CM - Lab #5.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Первій Михайло КА-96 20 Вар\n",
    "# Реалізувати метод Якобі\n",
    "\n",
    "import numpy as np\n",
    "import math\n",
    "EPS = 10**-5\n",
    "\n",
    "\n",
    "A = [[6.26, 0.94, 1.13, 1.24],\n",
    "     [0.94, 4.16, 1.3,  0.16],\n",
    "     [1.13,  1.3,  5.44,  2.1],\n",
    "     [1.24, 0.16, 2.1,   6.1]]\n",
    "\n",
    "A = np.array(A)\n",
    "\n",
    "y_0 = [1,1,1,1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Starting partial method on matrix A:\n",
      "|    6.2600000    0.9400000    1.1300000    1.2400000  |\n",
      "|    0.9400000    4.1600000    1.3000000    0.1600000  |\n",
      "|    1.1300000    1.3000000    5.4400000    2.1000000  |\n",
      "|    1.2400000    0.1600000    2.1000000    6.1000000  |\n",
      "\n",
      "                          eigenvalue radius\n",
      "Lambda 0 [9.57 6.56 9.97 9.6 ]\n",
      "lambda 1 [9.32546499 7.74121951 9.40209629 9.62639583]\n",
      "Lambda 2 [9.26581468 8.50277884 9.29039478 9.51552526]\n",
      "Lambda 3 [9.25951999 8.90798604 9.27801143 9.42434564]\n",
      "Lambda 4 [9.2667347  9.10682669 9.28361106 9.36735511]\n",
      "Lambda 5 [9.27533084 9.20255833 9.28930953 9.33467949]\n",
      "Lambda 6 [9.28217109 9.2489497  9.29264659 9.31658962]\n",
      "Lambda 7 [9.28693625 9.27175005 9.29424259 9.30672314]\n",
      "Lambda 8 [9.29004822 9.2831207  9.29489619 9.30137125]\n",
      "Lambda 9 [9.29200609 9.28886548 9.29511148 9.29846998]\n",
      "Lambda 10 [9.29320873 9.29180054 9.2951478  9.29689409]\n",
      "Lambda 11 [9.29393539 9.29331473 9.2951235  9.29603526]\n",
      "Lambda 12 [9.29436925 9.2941027  9.29508689 9.29556538]\n",
      "Lambda 13 [9.29462599 9.29451601 9.29505469 9.29530724]\n",
      "Lambda 14 [9.29477686 9.29473441 9.29503076 9.29516487]\n",
      "Lambda 15 [9.29486505 9.29485062 9.29501439 9.29508603]\n",
      "Lambda 16 [9.29491637 9.29491287 9.29500373 9.29504223]\n",
      "Lambda 17 [9.29494613 9.29494642 9.29499701 9.29501782]\n",
      "Lambda 18 [9.29496333 9.29496461 9.29499286 9.29500417]\n",
      "Lambda 19 [9.29497326 9.29497452 9.29499034 9.29499652]\n",
      "Lambda 20 [9.29497898 9.29497996 9.29498883 9.29499223]\n",
      "Calculated spectral radius (max eigenvalue):  9.294984998975313\n",
      "Vector assosiated:  [0.90035395 0.43810495 0.95640673 1.        ]\n",
      "------------------------------\n",
      "Starting partial method on matrix A:\n",
      "|    0.1734016   -0.0327077   -0.0172147   -0.0284646  |\n",
      "|   -0.0327077    0.2674539   -0.0657108    0.0222553  |\n",
      "|   -0.0172147   -0.0657108    0.2319053   -0.0746133  |\n",
      "|   -0.0284646    0.0222553   -0.0746133    0.1948235  |\n",
      "\n",
      "                          eigenvalue radius\n",
      "Lambda 0 [0.0950146  0.19129081 0.07436644 0.11400096]\n",
      "lambda 1 [ 0.0599257   0.23892536 -0.07349467  0.15977084]\n",
      "Lambda 2 [-0.16367393  0.28010643  1.04797989  0.26415971]\n",
      "Lambda 3 [0.66386579 0.30759887 0.43864963 0.34837685]\n",
      "Lambda 4 [0.38879626 0.32399097 0.38043445 0.36945474]\n",
