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Trefethen & Bau lecture notes in MATLAB
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Lecture 9: MATLAB/GS stability"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Comparing classical to modified Gram-Schmidt\n",
"\n",
"We're going to experiment to uncover the differences between classical GS and MGS in a numerical implementation. We'll start by constructing a matrix with particular singular values."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"[U,S,V] = svd(randn(80));\n",
"s = 2.^(-1:-1:-80);\n",
"A = U*diag(s)*V';\n",
"reset(0) % default graphical preferences"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The text explains why we should expect the diagonal entries of the $R$ in a QR factorization of $A$ to be close to the singular values of $A$. "
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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hH//4x7/+9S/b5Gy2PJbjhwHHRJradBWmav63r68ZJWnHa7rbGf2CZvQLqm1Q\nAAAATdGzzz47duxYe0cBOArS1CaqMEVjTTVdnrYwRaOMVTd0PAAAAE1EWVmZh4eHUeHIkSPNVtbr\n9UIIFxcXs2srKytdXett5g6zgRkFU1Ukdd+59SoqKtzc3Br6KLAvJqRpogpTrUpThRBMUQMAAFB3\neXl5Tz/9tL+/v6enp4+Pz+jRo8+cOSOvnT9//qBBg6TlKVOmPPPMM5cvX/7973/v5+fn4+MTExNz\n/vx5ufKlS5cmTZqkUql8fHyGDh16/vz5xx9/fP78+fLm48aNMzz0iRMnevTokZycbDawnTt39uvX\nr1mzZp6enn5+frNmzbpz5460auzYsQsXLiwuLp4zZ054ePiXX35ptO3rr7/eo0ePn376ybDwtdde\n69mzZ35+vuWdG6k27Bs3bkyfPl2lUjVv3jwmJmbjxo1WXls4I1pTmyhljLo47Zg1Nf3VcQ0dDAAA\ngM0U3qs4/9udB0Ja2vi4o0aNSktLmzdvXnh4eGZm5vvvv//www+fP38+MDBQCHHlyhU5Ec3IyCgq\nKho2bFh0dPTy5ctPnjz55Zdfjh49+pdfflEoFGfPnn3ooYcqKipmzJgRHBx84MCBoUOH6vX66Oho\neXOjVPDOnTvnzp27ffu2aVTbt2+fMmVKr169Fi1apFKpDhw48Mknn7i4uHz88cdCiIsXL7q5uc2Y\nMWPv3r3du3dv166d0eYDBw6Mj4/fsWPHa6+9JpXo9fqNGzd26tTJ39/f8s6NWA77119/feCBByor\nK5955hl/f/+9e/f+4Q9/yM3Nffnll6u9tnBGpKlNlPUDI9HjFwAANCZjN/07NeP2J5MibDmaxq+/\n/vrDDz/Mnj37zTfflEoeeOCBsWPHHjhwYMqUKab1z58///bbby9YsED6+Nxzz61fv/7UqVMPPvjg\nSy+9VFRU9OOPP4aHhwshFi5c+PTTT5tN/KyxYcMGb2/vf/7zn23atBFC/OEPf+jdu/fhw4flCsnJ\nyT169EhPT1epzEw3OGjQoE6dOmk0GjlNPXXqlFarXb58uTU7t94f//jHsrKyc+fOBQUFCSEWLFjw\nzDPPLFu27Mknn3RxcanRtYVToNNvE6UIVClj1EIIXZ7WQjWpjkSXq83XJGTOuZ9uwAAAwHnFdG7l\n6+Ue07mVLQ8aFBTk6+u7Z8+e5OTkiooKIcSYMWNKS0snTZpktr6Xl9dzzz0nfxw9erQQIicn5/r1\n61999dXEiROlHFUSHx9f68DWrFlz8uRJKY0UQuTn5xcUFBQVFckV7t27t3HjRrM5qhDCxcXlySef\n/OWXX3788UepJDEx0cvLa/z48dbs3EoFBQVffPHFzJkzpRxV8sorr1RWVm7ZvYfdEQAAIABJREFU\nsqWm1xZOgdbUpstfHVd0Ikm4Xr17ZrHpWmneVKnHry5XW5iqkbPTfE1CYYrGKzK6ZezEmk5XAwAA\nYF/LRoQuGxFq44MqFIr169c/++yzo0eP9vX1HTBgwLBhw6ZOnVpVr9QOHTq4u//fF3VpWCCdTnfp\n0iW9Xt+jRw/DyqGhoc2bN69dYP379798+fK6det++OGH8+fPnzt3rrS0NCAgQK4QFBTUs2dPC3uY\nNm3aypUrNRpNr169hBBffPHF2LFjlUqlNTu3ktQdeuvWrUlJSUarfv7555peWzgF0tSmSxGoEkJU\nFmeXFiearnX1cZHq3NwTKoRwaSb+d/aaXCGSSjKSSjKE35jMWhz90o2SLq29ahU4AACA85k8efLI\nkSOTk5MPHjyYmpr61VdfvfLKK3v27Bk2bJhpZcMc1VBZWZn47/C/Mr1eX15ebuHQRvUNrVq1atmy\nZV26dBk+fPj06dN79+69atWq77//Xq7Qvn17y+fVqVOnQYMGaTSa119//fTp05mZmevXr7dy55bJ\nYet0OiFERESEUX4+cuTI++67T9Tw2sIpkKY2aX5jMo1aSoUQigCVMlatjFErAlW6XK0QovKO3iti\ngunmUotrSdrxmrapztzx8+bvct4eG7ZgsPkOJAAAAI1JSUlJQUGBv7//lClTpkyZotfrExMT1Wp1\nfHx8jVKpLl26CCGMRtb96aefSktL5Y+urq6VlZWGFbKzs83uLTs7++WXXx49evSePXvkKV6MMl6F\nQlFtVNOnT3/qqadOnz6dmJjYpk2b4cOHW7lzQxbCjoiIEEIMHDhQfgNWCKHX67Ozs318fOrr2sKh\n8G5qU6cIVPmr40LfO9l23t8UASp/dVzo+yf91XFSW6uUvnpFTGjee43pf1LuWpCys6YH7diqmRCi\no1+z+j4bAAAAR3Ts2LGgoKBPP/1U+uji4jJhwoSQkJCavqipUqn69u37xRdfXLlyRSrR6/WrVq0y\nrBMYGHjx4sXr16/LJVu2bDG7t/T0dCHEkCFD5DQyJyfn5MmTFlpfzZowYYK3t7dGo0lMTJw8ebK0\nt5ru3ELYAQEBDzzwwJYtW+7evSuv3bZtW4cOHfbs2VNf1xYOhdZUCCGEIlClCFSZDupbmKr5376+\nZpSkHZeXpbbZwhSNMlZtYSYbu7wTAgAAYC/R0dEdOnRYuHBhQUFBv379ysrKdu3adeXKlSVLltRo\nPy4uLm+++ebIkSOHDBkye/bs1q1b7969Oz093XCqmIcffjgpKWnKlCnz5s2TZn+RpjA11a1bNzc3\nt/Xr13fq1KlNmzbnzp1bvXq1m5tbQUFBSkpKbGyslVG1aNFi3LhxH3zwQUFBwZNPPlm7nVsOe/Xq\n1UOHDu3Xr9/ixYtDQkKOHj26bt26sLCwJ554wtXVtV6uLRyLHk1D165da7pJwaGd6U8E5Sd1vHN6\nkdkKd04vyk/qmP5EUMGhnWW//Xpj55vpTwTJ/11+tn/OuvnFPx2rc+wAAKDRqsVXFCd1/vx5w9zM\n19d30aJF5eXl0tqxY8f6+flJy/fff39kZKThtgcOHBBCbN++Xfp48uTJQYMGeXt7N2vWLCYm5vLl\ny8HBwVJ/V71ef+/evVmzZrm4uAghPDw8xo8fL70OKm9ueKxNmza1avWfQY8DAwO3bNmi0WhcXFwC\nAgL0ev19990XHR1tzdnt379fCNG9e3fDQss7NwzDmrCPHz8eFRUlVRBCjBo16vLly9ZcW1N1fOqa\nzkNrRy76Grbpw0mFh4dLXS+sl71sfHHasdbTPD1V45v3XmNa4e6ZxaXaxBtbS6ttca3dMEsAAKDR\nq8VXFKd269atq1evenl5qVQqafzeWquoqKioqPDw8CgsLPT391+yZMmrr74qry0uLr5y5UqnTp08\nPav5nlZaWpqVldWiRQu5Sfbq1astWrSQRuuto5ruvNqw79y5c+XKFZVKZboH669tHZ+6pvbQ2gWd\nflElZYy6OO2YlZXrfZglAACAxqdVq1ZyA2PtDB8+vKio6OjRo25ubtJrn2+//XZ5efmgQYMMq3l7\ne0sjD1XL09PTcBZWYcXovtar6c6rDdvHxycyMtLsqrpfWzgO0lRUqUaJpTTMktlVpdrEgpSdpKkA\nAAB1N3ny5FmzZg0ePHjq1Kmenp4pKSmffvrpqFGjpPF1gcaBNBVVUgSqlDFqIZJ0edo67spwmCUA\nAADU2owZM1q3bv3uu++uXLmyoqIiLCxszZo1CxYssHdcQH0iTYUl/uq4ohNJwvXq3TOLTddKHXqt\n2Y8uTysN/1vfAQIAADQ5jz766KOPPmrvKIAGRJoKS6TZUyuLs0uLE03XWpmjSvI1CaSpAAAAAKrl\nau8A4Oj8xmS2eOCI/t5zN7aWyv8VJAfq7z3X4oEj7i3+auV+LEyjatbun/Jc4g7t/imv5iEDAAAA\ncGK0pqJ6ikCVvzpOGaMuSTsuNYrWNOcUQtS0KfXfV+9I/3+se0BNjwUAAADAeZGmwlqKQJUiUGWU\nbVo5zJIypsbdfZeNCB0b1bpXuxY1DhQAAACAMyNNRV1ZM8xSLVpfhRDkqAAAAEATRJqKurJmmCWp\nDgAAAMz6+OOPMzIyXn/9dac4im2iRVPGEEqoB5aHWfIbk2nvAAEAABzaP/7xjw0bNjjLUWwTLZoy\nWlNRP+plmCUAAAA0nGeffXbs2LH2jgKoHmkq6pPZYZYAAABgqLy83N3dqu/hZWVlHh4etVhrdtXI\nkSNrEY/lGKql1+tdXFxqvTmaJjr9AgAAoKnQ5WrzNQmZc+7P1yTY/uharXbixIkdOnTw8PBo167d\nwoULb968abbmzp07+/Xr16xZM09PTz8/v1mzZt25c0dem5eX9/TTT/v7+3t6evr4+IwePfrMmTPV\nrhJCzJ8/f9CgQVbGYzkGC8aOHbtw4cLi4uI5c+aEh4d/+eWXNbpKgKA1FQAAAE2BLldbmKqRs9N8\nTUJhisYrMrpl7ESvyGgbBJCWlhYTE1NaWvrkk0926dLl0KFD69atO378+OHDhz09PQ1rbt++fcqU\nKb169Vq0aJFKpTpw4MAnn3zi4uLy8ccfSxVGjRqVlpY2b9688PDwzMzM999//+GHHz5//nxgYKCF\nVUKIK1eunD9/3pp4qo3BgosXL7q5uc2YMWPv3r3du3dv165dfV9LNAF6NA1du3a1dwj178zVopj1\nP+w6l2vvQAAAQC3Z7CtK+hNB0n+Xn+1fcGindukTcoltAhgzZoxCofjhhx/kkueee04IkZSUpNfr\nx44d6+fnJ5UPHjzY29v7+vXrcs1evXp17txZWr5y5YoQYvbs2fLapKQkIcSnn35qYZX00fAoluOx\nHIPhfkzdd999zZo169+//6+//lqTK2Q7dXzqGuX3akdDayqcWNK5vNSM275eise6B9g7FgAA4KB0\nudrMufcLIaThHqV58pSxarl99eL4dqoVXzRom2p+fv6ePXsmTJjQp08fuXDlypVDhgzp3LmzUeU1\na9Z4e3u3adNG3ragoODu3bvSx6CgIF9f3z179jz22GPDhw93c3MbM2ZMaWmpm5tbZWVlVatqGo/l\nGKp17969jRs3qlRMSYha4t1UOLFlI0LfHhv2yaQIewcCAAAcl9TRVxmj9lfHGc7lLs1TIM1NUJCy\ns0FjuHTpkhDCMCcUQrRq1Wr8+PGRkZFGlfv37+/t7b1u3boZM2b0798/ODg4Kyvr/8JWKNavX3/3\n7t3Ro0e3bt36kUceeeutt27fvu3m5mZhVU3jsRxDtYKCgnr27Gl9fcAIaSqc24LBKl8vOgUAAIAq\nFaZqhBBVTZWnjFELIUrSjjdoDEVFRUIILy8vayqvWrWqa9eu69evVyqV06dPP3jw4KhRowwrTJ48\nOSsr69NPPx03btzPP/8cFxcXGhp68OBBy6tqFE+1MVjWvn176ysDpvh+DwAAgEarMEUjhJD7+ppS\nBKq8IwcUpx0rTNE03KR6Uk/aixcvGhZeu3bt1VdfHTlypOFcptnZ2S+//PLo0aP37Nkjt4KWl5fL\nFUpKSgoKCvz9/adMmTJlyhS9Xp+YmKhWq+Pj4wcMGFDVqmHDhlkfT9++fS3HUC2FQmF9ZcAUrakA\nAABotKSm1JaxEy3UkRpaG3SKmpCQkLCwsM8//zw/P18ufOuttz744AOjCUvT09OFEEOGDJHzw5yc\nnJMnT+r1eunjsWPHgoKCPv30U+mji4vLhAkTQkJCioqKLKyqUTzVxgA0NFpTAQAA0GgpY9TFacfy\nNQnBKxKrqiMlqFX1Cq4Xrq6uq1evHjdu3KBBg1588cXWrVvv2rXrs88+GzBgwIgRIwxrduvWzc3N\nbf369Z06dWrTps25c+dWr17t5uZWUFCQkpISGxsbHR3doUOHhQsXFhQU9OvXr6ysbNeuXVeuXFmy\nZImFVTWKJy8vz3IMtbgCXl5eO3bsGDt2rLxQ+6uJpsC+Aw3DZhg4GwAAOKCG/opS9tuv0qwzxT8d\ns1yhQcOQfPXVV/K4vi4uLpMmTZJnfDGc4mXTpk2tWrWSqgUGBm7ZskWj0bi4uAQEBEgVzp8/b5gr\n+vr6Llq0qLy83PIqvclEMhbisRxDtRPSREdHGxU+++yzZ86cMVywIyakcXwuetrum4bw8HCp/wYA\nAIDjsMFXlOvvLihM1Shj1G2f+1tN1zaEW7duXb9+vWPHjhZGMCotLc3KymrRokW7du2kkqtXr7Zo\n0UKpVBru5+rVq15eXiqVysPDw+gQVa2yPh5rYnBSdXzq+F5tA7ybCgAAgMZM6s1bknY8X5Ogy9XK\n5bpcbb4mwfI4wA2hVatWERERlkf99fT0DA8Pl/NDIUT79u2N8sNWrVp17969c+fOpomohVXWx2NN\nDNbr379/SkqK4QJgAe+movFb8XXm2KiAXu187B0IAACwA0WgqmvitYvj2+VrEvI1Cd6RA/zVcQUp\nO6UEVQjRNfGafSNsCs6dO1dYWGi4AFhAmopGbtOpnOX7M1MzbqXM7VN9bQAA0EiFvneyMFWTr0ko\nTjtWvOyYEEIRoFLGqqV5UwE4FNJUNHLjewT+I+3G9H5t7R0IAACwJ0Wgyl8dp4xRS71/lbFqW3b0\nBVAjpKmwNV2utjBVI82gbYNfD8pmbrtmRjX0UQAAgFNQBKoUgSplLC2ogEMjTYXtSAmqPHd2viah\nMEXjFRndMnaiV2S0aU2bpbIAAAAAHAdpKmzk5p5QIYRLM9F6mqdBca4QSSUZSSUZwm9MpqhJKgsA\nAACgUSJNhS1Ig79X3tF7RUwwXVvy899dfVxK0o6XZEwWVqSyAAAAABox0lTYQr4mwaOD8IqY0Lz3\nGrMVSrWJBSk7PTpUn8rSpgoAAOB09Hq90QJgAWkqbKEwVfO/DaRV1qk2lSVNBQAAcCIlJSUHDx4s\nLS0NCwvbu3dvaWlpVBTDW6IarvYOAI1fYYqmvnZVkna8vnYl0+Vq8zUJmXPul1+IBQAAQH25dOnS\ns88+u3jx4oqKCmmhU6dO9g4Kjo7WVDS4wtR6S1N1eVpp+N/62RvDNQEAADSwqKio7OxsaVleACwj\nTUWDU8aoi9OO1dfepPm4674fK0ceBgAAAGBjpKlocPXbMlkv06gajTxcVlGZeDa3g2+zgaG+guGa\nAAAAALsiTUWDUwSqlDFqIZJ0edq6761emlKNRh5uLsQzv/ufCgzXBAAAANgLaSpswV8dV3QiSbhe\nvXtmselaqfXSO3KAED9YTmWVMfXzVqo1Iw83xHBNAAAAAKpFmgpbUASqhBCVxdmlxYmma119XIQQ\nbea9XXRisOVUtl56/Fo58nD9DtcEAAAAwEqkqbARvzGZRiPrCiEUASplrFoZo5byWFFdKitXE/8d\np1fKJGuUvlo/8nB9DdcEAABw4MCBffv2CSFWrlypVCpNKyQlJaWkpAgh3nzzTXf3//mWfuTIkQMH\nDpw6dSorKys4OLhPnz7PPfdcSEiIYZ3KysqkpKRPP/300qVLFRUVHTt2fPzxx6dNm2a0K8Ap8NTC\ndhSBKn91nDJGXZJ2XMoAjdJLK1PZOk4kY/3Iw/XSeAsAACCE+O6779555x0hRFRU1FNPPWVaYcmS\nJRcuXBBC/PWvf5Vzy8rKyvj4+DfeeMPV1TUqKioqKiorK+vNN9/829/+9u677/7hD3+QqpWXl48Z\nMyY5OTkkJKRv375Xr15NTk7et2/fBx98cOTIkWbNmtnqLIH6QZoKW1MEqhSBqqpaKatNZWs0kYzZ\nFlfrB0aiKRUAANQvDw+PzZs3m6apJ06cuHDhgoeHR1lZmWH5008//cknnzz66KPbtm3z9fWVCtPS\n0p544om5c+d26NBh1KhRQog1a9YkJye/9NJLr732mouLixAiPz9/2bJl69evj4+PT0hIEIBTcbV3\nAIAZUh4b+v5JoxxVnkjGUzXe9L/KO3rx36GPdLnafE1C5tz78zUJujxtviYhc879199dUJJ2/L8j\nDwvbDNcEAAAgGzt27NGjRy9dumRUvnnzZqVSGRsba1h49uzZzZs3Dx8+fNeuXXKOKoSIjIyU+g9L\nzbNCiAMHDri5ub300ktSjiqE8Pf3X7t2bWBgYHJyspWx6fX62p0UUO9oTYUzMZpIxpQ0kUxJxmRh\nscXVX32k2pGH6fELAADq3eTJk3fv3r1ly5ZXX31VLiwpKdmxY8eECRNu3rxpWPnVV1/V6/VLly41\nfb+0c+fOS5cuvXbtWnl5ubu7e0lJSWVlZW5ubmhoqFzH1dV106ZNV69erSqYsWPHdurUadWqVXFx\ncYcOHXr99defeOKJejpRoE5IU+FMrJxIxqODqLyj94qYYGbtz3939XEpz8sWNRmuSQhxp7SixUuH\nfb3cb702uPYnAAAAmrbAwMBHHnlk69atK1ascHX9T8fG3bt3FxQUzJgx46233jKsfOzYsdDQ0Acf\nfNDsrpYtWyYvjx49+sSJEw899NBrr702evToli1bSuWPPPKIhWAuXrzo5uY2Y8aMvXv3du/evV27\ndnU6N6D+kKbCaVg/kYwQntW2uLZ9zqrhmgAAQCMgjW1hgeHYFg1txowZu3fvTklJGTZsmFSyefPm\nLl26DBw40DBNvXv37rVr1x566CHDba9du1ZcXGxY0r59ey8vr/j4+N9++23Dhg2TJ092c3Pr0aPH\nwIEDR44cOWLECDc3NwvBJCcn9+jRIz09XaXiyw8cCGkqnIb1E8lUS3p/tdrhmgz5eLrpE4bWVwAA\nAMDGXL2DFf4PmJbr8k9UFmfbMpLRo0cHBARs3rxZSlOzs7O/+eablStXGlXLyckRQvj7+xsWzp49\nW3orVXb48OHBgwe7urq+++67S5Ys2bVr16FDh44dO7Zu3bp169Z16dLlww8/jImJqSqYe/fubdy4\nkRwVjoY0FU7D+olkqqXL00rD/4rqRh4GAACNg8L/AbM9re6eWWz2JaAGjEShmDJlysaNG997770W\nLVps3bpVCDFt2jSjah06dHB3d8/M/J9m3oULF44fP15aPnTo0LZt2wzXBgcHP//8888//7wQ4ty5\nc5s2bVq7du2YMWNOnz7dpUsXs8EEBQX17Nmzvk4NqC+M9AunYf1EMtYw7OsLAABgSzNmzCguLtZo\nNEKIzZs3Dx061LQ908PDo0uXLufPn79z545cOGzYsBn/Jb9KqtVq//rXv548edJw86ioqLfffnvF\nihVFRUVGDbCG2rdvX29nBdQf0lQ4DSsnkrESA/kCAAB76dmzZ69evTZv3vztt9/+8ssvM2bMMFtt\n6tSpd+7cWbt2remq8vLyf/zjH9KyQqFYsmTJe++9Z1rt/vvvF//tP2yWQqGoRfxAQyNNhTP5T27p\nevXumcWm/5X8/Hfrd9UQvXz/M1nrnPtpqgUAAJbNmDHj6NGjS5cuVSqVjz/+uNk6cXFxoaGhf/nL\nX7766ivD8srKypdeeun8+fPSx7Zt24aFhSUmJn733XdGe5BS2SFDhjTAGQANiHdT4UykAXgtTySj\njFELkWS5xVVqla1HRoMG52sSClM0XpHRLWMn1m9fZQAA0DhMmTJl8eLFhw4devrpp728vMzWadas\n2c6dO5944onf//73M2bMGDJkSIcOHdLS0j7//PPvvvvu//2///f5559LNbdv3z5o0KCYmJinn356\n2LBhAQEB165dS0xM3LFjx/Dhw+Uhhavl5eW1Y8eOsWPHygv1c7ZADZGmwsn4jalmIhldrrboRJLU\n4mq6uTRvav32+JXGuHdpJv53TtdcIZJKMpJKMmw6xj0AAHAKrVu3fuSRR3bv3l1Vj19Jv379fvzx\nx+eff37Xrl2bNm2SCrt163b06NHw8PC///0/Xcl+97vf/etf/1q+fPnatWvlTsL+/v5/+tOfVqxY\n4eHhYWVUM2bMCAkJMVwA7MJFr9fbOwbYQnh4eHp6ur2jqE+6XG1VE8nU+9xoUmIsDQ5sdCxdrrbo\nxODKO3qviAmmG0pZsVfn7bSpAgBglm2+otzcE+qpGl/lSL/aRMf/m7Jer8/IyLh27VpUVFSrVq2q\nqnb79u3Lly/n5+e3b9++S5cu1ieoTUodn7rG973aAdGaCmdlYSKZaltcrT9Ktb158zUJHh2EV8QE\n6Tdf7HunhRApc/vIeyjVJhak7CRNBQDAvnT5J8z2tNLln7B9MLXg4uLSpUuXquaVkfn6+vbp08dy\nHQv69+//17/+NTY2Vl6o9a6AuiBNReOkCFT5q+OUMeqqWlytYU1v3sJUjeHa1IzbpvspSTtei1MA\nAAD1qKqxLWDo3LlzhYWFhguAXZCmojGz0OJaLV2uVghhuTfv9XcXGJWn/el+M7vK00odhoXF/sMA\nAKCBOH6fXgCGSFMB84x685oq1SYWpmqMCru1aV7V3rwioxkNGAAAAKgWaSpgnlFv3jpqOSq36MRg\nRgMGAAAAqkWaCphRmGLcTFp3lvsPl6Qdp00VAAAAEKSpgFmmvXkt0+Vpq61juf8wowEDAAAAEtJU\nwAxljLo47Zg1Nf3VcUK8K1yvmh3jXmoptWY/jAYMAAAASEhTATOsb9j0V8fd3PNuVWPcW5mjiv8d\nDRgAAMAu9Hq90QJgF672DgBwRIpAlTJGLarrzSvV8RuT2eKBI/p7z93YWir/V5AcqL/3XIsHjhT/\n0NfKg8qDAAMAANhYSUnJ3r17S0tLw8LCpIWoqCh7B4Wmi9ZUwDx/dVzRiSTLvXnliU8VgSp/dZwy\nRl2Sdjxfk2A4J6pp/+EVX2dm3br3yaQI0yM2wHkAAABU79KlS88+++zixYsrKiqkhU6dOtk7KDRd\npKmAeYpAlRDCcm9eqY7hJopAlVHHXdP+w8v3Zwoh4h/q2KW1l2E5PX4BAIC9REVFZWdnS8vyAmAv\npKlAlfzGZOpytYWpGsPuuIoAlTJWrYxRG+WoVflv/+Ekuf/wrplRWTfvGeeoMeSoAAAAgBCkqYBl\nFnrzWs+o//DDQogW4u6Z/6w16j8MAAAANHGkqUD1zPbmrdHmoob9hwEAAIAmizQVsIV66T8MAAAA\nNAWkqYCN1Ev/YQAAGp/w8HB7hwDAsZCmAjZVx/7DAAA0Munp6fYOAYDDcbV3AAAAAAAA/B/SVAAA\nAACAAyFNBRxR1s17m7/LsXcUAAAAgB3wbirgiHq/dep2Sfl9gd4PhLS0dywAAACATZGmAo5owSDV\nj9fudGvjY+9AAAAAAFsjTQUc0bIRofYOAQAAALAP3k0FAAAAADgQ0lQAAAAAgAMhTQUAAAAAOBDS\nVMDh6HK1+ZqEzDn352sS7B0LAAAAYGsMoQQ4EF2utjBVI2en+ZqEwhSNV2R0y9iJXpHR9o0NAAAA\nsA3SVMBR3NwTKoRwaSZaT/M0KM4VIqkkI6kkQ/iNybRXbAAAAIDNkKYCDkGXqxVCVN7Re0VMMF1b\n8vPfXX1cStKOG7WpXrpR0qW1l41CBAAAAGyCNBVwCPmaBI8OwitiQvPea8xWKNUmFqTsNExTx31y\nbvdPecdf6PtASEtbhQkAAAA0OIZQAhxCYaqm2jolaccNP3b0ayaECG7ZrKFiAgAAAOyBNBWwv8KU\n6nNUIYQuT2tY8+2xYfqEocG+nhY2AQAAAJwOaSpgf9Y0pUqYogYAAACNHmkqYH/KGLWVNf3VcQ0a\nCQAAAGB3pKmA/Vk/J6oy1tqEFgAAAHBSpKmA/SkCVVKDqi5Pa6Ga9Y2uAAAAgPNiQhrAIfir44pO\nJAnXq3fPLDZdK82batjjV5erLUzVFKZolLFqegIDAACgMSFNBRyCIlAlhKgszi4tTjRd6+rjIteR\nElR5LKV8TUJhisYrMrpl7ETrOw8DAAAADstFr9fbOwbYQnh4eHp6ur2jQDWMUlAhhCJApYxVK2PU\nUo56c0+o5T34jcls2BABAACaNr5X2wCtqYADUQSq/NVxyhh1SdrxfE2CUYdeXa5WCFF5R+8VMcF0\nW6ljcEnaccM21RVfZy7fn7lrZtRj3QNsED8AAABQd6SpgMNRBKoUgSrTQX3zNQkeHYRXxITmvdeY\n3bBUm1iQstMwTf3x2h0hRNbNew0XLQAAAFC/SFMBp1GYqmk9zdNynZK044Yfd82MunSjpEtrr4aM\nCwAAAKhPTEgDOIfCFI011XR5WqOa5KgAAABwLqSpgHMoTLUqTRVCGI7ABAAAADgd0lTAOShjjF9V\nrQrTqAIAAMCpkaYCzsH6OVFNx14CAAAAnAhpKuAcFIEqqUFVl6e1UM36RlcAAADAMTHSL+A0/NVx\nRSeShOvVu2cWm66V5k2lxy8AAACcHWkq4DQUgSohRGVxdmlxoulaVx8XuQ4AAADgvEhTAWfiNyZT\nl6stTNUYDuerCFApY9XKGHXtctQfr91ZuPvi24917dXOp/4iBQAAAGqJNBVwMopAlb86ThmjLkk7\nnq9JUMaq69jR950j2tSM2+8c0X4yKaK+ggQAAABqjTQVcEqKQJUiUFWgYqkgAAAgAElEQVQvg/q+\nPTasY6tm8