lectures/lecture9/lecture9.ipynb

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   "source": [
    "# Лекция 9. Введение в итерационные методы"
   ]
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   "source": [
    "## На прошлой лекции...\n",
    "- Разреженные матрицы и форматы их хранения\n",
    "- Прямые методы решения больших разреженных линейных систем\n",
    "- Вектор Фидлера и сепаратор графа"
   ]
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   "source": [
    "## План на сегодня\n",
    "\n",
    "Концепция **итерационных методов** для линейных систем:\n",
    "- Метод Ричардсона и его сходимость\n",
    "- Метод Чебышёва и его сходимость "
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   "source": [
    "## Итерационные методы\n",
    "\n",
    "- Если мы хотим достичь сложности $\\mathcal{O}(N)$ в решении разреженных линейных систем, то прямые методы нам не подходят.\n",
    "\n",
    "- Если нам нужно решить частичную задачу на собственные значения, то полное спектральное разложение оказывается слишком дорогой операцией.\n",
    "\n",
    "- Для обеих задач мы будем использовать итерационные методы, основанные на Крыловских подпространствах, которые рассматривают матрицу как **линейный оператор** для преобразования одного вектора в другой (\"чёрный ящик\"), а не как двумерный массив с числами."
   ]
  },
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   "source": [
    "## Матрица как \"чёрный ящик\"\n",
    "\n",
    "- Сейчас рассмотрим альтернативный взгляд на то, что такое матрица: далее матрица – это линейный оператор, который действует на вектор, и результат этого действия можно вычислить за $\\mathcal{O}(N)$ операций.\n",
    "\n",
    "- **Единственная информация, которая нам доступна:** мы знаем о матрице только то, как умножать её на вектор!\n",
    "\n",
    "- Можно ли решить линейную систему, используя только процедуру умножения матрицы на вектор?\n",
    "\n",
    "- Конечно, мы можем умножить матрицу на столбцы единичной матрицы и восстановить матрицу, но это не то, что нам нужно."
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   "source": [
    "## Метод Ричардсона (a.k.a метод простой итерации)\n",
    "\n",
    "Простейшая идея приводит нас к **методу Ричардсона**.\n",
    "\n",
    "$$Ax = f,$$\n",
    "$$\\tau  (Ax - f) = 0,$$\n",
    "$$x - \\tau (Ax - f) = x,$$\n",
    "$$x_{k+1} = x_k - \\tau (Ax_k - f),$$\n",
    "   \n",
    "где $\\tau > 0$ параметр метода, который надо выбрать так, чтобы метод сходился."
   ]
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   "source": [
    "## Связь с обыкновенными дифференциальными уравнениями (ОДУ)\n",
    "\n",
    "- Метод Ричардсона имеет глубокую связь с ОДУ.\n",
    "\n",
    "- Рассмотрим задачу, зависящую от времени ($A=A^*>0$)\n",
    "\n",
    "$$\\frac{dy}{dt} + A y = f, \\quad y(0) = y_0.$$\n",
    "\n",
    "Тогда $y(t) \\rightarrow A^{-1} f$ при $t \\rightarrow \\infty$, и **схема Эйлера** выглядит следующим образом\n",
    "\n",
    "$$\\frac{y_{k+1} - y_k}{\\tau} = -A y_k + f,$$\n",
    "\n",
    "что также приводит к методу Ричардсона\n",
    "\n",
    "$$ y_{k+1} = y_k - \\tau(Ay_k -f). $$"
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   "source": [
    "## Сходимость метода Ричардсона\n",
    "\n",
    "- Пусть $x_*$ решение, введём вектор ошибки $e_k = x_{k} - x_*$, тогда  \n",
    "\n",
    "$$ e_{k+1} = (I - \\tau A) e_k, $$\n",
    "\n",
    "поэтому если $\\Vert I - \\tau A \\Vert < 1$ в некоторой норме, метод сходится. \n",
    "\n",
    "- Для случая симметричной положительно определённой матрицы всегда можно выбрать $\\tau$ так, чтобы метод сошёлся.\n",
    "\n",
    "- Поведение метода для несимметричной матрицы будет рассмотрено далее"
   ]
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   "source": [
    "## Выбор оптимального параметра\n",
    "\n",
    "- Параметр $\\tau$, который минимизирует $\\|I - \\tau A\\|_2$ для $A = A^* > 0$, выражается следующим образом (получите это выражение!)\n",
    "\n",
    "$$ \\tau_\\mathrm{opt} = \\frac{2}{\\lambda_{\\min} + \\lambda_{\\max}}, $$\n",
    "\n",
    "где $\\lambda_{\\min}$ – минимальное собственное значение, а $\\lambda_{\\max}$ – максимальное собственное значение матрицы $A$.\n",
    "\n",
    "- Итак, чтобы найти оптимальный параметр нам нужно знать **границы спектра** матрицы $A$, и мы можем вычислить их с помощью **степенного метода**."
   ]
  },
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   "source": [
    "## Число обусловленности и скорость сходимости\n",
    "\n",
    "- Даже с оптимальным выбором значения параметра $\\tau$, ошибка на следующей итерации связана с ошибкой на текущей итерации следующим образом\n",
    "\n",
    "$$\\|e_{k+1}\\|_2 \\leq q \\|e_k\\|_2 , \\quad\\rightarrow \\quad \\|e_k\\|_2 \\leq q^{k} \\|e_0\\|_2,$$\n",
    "\n",
    "где \n",
    "\n",
    "$$ q = \\frac{\\lambda_{\\max} - \\lambda_{\\min}}{\\lambda_{\\max} + \\lambda_{\\min}} = \\frac{\\mathrm{cond}(A) - 1}{\\mathrm{cond}(A)+1}, $$\n",
    "\n",
    "$$\\mathrm{cond}(A) = \\frac{\\lambda_{\\max}}{\\lambda_{\\min}} \\quad \\text{для} \\quad A=A^*>0$$\n",
    "\n",
    "число обусловенности матрицы $A$.\n",
    "\n",
    "- Далее покажем на примере сходимость метода Ричардсона..."