      "Lambda 5 [0.35176257 0.33365205 0.36362437 0.3668945 ]\n",
      "Lambda 6 [0.34367638 0.33948534 0.35697077 0.36125773]\n",
      "Lambda 7 [0.34284448 0.34309078 0.35377044 0.35687442]\n",
      "Lambda 8 [0.34390845 0.34534996 0.3520176  0.35398363]\n",
      "Lambda 9 [0.34525332 0.34677431 0.35097981 0.35216816]\n",
      "Lambda 10 [0.34641032 0.3476743  0.35033911 0.3510453 ]\n",
      "Lambda 11 [0.3472894  0.34824328 0.34993525 0.35035305]\n",
      "Lambda 12 [0.347919   0.34860302 0.34967822 0.34992573]\n",
      "Lambda 13 [0.34835502 0.34883049 0.34951396 0.34966118]\n",
      "Lambda 14 [0.34865064 0.34897434 0.34940885 0.34949687]\n",
      "Lambda 15 [0.34884817 0.34906534 0.34934158 0.3493945 ]\n",
      "Lambda 16 [0.34897879 0.34912293 0.34929854 0.34933054]\n",
      "Lambda 17 [0.34906451 0.34915939 0.34927103 0.34929048]\n",
      "Lambda 18 [0.34912042 0.34918248 0.34925345 0.34926534]\n",
      "Lambda 19 [0.34915673 0.34919711 0.34924223 0.34924952]\n",
      "Lambda 20 [0.34918022 0.34920639 0.34923507 0.34923957]\n",
      "Lambda 21 [0.34919536 0.34921227 0.3492305  0.34923329]\n",
      "Lambda 22 [0.34920511 0.34921601 0.34922759 0.34922932]\n",
      "Lambda 23 [0.34921137 0.34921837 0.34922574 0.34922681]\n",
      "Calculated inverse spectral radius (min eigenvalue):  2.863519720664468\n",
      "Vector assosiated:  [-0.19689873  1.         -0.93257941  0.63109883]\n"
     ]
    }
   ],
   "source": [
    "# Часткова задача (Степеневим методом)\n",
    "def vnorm(vec: list):\n",
    "    # vector norm\n",
    "    sum = 0\n",
    "    for i in range(len(vec)):\n",
    "        sum += vec[i]**2\n",
    "    return sum ** 0.5\n",
    "\n",
    "def matrix_pow(M, n):\n",
    "    res = np.identity(len(M))\n",
    "    for i in range(n):\n",
    "        res = np.matmul(M, res)\n",
    "    return res\n",
    "\n",
    "def matrix_print(M):\n",
    "    for i in range(len(M)):\n",
    "        print(\"|\", end=\"  \")\n",
    "        for j in range(len(M[i])):\n",
    "            print(\"%11.7f\" % (M[i][j]) , end=\"  \")\n",
    "        print(\"|\")\n",
    "\n",
    "\n",
    "def solvePart(A, max=True):\n",
    "    if not max:\n",
    "        A = np.linalg.inv(A)\n",
    "        \n",
    "    print(\"Starting partial method on matrix A:\")\n",
    "    matrix_print(A)\n",
    "    print()\n",
    "    \n",
    "    print(\" \"*25, \"eigenvalue radius\")\n",
    "    y_0 = [1]*len(A)\n",
    "    \n",
    "    y_cur = np.matmul(A, y_0)\n",
    "    y_next = np.matmul(A, y_cur)\n",
    "    \n",
    "    \n",
    "    lam = np.divide(y_cur,y_0)\n",
    "    print(\"Lambda 0\", lam)\n",
    "    \n",
    "    lam_next = np.divide(y_next,y_cur)\n",
    "    print(\"lambda 1\", lam_next)\n",
    "    \n",
    "    i = 2\n",
    "    while vnorm(lam_next - lam) >= EPS:\n",
    "        y_cur = y_next\n",
    "        y_next = np.matmul(A, y_cur)\n",
    "        \n",
    "        lam = lam_next\n",
    "        lam_next = np.divide(y_next, y_cur)\n",
    "        print(\"Lambda\", i, lam_next)\n",
    "        i += 1\n",
    "    \n",
    "    result = 0\n",
    "    \n",
