Orthonormalization Algorithm.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 610,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import sys\n",
    "SIZE = 183"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reading fs_183_6 Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 611,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([['1 1  1.8470335834570e-01'],\n",
       "       ['2 1 -3.7192762029580e-07'],\n",
       "       ['20 1 -4.4616731475320e-09'],\n",
       "       ...,\n",
       "       ['182 182  2.2361846869070e+03'],\n",
       "       ['118 183 -2.2359999999970e+03'],\n",
       "       ['183 183  2.2361846869070e+03']], dtype=object)"
      ]
     },
     "execution_count": 611,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = pd.read_table('fs_183_6.mtx')\n",
    "df.values[1:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 612,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0., 0., 0., ..., 0., 0., 0.],\n",
       "       [0., 0., 0., ..., 0., 0., 0.],\n",
       "       [0., 0., 0., ..., 0., 0., 0.],\n",
       "       ...,\n",
       "       [0., 0., 0., ..., 0., 0., 0.],\n",
       "       [0., 0., 0., ..., 0., 0., 0.],\n",
       "       [0., 0., 0., ..., 0., 0., 0.]])"
      ]
     },
     "execution_count": 612,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.zeros((SIZE,SIZE))\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 613,
   "metadata": {},
   "outputs": [],
   "source": [
    "c = np.array(df.values[1:])\n",
    "for j in range(c.shape[0]):\n",
    "    for i in c[j]:\n",
    "        k = i.strip().split()\n",
    "        a[int(k[0])-1][int(k[1])-1] = float(k[2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 614,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = a.transpose()\n",
    "n = a.shape[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrix Conditions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 615,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 615,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tr = [1,2,3]\n",
    "tr[0:0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 616,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "norm(A):  1180838892.1872454\n",
      "condition_No(A):  173678181547.5921\n",
      "float_pt_precision:  2.220446049250313e-16\n"
     ]
    }
   ],
   "source": [
    "a_norm = np.linalg.norm(a,ord=2)\n",
    "a_cond = a_norm/np.linalg.norm(a,ord=-2)\n",
    "print(\"norm(A): \",a_norm)\n",
    "print(\"condition_No(A): \",a_cond)\n",
    "print(\"float_pt_precision: \",np.finfo(float).eps)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Initalising Values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 617,
   "metadata": {},
   "outputs": [],
   "source": [
    "qt = np.zeros((n,n)) ## Q - orthogonal matrix is initialised as zero matrix\n",
    "y =[] ## list of loss of orthogonality in CGS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Classical Graham Schmidt Algorithm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 618,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(0,n):\n",
    "    w = a[i]\n",
    "    for j in range(0,i):\n",
    "        w = w - np.dot(np.dot(a[i],qt[j]),qt[j])\n",
    "    qt[i] = w/(np.linalg.norm(w,ord=2))\n",
    "    I = np.identity(i+1)\n",
    "    Qj = qt[0:i+1]\n",
    "    r = np.dot(Qj,np.transpose(Qj)) ## calculating transpose(Q).Q at each iteration i\n",
    "    y.append(np.linalg.norm(I-r)) ## calculating loss of orthogonality at each iteration i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 619,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 619,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y ## printing value of L.O for CGS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 620,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00000000e+00,  2.01364840e-06,  1.14050301e-14, ...,\n",