   ]
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   "outputs": [
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     "name": "stdout",
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     "text": [
      "Maximum eigenvalue = 3.682507065662361, minimum eigenvalue = 0.08101405277100546\n",
      "Condition number = 45.45516413147909\n"
     ]
    },
    {
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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 matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "plt.rc(\"text\", usetex=True)\n",
    "import scipy as sp\n",
    "import scipy.sparse\n",
    "import scipy.sparse.linalg as spla\n",
    "import scipy\n",
    "from scipy.sparse import csc_matrix\n",
    "n = 10\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.ones(n)\n",
    "ev1, vec = spla.eigsh(A, k=2, which='LA')\n",
    "ev2, vec = spla.eigsh(A, k=2, which='SA')\n",
    "lam_max = ev1[0]\n",
    "lam_min = ev2[0]\n",
    "\n",
    "tau_opt = 2.0/(lam_max + lam_min)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "plt.close(fig)\n",
    "\n",
    "niters = 100\n",
    "x = np.zeros(n)\n",
    "res_richardson = []\n",
    "for i in range(niters):\n",
    "    rr = A.dot(x) - rhs\n",
    "    x = x - tau_opt * rr\n",
    "    res_richardson.append(np.linalg.norm(rr))\n",
    "#Convergence of an ordinary Richardson (with optimal parameter)\n",
    "plt.plot(res_richardson)\n",
    "plt.yscale(\"log\")\n",
    "plt.xlabel(\"Number of iterations, $k$\", fontsize=20)\n",
    "plt.ylabel(\"Residual norm, $\\|Ax_k - b\\|_2$\", fontsize=20)\n",
    "plt.xticks(fontsize=20)\n",
    "plt.yticks(fontsize=20)\n",
    "print(\"Maximum eigenvalue = {}, minimum eigenvalue = {}\".format(lam_max, lam_min))\n",
    "print(\"Condition number = {}\".format(lam_max.real / lam_min.real))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Вывод о влиянии числа обусловленности на сходимость\n",
    "\n",
    "- Для **плохо обусловленных** матриц норма вектора ошибки в методе Ричардсона убывает **очень** медленно.\n",
    "- Это ещё одна причина, по которой число обусловленности очень важно: кроме того, что оно показывает ошибку при решении, оно также даёт оценку числа итераций для сходимости итерационных методов.\n",
    "- Основной вопрос в разработке итерационных методов – как улучшить обусловленность матрицы?\n",
    "- Ответом является использование <font color='red'> предобуславливателей (preconditioners) </font>. Эта тема будет рассмотрена далее в курсе."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Рассмотрим несимметричную матрицу $A$\n",
    "\n",
    "Возможные варианты поведения метода Ричардсона:\n",
    "- сходимость\n",
    "- расходимость\n",
    "- почти стабильная траектория, то есть отсутствие сходимости, но и отсутствие существенного увеличения нормы вектора ошибки\n",
    "\n",
    "###### Как мы можем определить поведение метода Ричардсона для несимметричной матрицы **до** запуска метода?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1  2]\n",
      " [-1  0]]\n",
      "Spectrum of the matrix = [0.5+1.32287566j 0.5-1.32287566j]\n"
     ]
    }
   ],
   "source": [
    "# B = np.random.randn(2, 2)\n",
    "B = np.array([[1, 2], [-1, 0]])\n",
    "#B = np.array([[0, 1], [-1, 0]])\n",
    "print(B)\n",
    "x_true = np.zeros(2)\n",
    "f = B.dot(x_true)\n",
    "eigvals = np.linalg.eigvals(B)\n",
    "print(\"Spectrum of the matrix = {}\".format(eigvals))\n",
    "\n",
    "# Run Richardson iteration\n",
    "x = np.array([0, -1])\n",
    "tau = 1e-2\n",
    "conv_x = [x]\n",
    "r = B.dot(x) - f\n",
    "conv_r = [np.linalg.norm(r)]\n",
    "num_iter = 1000\n",
    "for i in range(num_iter):\n",
    "    x = x - tau * r\n",
    "    conv_x.append(x)\n",
    "    r = B.dot(x) - f\n",
    "    conv_r.append(np.linalg.norm(r))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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TswTkd4PZwHicWlCVtTblVu5UAko+1lre2NXMk2t3UvNOI1v3HeTQ0WP0y0hncG5fThmay7SReUwb5WPGaB/9MtOjeu+/btzLT/6yhertDRQOyuHf507moolDonYPkWQQ8wTUGygBSSSstfx1016+8/SbbN13kMumDuO7V00lPyd4k6BIqgknAUUyCKHj4iG349Z+QdLbGWO4eOIQzikayEMvbuWHqzdRta2Bu+dP58KPDPY6PJGEEMlipKcbYzYDDZ08RARnlN7NF0/g9zefS16/DP71Z69y3+rNtLer5UEk0sVIBwJ348z7EZEunDYij6e/fB7ffHwD967exNvvNfP9+dPJ6RtxI4RI0ot0ImqptfahaAcjksqyMtK5Z8F0Jo8YwF3PvsXOxsP8/PpZEY26E0kFkYwPXQ1oMxaRCBhjuOH8Qh78bAkb32thfsXL7Go87HVYIp6IJAEtBcqNMZ/sbCCCiIR2yaSh/OoLZ7K3+QjzH3iZd+sPeR2SSNxFkoD+BBQBK4EG7YgqEpkzxhfwcOlZHDhyjM889E/ea2r1OiSRuNKOqCIemjIyj//9/Blc+9A/+fRDr/BI6dkMzlWfkPQOmojaBU1ElXhYs62ez/70VcYPyuHRG8+mv0bHSZILZyKqFqkSSQCzxhVw/7XFbHy/hS//toZj2mpcegElIJEEcdHEIXznytP4y8a9/OfTb2qrcUl5queH4Lcdg9ehSC/ymTPH8s7+Q1S8UEfh4Bw+d+54r0MSiRnVgEKw1j5trS3Ny8vr+mSRKFp66anMnjSU7/7hLV7dqil3krqUgEQSTFqa4QcLpzO6IJubf1vDnmYNz5bUpAQkkoAGZGXwwLUzOdB6jJt/W0ObBiVIClICEklQE4fl8r2rp7JmWwPLn3vb63BEok4JSCSBXTljJNedNZYHX9zKC5v2eh2OSFR1moCMMe1Bltrp6qGleESi6I7LJ3HKkP58/dF17DtwxOtwRKKmq2HYd6Nld0Q8lZWRzo8/fTpX/OQf3LpiHT+7fhbGGK/DEumxThOQtXZpvAIRkdBOHTaAOy6bxH889Qa/eGmb5gdJSlAfkEiS+OzZY/mXU4ew7I9vs/G9Fq/DEemxiFdCMMZ8CigM9py19vsRRyQiQRljWD5vGpf+8AW++rvXePJL59K3T7rXYYlErNsJyBgzHqgC8t1DFvBvkLaAEpBIDAzq35fyq6fxhf+t4p7nN/HNyyZ5HZJIxCJpgqsAtgIzrbVpQBNQDBTgbNddFr3wRCTQJZOG8pkzx/Dgi3W8tGWf1+GIRCySBDQbWGKtfc39uQ7AWtsIlKMEJBJzd1w+ifEDc7hlxTqaDrV5HY5IRCJJQI2A/wqddUDHpkM+nNqQiMRQdmYffrhoBntbjnDH7zdo6wZJSpEkoNXAIr+fHwWWuoMSynESlIjE2LRRPr42+xSeWb+bJ9fu8jockW6LJAEtxekDAsBauxLYBqzE6QdaHJXIRKRLN100gZKx+XzrydfZ0XDI63BEuqXbCchau9Vae1vAsTlAkbW2wFr7eNSiiwFjzCqvYxCJlvQ0w70LZ2At3PLoOo63qylOkkfUJqJaa7d2fZZ3jDGzjTGlOIMoRFLG6IJsvn3Fafxzaz0PvljndTgiYYtkHtCArs6x1jZ343rzgEZr7eogzxXjDHCow5n0WhfsvHB0vM4YUxHJ60US2dXFI/nz2+9zz/MbOW/CIKaM1E6+kvgiHQXX0MUjLMaY2cCDIZ4rBMqttZXW2tXW2kqgzD0uIn6MMXz3qqkU5GTytUfW0tp23OuQRLoUSQK6McTjIZwVEbqcB2SMKXRrIoVAqE3vy3AmvfqrwBlpJyIB8nMy+f786WzZc4Dv/VEb2Eni63YTnFsTCeZBY8wWoCiMa9ThJipjTKgVt+dxcgKqAj4YROD26XR2v1WRNtmJJKPzTxnM588dz8/+sZWLJg7moolDvA5JJKSIFyMNYTXOvKDbe3IRY4yPILUja22jMQZjTKG1tq6TZCjSay25dCJ/37KXW1eu59mvnM/g3L5ehyQSVLS3Y5hNiBWyu6kAPljeJxj1A4mEkJWRzn2LTqeltY0b/ncNh45qk2JJTN1OQJ1t043TP1MThbh8UbjGCYwxxcaYJe7fy90BECIpadLwAfz4mmI27GziKw+/Rtvxdq9DEjlJJE1wnW3TXYvTBJdwrLU1OMlxeVfnun1LpQBjxoyJcWQisTFn8lD+84rT+NaTb3DTr6v5yaeLycrQ/kGSOCIZhBC3bbqNMb5OmuFixu1bqgQoKSnR1HJJWtedPQ6M4Vu/f53rf/4qP/l0MYP696xPaP+BI1Rtb6DmnQberT/E3pYjNB12VuQ2GPKyMxiS25fCwf05bcQAisfkqx9Kgor2IIRo6ZjOXYDf4qbu4AT/50WkC9edNZbcvn1Y+th6LrvvRb5z5RQ+dtpQjDFdvtZaS92+g1Rva6Bqez1V2xqo23cQgMz0NEYX9GNwbl/GD8ohzRiOt1saD7WxYWcTz27YTcfKQFNH5jFn8lCunjmKkb5+sSyuJJFOE1A4qx4E052VEEK8vtEYU8fJfUEFOKsmKAGJdMNVp49k4rBcvva7tdz462pmjPaxaNZozp0wiFH5/TDGYK2l4VAbb+5qZsPOJmreaaBmewP7Dx4FwJedQcnYfOaXjGbWuHymjMzrtEnv8NHjvLm7mVfq9vPnt/fwg1WbuHf1Ji44ZTBfvKiIMwsHxqv4kqBMZ/uIGGPaCd3fE5K1NuyGZmNMLVAWOF/Hnaha7T/U2l22Z6G1dn53Y+ouY8xcYO6ECRMWb968Oda3E4mLY8fbeaTqXX76963U7XVqMlkZaWRn9uFA6zGO+g1WGDswm5KxBZSMy2fWuHwKB/UnLa3rWlMo79YfYmX1Dn7zz+3sO3CUM8YXcPvHT+X0Mfk9LpckHmNMtbW2pNNzukhApZycgHw4o92WcuLePxOAW3F2S/1+N4JsABa72zr4H/cBK9yVtjuOrcJJVnGrAZWUlNiqqqp43U4kLqy1vLGrmbXvNrJ9/0EOHT1O/6w+DO7fl4nDcpkyIo/8nMyY3Pvw0eM8/Oo73P+3Wva2HGHRrNEsufRUCmJ0P/FGjxNQiIvej1MzeSjIc7cCBdbaTieiusnldpz5PPNw+nRW46xcsNLvvGJgIbDGPbcm3isbKAGJxMaBI8f40Z8287O/byWvXwbL503jkklDvQ5LoiRWCageuCHYvj/GmEuAB6y1p3TroglMCUgktja+18LXHlnLW7ubue6ssdxx+SQNF08B4SSgSFdCKA1xfD7uKgbJzhgz1xhT2dTU5HUoIilt4rBcfn/zOSw+fzy/emU7iypfYU9zq9dhSRxEkoC+B3zUGPOcMeaTxpgZxph/McY8gpOYUmJ9Nmvt09ba0rw87asiEmt9+6Rzx+WTeeDamWx6v4UrfvIPNuzQl79UF8mW3MuB24AzgMeAapz+mznA0q76f0REQrl0yjAeu+kc0tMMiypf5qUt+7wOSWIooiY4a+1ya20Bzm6lC4CZ1toCa+3dUY1ORHqdScMH8MQXz2FUfjbX/2INz7/xntchSYz0aDVsa22NtfYxa+1r0QpIRGTIgCweKTuLycMHcNNvanju9d1ehyQx0GkCMsY8b4w5FnAs5GrY7iMl1n7XIAQRb/myM/nNDWcyfVQeX374Nf66cY/XIUmUdTURdQlQYq1d4HesnC5WR7DW3ha1CD2mYdgi3mo63MY1la9Qu/cA//v5MzhLS/gkhZjMA+ptlIBEvLf/wBEWVr7Ce02tPHbTOUwclut1SNKFmMwDMsaMM8bMCDj2DWPMI8aYL3T3eiIiXRnYvy+//PwZZGem8/lfrGFPi+YJpYJIBiFUAGUdPxhjnsdZG64IqDTG3BWl2EREPjDC14+f/uss6g8eZfEvqzl89LjXIUkPRZKA5gCrAIwxpwOzceb/lOCs7xbzlapFpHeaOiqP+xbNYP2ORm5duQ51ISS3SIdhd6xGPRtnQELH6ge1OIuGJj2NghNJTB89bRi3fmwiz6zfzc//sc3rcKQHIklAdcACd7O6hTgrVHdsQFdIiuxWqqV4RBLXTRcWMWfyUO569i3WbKv3OhyJUCQJ6Db30QAU4+wL1KEMWBnsRSIi0WKM4Z4F0xmV34+bf1OjQQlJKpK14FbiDDhYAORba//s93Q5zmKlIiIxNSArg/uvnUlzaxtffXgtx9vVH5RsIu0DsjjrwD3iNsVhjMnDqRHpX4GIxMWk4QP4zhVTeLluP5UvpETrf68SyTygS3AGG8xxHwUA1tomnJqRakAiEjfzS0Zx+dTh3PP8Rta92+h1ONINkdSAyoGV7rBrE/BcJc7ABBGRuDDGcNcnpzIkty9f/d1rHDySEstR9gqRJKBi4JEQzzUAvoijSSAahi2SPPKyM7h34Qy21x/i20+94XU4EqZIElANTv9PMGU4m9MlPQ3DFkkuZxYO5OaLJrCiegd/3KDtG5JBnwhe8z3gUWOMwRlwkOeuDVeOMzF1ZvTCExEJ31dnn8ILm/dy+xMbKB6bz9ABWV6HJJ2IdBj2ApzajsGpEdUAs4AF1tq10QxQRCRcGelp3LtwBq1tx7l15Xot1ZPgIt2Se6W7JfdMTtyS+7GoRici0k1Fg/tzx2WTeGHTXn758navw5FO9HRL7tcCt+QO3KpBRCTerj1rLBdNHMxdz77Flj0tXocjIXSZgIwxA9y9fvYbYzaH2vPHPe9RoDrqUYqIdIMxhuVXTyM7M52vPbKWo8favQ5JgginBlSDM+H0Tzh9PpXGmFv8TzDGLMMZgj0bZ504ERFPDRmQxbJPTeP1nc3c96dNXocjQXSagIwxt7p/HW+tXWCtnQCsBZYbY3KNMTcYY/bjDEi4ze0Huju2IceH5gGJJL9Lpwxj/sxR3P/XWqq0anbC6aoGtBD4nrvMToelODWhRpyVDx7ESVApkXg6aB6QSGr4jytOY2R+P/7t0bW0tLZ5HY746SoBFQNVAcfWuH9W4qyGfVtAghIRSRj9+/bh3gUz2NlwmO88/abX4YifcPqAGv1/8Es2FUo8IpIMSsYVcNNFRayo3sFzr2uVhETRo2HYIiLJ4quXfIQpIwdw++Mb2NOsDewSQThL8VxtjAms6VicbbmDrglnrX2ox5GJiERRZp80frhwBpf/6O8seWw9P79+Fs6KYuKVcBJQqMEFoYZbW0AJSEQSzoQhuXzzskn8x1Nv8KtXtvPZs8d5HVKv1lUC0sKiIpJSPnv2WP66cQ//9cybnDIkl7OLBnodUq9ltFhf50pKSmxVVeBAQBFJZk2H25h3/0u839zK4188lwlD+nsdUsoxxlS7G5eGpEEIItLr5PXL4GfXzyKzTxr/+rNXebf+kNch9UpKQCLSK40uyObn15/BgSPHWFDxMrV7D0T9Hi2tbdTtPcAbu5qo3t7Am7ua2dFwiNa241G/VzJSE1wIxpi5wNwJEyYs3rx5s9fhiEiMvLW7mWsf+idtx9u5Z8EM5kwe2u1rWGvZ0XCYmncaeO2dRtbvaGTb/kPUHzwa9HxjYHR+NqcOc/qgzpswiFOG5va0KAklnCY4JaAuqA9IJPW9W3+Im35Tzes7m5k7fQS3fnQiYwZmd/qa3U2HeWnLfl6q3c9LtfvY3eTMLcrOTGfKyDyKBucwpiCH4XlZ9MtMp2+fNA4fPU5zaxu7m1rZsucA63c08Y7b/DdxaC5Xnj6ChSWjGdi/b8zLHGtKQFGgBCTSO7S2Hed//rKFihfqOHq8nbMLB3J24UDGD86hX0Y6La3H2N3Uypu7m9ng1nAACnIyObtwIGcWFlA8Jp9Th+XSJz383o0dDYf489t7eGrtLqq2N5CVkca8maMou6CI0QWdJ8FEpgQUBUpAIr3L7qbDPLLmXZ5Zv5ste07uFxrp68fkEQM4c3wB5xQN4tRhuaSlRWdC65Y9LVS+UMfvX9sFBr5w3ni+eFERuVkZUbl+PCkBRYESkEjv1dzaxu7GVg63HSc3qw+Dc/syIA7JYHfTYe5+biOPv7aTwbl9ueuTUyPqm/KSElAUKAGJiFfWvtvI7Y9v4K3dzcybOYp/nzs5LgkwGjQPSEQkic0Y7ePJm8/lSxdP4PGaHVz5k3+w6f0Wr8OKGiUgEZEEltknjW98bCK/Kz2bltZjXPXf/+DZDamxpYQSkIhIEjhjfAHPfPk8Jg7L5Yu/qeGhF+u8DqnHlIBERJLEsLwsHl58Fh+fMow7//AWy559i/b25O3HVwISEUkiWRnp/OTTxVx31lgqXqjjW0++TrIOJgtnPyAREUkg6WmG71x5Gjl9+/DA32rpk2b49hWnJd0Ge0pAIiJJyBjD0ksncry9nQdf3Eqf9DT+3+WTkioJKQGJiCQpYwzfvGwSx9otP/37VgpyMrn54glehxU2JSARkSRmjOFbl0+m/uBR7v6/jYz09eOq00d6HVZYNAghBGPMXGNMZVNTk9ehiIh0Ki3NsHzeNM4qLODWlet4acs+r0MKixJQCNbap621pXl5eV6HIiLSpb590qm4roTxg3K48dfVbN9/0OuQuqQEJCKSIvL6ZfDQZ2dhjKHsV9UcOnrM65A6pQQkIpJCxgzM5sfXnM6m91tY+tiGhJ4jpAQkIpJiLvjIYL7xsYk8vW4XP/37Vq/DCUkJSEQkBd10YREfnzKMZX98m5p3GrwOJyglIBGRFGSMoXzeNIbnZfGVh1+jubXN65BOogQkIpKiBmRl8KNrTmd3UyvffDzx+oOUgEREUljxmHy+PucjPLN+NyuqdngdzgmUgEREUtxNFxZx7oSB/MdTb1C394DX4XxACUhEJMWlpRl+sGAGfTPS+Pqj6zh2vN3rkAAlIBGRXmHogCz+68oprH23kYoXEmM3VSUgEZFeYu70EVw+bTg/XL2JN3c1ex2OEpCISG9y55VT8GVn8vVH13Lk2HFPY1ECEhHpRfJzMvnep6by9nst3Ld6s6exKAGJiPQyl0wayoKSUTzwt1qqt3u3SoISkIhIL/StT0xmeF4/vrFinWerZisBiYj0QrlZGdw9fxpb9x1k2bNvexJDr0lAxphiY0ypMWaJMWaFMabQ65hERLx0TtEgbjhvPL96ZTt/2bgn7vfvFQnIGOMDSqy1ldba5UAFsMrbqEREvPeNj01k4tBclqxcT/3Bo3G9t+cJyBgzzxgzO8RzHbWW2R1/RnibQmCp389VQKGbmEREeq2sjHTuXTiDxkNH475gqacJyE0oD4Z4rhAod2stq621lUBZJE1n1toaYI7foRKg0VrbGEHYIiIpZfKIAdzy0Yk898Z7PFazM2739SQBGWMKjTEVODWT+hCnleE0lfmrAMojuae11n/tiTJgcSTXERFJRYvPL+SMcQV8+6k32LrvYFzuabzeH8IYUwuUWWtXBzk+xz9xuE1mDdZa4/5cChR1cvlVQa5bCtRba1eGE19JSYmtqqoKqywiIslsZ+NhLv/RiwwbkMUTXzyXfpnpEV/LGFNtrS3p7Jw+EV89htxEc1LtyFrbaIzBGFNora1zm+W6c93ZQF1gUhIRERjp68cPF87gc79Ywx1PbOCeBdMxxsTsfp4PQgihAJyEE+L5bvcDGWOKcWo+q92f50UcnYhIirpo4hC+eskpPP7aTl6u3R/TeyVkDQjwRfNi7sCFPwE+v2xeBwRthnOb6UoBxowZE81QREQS3lf+5RSmjszj7KKBMb1PotaAosptrsu31hq/R8i+I3fkXYm1tmTw4MHxDFVExHNpaYZLJg2NafMbJHgC0jwdEZHUlagJqGPkW4H/Qb+ElBjb+YmISMQSMgG5gw/qOLkvqABnAmnME5AxZq4xprKpqSnWtxIR6ZUSMgG5VuOsWOCv2D0ec9bap621pXl5efG4nYhIr5MICaiA4KPelgLzA46VceKabiIikqQ8GYbt9uXcjjOfxweUG2Pm4KxcsBI+mHS61BhTDqxxzy2PR/ObiIjEnicJyO3j6bIm4y4iWhPzgEREJO4SdSKq54wxc4G5QLMxZnOElxkE7IteVElBZe4dVObU19Pyju3qBM8XI01lxpiqrhbjSzUqc++gMqe+eJQ3EQYhiIhIL6QEJCIinlACiq1ubReRIlTm3kFlTn0xL6/6gERExBOqAYmIiCc0DDsG3M3vSnDWsyskiXdhdTfuK8TZ+rwQqAjczjyc8ibre+JOmi631pYFHE+5MrvxLgT2AwNxftd1Ac+nTJndHZKL3R8HArWBuywne5nd/7+NweKJVtl6VH5rrR5RfLi/gFUBx1YAhV7HFkFZ5gHFfj/7gFqgtDvlTeb3BKgAVnT3d5xsZXZ/1xWBZU/VMuMkntlB3oOU+bcNzAYaAssZzbL1tPxqgou+MpwPLX8VQLkHsfRUoXVWowA+WMGinBPLF055k/I9cXfSLQjyVEqV2a3lPWj9annuN+fZfqelVJmBMhvwLd06Nfs5/ueQhGU2xhQaYypwkkN9iNOiVbaeld/rLJ1qD5waQmHAMZ/zVnsfXzfK4QOqAV/A8ULAdpQxnPIm63uCsy17KSfXgFKqzO6HRXmQ4/7fdFOtzNWBsbrHV/n9PenL7MYXrAYUlbL1tPyqAUWR+03ypG8d1qk5dHyjTgpuzIXuI6hwypus74nbP/BokOM+Uq/MpTgL/p7Auv0/KVrm1cAqt/8C+KDWt8L9u4/UKzMQvbJFo/wahBBdBfDhLyCIQpJoN1drbX6Qw7NxNwXs+AfWRXkJ45xEfE981lmRPfB4OL9jwjgnkcrsA+qMMaU4HyYFQL39cLBJypXZWrvU/ZJRbYxZihub/XAQQsqV2U+0yhbOOZ2WXwkounxeBxAHZcAy9+++MM4P55yEYoyZZwNG+vnxhXGJcM5JCH7fUgv9PnwxxpQbYwrcY74wLhXOOQnFWjvTGLMKpwmyBrjE72lfGJcI55xE5IvjOZ1SE5yEreMbsrV2udexxIrbrNDocRjx5Atx/BESc/BA1Lj/npfiDDwoxKkNJWSzWapSAooB90Mspbj/McustXOCPOcL4/VdnpMgFtgw5jCkUJnrAv4EPtiLy+f/gZxCZcYdJbbaWlvj/r7H47wH1QHn+cK4VpfnJKJola0n5VcCiq6O/8QnDN31+wUlYntwuMo5sYkCwitv0rwnbod0VRenpVSZ/drvG0Oc4t+OnxJl7ojJ+k2ytdY2ul+u6ty+oZQqc4Bola3H5VcfUBS5ndZ1nNysUYDbcR//qHrO/ba4NLCzMdzyJtF7UgIUGWMW+h0rBgo7toa31q5MsTKD3wz2YM+l6O+5NsRzFXw4ACWVyvyBaJatp+VXDSj6VuP8A/dX7B5POm47ebk9cUmW2X5NM+GUNyneE2ttpbV2qf8DWAXUuD93DExImTK7KvhwSRrgg9qg/4dIKpW5DmdpqWB8OAMSILXKHChaZetZ+eMxGao3PXD+AQcuTbGKBFiaI4KyzMOZI1Ls95jNiUu0dFneZH5PcJoeAyeiplSZ3Vhrg8Q6L4XLXMHJS/EU4jchNxXKjLMUz7wgx6NStp6WX9sxxIDfoo5rcP5Rd3R0Jg23HbchxNN11toiv3O7LG+yvSduDW8psADnP1klTuKtcZ9PqTL7lbcWp3awIpLyJFmZS3HKut891GiDL0aaVGV2/+/e7sYyD6fGtxonUaz0Oy8qZetJ+ZWARETEE+oDEhERTygBiYiIJ5SARETEE0pAIiLiCSUgERHxhBKQiIh4QglIREQ8oQQkMeMu2WPdR2mQ5wtDPRfjuMqNMQk/Ac4YU2GMaejqPTLGLHH3tUlYiRij++9vlfv+VngdT2+kxUglXpbirCYgYTDGrMBZ+mixe6izhR1n4SyRlMgSLkbrrHM3x/0yklDJsbdQDUjioRJnRel5XgeSRObhrEC+0n3UhDrRWjvfWnvC3uFuLS/um6uFum+wGBOBu4wMJP7ioSlJCUjiYQXON/jbvQ4kGfh9gDdG+HofsISAFa5jzav79tBs3C0nvA6kN1ICkngpB4r9vnGKJII5qPbjGSUgiQt3leFGnEQUUrABAsaYUrej2BdwXoMxptgYU+0+X92xV5Ff53JtqKY/97Wr3OvUujthBp4zL/D6gTG4f69wr+ELvEYn9+24Zqnfc+V8uFlaxzmdNl36v2du31HHKuYr3Nc3BJzf7TK57+kKv0ERq8yJ23V3et8Qv9eQ70NgPEHOjdYXmdm4/T9uOZe4j07/nUp0KAFJPC0D/Dez6ykfTvPeMqAMZyn4FX6P+e55K0K8/kGcfWGWuj8HfqjOc1+72r1WXeA57nnVOJtyVXTVlONes9q91hzgEaDCbxTWMvc4blxF/kvoh2Gx3+vLcLYbmBmFMpUFXL/ALUdY9w0Uxvvgz+fGXMGJv+ce8e//cb84LLDWLndjV39lPHi9YZIeqfvA+XZpcTf+wvkgsbgb2uF8kFig1O815c4/yxOuU+qe5ws8D79NxfyOLfE7Ns89VtzFPQr9Y3OPNeC3QZl7rLrjmN/9VoV6D4K8Jw3+9wjxPnXEctJGYiGueUJ5Ont9tMqE089zwj26uG9gjF2+DwHxzAtyzNdZjGG8b0twapu+gH+Dvp5eW4/wHqoBSdxY55t0JVAaTlNVmKr8/t7RdOXfpt8xfLmgi9g6Nu0qgQ++HfuAJebDuUyWD3eG9VdGGNymLh/ON3n/e6/GaZ6cf/KroieaZbIfjsrrdm02wvfB/3daG+T5SHTU2BZYv43orLWNVoMS4kLzgCTeynFqNKVAd5qWggrxQRHsWDjq+HB/+44P1plBrlcfEENnc3T8dVwz2Pl1RPBh3k09KpPbbLbQvU5PYu32+xCjhDAbWA7MdJscF9tOhrtL9CkBSVxZa+uMMStxhmQn2uijQj78UPzgw7EbCaYrHdcpBAI/6Ao5sTYXCxGXyV3FoBDnC0RHnNWhXxFWHF69D/79P8ustY3GmCU4/UpFnbxMokxNcOKFZThNMAvDPD9kZ3a0uE2Cs3GTovtNuJEgo/Z60HzY8cF6QvOWW7PwEYWOdVd9sIORlskdoDAbZ2JspXudYAks6H2DiNf70JnA+T91uM207mg4DUKIA9WAJO6stTXGmBqcTuBAteCsHYaTDGbjNNdFnTt0eCkffrNvxEmOHRbjDCnuGIHlw/nQrCPMfh9/7jftMpzRXgU4I7867r3S7QPpMfc+AGUdf1prO/o7ul0mt9baCJS712vkw5GD4d438Lyovg9ukizHqdGE04wWbP5PR1I9oU9IYkc1IPHKSR9grkdxmmXKcb4JDySCD/swPYIzB2QVzrf3mf59DdYZ/jwT50N6Fc6w7TpCx94l94NtDh8OJS7DqVlEewDCcpzkXY5fM1cPyrQYp4awgg9/N2HfN1AM3gcfzojHki7O61CA3yAI932pc7/4hFuTkx4y1ib8osAiIl0yxtQC8zWQIHmoBiQiKUPJJ7koAYlI0nMHUUTcNCreUBOciIh4QjUgERHxhBKQiIh4QglIREQ8oQQkIiKeUAISERFPKAGJiIgnlIBERMQT/x+z3ohO38avOQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(conv_r)\n",
    "plt.yscale(\"log\")\n",
    "plt.xlabel(\"Number of iteration, $k$\", fontsize=20)\n",
    "plt.ylabel(\"Residual norm\", fontsize=20)\n",
    "plt.yticks(fontsize=20)\n",
    "_ = plt.xticks(fontsize=20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "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"
    }
   ],
   "source": [
    "plt.scatter([x[0] for x in conv_x], [x[1] for x in conv_x])\n",
    "plt.xlabel(\"$x$\", fontsize=20)\n",
    "plt.ylabel(\"$y$\", fontsize=20)\n",
    "plt.xticks(fontsize=20)\n",
    "plt.yticks(fontsize=20)\n",
    "_ = plt.title(\"$x_0 = (0, -1)$\", fontsize=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Возможные улучшения метода простой итерации\n",
    "\n",
    "- Наискорейший спуск\n",
    "- Метод Чебышёва"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Улучшение 1: метод наискорейшего спуска\n",
    "\n",
    "- Пусть мы можем изменять значения $\\tau$ на каждой итерации, то есть \n",
    "\n",
    "$$ x_{k+1} = x_k - \\tau_k (A x_k - f). $$\n",
    "\n",
    "- Возможный выбор $\\tau_k$ такой что, $\\tau_k$ минимизирует нормы текущего вектора остатков, то есть\n",
    "\n",
    "$$\\tau_k = \\arg\\min_{\\tau > 0} \\|A(x_k - \\tau (A x_k - f)) - f \\|^2_2 $$\n",
    "\n",
    "- Используя технику, аналогичную той, что использовалась для получения нормального уравнения, получим выражение для $\\tau_k$\n",
    "\n",
    "$$\\tau_k = \\frac{r_k^{\\top}r_k}{r_k^{\\top}Ar_k}, \\quad r_k = Ax_k - f$$\n",
    "\n",
    "- Такой метод называется **методом наискорейшего спуска**.\n",
    "\n",
    "- Однако он сходится так же, как и метод Ричардсона."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Улучшение 2: метод Чебышёва\n",
    "\n",
    "- Другой метод нахождения $\\tau_k$ заключается в рассмотрении следующего выражения\n",
    "\n",
    "$$e_{k+1} = (I - \\tau_k A) e_k = (I - \\tau_k A) (I - \\tau_{k-1} A)  e_{k-1} = \\ldots = p(A) e_0, $$\n",
    "\n",
    "где $p(A)$ – **матричный полином** (простейшая матричная функция)  \n",
    "\n",
    "$$ p(A) = (I - \\tau_k A) \\ldots (I - \\tau_0 A), $$\n",
    "\n",
    "и $p(0) = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Оптимальный выбор шагов по времени\n",
    "\n",
    "- Вектор ошибки можно записать в виде\n",
    "\n",
    "$$e_{k+1} = p(A) e_0, $$\n",
    "\n",
    "тогда \n",
    "\n",
    "$$\\|e_{k+1}\\| \\leq \\|p(A)\\| \\|e_0\\|, $$\n",
    "\n",
    "где $p(0) = 1$ и $p(A)$ – **матричный полином**. \n",
    "\n",
    "- Для получения скорейшего уменьшения нормы вектора ошибки, нам нужно минимизировать\n",
    "\n",
    "$$\\Vert p(A) \\Vert$$ \n",
    "\n",
    "по всем возможным полиномам $p(x)$ степени $k+1$, таких что $p(0)=1$. \n",