    "    for j in lam_next:\n",
    "        result += j\n",
    "    result /= len(lam_next)\n",
    "    if not max:\n",
    "        result = 1/result\n",
    "        eigenvector = y_next * (result ** i)\n",
    "    else:\n",
    "        eigenvector = y_next/(result ** i)\n",
    "    \n",
    "    return result, eigenvector\n",
    "\n",
    "res_max, eigenMax = solvePart(A)\n",
    "print(\"Calculated spectral radius (max eigenvalue): \", res_max)\n",
    "print(\"Vector assosiated: \", eigenMax/max(eigenMax))\n",
    "print(\"------------------------------\")\n",
    "res_min, eigenMin = solvePart(A, False)\n",
    "print(\"Calculated inverse spectral radius (min eigenvalue): \", res_min)\n",
    "print(\"Vector assosiated: \", eigenMin/max(eigenMin))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Jacobi stated with matrix A\n",
      "|    6.2600000    0.9400000    1.1300000    1.2400000  |\n",
      "|    0.9400000    4.1600000    1.3000000    0.1600000  |\n",
      "|    1.1300000    1.3000000    5.4400000    2.1000000  |\n",
      "|    1.2400000    0.1600000    2.1000000    6.1000000  |\n",
      "\n",
      "Max element absolute = 2.1  on position  (3, 4)\n",
      "\n",
      "Iteration # 1\n",
      "|    6.2600000    0.9400000    0.0529275    1.6768121  |\n",
      "|    0.9400000    4.1600000    0.8840308    0.9664831  |\n",
      "|    0.0529275    0.8840308    3.6442296    0.0000000  |\n",
      "|    1.6768121    0.9664831    0.0000000    7.8957704  |\n",
      "Max element absolute = 1.676812058573542 on position (1, 4)\n",
      "a_ij= 1.676812058573542  a_ii= 6.26  a_jj= 7.895770448566825\n",
      "\n",
      "Iteration # 2\n",
      "|    5.2122392    0.2850229    0.0448854    0.0000000  |\n",
      "|    0.2850229    4.1600000    0.8840308    1.3177448  |\n",
      "|    0.0448854    0.8840308    3.6442296    0.0280468  |\n",
      "|    0.0000000    1.3177448    0.0280468    8.9435312  |\n",
      "Max element absolute = 1.3177448450652627 on position (2, 4)\n",
      "a_ij= 1.3177448450652627  a_ii= 4.16  a_jj= 8.943531203529949\n",
      "\n",
      "Iteration # 3\n",
      "|    5.2122392    0.2760358    0.0448854    0.0710090  |\n",
      "|    0.2760358    3.8210159    0.8491689   -0.0000000  |\n",
      "|    0.0448854    0.8491689    3.6442296    0.2474049  |\n",
      "|    0.0710090    0.0000000    0.2474049    9.2825153  |\n",
      "Max element absolute = 0.849168948900071 on position (2, 3)\n",
      "a_ij= 0.849168948900071  a_ii= 3.821015856643272  a_jj= 3.6442295514331757\n",
      "\n",
      "Iteration # 4\n",
      "|    5.2122392    0.2350931   -0.1514652    0.0710090  |\n",
      "|    0.2350931    4.5863798    0.0000000    0.1656381  |\n",
      "|   -0.1514652   -0.0000000    2.8788656    0.1837749  |\n",
      "|    0.0710090    0.1656381    0.1837749    9.2825153  |\n",
      "Max element absolute = 0.2350930955280726 on position (1, 2)\n",
      "a_ij= 0.2350930955280726  a_ii= 5.212239245036876  a_jj= 4.586379843512624\n",
      "\n",
      "Iteration # 5\n",
      "|    5.2907093    0.0000000   -0.1436731    0.1197989  |\n",
      "|    0.0000000    4.5079098    0.0479556    0.1346346  |\n",
