Time_integration.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Time integration Study Session"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Libraries\n",
    "import numpy as np # Used to do some math\n",
    "from scipy.integrate import odeint # Used to solve directly ODE\n",
    "import matplotlib.pyplot as plt # Used to plot graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 1 : Forward and Backward Euler method\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As a remainder, Forward and Backward Euler method let us solve ordinary differential equation. They have both their specificities that I will not\n",
    "discuss here (see chapter 2 on time integration). Two things to keep in mind to know if we need to chose between Forward and Backward: Forward/Explicit Euler method is easier but not stable for any time step and Backward/Implicit Euler method is harder but more stable. I will also try to show that here.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exo 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Find approximate solution to the initial value problem,\n",
    "\n",
    "$\\displaystyle \\frac{dy}{dt} = -3y + 7e^{4t},\\quad y(0)=2$,\n",
    "\n",
    "on the interval $0 \\le t \\le 1$ using **explicit** and **implicit Euler methods**.\n",
    "\n",
    "Compare approximate solutions with exact solution given by\n",
    "\n",
    "$\\displaystyle y(t)=e^{4t} + e^{-3t}$.\n",
    "\n",
    "Show graphically that both explicit and implicit Euler methods are first-order accurate for a finite time interval."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Forward Euler:\n",
    "\n",
    "$y^{n+1} = y^n + (-3y^n + 7e^{4t^n})dt$\n",
    "\n",
    "Backward Euler:\n",
    "\n",
    "$y^{n+1} = y^n + (-3y^{n+1} + 7e^{4t^{n+1}})dt$\n",
    "\n",
    "or\n",
    "\n",
    "$y^{n+1}(1 + 3dt) = y^n + 7e^{4t^{n+1}}dt$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"\\nFor 'nt' we use int() wich will transforme our value inside the braket into an integer.\\nFor exemple : int(1.2) = 1\\n\""
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Initialise our parameters\n",
    "ti = 0.0 # Initial time condition\n",
    "tf = 1.0 # Final time condition\n",
    "yi = 2.0 # Initial position condition\n",
    "dt = 0.1 # Size of a time step\n",
    "nt = int((tf-ti)/dt) # Number of time step to go from 'ti' to 'tf' with a size 'dt' \n",
    "\n",
    "'''\n",
    "For 'nt' we use int() wich will transforme our value inside the braket into an integer.\n",
    "For exemple : int(1.2) = 1\n",
    "'''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creating our containers to store our solutions\n",
    "\n",
    "#------------------Forward---------------------\n",
    "y_forward = np.empty(nt+1) # Create an empty array of size nt + 1\n",
    "y_forward[0] = yi # Set up the first element of the list 'y_forward' to be the initial condition\n",
    "\n",
    "'''\n",
    "Here We've created a numpy array to store de solution of our equation at each time step. The size of this array is the number of time step plus one\n",
    "to include the initial condition. We've used np.empty(x) wich create an empty array of size 'x'. Then we've setted up the first element of the list to be the \n",
    "initial condition.\n",
    "remark : we could use a python list instead of a numpy array but it is faster to do operation with numpy.\n",
    "'''\n",
    "\n",
    "t = np.arange(nt+1) * dt # Create the time array\n",
    "\n",
    "'''\n",
    "We've used np.arange(x) to create the time variable. np.arange(x) create an numpy array of size 'x' that look like this : [0,1,2,3,...,x].\n",
    "So here, with np.arange(nt+1), we've created an array that take values from 0 to nt + 1. \n",
    "'''\n",
    "\n",
    "for i in range(nt): # For loop to get the solution at each time\n",
    "    y_forward[i+1] = y_forward[i] + (-3 * y_forward[i] + 7 * np.exp(4 * t[i])) * dt # Compute the Forward Euler Scheme for my ODE\n",
    "\n",
    "'''\n",
    "Here we've compute the forward Euler scheme. We've used a for loop that take an argument ('i') that goes from 0 to nt. For each argument we get the\n",
    "solution of y for next step using the value for the current one that we have compute before.\n",
    "\n",
    "When i = 0, we have :\n",
    "y_forward[1] = y_forward[0] + (-3 * y_forward[0] + 7 * np.exp(4 * t[0])) * dt\n",
    "So we have found the value of y at the first time step, meaning at t = ti + 1 * dt = 0.1. \n",
    "To find this value we needed the value at the 0th time step, meaning at t = ti + 0 * dt = ti. But this value is the initial condition yi = 2.\n",
    "\n",
    "When i = 1, we have :\n",
    "y_forward[2] = y_forward[1] + (-3 * y_forward[1] + 7 * np.exp(4 * t[1])) * dt\n",
    "So we have found the value of y at the second time step, meaning at t = ti + 2 * dt = 0.2. \n",
    "To find this value we needed the value at the first time step, meaning at t = ti + 1 * dt. But this value has been compute at the previous loop.\n",
    "This is the value we found above at i = 1\n",
    "\n",
    "Etc for each value of 'i' until i = nt where we found the value at nt + 1 wich is tf. \n",
    "\n",
    "At each loop we've stored the solution to the numpy array. So we have at the end an array of size nt + 1 and each value is the solution at the corresponding\n",
    "time step.\n",
    "So y_forward[0] is the initial condition, y_forward[1] is the solution at t = 0.1, ..., y_forward[nt + 1] is the solution at tf.\n",
    "\n",
    "'''\n",
    "\n",
    "#-----------------Backward---------------------\n",
