    "Существующие подходы:\n",
    "\n",
    "- Спектральное разбиение (использует собственные векторы **Лапласиана графа**) – подробности далее\n",
    "- Геометрическое разбиение (для сеток с заданными координатами вершин) [обзор и анализ](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.31.4886&rep=rep1&type=pdf)\n",
    "- Итеративные перестановки ([(Kernighan-Lin, 1970)](http://xilinx.asia/_hdl/4/eda.ee.ucla.edu/EE201A-04Spring/kl.pdf), [(Fiduccia-Matheysses, 19820](https://dl.acm.org/citation.cfm?id=809204))\n",
    "- Поиск в ширину [(Lipton, Tarjan 1979)](http://www.cs.princeton.edu/courses/archive/fall06/cos528/handouts/sepplanar.pdf)\n",
    "- Многоуровневая рекурсивная бисекция (наиболее практичная эвристика) ([обзор](https://people.csail.mit.edu/jshun/6886-s18/lectures/lecture13-1.pdf) и [статья](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.499.4130&rep=rep1&type=pdf)). Пакет для подобного рода разбиений называется METIS, написан на C, и доступен [здесь](http://glaros.dtc.umn.edu/gkhome/views/metis)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Спектральное разбиение графа\n",
    "\n",
    "- Идея спектрального разбиения восходит к работам Мирослава Фидлера, который изучал связность графов ([статья](https://dml.cz/bitstream/handle/10338.dmlcz/101168/CzechMathJ_23-1973-2_11.pdf)).\n",
    "\n",
    "- Нам нужно разбить вершинеы графа на 2 множества\n",
    "\n",
    "- Рассмотрим метки вершин +1/-1 и **функцию потерь**\n",
    "\n",
    "$$E_c(x) = \\sum_{j} \\sum_{i \\in N(j)} (x_i - x_j)^2, \\quad N(j) \\text{ обозначает множество соседей вершины } j. $$\n",
    "\n",
    "Нам нужно сбалансированное разбиение, поэтому\n",
    "\n",
    "$$\\sum_i x_i =  0 \\quad \\Longleftrightarrow \\quad x^\\top e = 0, \\quad e = \\begin{bmatrix}1 & \\dots & 1\\end{bmatrix}^\\top,$$\n",
    "\n",
    "и поскольку мы ввели метки +1/-1, то выполнено\n",
    "\n",
    "$$\\sum_i x^2_i = n \\quad \\Longleftrightarrow \\quad \\|x\\|_2^2 = n.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Лапласиан графа\n",
    "\n",
    "Функция потерь $E_c$ может быть записана в виде (проверьте почему)\n",
    "\n",
    "$$E_c = (Lx, x)$$\n",
    "\n",
    "где $L$ – **Лапласиан графа**, который определяется как симметричная матрица с элементами\n",
    "\n",
    "$$L_{ii} = \\mbox{степень вершины $i$},$$\n",
    "\n",
    "$$L_{ij} = -1, \\quad \\mbox{если $i \\ne j$  и существует ребро},$$\n",
    "\n",
    "и $0$ иначе.\n",
    "\n",
    "- Строчные суммы в матрице $L$ равны нулю, поэтому существует собственное значение $0$, которое даёт собственный вектор из всех 1.\n",
    "- Собственные значения неотрицательны (почему?)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Разбиение как задача оптимизации\n",
    "\n",
    "- Минимизация $E_c$ с упомянутыми ограничениями приводит к разбиению, которое минимизирует число вершин в сепараторе, но сохраняет разбиение сбалансированным\n",
    "\n",
    "- Теперь мы запишем релаксацию целочисленной задачи квадратичного программирования в форме непрерывной задачи квадратичного программирования\n",
    "\n",
    "$$E_c(x) = (Lx, x)\\to \\min_{\\substack{x^\\top e =0, \\\\ \\|x\\|_2^2 = n}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Вектор Фидлера\n",
    "\n",
    "- Решение этой задачи минимизации – собственный вектор матрицы $L$, соответствующий **второму** минимальному собственному значению (он называется вектором Фидлера)\n",
    "- В самом деле,\n",
    "\n",
    "$$\n",
    "    \\min_{\\substack{x^\\top e =0, \\\\ \\|x\\|_2^2 = n}} (Lx, x) = n \\cdot \\min_{{x^\\top e =0}} \\frac{(Lx, x)}{(x, x)} = n \\cdot \\min_{{x^\\top e =0}} R(x), \\quad R(x) \\text{ отношение Релея}\n",
    "$$\n",
    "\n",