    "- Мы будем использовать спектральную норму $\\|\\cdot\\|_2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##  Полиномы, наименее отклоняющиеся от 0\n",
    "\n",
    "- Важный частный случай: $A = A^* > 0$.\n",
    "\n",
    "- Тогда $A = U \\Lambda U^*$, \n",
    "\n",
    "и \n",
    "\n",
    "$$\\Vert p(A) \\Vert_2 = \\Vert U p(\\Lambda) U^* \\Vert_2 = \\Vert p(\\Lambda) \\Vert_2 = \\max_i |p(\\lambda_i)| \\overset{!}{\\leq}\n",
    "\\max_{\\lambda_\\min \\leq \\lambda {\\leq} \\lambda_\\max} |p(\\lambda)|.$$\n",
    "\n",
    "- Последнее неравенство является приближением: мы делаем <font color='red'> ключевое предположение </font> о не использовании распределения спектра между $\\lambda_\\min$ и $\\lambda_\\max$ для получения более быстрой сходимости.\n",
    "\n",
    "- Таким образом, нам нужно найти такой полином заданной степени, что $p(0) = 1$ и отклонение от 0 является минимальным на отрезке $[\\lambda_\\min, \\lambda_\\max]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "##  Масштабирование задачи с $[\\lambda_\\min, \\lambda_\\max]$ на $[-1, 1]$\n",
    "\n",
    "- Мы можем сделать аффинное преобразование отрезка $[\\lambda_\\min, \\lambda_\\max]$ в отрезок $[-1, 1]$:\n",
    "\n",
    "$$ \\xi = \\frac{{\\lambda_\\max + \\lambda_\\min - (\\lambda_\\min-\\lambda_\\max)x}}{2}, \\quad x\\in [-1, 1]. $$\n",
    "\n",
    "- Теперь задача свелась к задаче поиска полинома, наименее отклоняющегося от 0 на отрезке $[-1, 1]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Точное решение: полиномы Чебышёва\n",
    "\n",
    "- Точным решением этой задачи являются знаменитые **полиномы Чебышёва**, которые можно записать в виде\n",
    "\n",
    "$$T_n(x) =  \\cos (n \\arccos x), \\quad x \\in [-1, 1]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Что вам нужно знать о полиномах Чебышёва\n",
    "\n",
    "1. Это полиномы!\n",
    "\n",
    "2. Можно выразить $T_n$ из $T_{n-1}$ и $T_{n-2}$: \n",
    "\n",
    "$$T_n(x) = 2x T_{n-1}(x) - T_{n-2}(x), \\quad T_0(x)=1, \\quad T_1(x)=x$$\n",
    "\n",
    "3. $|T_n(x)| \\leq 1$ на $x \\in [-1, 1]$.\n",
    "\n",
    "4. У них есть $n+1$ **точка альтернанса**, где максимальное абсолютное значение достигается (это необходимое и достаточное условие для **оптимальности**: об этом говорит теорема Чебышёва об альтернансе).\n",
    "\n",
    "5. Корни записываются как \n",
    "\n",
    "$$n \\arccos x_k = \\frac{\\pi}{2} + \\pi k, \\quad \\rightarrow\\quad x_k = \\cos \\frac{\\pi(2k + 1)}{2n}, \\; k = 0, \\ldots,n-1$$\n",
    "\n",
    "Мы можем их нарисовать!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Interval $x\\\\in[-1.1, 1.1]$')"
      ]
     },
     "execution_count": 5,
     "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": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "x1 = np.linspace(-1, 1, 128)\n",
    "x2 = np.linspace(-1.1, 1.1, 128)\n",
    "p = np.polynomial.Chebyshev((0, 0, 0, 0, 0, 0, 0, 0, 0, 1), (-1, 1)) #These are Chebyshev series, a proto of \"chebfun system\" in MATLAB\n",
    "fig, (ax1, ax2) = plt.subplots(1, 2)\n",
    "ax1.plot(x1, p(x1))\n",
    "ax1.set_title('Interval $x\\in[-1, 1]$')\n",
    "ax2.plot(x2, p(x2))\n",
    "ax2.set_title('Interval $x\\in[-1.1, 1.1]$')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Сходимость метода Чебышёва\n",
    "\n",
    "- Заметим, что $p(x) = (1-\\tau_n x)\\dots (1-\\tau_0 x)$, поэтому корни $p(x)$ задаются как $1/\\tau_i$ и нам осталось дополнительно отобразить их из $[-1,1]$ в $[\\lambda_\\min, \\lambda_\\max]$.\n",
    "\n",
    "- Это даёт следующий результат\n",
    "\n",
    "$$\\tau_i = \\frac{2}{\\lambda_\\max + \\lambda_\\min - (\\lambda_\\max - \\lambda_\\min)x_i}, \\quad x_i = \\cos \\frac{\\pi(2i + 1)}{2n}\\quad i=0,\\dots,n-1$$\n",
    "\n",
    "- Сходимость теперь будет такой (без доказательства)\n",
    "\n",
    "$$ e_{k+1} \\leq C q^k e_0, \\quad q = \\frac{\\sqrt{\\mathrm{cond}(A)}-1}{\\sqrt{\\mathrm{cond}(A)}+1}, $$\n",
    "\n",