    "y_backward = np.empty(nt+1) # Create an empty array of size nt + 1\n",
    "y_backward[0] = yi # Set up the first element of the list 'y_forward' to be the initial condition\n",
    "\n",
    "'''\n",
    "For the Backward Euler method, we've done the same as above with the Forward Euler method but the algorithm we've computed is different. All the \n",
    "python is the same so we will not go into details again.\n",
    "Remark : We have already define 't' so we will not doing it again but you also need 't' for Backward Euler method. If you are doing an exercise using\n",
    "only Backward Euler don't forget it.\n",
    "'''\n",
    "\n",
    "mult = 1 + 3 * dt # define a coefficient that we will use. just do that to not copy it every time\n",
    "for i in range(nt): # For loop to get the solution at each time\n",
    "    y_backward[i + 1] = y_backward[i] / mult + dt * 7 * np.exp(4 * t[i + 1]) / mult # Compute the Backward Euler Scheme for my ODE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"\\nodeint is a function of the Scipy library that solve ODE. Scipy is a library that contain a lot of algorithm to solve problem numericaly using \\npython (ODE, integration, Eigenvalue problem, ...). The function odeint solve ODE, it is based on FORTRAN library odepack wich is really fast and \\npowerfull.\\nIt takes multiple parameters as arguments : the function you want to integrate to solve the ODE, the initiale condition, the time at wich we want to solve y\\nand other secondary parameters like 'full_output' that return 'info' wich give us acces to additional information like the order or else.\\nodeint return an array containing the solution of the ODE at each time 't', here called 'y_odeint'.  \\n\""
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solve the ODE with the function odeint of scipy library\n",
    "def func(y, t): # define the right hand side of my function\n",
    "    RHS = -3 * y + 7 * np.exp(4 * t)\n",
    "    return RHS\n",
    "\n",
    "'''\n",
    "Here we want to use the function odeint of Scipy library to solve my ODE. To do that odeint need as a parameter the rigth hand side (RHS) of my \n",
    "function. So we are computing the RHS using python definition.\n",
    "'''\n",
    "\n",
    "y_odeint, info = odeint(func, yi, t, full_output=True)\n",
    "\n",
    "'''\n",
    "odeint is a function of the Scipy library that solve ODE. Scipy is a library that contain a lot of algorithm to solve problem numericaly using \n",
    "python (ODE, integration, Eigenvalue problem, ...). The function odeint solve ODE, it is based on FORTRAN library odepack wich is really fast and \n",
    "powerfull.\n",
    "It takes multiple parameters as arguments : the function you want to integrate to solve the ODE, the initiale condition, the time at wich we want to solve y\n",
    "and other secondary parameters like 'full_output' that return 'info' wich give us acces to additional information like the order or else.\n",
    "odeint return an array containing the solution of the ODE at each time 't', here called 'y_odeint'.  \n",
    "'''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"\\nWe' ve computed the exact solution to compare it with Forward Euler method, Backward Euler method and odeint.\\n\""
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the exact solution\n",
    "y_exact = np.exp(4 * t) + np.exp(-3 * t) \n",
    "\n",
    "'''\n",
    "We' ve computed the exact solution to compare it with Forward Euler method, Backward Euler method and odeint.\n",
    "'''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\nRemark : if you want information about matplotlib, go check the library documentation or the previous notebook about Taylor expansion.\\n'"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 700x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot everything\n",
    "fig, ax = plt.subplots(1, 1, figsize=(7, 6))\n",
    "\n",
    "ax.plot(t, y_exact, '*', label=\"Exact solution\")\n",
    "ax.plot(t, y_forward, label=\"Forward Euler\")\n",
    "ax.plot(t, y_backward, label=\"Backward Euler\")\n",
    "ax.plot(t, y_odeint, label=\"SciPy\")\n",
    "\n",
    "ax.set_xlabel(\"$t$\", fontsize=18)\n",
    "ax.set_ylabel(\"$y(t)$\", fontsize=18)\n",
    "\n",
    "ax.legend()\n",
    "\n",
    "'''\n",
    "Remark : if you want information about matplotlib, go check the library documentation or the previous notebook about Taylor expansion.\n",
    "'''"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will now show that the two Euler method are first order accurate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'\\nHere we\\'ve computed the algorithm to find the final solution of my ODE for different size of time step for Forward and Backward Euler methods.\\nTo do that we use a for loop with \"in enumerate(x)\". This take one argument, a list and give back two things that we usualy call count and value. \\nBefore we try to understand the loop above let\\'s look at an exemple of \"in enumerate(x)\" :\\n\\n---------------------------------Exemple--------------------------------------\\nx = [1,2,3,4] # Create a list \\n\\nfor count, value in enumerate(x): # Our for loop\\n    print(count, value)\\n\\nThis will give us :\\n0 1\\n1 2\\n2 2\\n3 4\\n\\nThe first values (the column) are the \\'count\\' values and they tell us the number of iteration in our for loop. Then the second values (the second column)\\nare the \\'value\\' values that are the value inside de list \\'x\\'. \\nSo at the first iteration of our for loop : \\ncount = 0 because it is the first iteration (first mean 0 in python) and value = x[0] which here is 1. That is why when we print(count, value)\\nwe obtained 0 1.