    "- Поскольку $e$ – собственный вектор, соответствующий наименьшему собственному значению, то на подпространстве $x^\\top e =0$ мы получим второе минимальное собственное значение.\n",
    "\n",
    "- Знак $x_i$ обозначает разбиение графа.\n",
    "\n",
    "- Осталось понять, как вычислить этот вектор. Мы знаем про степенной метод, но он ищет собственный вектор для максимального по модулю собственного значения.\n",
    "- Итерационные методы для задачи на собственные значения будут рассмотрены далее в курсе...\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of vertices = 34\n",
      "Number of edges = 78\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import networkx as nx\n",
    "kn = nx.read_gml('karate.gml')\n",
    "print(\"Number of vertices = {}\".format(kn.number_of_nodes()))\n",
    "print(\"Number of edges = {}\".format(kn.number_of_edges()))\n",
    "nx.draw_networkx(kn, node_color=\"red\") #Draw the graph"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The 2 smallest eigenvalues = [-4.14058418e-15  4.68525227e-01]\n"
     ]
    }
   ],
   "source": [
    "Laplacian = nx.laplacian_matrix(kn).asfptype()\n",
    "plt.spy(Laplacian, markersize=5)\n",
    "plt.title(\"Graph laplacian\")\n",
    "plt.axis(\"off\")\n",
    "plt.show()\n",
    "eigval, eigvec = spsplin.eigsh(Laplacian, k=2, which=\"SM\")\n",
    "print(\"The 2 smallest eigenvalues =\", eigval)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sum of elements in Fiedler vector = 8.743006318923108e-16\n"
     ]
    }
   ],
   "source": [
    "plt.scatter(np.arange(len(eigvec[:, 1])), np.sign(eigvec[:, 1]))\n",
    "plt.show()\n",
    "print(\"Sum of elements in Fiedler vector = {}\".format(np.sum(eigvec[:, 1].real)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nx.draw_networkx(kn, node_color=np.sign(eigvec[:, 1]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Резюме по примеру использования спектрального разбиения графа\n",
    "\n",
    "- Мы вызвали функцию из SciPy для поиска фиксированного числа собственных векторов и собстенных значений, которые минимальны (возможны другие опции)\n",
    "- Детали методов, которые реализованы в этих функциях, обсудим уже скоро\n",
    "- Вектор Фидлера даёт простой способ разбиения графа\n",
    "- Для разбиения графа на большее количество частей следует использовать собственные векторы Лапласиана как векторы признаков и запустить какой-нибудь алгоритм кластеризации, например $k$-means"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Вектор Фидлера и алгебраическая связность графа\n",
    "\n",
    "**Определение.** Алгебраическая связность графа – это второе наименьшее собственное значение матрицы Лапласиана графа.\n",
    "\n",
    "**Утверждение.** Алгебраическая связность графа больше 0 тогда и только тогда, когда граф связный."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Minimal degree orderings\n",
    "\n",
    "- Идея в том, чтобы исклоючить строки и/или столбцы с малым числом ненулей, обновить заполнение и повторить.\n",
    "\n",
    "- Эффективная реализация является отдельной задачей (добавление/удаление элементов).\n",
    "\n",
    "- На практике часто лучше всего для задач среднего размера\n",
    "\n",
    "- SciPy [использует](https://docs.scipy.org/doc/scipy-1.3.0/reference/generated/scipy.sparse.linalg.splu.html) этот подход для различных матриц ($A^{\\top}A$, $A + A^{\\top}$) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Резюме по второй части\n",
    "\n",
    "- Плотные матрицы большого размера и распределённые вычисления\n",
    "- Разреженные матрицы, приложения и форматы их хранения\n",
    "- Эффективные способы умножения разреженной матрицы на вектор\n",
