Numerical derivatives.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical Derivatives\n",
    "\n",
    "The goal is to compute the derivative $$\\frac{df}{dx} = \\lim_{h\\to 0} \\frac{f(x+h) - f(x)}{h}$$ of a function $f(x)$ where you are given *only* a \"black box\" that computes the function $f$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = sin(3x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4121184852417566"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.4121184852417566"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function f(x)\n",
    "    return sin(3x)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's try to *approximately* compute the derivative $f'(2)$, for which the exact answer is $3\\cos(6)$.\n",
    "\n",
    "Exact answer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.880510859951098"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "3cos(6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finite-difference approximation\n",
    "\n",
    "A simple approximation is the *forward difference*:\n",
    "\n",
    "$$\n",
    "f'(x) \\approx \\frac{f(x+h)-f(x)}{h}\n",
    "$$\n",
    "for some small number $h$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.880636553720977"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(f(2.0001) - f(2)) / (2.0001 - 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "deriv (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deriv(f, x, h) = (f(x+h) - f(x))/h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.880636553727056"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deriv(f, 2, 0.0001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.001"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1e-3 # same as 10⁻³"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.8414979985216231"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deriv(cos, 1, 1e-4) # should -sin(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.8414709848078965"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-sin(1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## How fast does this converge?\n",
    "\n",
    "How fast does the error go to zero as $h \\to 0$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00012569377595816889"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abs(deriv(f, 2, 1e-4) - 3cos(6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2573322953013388e-5"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abs(deriv(f, 2, 1e-5) - 3cos(6))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.996862120529915"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.00012569377595816889 / 1.2573322953013388e-5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That it, it looks like the error decreases *proportional to h*:  if we decrease $h$ by a factor of 10, the error decreases by a factor of 10:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.996528703829966"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abs(deriv(f, 2, 1e-5) - 3cos(6)) / abs(deriv(f, 2, 1e-6) - 3cos(6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But something weird starts to happen when $h$ becomes extremely small: the error actually increases as we decrease $h$!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.018400795812165596"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abs(deriv(f, 2, 1e-10) - 3cos(6)) / abs(deriv(f, 2, 1e-11) - 3cos(6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting the convergence rate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This sort of thing is a lot easier to see if we plot it.   To plot, we'll use a plotting package for Julia called [PyPlot](https://github.com/JuliaPy/PyPlot.jl) that calls the full-featured Python Matplotlib package for plotting:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "using PyPlot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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+fL4MNaBgIyIi4tlWrYIuXS4emfD++2bPmnwq//7mIiIinm79eujQAc6fh06dYOZM8PW1uypbKdiIiIh4ou+/h7Zt4exZaNUK5s0Df3+7q7Kdgo2IiIin2bnThJmEBIiIMIdbBgTYXZVbULARERHxJLt3m0MsT52C+vVhyRIoWNDuqtyGgo2IiIin+O03aNYMjh+HW2+FL7+EoCC7q3IrCjYiIiKeIDYWmjaFw4ehRg346isoWtTuqtyOgo2IiIi7O3rUhJo//oCqVeHrr6FkSburcksKNiIiIu7s77/N8NPevVChAqxeDaVL212V27I12Kxbt4727dsTFhaGw+EgOjo6y3sff/xxHA4H48ePz8MKRUREbHT6NLRoAbt2QViYCTXh4XZX5dZsDTZJSUnUrl2biRMnXvG+6OhoNm/eTFhYWB5VJiIiYrPERGjTBn74Aa67zoSaKlXsrsrt+dn5w1u3bk3r1q2veM+hQ4fo168fX331FW3bts2jykRERGx09iy0bw8bN0KxYubYhGrV7K7KI7j1HJv09HQiIyN5+umnqVGjht3liIiI5L7kZOjcGWJizFLur76C2rXtrspj2NpjczWvv/46fn5+DBgwINuvSU5OJjk5OeM6Pj4+N0oTERFxvZQUePBBWLHCbLr3xRdQr57dVXkUt+2x2bZtGxMmTGDmzJk4nDh6fcyYMYSEhGQ8wjXJSkREPEFaGvTqBYsXm+MRliyBRo3srsrjuG2wWb9+PceOHaN8+fL4+fnh5+fHH3/8wdChQ6lYsWKWrxsxYgRxcXEZj9jY2LwrWkREJCfS0+Hxx81Bln5+8NlnZom3OM1th6IiIyNp9q//UFu2bElkZCS9e/fO8nUBAQEE6CAwERHxFJYFgwbBhx+Cjw/MnQvt2tldlceyNdgkJiayb9++jOv9+/ezY8cOihcvTvny5SlRokSm+/39/SldujQ33nhjXpcqIiLiepYFI0bAu++a6xkz4D//sbcmD2drsNm6dSuNGzfOuB4yZAgAvXr1YubMmTZVJSIikkdefhlef908nzwZeva0tx4vYGuwiYiIwLKsbN9/4MCB3CtGREQkL731Fjz3nHk+bhz07WtvPV7CbScPi4iIeK3Jk+Gpp8zzl16CwYPtrceLKNiIiIjkpVmz4MknzfMRI+C//7W3Hi+jYCMiIpJXFiyAPn3M8wED4JVXwIm92uTqFGxERETywtKl0KOH2bPmkUdg/HiFmlygYCMiIpLbVq2CLl0gNRW6d4cpUxRqcomCjYiISG5atw46dIDz583hljNngq+v3VV5LQUbERGR3LJ5M7RtC2fPQps2F49MkFyjYCMiIpIbduyAVq0gMRGaNDHnPxUoYHdVXk/BRkRExNV274bmzeH0aWjQAD7/HAoWtLuqfEHBRkRExJX27YOmTeHvv6FuXfjySyhSxO6q8g0FGxEREVc5eNCEmiNH4Oab4auvICTE7qryFQUbERERVzhyxISagwfhhhvMEu8SJeyuKt9RsBEREblWx49Ds2ZmGKpiRVi9GkqXtruqfEnBRkRE5FqcPg0tW8LPP0PZsibUlCtnd1X5loKNiIhITiUkQOvW8MMPEBoKX38NlSvbXVW+pmAjIiKSE2fOQPv2sGkTFCtm5tRUq2Z3Vfmego2IiIizkpPN8Qhr10JwMKxcCbVq2V2VoGAjIiLinJQU6NrVLOUuVMjsU3PbbXZXJf9PwUZERCS70tKgZ0+zk3BAACxZAg0b2l2V/IOCjYiISHakp8Ojj8L8+eDvDwsXmn1rxK0o2IiIiFyNZcGAATBjBvj4wNy55tRucTsKNiIiIldiWfDMM/Dee+BwwKxZ0KWL3VVJFhRsREREruSll2DsWPN8yhTo0cPeeuSKFGxERESy8uab8Pzz5vnbb8Njj9lbj1yVgo2IiMjlTJoETz9tnr/yCgwaZG89ki0KNiIiIv82YwZERZnnzz5rHuIRFGxERET+acECeOQR83zQIHj5ZXvrEaco2IiIiFzw+edmcnB6uplPM26cWQklHkPBRkREBMwRCfffD6mpJtxMnqxQ44EUbERERNauhU6d4Px5uO++ixvxicfRf2oiIpK/bdoE7drB2bNmN+G5c8HPz+6qJIcUbEREJP/asQNat4bERHPu02efQYECdlcl10DBRkRE8qeff4bmzeH0aXNC9+efQ2Cg3VXJNVKwERGR/GffPmjWDP7+G267Db74AgoXtrsqcQFbg826deto3749YWFhOBwOoqOjM76WkpLC8OHDqVmzJoULFyYsLIyePXty+PBhGysWERGP98cfZtjpyBGoWRNWrICQELurEhexNdgkJSVRu3ZtJk6ceMnXzpw5w/bt2xk1ahTbt29n0aJF/Prrr9x77702VCoiIl7h8GETag4ehBtvhFWroEQJu6sSF3JYlmXZXQSAw+Fg8eLFdOzYMct7tmzZwu23384ff/xB+fLls/V94+PjCQkJIS4ujuDgYFeVKyIinub4cbjnHti9GypVgnXroFw5u6uSLOT077dHzbGJi4vD4XBQtGhRu0sRERFPcuoUtGhhQk25crB6tUKNl/KYhfrnzp3jmWeeoVu3bldMbsnJySQnJ2dcx8fH50V5IiLirhISzJLuHTsgNBS+/tr02IhX8ogem5SUFB544AHS09OZNGnSFe8dM2YMISEhGY/w8PA8qlJERNzOmTNm873Nm6F4cRNqbrzR7qokF7l9sElJSeH+++9n//79rFq16qrjbCNGjCAuLi7jERsbm0eVioiIW0lONsckrFsHwcGwcqVZBSVeza2Hoi6Emr1797JmzRpKZGPmekBAAAEBAXlQnYiIuK2UFHOg5cqVZn+a5cuhbl27q5I8YGuwSUxMZN++fRnX+/fvZ8eOHRQvXpywsDC6dOnC9u3bWbZsGWlpaRw9ehSA4sWLU0BbXouIyOWkpUFkJCxZAgEB5t8GDeyuSvKIrcu9Y2JiaNy48SXtvXr1YvTo0VTKYnLXmjVriIiIyNbP0HJvEZF8JD0dHn4YZs4Ef3+IjoY2beyuSnIgp3+/be2xiYiI4Eq5yk222BEREU9gWdC/vwk1vr4wb55CTT7k9pOHRURErsqyYNgwmDQJHA6YNQvuu8/uqsQGCjYiIuL5XngB3nzTPH//feje3d56xDYKNiIi4tnGjjXBBmDCBHj0UXvrEVsp2IiIiOeaOBGGDzfPX30VBgywtx6xnYKNiIh4punTzWRhgJEjYcQIe+sRt6BgIyIinmfePHjkEfN88GB48UV76xG3oWAjIiKeJTrabMBnWdC3L7z1llkJJYKCjYiIeJIVK6BrV7O7cM+e8N57CjWSiYKNiIh4hpgYc6jl+fPwn//Ahx+Cj/6MSWb6RIiIiPvbtAnatYNz58y/c+aAn1uf4yw2UbARERH3tn07tGoFSUnQrBl8+inoIGTJgoKNiIi4r127oEULiIuDu+4yE4cDA+2uStyYgo2IiLinvXtND82JE1CvHnzxBRQubHdV4uYUbERExP388Qc0bQpHj0KtWmY1VHCw3VWJB1CwERER93L4MDRpArGxUK0arFoFxYvbXZV4CAUbERFxH8eOmZ6a33+HypXh668hNNTuqsSDKNiIiIh7OHnSTBTeswfKlYPVq6FsWburEg+jYCMiIvaLj4fWrWHnTihVyoSaihXtrko8kIKNiIjY68wZs+ne999DiRJm+OmGG+yuSjyUgo2IiNjn3Dno2BHWr4eQEFi5Em6+2e6qxIMp2IiIiD1SUuD++82qp8KFYflyuPVWu6sSD6dgIyIieS8tDXr0gKVLzU7CS5dC/fp2VyVeQMFGRETyVno6PPwwfPIJ+PvD4sXQuLHdVYmXULAREZG8Y1nQrx/MmgW+vrBggTngUsRFFGxERCRvWBY8/TRMngwOB8yeDZ062V2VeBkFGxERyRujR8Nbb5nnU6dCt262liPeScFGRERy3+uvw4svmufvvGPm2IjkAgUbERHJXe++C888Y56/9hr0729vPeLVFGxERCT3fPghDBhgno8aBcOH21uPeD0FGxERyR1z58Kjj5rnQ4fCCy/YW4/kCwo2IiLieosXQ8+eZiXUE0/AG2+YlVAiuUzBRkREXGv5cuja1ewu3KsXTJyoUCN5RsFGRERcJyYGOne+eA7Uhx+Cj/7USN7Rp01ERFxj40Zo186c2N2+PcyZY3YXFslDtgabdevW0b59e8LCwnA4HERHR2f6umVZjB49mrCwMAoWLEhERAS7du2yqVoREcnS9u3maISkJGje/OI5UCJ5zNZgk5SURO3atZk4ceJlvz527FjGjRvHxIkT2bJlC6VLl6Z58+YkJCTkcaUiIpKl//0PWrSA+Hho1Aiio82J3SI28LPzh7du3ZrWrVtf9muWZTF+/Hj++9//0rlzZwBmzZpFqVKlmDt3Lo8//nhelioiIpfz66/QrBmcOAG33w7LlkGhQnZXJfmY286x2b9/P0ePHqVFixYZbQEBAdxzzz1s2LAhy9clJycTHx+f6SEiIrngwAFo2hT++gtq14YVKyA42O6qJJ9z22Bz9OhRAEqVKpWpvVSpUhlfu5wxY8YQEhKS8QgPD8/VOkVE8qVDh0yo+fNPqFYNVq6EYsXsrkrEfYPNBY5/7X1gWdYlbf80YsQI4uLiMh6xsbG5XaKISP5y7JgZfvr9d6hSBVavhtBQu6sSAXIQbGbNmsUXX3yRcT1s2DCKFi1KgwYN+OOPP1xWWOnSpQEu6Z05duzYJb04/xQQEEBwcHCmh4iIuMjJk2bV0549EB5uQk1YmN1ViWRwOti8+uqrFCxYEICNGzcyceJExo4dS8mSJRk8eLDLCqtUqRKlS5dm1apVGW3nz59n7dq1NGjQwGU/R0REsik+3izp/vFHKF3ahJoKFeyuSiQTp1dFxcbGcv311wMQHR1Nly5deOyxx2jYsCERERFOfa/ExET27duXcb1//3527NhB8eLFKV++PIMGDeLVV1+latWqVK1alVdffZVChQrRrVs3Z8sWEZFrkZQEbdvCli1QogR8/TVUrWp3VSKXcDrYFClShBMnTlC+fHlWrlyZ0UsTGBjI2bNnnfpeW7dupXHjxhnXQ4YMAaBXr17MnDmTYcOGcfbsWZ588klOnTrFHXfcwcqVKwkKCnK2bBERyalz56BjR/j2WwgJgVWroEYNu6sSuSyHZVmWMy/o3r07e/bs4ZZbbmHevHkcPHiQEiVKsGTJEp599ln+97//5VatORIfH09ISAhxcXGabyMi4qzz5+G++8z+NEWKmFBz5512VyX5QE7/fjs9x+a9996jfv36HD9+nIULF1KiRAkAtm3bxoMPPujstxMREXeVmgo9ephQExho/lWoETfndI+Np1GPjYhIDqSnQ+/eMHs2FCgAS5ZAy5Z2VyX5SE7/fmdrjs2PP/7IzTffjI+PDz/++OMV761Vq1a2f7iIiLghy4KoKBNqfH3NgZYKNeIhshVs6tSpw9GjRwkNDaVOnTo4HA7+2dFz4drhcJCWlpZrxYqISC6zLBg6FKZMAYcDPvoIOnSwuyqRbMtWsNm/fz/XXXddxnMREfFSzz0Hb79tnk+bBpo7KR4mW8Gmwj82YKpwhc2YvHy6joiId3vtNXj5ZfN84kTo08feekRywOlVUZGRkSQmJl7SfuDAAe6++26XFCUiInnsnXdgxAjzfOxYM8dGxAM5HWx+/vlnatasyXfffZfRNmvWLGrXrn3FM5xERMRNTZsGAwea588/D08/bW89ItfA6Z2HN2/ezMiRI2nSpAlDhw5l7969rFixggkTJtBH3ZYiIp7l44/hscfM86eeMsFGxIM5HWz8/Px47bXXCAgI4KWXXsLPz4+1a9dSv3793KhPRERyy6JF0KuXWQn15JNmCMrhsLsqkWvi9FBUSkoKQ4cO5fXXX2fEiBHUr1+fTp068eWXX+ZGfSIikhu+/BIeeADS0uChh+DddxVqxCs43WNz2223cebMGWJiYrjzzjuxLIuxY8fSuXNn+vTpw6RJk3KjThERyam0NFi/Ho4cgTJlICXFnP+UkgJdu5o5Nj5O//9cEbeUo2DzzjvvULhwYcBszjd8+HBatmxJjx49XF6Mf1HjAAAgAElEQVSgiIhcg0WLzMTgP/+82OZwmOGnDh3MBny+vvbVJ+JiLj0rKjk5mYCAAFd9O5fQWVEikm8tWgRdupgQcznz55seGxE3lGene//T2bNniY+Pz3gkJydfy7cTERFXSUszPTVZhRqHwyzr1jE44mWcDjZJSUn069eP0NBQihQpQrFixTI9RETEDaxfn3n46d8sC2JjzX0iXsTpYDNs2DC++eYbJk2aREBAANOmTeOFF14gLCyM2bNn50aNIiLirCNHXHufiIdwevLw0qVLmT17NhEREfTp04dGjRpx/fXXU6FCBT7++GO6d++eG3WKiIgzUlOzd1+ZMrlbh0gec7rH5uTJk1SqVAmA4OBgTp48CcBdd93FunXrXFudiIg4b+5ceOKJK9/jcEB4ODRqlDc1ieQRp4NN5cqVOXDgAADVq1fnk08+AUxPTtGiRV1anIiIOOHMGXj0UejeHZKSoHp1E2D+vfHehevx47XUW7yO08Gmd+/e7Ny5E4ARI0ZkzLUZPHgwT+vgNBERe+zeDXfcYTbbczhg1CjYuRM++wzKls18b7lypr1zZ3tqFclF17yPzcGDB9m6dStVqlShdu3arqrLZbSPjYh4vVmzzFlPZ85AqVIwZw40a3bx6//eebhRI/XUiNvL6d9vpycP/1v58uUpX778tX4bERFxVlISREWZYAPQtKkJNaVLZ77P1xciIvK8PBE76HAQERFP9L//Qb16JtT4+MCLL8JXX10aakTymWvusRERkTxkWTB9OvTrB+fOmaGlefPgnnvsrkzELajHRkTEUyQkQGQkPPKICTUtW8KOHQo1Iv+gYCMi4gl27IDbboOPPzZzZsaMgS+/hNBQuysTcStOB5uIiAhmz57N2bNnc6MeERH5J8uCyZPhzjvh11/NUu21a+GZZ8zcGhHJxOn/VtStW5dhw4ZRunRpHn30UTZt2pQbdYmISFwcPPCAWcqdnAzt2pmem4YN7a5MxG05HWzeeustDh06xOzZszl+/Dh333031atX58033+Svv/7KjRpFRPKfbdvg1lvhk0/Azw/efBOWLIESJeyuTMSt5agf09fXlw4dOhAdHc2hQ4fo1q0bo0aNIjw8nI4dO/LNN9+4uk4RkfzBsuDdd6FBA/j9d6hQwWyuN3TopUcjiMglrmmA9vvvv+e5557jzTffJDQ0lBEjRhAaGkr79u156qmnXFWjiEj+cOoU3HcfDBgA589Dx47www9mfo2IZIvT+9gcO3aMjz76iBkzZrB3717at2/P/PnzadmyJY7//38T999/Px07duTNN990ecEiIl5p82Yzn+bAAfD3N0NP/furl0bESU4Hm3LlylGlShX69OnDQw89xHXXXXfJPbfffjv16tVzSYEiIl7NsuDtt2H4cEhNhcqVYcECs7RbRJzm9FDU6tWr2b17N08//fRlQw1AcHAwa9asuebiUlNTGTlyJJUqVaJgwYJUrlyZF198kfT09Gv+3iIitjtxAu6918yfSU2FLl1g+3aFGpFr4HSPTaNGjXKjjst6/fXXmTJlCrNmzaJGjRps3bqV3r17ExISwsCBA/OsDhERl/vuO3jwQYiNhYAA02vTt6+GnkSukVufFbVx40Y6dOhA27ZtAahYsSLz5s1j69atNlcmIpJD6ekwdiyMHAlpaVC1qlnSXaeO3ZWJeAW33rbyrrvuYvXq1fz6668A7Ny5k2+//ZY2bdpk+Zrk5GTi4+MzPURE3MLx49C2LYwYYUJNt25mvxqFGhGXcesem+HDhxMXF0e1atXw9fUlLS2NV155hQcffDDL14wZM4YXXnghD6sUEcmGdevM0NPhwxAYaPaqefhhDT2JuNg19dicO3fOVXVc1oIFC5gzZw5z585l+/btzJo1izfffJNZs2Zl+ZoRI0YQFxeX8YiNjc3VGkVErigtDV5+GRo3NqGmWjX4/ntzQrdCjYjLOR1s0tPTeemllyhbtixFihTh999/B2DUqFF8+OGHLi3u6aef5plnnuGBBx6gZs2aREZGMnjwYMaMGZPlawICAggODs70EBGxxV9/QcuWMGqUmVvTqxds3Qo1a9pdmYjXcjrYvPzyy8ycOZOxY8dSoECBjPaaNWsybdo0lxZ35swZfP51eq2vr6+We4uI+1u9GmrXNv8WKgQzZ5pH4cJ2Vybi1ZwONrNnz+aDDz6ge/fu+Pr6ZrTXqlWLPXv2uLS49u3b88orr/DFF19w4MABFi9ezLhx4+jUqZNLf46IiMukpcHzz0Pz5qbHpkYN2LLF9NaISK5zevLwoUOHuP766y9pT09PJyUlxSVFXfDuu+8yatQonnzySY4dO0ZYWBiPP/44zz33nEt/joiISxw+DN27Q0yMuX7kEZgwwfTYiEiecDrY1KhRg/Xr11OhQoVM7Z9++im33HKLywoDCAoKYvz48YwfP96l31dExOW++goiI82S7iJF4P33zXJuEclTTgeb559/nsjISA4dOkR6ejqLFi3il19+Yfbs2Sxbtiw3ahQRcV+pqfDcc3BhUUPt2mbDvRtusLcukXzK6Tk27du3Z8GCBXz55Zc4HA6ee+45du/ezdKlS2nevHlu1Cgi4p5iYyEi4mKo6dsXNm1SqBGxkcOyLMvuInJTfHw8ISEhxMXFaem3iLjOF19Az55w8iQEBcG0aXD//XZXJeI1cvr3O8c7D58/f55jx45dsvS6fPnyOf2WIiLuLyUFnn0W3nzTXNetCwsWQJUq9tYlIkAOgs3evXvp06cPGzZsyNRuWRYOh4O0tDSXFSci4lb++AO6doXNm811//7wxhvmdG4RcQtOB5uHHnoIPz8/li1bRpkyZXBoS3ARyQ+io6F3bzh9GooWhenTQXtqibgdp4PNjh072LZtG9WqVcuNekRE3Mv58zBsmNmPBuD2283QU8WKtpYlIpfn9Kqo6tWr8/fff+dGLSIi7uX336Fhw4uhZuhQWL9eoUbEjWUr2MTHx2c8Xn/9dYYNG0ZMTAwnTpzI9LX4+PjcrldEJG989hnccos5tLJ4cViyxEwY/scZeSLifrI1FFW0aNFMc2ksy6Jp06aZ7tHkYRHxCufOmZ6ZSZPMdYMGMG8eaMWniEfIVrBZs2ZNbtchImK/vXvNXjQ7dpjr4cPhpZfA39/eukQk27IVbO65556M5wcPHiQ8PPyS1VCWZREbG+va6kRE8sq8efDYY5CYCCVLwkcfQatWdlclIk5yevJwpUqVOH78+CXtJ0+epFKlSi4pSkQkz5w9awJNt24m1Nx9t+mxUagR8UhOB5sLc2n+LTExkcDAQJcUJSKSJ/bsMcu3p04FhwNGjoTVq6FsWbsrE5EcyvY+NkOGDAHA4XAwatQoChUqlPG1tLQ0Nm/eTJ06dVxfoYhIbpg9G554As6cgVKlYM4caNbM7qpE5BplO9j88MMPgOmx+emnnyjwjyWPBQoUoHbt2jz11FOur1BExJWSkqBfP5g501w3aQIffwylS9taloi4RraDzYWVUb1792bChAk6KVtEPM+uXWbV088/g48PPP88/Pe/4Otrd2Ui4iJOH6kwY8aM3KhDRCT3WJY526l/fzNZuEwZmDsXIiLsrkxEXMzpYCMi4lESEsxcmo8/NtctWpil3KGh9tYlIrnC6VVRIiIeY+dOuO02E2p8feHVV2H5coUaES+mHhsR8T6WBR98AAMHQnKyWb49fz7cdZfdlYlILlOwERHvEh8Pjz4Kn3xirtu2NSugSpa0tSwRyRsaihIR77FtG9x6qwk1fn7wxhvmVG6FGpF8Qz02IuL5LAsmToSnnoLz581J3AsWwJ132l2ZiOQxBRsR8WynT8PDD8OiRea6QweztLt4cXvrEhFbaChKRDzX99/DLbeYUOPvD+PHw+LFCjUi+ZiCjYh4HsuCceOgYUM4cAAqVYLvvjOroC5zSK+I5B8aihIRz3LyJDz0ECxdaq67dIFp0yAkxNayRMQ9qMdGRDzHhg1Qp44JNQUKwHvvmRVQCjUi8v8UbETE/aWnw9ixcPfdEBsL118PmzbBk09q6ElEMtFQlIi4t+PHoVcvcxQCwIMPwvvvQ1CQvXWJiFtSsBER97VunQkyhw9DYCC88w488oh6aUQkSxqKEhH3k5YGL78MjRubUHPjjbB5szkqQaFGRK5APTYi4l7++gt69ICvvzbXkZEwaRIUKWJvXSLiEdy+x+bQoUP06NGDEiVKUKhQIerUqcO2bdvsLktEcsM335hVT19/DQULwowZMHu2Qo2IZJtb99icOnWKhg0b0rhxY5YvX05oaCi//fYbRYsWtbs0EXGltDR48UV46SWz+V6NGmYZd/XqdlcmIh7GrYPN66+/Tnh4ODNmzMhoq1ixon0FiYjrHT4M3btDTIy5fvhhM0m4UCFbyxIRz+TWQ1FLlizhtttu4z//+Q+hoaHccsstTJ069YqvSU5OJj4+PtNDRNzUypVm6CkmBgoXhjlzzC7CCjUikkNuHWx+//13Jk+eTNWqVfnqq6/o27cvAwYMYPbs2Vm+ZsyYMYSEhGQ8wsPD87BiEcmW1FT473+hVSuzT02tWrBtm+m5ERG5Bg7Lsiy7i8hKgQIFuO2229iwYUNG24ABA9iyZQsbN2687GuSk5NJTk7OuI6Pjyc8PJy4uDiCg4NzvWYRuYo//zR703z7rbnu29ccaFmwoL11iYhbiY+PJyQkxOm/3249x6ZMmTJU/9fkwZtuuomFCxdm+ZqAgAACAgJyuzQRyYkvv4SePeHECbNz8NSp0LWr3VWJiBdx66Gohg0b8ssvv2Rq+/XXX6lQoYJNFYlIjqSkwLBh0LatCTW33grbtyvUiIjLuXWwGTx4MJs2beLVV19l3759zJ07lw8++ICoqCi7SxOR7PrjD3N45RtvmOv+/c0p3ddfb29dIuKV3HqODcCyZcsYMWIEe/fupVKlSgwZMoRHH30026/P6RidiLjA559D795w6hSEhMD06dC5s91ViYgHyOnfb7cPNtdKwUbEBufPm6GnCRPMdb16sGABVKpkb10i4jFy+vfbrYeiRMQD/f47NGx4MdQMHmxWQCnUiEgecOtVUSLiYRYuhD59ID4eihWDmTPh3nvtrkpE8hH12IjItTt3Dvr1gy5dTKipXx927FCoEZE8p2AjItdm715o0ADee89cDxsGa9dC+fL21iUi+ZKGokQk5+bPh8ceg4QEKFkSZs+G1q3trkpE8jH12IiI886ehccfN0cjJCRAo0Zm6EmhRkRspmAjIs7ZswfuuAM++AAcDhg5Er75BsqWtbsyERENRYmIEz76CJ54ApKSIDQU5syB5s3trkpEJIN6bETk6pKSzDLunj3N88aNzdCTQo2IuBkFGxG5sl274PbbYcYMM/Q0ejSsWgVlythdmYjIJTQUJSKXZ1kmzPTrZyYLly4Nc+ea3hoRETelYCMil0pMNHNp5swx182bm+ehofbWJSJyFRqKEpHMfvwR6tY1QcbHB155BVasUKgREY+gHhsRMSzLLOEeOBCSk83y7XnzzB41IiIeQsFGRMz5To89BgsWmOs2bWDWLLObsIiIB9FQlEh+t327GXpasAD8/GDsWFi6VKFGRDySemxE8ivLMgdXDh0K58+bQyvnzzcnc4uIeCgFG5H86PRpePhhWLTIXN97r1naXby4vXWJiFwjDUWJ5Dfffw+33GJCjb8/vP02REcr1IiIV1CPjUh+YVkwfjwMHw4pKVCpkplXU6+e3ZWJiLiMgo1IfnDyJPTuDUuWmOv77oNp06BoUXvrEhFxMQ1FiXi7jRuhTh0TagoUgIkT4dNPFWpExCsp2Ih4q/R0s3S7USOIjYUqVUzIiYoyh1mKiHghDUWJeKO//4aePWH5cnPdtavZVTg42N66RERymXpsRLzN+vVm6Gn5cggMhPffN0cjKNSISD6gYCPiLdLTzYGVERFw6BDceCNs3myOStDQk4jkExqKEvEGf/0FkZGwapW5joyESZOgSBF76xIRyWMKNiKe7ptvoHt3OHoUChY0xyQ89JB6aUQkX9JQlIinSkuD0aOhWTMTaqpXhy1bzH41CjUikk+px0bEEx05Ynpp1qwx1336wLvvQqFC9tYlImIzBRsRT7NqFfToAceOQeHCMHmymVMjIiIaihLxGKmpMHIktGxpQk3NmrB1q0KNiMg/qMdGxBP8+Sd062b2qAF4/HFzKnfBgvbWJSLiZjyqx2bMmDE4HA4GDRpkdykiuSMtDWJizIZ6MTHm+ssvzYZ769dDUJD52pQpCjUiIpfhMT02W7Zs4YMPPqBWrVp2lyKSOxYtgoEDTe/MBUWKQGKieX7LLbBgAVStak99IiIewCN6bBITE+nevTtTp06lWLFidpcj4nqLFkGXLplDDVwMNa1bw4YNCjUiIlfhEcEmKiqKtm3b0qxZM7tLEXG9tDTTU2NZWd/zv/+Bv3/e1SQi4qHcfihq/vz5bN++nS1btmTr/uTkZJKTkzOu4+Pjc6s0EddYv/7Snpp/i40190VE5ElJIiKeyq17bGJjYxk4cCBz5swhMDAwW68ZM2YMISEhGY/w8PBcrlLkGpw6ZSYCZ8eRI7lbi4iIF3BY1pX6v+0VHR1Np06d8PX1zWhLS0vD4XDg4+NDcnJypq/B5XtswsPDiYuLIzg4OM9qF7mi336D8eNh+nQ4cyZ7r1mzRj02IpJvxMfHExIS4vTfb7ceimratCk//fRTprbevXtTrVo1hg8ffkmoAQgICCAgICCvShTJPssyE4Dfeguioy/OqalZ0wxFnT59+Xk2DgeUKweNGuVtvSIiHsitg01QUBA333xzprbChQtTokSJS9pF3FZqqln19NZb8P33F9tbt4ahQ6FJE1i82KyKcjgyh5sLh1mOHw+XCfIiIpKZW8+xEfFo8fEwbhxUqQJdu5pQExAAjzwCu3aZjfeaNjXhpXNn+OwzKFs28/coV860d+5sz+8gIuJh3HqOjSvkdIxOJMcOHoQJE2DqVEhIMG0lS0JUFDz5JISGZv3atDSz+unIEShTxgw/qadGRPIhr5xjI+JRtmwxw02ffWYCCkC1ajBkiDmNOztHIPj6aoKwiMg1ULARuRZpabB0qQk03357sb1pUxNoWrUCH434iojkFQUbkZxISoKZM82k3n37TJu/Pzz4IAwebA6tFBGRPKdgI+KMw4fh3Xfh/ffN5noAxYpB377Qrx+Ehdlbn4hIPqdgI5IdO3aYFU7z50NKimmrUsX0zjz0EBQubGt5IiJiKNiIZCU9HVasMPNnvvnmYnujRmb+TPv2WrEkIuJmFGxE/u3sWZgzB95+G3bvNm2+vvCf/5hAU6+evfWJiEiWFGxELjh2DCZNMo/jx01bcDA8+ij07w8VKthbn4iIXJWCjcjPP5v5M3PmwIUDVCtUgIED4eGHTbgRERGPoGAj+ZNlwerVZv7MihUX22+/3Zzf1Lkz+Om/HiIinkb/yy35S3KyWdk0bhz8+KNpczigUyczf6ZBg4sHT4qIiMdRsJH84cQJs/fMxInmHCYwS7T79DFDTlWq2FufiIi4hIKNeLe9e83uwDNmmNVOYDbRGzAAHnvMbK4nIiJeQ8FGvI9lmROy33rLnON04QD7OnXM/Jn774cCBeytUUREcoWCjXiPlBRzsva4cbB168X2du3M/JmICM2fERHxcgo24vni4mDqVHjnHYiNNW2BgdCrFwwaBNWq2VufiIjkGQUb8Vz795swM20aJCaattBQcxhl375w3XX21iciInlOwUY8z6ZNZrhp4UJznhNAjRpmuKlbN9NbIyIi+ZKCjXiGtDSIjjYTgjduvNjevLmZENyihebPiIiIgo24uYQEs1R7/Hgz9ARmRVP37jB4MNSsaW99IiLiVhRsxD39+Se8+67ZVC8uzrSVKAFPPAFRUVC6tL31iYiIW1KwEfeyfbuZP7NgAaSmmrYbbjC9Mz17QqFC9tYnIiJuTcFG7JeeDl98YQJNTMzF9ogIMyG4bVvw8bGrOhER8SAKNmKfM2dg9mx4+2349VfT5ucHXbuaHpq6de2tT0REPI6CjeS9o0fhvfdg8mRzOCVASAg8/rjZgyY83N76RETEYynYSN756SfTO/Pxx3D+vGmrVMnsDty7NwQF2VufiIh4PAUbyV2WBStXmvkzK1debK9f3+w/07Ej+PraV5+IiHgVBRvJHefOwdy5JtDs2mXafHzgvvvM/Jn69e2tT0REvJKCjbjW33+buTMTJ8KxY6atSBF45BEYMMAMPYmIiOQSBRtxjT17zO7As2aZ3hqAcuVg4EATaooWtbc+ERHJFxRsJOcsy+w7M24cLFt2sb1uXTN/pksX8Pe3rTwREcl/FGzEeefPwyefmEDzww+mzeGA9u1NoGnUSAdSioiILRRsJPtOnYIPPjBnOB06ZNoKFjRLtQcONEcfiIiI2EjBRq7ut99gwgSYPh2Skkxb6dLQv7/ZVK9ECXvrExER+X9ufwDPmDFjqFevHkFBQYSGhtKxY0d++eUXu8vyfpYF331nlmdXrWp6aZKSoGZNmDkTDhyAZ59VqBEREbfi9sFm7dq1REVFsWnTJlatWkVqaiotWrQg6ULPgbhWaqqZP1O/Ptx1FyxaZEJOq1awahXs3Am9ekFAgN2VioiIXMJhWZZldxHOOH78OKGhoaxdu5a77777qvfHx8cTEhJCXFwcwcHBeVChh4qPhw8/NENOf/xh2gICoEcPs6FejRr21iciIvlKTv9+e9wcm7i4OACKFy9ucyVe4uBBeOcdmDrVhBuAkiXhySfNo1Qpe+sTERFxgkcFG8uyGDJkCHfddRc333zzZe9JTk4mOTk54zr+wh9ryWzLFrNc+9NPIS3NtFWrZnpnIiPNaicREREP41HBpl+/fvz44498++23Wd4zZswYXnjhhTysyoOkpcHSpSbQrF9/sb1JE7P/TKtW5jwnERERD+Uxc2z69+9PdHQ069ato9IVzhu6XI9NeHh4/p5jk5RkVjKNHw/79pk2Pz948EEYMgTq1LG1PBERkX/z2jk2lmXRv39/Fi9eTExMzBVDDUBAQAABWrFjHD5sDqOcMsVsrgfmzKa+faFfPyhb1t76REREXMztg01UVBRz587l888/JygoiKNHjwIQEhJCQc0DubydO81w07x5kJJi2qpUgUGD4KGHzGnbIiIiXsjth6IcWZw5NGPGDB566KGrvj7fLPdOT4cVK0ygWb36Yvtdd5nhpnvvBV9f++oTERFxglcPRckVnD0Lc+bA22/D7t2mzdfXnKw9ZAjcfru99YmIiOQhtw82koVjx2DSJPM4fty0BQXBo4/CgAFQoYK99YmIiNhAwcbT/Pyz6Z356CO4sPqrfHlzuvYjj4A3D7eJiIhchYKNJ7As+OYbeOstWL78Ynu9emb/mfvuM8u3RURE8jn9NXRn58+blU3jxsGPP5o2hwM6djTzZxo2NNciIiICKNi4p5Mnzd4zEyfCkSOmrVAh6NPHLNmuUsXe+kRERNyUgo072bvX7A48cyacOWPawsKgf394/HEoVszW8kRERNydgo3dLAu+/dbMn1myxFyDOeZgyBDo2hUKFLC3RhEREQ+hYGOXlBT47DMzf2br1ovtbduaQNO4sebPiIiIOEnBJq/FxcHUqfDOOxAba9oCA6FnTzN/5qab7K1PRETEgynY5JUDB2DCBJg2DRITTVtoKERFwRNPwHXX2VqeiIiIN1CwyW2bN5v5MwsXmvOcAKpXN8NN3bub3hoRERFxCQWbnEpLg/XrzXLsMmWgUaOLh0ympcHnn5tAs2HDxdc0b24CTcuWmj8jIiKSCxRscmLRInOEwZ9/XmwrVw5eew1OnDBLtvfvN+3+/qZnZvBgqFXLnnpFRETyCQUbZy1aZE7O/vep43/+CT16XLwuXtzMnYmKMj06IiIikusUbJyRlmZ6av4dav7Jz8/02PTubXYLFhERkTzjY3cBHmX9+szDT5eTmgo1aijUiIiI2EDBxhkXzm1y1X0iIiLiUgo2zsjuXBnNqREREbGFgo0zGjUyq5+yWqrtcEB4uLlPRERE8pyCjTN8fc3uwXBpuLlwPX78xf1sREREJE8p2Dirc2dzeGXZspnby5Uz7Z0721OXiIiIaLl3jnTuDB06ZL3zsIiIiNhCwSanfH0hIsLuKkREROQfNBQlIiIiXkPBRkRERLyGgo2IiIh4DQUbERER8RoKNiIiIuI1FGxERETEayjYiIiIiNdQsBERERGvoWAjIiIiXsPrdx62LAuA+Ph4mysRERGR7Lrwd/vC3/Hs8vpgk5CQAEB4eLjNlYiIiIizEhISCAkJyfb9DsvZKORh0tPTOXz4MEFBQTgcDpd+7/j4eMLDw4mNjSU4ONil39vb6L3KPr1X2af3Kvv0XmWf3qvsy833yrIsEhISCAsLw8cn+zNnvL7HxsfHh3LlyuXqzwgODtaHP5v0XmWf3qvs03uVfXqvsk/vVfbl1nvlTE/NBZo8LCIiIl5DwUZERES8hu/o0aNH212EJ/P19SUiIgI/P68f1btmeq+yT+9V9um9yj69V9mn9yr73O298vrJwyIiIpJ/aChKREREvIaCjYiIiHgNBRsRERHxGgo2IiIi4jUUbLKwbt062rdvT1hYGA6Hg+jo6Ku+Zu3atdStW5fAwEAqV67MlClT8qBS+zn7XsXExOBwOC557NmzJ48qts+YMWOoV68eQUFBhIaG0rFjR3755Zervm7hwoVUr16dgIAAqlevzuLFi/OgWnvl5L2aOXPmZT9b586dy6Oq7TF58mRq1aqVsUla/fr1Wb58+RVfkx8/U+D8e5VfP1OXM2bMGBwOB4MGDbrifXZ/thRsspCUlETt2rWZOHFitu7fv38/bdq0oVGjRvzwww88++yzDBgwgIULF+ZypfZz9r264JdffuHIkSMZj0s9eksAAAtWSURBVKpVq+ZShe5j7dq1REVFsWnTJlatWkVqaiotWrQgKSkpy9ds3LiRrl27EhkZyc6dO4mMjOT+++9n8+bNeVh53svJewVmB9R/fq6OHDlCYGBgHlVtj3LlyvHaa6+xdetWtm7dSpMmTejQoQO7du267P359TMFzr9XkD8/U/+2ZcsWPvjgA2rVqnXF+9zis2XJVQHW4sWLr3jPsGHDrGrVqmVqe/zxx60777wzN0tzO9l5r9asWWMB1qlTp/KoKvd17NgxC7DWrl2b5T3333+/1apVq0xtLVu2tB544IHcLs+tZOe9mjFjhhUSEpKHVbmvYsWKWdOmTbvs1/SZyuxK75U+U5aVkJBgVa1a1Vq1apV1zz33WAMHDszyXnf4bKnHxkU2btxIixYtMrW1bNmSrVu3kpKSYlNV7u2WW26hTJkyNG3alDVr1thdji3i4uIAKF68eJb3ZPXZ2rBhQ67W5m6y814BJCYmUqFCBcqVK0e7du344Ycf8qI8t5GWlsb8+fNJSkqifv36l71HnykjO+8V6DMVFRVF27Ztadas2VXvdYfPlntsE+gFjh49SqlSpTK1lSpVitTUVP7++2/KlCljU2Xup0yZMnzwwQfUrVuX5ORkPvroI5o2bUpMTAx333233eXlGcuyGDJkCHfddRc333xzlvdl9dk6evRobpfoNrL7XlWrVo2ZM2dSs2ZN4uPjmTBhAg0bNmTnzp1eP9T5008/Ub9+fc6dO0eRIkVYvHgx1atXv+y9+f0z5cx7lZ8/UwDz589n+/bt/F979x7S1P/GAfw9d8E0im8lGDQzu6yL2UWjqZWBURRJQfTXEM0osAsZWWpQFN1vlkkXiii6QFQWXcTSLi6yErMNLcuk1IIE+6PR1Wj2fP/4fju/r7qZ85ebbe8XHNzO+XzOec7Dg3vczjxlZWUdGt8daouNzW+kUqlaPJd//6lz6/W+zmAwwGAwKM+jo6Px5s0b7N6926cam2XLlqGiogL37t375VhHteVLddXRXBmNRhiNRuV5bGwsxo8fj9zcXOzfv7+rw/Qog8EAq9UKm82GvLw8JCUlwWw2O33B9uWaciVXvlxTb968wYoVK1BYWOjSNUWeri1+FPWbBAcHt+lIGxsbodFo0LdvXw9F9ecwGo2oqanxdBhus3z5cly5cgV37tzBgAED2h3rrLZa/1XkrVzJVWt+fn6YMGGCT9SWTqfDkCFDEBUVhW3btmHMmDHIyclxONbXa8qVXLXmSzVVXl6OxsZGREZGQqPRQKPRwGw2Y//+/dBoNGhubm4zpzvUFhub3yQ6OhpFRUUt1hUWFiIqKgpardZDUf05LBaLT3xcJyJYtmwZLl68iNu3b2PQoEG/nOOstmJiYroqzG6hM7lytA+r1eoTtdWaiODbt28Ot/lqTTnTXq4cjfWVmoqPj0dlZSWsVquyREVFwWQywWq1Qq1Wt5nTLWrLbZcp/2E+fvwoFotFLBaLAJDs7GyxWCxSX18vIiKZmZmSmJiojH/16pUEBATIypUrpaqqSo4dOyZarVYuXLjgqVNwG1dztXfvXrl06ZK8ePFCnjx5IpmZmQJA8vLyPHUKbpOamiq9e/eW4uJiaWhoUJYvX74oYxITEyUzM1N5XlJSImq1WrZv3y7Pnj2T7du3i0ajkYcPH3riFNymM7nasGGDXL9+XV6+fCkWi0UWLFggGo1GSktLPXEKbpOVlSV3796V2tpaqaiokLVr14qfn58UFhaKCGvqv1zNla/WlDOtvxXVHWuLjY0TP7+S3HpJSkoSEZGkpCSJi4trMae4uFjGjRsnOp1OQkND5dChQ+4P3ANczdWOHTtk8ODB4u/vL3/99ZdMmjRJ8vPzPRO8mznKEwA5fvy4MiYuLk7J3U/nz58Xg8EgWq1Whg8f7hNNYGdylZaWJiEhIaLT6SQoKEimT58u9+/fd3/wbpaSkiIDBw5Uzjs+Pl55oRZhTf2Xq7ny1ZpypnVj0x1rSyXy7xWuRERERH84XmNDREREXoONDREREXkNNjZERETkNdjYEBERkddgY0NEREReg40NEREReQ02NkREROQ12NgQ0W9VXFwMlUoFm83m6VD+b3V1dVCpVLBarZ4OhYg6iI0NEXXa1KlTkZaW5ukwuoxer0dDQwPCw8M9HQoRdZDG0wEQEXVXarUawcHBng6DiFzAd2yIqFOSk5NhNpuRk5MDlUoFlUqFuro6ZXt5eTmioqIQEBCAmJgYVFdXt5h/9epVREZGwt/fH2FhYdi4cSPsdrvDYzU1NWHUqFFYvHixsq62tha9e/fG0aNHncaYnZ2N0aNHIzAwEHq9HkuWLMGnT5+U7SkpKYiIiFDu7Pz9+3dERkbCZDIBaPtR1Pv372EymRAUFIQePXpg6NChOH78uGuJI6Ku5dY7UxGR17DZbBIdHS2LFi1S7rxtt9uVm6JOnDhRiouL5enTpzJ58mSJiYlR5l6/fl169eolJ06ckJcvX0phYaGEhobKhg0bnB7PYrGITqeTS5cuid1ul9jYWJkzZ067Me7du1du374tr169klu3bonBYJDU1FRl+8ePHyUsLEzS0tJERCQjI0NCQkLEZrOJiEhtba0AEIvFIiIiS5culbFjx0pZWZnU1tZKUVGRXLlypdM5JKLfj40NEXVa6zv9ivzvbu83b95U1uXn5wsA+fr1q4iITJ48WbZu3dpi3qlTp6R///7tHm/nzp3Sr18/Wb58uQQHB8u7d+9civfcuXPSt2/fFuvu378vWq1W1q1bJxqNRsxms7KtdWOTkJAgCxYscOmYRORevMaGiLpERESE8rh///4AgMbGRoSEhKC8vBxlZWXYsmWLMqa5uRlNTU348uULAgICHO5z1apVuHz5MnJzc1FQUIB+/fq1G8OdO3ewdetWVFVV4cOHD7Db7WhqasLnz58RGBgIAIiOjkZ6ejo2bdqEjIwMTJkyxen+UlNTMW/ePDx+/BjTp0/H3LlzERMT0+GcEFHX4zU2RNQltFqt8lilUgEAfvz4ofzcuHEjrFarslRWVqKmpgb+/v5O99nY2Ijq6mqo1WrU1NS0e/z6+nrMmjUL4eHhyMvLQ3l5OQ4cOADgn2tpfvrx4wdKSko6tM+ZM2eivr4eaWlpePv2LeLj45Gent5+IojIrdjYEFGn6XQ6NDc3uzxv/PjxqK6uxpAhQ9osfn7Ofy2lpKQgPDwcJ0+exJo1a1BVVeV07KNHj2C327Fnzx4YjUYMGzYMb9++bTNu165dePbsGcxmM27cuPHLi4GDgoKQnJyM06dPY9++fThy5EjHT5yIuhw/iiKiTgsNDUVpaSnq6urQs2dP9OnTp0Pz1q9fj9mzZ0Ov12P+/Pnw8/NDRUUFKisrsXnzZodzDhw4gAcPHqCiogJ6vR4FBQUwmUwoLS2FTqdrM37w4MGw2+3Izc1FQkICSkpKcPjw4RZjrFYr1q9fjwsXLiA2NhY5OTlYsWIF4uLiEBYW5jDuyMhIjBo1Ct++fcO1a9cwYsSIDp0zEbkH37Ehok5LT0+HWq3GyJEjERQUhNevX3do3owZM3Dt2jUUFRVhwoQJMBqNyM7OxsCBAx2Of/78OVavXo2DBw9Cr9cD+KfRsdlsWLduncM5Y8eORXZ2Nnbs2IHw8HCcOXMG27ZtU7Y3NTXBZDIhOTkZCQkJAICFCxdi2rRpSExMdPhOlE6nQ1ZWFiIiIjBlyhSo1WqcPXu2Q+dMRO6hEhHxdBBEREREvwPfsSEiIiKvwcaGiIiIvAYbGyIiIvIabGyIiIjIa7CxISIiIq/BxoaIiIi8BhsbIiIi8hpsbIiIiMhrsLEhIiIir8HGhoiIiLwGGxsiIiLyGmxsiIiIyGv8DWD5gU/z9rA7AAAAAElFTkSuQmCC",
      "text/plain": [
       "PyPlot.Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "PyObject Text(24,0.5,'the y axis')"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot([1,2,3,4], [1,4,9,16], \"ro-\")\n",
    "title(\"example plot\")\n",
    "xlabel(\"the x axis\")\n",
    "ylabel(\"the y axis\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the error vs. h:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-16:-1"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-16:-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16-element Array{Int64,1}:\n",
       " -16\n",
       " -15\n",
       " -14\n",
       " -13\n",
       " -12\n",
       " -11\n",
       " -10\n",
       "  -9\n",
       "  -8\n",
       "  -7\n",
       "  -6\n",
       "  -5\n",
       "  -4\n",
       "  -3\n",
       "  -2\n",
       "  -1"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(-16:-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.001"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "10.0^-3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16-element Array{Float64,1}:\n",
       " 1.0e-16\n",
       " 1.0e-15\n",
       " 1.0e-14\n",
       " 1.0e-13\n",
       " 1.0e-12\n",
       " 1.0e-11\n",
       " 1.0e-10\n",
       " 1.0e-9 \n",
       " 1.0e-8 \n",
       " 1.0e-7 \n",
       " 1.0e-6 \n",
       " 1.0e-5 \n",
       " 0.0001 \n",
       " 0.001  \n",
       " 0.01   \n",
       " 0.1    "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = 10.0 .^ (-16:-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100-element Array{Float64,1}:\n",
       " 1.0e-16    \n",
       " 1.41747e-16\n",
       " 2.00923e-16\n",
       " 2.84804e-16\n",
       " 4.03702e-16\n",
       " 5.72237e-16\n",
       " 8.11131e-16\n",
       " 1.14976e-15\n",
       " 1.62975e-15\n",
       " 2.31013e-15\n",
       " 3.27455e-15\n",
       " 4.64159e-15\n",
       " 6.57933e-15\n",
       " ⋮          \n",
       " 0.00215443 \n",
       " 0.00305386 \n",
       " 0.00432876 \n",
       " 0.00613591 \n",
       " 0.00869749 \n",
       " 0.0123285  \n",
       " 0.0174753  \n",
       " 0.0247708  \n",
       " 0.0351119  \n",
       " 0.0497702  \n",
       " 0.070548   \n",
       " 0.1        "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = logspace(-16, -1, 100) # 100 numbers from 1e-16 to 1e-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Int64,1}:\n",
       "   1\n",
       "   4\n",
       "   9\n",
       "  16\n",
       "  25\n",
       "  36\n",
       "  49\n",
       "  64\n",
       "  81\n",
       " 100"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[i^2 for i = 1:10] # array \"comprehension\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100-element Array{Float64,1}:\n",
       " 2.88051   \n",
       " 2.88051   \n",
       " 2.88051   \n",
       " 2.9668    \n",
       " 1.24465   \n",
       " 0.0297085 \n",
       " 0.267579  \n",
       " 0.064616  \n",
       " 0.253113  \n",
       " 0.0751118 \n",
       " 0.286808  \n",
       " 0.129822  \n",
       " 0.0287368 \n",
       " ⋮         \n",
       " 0.00268886\n",
       " 0.0037995 \n",
       " 0.00536181\n",
       " 0.00755221\n",
       " 0.0106085 \n",
       " 0.014843  \n",
       " 0.0206485 \n",
       " 0.0284812 \n",
       " 0.038784  \n",
       " 0.0517725 \n",
       " 0.0669179 \n",
       " 0.0817831 "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "err = [ abs(deriv(f, 2, δ) - 3cos(6)) for δ in h ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Int64,1}:\n",
       " 1\n",
       " 2\n",
       " 3"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[1,2,3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1×3 RowVector{Int64,Array{Int64,1}}:\n",
       " 1  2  3"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[1,2,3]' # transpose of a real vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\"Broadcasting\" operations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       " 0.321751  0.588003  0.785398\n",
       " 0.244979  0.463648  0.643501\n",