    "что лучше, чем у метода Ричардсона."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3.6819647144817442, 3.6776278649924494, 3.6689646138641425, 3.655995831627603, 3.63875276121517, 3.617276942693781, 3.591620113191315, 3.5618440822573145, 3.528020582958373, 3.4902310990668894, 3.448566668759527, 3.403127665298269, 3.3540235552224402, 3.301372634634225, 3.2453017442129966, 3.1859459636450103, 3.1234482862046065, 3.057959274270888, 2.9896366966097645, 2.9186451482951723, 2.8451556541851266, 2.769345256907857, 2.691396590350612, 2.6114974396786246, 2.529840288944203, 2.446621857375787, 2.3620426254641, 2.276306351987082, 2.1896195831371505, 2.1021911549333288, 2.0142316901169854, 1.9259530907432025, 1.8375680276901643, 1.7492894283163813, 1.6613299635000383, 1.5739015352962162, 1.487214766446285, 1.4014784929692667, 1.3168992610575794, 1.2336808294891637, 1.152023678754742, 1.0721245280827543, 0.9941758615255097, 0.9183654642482408, 0.8448759701381943, 0.7738844218236021, 0.7055618441624789, 0.6400728322287605, 0.5775751547883563, 0.5182193742203696, 0.46214848379914164, 0.4094975632109268, 0.36039345313509763, 0.3149544496738399, 0.27329001936647734, 0.23550053547499372, 0.20167703617605226, 0.17190100524205176, 0.14624417573958537, 0.12476835721819679, 0.10752528680576345, 0.09455650456922404, 0.08589325344091736, 0.08155640395162234]\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": [
    "niters = 64\n",
    "roots = [np.cos((np.pi * (2 * i + 1)) / (2 * niters)) for i in range(niters)]\n",
    "taus = [(lam_max + lam_min - (lam_min - lam_max) * r) / 2 for r in roots]\n",
    "x = np.zeros(n)\n",
    "r = A.dot(x) - rhs\n",
    "res_cheb = [np.linalg.norm(r)]\n",
    "\n",
    "print(taus)\n",
    "\n",
    "for i in range(niters):\n",
    "    x = x - 1.0 / taus[i] * r\n",
    "    r = A.dot(x) - rhs\n",
    "    res_cheb.append(np.linalg.norm(r))\n",
    "    \n",
    "plt.semilogy(res_richardson, label=\"Richardson\")\n",
    "plt.semilogy(res_cheb, label=\"Chebyshev\")\n",
    "plt.legend(fontsize=20)\n",
    "plt.xlabel(\"Number of iterations, $k$\", fontsize=20)\n",
    "plt.ylabel(\"Residual norm, $\\|Ax_k - b\\|_2$\", fontsize=20)\n",
    "plt.xticks(fontsize=20)\n",
    "_ = plt.yticks(fontsize=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Что сломалось в прекрасном методе Чебышёва?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3.6819647144817442, 3.6776278649924494, 3.6689646138641425, 3.655995831627603, 3.63875276121517, 3.617276942693781, 3.591620113191315, 3.5618440822573145, 3.528020582958373, 3.4902310990668894, 3.448566668759527, 3.403127665298269, 3.3540235552224402, 3.301372634634225, 3.2453017442129966, 3.1859459636450103, 3.1234482862046065, 3.057959274270888, 2.9896366966097645, 2.9186451482951723, 2.8451556541851266, 2.769345256907857, 2.691396590350612, 2.6114974396786246, 2.529840288944203, 2.446621857375787, 2.3620426254641, 2.276306351987082, 2.1896195831371505, 2.1021911549333288, 2.0142316901169854, 1.9259530907432025, 1.8375680276901643, 1.7492894283163813, 1.6613299635000383, 1.5739015352962162, 1.487214766446285, 1.4014784929692667, 1.3168992610575794, 1.2336808294891637, 1.152023678754742, 1.0721245280827543, 0.9941758615255097, 0.9183654642482408, 0.8448759701381943, 0.7738844218236021, 0.7055618441624789, 0.6400728322287605, 0.5775751547883563, 0.5182193742203696, 0.46214848379914164, 0.4094975632109268, 0.36039345313509763, 0.3149544496738399, 0.27329001936647734, 0.23550053547499372, 0.20167703617605226, 0.17190100524205176, 0.14624417573958537, 0.12476835721819679, 0.10752528680576345, 0.09455650456922404, 0.08589325344091736, 0.08155640395162234]\n",
      "[ 0 63 31 32 15 48 16 47  7 56 24 39  8 55 23 40  3 60 28 35 12 51 19 44\n",
      "  4 59 27 36 11 52 20 43  1 62 30 33 14 49 17 46  6 57 25 38  9 54 22 41\n",
      "  2 61 29 34 13 50 18 45  5 58 26 37 10 53 21 42] 64\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "niters = 64\n",
    "roots = [np.cos((np.pi * (2 * i + 1)) / (2 * niters)) for i in range(niters)]\n",
    "taus = [(lam_max + lam_min - (lam_min - lam_max) * r) / 2 for r in roots]\n",
    "x = np.zeros(n)\n",
    "r = A.dot(x) - rhs\n",
    "res_cheb_even = [np.linalg.norm(r)]\n",
    "print(taus)\n",
    "\n",
    "# Implementation may be non-optimal if number of iterations is not power of two\n",
    "def leb_shuffle_2n(n):\n",
    "    if n == 1:\n",
    "        return np.array([0,], dtype=int)\n",
    "    else:\n",
    "        prev = leb_shuffle_2n(n // 2)\n",
    "        ans = np.zeros(n, dtype=int)\n",
    "        ans[::2] = prev\n",
    "        ans[1::2] = n - 1 - prev\n",
    "        return ans\n",
    "\n",
    "good_perm_even = leb_shuffle_2n(niters)\n",
    "print(good_perm_even, len(good_perm_even))\n",
    "# good_perm_even = np.random.permutation([i for i in range(niters)])\n",
    "\n",
    "for i in range(niters):\n",
    "    x = x - 1.0/taus[good_perm_even[i]] * r\n",
    "    r = A.dot(x) - rhs\n",
    "    res_cheb_even.append(np.linalg.norm(r))\n",
    "\n",
    "    \n",
    "plt.semilogy(res_richardson, label=\"Richardson\")\n",
    "plt.semilogy(res_cheb_even, label=\"Chebyshev\")\n",
    "plt.legend(fontsize=20)\n",