      "|   -0.1436731    0.0479556    2.8788656    0.1837749  |\n",
      "|    0.1197989    0.1346346    0.1837749    9.2825153  |\n",
      "Max element absolute = 0.1837749342254263 on position (3, 4)\n",
      "a_ij= 0.1837749342254263  a_ii= 2.878865564563823  a_jj= 9.282515346886678\n",
      "\n",
      "Iteration # 6\n",
      "|    5.2907093    0.0000000   -0.1470479    0.1156316  |\n",
      "|    0.0000000    4.5079098    0.0440769    0.1359539  |\n",
      "|   -0.1470479    0.0440769    2.8735958   -0.0000000  |\n",
      "|    0.1156316    0.1359539    0.0000000    9.2877851  |\n",
      "Max element absolute = 0.14704785913877125 on position (1, 3)\n",
      "a_ij= -0.14704785913877125  a_ii= 5.290709290294143  a_jj= 2.8735958421987884\n",
      "\n",
      "Iteration # 7\n",
      "|    5.2996222   -0.0026667    0.0000000    0.1154197  |\n",
      "|   -0.0026667    4.5079098    0.0439961    0.1359539  |\n",
      "|   -0.0000000    0.0439961    2.8646829    0.0069959  |\n",
      "|    0.1154197    0.1359539    0.0069959    9.2877851  |\n",
      "Max element absolute = 0.13595385035136223 on position (2, 4)\n",
      "a_ij= 0.13595385035136223  a_ii= 4.5079097982553575  a_jj= 9.287785069251711\n",
      "\n",
      "Iteration # 8\n",
      "|    5.2996222   -0.0059445    0.0000000    0.1152974  |\n",
      "|   -0.0059445    4.5040460    0.0437796   -0.0000000  |\n",
      "|   -0.0000000    0.0437796    2.8646829    0.0082429  |\n",
      "|    0.1152974    0.0000000    0.0082429    9.2916489  |\n",
      "Max element absolute = 0.11529739192824788 on position (1, 4)\n",
      "a_ij= 0.11529739192824788  a_ii= 5.299622248874395  a_jj= 9.29164887727302\n",
      "\n",
      "Iteration # 9\n",
      "|    5.2962950   -0.0059421   -0.0002378   -0.0000000  |\n",
      "|   -0.0059421    4.5040460    0.0437796   -0.0001715  |\n",
      "|   -0.0002378    0.0437796    2.8646829    0.0082395  |\n",
      "|   -0.0000000   -0.0001715    0.0082395    9.2949761  |\n",
      "Max element absolute = 0.04377962072710896 on position (2, 3)\n",
      "a_ij= 0.04377962072710896  a_ii= 4.504045990234048  a_jj= 2.864682883618537\n",
      "\n",
      "Iteration # 10\n",
      "|    5.2962950   -0.0059463   -0.0000792   -0.0000000  |\n",
      "|   -0.0059463    4.5052143   -0.0000000    0.0000484  |\n",
      "|   -0.0000792   -0.0000000    2.8635146    0.0082411  |\n",
      "|   -0.0000000    0.0000484    0.0082411    9.2949761  |\n",
      "Max element absolute = 0.008241147810480163 on position (3, 4)\n",
      "a_ij= 0.008241147810480163  a_ii= 2.8635145700054143  a_jj= 9.294976114117071\n",
      "\n",
      "Iteration # 11\n",
      "|    5.2962950   -0.0059463   -0.0000792   -0.0000001  |\n",
      "|   -0.0059463    4.5052143   -0.0000001    0.0000484  |\n",
      "|   -0.0000792   -0.0000001    2.8635040    0.0000000  |\n",
      "|   -0.0000001    0.0000484    0.0000000    9.2949867  |\n",
      "Max element absolute = 0.005946303831536169 on position (1, 2)\n",
      "a_ij= -0.005946303831536169  a_ii= 5.296295012030344  a_jj= 4.505214303847172\n",
      "\n",
      "Iteration # 12\n",