\\nAt the second iteration of our loop :\\ncount = 1 and value = x[1] = 2 \\netc\\n-------------------------------------------------------------------------------\\n\\nNow back to our loop above. Here \\'j\\' is \\'count\\', \\'dt\\' is \\'value\\' and \\'x\\' is \\'dts\\'. \\nSo in our first iteration:\\nj = 0 and dt = dts[0]\\nin my second :\\nj = 1 and dt = dts[1]\\netc\\n\\nThen inside our for loop with \"in enumerate\", we have another for loop using in range that we\\'ve already seen before.\\n\\nLet\\'s try to understand the complete for loop for Euler forward (this is the same for Euler backward) :\\nAt the first iteration of our first for loop (the one with enumerate):\\n\\nj = 0 and dt = dts[0]\\nT = 0\\nWe enter the second for loop : for i in range(ns[0])\\ninside this loop we will compute Euler forward for \\'i\\' going from 0 t \\'ns[0]\\'\\nthen at the end we do : T += dt \\nwhich is the same as : T = T + dt\\nIn our first iteration, it gives us : T = 0 + dts[0]\\n\\nend of the first iteration and begining of the second one.\\nIt goes until we\\'ve reached the end of the \\'dts\\' list. \\n'"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Initialize our parameters \n",
    "ns = 4**np.arange(0,5)*10 #creating different number of time step\n",
    "dts = (tf - ti) / ns # For each number of time step, computing the size of the time step\n",
    "\n",
    "'''\n",
    "If we want to show that Euler methods are first order accurate, we need to compute the solution of our ODE using different size of time step. To do that\n",
    "I create different number of time step ('ns') using np.arange(). Note that I put the np.arange() as an exponential of 4, this does not matter. It is a good idea\n",
    "to use a exponential form for 'ns' because to compare the accuracy we are gonna use log scale when we will plot all of that. That is why I've done that.\n",
    "\n",
    "Then after I've been defining 'ns', I can compute the size of the time step 'dts' for each number of time step 'ns'.\n",
    "'''\n",
    "\n",
    "y_exact_final = np.exp(4 * tf) + np.exp(-3 * tf) # Computing my exact solution at tf\n",
    "y_forward_final = np.ones(ns.shape) * 2 # Creating containers for all final solution for each dts using Euler Forward \n",
    "y_backward_final = np.ones(ns.shape) * 2 # Creating containers for all final solution for each dts using Euler Backward\n",
    "\n",
    "'''\n",
    "Here we've computed the exact solution at the final time 'tf' so we can compare it with the ones from Euler methods. we've created containers to store\n",
    "the final solution for each size of the time step 'dts' when we use Euler Forward and Backward. To create the containers we've used np.ones(x), it\n",
    "creates an array of one of size 'x' ([1,1,1,...,1]). Here the size of our array of ones is ns.shape, which give us the shape of an array. For example,\n",
    "if we have a matrix, matrix.shape will give us (#rows, #columns). Here we have a 1 dimensional array 'ns' so ns.shape give us the number of element\n",
    "in ns which is 5.\n",
    "Then we multiply our array of ones of size ns.shape by 2 which is our initial condition. So in the general case we have a code like this :\n",
    "y_forward/backward_final = np.ones(ns.shape) * yi\n",
    "This is an array like this [yi,yi,yi,...,yi] of size ns.shape.\n",
    "'''\n",
    "\n",
    "#---------------------------------------------Forward---------------------------------------------------\n",
    "\n",
    "for j, dt in enumerate(dts): # Computing the Euler forward method to calculate the final solution for each dts\n",
    "    T = 0\n",
    "    for i in range(ns[j]):\n",
    "        y_forward_final[j] = y_forward_final[j] - dt * 3 * y_forward_final[j] + dt * 7 * np.exp(4 * T)\n",
    "        T += dt\n",
    "\n",
    "#-------------------------------------------------------------------------------------------------------\n",
    "\n",
    "#--------------------------------------------Backward---------------------------------------------------\n",
    "\n",
    "for j, dt in enumerate(dts): # Computing the Euler Backward method to calculate the final solution for each dts\n",
    "    T = 0\n",
    "    mult = 1 + 3 * dt\n",
    "    for i in range(ns[j]):\n",
    "        y_backward_final[j] = y_backward_final[j] / mult + dt * 7 * np.exp(4 * T) / mult\n",
    "        T += dt\n",
    "\n",
    "#-------------------------------------------------------------------------------------------------------\n",
    "\n",
    "'''\n",
    "Here we've computed the algorithm to find the final solution of my ODE for different size of time step for Forward and Backward Euler methods.\n",
    "To do that we use a for loop with \"in enumerate(x)\". This take one argument, a list and give back two things that we usualy call count and value. \n",
    "Before we try to understand the loop above let's look at an exemple of \"in enumerate(x)\" :\n",
    "\n",
    "---------------------------------Exemple--------------------------------------\n",
    "x = [1,2,3,4] # Create a list \n",
    "\n",
    "for count, value in enumerate(x): # Our for loop\n",
    "    print(count, value)\n",
    "\n",
    "This will give us :\n",
    "0 1\n",
    "1 2\n",
    "2 2\n",
    "3 4\n",
    "\n",
    "The first values (the column) are the 'count' values and they tell us the number of iteration in our for loop. Then the second values (the second column)\n",
    "are the 'value' values that are the value inside de list 'x'. \n",