wfTWxgAAAAOgTQVaOp8vdyXjQi1dxQAAADAfzAhDQAAAADAgZCmAgAAAAAcCGkqAAAA\nAMCB8G6qUyovL583b15lZaVc4urqumHDBjuGBCclTW9TmKKp+4jBAAAAQL0gTXVKGRkZqampkydP\nbtasmVTi6krDOGrGaP7VfE1CYYrGKzK6ZexEr8ho+8YGAACApow01SlduHChRYsWS5cudXFxsXcs\ncEo394QKIVyaidbTPA2Kc4VIKslIKskQfmMy7RUbAAAAmjjSVKd04cKFiIgIclTUji5XK4SovKP3\niphgurbk57+7+riUpB03bFO9XVL+zhHt2KiAXu18bBcoAAAAmiTSVKd04cKFVq1aLViw4MyZM0ql\ncsiQIfPmzfPy8rJ3XHAO+ZoEjw7CK2JC895rzFYo1SYWpOw0TFPfOaJdvj8zNeNWytw+tgoTAAAA\nTRQvNDqlCxcupKamBgYGvvDCC3379t20adPMmTMNR1QChBC6XG2+JiFzzv3yC6iSwlRNtduWpB03\n/Dh/sOqx7gFvP9a1nkMEAAAATNCa6tCKior27t0rfwwICHjooYcqKirmzJnTp0+f7t27CyGeeOKJ\nqKiol156af/+/SNHjrRfsHAgFoZHknr8Vr+HPK00/K/00dfLfdfMqIYKFwAAADBAmmoLBQUF48aN\nmzVr1tSpU03XfvPNN3v37r106ZKPj0/Xrl2feuqpkJAQadXNmzdXr14t1+zTp89DDz3k5uY2bdo0\nwz089thjK1asOHfuHGkqRHXDI1m/n3xNgpymAgAAADZDmmoLH3300dWrV+/evWu6atWqVVu3bvX0\n9OzWrVthYaFGo0lKSlq/fv3AgQOFECEhIWfOnDHa5MaNG5cvX+7bt6+bm5tU4ubm5u7ubnb/aGqs\nGR7Jyl0xjSoAAADsgjS1oVRUVGi12itXruzevfurr74yW+fkyZNbt27t0KHD5s2b27dvL4T4+uuv\nFy5cGB8f//XXX8tzohrJysp68sknP/74YymVFUL8+9//vnv3bkRERAOdC5yINcMjWbkrmlIBAABg\nFwyh1FAuX748YsSI2bNnV5WjCiG++OILIcSiRYukHFUIMWLEiJEjR16/fv3bb7+taqsePXp07do1\nPj7+wIEDOTk5R44cWbRoUUhIyLhx4+r9LOB0rBkeSaLLs/SSqjKGHBUAAAD2QWtqQwkKClq7dq20\nfPbs2Y8++si0zsmTJ93c3IYMGWJYOGzYsH379p04cWLYsGFm9+zh4bFhw4aVK1c+//zzer1eoVAM\nHjx4xYoVHh4elkMKDw+3sDY9Pd3y5nB8hSnW5qhCCOF69e6ZxabFUsdgevwCAIAmy/LXZtgAaWpD\n8fHxGTFihLTs7m7mOut0utzc3ODgYKPOvZ07dxZCZGdnW9h5u3btPvjgg5KSktzc3Pbt25vdvykS\n0UbP+qZUIURlcXZpsZkOwNLLq4pAVb2FBQAA4FQsf20mibUB0lS7KSwsrKysbNmypVG5VFJQUFDt\nHry8vOQxgQEhhDJGXZx2zJqa7i3+6hUZbThpjRBCEaBSxqqVMepa56gnrhR0a+OjbOZWu80BAAAA\nQZpqRzqdTphraJVKpLVAjXhFRltZUxoeyV8dp4xRl6Qdl+aeqWNH38SzuRO2/BTT2Tdlbp+67AcA\nAABNHGmq3UjTyZimo1KJPNkMYD1FoEoZoxYiyfrhkRSBKkWgql4G9X2gQ0tfL/eYzq3qshNdrrYw\nVVOYoql72gwAAAAnRZpqN15eXkKIkpISo3KpRFoL1JS/Oq7oRJJdhkcK9vW89drgWm8uJahyJ+R8\nTUJhisYrMrpl7ETrW4kBAADQCJCm2o2Pj0+LFi20Wm1FRYVh22lWVpYQIigoyG6RwZlJr5XW7/BI\nNmjhvLknVAjh0ky0nuZpUJwrRFJJRlJJhvAbk9kQxwUAAIADIk21p4iIiFOnTqWlpfXo0UMuPH36\ntBCiW7du9osLzs1vTKZRy6So7fBItmnh1OVqhRCVd/ReERNM10otwCVpx2lTBQAAaCJIU+1p+PDh\np06dWrNmzbZt26SSnJyc7du3u7m5DR061L6xwakpAlV1Hx7JZi2c+ZoEjw7CK2JC895rzFYo1SYW\npOwkTQUAAGgiSFPtadKkSTt27Dh16tTjjz/+yCOP/Pbbb/v27SsuLn7mmWfatWtn7+jg9OoyPJIt\nWzgLUzX/mwmbUZJ2vO4HAgA4GkbOcyj1dTu4rag70lR7UigUW7ZsWbly5TfffJOWliaEaN68eVxc\n3FNPPWXv0NDU2ayFszBFY001XZ5W+m1Xx8MBABwEI+c5lPq6Hc57W0mtHQ1pqi0MGzYsPT3d7KrW\nrVuvXbtWp9NlZWU1b948KCjIxcXFxuEBpmzWwlmYalWaKoSQei/X/YgAAEeQOfd+aUERoPJXxxWm\naorTjulStYWpmq6J1+wbWxNUX7fDGW+r86bWjRtpqkNQKBRhYWH2jgL4j1q0cNb6b5DKGHVx2jFr\navKnTQBoHHS5WimZkYZRkMb2U8aq5Wzh4vh2qhVfkCHYRn3dDue9rc6YWjcFrvYOAIDDqVELpy5X\nm69JyJx7f74mQZenzdckZM65//q7C8y2tWbfLnWJO9Q74ZRcYv2vK5pSAaBxkJqtlDFqf3Wc4fjz\n0vh/0h8lC1J22i0+W/nPL9A59xuOzG979XU7GuK2NvQl0uVqL45vJ4TwV8eFvncy9P2Tylh18IrE\n0PdOSgFfHN+O0THshdZUAMasb+FsOSq36MTguowGrAhUKWPUQiTp8rSWQ7ImHgCA45P+GFpVHxll\njDpfk9C4cwOH6mVaX7ejfm+rbS6RYWptWC6l1lIF5hqwF9JUAMZq9M9xjUYDDvb11CcYT7bkr44r\nOpEkXK/ePbO4qv3Q4xdAtRgBxSlI75VYmMdbEajyjhxQnHasEY+c5zi9TOvrdtT7ba3fS1TVPw78\nxcSRkaYCMGZlC6fE+tGAq/olIf1KqyzOLi1ONN2Jq4+LXAcAzHKotilYJiUGLWMnWqjjr44rXnbM\nqUfOq+pXnqO9wFlft6Meb2v9XiIL/zhIc+818b+YODLSVABmWNPCac1+pL9BVvsN0m9MplEdIYQi\nQKWMVVv4/QEAEsdpm0K1pPdK8jUJwSvM/GlSIv0ucNImccu/8hytl2l93Y56vK31e4ks/OPgHTlA\nNIG/mDgv0lQAZljTwmkNXZ725p5QIUS1769Kv36UMeqStOPS7wMn/YICwJYcrW0K1ZLuRXHaMaO3\nQv4/e/ceJVV95wt7Q9NcG5BrAGmx4xWETHSSSIwv3nLEs5J4ich4ZiXGyzJZ0RkHXmTe9SY5gzpx\nckaGI7rUnMwiozLrGNOSE/HVzDFxBkISbDQnJGpPgmLA6VYUaAItdIMlzfvHNmXbl+rd3bv23lX1\nPIvlanftrvrWtX+f+t3ycrubwvURSjQYFP7SJGujTAfwdPTYURzj0xrXQ9Tnh0NYTxl/Y1LqxFSg\nZ4V7ON+6f2kQRFpmKejP/NXqqbXVU2tLtGkCJC9rfVP0KZxX0rqxvrfnJf+cJl7aYEX50iRIb5Rp\nj/GyX09H4Y7iWJ7WGOe4RvlwCMr6G5NSZ0MaoFfhR3ndA1um3bw6/Fa47ttbwoXm+9WACOevdv8X\nZtdK2HUAKJI+O16CP84+yKwoW25kZOeSuITPVzh2JpwfGArvZuHnNMui7MgSRBhlmr+quBTeOi76\n0xFeQxAE1VNqp928evQZ5+T2NLVurG9acWW/rqeAiHNcgwgPUZQPh1Bv7ZDS/cakPOhNBfrQYw9n\njF0TGW9BAplV6mvGRln5qaRXhyqwct6p6954edGMlvpVLfWrRp9xzqTFyw5s+H5+1+4SnVEcZbRq\nkMYo08LjkKM8HZ33Fy0wun7wT2tcc1yjfzgEf4zWo884p+uijyX7jUl5EFOBgejXasCF5fY0ZbMF\nCWRcqa8ZG2XlpxJdHSpKuq57YEt+imDbis1Bp3klqdUdTY/xO3ouSnKUafTJ24Wfjuij6wf5tMY1\nxzX6h0P1lNqwh7klKJ9vTMqDmAoMUFyrAQdB0KUFWXfn5hMnjNxw01nxFAqUqdJdMzZKeJh28+o3\n719S+JzMrg4VJV2X4sp5BeJ39FwUBEFi83Kjx8vCT0f0ZY0G+bTGNXW5Xx8Oo874ZIl+Y1LexFRg\ngOJaDTj44F++N99+Z+e+wzv3HR58hUB5y/6asb2Neo0SHsKMWnKrQ/V37eXSWjmvQPyedvPqiLko\n+OMo085dr0UaZdrfVXN7fDoGMLp+ME9r+MAOciBufz8cSu4bk0ogpgIDV3g14CAI9q49O8r1dP5L\nNm3s8Ma/PnvcCJ9OFFFv4YHSku6asYVfRYVHvUacxNjnORmc21+uay/3Gb/DrxWi5KLE5uXGNXk7\n4dH1H5grO9CBuAP4cIgYrcNn/LunHWqpX+XPR1FV3XbbbWnXQBLuu+++v/zLv0y7CspQ1Zjxo884\nZ9z5i0eeOPfIzsYJn71xxv/zT6PPOKdqzPiqMeNzu5uqxm879s7YkSct6v67uTd/crT134dNuLLm\nE5d0Pj6lZvi4kWIqRZHb3bT/qTVv3HV9e+OzHW2t7Y3Ptm6oP7KzsWrM+N5acpUpfKDeum9pR1vr\n6DPOSbucXo048Yz9T63pONTa0dZaPaW2asz48HhY//6n1gRBMOOv/yl/PBZRXkWvfmn2e2Mgp9RO\nve6OjrbWIzsbj+xsbN1YXz2l9uDz/3vc+Yt7a/dXjRnf3vhsbk9Tn+cc2dlYPaV2RN0ZMd67KAq8\nPN646/qg98e8ekpt+HxN+OyNCdUakz0Prjiys3Hc+YunXndH57sW/hEMPrgcYJc/al2uoeYTl4w7\nf3H+WW7dWB8+jxM+e+O0m1fH+Frd89CK3J6mqdfdUeDDrXpqbVhA4Wfk4PP/+909zQUi3Fv3Lw1v\nK65X4+Afotg/HDq/8cdUHfPno9iGHDt2LO0aSMJpp522bdu2tKug4uR2N73dsGDo6JnVk+Z3vzSc\nvzp2/iaf7yQmXKwy+OCYvfCIpTJCPQ6RyPjSsvmnNZkVUAq/inrsdgu6PbCFp5U2r1jU1ri5ekpt\n3be39HZOe+OzTSuuLHxO7Aq/PFo31L95/5Jx5y+e9here7uG8K5Nu3l1qQz0DYVPet0DW3r8g5V/\n0oM/DinqPlo1fNA6X0Nud1NRR5mGT8foM84pMA45ytORv3e9vWjzJ8T+dhv8QxTjh0PnN/7KrXu+\ntvAsfz6KSkytFGIqadn3RF3hEyZeuiOZSqhwUcJDZhekSVIpJvkCsw+6h4rBDPmO8ioKFw7tLaqF\nQz2DIIgSHgYfMLrXP8jh7oVfHmFJhd9HqaTrQepX/A5H/wZJfWlSQIzx8s37lrRurO/tESh8abr6\n9eFQ4Eq6vPHDdrU/H0UlplYKMZUUxfJHYmA3avIheYUbUmF4yGYzKzGlnuT77HgZfEdxlFdR+HOU\nbrc+w0OUcyImn8Hf9ygvj3A2YOzpOnX9it8zb1+X/J+83sQVL8NnP3pHcdYMsle2+wPVuV3tz0eR\niKmVQkwldcUe3dT5hkpuyCIJiDJmr7R6eGJX9kl+8B3FEUd+Rul2K3Ba+ESEP8fVfzX4+x7l5RGu\nxxOkNDq0eF9NDmD0bGJ/8grrHC97+3IherxMeHR9dnR/43duV/vzUSRiaqUQU6kQxhjTozKeMhej\nMk7ysXQUx/UqCrvdgr4mMdbe/oOwd26QASOuTvKIL49wHeMkR4cm8NVkipMzYxFXvExlbFTqenzj\nd2lX+/NRDEPTLgAgNrndTUEQdBw8NqJ2Ufd/HQePBR9cjJHKEXFDhaDTxoaVJuLeFbk9TeGZQRDk\ndje11K/a8dWzs/+gdd4lpfMdDHdJCZ/6Axu+X/hKoryKwv0tCj8g4aXTbl6d29PUUr+qacWVzSsW\ntTc+++Z9S3bc9N6Deeq6N0ad8clT170RnrPjprN7PCdKNojnvkd+eYQnhB2JnT9vw1dL7PuCBkGQ\nf0Cqp9ROu3n1e6/SjfXhdwGxCHc3CXp/oIq69dHg1T2wJXzM2xo3N624MlxuetLiZXUP9O8rp/A1\nU/fAlmk3r37vGr69pcvrKnnF/iDy5yMtelMrhd5UysztT+8IgmDFwg/0nb5535LhJ6wfUbtozJkr\nu//Koa3LjzSte+c/LivdIYsMWFwrXpaxwc++G3D/VQIzyWPpKI7+KgqidbtF6ZsafP9VLPe9v2sj\nJTM6tL8dxYNfPSuu0bOpyMg45BglM8enxze+3tQE2JkQKEm3/biHmNq6sX7yNSMK/6Le1MoUNlna\nGje3Nz7bW3h4b7pgSo2M1Bf9Gnf+4rbGzS31qwpksLA5OGnxss57b+QnOuY2NrVurB/MGMKW+lWt\nG+qL0coMIvQEtjVuDh//3q4n+qsoCIIDG77f4zmdu93Cvqlx5y8uEB6inFNAXPe9Xy+PIAjqHtgS\nPrNtjZvbVmwOOqXr6MX3qXNHcefj4YMWnhA+EYN/pVVPrT113RsvL5qRn4VbcpMzq6fWVk+tLacc\nFdcHUWHZ//NRrgz6BUrS3Zedcvdlp3Q+kh+IWFjnIYuUnAEP7srymL337tRNZ7fUr3pvhOdXz37z\nviUJf6XSuSnWW51hUyzcbyMc+1f37S3jLlg88/Z1+VGFLy+aEb3