       "        -2.06792834e-11,  1.08510644e-09,  2.41916394e-11],\n",
       "       [-2.01364839e-06,  1.00000000e+00, -4.02616587e-15, ...,\n",
       "        -2.07791597e-09, -1.19470304e-09, -1.09481576e-10],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  9.99999994e-01, ...,\n",
       "        -1.08644872e-10, -6.24657158e-11, -5.72429767e-12],\n",
       "       ...,\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00, ...,\n",
       "         8.66213925e-01,  4.98049493e-01,  1.84420548e-04],\n",
       "       [-3.89899520e-09, -7.85120543e-15, -4.44681577e-23, ...,\n",
       "        -4.07967409e-01,  7.09541941e-01,  2.62733286e-04],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00, ...,\n",
       "        -6.38927168e-05, -4.07021603e-04,  9.99998241e-01]])"
      ]
     },
     "execution_count": 620,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Printing Orthogonalized matrix Q\n",
    "q = np.transpose(qt)\n",
    "q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Modified Graham Schmidt Algorithm "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 621,
   "metadata": {},
   "outputs": [],
   "source": [
    "qt = np.zeros((n,n))\n",
    "z = [] ## list of loss of orthogonality in MCGS/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 622,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(0,n):\n",
    "    w = a[i]\n",
    "    for j in range(0,i):\n",
    "        proj = np.dot(w,qt[j]) ## In MCGS a[i] is replaced with w\n",
    "        w = w - proj*(qt[j])\n",
    "    qt[i] = w/(np.linalg.norm(w))\n",
    "    I = np.identity(i+1)\n",
    "    r = np.dot(qt[0:i+1],np.transpose(qt[0:i+1]))\n",
    "    z.append(np.linalg.norm(I-r))  ## Loss of orthogonality at each iteration i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 623,
   "metadata": {},
   "outputs": [
    {
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       " 3.1465071940697196e-15,\n",
       " 3.162979298180242e-15,\n",
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       " 3.281122651924597e-15,\n",
       " 3.288706974759718e-15,\n",
       " 3.3398102100661394e-15,\n",
       " 3.359191053715068e-15,\n",
       " 3.388540566175177e-15,\n",
       " 3.3886322058619236e-15,\n",
       " 3.4164226048461744e-15,\n",
       " 3.4425626836590903e-15,\n",
       " 3.4501026451187477e-15,\n",
       " 3.540659988459151e-15,\n",
       " 3.559469281369857e-15,\n",
       " 3.5750183456040645e-15,\n",
       " 3.60659323632234e-15,\n",
       " 3.6090718736375155e-15,\n",
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       " 3.653883101646246e-15,\n",
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       " 3.7193979459307786e-15,\n",
       " 3.735157481345626e-15,\n",
       " 3.735157481345632e-15,\n",
       " 3.766455844879898e-15,\n",
       " 3.770942963415276e-15,\n",
       " 3.7953790093042286e-15,\n",
       " 3.8093153342842675e-15,\n",
       " 3.851682884859293e-15,\n",
       " 3.87174589384172e-15,\n",
       " 3.888937125056805e-15,\n",
       " 3.991944215146885e-15,\n",
       " 4.007723800776826e-15,\n",
       " 4.019776554361341e-15,\n",
       " 4.036492747584096e-15,\n",
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       " 4.137439314675158e-15,\n",
       " 4.1528579271291756e-15,\n",
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       " 4.181125339708382e-15,\n",
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       " 4.203142769247412e-15,\n",
       " 5.695871810907665e-15,\n",
       " 6.022146122913142e-15,\n",
       " 6.035296077793629e-15,\n",
       " 6.109005876174032e-15,\n",