      "|    5.2963397    0.0000000   -0.0000792   -0.0000005  |\n",
      "|   -0.0000000    4.5051696   -0.0000007    0.0000484  |\n",
      "|   -0.0000792   -0.0000007    2.8635040    0.0000000  |\n",
      "|   -0.0000005    0.0000484    0.0000000    9.2949867  |\n",
      "Max element absolute = 7.917193637203397e-05 on position (1, 3)\n",
      "a_ij= -7.917193637203397e-05  a_ii= 5.2963397059930974  a_jj= 2.8635040099786995\n",
      "\n",
      "Iteration # 13\n",
      "|    5.2963397    0.0000000    0.0000000   -0.0000005  |\n",
      "|    0.0000000    4.5051696   -0.0000007    0.0000484  |\n",
      "|   -0.0000000   -0.0000007    2.8635040   -0.0000000  |\n",
      "|   -0.0000005    0.0000484   -0.0000000    9.2949867  |\n",
      "Max element absolute = 4.8386432829155296e-05 on position (2, 4)\n",
      "a_ij= 4.8386432829155296e-05  a_ii= 4.505169609884417  a_jj= 9.294986674143786\n",
      "\n",
      "Iteration # 14\n",
      "|    5.2963397    0.0000000    0.0000000   -0.0000005  |\n",
      "|    0.0000000    4.5051696   -0.0000007    0.0000000  |\n",
      "|   -0.0000000   -0.0000007    2.8635040   -0.0000000  |\n",
      "|   -0.0000005    0.0000000   -0.0000000    9.2949867  |\n",
      "All eigenvalues of A = [5.296339708569595, 4.5051696093956215, 2.863504007402202, 9.294986674632584]\n",
      "Matrix of eigenvectors as colums\n",
      "|    0.7553937   -0.3670181    0.1296337    0.5271367  |\n",
      "|    0.2547458    0.6601673   -0.6584005    0.2565005  |\n",
      "|   -0.2562534    0.4937086    0.6140329    0.5599550  |\n",
      "|   -0.5466435   -0.4309625   -0.4155329    0.5854782  |\n",
      "Checking...\n",
      "A*v1 - lambda*v1 =  [-2.45202219e-07 -1.19291559e-07 -2.60444893e-07 -2.72340853e-07]\n",
      "A*v2 - lambda*v2 =  [-8.51603374e-08  4.32630149e-07 -4.03477012e-07  2.73025163e-07]\n",
      "A*v3 - lambda*v3 =  [ 2.41149764e-07 -4.33790040e-07 -3.24419528e-07  2.83165197e-07]\n",
      "A*v4 - lambda*v4 =  [-3.51367031e-07 -1.18478276e-07  1.19180465e-07  2.54275073e-07]\n"
     ]
    }
   ],
   "source": [
    "# Метод поворотів Якобі (20 вар)\n",
    "\n",
    "def final_condition(M):\n",
    "    \"\"\"\n",
    "    Returns true if sqrt of sum of squared is less then goal estimation EPS\n",
    "    \"\"\"\n",
    "    s = 0 # sum\n",
    "    for i in range(len(M)):\n",
    "        for j in range(len(M[i])):\n",
    "            if i < j:\n",
    "                s += M[i][j] ** 2\n",
    "    return math.sqrt(s) < EPS\n",
    "    \n",
    "    \n",
    "\n",
    "def Jacobi(M):\n",
    "    \"\"\"\n",
    "    Uses Jacobi rotation method to calculate eigenvalues & eigenvectors\n",
    "    :return: list of eigenvalues\n",
    "     \"\"\"\n",
    "    \n",
    "    EVec = np.identity(len(M))\n",
    "    \n",
    "    print(\"Jacobi stated with matrix A\")\n",
    "    matrix_print(M)\n",
    "    print()\n",
    "    \n",
    "    A_cur = M\n",
    "    i_max = j_max = 0\n",
    "    el_max = np.NINF\n",
    "    \n",
    "    for i in range(len(A_cur)):\n",
    "        for j in range(len(A_cur[i])):\n",
    "            if i < j and abs(A_cur[i][j]) > el_max:\n",