yBIZrBvGN2Yv4Khp9xjlBf0a9\nhskhHDxZ6Kb7OqdHcd336C+PsKWezOjQwkOIwycrLDiuV1pco2cZpNzuprDHPpYPosKy/OejvImp\nQElasqB2yYIPNHfyX2n3yeyRUjT4LBe2XcLwEE5j7nzNxZgyF1EyUa1PEZtioUFOdAySbWVGny/a\n5wsgyqvoQzffnZ9T2tu808SGhsZ13wfWUh9wuo4i+nTZfr3SCn8Xls3JmRUolhnX0WX2z0d5Mze1\nUpibStkLZ49MvmZE4bmpe9ceMXukFA1+O40u15OFMXtZ26c0yuy70OBXA05y85vYl2mN8irKyJqo\nMd73rE3OjD5dNuh9X5/Or7RsbmaW+nSAbEplWfLOb/wl/9+v77v+syU05LsU6U0FykT0ZoSMWlri\n7XaLd8zeIFeYTLhDoE/h7LsCS8tOu3l10M/VgHsTfbhmaDAPdexj9qK8ijLS7Rbjfe/z5ZFkL3HQ\nn47iINmBwXHJyHSADBrAsuSx6PzG/7sPtxnyXWx6UyuF3lQqQbjS79CRZx938aPdL7XSb4kqRrfb\n4Fe8jKXjJa19Sgv3zxToBnzr/qX9Wu61t3P6tQFpuA/nIB/qzj2Bve1T2t+ewFJZNzXe+56RXuKg\nPx3FUV5p4c8ZGdcQimsISfnp77rT8d56+MZ/4e7/e/YXlmb5jV8GxNRKIaZSTnprZOd2N73dsGDo\n6JnVk+Z3/6323z42tGbI2PmbTCIqLdnMcoNvQfYrqsU1BCB6uu4xg8W1r0+/Wpm5PU35UgfZWM/U\nkO+ExXvfMxLRC3+HFV4a9JUw+zUwOL7aC8nadICsycIGY9rVCRBTK4W3E+Whz0b2vifqev/tIAiC\niZfuKG6JxCqDWS6uFmQqHQKDTNdxTXTs1wakQXyN9ez0BCavLO97lI7iIAgivtKS/y6sN0nO3C5F\nsc82HwDt6gSYmwqUjH1P1L3dsGDIyPsmXzMi/2/8f949/IT17a/+eRhQJ166Y+z8TccO/8XetUfy\n/w78y9Rjh/9i7PxNMmqW9Tj5MK7tNPql8BS1uCaUxrj8bBSxTPGNa6Jj9N1Ngljn7mZkvmgqyvK+\nd54u2+OiyuGkwUj76CQ+0bGA/s7crjR2iKkQYipQGsIl4DsOHhtRu6j7v46Dx4I//tnOt8b2L77z\nyPgZ5dEaK28FVgrJYJaLqwXZ340oBymudN15Y4Yom4L2qF+b38TeWC/qLikZV5b3vcCKVtFfaQl/\nF1ZAWusDlRY7xFSCYWkXABBJS/2q4ScEo2Zf1eNmM0EQHGlad2DD9/Pjf6qn1p69ZVZw3F2vXPjJ\nSQnWyQCEo7OCD45EzW1sat1Y37kzpLfBXUXKcp2Ph1mu8wmFW5BtjZvDSa0FbitsQLdurO/8uu2x\nmIHemw/oM113iZ29CfuvXl40o6V+VUuwasATHcMn+r35jd0G9OavLZaHmvIWvj3Hnb+4x+mynV9p\n3QcG519pLfWrCgwMjvG7sD5FHELStmJzeGcTKCmDPvBBVD/wDyKyTG8qUBryf3sK6NLIXrKg9vK5\nU06ePKpoRTFYffZehlPCgqQGd0XpKQ3i63hJrEMg9v6Zwe/r0+fuJqPPOCfIUh8XGddbR3GcA4MT\nyYQJDyEpafFuMEbWiKlACYjYdO7SyL77slN+eN28ohVFDKKMRA2PD3KUaRQRs1wQXwsysY0oY5/i\nG9dExwKtTI11YjT4gcGJTXRMeDpASSvLGdfkGfQLlIAoXamhSh4EVYqijETN7W6KZZRpxGKijLWL\ndxBy3QNbwlVJ2xo3t63YHHRagrXf96EX485f3Na4OfZhjdVTa8MurALnFN7Xp/BwzSDB8d6Ut+gD\ng3sbgp7YtyEJTwcoA1E+iChFYipQAsJGdpQz9auUkIi9l+Hkw0xluSAIYmxB9hnV8gpHvgI6988k\nFvm6bArSUr+qdUN9j3u09tjK1Fgndr3lmaxNdMxUbIa02De1UtjfiZIW7oE2+ZoRI2oX9biE0qGt\ny480rdu79ohVE0rIADYOze1u6jPLDVj0vfiCP4bk3rZq7G0DxsHUVnjH4D4V3omx8KUDM8g9WoMP\n7orZ276psT/UVKysbS2bfwelHpvpTrs6AWJqpfB2otS9ed+S4SesHzry7OMufrT7pWFMfec/LqvY\n7c5LUeuG+jfvXzL6jHMK9F6GUXbazauTGdAVPctFb0EOuAu0s+iRr7eb6xz5ip2u83k+7CXuMV4W\n/nqiM411klTU78L6W0n02BzL5wzRaVcnQEytFN5OlLrc7qa3GxYMHT2zetL87pe2//axoTVDxs7f\npF+lhETvvUwsjUTPclFakIPvAg36E/mi3Fwyka9w2g/HVUbvuc1aHxckqc/YHMvnDP2lXZ0AMbVS\neDtRBvY9UVf4hImX7kimEuKS/EjUKPrVU1qgBTn4Ua9BfyJflJtLJvKFlfTWN5v/LiAcyB1Rdvq4\nIFNi+Zyhv7SrEyCmVgpvJ8pDMRrZb779zrSxw2MqkP5JciRqv6oa5MssxlGvESNfbk9T9JsrauQL\nx3IX/nIh4bHcFaKMh32W8V0bjHhH1yevpJ9W7eoEiKmVwtuJchJjI7vuzs079x1u/Ouz53xoTFzl\nlZO4mhGFryebkw8H8zKLa9Rr9MgXBEFcg2wHaQArYzFIZTzss4zv2uDFO7o+SWXwtGpXJ0BMrRTe\nTtCjCx741cZX9zf910/NPG5E2rVkS/RmROEIGuV6ym/yYVyjXqNHvlhuLhYZXBmr7JXxsM8yvmuD\nV4zR9ckog6dVuzoBYmql8HYC+mXAEx17W7MnyhK15TH5MMZRr9EjX3YG2WZwZawyVurDPgso47sW\nixIdXV82T6t2dQKGpl0AANmS290UZstJi5fVPbCl7ttbxl2weObt6+oe2BKmx5cXzWhvfDYIgh03\nnR22Kqqn1E67efXoM87J7Wlq3Vgfdu5Fv55Q9dTacRcsrvv2lpLOqEEQhCOWx1/wZwXOCe9j54Tf\no7Ct1ta4ufMD1Vlud1MY+2O5uVhUT60dd/7iIAgObPh+jyeEZYTnMEj5B3PS4mWdu9Sqp9ZOWrws\nfN57eyIyrozvWixi/JxJkqeV6MRUAD4gYjOizwjavGJRlOtJ9s4lIQxghZuG4aV9BvKIkS+um4tL\neENh33hud1P+eG53U0v9qrB5XepfRmRE4QczfPH09h1HxpXxXYtFjJ8zSfK0Ep2YCsAHRGlGhOcU\njqD5tWcLXE9ZNkeid4FGGYnXOfJ1vsLOkS/Gm4tF9dTaU9e9kdvT1FK/asdNZzevWNTe+Oyb9y3J\nd7+fuu6NEp11nCmtG957G/b2YFZPrX1vjMOG+uhXG760dnz17BR74Yp018pJvJ8zyfC00i9iKgDv\ni9iMCH8uHEGjXE9ZNkfiHfXaOfI1rbiyx8iXzUG2+a71tsbNTSuubN1YH85Prnsgcwu6lKjYh32+\nF1BvOrulftV73zJ89ew371uS/NdJJTqiNUmlOLre00q/iKkAvC9KMyJs91RPqe0zyuYDbY/KuDkS\n+6jXwpEvm4Nsw671uge2TLt59XvVfntLl+53BiP2YZ+FZ5snqURHtCYsylCLTD0+nlb6xUq/lcKK\nZDAwJb3/+ABEWVp2x1fPzu1pKrx6ZHg9hfdCyOAqlPGKfT/YwoshZ3P7WYonxkWVs7b+qvWioyuh\nN345Pa3a1QnQmwrQs+yMf0tSlPlO4aTTfEuoR+GluT1Nyc+bysLMulDso14LL4ZskG2liXHYZ9bW\nXy3FEa1pKaE3vqeVftGbWil86wP9su+JusInTLx0RzKVJO/N+5a0bqzvbTu+8NLw5z6/EQ+CoPD1\nFN70r7+i7OOavIT3gy2b7WeJInyvVU+pHXfB4t56Qese2NLnQOuwR663M/O3UmBwROziumsVolTe\n+GXztGpXJ0BMrRTeThBdbnfT2w0LOg4eGzX7qiAI/m37H/7jD4fPrRt/8uTRQRC0//axoTVDRp30\nSGb3Hx/kQOXOzYjRZ5yTv5udmxGjzzinrXFz4QganlP4euJtjuQHv4WdCa0b68MO26AUhpDBgA1y\n2Gc4RL/wd0ZpDdEvoRGtRFcGT6t2dQLE1Erh7QTd9Rbn3rxvyfAT1o+oXTTmzJXhke1720+ePCr8\n+dDW5Uea1r3zH5fF2A0Ylxi7Ews3I6J/I55McyRrM+sgST2+8bu8NwsII2jhN0h747NNK65MuEM1\nGPRdI5vK4GnVrk6AmFopvJ2gs8Jx7uVFMyZfM6JzTO0sjKkH/gtO1N4AACAASURBVGVqws21KGLs\nTozSjIgSQZNpjhQeQtxSv6qlflW8A4wpb6W4dtqAh31GWTgt3QXPSmVEK/1S0k+rdnUCxNRK4e0E\neX3OO9279kifMXXv2iOZWqK2SN2JhZsR0SNosZsjGZxZR4nK5gznoiqn9VchGdrVCRiWdgEAiQp3\nlczPO+0inHca8arC0BVncRH01snTeaHOzueHC3WGJxzY8P3+trOrp9aGq8v2dmmYivuMoIWvZ5Ba\nN9QHQVCgezbcx7WtcXP4uBWjBspGfvWvzkMSchubWjfWl2tIC9dfbd1Y39tHhPVXgeSJqUBlaalf\nNfyEYNTsq3rsKQ2C4EhTr8Peuoi9V7DwOMMunTwt9ataN9TnO3kK7+Q+7vzF3fd/j2tMY1EjaBTh\nfR9/wZ8VOGfS4mVtKzan8s0CpaLHIQnjLlicf+u9vGhGuc5wDgP5e983dRuOUfjjBaAYxFSgsrRu\nrJ98zYhYrirGwFM4goYHC3TyTLt5dRC5OzHMtIVvq7SMO39xW+PmlvpVBWbWhfdXO5sCijQkoSRU\nT609dd0bLy+aEU7kLt31V4GyIaYCFSQcHRpRbk9TgUvjHf9WeJxhn508b96/JIjcnZi/X2UzpjGM\nDW2Nm9sbn+1tZl24lJSuVAro75CE8lP3wJbwI6WtcXPbis1Bp9nmaZcGVBwxFagg+Z6BSIa+fmjr\n8u6Hw/mrcfXLRRlnWD2lNuirkyf8b5/diWFGLbMxjWbWMXhmOAf9mW0OUGxD0y4AIDn9Ciodbc1H\nmtZ1/xeusdSlLfv4S3sueOBXv37jYH9L6jzOsPN1hu3FsI2Yz5aF71TYndjjOfnuxD5v68CG7/f3\nLmRBWHzYtu4yBbelfpWZdfQp4gzn4I/v2TIWTjWv+/YWbxkgRWIqUEGi9xNOvHTH2Pmbjh3+i71r\nj+T/HfiXqccO/8XY+ZsmXrqjy/n3bGra+Or+9S/u6W9JfY4zzP9QuJMn/Lm3kNm5YV34tkp0TGM4\nsy63p6mlflXTiiubVyxqb3z2zfuW7Ljp7PC+n7rujVLZNZ5UhK//whHUDGeAxIipQAUJR4cG0ead\nhn2MdQ9smXbz6nAaZ9i90GPa+eF1H7n7slNWLOxjR9YuIo4zDIIgH0R7lG83F+5OjHJbuT1N/ZrB\nmyl1D7zX/9PWuLlpxZWtG+vfe+IesF0qfeg8w7nHE8xwBkjSkGPHjqVdA0mwDTGEcrub3m5YMHT0\nzOpJ87tfGs47HTt/UzI9b80rFrU1bi48HbR1Q/2b9y+pnlJb9+1es1Z4PdNuXh2upRQEQfeFOsNp\ndYVvq73x2aYVVxa+rezL7W4ys44BePO+Ja0b68edv3jaX6zu76VARdGuToAllIDKEubPjrbmI209\nrDbU47zT4omyk0qYM3N7mqIsY5vfbKb7Qp3tjc9WyK4tqe/jSonqvHfo6DPOyb/d7B0KkDwxFag4\nEy/d0WWf0qBTnEtyBmP0nVSCIIiyjG2fC3XatQV684G9QwN7hwKkSUwFKlFG9l2IuJNKOF73vVI7\nBeneOnl67E60awtEYe9QgCwwN7VSGEMP2RTumxq2g7uPMwyjY90DW6qn1r68aEZ40YA7eaLfVrz3\nEUqOGc5AAdrVCRBTK4W3E2RZlAga10DlwcddAKhk2tUJEFMrhbcTZFn0CDr4Tp6MzMsFgBKlXZ0A\nMbVSeDtB6va3v3vPpqa/WlB73Kie1wVIcpyhMY0AMDDa1QmwhBJAQm7/8Y7Vm5r2H3737stO6fGE\nJHdSsWsLAJBZYipAQr708em/fv3tL318etqFAABk2tC0CwCoCLndTbU//84/vfjV2p9/J+1aAAAy\nTW8qwGC9t3/phvoe53l2WbKopX5V64b6UWd8cvwFf9bj/qUAABXOEkqVwlRvKIYeV83tEkHzG8BU\nT6mdtHhZ68b6tsbN4REbwABAydGuToCYWim8nSB2+56oK3zC2Pmbdtx0dhAEkxYv67zdS+dwW3v7\nD/SpAkAJ0a5OgEG/AAOR290UBEHHwWOjZl/V/dL23z42tGZI84pFQRCMO7/rSODqqbXhkZb6VQc2\nfL/91T8vfFsTL90RW90AAJknpgIMREv9quEnBKNmXzXmzJU9nnCkaV1uT1MQBL3tSjru/MUt9ava\nG58dfkIwdPTM6knzu5+Ta2noaGuOsWwAgOwTUwEGonVj/eRrRvR5Wuexvl1UT60dfcY5bY2bg2BE\n9aT5PcbdQ1uXH2lbN9haAQBKig1pAPqtdUN9xDNHn3FOgUt77GhteO1A6+GjAykLAKAsiKkA/da6\nMWpM7bwIcJRLV29q+uS9/+dL3/v3AVYGAFD6xFSAfht3/uKIZ+b2NLU3PtvzRbub8jvT5H30+Jog\nCC6bO3kw5QEAlDQxFaDf+rWFzIEN3+/xeNiV2iXxnn/ShGOrLrz249MHUx4AQEkTUwH6rXpqbRgv\nw7V8exNOTG1vfDZc0Td/PLe7qaV+VThyuLd1gAEAKpaVfgEGYtLiZW83rA+Gvn5o6/Lul4b7pn7o\n5rurp9a+vGhGS/2qlmDV6DPOmbR42YEN389PbT113RvJVg0AUALEVICBCLeZ6Whr7nHDmKE1Q/Ln\n1D2wpXVjfUv9qrbGzW0rNgdBUD2ldtwFizsP9821NPQYd3MtDUWqHwAgs8RUgAGaeOmO3O6mMILm\nD+YjaH671OqptZMWLxt3/uJw9O+4CxZ3H+jbW9wFAKhAQ44dO5Z2DSThtNNO27ZtW9pVQHnK7W4q\nEEEBgHKiXZ0AvakAg1U9tbZ6au24C6LuUgMAQAFW+gUAACBDxFSAEtPw2oEhy/7tigdfTLsQAICi\nEFMBAADIEHNTAUrM/Fnjj626MO0qAACKRW8qAAAAGSKmAgAAkCEG/QIU0b4n6gqfMPHSHclUAgBQ\nKsRUgOIaOnpm9aT53Y/nWho62pqTrwcAIOPEVIDiqp40f8yZK7sfP7R1+ZG2dcnXAwCQceamAgAA\nkCFiKgAAABli0C9ACejvUkyth4/+yaotl8+dcvdlpxSzLgCA+ImpAKWhX0sxNR84vHPf4Yee3yWm\nAgAlR0wFKA39WoppzofGPHvLn84cPzKR0gAA4iSmApSn+bPGp10CAMBAiKkAxZVraTi0dXmPx+O9\nof7OXwUAyCYxFaC4OtqaE9sftV/zVwEAsklMBSiihDsw+zV/FQAgm8RUgApiYDAAkH1iKkBlMTAY\nAMg4MRWgNMS1FJOBwQBAxompAKWhGEsxXffob/9kRs2SBbXxXi0AwGCIqQAloBhTRrfvbX/o+V1B\nEIipAECmiKkAFerkyaPuvuyUEyeOTLsQAIAPEFMBykd/56/qRwUAMkhMBSgfxZi/CgCQMDEVoEzY\n8hQAKA9iKkBliWtjGwCAIhFTASqLgcEAQMaJqQAVxMBgACD7xFQABmLfE3WFTxCJAYCBEVMBKKTh\ntQNBEMyfNb77RUNHz6yeNL/78VxLQ0dbc9ErAwDKlJgKQCGfvPf/BEFwbNWF3S+qnjR/zJkrux8/\ntHW56a8AwICJqQAUcvncKWmXAABUFjEVgEJ+eN28tEsAACrL0LQLAAAAgPeJqQAAAGSImAoAAECG\niKkAAABkiCWUABigXEvDoa3LezyefDEAQNkQUwEYoI625sL7o+57oq7wNUy8dEesFQEA5UBMBWAg\nOifMg0eOHnzn6LSxw7ufNnT0zOpJ87sfz7U0dLQ1F7E+AKBkiakADFbt3/5if/u7u247t3tSrZ40\nf8yZK7v/yqGtywv3xAIAFcsSSgAM1okTRh43aljN8Kq0CwEAyoHeVAAGa+uyT6RdAgBQPsRUALpK\ncukjyywBAF2IqQD0IMmljyyzBAB0JqYC0IMklz6yzBIA0JkllAAAAMgQvakAFFGupeHQ1uU9Hk++\nGACgJIipABRRR1uzgbsAQL+IqQAUi0V6AYABMDcVgCRc8eCLE76xqXn/kbQLAQCyTkwFIAm/fuPt\n/e3vth55N+1CAICsM+gXgB7EvvTRi7ee/ebb75w8eVQCtwUAlDQxtQS88MILX/nKV/71X/919OjR\n+YPPP//8mjVrXn311YkTJ5577rlf+cpXRowYkWKRQJmJfemjmhFVJ4/oIaMW47YAgJImpmbdW2+9\n9c1vfnPfvn3Hjh3LH/zZz3524403fvzjH7/22mt37dq1Zs2a3/zmN2vWrBkyZEiKpQJlI8mljyyz\nBAB0IaZm12OPPbZ27drt27d3dHR0ueiuu+6aM2fOww8/PHTo0CAITj/99FtvvXXz5s2f+tSn0qgU\nAAAgNpZQyq558+Z9+ctfvuuuuxYtWtT5+N69e19++eXPfe5zYUYNguCSSy4ZPnz4z3/+8zTKBAAA\niJPe1Ow6/fTTTz/99CAIDh8+vG7d+7O2mpqagiA48cQT80eqq6uPP/74119/PfEaAQAAYqY3tfS0\ntbUFQTBmzJjOB0ePHn3o0KGUKgIAAIiN3tT0vf32208++WT+f6dMmfLpT3+6wPnhWkqdV1QK/7fL\nEQAAgFIkpsbmwIEDV1xxxfXXX/+FL3yh+6XPPPPMk08+uX379pqamlNPPfWGG26YNWtWeNG+ffvu\nuuuu/JlnnXVW4Zg6atSoIAja29s7H2xvb58+fXoMdwMgPQ89v+vh53etv/5Pxo2sSrsWACA1Ymps\n1qxZ8/rrr/c48vbOO+9cu3btiBEj5syZ09raWl9fv379+vvvv//cc88NgmDWrFlbt26NfkNhHN25\nc2f+yNGjR5ubmy3zC5S6h5/ftfHV/ete2H39J3zvBgCVS0wdlKNHjzY1Nb322muPP/74j370ox7P\n2bJly9q1a0844YSHHnro+OOPD4Lg6aefXrp06de//vWnn3565MiR/b3RGTNmTJ8+/Sc/+cm1114b\nHtm4cWMul/vYxz42iLsCkL4Hr57z+Et7ZFQAqHCWUBqU3//+9wsXLvzyl7/cW0YNguAHP/hBEAS3\n3nprmFGDIFi4cOEll1zy5ptv/uIXvxjY7V5zzTW//OUv77zzzpdffvnJJ5/85je/WVtbe8EFFwzs\n2gAy4sSJI5csqE27CgAgZXpTB2X69On33ntv+PMLL7ywZs2a7uds2bKlqqrqvPPO63zwoosueuqp\npxoaGi666KIB3O6XvvSlAwcOPPjgg2vXrg2CYN68ed/61rf67Jg97bTTCly6bdu2AVQCAABlpnCz\nmQSIqYNSU1OzcOHC8Odhw3p4MHO53O7du2fOnNklQ5500klBEDQ3N0e5lauuuuqqq67qfKSqqmrp\n0qW33HLL66+/PmHChLFjx0a5HkEUAAD6VLjZLMQmQEwtrtbW1o6OjvHjx3c5Hh45cODAYK68qqrq\nhBNOGMw1AAAAZI25qcWVy+WCnjpawyPhpQAAAOTpTS2uqqqqoKc4Gh4JLwVg8PY9UVf4hImX7kim\nEgBgkMTU4ho1alQQBO3t7V2Oh0fCSwGIxdDRM6snze9+PNfS0NEWaS0AACALxNTiqqmpGTt2bFNT\n09GjRzv3ne7cuTMIgunT7Q0I0D+/fuPg+hf3/NWC2uNGdf0TVj1p/pgzV3b/lUNblx9pW5dIdQBA\nDMxNLbrZs2e/8847jY2NnQ/+6le/CoJgzpw5KRUFUKqWPv7ybT/ecc+mprQLAQCKRW9q0V188cXP\nPffcypUr//mf/zk8smvXrkceeaSqqurCCy9MtzaAktB53ukPZgbBzCA4Eux74v0TzDsFgHIiphbd\n1Vdf/eijjz733HOf//znP/OZz7z11ltPPfVUW1vbjTfeOGPGjLSrAygN5p0CQOUQU4uuurr64Ycf\nvuOOO5555plw6O+YMWOWLVt2ww03pF0aQMkw7xQAKoeYGpuLLrpo27ZtPV40efLke++9N5fL7dy5\nc8yYMdOnTx8yZEjC5QEAAJQEMTU51dXVp5xyStpVAAAAZJqYCkCZyLU0HNq6vMfjyRcDAAyYmApA\nmehoazZPFQDKgJgKQDmwJw0AlI2haRcAAPFreO1A2iUAAAOkNxWAEtCveafXPfrbh57ftWRB7d2X\nWbgOAEqPmApACejXvNPL5k5+6Pld5510XFFLAgCKREwFIOv6O+/08rlTjq26sEjFAADFZm4qAAAA\nGaI3FQDet++JusInWFIYAIpNTAWADxg6emb1pPndj+daGjrampOvBwAqjZgKAB9QPWn+mDNXdj9+\naOvy6Ms4AQADZm4qAAAAGSKmAgAAkCFiKgAAABkipgJQiR5/ac+QZf+2dP0raRcCAHQlpgJQifa3\nv5t2CQBAz6z0C0Aluvbj06/9+PS0qwAAeiCmAsAH5FoaDm1d3uPx5IsBgAokpgLAB3S0NdsfFQBS\nJKYCwPsmXroj7RIAoNJZQgkAAIAMEVMBAADIEDEVAACADBFTAQAAyBBLKAFQKfY9UVf4hC7rJzXv\nPzLvH7Ys+b9qVyzs4xcHf1sAQJ6YCkAFGTp6ZvWk+d2P51oaOtqauxxs+I8D+9vf3fjqH1YE/Y6p\n/b0tACBPTAWgglRPmj/mzJXdjx/aurz7XqmLPjL12Vv+dM6HahK4LQAgT0wFgF7NnzU+7RIAoOJY\nQgkAAIAMEVMBAADIEDEVAACADBFTAQAAyBAxFQAAgAyx0i8AFSTX0nBo6/Iej5f0bQFAORFTAagg\nHW3Nse9Zet2jv71s7uTL505J4LYAoBKIqQBUiomX7oj9OhteO/DQ87seen7XsVUXFvu2AKBCiKkA\n0D/7nqjL/3xqELScFx58/wQZFQAGQ0wFgH4bOnpm9aT53Y/nWho62pqTrwcAyomYCgD9Vj1p/pgz\nV3Y/fmjrcvNRAWCQbEgDAABAhoipAAAAZIiYCgAAQIaIqQAAAGSImAoAAECGiKkAUHQ/3rav4bUD\naVcBAKXBhjQA0G+5loZDW5f3eLz7we172xf+46+DIDi26sKiVwYApU9MBYB+62hrjr4/6smTR10+\nd8qJE0cWtSQAKBtiKgD0z8RLd/T3V3543bxiVAIAZcncVAAAADJETAUAACBDxFQAAAAyREwFAAAg\nQyyhBACZtu+JusInDGBJJwDIMjEVALJu6OiZ1ZPmdz+ea2noaGtOvh4AKCoxFQAy4eCRo0EQ1Iyo\n6n5R9aT5Y85c2f34oa3Lo2/fCgClQkwFgEwY+7WfBkFwbNWFaRcCACkTUwEgE44b5Y8yAASBmAoA\nGfGHby5IuwQAyAQb0gAAAJAhelMBIDU2mwGA7sRUAEiTzWYAoAsxFQDSFGWzmVxLw6Gty7ufk2tp\nKG5xAJAGMRUAsq6jrdn+qABUDjEVADLN9FQAKo2VfgEAAMgQMRUASsaZq56b8I1NB48cTbsQACgi\nMRUASsbOPxze3/7uwXfEVADKmbmpAFAymv7rpw6+c3Ta2OFpFwIARSSmAkCa+rXZTM2IqpoRVcUv\nCgDSJKYCQJpsNgMAXYipAJAam80AQHeWUAIAACBDxFQAAAAyREwFAAAgQ8RUAAAAMkRMBYCysnpT\n0xUPvth6+GjahQDAAFnpFwDKyu0/3rG//d1/f+vg/Fnj064FAAZCTAWAsrLhprM2bv+DjApA6RJT\nAaCsfHRGzUdn1KRdBQAMnLmpAAAAZIiYCgAAQIaIqQAAAGSImAoAAECGiKkAAABkiJgKAABAhoip\nAFBxfv3Gwduf3pF2FQDQM/umAkDFWfr4yxtf3V87YeT1n5iedi0A0JWYCgAV568W1B43qnrRR6am\nXQgA9EBMBYCSt++JusInTLx0R+dzFgTBgknBuz8O9n3wnOJUBwD9I6YCQDkYOnpm9aT53Y/nWho6\n2pqjnwMAqRNTAaAcVE+aP+bMld2PH9q6/EjbuujnAEDqrPQLAABAhoipAAAAZIiYCgAAQIaIqQAA\nAGSImAoA9KzhtQNplwBAJRJTAYAeXPHgi5+89/+s3tSUdiEAVBwb0gBAOci1NBzaurzH4/06J++8\nk457/KU9Hz2+JsYiASAKMRUAykFHW3Ofe59GOSdvyYLaJQtqB11Xr/Y9UVf4hImX7ijerQOQZWIq\nAJS8KIkug6lv6OiZ1ZPmdz+ea2noaGtOvh4AMkJMBQDSUT1p/pgzV3Y/fmjr8ui9vgCUH0soAQAA\nkCFiKgAAABkipgIAAJAhYioAAAAZIqYCAACQIVb6BQAGaPWmpqXrX7n7slO67LBqT1QABkNMBQAG\n6ED7u/n/dhFlT9RcS8Ohrct7PCfWMgEoMWIqAPCe/vaCrlhYt2Jhz78SZU/UjrZm+6MC0J2YCgC8\nL0ovaCyM+wWgN2IqAPC+KL2gAFBUYioAkFGWYgKoTGIqAJBdiQ1CBiA7xFQAoB8S7uE0CBmgAomp\nAED/6OEEoKjEVACgf6L0cNoTFYABE1MBgPjZExWAARNTAYD3xdIL2nl66upNTUvXv3L53Ck/vG5e\nDPUBUAHEVADgfbH3gi5ZUDt/1rg5H6qJ8ToBKG9iKgDwniJtQzp/1vhiXC0A5UpMBQCyy1JMABVI\nTAUAsstSTAAVSEwFAPonsR7OIg1CBiDjxFQAoH/0cAJQVGIqANAPejgBKLahaRcAAAAA7xNTAYCU\ntR4+et2jv338pT1pFwJAJoipAEDKfvxyy0PP77pnU1PahQCQCeamAgApu/jUSbddXHfZvClpFwJA\nJoipAEDKxo2sWrGwLu0qAMgKg34BAADIEDEVAACADBFTAQAAyBAxFQAAgAwRUwEAAMgQMRUAAIAM\nEVMBgNKw7oXdzfuPpF0FAEUnpgIAJaDhtQNXPfzSvH/YknYhABTdsLQLAADo25wP1Zx/0nHnnzQh\n7UIAKDoxFQAoAeNGVm246ay0qwAgCQb9AgAAkCF6UwGA8rfvibrCJ0y8dEcylQDQJzEVAKgIQ0fP\nrJ40v/vxXEtDR1tz8vUA0BsxFQCoCNWT5o85c2X344e2Lj/Sti75egDojbmpAAAAZIje1BLwwgsv\nfOUrX/nXf/3X0aNHh0fefffdm2++uaOjI3/O0KFDv/Od76RUIACkxqRTgPIjpmbdW2+99c1vfnPf\nvn3Hjh3LH3z11Vc3btz453/+5yNHjgyPDB2qYxyACtV50uk7RzuGV733