    "plt.xlabel(\"Number of iterations, $k$\", fontsize=20)\n",
    "plt.ylabel(\"Residual norm, $\\|Ax_k - b\\|_2$\", fontsize=20)\n",
    "plt.xticks(fontsize=20)\n",
    "_ = plt.yticks(fontsize=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- Перестановки корней полинома Чебышёва очень сильно влияют на сходимость метода (почему?)\n",
    "- Об оптимальной перестановке можно прочитать в [этой статье](http://www.mathnet.ru/links/42ab1a270ad179d43359ef8fab53422c/zvmmf6870.pdf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Проект [Chebfun](http://www.chebfun.org/) \n",
    "\n",
    "- Проект с открытым исходным кодом для численных расчётов (с интерфейсом для [Python](https://github.com/chebpy/chebpy) и [Matlab](https://github.com/chebfun/chebfun))\n",
    "- Основан на алгоритмах, работающих с кусочно-полиномиальными интерполяциями и Чебышёвскими полиномами\n",
    "- Авторами проекта являются Nick Trefethen и его студент Zachary Battles, подробности см. в [этой статье](http://www.chebfun.org/publications/chebfun_paper.pdf)\n",
    "- Chebfun нацелен на решение следующих задач\n",
    "    - Аппроксимация\n",
    "    - Квадратуры\n",
    "    - ОДУ\n",
    "    - Уравнения с частными производными\n",
    "    - Поиск корней\n",
    "    - Одномерная глобальная оптимизация"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Улучшения метода Чебышёва\n",
    "\n",
    "- Мы сделали важное предположение о спектре: он содержится на отрезке (и нам нужны границы)\n",
    "\n",
    "- Если спектр содержится в **двух отрезках** и мы знаем границы, также можно поставить задачу оптимизации для поиска оптимального полинома"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "## Если спектр матрицы лежит в нескольких отрезках\n",
    "\n",
    "- Для случая двух отрезков оптимальными полиномами являются **[полиномы Золотарёва](https://www.encyclopediaofmath.org/index.php/Zolotarev_polynomials)** (выражаются через эллиптические функции). \n",
    "- Оригинальная работа опубликована в 1877, подробности см. [тут](http://gpntb.dlibrary.org/ru/nodes/4245-zolotarev-e-prilozhenie-ellipticheskih-funktsiy-k-voprosam-o-funktsiyah-naimenee-i-naibolee-otklonyayuschihsya-ot-nulya-spb-1877) \n",
    "- Для случая, когда интервалов больше двух, оптимальный полином может быть записан с помощью гиперэллиптических функций"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    }
   },
   "source": [
    "## Как ещё можно ускорить метод Чебышёва?\n",
    "\n",
    "- Реализация метода Чебышёва требует знания границ спектра.\n",
    "\n",
    "- Метод хранит вектор $x_k$ и вычисляет новую поправку\n",
    "\n",
    "$$r_k = A x_k - f.$$\n",
    "\n",
    "- Оказывается, что если мы будем использовать больше векторов (не только $x_k$ и $r_k$), то мы можем избавиться от необходимости знать спектр и получим более быструю сходимость!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "## Главная идея: Крыловские пространства\n",
    "\n",
    "- Метод Чебышёва даёт аппроксимацию вида\n",
    "\n",
    "$$x_{k+1} = x_0 + p(A) r_0,$$\n",
    "\n",
    "то есть $x_{k+1}$ лежит в Крыловском пространстве.\n",
    "- Крыловское пространство определяется для матрицы $A$ следующим образом\n",
    "\n",
    "$$ \\mathcal{K}_k(A, r_0) = \\mathrm{Span}(r_0, Ar_0, A^2 r_0, \\ldots, A^{k-1}r_0 ) $$\n",
    "\n",
    "- Наиболее естественный подход теперь – это искать вектор в этом пространстве, который минимизирует норму некоторого вектора"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "## Идея Крыловских методов\n",
    "\n",
    "Идея заключается в минимизации заданного функционала:\n",
    "- Энергетическая норма вектора ошибки для систем с эрмитовой положительно определённой матрицей (метод сопряжённых градиентов).\n",
    "- Норма вектора остатков для систем с матрицами общего вида (методы minres и GMRES).\n",
    "- Отношение Релея для частичных задач на собстенные значения (метод Ланцоша).\n",
    "\n",
    "Для того чтобы сделать метод практичным нужно следующее\n",
    "1. Ортогонализовать векторы $A^i r_0$ Крыловского подпространства для устойчивости (процесс Ланцоша).\n",
    "2. Получить рекуррентные формулы для уменьшения сложности.\n",
    "\n",
    "Эти методы рассмотрим в следующей лекции."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "## Выводы по лекции\n",
    "\n",
    "- Основная идея итерационных методов\n",
    "- Метод Ричардсона: эрмитов и неэрмитов случай\n",
    "- Метод Чебышёва\n",
    "- Определение Крыловских подпространств"
   ]
  }
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