       " 6.1406524549321805e-15,\n",
       " 6.228507549939554e-15,\n",
       " 6.234777536474091e-15,\n",
       " 6.377727036665696e-15,\n",
       " 6.381591205311079e-15,\n",
       " 6.3854530082408046e-15,\n",
       " 6.38545303046e-15,\n",
       " 6.386418117285865e-15,\n",
       " 6.3950973358206644e-15,\n",
       " 6.398950987562972e-15,\n",
       " 6.398950987571127e-15,\n",
       " 6.4028023199171785e-15,\n",
       " 6.403764791095126e-15,\n",
       " 6.412420530855297e-15,\n",
       " 6.412420530889651e-15,\n",
       " 6.412420530898172e-15,\n",
       " 6.413381558697355e-15,\n",
       " 6.677488278781759e-15,\n",
       " 6.901385342150923e-15,\n",
       " 7.131614522751275e-15,\n",
       " 7.2296124181096e-15,\n",
       " 7.638978837627034e-15,\n",
       " 8.044113785331637e-15,\n",
       " 8.357036456190815e-15,\n",
       " 8.480805106874998e-15,\n",
       " 8.986509785999048e-15,\n",
       " 9.166376175827036e-15,\n",
       " 9.429708655546971e-15,\n",
       " 9.535864940234154e-15,\n",
       " 9.668489947782381e-15,\n",
       " 9.766081956106807e-15,\n",
       " 9.886315681205389e-15,\n",
       " 9.999624573255425e-15,\n",
       " 1.009700720085578e-14,\n",
       " 1.0116203834736555e-14,\n",
       " 1.0287541620246706e-14,\n",
       " 1.0500839546298305e-14,\n",
       " 1.0518121092216408e-14,\n",
       " 1.0712662022595308e-14,\n",
       " 1.0715335985386484e-14,\n",
       " 1.0727692484423974e-14,\n",
       " 1.0943518712919154e-14,\n",
       " 1.0951089532499988e-14]"
      ]
     },
     "execution_count": 623,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 624,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00000000e+00,  2.01364840e-06,  1.14050301e-14, ...,\n",
       "        -2.06792834e-11,  1.08510644e-09,  2.41916394e-11],\n",
       "       [-2.01364839e-06,  1.00000000e+00, -4.02616587e-15, ...,\n",
       "        -2.07791606e-09, -1.19470336e-09, -1.09481580e-10],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  9.99999994e-01, ...,\n",
       "        -1.08644872e-10, -6.24657158e-11, -5.72429767e-12],\n",
       "       ...,\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00, ...,\n",
       "         8.66213925e-01,  4.98049493e-01,  1.84420548e-04],\n",
       "       [-3.89899520e-09, -7.85120543e-15, -4.44681577e-23, ...,\n",
       "        -4.07967409e-01,  7.09541941e-01,  2.62733286e-04],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00, ...,\n",
       "        -6.38927168e-05, -4.07021603e-04,  9.99998241e-01]])"
      ]
     },
     "execution_count": 624,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Printing Orthogonalized matrix Q\n",
    "q = np.transpose(qt)\n",
    "q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Classic CGS with Reorthogonalization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 625,
   "metadata": {},
   "outputs": [],
   "source": [
    "qt = np.zeros((n,n))\n",
    "w = np.zeros((3,183))\n",
    "u = [] ## list of loss of orthogonality in CGS with Reortho"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 626,
   "metadata": {},
   "outputs": [],
   "source": [
    "for j in range(0,n):\n",
    "    w[0] = a[j]\n",
    "    for i in range(1,3):\n",
    "        w[i] = w[i-1]\n",
    "        for k in range(0,j):\n",
    "            w[i] = w[i]-np.dot(np.dot(w[i-1],qt[k]),qt[k])\n",
    "    qt[j] =w[i]/np.linalg.norm(w[i],ord=2)\n",
    "    I = np.identity(j+1)\n",
    "    r = np.dot(qt[0:j+1],np.transpose(qt[0:j+1]))\n",
    "    u.append(np.linalg.norm(I-r))  ## Loss of orthogonality at each iteration i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 627,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2.220446049250313e-16,\n",