N9GkU4ASJaZm12OPPbZ2\n7drt27d37jUN/e53vxs7duzf/M3fDBkyJJXaACA78pNO33z7nem3/fy4UcP+8M0FgUmnACVLTM2u\nefPmffnLXw6CoKGhYd26D/yV/d3vfjd79mwZFQA6qxleddyoYSdOGJl2IQAMipiaXaeffvrpp58e\nBMHhw4e7x9QJEyYsWbJk69at48aNO++8826++eZRo0alVCkAZELNiKqwHxWAkiamlqTf/e53hw4d\nuvrqq2+55ZYXX3zxn/7pn375y18+8sgjZqgCQG9yLQ2Hti7v8XjyxQBQgJiavrfffvvJJ5/M/++U\nKVM+/elPFzj/6NGjX/3qV88666y5c+cGQXDllVfOmzfva1/72o9//ONLLrmk6OUCQGnqaGs2VRWg\nJIipsTlw4MAVV1xx/fXXf+ELX+h+6TPPPPPkk09u3769pqbm1FNPveGGG2bNmhVetG/fvrvuuit/\n5llnnVU4plZVVV1zzTWdj1x++eW33377iy++KKYCQI9sSwNQQsTU2KxZs+b1118/dOhQ94vuvPPO\ntWvXjhgxYs6cOa2trfX19evXr7///vvPPffcIAhmzZq1devW6De0d+/e3//+93/6p39aVVUVHqmq\nqho2bFiPNw0AAFBaxNRBOXr0aFNT02uvvfb444//6Ec/6vGcLVu2rF279oQTTnjooYeOP/74IAie\nfvrppUuXfv3rX3/66afzG59Gt3Pnzi9+8Yvf/e53w5QbBMFvfvObQ4cOzZ49ezD3BQBKlEmnAGVG\nTB2U3//+95/97GcLn/ODH/wgCIJbb701zKhBECxcuPCSSy556qmnfvGLX1x00UX9vdGPfOQjp556\n6te//vVvfOMbc+fOfeWVV/72b/921qxZV1xxxQDuAgCUOpNOAcqMmDoo06dPv/fee8OfX3jhhTVr\n1nQ/Z8uWLVVVVeedd17ngxdddNFTTz3V0NAwgJg6fPjw73znO3fcccdf/uVfHjt2rLq6esGCBbff\nfvvw4cML/+Jpp51W4NJt27b1txIASJ1Jp0DsCjebSYCYOig1NTULFy4Mfx42rIcHM5fL7d69e+bM\nmV0G95500klBEDQ3N0e5lauuuuqqq67qfGTGjBn/43/8j/b29t27dx9//PE93nR3gigABEFQd+fm\n40YO27rsE2kXAmRU4WazEJsAMbW4WltbOzo6xo8f3+V4eOTAgQODufJRo0bllwsGACLaue9w2iUA\nUIiYWly5XC7oqaM1PBJeCgAkaddt56ZdAgCFiKnFFe4Z0z2OhkfyO8oAAImZNraP1RwASNfQtAso\nc6NGjQqCoL29vcvx8Eh4KQAAAHlianHV1NSMHTu2qanp6NGjnY/v3LkzCILp06enUxYAAEBWialF\nN3v27HfeeaexsbHzwV/96ldBEMyZMyelogAAADLK3NSiu/jii5977rmVK1f+8z//c3hk165djzzy\nSFVV1YUXXphubQBA3r4n6gqfYI9WgGSIqUV39dVXP/roo88999znP//5z3zmM2+99dZTTz3V1tZ2\n4403zpgxI+3qAID3DR09s3rS/O7Hcy0NHW2RdjsHYPDE1KKrrq5++OGH77jjjmeeeSYc+jtmzJhl\ny5bdcMMNaZcGAHxA9aT5Y85c2f34oa3Lj7StS74egMokpsbmoosu2rZtW48XTZ48+d57783lcjt3\n7hwzZsz06dOHDBmScHkAQHRL17+yc9/hH143L+1CACqRmJqc6urqU045Je0qAIC+rd7UFATB9r3t\nJ0+2exxA0sRUAICuNtx05s59h7tnVMssASRATAUA6Or8kyYEJ/V8kWWWAIpNTAUA6AfLLAEU29C0\nCwAAAID36U0FAHhPrqXh0NblPR5PvhiAiiWmAgC8p6OtrJlJFQAAIABJREFU2cBdgNSJqQAAQWCR\nXoDMMDcVAACADNGbCgAwEBtf/cOvXz+4ZEHtAH43rv1X7eMKlCUxFQCgH/LLLDU+v6s6CF4fNvm4\nUdVB/5dZimv/Vfu4AuVHTAUA6If8Mkv/ZVoQBEGwNzgyoOuJa/9V+7gC5UdMBQCIKsoYWgNxAQZJ\nTAUAiJmBuACDIaYCAMTMQFyAwbAhDQAAABkipgIAAJAhYioAAAAZYm4qAEAK8vuvdj+eyvUAZIeY\nCgBQLAePHN3e0vbRGWO7X5Tff3WQ4roegOwQUwEAiuVz3/3Nxlf3P/aluYs+MrXz8bi2TrUFK1CW\nxFQAgJjlB+L+zczWPx/ZdsGRqYe2Dg0MxAWIRkwFAIhZfiDunwTBn0wOgl3BkbRLAighYioAQJzi\nGoi774m6ZG4IIGvEVACAjBo6emb1pPndj+daGjrampOvByAZYioAQEZVT5o/5syV3Y8f2ro8+uq+\nemWBkiOmAgCUOb2yQGkRUwEAylwsvbIJ0wkMlUxMBQAgi3QCQ8USUwEASFTEntJS7AQGYiGmAgCk\naen6V1ZvavrhdfMunzsl7VqSo6cUKEBMBQBI0/72d/P/rRx6SoECxFQAgDQ9ePXsB6+e3eNFuZaG\nQ1uX93i8yEUNnKWPgMETUwEAMqqjrTk7XYvR86cBvcnwjQBlTEwFAMiiGDNGXL2yEfNnxQ7oTT43\n+kaAciWmAgCUubh6ZSs2f0aXcG70jFCuxFQAgHJWuiM/E5uaG2MvaCnmRoOHySAxFQCALIqlEzhi\nBqvw0bMVfvfJIDEVAKDSJd+f1mdPaZRbjDGClmIvaIwq/O6TQWIqAABJ96fFNV22LCOoUbggpgIA\nZN3jL+1Zuv6Vuy875fK5U4p0E0lmuRhTVmJlJxwd4/rWQOKlRImpAABZt/6lvTv3HV7/0t7ixdQY\nJbb0UcKS7HCOMX6bd0opElMBALLuwatnf+WTM+bPGp9uGRHzZ1wDerMmlugYb/dmlGek5MY8QyCm\nAgCUhNQzahAtf1b4INIouTHG7s1y/UYAxFQAAPpW4fkzoii5Ma7uzaw9I1E6is2VJSIxFQCAShfL\nfNrSjVhxTSeO0lFsrixRiKkAAJSzKBmswkfP9nn3I/aCRukoLsW5sl3u/rMrux4p3W8oMktMBQCg\nVJfnjSWCVnjGiHj3K7wXtPPd/18//F+fv+Lz4c8VcveTJ6YCAFCq3YnlGkEz+K1BKfaCxqjz3f/m\n1U988Y73fq6Qu588MRUAoNKVaJZLuOwko2OM3xpkMPFCn8RUAIBysL/93ese/e2KhXUfnVGTdi3l\nKbEO53jjd4n2k1PhxFQAgHJwz6amx1/as789t+Gms9KupQzFGB2T7N4s0X5yEFMBAMrBXy2oDYLg\nsnlT0i6EPujezA77uGaWmAoAUA6OGzVsxcI+2tykrrxjT5SO4lg6k2OMlxW+gnFmiakAAEAMonQU\nx9WZ3Ge8jHGvV5InpgIAAH3osxc0SgdmlHNijJcx9pR2vvvf+LN38z9bMLlIxFQAAKAPSU6pjSte\nxthT2vnuf+ZjwZEmHa3FJaYCAACFJDylNmsDcbvc/dNOO23btm3Jl1FRhqZdAAAAALxPTAUAACBD\nxFQAgAqy7oXdzfuPpF0FQCHmpgIAVIp1L+y+6uGXzj/puA03nZV2LZAJsezjSuzEVACASjH/hPHn\nn3Tclz4+Pe1CYLDiipdJrmBMdGIqAEClmHncCP2olIco8TKWvV5JhZgKAABkSFzxUk9p6RJTAQCA\nDIklXuopLWliKgAAkBXiJYENaQAAAMgUMRUAAIAMEVMBAADIEDEVAIAPePPtd9IuAahoYioAAO/7\n97cOTb/t53V3bk67EKByiakAALxv3IhhQRCcOGFk2oUAlcuGNAAAvG/mcSOOrbow7SqAiqY3FQAA\ngAwRUwEAAMgQMRUAAIAMEVMBAADIEDEVAACADBFTAQAAyBAxFQAAgAwRUwEA6J+DR45O+MamM1c9\nl3YhQHkalnYBAACUmIPvHN3f/u7O4HDahQDlSUwFAKB/po0dvuu2c2uGV6VdCFCexFQAAPpt2tjh\naZcAlC1zUwEAAMgQMRUAAIAMEVMBAADIEDEVAACADBFTAQAAyBAxFQAAgAwRUwEAKIql619Zuv6V\ntKsASo+YCgBAUaze1LR6U1PaVQClZ1jaBQAAUJ5+eN28tEsASpKYCgBAUVw+d0raJQAlyaBfAAAA\nMkRMBQAAIEPEVAAAADJETAUAACBDxFQAAAAyREwFAAAgQ8RUAABS8/hLe1Zvakq7CiBb7JsKAEBq\nrnjwxSAIPjtn8smTR6VdC5AVYioAAKlZsqB2f/u7MirQmZgKAEBq7r7slLRLADLH3FQAAAAyREwF\nAAAgQ8RUAAAAMkRMBQAAIEPEVAAAADJETAUAACBDxFQAADLt4JGj2/e2p10FkBwxFQCATKv921+c\n8q1n//2tQ2kXAiRETAUAINMunzvluFHDxo0YlnYhQEK82wEAyLQHr579YDA77SqA5OhNBQAAIEPE\nVAAAADJETAUAACBDxFQAAAAyREwFAAAgQ8RUAAAAMkRMBQAAIEPEVAAASt4VD744ZNm/Nbx2IO1C\ngBiIqQAAAGTIsLQLAACAwfrhdfPSLgGIjd5UAAAAMkRMBQAAIEPEVAAAADJETAUAACBDxFQAAAAy\nREwFAAAgQ8RUAAAAMkRMBQCgIjz0/K4J39j06zcOpl0I0AcxFQCAirD+pb3729/duP0PaRcC9GFY\n2gUAAEASHv4vcxpeO3DxaRPTLgTog95UAAAqwriRVTIqlAQxFQAAgAwRUwEAAMgQMRUAAIAMEVMB\nAADIEDEVAACADBFTAQAAyBAxFQAA3vPrNw5e9+hvf/3GwbQLgYompgIAwHvu2dT00PO77tnUlHYh\nUNGGpV0AAABkxV8tqM3/F0iLmAoAAO/56IyaB6+enXYVUOkM+gUAACBD9KZm2k9/+tPvfe9727dv\nnzRp0n/6T//pmmuuGT58eHjR888/v2bNmldffXXixInnnnvuV77ylREjRqRbLQAAwODpTc2u733v\ne1/+8pdHjRp1ww03zJs375577rnllluOHTsWBMHPfvazL37xi21tbddee+3HP/7xNWvW3HTTTeFF\nAAAAJU1vaka1t7f/wz/8w5VXXvl3f/d34ZGTTjrptttua2xsnDt37l133TVnzpyHH3546NChQRCc\nfvrpt9566+bNmz/1qU+lWjUAAMBg6U3NqObm5oMHD372s5/NHznrrLOCINi5c+fevXtffvnlz33u\nc2FGDYLgkksuGT58+M9//vN0agUAAIiP3tSMOv7449etW3fyySfnj7z44otBENTV1TU1NQVBcOKJ\nJ+Yvqq6uPv74419//fXEywQAAIiZ3tSMGj169Lx580aNGhX+7yuvvPLf//t/P/vss88444y2trYg\nCMaMGdPl/EOHDqVQKABA5Vn3wu7Ww0fTrgLKlt7U9L399ttPPvlk/n+nTJny6U9/Ov+/x44de+yx\nx/7bf/tvH/7wh++5557wSP6/nU+zhBIAQAJWb2pauv6Vy+dO+eF189KuBcqTmBqbAwcOXHHFFddf\nf/0XvvCF7pc+88wzTz755Pbt22tqak499dQbbrhh1qxZ4UX79u2766678meeddZZ+Zj6xhtvfO1r\nX/vlL39544033nTTTdXV1UEQhF2s7e3tna+/vb19+vTpRbprAADknX/yhI/OqLls7uS0C4GyJabG\nZs2aNa+//nqPI2/vvPPOtWvXjhgxYs6cOa2trfX19evXr7///vvPPffcIAhmzZq1devW7r/1yiuv\nXHvttTNmzHjiiSc+/OEP54+HcXTnzp35I0ePHm1ubrbMLwBAAj46o2brsk+kXQWUM3NTB+Xo0aM7\nd+786U9/unTp0n/8x3/s8ZwtW7asXbv2hBNO+Jd/+ZdHH330Rz/60T333JPL5b7+9a8fPny4t2vu\n6Oi49dZbTz755P/5P/9n54waBMGMGTOmT5/+k5/8JH9k48aNuVzuYx/7WFz3CwAAIC16Uwfl97//\nfec9Y3r0gx/8IAiCW2+99fjjjw+PLFy48JJLLnnqqad+8YtfXHTRRT3+1rPPPvu73/3ur//6r597\n7rnOx+fOnXvcccddc801f//3f3/nnXdeddVVL7/88qpVq2pray+44II47hMAAECaxNRBmT59+r33\n3hv+/MILL6xZs6b7OVu2bKmqqjrvvPM6H7zoooueeuqphoaG3mLqb37zmyAIOs9ZDX33u98999xz\nv/SlLx04cODBBx9cu3ZtEATz5s371re+NXLkyMLVnnbaaQUu3bZtW+FfBwCASlC42UwCxNRBqamp\nWbhwYfjzsGE9PJi5XG737t0zZ87skiFPOumkIAiam5t7u+abbrrppptu6u3SqqqqpUuX3nLLLa+/\n/vqECRPGjh0bpVpBFAAA+lS42SzEJkBMLa7W1taOjo7x48d3OR4eOXDgwGCuvKqq6oQTThjMNQAA\nAGSNJZSKK5fLBT11tIZHwksBAADIE1OLq6qqKugpjoZHwksBACg/ew/pkIABElOLa9SoUUEQtLe3\ndzkeHgkvBQCgzDS8dmDK3/zsigdfTLsQKElianHV1NSMHTu2qanp6NGjnY/v3LkzCILp06enUxYA\nAMV34sQ+NmIAeiSmFt3s2bPfeeedxsbGzgd/9atfBUEwZ86clIoCAKCI5s8af2zVhXdfdkrahUBJ\nElOL7uKLLw6CYOXKlfkju