       " 0.197396  0.380506  0.54042 "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "atan.([1,2,3]', [3,4,5])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Int64,2}:\n",
       " 4  5  6\n",
       " 5  6  7\n",
       " 6  7  8"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[1,2,3]' .+ [3,4,5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Evaluate `deriv` elementwise by `deriv.(f, 2, h)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100-element Array{Float64,1}:\n",
       " 2.88051   \n",
       " 2.88051   \n",
       " 2.88051   \n",
       " 2.9668    \n",
       " 1.24465   \n",
       " 0.0297085 \n",
       " 0.267579  \n",
       " 0.064616  \n",
       " 0.253113  \n",
       " 0.0751118 \n",
       " 0.286808  \n",
       " 0.129822  \n",
       " 0.0287368 \n",
       " ⋮         \n",
       " 0.00268886\n",
       " 0.0037995 \n",
       " 0.00536181\n",
       " 0.00755221\n",
       " 0.0106085 \n",
       " 0.014843  \n",
       " 0.0206485 \n",
       " 0.0284812 \n",
       " 0.038784  \n",
       " 0.0517725 \n",
       " 0.0669179 \n",
       " 0.0817831 "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "err = abs.(deriv.(f, 2, h) - 3cos(6)) # call abs and deriv elementwise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "PyPlot.Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "PyObject <matplotlib.legend.Legend object at 0x149e2b908>"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loglog(h, err, \"b.-\")\n",
    "loglog(h, h, \"k-\")\n",
    "title(\"forward difference error\")\n",
    "xlabel(\"h\")\n",
    "ylabel(\"|error|\")\n",
    "legend([\"error\", \"h\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Two questions:\n",
    "\n",
    "1. Why does the error initially go down roughly proportional to $h$, as long as $h$ is not too small?\n",
    "\n",
    "2. Why does the error go *up* when $h$ decreases below about $10^{-8}$???"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Why does the error increase?\n",
    "\n",
    "Let's try the derivative of $g(x) = 2x$, which should be 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "g (generic function with 1 method)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(x) = 2x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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74O7uzvbt25k/fz63b9/Gy8uLoKAgZs2aJcmSuG9SwySEEEIIUQAZNiCEEEIIUYBynzD169cva0VyIYQQQghLyn3CNHbsWFauXKl1GEIIIYSwYeW+03fnzp3zXdwyL5mZmVy8eJHKlStrthSAEEIIIQpHURRu376Nt7d34SY0Vcqw2NhYpVevXkrt2rUVQNm0aZPZMZ999pny0EMPKS4uLkrLli2VXbt2mR0TExOj9O/fv1D3TkhIUADZZJNNNtlkk60MbgkJCYX63S/TNUypqam0aNGCl156if79+5u9vm7dOsaNG0d4eDiPPfZY1hICx48fp06dOoW6V3p6Ounp6Vn7yj+DCxMSEqxaJkIIIYQQ2ktOTubBBx/Md2FrS8p0whQUFJTvMg/z5s0jNDSU4cOHAzB//ny2bdvGwoULmTVrVqHuNWvWLGbMmGFW7u7uLgmTEEIIUcYUtjuN3Xb6vnv3LgcOHKBbt24m5d26dSMuLq7Q15syZQpJSUlZW0JCQnGFKoQQQggbV6ZrmPJz/fp1DAYDNWvWNCmvWbMmly9fztrv3r07Bw8eJDU1FR8fHzZt2kTr1q3Nrufi4oKLi0uJxy2EEEII22O3CZNR7io3RVFMyrZt21baIQkhhBCijLHbJjkvLy/0er1JbRLA1atXzWqdhBBCCCHyY7cJk7OzM/7+/kRHR5uUR0dH0759e42iEkIIIURZVKab5FJSUjh9+nTW/tmzZzl8+DBVq1alTp06jB8/niFDhtCqVSvatWvH4sWLuXDhAiNHjtQwaiGEEEKUNWU6Ydq/fz+dO3fO2h8/fjwAISEhLF++nEGDBnHjxg3effddLl26hK+vL1u3bqVu3bpahSyEEEKIMkinGGdgFIWSnJyMh4cHSUlJMg+TEEIIUUYU9ffbbvswCSGEEEIUF0mYhBBCCCEKIAmTDUpMhJgY9dHSfl5lQgghhCgZZbrTtz2KjISwMMjMBAcHGDIEVq3K3l+8WD0u5zGLF0NoqLZxCyGEEPZMOn0XUUl0+k5MhLp11USoMPR6OHcOfHyKJQwhhBDCbkmnbztw6lThkyUAgwFyTEclhBBCiGImCZMNadRIbWLLj06nbjk5OEDDhiUXlxBCCFHeScJkQ3x81P5Ier26r9dDSIjp/uefq5uxzOi330o3ViGEEKI8kT5MRVSSE1cmJqpNbA0bqklU7n3jMb//DgsWwDffgLs7rF8Pjo5qTZX0ZxJCCCHMFfX3WxKmIrKVmb7T06F7d4iNzS6TkXNCCCGEZdLpu5xycYFPPzUty8yEl1+WOZqEEEKI4iIJkx24ds28TEbOCSGEEMVHEiY7YGl0nV4vI+eEEEKI4iIJkx0wjq4zTjeg00FEhHT8FkIIIYqLJEx2IjQU3ntPfe7vLx2+hRBCiOIkCZMdGTxYfTx4EG7c0DYWIYQQwp5IwmRH6tSB5s3VUXLffWf5mMREiImREXRCCCFEYUjCZGd69VIft2wxf23JEnVx3y5d1MfIyNKNTQghhCirJGGyM089pT5+9x3cu5ddnpgIYWHZi/vKXE1CCCGE9SRhsjNt2kC1anDrFsTFZZf/8APkntM9r7mapNlOCCGEMCUJk53R66FnT/W5sVlOUWDhQsvH5p6rKTJSmu2EEEKI3CRhskPGZjljwrR8Ofz8Mzg5mU5w2aeP6VxNiYkwYoQ02wkhhBC5ScJkh7p3V2uPTpyAvXthwgS1/P334fx5eOstdf+77+DChezz/vMf65vthBBCiPJEEiY7VKUKBAaqz7t1g7/+ghYtYNw4tUZp5kz19bQ0GDVK7a80Y4blZjudDurXL934hRBCCFsjCZOdql5dfUxOVh979VKb5EBNgj75RH2MilL7K02frr721FNq7ZSRoqg1T9IJXAghRHmmU5TcjTDCGsnJyXh4eJCUlIS7u7vW4ZhITFQ7bBv7IoGaBJ07l91nKTFRnegy5399nU5tstPp1Ga4+Hi1VsrIwUFds06WXRFCCFFWFfX3W2qY7NCpU6bJEpj3RTp1yry/kqLAmTNqUtWpE/Tvn72gL0gncCGEEOVXuU6Y+vXrh6enJ88++6zWoRSrRo1MR8OB+RQC1hxjKamSTuBCCCHKo3KdMI0dO5aVK1dqHUax8/FRm86MfZH0eoiIMJ1CwJpjLCVVOp353E1CCCGEvSvXCVPnzp2pXLmy1mGUiNBQtc9STIz6aKnfUUHH5E6qABwdTZdcMZLZwYUQQtgzm02Ydu3aRe/evfH29kan07F582azY8LDw6lXrx6urq74+/vz448/ahCp7TL2RcpZa1TYY4xJ1Q8/QNu2kJFh2hEc8p4dXJIoIYQQRXXz5k1SU1O1DiOLzSZMqamptGjRgk8//dTi6+vWrWPcuHG89dZbHDp0iMDAQIKCgriQYyZGf39/fH19zbaLFy8WOp709HSSk5NNtvLCxwc6d4YlS9Qapm++gRUr1GRo3z7zRX3DwmDqVFliRQghRNFs376dZs2a8cYbb2gdSjalDACUTZs2mZQFBAQoI0eONClr0qSJMnny5EJdOyYmRunfv3+Bx02bNk0BzLakpKRC3a+smzBBUdSu4IXb9HpFSUjQOnohhBC2LCUlRRk1alTWb+zDDz+spKSkFOs9kpKSivT7bbM1TPm5e/cuBw4coFu3bibl3bp1Iy4urkTuOWXKFJKSkrK2hISEErmPrRs+vGjnyeg6IYQQ+YmLi8PPz4/w8HAARo8ezcGDB6lYsaLGkanKZMJ0/fp1DAYDNWvWNCmvWbMmly9ftvo63bt3Z8CAAWzduhUfHx/27duX57EuLi64u7ubbOVRXq2ZxtF0ej3MmVPwlAVCCCEEqF1epkyZQmBgIKdPn8bHx4fo6Gg++eQT3NzctA4vi6PWAdwPXc5ZFQFFUczK8rNt27biDsnuGacayD2L+J49kJqqJkU+PlC1qmnfpvDw/DufCyGEKH+OHDnCkCFDOHLkCADBwcF8/PHHVKlSRePIzJXJGiYvLy/0er1ZbdLVq1fNap1E8cpr/qbWrU1H24WGqhNfGr/zXl6ahCuEEMIGGQwGZs+eTatWrThy5AheXl5s3LiRFStW2GSyBGU0YXJ2dsbf35/o6GiT8ujoaNq3b69RVOWHNXM8AdSvD6+8oj5ftKi0ohNCCGHLTp8+TceOHZkyZQoZGRn07duXY8eO0a9fP61Dy5fNNsmlpKRwOkcv4bNnz3L48GGqVq1KnTp1GD9+PEOGDKFVq1a0a9eOxYsXc+HCBUaOHKlh1OWHj491TWxhYTB7NkRHqzVOjRqVfGxCCCFsj6IoLFq0iAkTJnDnzh3c3d1ZsGABwcHBhepOoxWbTZj2799P586ds/bHjx8PQEhICMuXL2fQoEHcuHGDd999l0uXLuHr68vWrVupW7euViELCx56CIKCYOtWtenuv//VOiIhhBClLTExkdDQULZv3w5Aly5dWLZsGXXq1NE4MuvpFCX38qrCGsnJyXh4eJCUlFRuR8xZa8sW6N1b7Qj+55/g6qp1REIIIUqDoiisWbOG0aNHc+vWLVxdXZkzZw6vvvoqDrmHU5eSov5+l8k+TKJsCQqCOnXgr7/g66+1jkYIIURpuH79OgMGDGDw4MHcunWL1q1bc+jQIcaMGaNZsnQ/yl7EoszR69W+TKD2Z5K15YQQwr7973//w9fXlw0bNuDo6Mi7775LXFwcTZo00Tq0IpOESZSKChXUx+PHZW05IYSwV8nJyQwfPpw+ffpw5coVmjZtyt69e3n77bdxdLTZbtNWkYRJlLjERJg4MXs/MxNefllqmoQQwp7s3LmT5s2bExkZiU6nY8KECezfv5+WLVtqHVqxkIRJlLhTp0xnBgdZW04IIexFWloa48ePp3Pnzpw/f56HHnqInTt38uGHH+JqR6N8ynb9mCgT8lpOJa+15RITs+dskuVUhBDCdu3fv5/g4GBOnDgBwIgRI5g7dy6VK1fWOLLiJzVMosQZl1PJOSgiIsJyMrRkidrHqUsX6eskhBC2KiMjg+nTp9O2bVtOnDhBrVq1iIqKYvHixXaZLIHMw1RkMg9T4X39NQwcqC6ZcuaM+euJier0Azm/kXq9uvyK1DQJIYRtOHHiBEOGDOHAgQMADBw4kPDwcKpVq6ZxZNaReZiEzXvkEfUxKcny66dOmSZLIH2dhBDCVmRmZvLRRx/x6KOPcuDAATw9PVm7di3r1q0rM8nS/ZA+TKLUeHmpjzdvqomQXm/6eqNGoNOZJ02OjupCv9KnSQghtHHu3Dleeukldu7cCUCPHj2IjIzE29tb28BKkdQwiVJTtar6mJkJt26Zv+7jA5ZGnwYGSp8mIYTQgqIoLF26lObNm7Nz504qVqzIwoUL2bp1a7lKlkASJlGKnJzAw0N9fv265WMqVVIf334bPv3U9DWZv0kIIUrP5cuX6dOnD6Ghody+fZvHHnuM+Ph4Ro4ciU6n0zq8UicJkyhVxma5vBKmq1fVx06dsvs85SR9moQQouStX78eX19ftmzZgrOzM3PmzCE2NpYGDRpoHZpmpA+TKFVeXuoIuRs3LL9+7Zr6WL06eHqaz98EcOAANGigJk7Sr0kIIYrPzZs3GTNmDKtXrwbAz8+PlStX0qxZM40j057UMIlSlV8Nk8GQnUhVr549f1PuzuETJqjTD0i/JiGEKD7bt2+nWbNmrF69GgcHB6ZOncrevXslWfqHJEyiVBlHnlpKmG7cyB4hZzwuNFSdhykmRn2cNMn0nMxMCAuDX35Rj5H+TUIIUTipqam8+uqrdO/enT///JNGjRqxe/duZs6cibOzs9bh2QxJmESpyq+GydgcV7Wq2kHcyMdH7dNUty706GF+XmYmtGkjNU5CCFFYcXFx+Pn5ER4eDsDo0aM5fPgwbdu21Tgy2yMJkyhV1iRM1avnfb5xXbq85DeSLjFRaqGEEAIgPT2dKVOmEBgYyOnTp/Hx8SE6OppPPvkENzc3rcOzSZIwiVJlTJgsdfq2JmHK3a/JUvJkaSRdZKTlNeokiRJClDdHjhwhICCA2bNnk5mZSXBwMEePHuWJJ57QOjSbJgmTKFX51TAZpxSoUSP/a+Ts1/Tzz5aTpoSE7OeJiWo/J+Nou8xMGD4cmjSRzuNCiPLDYDAwe/ZsWrVqxZEjR6hevTobN25kxYoVVKlSRevwbJ5MKyBKVX6dvq2pYTLy8cmeTmDxYrUZzmDIXlpl9Gjw9lb3160zn5oA4Lffsp8bm/K6d5dpCoQQ9ufUqVOEhISwZ88eAPr27UtERAQ1a9bUOLKyQxImUarutw+TJaGhaqJz+rRaY/TSS7BrF+RXu2xpzTpjU54kTEIIe6EoCosWLWLChAncuXMHd3d3FixYQHBwcLmcrft+SJOcKFW5F+DNydomOUuMI+nq1zdfUsXI2O9Jr4cPPjBvynNwgIYNC39vIYSwRYmJifTo0YNRo0Zx584dunTpwtGjRwkJCZFkqQikhkmUKuMCvIqiJk3GBAqKXsOUW17Lrqxdq167YUM1wapaNbspD9TyBx64v3sLIYTWFEVhzZo1vPrqqyQlJeHq6soHH3zA6NGjcchvmLHIl3xyolQ5OoKxb2HuxKa4EiZLUw/o9dCunVoLZWxyM3YeX7cOnJ3h998hKur+7i2EEFq6du0aAwcOZPDgwSQlJREQEMChQ4cYO3asJEv3ST49Uery6sdUXAlT7qkH9HqIiLDcN8nHBwYOhNdeU/cnTYJ79+7v/kIIoYVvv/0WX19f1q9fj6OjIzNnzmT37t00adJE69DsgiRMotRZmovJYMhOoIrShym33EuqhIbmf/zkyWoT3YkT8MYbMi+TEKLsSE5OJjQ0lL59+3L16lWaNm3K3r17mTp1Ko6O0vOmuJTbhOn27du0bt0aPz8/mjVrxueff651SOWGpRqmv/4yX0fufhk7glsz6q1KFXjySfX5vHkyL5MQomyIjY2lefPmLF26FJ1Ox4QJE9i/fz8tW7bUOjS7U25TTzc3N2JjY3Fzc+POnTv4+vryzDPPUK24fq1FnizNxWRsjvP0NF1HrrQkJsLXX2fvy7xMQghblpaWxptvvsn8+fMBqFevHitWrCAwMFDjyOxXua1h0uv1WevDSTvzAAAgAElEQVTl/P333xgMBpTcE/OIEmGphul+phQoDqdOmU9uaWmJFSGE0JqxBsmYLI0YMYL4+HhJlkqYzSZMu3btonfv3nh7e6PT6di8ebPZMeHh4dSrVw9XV1f8/f358ccfC3WPW7du0aJFC3x8fJg0aRJeOce4ixJjKWEqrg7fRWVpZJ1Ol/e8TLIGnRCitGVkZDB9+nTatm3LyZMnqV27NlFRUSxevJjKlStrHZ7ds9mEKTU1lRYtWvBpHrMQrlu3jnHjxvHWW29x6NAhAgMDCQoK4sKFC1nH+Pv74+vra7ZdvHgRgCpVqhAfH8/Zs2dZs2YNV65cyTOe9PR0kpOTTTZRNJY6fWudMOUeWQfqFAiW+ksuXmx5IV8hhCgpx48fp127dsyYMQODwcCgQYM4evQoPXv21Dq08kMpAwBl06ZNJmUBAQHKyJEjTcqaNGmiTJ48uUj3GDlypPLVV1/l+fq0adMUwGxLSkoq0v3Ks02bFAUUpV277LJp09SysDDNwlIURVESEhTlhx8Uxc9PjWf0aNPXz51Ty3Nuer16nhBCFDeDwaDMmzdPcXFxUQDF09NTWbt2rdZhlWlJSUlF+v222Rqm/Ny9e5cDBw7QrVs3k/Ju3boRFxdn1TWuXLmSVUuUnJzMrl27ePjhh/M8fsqUKSQlJWVtCQkJRX8D5Vx+nb616sNk5OMDnTvD3LnqfkQEnD+vPlcUGDXK/Bzp6ySEKAnnzp2jS5cujB8/nvT0dIKCgjh27BjPPfec1qGVS2UyYbp+/ToGg8FsleWaNWty+fJlq66RmJhIx44dadGiBR06dGD06NE0b948z+NdXFxwd3c32UTR2GIfpty6dFG3jAyYMUMtmzIFtm41P1bWoBNCFCdFUYiMjKR58+bExsZSsWJFIiIiiIqKwtvbW+vwyq0yPa1A7sUDFUWxekFBf39/Dh8+XBJhiQIYE6Zbt9RZtR0dbS9hAnj/fXU5leXL1XXvjOMOQkLgiy9MFw/+5Rd1pF2jRnlPQ5CYWPAxQojy7fLly4wYMYItW7YA8Nhjj7FixQoaNGigcWSiTNYweXl5odfrzWqTrl69albrJGyPp6c6As24AC9oP62AJW3bQosWapzGZOnZZ9UE6tw5+OEH6NVLnY6gf3/zTuA5R9JFRkpHcSFE/tavX4+vry9btmzB2dmZOXPmEBsbK8mSjSiTNUzOzs74+/sTHR1Nv379ssqjo6Pp27evhpEJaxgX4L15U22Wq17dNmuYEhPh6FHTsk2b1HIfH3V74AH45w9BQE2ewsLU9/PWW+q+sdLTOM2XTIophMjp5s2bjB49mjVr1gDg5+fHqlWr8PX11TgykZPNJkwpKSmcztGT9uzZsxw+fJiqVatSp04dxo8fz5AhQ2jVqhXt2rVj8eLFXLhwgZEjR2oYtbCWl1d2wmQwZE8xYEsJU36TWRoTnT//ND8vM1Pt72RkaT7U3NcRQpRP27dvZ9iwYfz55584ODgwZcoU3nnnHZydnbUOTeRiswnT/v376dy5c9b++PHjAQgJCWH58uUMGjSIGzdu8O6773Lp0iV8fX3ZunUrdevW1SpkUQheXmpCcv26uo6cMTGxpblDjZNZ5kya9HrTDt6WjrFG7usIIcqX1NRUJk6cyMKFCwFo3LgxK1eupE2bNhpHJvJis32YOnXqhKIoZtvy5cuzjhk1ahTnzp0jPT2dAwcO0LFjR+0CFoWSc/JKrdeRy0vuySz1enWagZy1QpaO+eADy7OG5ywLD5faJSHKq7i4OPz8/LKSpTFjxnDo0CFJlmyczdYwCfuWc2oBW+y/ZBQaqvY1On1arRGylORYOqZaNbWfksGQnWg9+aTaifzWLbVmSghRvqSnpzN9+nTmzJlDZmYmPj4+LF++nK5du2odmrCCJExCEzknr7TlhAmyO3gX5pi8Eq2+fWHFCoiKUifIFEKUD/Hx8QQHB3PkyBEAgoOD+fjjj6lSpYrGkQlr2WyTnLBvOWuYbHFKgeLg4wOdOpkmUk89pT5GRWkSkhCilN27d49Zs2bRunVrjhw5QvXq1dm4cSMrVqyQZKmMkRomoQlLfZhstYapOHXrpjbRnTwJf/wB9etrHZEQoqScOnWKkJAQ9uzZA8DTTz9NREQENeztr8NyQmqYhCbKSh+m4ubhAR06qM8tLbMihCj7FEUhPDwcPz8/9uzZg7u7O8uXL2fjxo2SLJVhkjAJTeTsw2SvTXJ5kWY5IexXYmIiPXr04NVXX+XOnTt06dKFo0ePEhISYvXSXcI2ScIkNFFea5ggO2GKiYHUVG1jEUIUD0VRWL16Nb6+vmzfvh1XV1cWLFhAdHQ0derU0To8UQwkYRKayLkA76VL6vPykjD961/w0EOQnq6uR1dYOdeoE0Jo79q1awwYMIDBgweTlJREQEAAhw4dYsyYMTjknpRNlFnyX1JowrgAL8CZM+pjeUmYdDro2VN9XthmOVnEVwjb8r///Y9mzZqxYcMGHB0dmTlzJrt376ZJkyZahyaKmSRMQhN6PVStqj6/d099LC99mCC7WW7rVstrzVmSmKgu7GtchsW4iK/UNAlR+pKTkwkNDaVPnz5cuXKFpk2bsnfvXqZOnYqjowxAt0eSMAnNGDt+G9nSOnIlrXNnqFABEhJg6VLrkp78FgMWQpSenTt30rx5c5YuXYpOp2PChAns37+fli1bah2aKEGSMAnN5EyQqlSxrXXkSlqFCtmL7w4fbl3zWqNG2c2YRjqdLOIrRGlJS0tj/PjxdO7cmfPnz1OvXj1iY2P58MMPcXV11To8UcIkYRKayZkwlafmOFBrlI4dy963pnmtRg01scypQgVwdy+ZGIUQ2Yw1SB999BEAYWFhxMfHExgYqHFkorRIwiQ0kzNhKi8dvo1OnTLvu1RQ89oXX8DNm2ri9H//p84SfucOzJ9fsrEKUZ5lZGQwffp02rZty8mTJ6lVqxZRUVFERERQuXJlrcMTpUgSJqGZnH2YylvC1KgR5B5t7OCQd/OawQAffKA+nzgRevSA999X9+fNUxMpIUTxOn78OO3atWPGjBkYDAYGDRrEsWPH6Gkc5irKFUmYhGbKc5Ocjw8sXqyOFsxZltfnsGkT/P67Oh3Dyy+rZQMHgq8vJCWpSZMQonhkZmYyb948WrZsyYEDB/D09GTt2rV8+eWXVMs9WkWUG5IwCc2U5yY5gNBQOHcO1q9X+yZduADvvWd+nKLA7Nnq89GjwdgK4OAAM2aoz+fNg82bZYoBIe7XuXPn6NKlC6+//jrp6ekEBQVx7NgxnnvuOa1DExqThEloprwnTKDWKvXvr9Y2gdrMFh5umvisXQsHDoCrK4wda3r+00/Dgw+qfZn69ZPJLIUoKkVRWLp0Kc2aNSM2NpaKFSsSERFBVFQU3t7eWocnbIAkTEIzkjBlGzAA2rRRR8u9+irUqaMmQ336wIsvqsekp8M335ied/GiaXIlk1kKUXiXL1+mT58+hIaGkpKSQocOHThy5AhhYWGyYK7IIgmT0EzOrgDl/d+kxETYty97X1HU5Oh//zMty50MFWW0nRAi2/r16/H19WXLli04OzszZ84cdu7cSf369bUOTdgYmb9daOa777KfDx6sNiuFhmoXj5YszeJtiTEZ8vFR942j7XKeq9fLZJZCFOTmzZuMHj2aNWvWAODn58eqVavw9fXVODJhq6SGSWgiMRHGj8/eL+9NSXlNM5C7LHcyZBxtl/OciIjshEoIYW7btm34+vqyZs0aHBwcmDp1Knv37pVkSeRLEiahCVkXzVTuaQb0enU/d5mlZCg0VO3wDTB5cvmtpROiIKmpqYwaNYoePXpw8eJFGjVqxO7du5k5cybOzs5ahydsnDTJCU1IU5K50FDo3l1NGhs2zE6MLJXl1qiR+njnTuHumZioJq+NGkmtlLBvcXFxBAcHc+bMGQDGjBnD7NmzcXNz0zgyUVZIDZPQhKUaFWlKUt9/p06mn4OlstyMo57//NP6e0VGqtMQdOki0xEI+5Wens6UKVMIDAzkzJkz+Pj4sGPHDhYsWCDJkigUqWESmsmrRkUUnjFhunjRuuMTEyEsLLuGz9iHrHt3+e8g7Ed8fDxDhgzh6NGjAAwZMoQFCxZQJfcq1kJYQRImoSkfH/mBLg6FTZjy60Mm/z1EWXfv3j0+/PBDpk2bRkZGBtWrVyciIoJ+xs5+QhRBuU6YHB0ds0ZFtGrViiVLlmgckRBFkzNhUpSC57Vq1Eg9JuccTuW9D5mwD6dOnSIkJIQ9e/YA0LdvXxYvXkyN8rZgpSh25TphqlKlCocPH9Y6DCHuW+3a6mN6Oty8CVWr5n+8sV9UTIy6r9NJHzJRtimKwsKFC5k4cSJ37tzB3d2dBQsWEBwcLLN1i2JRrhMmIeyFq6s6c/qNG2otU0EJE4CHR/bzJk1kOgJRdiUmJjJs2DCio6MB6NKlC8uWLaNOnToaRybsic2Oktu1axe9e/fG29sbnU7H5s2bzY4JDw+nXr16uLq64u/vz48//lioeyQnJ+Pv70+HDh2IjY0trtCF0ERh+zHlHFF34oT1c2AlJqo1U+V1klFhOxRF4YsvvsDX15fo6GhcXV1ZsGAB0dHRkiyJYmezCVNqaiotWrTg008/tfj6unXrGDduHG+99RaHDh0iMDCQoKAgLly4kHWMv78/vr6+ZtvFf35Rzp07x4EDB1i0aBHBwcEkJyeXynsToiQUNWGqWVN93LCh4HNkKgJhK65du8aAAQMYMmQISUlJtG7dmkOHDjFmzBgcck+RL0RxUMoAQNm0aZNJWUBAgDJy5EiTsiZNmiiTJ08u0j169Oih7Nu3L8/X//77byUpKSlrS0hIUAAlKSmpSPcTori99JKigKK8917Bx2ZkKIqDg3r8tGnqY+vW+Z+TkJB9jnHT69VyIUrTN998o9SoUUMBFEdHR+Xdd99VMjIytA5LlBFJSUlF+v0uk2n43bt3OXDgAN26dTMp79atG3FxcVZd4+bNm6SnpwNq+/fx48fzXZ161qxZeHh4ZG0PPvhg0d+AECWgMDVMV66o0wro9er8Szod7NsH58/nfY4sZyO0lpyczLBhw+jbty9Xr16ladOm7N27l7fffhtHR+mSK0pWmUyYrl+/jsFgoKaxLeEfNWvW5PLly1Zd48SJE7Rq1YoWLVrQq1cvPv74Y6rm01N2ypQpJCUlZW0JCQn39R6EKG6FSZiM/Y9q11a3jh3V/Y0b8z7HOBVBTjqdTEUgSsfOnTtp3rw5y5YtQ6fTMWHCBPbv30/Lli21Dk2UE2U6Jc89VFRRFKuHj7Zv3z5r9ldruLi44OLiUqj4hChNhUmYjP2XHnhAfezfH2JjYf16eO01y+f4+IC/P+zfn12mKJCQINMRiJKTlpbGm2++yfz58wGoV68eK1asIDAwUOPIRHlTJmuYvLy80Ov1ZrVJV69eNat1EqK8MCY/RUmYnnlGfYyLy/v8tDR1NB3AokUwcKD6fNw486Y6GUknioOxBsmYLIWFhREfHy/JktBEmUyYnJ2d8ff3z5pzwyg6Opr27dtrFJUQ2jLWMF26ZJ7A5GZMmIw1Qw88AO3aqc8/+MByorNtG6SmqqPjwsLg44+hUiX45RdYvTr7OBlJJ+5XRkYG06dPp23btpw8eZJatWoRFRVFREQElStX1jo8UU7ZbMKUkpLC4cOHs2biPnv2LIcPH86aNmD8+PEsWbKEpUuXcuLECV577TUuXLjAyJEjtQxbCM3UrKn2KTIY4Nq1/I/NXcME2cnTggWWEx3jtAPPPKPep1YteOsttWziRNi6VU2eLC3qKzVNwlrHjx+nXbt2zJgxA4PBwMCBAzl27Bg9e/bUOjRR3pXMoL37FxMTowBmW0hISNYxn332mVK3bl3F2dlZadmypRIbG1tq8RV1WKIQJalWLXW4/8GD+R/XubN63BdfqPsFTRmQnq4oHh5q+U8/ZV8nLU1RvLxMz7O0xcSUyNsVdsRgMChz585VXFxcFEDx9PRU1q5dq3VYwg4V9ffbZjt9d+rUCSXnyqAWjBo1ilGjRpVSRELYPm9vuHxZrUF69NG8j8tdw5TflAE+PvD995CUpI6oMzbdAVy/ri7Hkh8HBxlJJ/J37tw5hg4dmrXiQo8ePYiMjMTb2M4shA2w2SY5IUThWTNSTlGym8iMCVOjRmpik1PORMfYHNevn+lxp06p18st5zHu7lClivXvQZQfiqIQGRlJs2bNiI2NpWLFiixatIitW7dKsiRsjiRMQtgRaxKmpCS4c0d9bkyYfHxg8WJ1IksjJydIT4d798C4lGP//qbXspRo6fXw88/w3Xfw4INw6xbMnFn09yTs0+XLl+nTpw/Dhw8nJSWFDh06EB8fz8svv2z19DBClCZJmISwI9YkTMbmOE9PcHPLLg8NhXPnYMcOCAhQk6UXXlCb427cgGrVsie4NMqdaOn1EBEBrVtD9+6wcKFaPm9e9pQEQqxfvx5fX1+2bNmCs7Mzc+bMYefOnTRo0EDr0ITIkyRMQtgRa+ZisjRCzsjHB7p2ha+/VpvRfvkFnn9efe2JJ8DS6hPGRCsmRn0MDc1+7amnoE8ftZZq+HD44QcZMVee3bx5kxdffJEBAwZw48YN/Pz8OHDgABMnTkSfs3pTCBskCZMQdqQwNUyWEiajOnXg88/V5zdvqo9ffZX3nEo+PtCpk+UZv+fPVxOtuDg1GZO5mcqnbdu24evry5o1a3BwcGDq1Kns3bsXX19frUMTwiqSMAlhR4orYQJo29Z07ThFKdqcSk5O6og7I5mbqXxJSUnhlVdeoUePHly8eJHGjRsTFxfHzJkzcXZ21jo8IawmCZMQdsSYMF29ChkZlo/JPUIuL5ZGwBmnGiiMvK5z8KAsoWLvdu/ejZ+fH4sWLQJg9OjRHDp0iDZt2mgcmRCFJwmTEHbEy0tt/lIUdT4mS3Ivi5KXvEbAFXZOJUvXAXjxRbXpT5ZQsT/p6elMnjyZjh07cubMGR588EF27NjBJ598glvOkQZClCGSMAlhRxwc1MklIe9mOWub5PIaAVdQolXQdRwcoHp1SEnJrnmSZjr7ER8fT+vWrfnggw/IzMwkJCSEo0eP0rVrV61DE+K+SMIkhJ0pqB+TtQkT5D8CrjByXuf8eVi1yvyYojT3Cdtx7949Zs2aRevWrTl69CheXl5s3LiR5cuX4+HhoXV4Qtw3m10aRQhRNPlNLXD3rtq/KedxBfHxKXytkjXXcXAwXY6lKM19wjacOnWK4OBgfv75ZwD69u3L4sWLqVGjhsaRCVF8pIZJCDuTXw3TpUvqo7Oz2t9JKz4+8E8/YEBNnorS3Ce0lZmZyWeffUaLFi34+eefcXd3Z/ny5WzatEmSJWF3JGESws7klzAZm+O8vU2nDNDCiBHq3E0A06YVvblPaCMxMZEePXowevRo0tLS6Nq1K0ePHiUkJESWNhF2SRImIexMfgmTsVO1rdTkdO6sPv72m7ZxCOspisIXX3yBr68v0dHRuLq6smDBArZv306dOnW0Dk+IEiN9mISwM9bUMFnbf6mktWunPu7Zo20cwjrXrl3jlVdeYcOGDQAEBASwcuVKHn74YY0jE6LkSQ2TEHbGmDAZk6OcbC1hCghQmwbPnoUrV7SORuTn22+/xdfXlw0bNuDo6MjMmTPZvXu3JEui3JCESQg7Y0yYbt6EtDTT12wtYfLwgEceUZ//M8BK2Jjk5GSGDRtG3759uXr1Kk2bNmXv3r1MnToVR0urMQthpyRhEsLOVKkCrq7qc+OoOCNbS5hAXbMOJGGyRTExMTRr1oxly5ah0+mYMGEC+/fvp2XLllqHJkSpk4RJCDuj0+U9F5O1y6KUJunHZHvS0tJ47bXX6NKlCxcuXKBevXrExsby4Ycf4mrMxoUoZyRhEsIOGZvltm/PHhmnKLZdw7RvH9y7p20sAvbt20fLli2ZP38+AGFhYcTHxxMYGKhxZEJoSxImIezQ33+rjzNnZi9se+MGpKer5caEyhb8619qX6Y7d+DoUa2jKb8yMjKYPn067dq14+TJk9SuXZuoqCgiIiKoXLmy1uEJoTlJmISwM4mJsH9/9r5xYduDB9X96tXVmb5thYMDtGmjPpd+TNo4fvw47dq1Y8aMGRgMBgYNGsTRo0fp2bOn1qEJYTMkYRLCzpw6pTa/5WQwZCdMttQcZ2RslpN+TKXLYDAwd+5cWrZsyYEDB/D09GTt2rV8+eWXVKtWTevwhLApMiZUCDvTqJHlhW0d/vnzyJY6fBsZO35bqmFKTFSTwEaN1Nhz74uiOXv2LEOHDmXXrl0ABAUFsWTJErxtqb1WCBsiNUxC2BkfH1i8ODtBAggKUvsIgW3WMBmb5E6dguvXs8sjI9U+WF26qI9PP226HxmpTbxlmaIoREZG0rx5c3bt2kXFihWJiIggKipKkiUh8iEJkxB2KDQUzp+HN99U97dsgdWr1eeVKmkXV148PaFJE/X53r3q4/nz6gK9xpqyzEz45hvT/Zdfzh4FKAp2+fJlevfuzfDhw0lJSaFDhw4cOXKEsLAwWTBXiAJIk5wQdsrHB95/H5KS4LPP4PRptXzePHVkWmiotvHl1rYtnDyp9mMKDIRBg8z7YuVmMKjvS5rmCvb111/zyiuvcOPGDZydnXn//fd57bXX0Ov1WocmcjEYDGRkZGgdRpnm7OyMg0Px1gnpFKWgf5Ls02+//cagQYNM9teuXcvTTz9t1fnJycl4eHiQlJSEu7t7SYUpxH374w9o0MC0TK+Hc+dsK9FYvFitMWrQQI3v99+tO+/IEWjWrPD3Ky99oW7evMno0aNZs2YNAH5+fqxatQpfX1+NIxO5KYrC5cuXuXXrltahlHkODg7Uq1cPZwtDgov6+11ua5gefvhhDh8+DEBKSgoPPfQQTz75pMZRCVH8zp83L7PFmhnjrORnzqiPHh4wejTMnq3Gq9fD4MHwxRfqvtHo0bBsmfo+rU1+IiMhLExt1nNwUJM1W6txKw7btm1j2LBhXLx4EQcHB6ZMmcI777xj8UdEaM+YLNWoUQM3NzdpJi2izMxMLl68yKVLl6hTp06xfY7ltoYppzVr1vDNN9+wbt06q8+RGiZRViQmqh2kc4+as6UapvxiBDW5a9gwe5Tc6dPqsf36QXKyuhyMouSf/BhrlCpVUpv/bPnzuF8pKSlMnDiRRYsWAdC4cWNWrlxJG2PvemFzDAYDv//+OzVq1JApHYpBUlISFy9epGHDhjg5OZm8VtTfb5vt9L1r1y569+6Nt7c3Op2OzZs3mx0THh5OvXr1cHV1xd/fnx9//LFI9/rqq69MmueEsCfGUXPGrip6PURE2FZycOqUaQIDprVgnTplx2vc79IFPv1ULTP+2ZdXR/Cco+0CAvK+lz3YvXs3fn5+WcnSmDFjOHTokCRLNs7YZ8nNzU3jSOyDsRbVkLM6+j7ZbMKUmppKixYt+NT4L2Iu69atY9y4cbz11lscOnSIwMBAgoKCuHDhQtYx/v7++Pr6mm0Xc6xImpyczO7du2VGW2HXQkPVGpSYGPXR1pqfjHNH5aTXq7VK+bGU9OVOfhITs5vf8mLNvWxdeno6kydPpmPHjpw5c4YHH3yQHTt2sGDBAvkRLkOkGa54lMTnaLN9mIKCgggKCsrz9Xnz5hEaGsrw4cMBmD9/Ptu2bWPhwoXMmjULgAMHDhR4n2+++Ybu3bsXuAJ3eno66caFuFATLSHKEh8f26pVyslYC/byy9n9laypBctrks6cyY+l2iswPW/RItv9bKxx+PBhgoODOfrPYnwhISF8/PHHeHh4aByZEPbDZmuY8nP37l0OHDhAt27dTMq7detGXFxcoa5lbXPcrFmz8PDwyNoefPDBQt1HCJG/otSCWZqkc/580+TH0h+aej3s3Jn92lNP3UfgGrp37x6zZs0iICCAo0ePUr16dTZt2sTy5cslWRKimBWqhqlz585FquYaOnQowcHBhT4vL9evX8dgMFCzZk2T8po1a3L58mWrr5OUlMQvv/zChg0bCjx2ypQpjB8/Pms/OTlZkiYhillRasFCQ+HJJ6FDB0hIgJQU09c//1x9NHYMN9ZeBQaqk2WeOAGHD0Pt2pavb6vTD5w6dYrg4GB+/mc9mb59+7J48WJq1KihcWRC2KdCJUxDhw4t0k1atGhRpPMKkjt5UxSlUAmdh4cHV65csepYFxcXXFxcChWfEKJ01KkD770HISHqxJxjx4KbmzoR5pdfqsdERUGFCtmj7QD8/NSEKT5eXT4mN0vTD3Tvrm0ClZmZycKFC5k4cSJpaWm4u7vzySefMGTIEOn/IkQJKlTCFBISUlJxFIqXlxd6vd6sNunq1atmtU5CiPLh+edh2jS1OS8yEsaMUZOozEzo08dyQtSiBaxdq9Yw5Za7s3hmprpUi06n3fxNCQkJhIaGEh0dDUDXrl1ZunQpderUKb0gRJlgqzWjiqJgMBhwdDRNPzIyMsyG/9/P9UpCkfowZWRkUL9+fY4fP17c8VjF2dkZf3//rH80jKKjo2nfvr0mMQkhtOXkBG+8oT7/8EM4dkxNhkBNpCzx81Mf4+PNX7PUWVxRtFnLTlEUVq1aRbNmzYiOjqZChQosWLCA7du3S7JkxxQFUlMLv4WHmy5SHR5euPMLOzujoijMmTOH+vXrU6FCBVq0aMH69esB2LlzJzqdjm3bttGqVStcXFz48ccfmT59On5+fixdupT69evj4uKCoiikp6czduxYatSogaurKx06dGDfvn1Z98rreqVCKSJvb2/l+PHjRT29QLdv31YOHTqkHDp0SAGUefPmKYcOHVLOnz+vKIqifPnll4qTk5MSGRmpHD9+XBk3bpxSsWJF5dy5cyUWU05JSUkKoCQlJZXK/YQQBUtLU5TatRUFFMXLS33s3Tvv4y9dUlknJ1AAACAASURBVI9xcFCU1FTT1xIS1NcK2mJiSvQtKVevXlWeeeYZBVAAJSAgQDl58mTJ3lSUurS0NOX48eNKWlpaVllKinXfweLeUlIKF/ubb76pNGnSRPnuu++UM2fOKMuWLVNcXFyUnTt3KjExMQqgNG/eXNm+fbty+vRp5fr168q0adOUihUrKt27d1cOHjyoxMfHK5mZmcrYsWMVb29vZevWrcqvv/6qhISEKJ6ensqNGzcURVHyvJ41n6dRUX+/i5wwzZo1SwkJCVEyMjKKeol8GT+U3FtISEjWMZ999plSt25dxdnZWWnZsqUSGxtbIrFYIgmTELZpwADTf/ynTs3/+Jo11eP27jUtT05WFJ0u+zoODqb7xrKEhJJ7L998841So0YNBVAcHR2V9957r8T+zRXaKqsJU0pKiuLq6qrExcWZlIeGhirPP/981m/55s2bTV6fNm2a4uTkpFy9etXkWk5OTsrq1auzyu7evat4e3src+bMURRFyfN61nyeRkX9/S5yo9/evXv5/vvv2b59O82aNaNixYomr2/cuLGolwagU6dOKAXUC44aNYpRo0bd132EEPYjMRFyD3qdNUttOsurH0eLFrB9u9qPKSAgu3zXLvXno04dWLFC7Sy+bVv2XFEA9erBAw8U//tITk5m3LhxLFu2DICmTZuyatUqHn300eK/mbBZbm7moz4L8uef8K9/mc9Ndvy49d/Vwsxzevz4cf7++2+ztVjv3r1r8n1t1aqV2bl169alevXqWftnzpwhIyODxx57LKvMycmJgIAATpw4YXKupeuVtCInTFWqVKF///7FGYsQQtyXgpZYscTPT02Ycvdj+v579bF7d3UpFlA7eHfvDnv2qCPyzpyB1avVRYGLS0xMDEOHDuXChQvodDpef/11Zs6cWeDkusL+6HSQqy6iQI0bW54EtnHjkokx85//4aKionggV0bm4uLCmX9W085dqWKpzFhJYs0IeEvXK2lFTpiMf/kIIYStsGbm79yMs57kHilnTJi6djUt9/GBAQPUZGnKFJg0Cfr2hcqVs48pygiltLQ0pkyZwscffwxAvXr1WLFiBYGBgdZdQIh/GBP7nItWl5RHHnkEFxcXLly4wOOPP272ujFhskbDhg1xdnbmp59+4oUXXgDUQWb79+9n3LhxxRZzUd33OLxr167x22+/odPpaNy4sUn1mhBClKaiLLFiHCl35Ej2dAHXrqn7AJ07Wz7vtdfU6QtOn1ZH5w0YoCZI27aZz91U0NQD+/btIzg4mJMnTwIQFhbGf//7XyrnzMKEKITSWgqpcuXKTJgwgddee43MzEw6dOhAcnIycXFxVKpUibp161p9rYoVK/LKK68wceJEqlatSp06dZgzZw537twh1AYWwCxywpSamsqYMWNYuXJlVpWcXq8nODiYTz75RBZ7FEJoorB/XTduDC4ual+RP/5Qz4mJUV9r1gzymjjbxUVdhqVXL1i4UN1yM0490L275TgyMjJ47733eP/99zEYDNSuXZslS5bIYuCiTJk5cyY1atRg1qxZ/PHHH1SpUoWWLVvy5ptvZuUH1po9ezaZmZkMGTKE27dv06pVK7Zt24anp2cJRV8IheoinkNYWJhSv359ZevWrUpSUpKSlJSkREVFKQ0aNFBGjhxZ1MuWGTJKTgj70aqVOjpo/Xp1PyxM3R83Lv/z7mfqgV9//VVp2bJl1gjg5557LmvotCh/8hvVJQqvJEbJFXnx3Q0bNhAZGUlQUBDu7u64u7vTs2dPPv/886wJq4QQoizI3Y8pr/5LuZ06VfC1c/ehMhgMzJ07l5YtW3Lw4EE8PT1Zu3Yta9eupWrVqoUPXghRKorcJHfnzh2Ly5DUqFGDO3fu3FdQQghRmnLO+H3+vNqhW6+Hjh3zP89SJ3OdLnv5FIDPPstujjt79ixDhw5l165dAPTo0YPIyEi8vb2L+R0JIYpbkWuY2rVrx7Rp0/j777+zytLS0pgxYwbt2rUrluCEEKI05Kxh+uEH9Xnr1uDunv95xk7mer26r9fD55/D2bPg4aGW/etf6rDoJUuW0Lx5c3bt2kXFihWZPTuCiRO3kpkpyZIQZUGRa5jmz59PUFAQPj4+tGjRAp1Ox+HDh3F1dWXbtm3FGaMQQpSo5s3Vx4QE+Ppr9XlBzXFGeXUy79lTXctu48ZLzJkzgqioKAACAwPp2XM5b75ZX7NFfIUQhadTlMIus5ctLS2NL774gpMnT6IoCo888ggvvvgiFSpUKM4YbVJycjIeHh4kJSXhXtCfoUIIm1e/vlozZPTDD3lPKWCNZctg2LCvcXQcyb17f+Hs7Mx7773HwIHjqV9fb9KM5+AA586pTXm2uMK8KHl///03Z8+epV69ejJJaTHI7/Ms6u93kWqYMjIyCAsL4+2332bEiBFFuYQQQtgUP7/shMnVFe6nZ8HNmzf59tvRwBru3YPmzR9l9eqV+Pr6EhNjPht5Zqa6svwff5jO39S9uyRQQtiKIvVhcnJyYtOmTcUdixBCaMbYjwnA319Nmopi27Zt+Pr6snnzGkAPTOXNN3/G19cXUJOfXKs8AGqTnjGRysyEESPUdey6dIG6ddVJMoUQ2ilyp+9+/fqxefPm4oxFCCE0c+1a9vO4uMInKCkpKbzyyiv06NGDixcv0rhxYwYM2A3MZOdO56zjqlUzTcb0eggONr+ecRYnyJ4AMzGxcDEJIYpPkTt9N2zYkJkzZxIXF4e/v7/ZQnhjx4697+CEEKI0JCaaztStKPnP0J3b7t27CQ4O5o8//gBgzJgxzJ49m++/d+PrryE6OvvY9eshLU1dOX7lyuxFUb/4wrypLieDQV12RZrphNBGkROmJf/f3r3HRVmm/wP/DMNBCMXIQ/ADDVu11eGgKCqKK2gSlpusbmaleBa/eSDXsNI8pqXmpkmYeAJpSy1DtzygKAnqekQM1NRdMZgVRRcdRG2U4fn98TScBhgOM/PMDJ/368VrZu555pnrjpy5uO/7ue6NG9GyZUucPXsWZ8+erfScTCZjwkREFuPqVd1kRaMRp8lqS0rUajUWLFiAFStWQBAEeHp6YsuWLRj4+yV2AwaII0j/+Y+4PsrLS1ybBABTp4rTbVoV98Czsak8wqQ1ZYoYpyDw6joyHwMGDICfnx9Wr14tdShG1eCEKafi5SRERBasugKUVSt0V5WZmYkxY8YgKysLABAREYE1a9bARVuACUDz5kDv3sCxY+IoU1AQcPSoeO5x4yqfr2p5guTkypsId+wI/L43LwD9+9QRkWE1aA3TkydP0KFDB1y8eNHQ8RARmVx1BSjXr68+ESkpKcGyZcsQEBCArKwstG7dGklJSYiPj6+ULGm9+KJ4e/CgWNQSEDfsra64t4eHOCrl4SEmUNevixsBX78OxMToHq8dBSMi42vQCJOdnR3UajVk1V3qQURkgWoqQFnR1atXMWbMGJw4cQIAMGzYMKxfvx5t2rSp8bwvvggsXCjuT6f9yKxrNRYPj8px1HcUjCybIAiSbDXm5ORU7+/30tJSREdHY+PGjbC3t0dkZCQWLlxonAAl0uApuenTp2P58uXYuHEjbG0bfBoiIrNRNUHRKi0tRWxsLKKjo/Ho0SO0aNECa9euxejRo/V+sQQEiFus3L0rPn72WeCllxoWW1wcMHlyedK0cCGn46zZw4cP4ezsbPL3LS4u1rmQS5+EhATMmjULJ0+exL/+9S+MHTsWffv2xYvaIVYr0OBM5+TJkzh06BAOHDgAb29vnf+433//faODIyKSWl5eHsaPH4+UlBQAwMCBA7F582a0a9euTq+3tRWriGdmio9v3QLi4xu2WFs7Cvb66+K6qLy8+p+DyBh8fHywYMECAEDHjh0RExODQ4cOMWECgJYtW2L48OGGjIWIyGwIgoCvvvoK06dPh0qlgqOjI5YvX463334bNjZ1X/6pVALnz1c8b+MWa3t4AB9/DPTvL5Yl+OgjoHXr+p+HzJ+TkxOKi4sled/68tFuyPg7Nzc3FBQUGCoks9DghGnLli2GjIOIyGzcvn0bkZGRZSPlvXr1wtatW9FJWzSpHq5e1S0PUJeSBbXp1w/o2RM4fRqIjQV+/8OerIxMJqv31JhU7OzsKj2WyWQora2wmAVqcKVvQLxaJCUlBevXr8f9+/cBADdu3JAkIyYiMoTdu3dDoVDg+++/h62tLT766CMcPXq0QckSUF6yoKLGLtaWyYBZs8T7X3whFsIkIuNqcML066+/wtvbG6+++irefvtt3P59X4EVK1Zg9uzZBguQiMgUioqKMH78eAwbNgwFBQVQKBQ4deoU5s6d26gLW+pTsqA+RowQ95q7fRtYu1YsP1Db1ilKpf5jiKhmDU6YZs6ciR49euDu3btwdHQsaw8PD8ehQ4cMEhwRkSmkpqbC29sbW7ZsgUwmw7vvvovTp0+jW7duBjl/1ZpKhqjObWsLzJwp3p8zp/ZNejduFJ/jRr5EDScThKqz63XTqlUrHDt2DJ07d0bz5s1x/vx5dOjQAdevX0eXLl0kqR1hSkVFRXBxcYFKpUKLFi2kDoeIGuDRo0d4//33sWbNGgCAl5cXEhISEBQUJHFkdXPpEtClS+U2uVxMyrQjWHl5YpJU8ZO+6jEkvd9++w05OTnw8vJCs4q7M1OD1Pbfs6Hf3w0eYSotLYVGo9FpVyqVaN68eUNPS0RkEqdPn0b37t3LkqXJkyfj/PnzFpMsAcDNm7ptVat/L1pU86JzIqq7BidML774YqWN9mQyGYqLi7FgwQIMGTLEIMERERnakydPMH/+fPTp0we//PIL3NzcsHfvXqxfv97i/tirbkG5jU35gvLY2Oqn3yoeQ0R10+CE6bPPPsORI0fQpUsX/Pbbb3jjjTfw3HPP4b///S+WL19uyBiN5tNPP0XXrl2hUCjw1VdfSR0OERnZhQsX0Lt3byxZsgQajQavv/46srKyEBYWJnVoDVJ1QTkgjiZt2wbMnQu8/bbY9sorlY9p2RKQoIA0kUVr8BomQJz//+abb5CRkYHS0lJ0794db775ZqVF4OYqKysLEREROH78OACxeu+ePXvQsmXLOr2ea5iILEdpaSlWr16NDz74AGq1Gq6uroiNjcXIkSOlDs0glErg8mUxedqxo/JzISFASgrw3/8CP/8MTJ0K5OYCw4YBM2aIo1RcyyQ9rmEyLGOsYWrUJnCOjo4YP348xo8f35jTSOLSpUsIDAws+w/p5+eH/fv34/XXX5c4MiIypJycHIwdOxZpaWkAgLCwMGzcuBHu7u4SR2Y42j3wOnUCvv228pqlI0fEZEl7zPbtQGAgsGuX+GNjIyZahrhyjxrP2oo9SqURY0E1qlfC5OXlVe8djAEgKioKM2bMqNdr0tLSsHLlSpw9exb5+flISkrCsGHDKh0TGxuLlStXIj8/H127dsXq1avrvGBToVBg0aJFuHfvHgDg8OHD6NChQ71iJCLzJQgCNm3ahHfeeadsM9G///3vmDRpUoM+xyzBv/+tv6p41dGk0tKat2pRKsVK5RyFMj57e3vY2Njgxo0baN26Nezt7a32/1NjEwQBt2/fhkwm06lA3hj1Spji4+Mb9CbPPfdcvV/z4MED+Pr6Yty4cdXuWbd9+3ZERUUhNjYWffv2xfr16xEWFoaLFy+WbYrp7+8PtVqt89oDBw6gS5cumDFjBkJCQuDi4oKePXvWWpxOrVZXOldRUVG9+0REppGfn49JkyZhz549AICgoCDEx8db/R9F2kXgFQcpqlYVr2mrlqtXKydFmzYBkyeL5+IolPHZ2NjAy8sL+fn5uHHjhtThWDyZTAYPDw/IKy7ea+w5G7OGyVRkMpnOCFOvXr3QvXt3rFu3rqztj3/8I4YNG4aPP/643u8xceJEhIeH4+WXX672+YULF2LRokU67VzDRGReduzYgalTp6KwsBD29vZYunQp3nnnHYN+cJqzTZvEESONpryqeMVER6kU6zJVnfl5/XVgyRKxbpOTkzhtVzXxYu0m4xMEASUlJdWW7aG6s7Ozq/HffEPXMFlkwvT48WM4OTnh22+/RXh4eNlxM2fORGZmJo4cOVKn8xYUFKBNmza4fPky/vrXvyIjI6PGUabqRpg8PT2ZMBGZicLCQkybNg3ffPMNAKBbt27YunUrFAqFxJGZnlIpTsP94Q/VJzgVkyrtrE9dvglSU4EBAwwaKpHJSbLoWyp37tyBRqNB27ZtK7W3bdsWN6ur5FaDYcOG4d69e3jqqaewZcuWWqfkHBwc4ODg0OCYich4kpOTMX78eNy4cQNyuRwffPAB5s2bB3t7e6lDk4R2gXdNJkwQ1yxpk6ojR4C33qr9nI3dMJjI0llkwqRVdUGcIAj1WiSnLSlARJapuLgY7777Lr788ksAQKdOnZCYmIiAgACJIzN/FZOqmi4YrLge6rPPOB1HTVuDC1dKqVWrVpDL5TqjSQUFBTqjTkRknY4dOwZfX9+yZGn69Ok4d+4ck6UGqK5iuFwOnDgBeHqKj62oCgNRg1hkwmRvbw9/f38cPHiwUvvBgwcRGBgoUVREZApqtRpz5sxBUFAQrl27Bk9PT6SkpODzzz+Hk5OT1OFZpKoVw7WLxXv2BLQXKe/bJ118RObAbKfkiouL8e8Ku0Pm5OQgMzMTrq6uaNeuHWbNmoXRo0ejR48e6NOnD+Li4pCbm4vIyEgJoyYiY8rMzMSYMWOQlZUFAIiIiMCaNWvg4uIicWSWr+q6Ju3020svAatXA/v3iwvDWRqImiqzvUrup59+QnBwsE57REREWT2o2NhYrFixAvn5+VAoFPjss8/Qv39/k8THrVGITKekpAQrVqzAwoUL8eTJE7Ru3RpxcXE6xWzJ8B49Alxdgd9+A7KygCZ40SFZGasuK2COmDARmcbVq1cxZswYnDhxAgAQHh6OL7/8Em3atJE4sqZjyBBxSm7lSmD27MrPsRo4WZqGfn9b5BomIrJ+paWliImJga+vL06cOIEWLVogISEBO3fuZLJkYi+9JN7u31+5fdMmsQhmSIh4u2mT6WMjMhWOMDUQR5iIjCcvLw/jx49HSkoKAGDgwIHYvHlz2bZHZFpXrgCdOwP29sD//gc4O1dfMZzVwMkScISJiCyeIAhITEyEt7c3UlJS4OjoiM8//xwHDhxgsiShjh0BLy/g8WOx2jcgTsNV3V5Fu9EvkTViwkREZuH27dsYPnw4xowZA5VKhV69eiEzMxPTp0+HTdUiQWRSMpnutFx1Vb9ZDZysGT+FiEhyu3fvhkKhQFJSEmxtbfHRRx/h6NGj6NSpk9Sh0e/CwsTbffvE8gK7dukes349p+PIejFhIiLJqFQqjBs3DsOGDUNBQQEUCgVOnTqFuXPn1rq3I5lecDBgZwfk5ABz5wKzZont48eLt3Z2QESEdPERGRsTJiKSRGpqKnx8fBAfHw+ZTIZ3330Xp0+fRrdu3aQOjarh7Aw8/7x4/+OPgZISoHt3sUJ4s2bAkyfigm8ia8WEiYhM6uHDh5g5cyZCQkKQm5sLLy8vHDlyBCtWrECzZs2kDo9qoFQCly9Xbjt/HsjPF6+gA4BffjF9XESmwoSJiEzm1KlT6N69Oz7//HMAwOTJk3H+/HkEBQVJHBnpc/WquHapIu1VcX/8o/j40iXTx0VkKkyYiMjonjx5gvnz5yMwMBCXL1+Gm5sb9u7di/Xr16N58+ZSh0d10LEjUPViRe1VcS+8ID7mCBNZMyZMRGRUFy5cQO/evbFkyRJoNBqMGjUK2dnZCNNedkUWwcNDXK8kl4uP5fLyq+I4wkRNAS9DISKj0Gg0WL16NebOnQu1Wg1XV1fExsZi5MiRUodGDTRhAhAaKk7D/eEP5SUEKo4wCYJYt6kq7jlHlo4JExEZXE5ODsaOHYu0tDQAwJAhQ7Bx40a4ublJHBk1loeHbsLTsaOYJN29CxQUAG3bVn5+0yZg8mSxMriNjThSNWGC6WImMgROyRGRwQiCgA0bNsDHxwdpaWlwdnZGXFwcfvzxRyZLVszRUdw6BdBdx6RUlidLgHg7ZYrYTmRJmDARkUHk5+fjlVdeweTJk1FcXIygoCCcP38ekyZNgqy6ORqyKjUt/K7PnnNKpbhXHZMpMkdMmIio0Xbs2AGFQoG9e/fC3t4eK1euRGpqKjp06CB1aGQiNS381ha7rKi6Pec2bQLatwdCQsTbTZuqfx8mVSQVJkxE1GCFhYUYNWoURo4cicLCQnTr1g0ZGRmYPXs25NrLqahJqGmE6d493WPXrq28Dqqu03Z1TaqIjIEJExE1yP79+6FQKLBt2zbI5XJ8+OGHOHHiBLp27Sp1aCQBbcJUdYTp0CHxtm/f8sXgT55UPiYjQ/+0nVIJTJrEtVAkHSZMRFQvxcXFmDp1KsLCwpCfn4/OnTvj+PHjWLx4Mezt7aUOjySinZLLzQUePChvT0kRb4cNAxYtEu8vXw789pt4XxCAdet0z1d12i4lpeZK40SmwISJiOrs6NGj8PX1xZdffgkAmDlzJjIyMhAQECBxZCS1Z54BWrUS71+5It4+eQIcOSLeHzgQGDtWnIq7cQPYskVsX78e2L9fTJAqVhIfPLh82u6338Qkq6rq1kIRGQsTJiLSS61WY86cOejfvz+uXbsGT09PHDp0CKtXr4aTk5PU4ZGZqLrw++RJcbTpmWcAX1/AwQGYM0d8bskSMVmaOVN8vGIF8OuvYjsAJCcDp0+L96OjxbVRzs6Vk6ply1gEk0yHCRMR1SozMxM9evTAihUrIAgCxo4di6ysLISEhEgdGpmZqgu/teuXQkLKE52JEwEXFyA/H4iMBB4/Bry9gagoMfmZNw944w1xjdKbbwILF4qLxAFgxw4xqereXXxcWFh9HLySjoyBCRMRVaukpARLly5FQEAAsrOz0bp1a+zatQtbtmyBi4uL1OGRGaq68FubMA0aVH7MnTtAUVHl1128KE7TaX3+OdC8uVjDSbvu6cUXgbAwMan68EOxbcMG4NGjyufilXRkLEyYiEjHlStX0K9fP8ybNw9PnjxBeHg4srOz8eqrr0odGpkx7ZTcL78AxcXAv/4lPh44sPyYq1f1L95+9Eh8fUWHD5ePGA0dKiZDhYXAN9+UH8Mr6ciYmDARUZnS0lLExMTAz88PJ0+eRIsWLbB161bs3LkTbdq0kTo8MnPaEaYrV4CffgJKSoDnngMq1i/t2LHyOiRAd/G2vqRKLgf+7//E+2vXlh/7xRe8ko6MhwkTEQEA8vLyEBoaiunTp+PRo0cYOHAgsrOzMXr0aG5tQnXSvj3QrJm4LmnjRrFt4EBxY14tDw9x811tXVO5XFz8XXHxdl2SqgkTxPfKzARiYsTSBNVdSSeTVU7YiBqKCRNREycIAhITE+Ht7Y2UlBQ4Ojpi7dq1OHDgADw9PaUOjyyIjQ3QubN4/4cfxNuK65e0JkwArl8XF2Zfvy4+rqguSdUzzwA9eoj3Z8wQR5wEAQgKKn8dILZ9/jkXgVPjNYmEKTw8HE8//TRGjBhRr+eIrN3t27cxfPhwjBkzBiqVCr169UJmZiamTZsGm6p/4hPVgXZaTruOqKaLKT08gAEDai4LoC+pUiqB48crt8lkQGJi+etWrhTbV63iInBqvCbxiThjxgxs3bq13s8RWbPdu3dDoVAgKSkJtra2+Oijj3D06FF06tRJ6tDIgmkXfgNi8tSYpW+1JVVXr+pupyIIQE5O+etef73ydCAXgVNjNImEKTg4GM2bN6/3c0TWSKVSYdy4cRg2bBgKCgqgUChw6tQpzJ07F7a2tlKHRxauYnmAy5eNN6JjiMXjRPUhecKUlpaGoUOHwt3dHTKZDLt27dI5JjY2Fl5eXmjWrBn8/f2Rnp4uQaREli81NRU+Pj6Ij4+HTCZDdHQ0zpw5g27dukkdGlkBpbJ8sTcgJivGGtFp6OJxAHj2WcPHQ9ZP8oTpwYMH8PX1RUxMTLXPb9++HVFRUZg7dy7OnTuHoKAghIWFITc3t+wYf39/KBQKnZ8bFf/UaSS1Wo2ioqJKP0SW4uHDh4iKikJISAhyc3PRoUMHpKWlYfny5XBwcJA6PLIS1U2TGXNEp76Lx7WiosSpOy4Ep3oRzAgAISkpqVJbQECAEBkZWanthRdeEN577716nTs1NVUYPnx4vZ/TWrBggQBA50elUtUrDiJTO3nypNC5c+ey/2enTJki3L9/X+qwyArl5QmCjY0giGNL4o9cLrZLHVdqqiD88IMgODmJcclk4q2NjSBs3ChtfGRaKpWqQd/fko8w1ebx48c4e/YsBg8eXKl98ODBOF718ggje//996FSqcp+8vLyTPr+RPX1+PFjzJ8/H4GBgbh8+TLc3Nywd+9efPnll3B2dpY6PLJCdZkmkyquAQOAV14BYmPFNu3apooLwbkHHdXGrFd43rlzBxqNBm3btq3U3rZtW9y8ebPO5wkNDUVGRgYePHgADw8PJCUloWfPnnqfq8jBwYFTF2QxLly4gDFjxiAjIwMAMGrUKMTExMDV1VXiyMjaTZgAhIaK03B/+IP0yVJV7drptmk0wIIFQHy8mEDZ2IiJX9UpPmrazDph0qpaZVgQhHpVHk5OTm7Qc0SWRqPRYPXq1Zg7dy7UajVcXV2xbt06vPbaa1KHRk2Ih4f5JUpa2oXgVddabd5cfl876hQaar79INMz6ym5Vq1aQS6X64wmFRQU6Iw6ETV1OTk5CA4OxuzZs6FWqzFkyBBkZ2czWSKqoOq0oY0N8Pzzusex/ABVZdYJk729Pfz9/XHw4MFK7QcPHkRgYKBEURGZF0EQsHHjRvj4+CA9PR3Ozs6Ii4vDjz/+CDc3N6nDIzI7Fa+u+/VXcaNgfTWdiCSfkisuLsa/K6TxOTk5yMzMhKurK9q1a4dZs2Zh9OjR6NGjB/r06YO4uDjk6aT7VwAAIABJREFU5uYiMjJSwqiJzMPNmzcxceJE7NmzBwAQFBSE+Ph4dOBuo0S1qjptGBcHTJwo3pfJal6srlSK5RM6duR0XZNjlGv26iE1NbXay/UjIiLKjvniiy+E9u3bC/b29kL37t2FI0eOSBfw7xp6WSKRoWzfvl1wdXUVAAj29vbCp59+KpSUlEgdFpHFWrlSLDXg5SUIpaW6z2/cWF42geUILFdDv79lglC1cDzVRVFREVxcXKBSqdCiRQupw6EmpLCwENOmTcM333wDAOjWrRsSExPRtWtXiSMjsmzFxYCbm3iblgYEBZU/p1SKm/dWXCwul4tTexxpsiwN/f426zVMRFRZcnIyvL298c0330Aul+PDDz/EiRMnmCwRGYCzMzBypHi/6h54v/xi2irmZH6YMBFZgOLiYkydOhUvvfQSbty4gc6dO+P48eNYvHgx7O3tpQ6PyGpoay99+y1QcQesY8d0j+XC8KaFCRORmTt27Bh8fX3x5ZdfAgBmzpyJjIwMBAQESBwZkfXp3Rt44QXg4UNg2zaxLScHWLGi8nG1LQwn68SEichMqdVqzJkzB0FBQbh27Ro8PT1x6NAhrF69Gk5OTlKHR2SVZLLyUaZNm8QtVCZPFhOoP/0J+Ppr8bnmzYG33pIuTjI9JkxEZigzMxM9e/bEihUrIAgCIiIikJWVhZCQEKlDI7J6Y8YAtrbAqVPAa68BKSlAs2bAhg3i4//3/8Tpur17pY6UTIkJE5EZKSkpwdKlSxEQEICsrCy0adMGu3btQnx8PFxcXKQOj6hJaNMG8PYW73/3nXg7dKhYe0kuB954Q2z76itp4iNpMGEiMhNXrlxBv379MG/ePDx58gTh4eHIzs7Gq6++KnVoRE2KUglkZlZu+/57sR0on4r78Ufg7t3qX5+aWn48WQcmTEQSKy0tRUxMDPz8/HDy5Em4uLhg69at2LlzJ1q3bi11eERNztWr4tqliiqWEPDxEUegHj8uH4HS2rRJrNcUEiLeVi1PQJaLCRORhPLy8hAaGorp06fj0aNHGDRoELKysjB69GjIZDKpwyNqkjp21L+3nHaUqeK0nFIpLhDX1msqLQWmTOFIk7VgwkQkAUEQkJiYCG9vb6SkpMDR0RExMTFITk6Gp6en1OERNWkeHuLecnK5+Fgu1y0hMGqUeEVdWpq4gS8AfPMNi1taM8k33yVqam7fvo0pU6YgKSkJANCrVy9s3boVnTp1kjgyItKaMAEIDRWTnT/8QbfekqcnMGCAuFZp8WKgVStg5Urd87C4pfVgwkRkQrt378bkyZNRUFAAOzs7LFy4ENHR0bC15T9FInPj4VF7Ycr27cXbzZvL2/r1E6uCC0L9ilsqleLaqY4dWQzTXHFKjsgEVCoVxo0bh2HDhqGgoAAKhQKnTp3CBx98wGSJyAIplcDWrZXbbGzEablDh8THMplYjkAfLhS3DEyYiIwsNTUVPj4+iI+Ph0wmQ3R0NM6cOQM/Pz+pQyOiBrp6VXe9UmmpOIUXHAz07Ck+1lYGrwkXilsOJkxERvLo0SNERUUhJCQEubm56NChA9LS0rB8+XI4ODhIHR4RNYK+K+kiIsTbhITaz/PLL1wobimYMBEZwenTp9G9e3esWbMGABAZGYnz58+jX79+EkdGRIag70q6118H7OzEApg//6z7eqUS2LEDePdd3edsbLhQ3BwxYSIyoCdPnmD+/Pno06cPfvnlF7i5uWHv3r1Yt24dnJ2dpQ6PiAxowgTg+nXxSrnr18s37QWAZ54pX79UdZRJu2Zp5EgxobK1rTxa5e3Nhd/miAkTkYFcuHABvXv3xpIlS6DRaDBq1ChkZ2cjLCxM6tCIyEg8PMTyAtUlONppuX/8AygpEe/n5QGTJlWehistBU6cEBeRy2TA+fPA6dNGD53qiQkTUSNpNBqsWrUK/v7+yMjIgKurK7Zv346vv/4arq6uUodHRBIJCwNatwZu3QKSk8WtVKZM0d12pbQUePAAGD26vIL4woUmD5f0YMJE1Ag5OTkIDg7G7NmzoVarMWTIEGRnZ+O1116TOjQikpidHfDmm+L9RYuA/v2Bfft0j6u4WPzDD8XHe/cCJ0+aLlbSjwkTUQMIgoCNGzfCx8cH6enpcHZ2RlxcHH788Ue4ublJHR4RmYkWLcTb06fFBMjBAZg5s+bF4h07iiNNAPDee+L6qIolBpRK3TYyDZkgVB0cpLooKiqCi4sLVCoVWmj/RVCTkJ+fj0mTJmHPnj0AgKCgICQkJMDLy0viyIjInCiV4uLuiuuVbGzK956raduV//xHTJy03842NsC6dcDdu8AHH4jns7ERr9ILDWWF8Ppq6Pc3SwwT1cOOHTswdepUFBYWwt7eHsuWLUNUVBTk2j8XiYh+V1txy5oWigPiKFTV10yZots2caK4SFwQyhOoilfqkWFxSo6oDgoLCzFq1CiMHDkShYWF6N69OzIyMvC3v/2NyRIRVUtfccuaXL2quzC8JtrjWCHc+JgwEemxf/9+KBQKbNu2DXK5HPPnz8eJEyfQtWtXqUMjIjOmr7hlTapLtGxsdNuqYoVw42LCRFSD4uJiREZGIiwsDPn5+ejcuTOOHz+ORYsWwc7OTurwiMgC1FbcsibVJVpxcZXbbGzE6biK6jJ6RQ3HNUxE1Th69CgiIiJw7do1AMCMGTPwySefwNHRUeLIiMjSeHjUf0H2hAnigu6qC8MrtiUni9NwGo343Lx5XPhtTE1ihCk8PBxPP/00RowYUan9/v376NmzJ/z8/ODt7Y0NGzZIFCGZC7VajTlz5qB///64du0a2rVrh0OHDmHNmjVMlojIpKqrIl6xTTt61b+/+NzduxIE2YQ0ibICqampKC4uRkJCAr777ruydo1GA7VaDScnJzx8+BAKhQKnT5/GM888o/ecLCtgfTIzMzF69GhkZ2cDAMaOHYvVq1fDxcVF4siIiGq2bx8wZAjg6grcuKF7lR1V1tDv7yYxwhQcHIzmzZvrtMvlcjg5OQEAfvvtN2g0GjSB/JGqKCkpwbJlyxAQEIDs7Gy0adMGu3btwpYtW5gsEZHZGzxYHHEqLAR27ZI6GuslecKUlpaGoUOHwt3dHTKZDLuq+W3HxsbCy8sLzZo1g7+/P9LT0w32/vfu3YOvry88PDwQHR2NVq1aGezcZP6uXLmCoKAgzJ07F0+ePEF4eDiys7Px6quvSh0aEVGdyOXA2LHi/U2bJA3FqkmeMD148AC+vr6IiYmp9vnt27cjKioKc+fOxblz5xAUFISwsDDk5uaWHePv7w+FQqHzc+PGDb3v37JlS5w/fx45OTn4+uuvcevWrWqPU6vVKCoqqvRDlqu0tBRffPEF/Pz8cOLECbRo0QJbt27Fzp070bp1a6nDIyKql3HjxNuUlPJK4mRgghkBICQlJVVqCwgIECIjIyu1vfDCC8J7771Xr3OnpqYKw4cPr/WYyMhIYceOHdU+t2DBAgGAzo9KpapXHCS93NxcYdCgQWW/w0GDBgm5ublSh0VE1CgDBwoCIAgLFlRuz8sThMOHxVsSBJVK1aDvb8lHmGrz+PFjnD17FoMHD67UPnjwYBw/frzR579161bZSFFRURHS0tLQuXPnao99//33oVKpyn7y8vIa/f5kWoIgIDExEd7e3khJSYGjoyNiYmKQnJwMT09PqcMjImoUbY2nDRvEkSalUpyia98eCAkRbzll13BmXYfpzp070Gg0aNu2baX2tm3b4ubNm3U+T2hoKDIyMvDgwQN4eHggKSkJPXv2hFKpxIQJEyAIAgRBwLRp0+Dj41PtORwcHODASw8s1u3btxEZGYnvv/8eANCrVy9s3boVnTp1kjgyIiLDCA8HnJzEK+VefFH3ee32KaGhrNfUEGadMGnJqpQzFQRBp602ycnJ1bb7+/sjMzOzUbGR+du9ezcmT56MgoIC2NraYtGiRYiOjoatrUX8709EVCd37gCPHtV+jEYj7lUHiLcdOzJ5qiuz/sZo1aoV5HK5zmhSQUGBzqgTUVUqlQozZ85EQkICAEChUCAxMRF+fn4SR0ZEZHh13bR34kSx4GVpqbjFSlxc9Vu2KJVMqioy6zVM9vb28Pf3x8GDByu1Hzx4EIGBgRJFRZbg8OHD8PHxQUJCAmQyGaKjo3HmzBkmS0RktarbtFcmK99/TiYDbG2Ba9fEZAkon6ZTKiu/jmufdEmeMBUXFyMzM7NsaiwnJweZmZllZQNmzZqFjRs3YvPmzbh06RLeeecd5ObmIjIyUsqwyUw9fPgQM2fOxMCBA5Gbm4sOHTogLS0Ny5cv5xo0IrJq1W3au2FD+ea/ublAfLzu6zQacX86LaUSmDxZf1LV1Eg+JXfmzBkEBweXPZ41axYAICIiAvHx8Rg5ciT+97//YfHixcjPz4dCocDevXvRvn17qUImM3Xq1CmMGTMGly9fBgBERkZi5cqVcHZ2ljgyIiLTqGnTXu3tn/4kjkJpkyGtY8fE6TyZDFi1Svd5bVLVlKfmmsRecsbAveTMx5MnT7BkyRIsW7YMGo0G7u7u2LRpE1566SWpQyMiMjubNokjRhpN3V8jl4sjVdaQMHEvOWqSLly4gN69e2PJkiXQaDQYNWoUsrKymCwREdVgwoTyabpjx8RRpYpkMiAysrxdJgPWr7eOZKkxmDCRRdJoNFi1ahX8/f2RkZEBV1dXbN++HV9//TVcXV2lDo+IyKx5eAADBgBqte6VdYIAjBwJLFkiPu7Tp/qr6JoaJkxkcXJychASEoLZs2dDrVbj5ZdfRnZ2Nl577TWpQyMisijVXVknl4vrn0aMEB+fPau/vlNTwISJLIYgCNi4cSN8fHyQlpYGZ2dnbNiwAT/88APc3NykDo+IyOJUd2WddvqtUyfxVq0Wp+6aOiZMZBHy8/MxdOhQTJo0CcXFxejfvz9+/vlnTJw4sV5V34mIqLKKa5quXy+ffpPJgEGDxPspKVJFZz6YMJHZ27FjBxQKBfbs2QMHBwesWrUKqamp8PLykjo0IiKroF3TVHVh98CB4i0TJjOow0RUk8LCQrz99tvYtm0bAKBbt25ITExE165dJY6MiKhp0CZMGRlAYSHQlK+p4QgTmaX9+/dDoVBg27ZtkMvlmD9/Pk6ePMlkiYjIhNzcgC5dxCvnUlOljkZaTJjIrBQXFyMyMhJhYWHIz89H586dcfz4cSxatAh2dnZSh0dE1ORo1zEdOlS345VKMbmytq1UmDCR2Th69Ch8fX2xfv16AMCMGTNw7tw5BAQESBwZEVHTVZ+F39a8aS8TJpKcWq3GnDlz0L9/f1y7dg3t2rXDoUOHsGbNGjg6OkodHhFRk/anP4nlBq5eFTfwrUg7mvTrr8C33wKTJlnvpr1MmEhSmZmZ6NGjB1asWAFBEDB27Fj8/PPPCAkJkTo0IiIC0KIFoB3orzgtV3E06bnngNde060art201xowYSJJlJSUYOnSpejZsyeys7PRpk0b7Nq1C1u2bIGLi4vU4RERUQVVp+WuXq08mlQTbdVwa8CEiUzuypUr6NevH+bNm4eSkhKEh4cjOzsbr776qtShERFRNbTlBfbtA777TqzZVHU0CQBmz668mW9Nm/Za4sJwJkxkMqWlpfjiiy/g5+eHkydPwsXFBYmJidi5cydat24tdXhERFSDS5fE27t3gb/+FbhxQ/cYuRyYORNITy9v691b97iNGy1zYTgTJjKJvLw8hIaGYtq0aXj06BEGDRqErKwsvPXWW9zahIjIjCmVwNtvV26TyYDFi6vfg65vX+AvfxHbP/tM91yTJ1vmwnAmTGRUgiAgMTER3t7eSElJgaOjI9auXYvk5GR4enpKHR4REelx9aruWiVBAIKCqt+DDgD+9jfxNjERuHWrvH3TJstdGM6EiYzm9u3bGD58OMaMGQOVSoXevXsjMzMT06ZNg40N/9cjIrIEHTsCVT+ytYu5a9qDrk8foFcv4PFjIDZWbMvKApYv1z2/pSwM57cWGcXu3buhUCiQlJQEOzs7LF26FOnp6ejUqZPUoRERUT14eABxcdVPv9VEJisfZVq7FtixA3j5ZeDRI+CPfyw/zsZG/7nMhUwQqlvnTvoUFRXBxcUFKpUKLVq0kDocs6FSqRAVFYX4+HgAgEKhQGJiIvz8/KQNjIiIGkWpFKfOtCNL+pSUAG3bipv2arVqBVy+DHz8MfDpp8CQIcCePcaLuToN/f7mCBMZTGpqKnx8fBAfHw+ZTIY5c+bgzJkzTJaIiKxATdNvNbl5U7yqrqLCQuDhQ0BbRebUqerLE5gjJkzUaI8ePUJUVBRCQkKQm5uL559/Hunp6fjkk0/g4OAgdXhERCSBq1d1k6HSUnGUKiAAcHIC7twBLlyQJr76YsJEjXLq1Cl069YNa9asAQBERkYiMzMTffv2lTgyIiKSUm2Lxe3txfIDAPDTTyYPrUGYMFGDPH78GPPnz0dgYCAuX74Md3d37Nu3D+vWrYOzs7PU4RERkcT0LRYfMEC8TU2VJLx6s5U6ALI8Fy5cwOjRo3Hu3DkAwKhRoxATEwNXV1eJIyMiInMyYQIQGlr9YvHgYPH2p5/EqTpzrzZj5uGROdFoNPj000/h7++Pc+fOwdXVFdu3b8fXX3/NZImIiKpV02LxHj2Ap54SF4JnZ+u+ztz2m2PCRHVy7do1BAcH491334VarcbLL7+M7OxsvPbaa1KHRkREFsjODujXT7xfdVpuwwagXTvz2m+uSSRM4eHhePrppzFixAid52xtbeHn5wc/Pz9MnDhRgujMmyAI2LBhA3x8fJCeng5nZ2ds2LABP/zwA9zc3KQOj4iILJh2Wq5iwqRUivvLaa+wM5f95prEGqYZM2Zg/PjxSEhI0HmuZcuWyMzMlCAq85efn4+JEydi7969AICgoCAkJCTAy8tL4siIiMgaaBOmtLTydUzffVfzfnNSVgRvEiNMwcHBaN68udRhWJQdO3ZAoVBg7969cHBwwKpVq/DTTz8xWSIiIoPp3h1o3lwscHn+vFiXyVz3m5M8YUpLS8PQoUPh7u4OmUyGXbt26RwTGxsLLy8vNGvWDP7+/khPTzfY+xcVFcHf3x/9+vXDkSNHDHZeS1VYWIhRo0Zh5MiRKCwsRPfu3XH27FnMmjWLG+YSEZFB2doCQUHi/cOHgXHjxArhbdvWb+86U5B8Su7Bgwfw9fXFuHHjMHz4cJ3nt2/fjqioKMTGxqJv375Yv349wsLCcPHiRbRr1w4A4O/vD7VarfPaAwcOwN3dvdb3v379Otzd3ZGdnY2XX34ZWVlZTXZvuP3792P8+PHIz8+HXC7H3LlzMW/ePNjZ2UkdGhERWangYGDvXmDBAuDBA8DBAdi/X9x3rj571xmb5AlTWFgYwsLCanz+73//OyZMmFC2IHv16tVITk7GunXr8PHHHwMAzp492+D31yZUCoUCXbp0wZUrV9CjRw+d49RqdaWkrKioqMHvaW6Ki4sxe/ZsrF+/HgDQuXNnbN26FQEBARJHRkRE1k6lEm8fPBBvhw8HtFuQmkOipGXWcyyPHz/G2bNnMXjw4ErtgwcPxvHjxxt9/rt375YlQUqlEhcvXkSHDh2qPfbjjz+Gi4tL2Y+np2ej398cHDt2DL6+vmXJ0syZM3Hu3DkmS0REZHRKJbBsWeW27dulvyKuOmadMN25cwcajQZt27at1N62bVvcvHmzzucJDQ3FX//6V+zduxceHh44ffo0AODSpUvo0aMHfH198corr2DNmjU1FmB8//33oVKpyn7y8vIa3jEzoFarMWfOHAQFBeHatWto164dDh8+jNWrV8PR0VHq8IiIqAm4elW8Oq4i7RVx5kbyKbm6kMlklR4LgqDTVpvk5ORq2wMDA5GVlVWnczg4OMDBwaHO72nOMjMzMXr0aGT/Xlp13LhxWL16dZNdu0VERNLQbtBbMWkyhyviqmPWI0ytWrWCXC7XGU0qKCjQGXUi/UpKSrB06VL07NkT2dnZaNOmDXbv3o3NmzczWSIiIpPTt0GvOTHrhMne3h7+/v44ePBgpfaDBw8iMDBQoqgs05UrV9CvXz/MmzcPJSUl+Mtf/oLs7Gz8+c9/ljo0IiJqwiZMAK5fF6t9X78uPjZHkk/JFRcX498VJitzcnKQmZkJV1dXtGvXDrNmzcLo0aPRo0cP9OnTB3FxccjNzUVkZKSEUVuO0tJSxMbGIjo6Go8ePYKLiwvWrl2Lt956q17TmkRERMbi4WGeo0oVSZ4wnTlzBsHa2ugAZs2aBQCIiIhAfHw8Ro4cif/9739YvHgx8vPzy6pPt2/fXqqQLUZeXh7Gjx+PlJQUAMCgQYOwefNmq7nCj4iIyFRkglB1xxaqi6KiIri4uEClUpnd+h9BEPDVV19h+vTpUKlUcHR0xMqVKzF16lRW6yYioiatod/fko8wkWHdvn0bU6ZMQVJSEgCgd+/e2Lp1Kzp27ChxZERERJaLww1WZPfu3ejatSuSkpJgZ2eHZcuWIT09nckSERFRI3GEyQqoVCrMnDkTCQkJAMRtXhITE+GnrS1PREREjcIRJgt3+PBheHt7IyEhATKZDNHR0Thz5gyTJSIiIgPiCJOFevjwId5//318/vnnAIDnn38eCQkJ6Nu3r8SRERERWR8mTBbo1KlTGDNmDC5fvgwAmDp1KlasWAFnZ2eJIyMiIrJOnJKzII8fP8b8+fMRGBiIy5cvw93dHfv370dsbCyTJSIiIiPiCJOFyM7OxpgxY3Du3DkAwBtvvIGYmBg8/fTTEkdGRERk/TjCZOY0Gg0+/fRT+Pv749y5c3jmmWewY8cO/OMf/2CyREREZCIcYTJj165dw9ixY5Geng4AeOWVV7BhwwY8++yzEkdGRETUtHCEyQwJgoC4uDj4+PggPT0dzs7O2LBhA/75z38yWSIiIpIAR5jMzI0bNzBx4kTs27cPANC/f3/Ex8fDy8tL4siIiIiaLo4wmZmMjAzs27cPDg4OWLVqFVJTU5ksERERSYwjTGbmlVdewbJly/DnP/8ZXbt2lTocIiIiAiATBEGQOghLVFRUBBcXF6hUKrRo0ULqcIiIiKgOGvr9zSk5IiIiIj2YMBERERHpwYSJiIiISA8mTERERER6MGEiIiIi0oMJExEREZEeTJiIiIiI9GDCRERERKQHEyYiIiIiPZgwEREREenBhImIiIhIDyZMRERERHowYSIiIiLSw1bqACyVIAgAxF2PiYiIyDJov7e13+N1xYSpge7fvw8A8PT0lDgSIiIiqq/79+/DxcWlzsfLhPqmWAQAKC0txY0bN9C8eXPIZDKpwzGIoqIieHp6Ii8vDy1atJA6HKNjf60b+2v9mlqf2V/DEAQB9+/fh7u7O2xs6r4yiSNMDWRjYwMPDw+pwzCKFi1aNIl/jFrsr3Vjf61fU+sz+9t49RlZ0uKibyIiIiI9mDARERER6SFfuHDhQqmDIPMhl8sxYMAA2No2jdla9te6sb/Wr6n1mf2VDhd9ExEREenBKTkiIiIiPZgwEREREenBhImIiIhIDyZMRERERHowYSIiIiLSgwkT1Vl4eDiefvppjBgxQuc5W1tb+Pn5wc/PDxMnTpQgOsOrqb/3799Hz5494efnB29vb2zYsEGiCA2rtt9vbc9Zi08//RRdu3aFQqHAV199JXU4RnX58uWyf69+fn5wdHTErl27pA7LqKzxM6om1voZVRtTfEaxrADVWWpqKoqLi5GQkIDvvvuu0nOtWrXCnTt3JIrMOGrqr0ajgVqthpOTEx4+fAiFQoHTp0/jmWeekTDaxqvt91vbc9YgKysLEREROH78OABg4MCB2LNnD1q2bClxZMZXXFyM5557Dr/++iueeuopqcMxGmv8jKqJtX5G1cYUn1EcYaI6Cw4ORvPmzaUOw2Rq6q9cLoeTkxMA4LfffoNGo4E1/N1R2+/X2n/3ly5dQmBgIJo1a4ZmzZrBz88P+/fvlzosk/jnP/+JgQMHWnWy1NRY62dUbUzxGcWEyUqkpaVh6NChcHd3h0wmq3Z4PTY2Fl5eXmjWrBn8/f2Rnp5usPcvKiqCv78/+vXrhyNHjhjsvDWRur/37t2Dr68vPDw8EB0djVatWhns3NWRur9SM3b/FQoFUlNTce/ePdy7dw+HDx/Gf//7X0N2oV5M+fvesWMHRo4c2diQG8UU/TX1Z1RtTNFfU39G1cZaPr+krzVOBvHgwQP4+vpi3LhxGD58uM7z27dvR1RUFGJjY9G3b1+sX78eYWFhuHjxItq1awcA8Pf3h1qt1nntgQMH4O7uXuv7X79+He7u7sjOzsbLL7+MrKwso+6mLXV/W7ZsifPnz+PWrVv4y1/+ghEjRqBt27aG6Vw1pO6v1Izd/y5dumDGjBkICQmBi4sLevbsKelWDKb6fRcVFeHYsWPYtm2bcTukhyn6a+rPqNqYor+m/oyqjdV8fglkdQAISUlJldoCAgKEyMjISm0vvPCC8N5779Xr3KmpqcLw4cNrPeall14STp8+Xa/zNobU/Y2MjBR27NhRr/M2hlT9rct/C1MwZv+1JkyYIPz4448NjtGQjNnfrVu3Cm+++WajYzQkU/x+Tf0ZVRtT9NfUn1G1kfrzujE4JdcEPH78GGfPnsXgwYMrtQ8ePLhskWtj3L17tyzzVyqVuHjxIjp06NDo8zaUsft769YtFBUVARD/Qk9LS0Pnzp0bfd6GMnZ/zZ2h+l9QUABAvILs1KlTCA0NNWichmLI37c5TMfpY4j+mttnVG0M0V9z+4yqjSV9fnFKrgm4c+cONBqNznBs27ZtcfPmzTqfJzQ0FBkZGXjw4AE8PDyQlJSEnj174tKlS5gyZQpsbGwgk8mwZs0auLq6GrobdWbs/iqVSkyYMAGCIEAQBEybNg0+Pj6G7kadGbu/+p6TmqH6P2zYMNy7dw/CF9uBAAAC2UlEQVRPPfUUtmzZYha7o1fHUP1VqVQ4deoUdu7caegQDcoQ/TW3z6jaGKK/5vYZVRtTfH4Zinl+IpBRyGSySo8FQdBpq01ycnK17YGBgcjKympUbMZgrP76+/sjMzOzUbEZg7H6q+85c9HY/pvbX7P6NLa/Li4uuHXrlqHDMprG9NdcP6Nq05j+mutnVG2M+fllKJySawJatWoFuVyuk60XFBRItgjQmNhfkbX2t6qm1n/2V8T+WgdL6i8TpibA3t4e/v7+OHjwYKX2gwcPIjAwUKKojIf9FVlrf6tqav1nf0Xsr3WwpP5ySs5KFBcX49///nfZ45ycHGRmZsLV1RXt2rXDrFmzMHr0aPTo0QN9+vRBXFwccnNzERkZKWHUDcf+Wnd/q2pq/Wd/2V/21wz7a7Tr78ikUlNTBQA6PxEREWXHfPHFF0L79u0Fe3t7oXv37sKRI0ekC7iR2F/r7m9VTa3/7C/7y/6aX3+5lxwRERGRHlzDRERERKQHEyYiIiIiPZgwEREREenBhImIiIhIDyZMRERERHowYSIiIiLSgwkTERERkR5MmIiIiIj0YMJEREREpAcTJiKi3w0YMABRUVFSh0FEZogJExEREZEeTJiIiIiI9GDCRERUQWlpKaKjo+Hq6opnn30WCxculDokIjIDTJiIiCpISEjAU089hZMnT2LFihVYvHgxDh48KHVYRCQxmSAIgtRBEBGZgwEDBkCj0SA9Pb2sLSAgACEhIfjkk08kjIyIpMYRJiKiCnx8fCo9dnNzQ0FBgUTREJG5YMJERFSBnZ1dpccymQylpaUSRUNE5oIJExEREZEeTJiIiIiI9GDCRERERKQHr5IjIiIi0oMjTERERER6MGEiIiIi0oMJExEREZEeTJiIiIiI9GDCRERERKQHEyYiIiIiPZgwEREREenBhImIiIhIDyZMRERERHowYSIiIiLSgwkTERERkR7/H1fMCKiVXEVQAAAAAElFTkSuQmCC",
      "text/plain": [
       "PyPlot.Figure(PyObject <Figure size 640x480 with 1 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "PyObject <matplotlib.legend.Legend object at 0x14e42a320>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "loglog(h, abs.(deriv.(g, 2, h) - 2), \"b.-\")\n",
    "loglog(h, h, \"k-\")\n",
    "title(\"forward difference error in g\")\n",
    "xlabel(\"h\")\n",
    "ylabel(\"|error|\")\n",
    "legend([\"error\", \"h\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's break this into pieces, for $h = 10^{-8}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.00000002"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(2+1e-8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.999999987845058e-8"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(2+1e-8) - g(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.999999987845058"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(g(2+1e-8) - g(2)) / 1e-8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Weird.\n",
    "\n",
    "Let's try it for $h = 1e-100$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g(2+1e-100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wait, that's not right!  It should be $4 + 2 \\times 10^{-100}$!  Let's look at the argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0e-100"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1e-100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "2 + 1e-100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The issue here is **roundoff error**: by default, in most computer languages, real numbers are stored to only about 15 decimal digits.   This corresponds to 64 bits (8 bytes) in memory, which is the most accurate **floating point number** that the computer can do operations on.\n",
    "\n",
    "We can work with more digits, but they are much slower."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "308.25471555991675"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "setprecision(BigFloat, 1024) # 1024 binary (base-2 digits)\n",
    "log10(2)*1024 # this many decimal digits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000099999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999997"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "big(2) + BigFloat(\"1e-100\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.000000000000000055511151231257827021181583404541015625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-01"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "big(0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.551115123125782702118158340454101562499999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999860932883843299914e-18"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "big(0.1) - 1//10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1//10"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1//10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 + 1e-100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Cancellation error:\n",
    "\n",
    "When we compute $f(x + h) - f(x)$ for very small $h$, we are subtracting two nearly equal numbers, and cancelling most of the significant digits.  What is left over has few or no accurate digits, and smaller h makes this worse.\n",
    "\n",
    "In differentiating the linear function $g(x) = 2x$, it is easy to see that the forward-difference formula is *exact* regardless of the value of $h$ *if* arithmetic were exact.  So *all* of the error is due to cancellation error, which increases as we decrease $h$.\n",
    "\n",
    "With our $f(x) = \\sin(3x)$, the finite-difference formula is no longer exact in exact arithmetic.  So, cancellation error is not the dominant source of error in that case until $h$ becomes very small."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Floating-point math is weird\n",
    "\n",
    "Because the result of every operation is rounded, `+` is no longer associative: $(x+y)+z = x +(y+z)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "-1 + (1+1e-100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0e-100"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(-1 + 1) + 1e-100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The other error: finite-difference error\n",
    "\n",
    "Forgetting about floating-point math, supposing we have infinite precision, how big is the error in our forward-difference approximation?  How fast does it decrease with $h$?\n",
    "\n",
    "The key thing is to figure out the behavior of $f(x+h)$ for small $h$.  The tool (for smooth functions) is the Taylor series:\n",
    "\n",
    "$$\n",
    "f(x+h) = f(x) + f'(x) h + \\frac{f''(x)}{2} h^2 + \\frac{f'''(x)}{6} h^3 + \\cdots\n",
    "$$\n",
    "\n",
    "Let's plug this into our forward-difference approximation $\\frac{f(x+h)-f(x)}{h}$.  Doing thing, we obtain:\n",
    "\n",
    "$$\n",
    "\\frac{f(x+h)-f(x)}{h} \\approx f'(x) + \\frac{f''(x)}{2} h\n",
    "$$\n",
    "\n",
    "where we have dropped higher-order terms in $h$ (which are negligible for small $h$).   The $f'(x)$ term is the correct answer — the derivative we want! — and the $\\frac{f''(x)}{2} h$ term is the [leading-order term](https://en.wikipedia.org/wiki/Leading-order_term) in the error.  It shows that, for small enough $h$ (but not so small that we hit the the limits of the floating-point precision!) we should expect the forward-difference approximation to have **an error proportional to h**.  This is exactly what we observed in our calculations above!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Home exercise: Centered differences\n",
    "\n",
    "Instead of the forward-difference approximation we tried above, try a different approximation, called a \"center-difference\" approximation:\n",
    "\n",
    "$$\n",
    "f'(x) \\approx \\frac{f(x+h) - f(x-h)}{2h}\n",
    "$$\n",
    "\n",
    "How does this approximation compare for accuracy?  (What happens when you plug the Taylor series in?)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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