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       " 3.510356580232672e-15,\n",
       " 3.5432112729161656e-15,\n",
       " 3.571963206693976e-15,\n",
       " 3.589972916736056e-15,\n",
       " 3.596968042683953e-15,\n",
       " 3.6125855188603135e-15,\n",
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       " 3.647066034344095e-15,\n",
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       " 3.683956115482438e-15,\n",
       " 3.695956543095082e-15,\n",
       " 3.6969896639111166e-15,\n",
       " 3.7047762008612445e-15,\n",
       " 3.7073909912261094e-15,\n",
       " 3.7142791581615275e-15,\n",
       " 3.714280400781604e-15,\n",
       " 3.714494780638716e-15,\n",
       " 3.71845167418737e-15,\n",
       " 3.7184518220161675e-15]"
      ]
     },
     "execution_count": 627,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 628,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00000000e+00,  2.01364840e-06,  1.14050301e-14, ...,\n",
       "        -2.06789756e-11,  1.08510756e-09,  2.41918078e-11],\n",
       "       [-2.01364839e-06,  1.00000000e+00, -4.02616587e-15, ...,\n",
       "        -2.07791617e-09, -1.19470387e-09, -1.09481546e-10],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  9.99999994e-01, ...,\n",
       "        -1.08644874e-10, -6.24657272e-11, -5.72429777e-12],\n",
       "       ...,\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00, ...,\n",
       "         8.66213925e-01,  4.98049493e-01,  1.84420548e-04],\n",
       "       [-3.89899520e-09, -7.85120543e-15, -4.44681577e-23, ...,\n",
       "        -4.07967409e-01,  7.09541941e-01,  2.62733286e-04],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00, ...,\n",
       "        -6.38927168e-05, -4.07021603e-04,  9.99998241e-01]])"
      ]
     },
     "execution_count": 628,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Printing Orthogonalized matrix Q\n",
    "q = np.transpose(qt)\n",
    "q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## MCGS with Reorthogonalization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 629,
   "metadata": {},
   "outputs": [],
   "source": [
    "qt = np.zeros((n,n))\n",
    "w = np.zeros((3,183))\n",
    "v = [] ## list of loss of orthogonality in CGS with Reortho"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 630,
   "metadata": {},
   "outputs": [],
   "source": [
    "for j in range(0,n):\n",
    "    w[0] = a[j]\n",
    "    for i in range(1,3):\n",
    "        w[i] = w[i-1]\n",
    "        for k in range(0,j):\n",
    "            w[i] = w[i]-np.dot(np.dot(w[i],qt[k]),qt[k])\n",
    "    qt[j] =w[i]/np.linalg.norm(w[i],ord=2)\n",
    "    I = np.identity(j+1)\n",
    "    r = np.dot(qt[0:j+1],np.transpose(qt[0:j+1]))\n",
    "    v.append(np.linalg.norm(I-r))  ## Loss of orthogonality at each iteration i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 631,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2.220446049250313e-16,\n",
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       " 7.771561172378148e-16,\n",
       " 7.771561172378148e-16,\n",
       " 7.771561172378148e-16,\n",
       " 7.771561172378147e-16,\n",
       " 7.771561172378148e-16,\n",
       " 8.082545620882502e-16,\n",
       " 8.382000022147355e-16,\n",
       " 8.382000022147354e-16,\n",
       " 8.671119018264573e-16,\n",
       " 8.741904837809515e-16,\n",
       " 9.35490548712708e-16,\n",
       " 9.420554752109266e-16,\n",
       " 9.48574968054168e-16,\n",
       " 1.0063617167877157e-15,\n",
       " 1.0063617167877157e-15,\n",
       " 1.006361716787716e-15,\n",
       " 1.030566694393058e-15,\n",
       " 1.0542160680660053e-15,\n",
       " 1.1487298361257862e-15,\n",
       " 1.1551498266686683e-15,\n",
       " 1.2061507557430754e-15,\n",