    "                el_max = abs(A_cur[i][j])\n",
    "                i_max = i\n",
    "                j_max = j\n",
    "    print(\"Max element absolute =\", el_max, \" on position \", (i_max+1, j_max+1))\n",
    "    \n",
    "    U = np.identity(len(M))\n",
    "    U_T = np.identity(len(M))\n",
    "    if A[i_max][i_max] != A[j_max][j_max]:\n",
    "        phi = 0.5 * math.atan(2 * A_cur[i_max][j_max] / (A_cur[i_max][i_max] - A_cur[j_max][j_max]))\n",
    "    else:\n",
    "        phi = math.pi/4\n",
    "\n",
    "    U[i_max][i_max] = U[j_max][j_max] = U_T[i_max][i_max] = U_T[j_max][j_max] = math.cos(phi)\n",
    "    U[i_max][j_max] = U_T[j_max][i_max] = -math.sin(phi)\n",
    "    U[j_max][i_max] = U_T[i_max][j_max] = math.sin(phi)\n",
    "    EVec = np.matmul(EVec, U)\n",
    "    \n",
    "    A_next = np.matmul(np.matmul(U_T, A_cur), U)\n",
    "    print(\"\\nIteration # 1\")\n",
    "    matrix_print(A_next)\n",
    "    \n",
    "    iteration = 2\n",
    "    while not final_condition(A_next):\n",
    "        A_cur = A_next\n",
    "        i_max = j_max = 0\n",
    "        el_max = np.NINF\n",
    "\n",
    "        for i in range(len(A_cur)):\n",
    "            for j in range(len(A_cur[i])):\n",
    "                if i < j and abs(A_cur[i][j]) > el_max:\n",
    "                    el_max = abs(A_cur[i][j])\n",
    "                    i_max = i\n",
    "                    j_max = j\n",
    "        print(\"Max element absolute =\", el_max, \"on position\", (i_max+1, j_max+1))\n",
    "        print(\"a_ij=\", A_cur[i_max][j_max], \" a_ii=\", A_cur[i_max][i_max], \" a_jj=\", A_cur[j_max][j_max])\n",
    "\n",
    "        U = np.identity(len(M))\n",
    "        U_T = np.identity(len(M))\n",
    "\n",
    "        phi = 0.5 * math.atan(2 * A_cur[i_max][j_max] / (A_cur[i_max][i_max] - A_cur[j_max][j_max]))\n",
    "\n",
    "        U[i_max][i_max] = U[j_max][j_max] = U_T[i_max][i_max] = U_T[j_max][j_max] = math.cos(phi)\n",
    "        U[i_max][j_max] = U_T[j_max][i_max] = -math.sin(phi)\n",
    "        U[j_max][i_max] = U_T[i_max][j_max] = math.sin(phi)\n",
    "        EVec = np.matmul(EVec, U)\n",
    "\n",
    "        A_next = np.matmul(np.matmul(U_T, A_cur), U)\n",
    "        print(\"\\nIteration #\", iteration)\n",
    "        matrix_print(A_next)\n",
    "        iteration += 1\n",
    "    res = []\n",
    "    for i in range(len(A_next)):\n",
    "        res.append(A_next[i][i])\n",
    "    return res, EVec\n",
    "\n",
    "eigenvalues, eigenvectors = Jacobi(A)\n",
    "print(\"All eigenvalues of A =\", eigenvalues)\n",
    "print(\"Matrix of eigenvectors as colums\")\n",
    "matrix_print(eigenvectors)\n",
    "print(\"Checking...\")\n",
    "v1, v2, v3, v4 = np.transpose(eigenvectors)\n",
    "print(\"A*v1 - lambda*v1 = \", (np.matmul(A, v1) - eigenvalues[0]*v1) )\n",
    "print(\"A*v2 - lambda*v2 = \", (np.matmul(A, v2) - eigenvalues[1]*v2) )\n",
    "print(\"A*v3 - lambda*v3 = \", (np.matmul(A, v3) - eigenvalues[2]*v3) )\n",
    "print(\"A*v4 - lambda*v4 = \", (np.matmul(A, v4) - eigenvalues[3]*v4) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}