3bteuSRR6qqqi688ML06gIAAMgiG9IU3dVXX/3oo48+99xzn//85z/z\nmc+89dZbTz31VFtb24033jhjxoy0qwMAAMgWMbXoqqurH3744TvuuOOZZ54Jh/6OGTNm2bJlN9xw\nQ9qlAQAAZM6QY8eOpV1Dpcjlcjt37hwzZsz06dOHDBmS8K2fdtpp27ZtS/hGAQCgzGhXJ0BvanKq\nq6tPOcU0egAAgEIsoQQAAECGiKkAAABkiJgKAAApaN5/ZMI3Nl3x4ItpFwKZI6YCAEAKWo+8u7/9\n3V+/8XbahUDmWEIJAABSMOdDY175fz85bezwtAuBzBFTAQAgHSdPHpV2CZBFBv0CAACQIWIqAAAA\nGSKmAgAAkCFiKgAAABkipgIAAJAhYioAAAAZIqYCAEB2XfHgi0vXv5J2FZAoMRUAADJq+972x1/a\ns3pTU9qFQKKGpV0AAADQs5Mnj3rw6tnHjdJop7J4xQMAQHZd+/HpaZcASTPoFwAAgAwRUwEAAMgQ\nMRUAAIAMEVMBAADIEDEVAACADBFTAQAAyBAxFQAAStvjL+15/KU9aVcBsbFvKgAAlLYrHnwxCIJj\nqy5MuxCIh5gKAAClbcmC2rRLgDiJqQAAUNruvuyUtEuAOJmbCgAAQIaIqQAAAGSImAoAAECGiKkA\nAABkiJgKAABAhoipAAAAZIiYCgAA5e/Nt9/Zvrc97SogEjEVAADK3+y/bzjlW8+++fY7aRcCfRuW\ndgEAAEDRnThh5P5R704bOzztQqBvYioAAJS/rcs+kXYJEJVBvwAAAGSImAoAAECGiKkAAABkiJgK\nAP9/e3ceXVV99wt4Q0IgJoCA0QYoqNAgEKgMvWpVvAIKDhWRtkurUjEOEAeQglpBr8NSccBX2sWq\nUrDocqAoFBBQERAcGBThIoNIEKKESVEhIAFCcu4fu83KCujrfXOSswnP84eL89s/wpePLpPP2fvs\nDQBEiJoKAABAhKipAAAARIiaCgAAQISoqQAAQBAEQafRH9b60/yCXQcSPQjHOjUVAAAIgiDYtf9Q\nEASFBw4lehCOdcmJHgAAAIiETSN+ve9g6XEpTmWRYP4TBAAA/k1HJQr8VwgAAECEqKkAAABEiJoK\nAABAhKipAAAARIiaCgAAQISoqQAAAESImgoAAPxUT7+7+ZSHFxXsOpDoQajJ1FQAAOCnmr766/xv\n9y/5cneiB6EmS070AAAAwFFj+vW/nLP+m992PDHRg1CTOZsKAAD8VA3qJemoVDU1FQAAgAhRUwEA\nAIgQNRUAAIAIUVMBAACIEDUVAACACFFTAQAAiBA1FQAAiKf/u3XvHdPzdhUdSvQgHK2SEz0AAABQ\no9wxbf2Cz3cdXy/5//Q6JdGzcFRSUwEAgHj6r8uznv9o2+BuP0/0IByt1FQAACCeTm+afnqfXyR6\nCo5iPpsKAABAhKipAAAARIiaCgAAQISoqQAAAESImgoAAECEqKkAAABEiJoKAABUt8L9Ja998lXh\n/pJED0IUqakAAEB1Gzxt/e+eXz142vpED0IUqakAAEB165N9wsmN6/XJPiHRgxBFyYkeAAAAOOZc\nnp1xeXZGoqcgopxNBQAAIELUVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAAImrf\nwdJEj0ACqKkAAEAUTfxoW9qfFwyY9GmiB6G6qakAAEAUHZ+aXPZPjin+lQMAAFF0eXZGbHT3RE9B\nAjibCgAAQISoqQAAAESImgoAAECEqKkAAABEiJoKAABAhKipAAAARIiaCgAAQISoqQAAwNGqYNeB\nRiPfvWN6XqIHIZ7UVAAA4Gi15Mvdu4oO5X+7P9GDEE/JiR4AAADgf+i3HU/M+/NZP6ufkuhBiCc1\nFQAAOIq1PiE10SMQZy76BQAAIELUVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAA\narLC/SV9/7Fq4kfbEj0IP5WaCgAA1GRrd+ydtvrrAZM+TfQg/FTJiR4AAACgCp3ZsuF/9fnF/27d\nKNGD8FOpqQAAQA03pNvPEz0C/x9c9AsAAECEqKkAAABEiJoKAABAhKipAAAARIiaCgAAQISoqQAA\nAESImgoAABA8/e7maau/TvQUBIHnpgIAAGzYWXTH9LwgCGKjuyd6FtRUAADgmNf6hNTLszNOblwv\n0YMQBGoqAABAEAT/GtAh0SPwbz6bCgAAQISoqQAAAESImgoAAECEqKkAAABEiJoKAABAhKipAAAA\nRIiaCgAA8JNs33PwUL0GiZ6i5vPcVAAAgJ8k8/73g56PJXqKms/ZVAAAgJ/k9KbpdQsLEj1Fzaem\nAgAA/CQr/vS/Wr77cKKnqPnUVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAAIELU\nVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAAIELUVAAAACJETQUAACBC1FQAAAAi\nRE0FAAAgQtRUAAAAIkRNBQAAIEKSEz0AP2bhwoWvvPLKhg0bmjRpcsEFF/Tv3z8lJSUIgkOHDt1y\nyy2lpaVlO2vXrv3ss88mblIAAID4cDY1ul555ZWbbropNTU1JyenQ4cOY8aMuf3222OxWBAEn3/+\n+YIFC5o3b966nETPy7+1adMm0SMcK0RdbURdbURdbURdbURdbURNTeJsakQVFRU9+eST/fr1e+SR\nR8KVVq1a3X///WvWrMnOzl63bl39+vXvu+++WrVqJXZOAACA+HI2NaIKCgr27t176aWXlq107tw5\nCIL8/PwgCNatW9e2bVsdFQAAqHmcTY2oZs2avfbaa+Uv5V21alUQBKecckoQBOvWrWvUqNGQIUNW\nrFjRoEGD884775ZbbklNTU3YuAAAAHHibGpEHXfccR06dChrnnl5eU899dQZZ5zRvn37IAjWrVu3\nYMGCE0888fbbb+/Spctzzz03YMCA8ndUAgAAOEo5m5p4e/bsmTlzZtnLjIyMnj17lr2MxWKvvvrq\nqFGjTj311DFjxgRBUFJSMmjQoM6dO2dnZwdB0K9fvw4dOtxzzz1z5szp3bt39c8PAAAQR2pq3Oze\nvbtv377XX3/9Nddcc/jRuXPnzpw5c8OGDenp6VlZWTk5OS1btgwPffvtt48//njZzs6dO5fV1K1b\nt95zzz3Lli278cYbc3Nz69SpEwRBUlJS//79y3/xyy+//IEHHli1apWaCgAAHO3U1LgZP378li1b\nvv/++8MPPfzwwy+88ELdunXbtWtXWFg4efLk6dOnjx079pxzzgmCoGXLlitWrDj8d+Xl5V133XVN\nmzadMWPGqaeeWra+c+fOjRs3dunSJSkpKVxJSkpKTk4+4h8NAABwdPHZ1EopKSnJz89fuHDhHXfc\nMW7cuCPuWbp06QsvvNCiRYs33nhj0qRJs2fPHjNmTHFx8YgRI/bv3/9DX7m0tHTYsGGtW7d+6aWX\nynfUIAjy8/OvvfbaxYsXl62sXLny+++/b9u2bbz+XgAAAInibGqlbNy4sfwzY45oypQpQRAMGzas\nWbNm4UqvXr169+49a9asDz74oEePHkf8XYsXL163bt2dd9754Ycfll/Pzs7u2LFjVlbWiBEjRo4c\nmZ2dnZeX99BDD7Vs2bJv377x+DsBAAAkkppaKZmZmX/5y1/CX3/yySfjx48/fM/SpUuTkpLOO++8\n8os9evSYNWvWkiVLfqimrly5MgiC8p9ZDU2YMOGcc8559tlnH3zwwdtuuy0Wi9WpU6dbt24PPPBA\nSkrKj0/bpk2bn/j3opJEXW1EXW1EXW1EXW1EXW1EXW1ETY2hplZKenp6r169wl8nJx8hzOLi4q++\n+qp58+b16tUrv96qVasgCAoKCn7oK+fm5ubm5v7Q0aZNmz7zzDNFRUVfffVVs2bNjvhHV/DZZ5/9\nt3sAAAASTk2tWoWFhaWlpQ0bNqywHq7s3r27Ml88NTW17HbBAAAANYNbKFWt4uLi4EgnWsOV8CgA\nAABl1NSqFT4z5vA6Gq6UPVEGAACAkJpatVJTU4MgKCoqqrAeroRHAQAAKKOmVq309PT69etv3ry5\npKSk/Hp+fn4QBJmZmYkZCwAAIKrU1CrXtm3bgwcPrlmzpvzi8uXLgyBo165dgoYCAACIKDW1yl14\n4YVBEDzxxBNlK9u2bXv55ZeTkpK6d++euLkAAACiyANpqtyVV145adKkDz/88Iorrrjkkkt27Ngx\na9asffv23XjjjU2bNk30dAAAANGipla5OnXqPP/88w8++ODcuXPDS3/T0tL+9Kc/5eTkJHo0AACA\nyKkVi8USPcOxori4OD8/Py0tLTMzs1atWokeBwAAIIrUVAAAACLERb813Ny5c2fOnLlhw4b09PSs\nrKycnJyWLVsmeqiaY/fu3X379r3++uuvueaaw48KPy6mTZu2cOHC9evXN2jQIDs7+4YbbjjppJMq\n7BF15ZWUlLz00kuLFi3atGnTCSec8Mtf/nLAgAEZGRkVtok6voqKiq699trGjRuPGzeuwiFRV97s\n2bPfeeedCotNmjS5++67y6+IOi62bt06d+7cJUuWbNq06ec///lvf/vb8BaS5Ym6ku699979+/cf\n8VB2dvYf//jHspeirrxYLDZnzpx//etfX3zxxfHHH9+6devrrruuVatWFbaJuuo4m1qTPfzwwy+8\n8ELdunXbtWtXWFi4cePGunXrjh079pxzzkn0aDXE6NGjx40bN3To0JtvvrnCIeFXXklJyW233TZv\n3rz09PT27dtv3779iy++qFev3oQJE7p27Vq2TdSVd+jQoeuuu+6jjz5q0qRJq1atVq1aVVRUlJaW\nNmXKlFNOOaVsm6jj7t577508eXL79u2nTp1afl3UcXHLLbfMnz8/LS2t/GLTpk1nzJhR9lLUcfHl\nl19eddVVO3fuPOmkk5o0abJ27dogCHJzcwcPHly2R9SV16lTp3379h3xUM+ePceOHRv+WtRxceed\nd06fPj0tLS07O/uLL77Yvn17cnLy008/fcEFF5TtEXXVilFDLVmyJCsrq2fPngUFBeHKm2++2bZt\n227duhUVFSV2tqPaoUOHNm3atGDBgiFDhmRlZWVlZT3zzDMV9gg/Ll566aXwjcmy0CZPnpyVlXXu\nuecePHgwXBF1XEycODErK+uuu+4qLi6OxWL79+9/9NFHs7Ky+vfvX7ZH1HE3b968du3adezYsW/f\nvuXXRR0vPXr0+MMf/vAjG0QdF3v37j3//POzs7Pff//90tLSWCy2cePGrl27nnbaaWXBijou9h3J\nY4891r59++XLl4d7RB0X8+bNy8rK6tOnz3fffReLxUpLS2fPnp2VlXXWWWcdOnQo3CPqqua5qTXW\nlClTgiAYNmxYs2bNwpVevXr17t17+/btH3zwQUJHO7pt3LixV69eN9100+zZs39oj/DjYuLEicnJ\nyY888ki9evXCld/97nfnnnvujh071q9fH66IOi6mT59+3HHHjRw5Mjk5OQiCunXr3nTTTSkpKStX\nriwtLQ33iDq+du7cOWLEiEGDBjVu3LjCIVHHxb59+woKCtq2bfsje0QdF3PmzNmyZcvw4cPPPvvs\n8A6Rp5xySk5OTqNGjVatWhXuEXVcpB7mvffee+6550aOHNmpU6dwj6jj4uOPPw6CICcn5/jjjw+C\noFatWhdddFHr1q2/+eabjRs3hntEXdXU1Bpr6dKlSUlJ5513XvnFHj16BEGwZMmSBA1VE2RmZv7l\nP2644YYj7hF+5cVisa1bt7Zp0+bEE08svx5eg1pQUBC+FHXllZaWfvPNN506dUpPTy9bbNy4cYMG\nDVJSUmL/+WCIqONr5MiRmZmZgwYNOvyQqONi/fr1sVjstNNO+5E9oo6LmTNnJiUlXXHFFeUXBw4c\nuGjRot69e4cvRV0V8vLy7rrrrj59+lx55ZVli6KOi8PfQAyCoKSkpHbt2ieccEL4UtRVzS2Uaqbi\n4uKvvvqqefPmZaehQuEnv8t+xOd/ID09vVevXuGvw1NPFQg/LkpKSh544IHD75a0YcOGIAhOPvnk\nQNRxUrt27YULF1ZYfP3113fu3NmvX7+kpKRA1PE2adKkDz74YOrUqWG85Yk6Xj777LMgCH72s589\n8cQTq1evTk1NbdOmzbXXXlv2I6ao4+XDDz/s2LGNzd+hAAAK50lEQVRjenp6Xl7eihUrtmzZ0rp1\n67POOkvUVe3+++8PgmD48OFlK6KOl549e44ePXrixIndu3cPP98+Z86cTZs2nXnmmY0aNQpEXS3U\n1JqpsLCwtLS0YcOGFdbDld27dydiqGOF8OMiOTm5X79+FRYXLVq0ePHi1q1bt27dOhB1FVi9evW8\nefPWrFnz7rvvXnjhhSNHjgzXRR1H+fn5jz322ODBg3/xi18cflTU8RLW1CFDhhw4cODkk0/evHnz\nO++88/LLL48ZM+bXv/51IOo42bNnz8GDBzMzM8ePH//EE0+UrTdo0ODRRx/t2bNnIOqq8cYbbyxb\ntmzgwIFlbwcEoo6fli1bTpw48fbbb+/evXuXLl02bty4adOms88+++mnnw43iLoauOi3ZiouLg6O\ndK4vXAmPUkWEX0Xefvvt3NzcunXrPvTQQ2Wn+AJRx9XKlSsnTJiwcOHCWCzWqFGjkpKScF3U8VJS\nUjJ8+PDTTjvt+uuvP+IGUcfLZ599VqtWrRtuuGHZsmWvv/76xx9/PGTIkMLCwrvvvnvv3r2BqOPk\nm2++CYJgyZIlf/3rX++9997333///fffHzFixP79+4cOHbply5ZA1FXgwIEDjz/+eMOGDSt8+EjU\ncbRnz57S0tJdu3YtWLBg06ZNQRDs27fvu+++C4+KuhqoqTVT+R/iywtXDr/MjDgSftwVFhb++c9/\nvvXWW9PS0iZMmNC5c+dwXdRxd/XVV3/yyScfffTRnXfe+dprr1100UXffvttIOr4GTt27IYNG0aN\nGlW79pG//4o6Xp5++un58+cPHDiwbt26QRAkJSUNGjTosssu27Fjx1tvvRWIOk7CuL799tvhw4df\nc801GRkZGRkZ/fv3Hzx48IEDB8aPHx+Iugq89dZbW7du/c1vflO/fv3y66KOlxdffDE3NzcjI+OF\nF14Ivy2OGDFi7dq1ffr0Ca/UEHU1UFNrptTU1CAIioqKKqyHK+FRqojw42v+/PkXX3zx1KlTL7nk\nkpkzZ5Z/Yqqoq0iDBg1ycnIGDBjw9ddfv/fee4Go42TVqlXPPPPMwIEDGzduvOc/SktLS0tL9+zZ\nEz4OUdTxkpGR0bRp0wqLF154YRAEeXl5gajjJLwPahAEffv2Lb9+6aWXBkHw6aefBqKuAtOmTQv+\nE3J5oo6XKVOm1KlTZ9y4cWeccUZycnKDBg369+8/dOjQoqKit99+OxB1tVBTa6b09PT69etv3ry5\n7Jq9UH5+fhAEmZmZiRnr2CD8OPrHP/4xaNCglJSU559//qmnngrvW1BG1HGxdu3a++67b+7cuRXW\nw7sXLlq0KBB1nKxevbqkpOSpp57qWs727ds//fTTrl27XnXVVYGo4yd8gGeFxfB21uF1qqKOi+OP\nP75OnTrHHXdceJuZMuH/rsPLMUQdXzt27Fi8eHHTpk1PP/30CodEHRfffffdp59+mpWVVfakmVD4\nPld4F19RVwM1tcZq27btwYMH16xZU35x+fLlQRC0a9cuQUMdK4QfF/Pnzx81alTXrl1nzJhx5pln\nHnGPqCsvFov985//HDt2bIX18OfLJk2ahC9FXXnt27fPPUx6enpGRkZubu7vf//7cJuoKy8/P79t\n27ZDhw6tsB5erRfegy0QdTzUqVPnzDPP3LdvX/gx1DLhXdlbtmwZvhR1HM2dO7e0tLRnz57hU2or\nEHXlpaWlJSUl7dmzp8L6rl27gv+8BROIuuqpqTVW+JZP+dvubdu27eWXX05KSurevXvi5jomCL/y\nSkpKRo0aVb9+/Weffbb88zwrEHXltWrVqmHDhmvXrl27dm3Z4oEDB/7+978HQdClS5dwRdSV17Fj\nx8GHadCgwYknnjh48OCrr7463CbqymvZsmXjxo3nz5+/bt26ssVdu3ZNmDAhOTn5ggsuCFdEHRdh\njGPGjCk7fV1aWhq+81V2Vaqo4yh8t+WHngks6spLSUlp167dl19+GX6OPRSLxZ555pnAt8VqVOvw\nS2KoGYqLiy+//PINGza0b9/+kksu2bFjx6xZs3bu3HnjjTcOGzYs0dPVEPPmzcvNzR06dOjNN99c\nfl34lffZZ59ddtllLVq06Nat2+FHBwwY0Lx580DUcTJjxozhw4enp6dfc801p5566vbt2ydPnlxQ\nUHDRRReV3Xxf1FXk/PPPb9So0dSpU8tWRB0Xb7311u23356amnrVVVe1adNm27ZtL7300tdffz14\n8ODc3Nxwj6jjoqSk5Lbbbps3b94ZZ5wR/uD+5ptvfvTRR126dHnxxRfDu4WJOo6uuuqq5cuXT5o0\nqVOnTocfFXVcrFmz5sorrzx48GB2dvZFF12UkpIye/bsFStWdOjQ4ZVXXqlTp04g6qqnptZkO3fu\nfPDBB+fOnRteN5+WljZw4MCcnBz3H4uXH6qpgfAr7dVXXy17aOfhyn97FnVczJ8//7HHHgs/VBME\nQaNGjW644YZrr702vEtqSNRV4fCaGog6TubPn//kk09+/vnnQRDUqlWrRYsWw4cPLzuVGhJ1XBw4\ncODRRx999913w0t/GzdufOmll955553hT/MhUcfLr371q8LCwmXLllW4zW8ZUcfFypUrR48evXTp\n0vBlSkrKlVdeeeutt5Z/Vqqoq5SaWvMVFxfn5+enpaVlZmYe8WMMVB3hVxtRx0VhYWFBQUH4SIkf\n2iPqaiPquNi1a9e2bdtatGhR4R4/5Yk6XrZu3VqrVq0fuX+MqKuNqOOiqKiooKCgbt26zZo1+6Hy\nKeoqoqYCAAAQIW6hBAAAQISoqQAAAESImgoAAECEqKkAAABEiJoKAABAhKipAAAARIiaCgAAQISo\nqQAAAESImgoAAECEqKkAAABEiJoKAImXl5f34osvrlixItGDAEDiqakAkHjLli176KGH5s6dm+hB\nACDx1FQASLzatWunpKQkJSUlehAASLxasVgs0TMAAADAvzmbCgAAQISoqQCQeNOmTevWrdvf/va3\nRA8CAImnpgJA4hUVFe3YsWPv3r2JHgQAEk9NBQAAIELUVAAAACJETQUAACBC1FQAAAAiRE0FAAAg\nQtRUAAAAIkRNBQAAIELUVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAEi8wsLCIAiSkpISPQgA\nJF5yogcAgGNaYWHhunXr5syZEwRB8+bNEz0OACSemgoAifTxxx8PHDgwCIIWLVpcfPHFiR4HABKv\nViwWS/QMAHDs2rx58+uvv961a9euXbvWru3DOACgpgIAABAl3rUFAAAgQtRUAAAAIkRNBQAAIELU\nVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAAIELUVAAAACJETQUAACBC1FQAAAAi\nRE0FAAAgQtRUAAAAIkRNBQAAIELUVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAA\nIELUVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAAIELUVAAAACJETQUAACBC1FQA\nAAAiRE0FAAAgQtRUAAAAIkRNBQAAIELUVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRN\nBQAAIELUVAAAACJETQUAACBC1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAAIELUVAAAACJETQUAACBC\n1FQAAAAiRE0FAAAgQtRUAAAAIkRNBQAAIELUVAAAACLk/wG0iH/w3yXevgAAAABJRU5ErkJggg==\n",
"text/plain": [
"<IPython.core.display.Image object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"semilogy(s,'.')\n",
"[Qc,Rc] = gs(A); % classical\n",
"hold on, semilogy(diag(Rc),'o')\n",
"[Qm,Rm] = mgs(A); % modified\n",
"hold on, semilogy(diag(Rm),'s')\n",
"legend('singular values','classical r_{jj}','MGS r_{jj}')\n",
"title('Gram-Schmidt in double precision'), xlabel('j')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The modified version stays close to the line until around $10^{-16}$. This is the limit of machine precision. In the classical case the departure is closer to $10^{-8}$. We can get a little more information by recasting the problem into \"single precision,\" which has a limit of about $10^{-8}$ relative precision. "
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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TbdrdI4GEEFkr0u4VJ9biUkJSa/JTa/JJ27EFpm7zxt2Hhy+XxQ9RCFdNAAAA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\nPK58ErWNU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"text/plain": [
"<IPython.core.display.Image object>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"semilogy(s,'.')\n",
"[Qc,Rc] = gs(single(A)); % classical\n",
"hold on, semilogy(diag(Rc),'o')\n",
"[Qm,Rm] = mgs(single(A)); % modified\n",
"hold on, semilogy(diag(Rm),'s')\n",
"legend('singular values','classical r_{jj}','MGS r_{jj}')\n",
"title('Gram-Schmidt in single precision'), xlabel('j')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A reasonable hypothesis (based on two data points) is that roundoff at precision $\\epsilon$ affects MGS at the level $O(\\epsilon)$, but classical GS at $O(\\sqrt{\\epsilon})$. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Loss of orthogonality\n",
"\n",
"A different sort of instability remains in the MGS algorithm. It can be seen in a $2\\times 2$ toy example. Mine is different from the text's, but constructed using the same principle: the columns of $A$ are nearly parallel."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A =\n",
"\n",
" 3.1416 1.4142\n",
" 3.1416 1.4142\n"
]
}
],
"source": [
"A = [pi, sqrt(2); 355/113, sqrt(2)]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"When we subtract off the projection of the second column onto the first, the remainder is quite small (relative to the original components). "
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"v =\n",
"\n",
" 1.0e-07 *\n",
"\n",
" 0.6004\n",
" -0.6004\n"
]
}
],
"source": [
"Q = A(:,1)/norm(A(:,1));\n",
"v = A(:,2) - Q(:,1)*(Q(:,1)'*A(:,2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Of the original 16 significant digits, about 7 have been canceled out in subtraction, never to return."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans =\n",
"\n",
" 3.7693e-09\n"
]
}
],
"source": [
"Q(:,2) = v/norm(v);\n",
"norm(Q'*Q - eye(2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Much later, when we start doing iterative methods, we'll be coming back to the MGS algorithm. This loss of orthogonality is so important that practical implementations take measures to re-orthogonalize their results regularly. "
]
}
],
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"help_links": [
{
"text": "MetaKernel Magics",
"url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
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