       " 1.2264189546046889e-15,\n",
       " 1.226418954604689e-15,\n",
       " 1.2314338812367241e-15,\n",
       " 1.4000584498677739e-15,\n",
       " 1.550550397520993e-15,\n",
       " 1.585919323310368e-15,\n",
       " 1.6469229873781206e-15,\n",
       " 1.872238347914543e-15,\n",
       " 1.890438953895861e-15,\n",
       " 1.8910628079074533e-15,\n",
       " 1.920169812346691e-15,\n",
       " 2.0357876719642005e-15,\n",
       " 2.036550440534461e-15,\n",
       " 2.0643950267921376e-15,\n",
       " 2.0910907181279393e-15,\n",
       " 2.1343947991048005e-15,\n",
       " 2.1459135502209626e-15,\n",
       " 2.216549966981096e-15,\n",
       " 2.2276439488168247e-15,\n",
       " 2.2524056312078705e-15,\n",
       " 2.2768980416991953e-15,\n",
       " 2.2877035705223815e-15,\n",
       " 2.299665354109705e-15,\n",
       " 2.3105921302285653e-15,\n",
       " 2.310592130228565e-15,\n",
       " 2.3212376949215445e-15,\n",
       " 2.3318336653588733e-15,\n",
       " 2.3423817044788648e-15,\n",
       " 2.3659428594693532e-15,\n",
       " 2.366060302950381e-15,\n",
       " 2.377090297165417e-15,\n",
       " 2.4009574747422724e-15,\n",
       " 2.4113057673933445e-15,\n",
       " 2.4215076523605544e-15,\n",
       " 2.4215076523605544e-15,\n",
       " 2.4240547176198275e-15,\n",
       " 2.4643977966879044e-15,\n",
       " 2.4668973330814315e-15,\n",
       " 2.4768702546844457e-15,\n",
       " 2.5383134262030273e-15,\n",
       " 2.5407402467168164e-15,\n",
       " 2.5504397246247725e-15,\n",
       " 2.5506057597961925e-15,\n",
       " 2.553050833460777e-15,\n",
       " 2.562711835243711e-15,\n",
       " 2.573496802784712e-15,\n",
       " 2.5957502045707906e-15,\n",
       " 2.6335474253193234e-15,\n",
       " 2.6335830320345562e-15,\n",
       " 2.6431157352409562e-15,\n",
       " 2.653640194800557e-15,\n",
       " 2.6537057566997332e-15,\n",
       " 2.6919710096003702e-15,\n",
       " 2.6919710096003706e-15,\n",
       " 2.7124972776623285e-15,\n",
       " 2.7486099719971906e-15,\n",
       " 2.7486570266868474e-15,\n",
       " 2.776067238539911e-15,\n",
       " 2.7855648202777423e-15,\n",
       " 2.8176307266902318e-15,\n",
       " 2.817865265905025e-15,\n",
       " 2.8186697979610085e-15,\n",
       " 2.8274022063572535e-15,\n",
       " 2.827402206357254e-15,\n",
       " 2.862065419032053e-15,\n",
       " 2.862299697583539e-15,\n",
       " 2.862440199365212e-15,\n",
       " 2.866779833041913e-15,\n",
       " 2.8863316614338338e-15,\n",
       " 2.886544964015942e-15,\n",
       " 2.896037844909888e-15,\n",
       " 2.999339779641903e-15,\n",
       " 3.0077785026145533e-15,\n",
       " 3.0081117537041717e-15,\n",
       " 3.0103596797876003e-15,\n",
       " 3.045345967719391e-15,\n",
       " 3.047798407945047e-15,\n",
       " 3.050481533769501e-15,\n",
       " 3.050820834016626e-15,\n",
       " 3.0693398921279057e-15,\n",
       " 3.0693398921279057e-15,\n",
       " 3.0713471515040156e-15,\n",
       " 3.073353099905318e-15,\n",
       " 3.105488735177364e-15,\n",
       " 3.114892155496392e-15,\n",
       " 3.116877115180471e-15,\n",
       " 3.134622230591579e-15,\n",
       " 3.1425684491542125e-15,\n",
       " 3.1740144590983186e-15,\n",
       " 3.1759555761889645e-15,\n",
       " 3.198258999783302e-15,\n",
       " 3.2059576477604505e-15,\n",
       " 3.213637852336844e-15,\n",
       " 3.213637853019979e-15,\n",
       " 3.215555038301766e-15,\n",
       " 3.23275853864145e-15,\n",
       " 3.240375221482753e-15,\n",
       " 3.2403752214981964e-15,\n",
       " 3.247974042797082e-15,\n",
       " 3.2498709719556555e-15,\n",
       " 3.2668937691895683e-15,\n",
       " 3.2668937692528444e-15,\n",
       " 3.266893769257981e-15,\n",
       " 3.268779719057831e-15,\n",