    "- LU разложение разреженной матрицы: заполнение и перестановки строк\n",
    "- Минимизация заполнения: сепараторы и разбиение графа\n",
    "- Nested dissection\n",
    "- Спектральное разбиение графа: Лапласиан графа и вектор Фидлера"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Alegreya+Sans:100,300,400,500,700,800,900,100italic,300italic,400italic,500italic,700italic,800italic,900italic' rel='stylesheet' type='text/css'>\n",
       "<link href='http://fonts.googleapis.com/css?family=Source+Code+Pro:300,400' rel='stylesheet' type='text/css'>\n",
       "<style>\n",
       "    @font-face {\n",
       "        font-family: \"Computer Modern\";\n",
       "        src: url('http://mirrors.ctan.org/fonts/cm-unicode/fonts/otf/cmunss.otf');\n",
       "    }\n",
       "    div.cell{\n",
       "        /*width:80%;*/\n",
       "        /*margin-left:auto !important;\n",
       "        margin-right:auto;*/\n",
       "    }\n",
       "    h1 {\n",
       "        font-family: 'Alegreya Sans', sans-serif;\n",
       "    }\n",
       "    h2 {\n",
       "        font-family: 'Fenix', serif;\n",
       "    }\n",
       "    h3{\n",
       "\t\tfont-family: 'Fenix', serif;\n",
       "        margin-top:12px;\n",
       "        margin-bottom: 3px;\n",
       "       }\n",
       "\th4{\n",
       "\t\tfont-family: 'Fenix', serif;\n",
       "       }\n",
       "    h5 {\n",
       "        font-family: 'Alegreya Sans', sans-serif;\n",
       "    }\t   \n",
       "    div.text_cell_render{\n",
       "        font-family: 'Alegreya Sans',Computer Modern, \"Helvetica Neue\", Arial, Helvetica, Geneva, sans-serif;\n",
       "        line-height: 1.2;\n",
       "        font-size: 120%;\n",
       "        /*width:70%;*/\n",
       "        /*margin-left:auto;*/\n",
       "        margin-right:auto;\n",
       "    }\n",
       "    .CodeMirror{\n",
       "            font-family: \"Source Code Pro\";\n",
       "\t\t\tfont-size: 90%;\n",
       "    }\n",
       "/*    .prompt{\n",
       "        display: None;\n",
       "    }*/\n",
       "    .text_cell_render h1 {\n",
       "        font-weight: 200;\n",
       "        font-size: 50pt;\n",
       "\t\tline-height: 110%;\n",
       "        color:#CD2305;\n",
       "        margin-bottom: 0.5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "    }\t\n",
       "    .text_cell_render h5 {\n",
       "        font-weight: 300;\n",
       "        font-size: 16pt;\n",
       "        color: #CD2305;\n",
       "        font-style: italic;\n",
       "        margin-bottom: .5em;\n",
       "        margin-top: 0.5em;\n",
       "        display: block;\n",
       "    }\n",
       "    \n",
       "    li {\n",
       "        line-height: 110%;\n",
       "    }\n",
       "    .warning{\n",
       "        color: rgb( 240, 20, 20 )\n",
       "        }  \n",
       "\n",
       "</style>\n",
       "\n",
       "<script>\n",
       "    MathJax.Hub.Config({\n",
       "                        TeX: {\n",
       "                           extensions: [\"AMSmath.js\"]\n",
       "                           },\n",
       "                tex2jax: {\n",
       "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
       "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
       "                },\n",
       "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
       "                \"HTML-CSS\": {\n",
       "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
       "                }\n",
       "        });\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = open(\"./styles/custom.css\", \"r\").read()\n",
    "    return HTML(styles)\n",
    "css_styling()"
   ]
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