       " 3.2930111236779997e-15,\n",
       " 3.3138234726531807e-15,\n",
       " 3.3218780556648964e-15,\n",
       " 3.341365875455633e-15,\n",
       " 3.3620609055767613e-15,\n",
       " 3.3635101912305304e-15,\n",
       " 3.366301940625414e-15,\n",
       " 3.390784225268913e-15,\n",
       " 3.4247979291297945e-15,\n",
       " 3.454532088839978e-15,\n",
       " 3.4617820118533775e-15,\n",
       " 3.469035338476134e-15,\n",
       " 3.4852681072042453e-15,\n",
       " 3.4924054911079405e-15,\n",
       " 3.5057420242125063e-15,\n",
       " 3.5343265148701983e-15,\n",
       " 3.5441034608956296e-15,\n",
       " 3.556575587183076e-15,\n",
       " 3.587318861134003e-15,\n",
       " 3.591317920436383e-15,\n",
       " 3.594015576939743e-15,\n",
       " 3.59597071081469e-15,\n",
       " 3.595972214596528e-15,\n",
       " 3.597793882691807e-15,\n",
       " 3.6065656374595434e-15,\n",
       " 3.606565818820465e-15]"
      ]
     },
     "execution_count": 631,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 632,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.00000000e+00,  2.01364840e-06,  1.14050301e-14, ...,\n",
       "        -2.06789756e-11,  1.08510756e-09,  2.41918078e-11],\n",
       "       [-2.01364839e-06,  1.00000000e+00, -4.02616587e-15, ...,\n",
       "        -2.07791617e-09, -1.19470387e-09, -1.09481546e-10],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  9.99999994e-01, ...,\n",
       "        -1.08644874e-10, -6.24657272e-11, -5.72429777e-12],\n",
       "       ...,\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00, ...,\n",
       "         8.66213925e-01,  4.98049493e-01,  1.84420548e-04],\n",
       "       [-3.89899520e-09, -7.85120543e-15, -4.44681577e-23, ...,\n",
       "        -4.07967409e-01,  7.09541941e-01,  2.62733286e-04],\n",
       "       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00, ...,\n",
       "        -6.38927168e-05, -4.07021603e-04,  9.99998241e-01]])"
      ]
     },
     "execution_count": 632,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Printing Orthogonalized matrix Q\n",
    "q = np.transpose(qt)\n",
    "q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setting x-axis values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 633,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.arange(1,184)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ploting Loss of Orthogonality"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 634,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<function matplotlib.pyplot.show(*args, **kw)>"
      ]
     },
     "execution_count": 634,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.figure(figsize=(10,10))\n",
    "plt.yscale('log') ## y is scaled to log axis for visualization\n",
    "plt.plot(x,y,'k--',label = 'CGS') ## CGS\n",
    "plt.plot(x,z,'b:',label = 'MCGS') ## MGS\n",
    "plt.plot(x,u,'g-',label = 'CGS with reortho') ##CGS with reortho\n",
    "plt.plot(x,v,'r-.',label = 'MCGS with reortho') ## MCGS with reortho\n",
    "plt.ylabel('Loss of Orthogonality')\n",
    "plt.xlabel('Iteration Step')\n",
    "legend = plt.legend(loc='upper left', shadow=True)\n",
    "legend.get_frame().set_facecolor('C0')\n",
    "plt.show"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The loss of orthogonality in the QR factorization for different Gram-Schmidt orthogonalization variants: \n",
    "    CGS algorithm (dashed line) (black)\n",
    "    MGS algorithm (dotted line) (blue)\n",
    "    CGS algorithm with reorthogonalization (solid line) (green)\n",
    "    MGS algorithm with reorthogonalization (dotted-solid line) (red)"
   ]
  }
 ],
 "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": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}