    "So at the first iteration of our for loop : \n",
    "count = 0 because it is the first iteration (first mean 0 in python) and value = x[0] which here is 1. That is why when we print(count, value)\n",
    "we obtained 0 1.\n",
    "At the second iteration of our loop :\n",
    "count = 1 and value = x[1] = 2 \n",
    "etc\n",
    "-------------------------------------------------------------------------------\n",
    "\n",
    "Now back to our loop above. Here 'j' is 'count', 'dt' is 'value' and 'x' is 'dts'. \n",
    "So in our first iteration:\n",
    "j = 0 and dt = dts[0]\n",
    "in my second :\n",
    "j = 1 and dt = dts[1]\n",
    "etc\n",
    "\n",
    "Then inside our for loop with \"in enumerate\", we have another for loop using in range that we've already seen before.\n",
    "\n",
    "Let's try to understand the complete for loop for Euler forward (this is the same for Euler backward) :\n",
    "At the first iteration of our first for loop (the one with enumerate):\n",
    "\n",
    "j = 0 and dt = dts[0]\n",
    "T = 0\n",
    "We enter the second for loop : for i in range(ns[0])\n",
    "inside this loop we will compute Euler forward for 'i' going from 0 t 'ns[0]'\n",
    "then at the end we do : T += dt \n",
    "which is the same as : T = T + dt\n",
    "In our first iteration, it gives us : T = 0 + dts[0]\n",
    "\n",
    "end of the first iteration and begining of the second one.\n",
    "It goes until we've reached the end of the 'dts' list. \n",
    "'''"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x136b283df40>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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KK5qN19bWNrsO0N/fv9l/RwAAOpLXhL0dO3Zo2rRprucrVqyQJC1cuFDr1q3T/PnzVVpaqocffliFhYUaNmyY3nnnHcXHx5tVcqvdlHSTkkKTms3knfH8jc9rSPgQj9Vy5nq2MywWS7OxM9dANjY2XnRf376G8lx69OjRxgrbJjY29rynhH18fGQYRrOxC516bmxslNVq1c6dO1sExrMXnPTo0aNVnx0AAHfwmrA3derUFn8xf9sPfvAD/eAHP/BQRR3DIosMGa7HruLTTz9VXFycJOn48eP66quvdOWVV170fb169VJCQoI++uijZmH+fPz9/SU1rXJtr759++rAgQPNxvbs2dMi8J4xevRoOZ1OFRcXy2aztfv4AAC4g9eEPbOkp6crPT3dLeHiQsICwxQeGK5+Qf00b+A8vfL1KyqqKlJYYFiHHtddHn74YYWHhysqKkoPPfSQIiIiWt3Db+XKlbr77rsVGRmpmTNn6uTJk9q8ebPuu+++FttGRkaqR48eeu+99xQTE6PAwMALXktZWlraYuFO7969FRgYqGuuuUa///3v9cwzz2jSpEn629/+pgMHDmj06NHn3NcVV1yhO++8UwsWLNAf/vAHjR49WiUlJfrnP/+p4cOH64YbbmjV5wUAwJ26xWrcjrRs2TIdPHhQ27dv79Dj9Avqpw9u/UB/v/Hvun3Q7fr7jX/XB7d+oH5Bl95jzpN++9vf6kc/+pGSk5NVWFioN954wzULdzELFy7U448/rtWrV2vo0KG66aabmq1uPZuvr6/+9Kc/6a9//auio6MvugDn2muvVf/+/Zv9nGnjMmPGDP3iF7/QAw88oHHjxunkyZNasGDBBfe3du1aLViwQD/5yU80aNAgzZ49W5999hl3WAEAmMZiXOzcJ1rlzGrc8vJyhYSENHutpqZGDodDiYmJCgwMNKlCc3zyySeaNm2ajh8/3uyWaLiw7vxnBgBwcRfKHd/GzB4AAIAXI+wBAAB4McIeOtSZVdKcwgUAdBtHdknrbmp67AQIe+3kidulAQCALmTvC1KOXdq33uxKJBH22q0tq3FZC4PW4s8KAHQxJ/Kkgt1SwR4p6/R92Q+83PS8YHfT6yahz54HnLmbQl1dXYffEQLeobq6WlLLO5YAADqpx4ef9eT0XZKqSqQ1V/97eGW5R0s6g7DnAb6+vurZs6eOHTsmPz8/+fgwoYpzMwxD1dXVKi4uVu/evS/5Pr0AAA+blyG9do/U2CC57nB1+tHHV5rzhFmVEfY8wWKxqH///nI4HMrNzTW7HHQBvXv3Vr9+XaNhNgBA0ojbpYgrms/knbH0Iyl6lMdLOoOw5yH+/v4aOHCg6urqzC4FnZyfnx8zegDQpflIajzr0VyEvXZqy71xfXx8uBsCAADeKqivFBwphQyQxiyQdj0jVRxpGjcRt0tzk7bctgQAAHiphlrJ6i9ZLJJhSM46yTfA7YdpS+5gZg8AAMBdzg52FkuHBL22YlkoAACAFyPsAQAAeDHCHgAAgBcj7AEAAHgxwl47paena8iQIRo3bpzZpQAAALRA6xU3ofUKAADwlLbkDmb2AAAAvBhhDwAAwIsR9gAAALwYYQ8AAMCLEfYAAAC8GGEPAAAPyCrJ0pL3lyirJMvsUtDNEPYAAPCAN755Q9uKtunN7DfNLgXdjK/ZBXR16enpSk9Pl9PpNLsUAEAnU1BZoOO1x2WRRe/lvCdJetfxrmZfNluGDPUJ6KPo4GiTq4S3o6mym9BUGQDwbcOfHn7RbfYv3O+BSuBtaKoMAEAnsMq2SlaL9ZyvWS1WrbKt8nBF6I44jQsAQAe5KekmJYUmaf5b81u89vyNz2tI+BATqkJ3w8weAAAeYJGl2SPgKczsAQDQgcICwxQeGK5+Qf00b+A8vfL1KyqqKlJYYJjZpaGbYIGGm7BAAwBwPnXOOvn5+MliscgwDNU31svf6m92WejC2pI7mNkDAKCDnR3sLBYLQQ8exTV7AAAAXoywBwAA4MUIewAAAF6MsAcAAODFCHvtlJ6eriFDhmjcuHFmlwIAANACrVfchNYrAADAU7g3LgAAACQR9gAAALwaYQ8AAMCLEfYAAAC8GGEPAADAixH2AAAAvBhhDwAAwIsR9gAAALwYYQ8AAMCLEfYAAAC8GGEPAADAixH2AAAAvBhhDwAAwIsR9gAAALwYYa+d0tPTNWTIEI0bN87sUgAAAFqwGIZhmF2EN6ioqFBoaKjKy8sVEhJidjkAAMCLtSV3MLMHAADgxQh7AAAAXoywBwAA4MUIewAAeMKRXdK6m5oeAQ8i7AEA4Al7X5By7NK+9WZXgm7G1+wCAADwWifypOpSSRYp65WmsQMvSyP/Q5Ih9QyXeseZWSG6AcIeAAAd5fHhZz2xND1UlUhrrv738Mpyj5aE7ofTuAAAdJR5GZLPmXkVo/mjj2/T60AHY2YPAICOMuJ2KeKK5jN5Zyz9SIoe5fGS0P0wswcAgEf4fOsR8Az+xAEA0JGC+krBkVL0SOmmx5oegyObxgEP4DQuAAAdKXSAtPyAZPWXLBYpebHkrJN8A8yuDN0EYQ8AgI52drCzWAh68ChO4wIAAHgxwh4AAIAXI+wBAAB4McIeAACAFyPsAQAAeDHCHgAAgBcj7AEAAHgxwt5Z3nrrLQ0aNEgDBw5UZmam2eUAAAC0G02VT2toaNCKFSv08ccfKyQkRGPGjNG8efMUFhZmdmkAAACXjJm907Zt26ahQ4dqwIAB6tWrl2644Qa9//77ZpcFAADQLl4T9jZu3KhZs2YpOjpaFotFr732WottVq9ercTERAUGBio5OVl2u931WkFBgQYMGOB6HhMToyNHjniidADdWFZJlpa8v0RZJVlmlwLAS3lN2KuqqtLIkSP1l7/85Zyvr1+/XsuXL9dDDz2k3bt3y2azaebMmcrLy5MkGYbR4j0Wi6VDawaAN755Q9uKtunN7DfNLgWAl/Kaa/ZmzpypmTNnnvf1Rx99VEuWLNHSpUslSY8//rjef/99PfHEE1q1apUGDBjQbCbv8OHDmjBhwnn3V1tbq9raWtfziooKN3wKAN1BQWWBjtcel0UWvZfzniTpXce7mn3ZbBky1Cegj6KDo02uEoC38JqwdyF1dXXauXOnHnzwwWbj06dP15YtWyRJ48eP14EDB3TkyBGFhITonXfe0S9/+cvz7nPVqlX69a9/3aF1A/BOM16e0WKsrKZM89+a73q+f+F+T5YEwIt5zWncCykpKZHT6VRUVFSz8aioKBUVFUmSfH199Yc//EHTpk3T6NGj9V//9V8KDw8/7z5/9rOfqby83PWTn5/foZ8BgPdYZVslq8V6ztesFqtW2VZ5uCIA3qxbzOyd8e1r8AzDaDY2e/ZszZ49u1X7CggIUEBAgFvrA9A93JR0k5JCk5rN5J3x/I3Pa0j4EBOqAuCtusXMXkREhKxWq2sW74zi4uIWs30A4EkWWZo9AoC7dYuw5+/vr+TkZG3YsKHZ+IYNG3TVVVe1a9/p6ekaMmSIxo0b1679AOhewgLDFB4YriHhQ/SLib/QkPAhCg8MV1ggjdwBuJfFOFfPkS6osrJShw4dkiSNHj1ajz76qKZNm6awsDDFxcVp/fr1Sk1N1f/93/9p0qRJWrNmjTIyMpSVlaX4+Ph2H7+iokKhoaEqLy9XSEhIu/cHwPvVOevk5+Mni8UiwzBU31gvf6u/2WUB6ALakju85pq9HTt2aNq0aa7nK1askCQtXLhQ69at0/z581VaWqqHH35YhYWFGjZsmN555x23BD0AuBRnBzuLxULQA9AhvGZmz2zM7AEAAE9pS+7oFtfsdSSu2QMAAJ0ZM3tuwsweAADwFGb2AAAAIImwBwAA4NUIewAAAF6MsAcAAODFCHvtxGpcAADQmbEa101YjQsAADyF1bgAAACQRNgDAADwaoQ9AAAAL0bYAwAA8GKEvXZiNS4AAOjMWI3rJqzGBQAAnsJqXAAAAEgi7AEAAHg1wh4AAIAXI+wBAAB4McIeAACAFyPstROtVwAAQGdG6xU3ofUKAADwFFqvAAAAQBJhDwAAwKsR9gAAALwYYQ8AAMCLEfYAwExHdknrbmp6BIAOQNgDADPtfUHKsUv71ptdCQAv5Wt2AQDQ7ZzIk6pLJVmkrFeaxg68LI38D0mG1DNc6h1nZoUAvAhhr53S09OVnp4up9NpdikAuorHh5/1xNL0UFUirbn638Mryz1aEgDvxWncdlq2bJkOHjyo7du3m10KgK5iXobkc+bf2kbzRx/fptcBwE2Y2QMATxtxuxRxRfOZvDOWfiRFj/J4SQC8FzN7AGAqn289AoB78dsFAMwQ1FcKjpSiR0o3Pdb0GBzZNA4AbsRpXAAwQ+gAafkByeovWSxS8mLJWSf5BphdGQAvQ9gDALOcHewsFoIegA7BaVwAAAAvRtgDAADwYoQ9AAAAL0bYAwAA8GKEPQAAAC9G2Gun9PR0DRkyROPGjTO7FAAAgBYshmEYF98MF1NRUaHQ0FCVl5crJCTE7HIAAIAXa0vuYGYPAADAixH2AAAAvBhhD+iEskqytOT9JcoqyTK7FABAF0fYAzqhN755Q9uKtunN7DfNLgUA0MVxb1ygkyioLNDx2uOyyKL3ct6TJL3reFezL5stQ4b6BPRRdHC0yVUCALoawh7QScx4eUaLsbKaMs1/a77r+f6F+z1ZEgDAC3AaF+gkVtlWyWqxnvM1q8WqVbZVHq4IAOANmNkDOombkm5SUmhSs5m8M56/8XkNCR9iQlUAgK6OmT2gE7LI0uwRAIBLxcwe0ImEBYYpPDBc/YL6ad7AeXrl61dUVFWksMAws0sDAHRR3C7NTbhdGtylzlknPx8/WSwWGYah+sZ6+Vv9zS4LANCJtCV3MLMHdDJnBzuLxULQAwC0C9fsAQAAeDHCHgAAgBcj7LVTenq6hgwZonHjxpldCgAAQAss0HATFmgAAABPaUvuYGYPAADAixH2AAAAvBhhDwAAwIsR9gAAALwYYQ8AAMCLEfYAAAC8GGEPAADAixH2AAAAvBhhDwAAwIsR9gAAALwYYQ8AAMCLEfYAAAC8mG97d5CXlydJioyMVGBgYLsLAgAAgPu0e2YvISFBSUlJKisrc0c9AAAAcKN2z+wFBwfLz89P0dHR7qgHAAAAbuSWmb3q6mo5nU531AMAAAA3anfYmzNnjurq6vT222+7ox4AAAC4UbvD3k9/+lNdfvnluueee7Rv3z531AQAAAA3afc1ey+//LK+//3va+XKlRo7dqyuv/56paSkKDIyUlar9bzvW7BgQXsPDQAAgIuwGIZhtGcHPj4+slgskiTDMFz/+4IHtVjU0NDQnsN2OhUVFQoNDVV5eblCQkLMLgcAAHixtuSOds/sxcXFtSrgAQAAwPPaHfZycnLcUEbnMHfuXH3yySf6zne+o5deesnscgAAANqN26Wd5Yc//KGeeeYZs8sAAABwG8LeWaZNm6ZevXqZXQYgHdklrbup6REAgHZwe9jLysrS008/rd///vf63//9Xz399NM6ePBgu/e7ceNGzZo1S9HR0bJYLHrttddabLN69WolJiYqMDBQycnJstvt7T4uYIq9L0g5dmnferMrAQC0kbPR0Lv7C3Wypt7sUiS54Zq9M95//3098MADOnDgwDlfHz58uH73u99p+vTpl7T/qqoqjRw5UosXL9Ytt9zS4vX169dr+fLlWr16tVJSUvTXv/5VM2fO1MGDBxUXFydJSk5OVm1tbYv3fvDBB22+3VttbW2zfVVUVLTxEwHfciJPqi6VZJGyXmkaO/CyNPI/JBlSz3Cpd5yZFQIALqCqtkEv7sjXU5tzlFdWrZ/fOFhLbUlml9X+1iuS9Je//EU/+tGPJDW1X7FarYqIiJAklZaWutqsWCwW/elPf9KyZcvadTyLxaJXX31Vc+bMcY1NmDBBY8aM0RNPPOEaGzx4sObMmaNVq1a1et+ffPKJ/vKXv1x0gcbKlSv161//usU4rVdwyVaGnvXEIsk46/HMNuWerQkAcFHFFTVatyVHz32Wp/JTTbN5vXv66UffGajFKYkdcsy2tF5p92ncvXv3avny5TIMQ+PHj9c777yjyspKFRYWqrCwUCdPntQ777yjSZMmyTAMLV++3O132qirq9POnTtbzBpOnz5dW7ZsceuxzvjZz36m8vJy109+fn6HHAfdyLwMyefMZLvR/NHHt+l1AECn8WXRSd3/4l6l/L9/avUn36j8VL0Swnvqv28eqi0PXtNhQa+t2n0a99FHH1VjY6NmzZqlV155pcVdMwICAnT99dfruuuu07x58/Tmm2/qscce09q1a9t7aJeSkhI5nU5FRUU1G4+KilJRUVGr9zNjxgzt2rVLVVVViomJ0auvvqpx48adc9uAgAAFBAS0q26gmRG3SxFXSGuubvna0o+k6FEeLwkA0JxhGNp0qEQZdoc2fnXMNT42vo/SpiTp2sFRsvp0rv7D7Q57//rXv2SxWPTHP/7xgrdHs1qtevzxx/Xmm2/q448/bu9hz+nbzZ1be0ePM95//313lwRcIh9JjWc9AgDMVNfQqDf3FijDnq0vik5Kknws0vXD+mmpLUlj4vqYXOH5tTvsHT16VKGhoUpISLjotomJierdu7eOHj3a3sM2ExERIavV2mIWr7i4uMVsn7ulp6crPT1dTqezQ4+DbiKorxQcKYUMkMYskHY9I1UcaRoHAHhceXW9nt+Wp3VbHDpa0bQws6e/VbePjdX3UhIVF97T5Aovrt1hr0ePHqqurlZDQ4N8fS+8u4aGBlVXV6tnT/f+h/H391dycrI2bNiguXPnusY3bNigm2++2a3H+rZly5Zp2bJlrgslgXYJHSAtPyBZ/SWLRUpeLDnrJF8uGQAAT8ovq9aTmxz6x458Vdc1TehE9grQopQE3Tk+XqE9/UyusPXaHfYGDx6sTz/9VC+99JLuuOOOC2774osvqq6uTsnJyW0+TmVlpQ4dOuR67nA4tGfPHoWFhSkuLk4rVqxQamqqxo4dq0mTJmnNmjXKy8vT3Xff3eZjAaY6O9hZLAQ9APCg3XnHlWl36N0DhWo8vUbuyn69tNSWpNkjo+Xv2/XuR9HusHfbbbdp69at+sEPfqCIiAhde+2159zuww8/1A9+8ANZLBbdfvvtbT7Ojh07NG3aNNfzFStWSJIWLlyodevWaf78+SotLdXDDz+swsJCDRs2TO+8847i4+Mv7YMBAIBuwdlo6MPPjypjY7Z25B53jdsGRijNliTbwIg2rQHobNrdZ6+2tlZjx45VVlaWLBaLJk2apGuvvVYDBgyQxWJRfn6+PvroI23dulWGYWjYsGHasWOH/P393fUZOoW29LsBAADmO1Xn1Eu7DutJe7ZySqslSX5Wi2aPHKCltkQN7t95/z5vS+5wS1PlgoICzZs3T9u2bWva6TlWxUpNjY9ffvnlNt+tojM7e4HGV199RdgDAKCTO3ayVs9szdHfPs3V8eqmJsghgb66c2K8Fl2VoKiQQJMrvDiPhz1Jamxs1EsvvaT169drx44dKi4uliRFRkZq7NixuuOOO3TLLbfIx6frnetuDWb2AADo3L4+elKZdode3XNEdQ1Nba1iw3poSUqibhsbq6AAt91FtsOZEva6O8IeAACdj2EY2vpNqTLs2fr4y383QR4V21t3TUnSjKH9Ol0T5NZoS+5od4T18fGRj4+PvvjiC11++eXt3R0AAEC71Tsb9fa+QmXYs5VVUCGpqcHB9CFRSrMlKTm+T5dedNEWbumz5+fnR9ADAACmq6ip1wvb8rR2c44Ky2skSYF+ProtOVbfm5yoxIggkyv0vHaHvZiYGB0+fNgdtXRJ3EEDAADzHTlxSms3OfTC9nxV1jZIkiKCA7RwUrz+c2K8+gR5VxeQtmj3NXsrVqzQH//4R/3zn//U1Vef4wbu3QTX7AEA4Hn7Dp9Qht2hd/YXynm6C/LAyGAttSXq5lEDFOhnNbnCjuHRBRrHjh3T8OHDFRYWpo8++kj9+/dvz+66rI4Oe1klWXp056NakbxCQyOGun3/AAB0FY2Nhv75RbEy7Nn6zFHmGk+5PFxLbUm6emBf+XTBRRdt4dEFGp9//rl+85vf6Mc//rGGDBmi1NRUpaSkKDIyUlbr+dP0lClT2nvobuWNb97QtqJtejP7TcIeAKBbqql36pVdR5S5KVvZx6okSb4+Fs0aGa2ltkQNjeYe9efS7pk9Hx+fNq9msVgsamhoaM9hO52OmNkrqCzQ8drjssiiez68R2U1ZQoLDNMT1z4hQ4b6BPRRdLD3NKgGAOBcSitr9eynuXp2a65Kq+okSb0CfPXdCXFalJKg/qE9TK7Q8zw6syf9+w4ZHbV9dzXj5RktxspqyjT/rfmu5/sX7vdkSQAAeMw3xyr15CaHXt55WLWnmyAP6N1D35ucqPnjYhXchZogm6nd/5UaGxvdUUeX1ZGrcVfZVunnm34up9Fy31aLVY9MfsTtxwQAwEyGYWibo0wZdoc+/Pyoa3xETKjSbEmaOayffK3eeTeujtLu07h5eXmSmm6LFhjY+e8l11E6aoHGwdKDzWbyzlh/03oNCR/ituMAAGCmBmej3j1QpAx7tvYdLneNXzs4Smm2RI1PDOs2TZBbw6OncRMSEuTj46O8vDxFR3P9WEexyCJDhusRAABvUFnb4GqCfOTEKUlSgK+PbkmO0ZLJibqsb7DJFXZ97Q57wcHB8vPzI+h1kLDAMIUHhqtfUD/NGzhPr3z9ioqqihQWGGZ2aQAAXLLC8lNatzlHz2/L08mapkWb4UH+Sp0Ur9SJ8QoPDjC5Qu/R7tO4I0aM0Ndff63KysoLtlrxdh3ZZ6/OWSc/Hz9ZLBYZhqH6xnr5W7tvJ3AAQNeVVVCuTLtDb+4tUMPpJshJfYO0dHKS5o3x3ibI7ubR07hz5szRb37zG7399tuaPXt2e3eHczg72FksFoIeAKBLMQxDn3x1TJn2bG0+VOoan5AYprumJGnaoEivb4JspnbP7FVVVWnMmDGqrKzUu+++qxEjRrirti6F26UBANBcbYNTr+8uUIY9W18XV0qSrD4W3TC8v9JsiRoR09vcArswj87svfzyy/r+97+vlStXauzYsbr++utbdQeNBQsWtPfQAACgEzpeVafnPsvVui25KqmslSQFB/jqjnGxWpSSoJg+PU2usHtx6x00DMNo1bJob7qDxtl99r766itm9gAA3VZOSZWe3OTQizvzVVPf1Ie3f2igFqck6I7xcQoJ9DO5Qu/Rlpm9doe9hISES+p743A42nPYTofTuACA7mpnbpnWbMzWBweP6kyqGBodojRbkm4c0V9+NEF2O4+exs3JyWnvLgAAQBfjbDT0flZTE+TdeSdc49MG9VWaLUmTLgunCXInwU3lAABAq1XVNujFHfl6anOO8sqqJUn+Vh/NHT1AS22JGhjVy+QK8W0eD3uffvqp6urqNGXKFE8fGgAAXKLiihqt25Kj5z7LU/mpeklS755+Sp0Yr9RJ8Yrs1X1vmdrZtTns+fj4qH///jpy5EiL13784x+roqJCTz755HnfP3fuXB07dsxrFmgAAODNviiqUKbdodf3HFG9s+mCvITwnlpiS9KtY2LUw58myJ3dJc3snW9NxwsvvKDi4uILhr0LvR8AAJjPMAxtOlSiNRuzZf+6xDU+LqGPltqSdO3gKFlpgtxlcM0eAACQJNU1NOqNvQXKtGfri6KTkiQfizRzWH8ttSVqdFwfkyvEpSDsAQDQzZVX1+u5bbl6ekuOjlY0NUHu6W/V7WNjtWRyomLDaILclRH22unspsoAAHQl+WXVenKTQ//Yka/quqa/xyJ7BWhRSoLuHB+v0J40QfYGhL12WrZsmZYtW+ZqbggAQGe3O++4Mu0OvXugUI2nL6O/sl8vpdmSNGtktPx9aYLsTQh7AAB0A85GQx9+flQZG7O1I/e4a9w2MEJ3TUnS5MsjaILspQh7AAB4sVN1Tr2067CetGcrp7SpCbKf1aKbRzU1Qb6yH7f49HaEPQAAvNCxk7V6ZmuO/vZpro5XNzVBDgn01X9OjNfCqxIUFUIT5O7iksLe0aNHZbWev4nihV4zDINpYgAAOsjXR08q0+7Qq3uOqK6hUZIUG9ZDS1ISddvYWAUFMM/T3bi1qTIAAPA8wzC09ZtSZdiz9fGXx1zjo+N6K82WpBlD+9EEuRtrc9j71a9+1RF1AACANqp3NurtfYXKsGcrq6BCkmSxSNOHROmuKUlKjg8zuUJ0BhaDaTq3ONN6pby8XCEhXOwKAOg4FTX1emFbntZuzlFheY0kKdDPR7clNzVBTogIMrlCdLS25A5O3AMA0EUcOXFKazc59ML2fFXWNkiSIoIDtOiqeN05IV59gvxNrhCdEWEPAIBObt/hE8qwO/TO/kI5T3dBHhgZrDRbkmaPilag3/kXRgKEvXbidmkAgI7Q2Gjon18UK8Oerc8cZa7xlMvDtdSWpKlX9KW7BVqFa/bchGv2AADuUFPv1Cu7jihzU7ayj1VJknx9LJo1MlpLbYkaGs2tOcE1ewAAdDmllbV69tNcPbs1V6VVdZKkXgG++u6EOC1KSVD/0B4mV4iuirAHAICJvjlWqSc3OfTyzsOqPd0EeUDvHvre5ETNHxerYJogo534E9RVHNklbfildN3D0oAxZlcDAGgHwzC0zVGmDLtDH35+1DU+IiZUabYkzRzWT75WHxMrhDch7HUVe1+QcuzSvvWEPQDoohqcjXr3QJEy7Nnad7jcNX7t4Cil2RI1PjGMRRdwO8JeZ3YiT6oulWSRsl5pGjvwsjTyPyQZUs9wqXecmRUCAFqhsrZB67fn66lNDh05cUqSFODro1uSY7RkcqIu6xtscoXwZoS9zuzx4Wc9Of0vvaoSac3V/x5eWS4AQOdUWH5K6zbn6PlteTpZ09QEOTzIX6mT4pU6MV7hwQEmV4jugLDXmc3LkF67R2pskHSmQ87pRx9fac4TZlUGALiArIJyZdodenNvgRpON0FO6hukpZOTNG/MAJogw6MIe53ZiNuliCuaz+SdsfQjKXqUx0sCAJybYRj65KtjyrRna/OhUtf4hMQw3TUlSdMGRcrHh+vx4HmEvS7DR1LjWY8AgM6gtsGp13cXKMOera+LKyVJVh+LbhjeX2m2RI2I6W1ugej2CHudXVBfKThSChkgjVkg7XpGqjjSNA4AMM3xqjo991mu1m3JVUllrSQpOMBXd4yL1aKUBMX06WlyhUATwl5nFzpAWn5AsvpLFouUvFhy1km+XNQLAGbIKanSk5scenFnvmrqm8609A8N1OKUBN0xPk4hgX4mVwg0R9jrCs4OdhYLQQ8ATLAzt0xrNmbrg4NHdeau8kOjQ5RmS9KNI/rLjybI6KQIewAAnIez0dD7WU1NkHfnnXCNTxvUV2m2JE26LJwmyOj0CHsAAHxLVW2DXtyRr6c25yivrFqS5G/10dzRA7TUlqiBUb1MrhBoPcJeO6Wnpys9PV1Op9PsUgAA7VRcUaN1W3L03Gd5Kj9VL0nq3dNPqRPjlTopXpG9Ak2uEGg7i2GcufIA7VFRUaHQ0FCVl5crJCTE7HIAAG3wRVGFMu0Ovb7niOqdTX8tJoT31BJbkm4dE6Me/jRBRufSltzBzB4AoFsyDEObDpVozcZs2b8ucY2PS+ijpbYkXTs4SlaaIMMLEPYAAN1KXUOj3thboEx7tr4oOilJ8rFIM4f111JbokbH9TG5QsC9CHsAgG6hvLpez23L1dNbcnS0oqkJck9/q24fG6slkxMVG0YTZHgnwh4AwKvll1XryU0O/WNHvqrrmhbTRfYK0KKUBN05Pl6hPWmCDO9G2AMAeKXdeceVaXfo3QOFajy9FPHKfr2UZkvSrJHR8velCTK6B8IeAMBrOBsNffj5UWVszNaO3OOucdvACN01JUmTL4+gCTK6HcIeAKDLO1Xn1Eu7DutJe7ZySpuaIPtZLbp5VFMT5Cv70RIL3RdhDwDQZR07Watntubob5/m6nh1UxPkkEBf/efEeC28KkFRITRBBgh7AIAu5+ujJ5Vpd+jVPUdU19AoSYoN66ElKYm6bWysggL46w04g28DAKBLMAxDW78pVYY9Wx9/ecw1Pjqut9JsSZoxtB9NkIFzIOwBADq1emej3t5XqAx7trIKKiRJFos0fUiU7pqSpOT4MJMrBDo3wh4AoFOqqKnXC9vytHZzjgrLayRJgX4+ui25qQlyQkSQyRUCXQNhDwDQqRw5cUprNzn0wvZ8VdY2SJIiggO06Kp43TkhXn2C/E2uEOhaCHsAgE5h3+ETyrA79M7+QjlPd0EeGBmsNFuSZo+KVqCf1eQKga6JsAcAME1jo6F/flGsDHu2PnOUucZTLg/XUluSpl7RlybIQDsR9gAAHldT79Qru44oc1O2so9VSZJ8fSyaNTJaS22JGhodanKFgPcg7AEAPKa0slbPfpqrZ7fmqrSqTpLUK8BX350Qp0UpCeof2sPkCgHvQ9gDAHS4b45V6slNDr2887BqTzdBHtC7h743OVHzx8UqmCbIQIfh2wUA6BCGYWibo0wZdoc+/Pyoa3xETKjSbEmaOayffK0+JlYIdA+EPQCAWzU4G/XugSJl2LO173C5a/zawVFKsyVqfGIYiy4ADyLsAQDcorK2wdUE+ciJU5KkAF8f3ZIcoyWTE3VZ32CTKwS6J8Leafn5+UpNTVVxcbF8fX31i1/8QrfddpvZZQFAp1dYfkrrNufo+W15OlnT1AQ5PMhfqZPilToxXuHBASZXCHRvhL3TfH199fjjj2vUqFEqLi7WmDFjdMMNNygoiNvxAMC5ZBWUK9Pu0Jt7C9RwuglyUt8gLZ2cpHljBtAEGegkCHun9e/fX/3795ckRUZGKiwsTGVlZYQ9ADiLYRj65KtjyrRna/OhUtf4hMQw3TUlSdMGRcrHh+vxgM6kyyyD2rhxo2bNmqXo6GhZLBa99tprLbZZvXq1EhMTFRgYqOTkZNnt9ks61o4dO9TY2KjY2Nh2Vg0A3qG2wal/bM/X9Mc2avHa7dp8qFTW002Q37g3Reu/P0nfGRxF0AM6oS4zs1dVVaWRI0dq8eLFuuWWW1q8vn79ei1fvlyrV69WSkqK/vrXv2rmzJk6ePCg4uLiJEnJycmqra1t8d4PPvhA0dHRkqTS0lItWLBAmZmZF6yntra22b4qKira8/EAoFM6XlWn5z7L1botuSqpbPqdFxzgqzvGxWpRSoJi+vQ0uUIAF2MxDMMwu4i2slgsevXVVzVnzhzX2IQJEzRmzBg98cQTrrHBgwdrzpw5WrVqVav2W1tbq+uuu05paWlKTU294LYrV67Ur3/96xbj5eXlCgkJad0HAYBOKqekSk9ucujFnfmqqW9qgtw/NFCLUxJ0x/g4hQT6mVwh0L1VVFQoNDS0Vbmjy8zsXUhdXZ127typBx98sNn49OnTtWXLllbtwzAMLVq0SNdcc81Fg54k/exnP9OKFStczysqKjjtC6DL25lbpjUbs/XBwaM6MxUwNDpEabYk3Tiiv/xoggx0OV4R9kpKSuR0OhUVFdVsPCoqSkVFRa3ax+bNm7V+/XqNGDHCdT3gs88+q+HDh59z+4CAAAUE0E4AQNfnbDT0flZTE+TdeSdc49MG9VWaLUmTLgunCTLQhXlF2Dvj27+MDMNo9S+oyZMnq7GxsSPKAoBOqaq2QS/uyNdTm3OUV1YtSfK3+mju6AFaakvUwKheJlcIwB28IuxFRETIarW2mMUrLi5uMdvnbunp6UpPT5fT6ezQ4wCAuxRX1Gjdlhw991meyk/VS5J69/RT6sR4pU6KV2SvQJMrBOBOXhH2/P39lZycrA0bNmju3Lmu8Q0bNujmm2/u0GMvW7ZMy5Ytc10oCQCd1RdFFcq0O/T6niOqdzZdkJcQ3lNLbEm6dUyMevjTBBnwRl0m7FVWVurQoUOu5w6HQ3v27FFYWJji4uK0YsUKpaamauzYsZo0aZLWrFmjvLw83X333SZWDQDmMgxDmw6VaM3GbNm/LnGNj0voo6W2JF07OEpWeuMBXq3LhL0dO3Zo2rRprudnVsIuXLhQ69at0/z581VaWqqHH35YhYWFGjZsmN555x3Fx8ebVTIAmKauoVFv7C1Qpj1bXxSdlCT5WKSZw/prqS1Ro+P6mFwhAE/pkn32OqO29LsBgI5SXl2v57bl6uktOTpa0dQEuae/VbePjdWSyYmKDaMJMuANul2fPTOxQANAZ5BfVq0nNzn0jx35qq5r+n0U2StAi1ISdOf4eIX2pAky0F0xs+cmzOwBMMPuvOPKtDv07oFCNZ7+bX5lv15KsyVp1sho+fvSBBnwRszsAYAXczYa+vDzo8rYmK0ducdd47aBEbprSpImXx5BE2QALoQ9AOgiTtU59dKuw3rSnq2c0qYmyH5Wi24e1dQE+cp+nFUA0BJhDwA6uWMna/XM1hz97dNcHa9uaoIcEuir/5wYr4VXJSgqhCbIAM6PsNdOLNAA0FG+PnpSmXaHXt1zRHUNTbdzjA3roSUpibptbKyCAvgVDuDiWKDhJizQAOAOhmFo6zelyrBn6+Mvj7nGR8f1VpotSTOG9qMJMgAWaABAV1PvbNTb+wqVYc9WVkGFJMlikaYPidJdU5KUHB9mcoUAuirCHgCYqKKmXi9sy9PazTkqLK+RJAX6+ei25KYmyAkRQSZXCKCrI+wBgAmOnDiltZscemF7viprGyRJEcEBWnRVvO6cEK8+Qf4mVwjAWxD2AMCD9h0+oQy7Q+/sL5TzdBfkgZHBSrMlafaoaAX6WU2uEIC3IewBQAdrbDT0zy+KlWHP1meOMtd4yuXhWmpL0tQr+tIEGUCHIey1E61XAJxPTb1Tr+w6osxN2co+ViVJ8vWxaNbIaC21JWpodKjJFQLoDmi94ia0XgFwRmllrZ79NFfPbs1VaVWdJKlXgK++OyFOi1IS1D+0h8kVAujqaL0CACb45lilntzk0Ms7D6v2dBPkAb176HuTEzV/XKyCaYIMwAT85gGAdjAMQ9scZcqwO/Th50dd4yNiQpVmS9LMYf3ka/UxsUIA3R1hDwAuQYOzUe8eKFKGPVv7Dpe7xq8dHKU0W6LGJ4ax6AJAp0DYA4A2qKxtcDVBPnLilCQpwNdHtyTHaMnkRF3WN9jkCgGgOcIeALRCYfkprduco+e35elkTVMT5PAgf6VOilfqxHiFBweYXCEAnBthr51ovQJ4t6yCcmXaHXpzb4EaTjdBTuobpKWTkzRvzACaIAPo9Gi94ia0XgG8h2EY+uSrY8q0Z2vzoVLX+ITEMN01JUnTBkXKx4fr8QCYh9YrAHAJahucen13gTLs2fq6uFKSZPWx6Ibh/ZVmS9SImN7mFggAl4CwB6DbO15Vp+c+y9W6LbkqqayVJAUH+OqOcbFalJKgmD49Ta4QAC4dYQ9At5VTUqUnNzn04s581dQ3NUHuHxqoxSkJumN8nEIC/UyuEADaj7AHoNvZmVumNRuz9cHBozpz1fLQ6BCl2ZJ044j+8qMJMgAvQtgD0C04Gw29n9XUBHl33gnX+LRBfZVmS9Kky8JpggzAKxH2AHi1qtoGvbgjX09tzlFeWbUkyd/qo7mjB2ipLVEDo3qZXCEAdCzCHgCvVFxRo3VbcvTcZ3kqP1UvSerd00+pE+OVOilekb0CTa4QADyDsNdONFUGOpcviiqUaXfo9T1HVO9suiAvIbynltiSdOuYGPXwpwkygO6FpspuQlNlwDyGYWjToRKt2Zgt+9clrvFxCX201JakawdHyUoTZABehKbKALqFuoZGvbG3QJn2bH1RdFKS5GORZg7rr6W2RI2O62NyhQBgPsIegC6nvLpez23L1dNbcnS0oqkJck9/q24fG6slkxMVG0YTZAA4g7AHoMvIL6vWk5sc+seOfFXXNV0nG9krQItSEnTn+HiF9qQJMgB8G2EPQKe3O++4Mu0OvXugUI2nrzK+sl8vpdmSNGtktPx9aYIMAOdD2APQKTkbDX34+VFlbMzWjtzjrnHbwAjdNSVJky+PoAkyALQCYQ9Ap3KqzqmXdh3Wk/Zs5ZQ2NUH2s1p086imJshX9mO1OwC0BWEPQKdw7GStntmao799mqvj1U1NkEMCffWfE+O18KoERYXQBBkALgVhD4Cpvj56Upl2h17dc0R1DY2SpNiwHlqSkqjbxsYqKIBfUwDQHvwWBeBxhmFo6zelyrBn6+Mvj7nGR8f1VpotSTOG9qMJMgC4CWGvnbhdGtB69c5Gvb2vUBn2bGUVVEiSLBZp+pAo3TUlScnxYSZXCADeh9uluQm3SwPOr6KmXi9sy9PazTkqLK+RJAX6+ei25KYmyAkRQSZXCABdC7dLA9ApHDlxSms3OfTC9nxV1jZIkiKCA7ToqnjdOSFefYL8Ta4QALwfYQ+A2+07fEIZdofe2V8o5+kuyAMjg5VmS9LsUdEK9LOaXCEAdB+EPQBu0dho6J9fFCvDnq3PHGWu8ZTLw7XUlqSpV/SlCTIAmICwB6BdauqdemXXEWVuylb2sSpJkq+PRbNGRmupLVFDo0NNrhAAujfCHoBLUlpZq2c/zdWzW3NVWlUnSeoV4KvvTojTopQE9Q/tYXKFAACJsAegjb45VqknNzn08s7Dqj3dBHlA7x763uREzR8Xq2CaIANAp8JvZQAXZRiGtjnKlGF36MPPj7rGR8SEKs2WpJnD+snX6mNihQCA8yHsATivBmej3j1QpAx7tvYdLneNXzs4Smm2RI1PDGPRBQB0coQ9AC1U1ja4miAfOXFKkhTg66NbkmO0ZHKiLusbbHKFAIDWIuwBcCksP6V1m3P0/LY8naxpaoIcHuSv1EnxSp0Yr/DgAJMrBAC0FWEPgLIKypVpd+jNvQVqON0EOalvkJZOTtK8MQNoggwAXRhhD+imDMPQJ18dU6Y9W5sPlbrGJySG6a4pSZo2KFI+PlyPBwBdHWEP6GZqG5x6fXeBMuzZ+rq4UpJk9bHohuH9lWZL1IiY3uYWCABwK8Ie0E0cr6rTc5/lat2WXJVU1kqSggN8dce4WC1KSVBMn54mVwgA6AiEPcDL5ZRU6clNDr24M1819U1NkPuHBmpxSoLuGB+nkEA/kysEAHQkwl47paenKz09XU6n0+xSgGZ25pZpzcZsfXDwqIymNRcaGh2iNFuSbhzRX340QQaAbsFiGGf+GkB7VFRUKDQ0VOXl5QoJCTG7HHRTzkZD72c1NUHenXfCNT5tUF+l2ZI06bJwmiADgBdoS+5gZg/wAlW1DXpxR76e2pyjvLJqSZK/1UdzRw/QUluiBkb1MrlCAIBZCHtAF1ZcUaN1W3L03Gd5Kj9VL0nq3dNPqRPjlTopXpG9Ak2uEABgNsIe0AV9UVShTLtDr+85onpn05UYCeE9tcSWpFvHxKiHP02QAQBNCHtAF2EYhjYdKtGajdmyf13iGh+X0EdLbUm6dnCUrDRBBgB8C2EP6OTqGhr1xt4CZdqz9UXRSUmSj0WaOay/ltoSNTquj8kVAgA6M8Ie0EmVV9fruW25enpLjo5WNDVB7ulv1e1jY7VkcqJiw2iCDAC4OMIe0Mnkl1XryU0O/WNHvqrrmvo3RvYK0KKUBN05Pl6hPWmCDABoPcIe0EnszjuuTLtD7x4oVOPp7pdX9uulNFuSZo2Mlr8vTZABAG1H2ANM5Gw09OHnR5WxMVs7co+7xm0DI3TXlCRNvjyCJsgAgHYh7AEmOFXn1Eu7DutJe7ZySpuaIPtZLZo9sqkJ8uD+3IUFAOAehD3Ag46drNUzW3P0t09zdby6qQlySKCv7pwYr0VXJSgqhCbIAAD3IuwBHvD10ZPKtDv06p4jqmtolCTFhvXQkpRE3TY2VkEBfBUBAB2Dv2GADmIYhrZ+U6oMe7Y+/vKYa3xUbG/dNSVJM4b2owkyAKDDEfYAN6t3NurtfYXKsGcrq6BCkmSxSNOHRCnNlqTk+D4sugAAeAxhD3CTipp6vbAtT2s356iwvEaSFOjno9uSY/W9yYlKjAgyuUIAQHdE2APa6ciJU1q7yaEXtuersrZBkhQRHKCFk+L1nxPj1SfI3+QKAQDdGWEPuET7D5crw56tt/cXynm6C/LAyGAttSXq5lEDFOhnNblCAAAIe0CbNDYa+vjLYmXYs/Vpdplr/KrLwpU2JUlXD+wrHxZdAAA6EcIe0Ao19U69uvuIMu3Z+uZYlSTJ18eiWSOjtWRyooYNCDW5QgAAzo2wB1xAaWWtnv00V89uzVVpVZ0kqVeAr747IU6LUhLUP7SHyRUCAHBhhL3TTp48qWuuuUb19fVyOp364Q9/qLS0NLPLgkm+OVapJzc59PLOw6o93QR5QO8eWpySoPnjYtUr0M/kCgEAaB3C3mk9e/bUv/71L/Xs2VPV1dUaNmyY5s2bp/DwcLNLg4cYhqFtjjJl2B366IujMprWXGhETKiW2pJ0w7B+8rX6mFskAABtRNg7zWq1qmfPnpKkmpoaOZ1OGWf+todXa3A26t0DRcq0Z2vv4XLX+LWDI5VmS9L4xDCaIAMAuqwuM02xceNGzZo1S9HR0bJYLHrttddabLN69WolJiYqMDBQycnJstvtbTrGiRMnNHLkSMXExOiBBx5QRESEm6pHZ1RZ26AnNzl09e8/0X1/3629h8sV4Ouj/xgfpw9XXK3MheM0ISmcoAcA6NK6zMxeVVWVRo4cqcWLF+uWW25p8fr69eu1fPlyrV69WikpKfrrX/+qmTNn6uDBg4qLi5MkJScnq7a2tsV7P/jgA0VHR6t3797au3evjh49qnnz5unWW29VVFTUOeupra1ttq+Kigo3fVJ0tMLyU1q3JUfPf5ankzVNTZDDgvy1YFK8UifGKzw4wOQKAQBwH4vRBc9VWiwWvfrqq5ozZ45rbMKECRozZoyeeOIJ19jgwYM1Z84crVq1qs3HuOeee3TNNdfotttuO+frK1eu1K9//esW4+Xl5QoJCWnz8dDxsgrKlWl36M29BWo43QQ5KSJIS21JmjeGJsgAgK6joqJCoaGhrcodXWZm70Lq6uq0c+dOPfjgg83Gp0+fri1btrRqH0ePHlWPHj0UEhKiiooKbdy4Uffcc895t//Zz36mFStWuJ5XVFQoNjb20j4AOoxhGPrXV8eUYc/W5kOlrvEJiWFKsyXpmisjaYIMAPBqXhH2SkpK5HQ6W5xyjYqKUlFRUav2cfjwYS1ZskSGYcgwDN17770aMWLEebcPCAhQQACn+zqr2ganXt9doMxN2frqaKUkyepj0Q3D+yvNlqgRMb3NLRAAAA/xirB3xrcvpDcMo9UX1ycnJ2vPnj0dUBU86XhVnZ77LFdPb83VsZNN11QG+Vt1x/g4LU5JUEyfniZXCACAZ3lF2IuIiJDVam0xi1dcXHzeBRbwLrmlVXpyk0Mv7jisU/VOSVK/kEAtTknQHePjFNqDJsgAgO7JK8Kev7+/kpOTtWHDBs2dO9c1vmHDBt18880deuz09HSlp6fL6XR26HFwbjtzy5Sx0aH3Dxa5miAP6R+iu6Yk6cYR/eVHE2QAQDfXZcJeZWWlDh065HrucDi0Z88ehYWFKS4uTitWrFBqaqrGjh2rSZMmac2aNcrLy9Pdd9/doXUtW7ZMy5Ytc62KQcdzNhr6IKtIa+zZ2p13wjU+bVBfpdmSNOkyeuMBAHBGlwl7O3bs0LRp01zPz6yEXbhwodatW6f58+ertLRUDz/8sAoLCzVs2DC98847io+PN6tkuFlVbYNe3JGvpzbnKK+sWpLkb/XR3NEDtNSWqIFRvUyuEACAzqdL9tnrjNrS7wZtU1xRo3VbcvTcZ3kqP1UvSerd00+pE+OVOilekb0CTa4QAADP6nZ99szENXsd58uik8qwZ+v1PUdU72z6N0lCeE8tmZyoW5Jj1NOfP74AAFwMM3tuwsyeexiGoU2HSpRhd2jjV8dc42Pj+yhtSpKuHRwlK02QAQDdHDN76HLqGhr15t4CZdiz9UXRSUmSj0W6flg/LbUlaUxcH5MrBACgayLswVTl1fV6flue1m1x6GhFUxPknv5W3T42Vt9LSVRcOE2QAQBoD8IeTJFfVq0nNzn0jx35qq5rut4xsleAFqUk6M7x8QrtSRNkAADcgbAHj9qdd1yZdofePVCoxtNXi17Zr5eW2pI0e2S0/H1pggwAgDsR9tqJ1bgX52w09OHnR5Vpz9b2nOOucdvACKXZkmQbGEETZAAAOgircd2E1bgtnapz6qVdh/XUJoccJVWSJD+rRbNHNjVBHtyf/04AAFwKVuPCVMdO1urZrTl69tNcHa9uaoIcEuirOyfGa9FVCYoKoQkyAACeQtiD2xwqPqlMu0Ov7D6iuoZGSVJsWA8tSUnUbWNjFRTAHzcAADyNv33RLoZhaGt2qTLtDv3zi2LX+KjY3rprSpJmDO1HE2QAAExE2MMlqXc26p39hcqwZ+vAkQpJksUiTR8SpTRbkpLj+7DoAgCAToCw107dbTVuRU291m/L19rNDhWU10iSAv18dFtyrL43OVGJEUEmVwgAAM7Galw38fbVuEdOnNLaTQ69sD1flbUNkqSI4AAtnBSv/5wYrz5B/iZXCABA98FqXLjN/sPlyrBn6+39hXKe7oI8MDJYS22JunnUAAX6WU2uEAAAXAhhDy00Nhr6+MtiZdiz9Wl2mWv8qsvClTYlSVcP7CsfFl0AANAlEPbgUlPv1Ku7jyjTnq1vjjU1Qfb1sWjWyGgtmZyoYQNCTa4QAAC0FWEPKquq07Nbc/XspzkqqayTJPUK8NV3J8RpUUqC+of2MLlCAABwqQh73Vj2sUo9ucmhl3YeVu3pJsgDevfQ4pQEzR8Xq16BfiZXCAAA2ouw105drfWKYRjannNcazZm66MvjurMWuwRMaFaakvSDcP6ydfqY26RAADAbWi94iadvfVKg7NR72UVKWNjtvYeLneNXzs4Umm2JI1PDKMJMgAAXQStV+BSWdug9dvz9dQmh46cOCVJCvD10bwxMVoyOVGXRwabXCEAAOhIhD0vVVReo7VbHHr+szydrGlqghwW5K8Fk+KVOjFe4cEBJlcIAAA8gbDnZQ4WVCjTnq039hao4XQT5KSIIC21JWneGJogAwDQ3RD2vIBhGPrXV8eUaXdo06ES1/iExDCl2ZJ0zZWRNEEGAKCbIux1YbUNTr2+p0CZ9mx9dbRSkmT1seiG4f2VZkvUiJje5hYIAABMR9jrgk5U1+m5z/K0bkuOjp2slSQF+Vt1x/g4LU5JUEyfniZXCAAAOgvCXheSW1qlJzc59OKOwzpV39TXr19IoBanJOiO8XEK7UETZAAA0Bxhr5080VS5wdmo+/6+W+9lFbmaIA/pH6K7piTpxhH95UcTZAAAcB40VXaTjm6q/L112/XPL4o1bVBfpdmSNOmycJogAwDQTbUldxD23KSjw95XR09Kkq6I6uX2fQMAgK6FO2h4IUIeAAC4FFzsBQAA4MUIewAAAF6MsAcAAODFCHsAAABejLAHAADgxQh7AAAAXoywBwAA4MUIewAAAF6MsNdO6enpGjJkiMaNG2d2KQAAAC1wuzQ36ejbpQEAAJzRltzBzB4AAIAXI+wBAAB4McIeAACAFyPsAQAAeDHCHgAAgBcj7AEAAHgxwh4AAIAX8zW7AG9xpl1hRUWFyZUAAABvdyZvtKZdMmHPTU6ePClJio2NNbkSAADQXZw8eVKhoaEX3IY7aLhJY2OjCgoKdM0112jHjh0deqxx48Zp+/btnWZ/l/r+tr6vLdtfbNuKigrFxsYqPz+/293xxN1/ftrLE/XwnWn/dt31O9Mdvy8dcZzu9p3xxPfFMAydPHlS0dHR8vG58FV5zOy5iY+Pj2JiYuTr69vhvwitVqtbj9He/V3q+9v6vrZs39ptQ0JCutVfXJL7//y0lyfq4Tvjvv12t+9Md/y+dMRxuut3pqO/Lxeb0TuDBRputmzZsi53jPbu71Lf39b3tWV7T/z/0FV1tv82fGc67n2t3b6z/ZnoTDrbfxtP1cN3xr37NRuncdEtteUG0gD4zgBt0dm+L8zsoVsKCAjQr371KwUEBJhdCtAl8J0BWq+zfV+Y2QMAAPBizOwBAAB4McIeAACAFyPsAQAAeDHCHgAAgBcj7AEAAHgxwh5wASdPntS4ceM0atQoDR8+XBkZGWaXBHRq+fn5mjp1qoYMGaIRI0boxRdfNLskoNObO3eu+vTpo1tvvbVD9k/rFeACnE6namtr1bNnT1VXV2vYsGHavn27wsPDzS4N6JQKCwt19OhRjRo1SsXFxRozZoy+/PJLBQUFmV0a0Gl9/PHHqqys1NNPP62XXnrJ7ftnZg+4AKvVqp49e0qSampq5HQ6xb+PgPPr37+/Ro0aJUmKjIxUWFiYysrKzC0K6OSmTZumXr16ddj+CXvo0jZu3KhZs2YpOjpaFotFr732WottVq9ercTERAUGBio5OVl2u71Nxzhx4oRGjhypmJgYPfDAA4qIiHBT9YDneeI7c8aOHTvU2Nio2NjYdlYNmMeT35mOQthDl1ZVVaWRI0fqL3/5yzlfX79+vZYvX66HHnpIu3fvls1m08yZM5WXl+faJjk5WcOGDWvxU1BQIEnq3bu39u7dK4fDoeeff15Hjx71yGcDOoInvjOSVFpaqgULFmjNmjUd/pmAjuSp70yHMgAvIcl49dVXm42NHz/euPvuu5uNXXnllcaDDz54Sce4++67jX/84x+XWiLQqXTUd6ampsaw2WzGM888444ygU6jI/+e+fjjj41bbrmlvSWeEzN78Fp1dXXauXOnpk+f3mx8+vTp2rJlS6v2cfToUVVUVEiSKioqtHHjRg0aNMjttQKdgTu+M4ZhaNGiRbrmmmuUmpraEWUCnYY7vjOe4Gt2AUBHKSkpkdPpVFRUVLPxqKgoFRUVtWofhw8f1pIlS2QYhgzD0L333qsRI0Z0RLmA6dzxndm8ebPWr1+vESNGuK5tevbZZzV8+HB3lwuYzh3fGUmaMWOGdu3apaqqKsXExOjVV1/VuHHj3FYnYQ9ez2KxNHtuGEaLsfNJTk7Wnj17OqAqoPNqz3dm8uTJamxs7IiygE6rPd8ZSXr//ffdXVIznMaF14qIiJDVam3xr6vi4uIW/woDwHcGaKuu8p0h7MFr+fv7Kzk5WRs2bGg2vmHDBl111VUmVQV0XnxngLbpKt8ZTuOiS6usrNShQ4dczx0Oh/bs2aOwsDDFxcVpxYoVSk1N1dixYzVp0iStWbNGeXl5uvvuu02sGjAP3xmgbbziO9Mha3wBD/n4448NSS1+Fi5c6NomPT3diI+PN/z9/Y0xY8YY//rXv8wrGDAZ3xmgbbzhO8O9cQEAALwY1+wBAAB4McIeAACAFyPsAQAAeDHCHgAAgBcj7AEAAHgxwh4AAIAXI+wBAAB4McIeAACAFyPsAQAAeDHCHgCYZNGiRbJYLFq0aJHZpQDwYr5mFwAAaO7xxx/XiRMnNGfOHI0aNcrscgB0cYQ9AOhkHn/8ceXm5iohIYGwB6DdOI0LAADgxQh7AAAAXoywBwAd6LnnnlNKSop69eql0NBQTZgwQWvWrJFhGC22XblypSwWi3JzcyVJixcvlsViafYDAG3FNXsA0AEMw9CSJUu0du1aSZLFYlHv3r21Y8cObdu2TR9//LECAgKavSc4OFhRUVE6duyYGhsbFRISoh49ephRPgAvwsweAHSAP//5z66gd++996q4uFhlZWUqKyvTypUrtX79er3++uvN3nP//ferqKhIsbGxkqQ//vGPKioqavYDAG3FzB4AuFlNTY1+/etfS5JSU1P15z//2fVaaGiofvWrX6mmpka//e1vzSoRQDfCzB4AuNkHH3ygsrIySdIvf/nLc27z4IMPKjAw0JNlAeimCHsA4GY7duyQJMXGxuryyy8/5zahoaFKTk72ZFkAuinCHgC4WXFxsSRpwIABF9wuJibGE+UA6OYIewDQQWiVAqAzIOwBgJtFRkZKkg4fPnzB7Y4cOeKJcgB0c4Q9AHCzsWPHSpLy8/P1zTffnHObiooK7dy585yv+fg0/Wo+V+NlAGgrwh4AuNl1112nPn36SJL++7//+5zb/O53v9OpU6fO+VpISIgk6cSJEx1SH4DuhbAHAG7Wo0cP/eIXv5AkPf3001q+fLlKS0slNc3o/fd//7f+53/+R7179z7n+4cNGyZJeumll3T8+HGP1AzAe1kMzhMAgNs1NjZq0aJFevbZZyU1nZoNDQ1VRUWFnE6n7rjjDgUEBOjpp5/WwoULtW7dOtd7N27cqKlTp8owDFmtVkVGRsrf31+SlJOTY8KnAdCVMbMHAB3Ax8dHzzzzjJ555hlNnDhRPXr0UENDg8aMGaP/+7//0/PPP3/e906ZMkVvv/22rr32WoWGhuro0aPKzc1Vbm6uBz8BAG/BzB4AAIAXY2YPAADAixH2AAAAvBhhDwAAwIsR9gAAALwYYQ8AAMCLEfYAAAC8GGEPAADAixH2AAAAvBhhDwAAwIsR9gAAALwYYQ8AAMCLEfYAAAC8GGEPAADAi/1/pRJyNFlq5xkAAAAASUVORK5CYII=",
      "text/plain": [
       "<Figure size 700x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plot everything\n",
    "fig, ax = plt.subplots(1, 1, figsize=(7, 6))\n",
    "\n",
    "ax.loglog(dts, dts) # Plotting a first ord function\n",
    "ax.loglog(dts, np.abs(y_forward_final - y_exact_final), '*', label=\"Explicit Euler\") # Plotting the error for Euler Forward \n",
    "ax.loglog(dts, np.abs(y_backward_final - y_exact_final), '*', label=\"Implicit Euler\") # Plotting the error for Euler Backward\n",
    "\n",
    "'''\n",
    "To plot the error we've used .loglog to get log scale axis.\n",
    "'''\n",
    "\n",
    "ax.set_xlabel(\"dt\", fontsize=18)\n",
    "ax.set_ylabel(\"Error\", fontsize=18)\n",
    "\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Documentation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scipy : https://docs.scipy.org/doc/scipy/\n",
    "\n",
    "Scipy.odeint : https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.odeint.html\n",
    "\n",
    "np.empty : https://numpy.org/doc/stable/reference/generated/numpy.empty.html\n",
    "\n",
    "np.arange : https://numpy.org/doc/stable/reference/generated/numpy.arange.html\n",
    "\n",
    "np.ones : https://numpy.org/doc/stable/reference/generated/numpy.ones.html\n",
    "\n",
    "np.shape : https://numpy.org/doc/stable/reference/generated/numpy.shape.html\n",
    "\n",
    "Enumerate : https://www.programiz.com/python-programming/methods/built-in/enumerate\n",
    "\n",
    ".loglog : https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.loglog.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Part 2 : Runge-Kutta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exo 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solve the equation of the harmonic oscillator using the explicit RK4 method and show that the solution is not blowing up (choose appropriate time step).\n",
    "\n",
    "System of ODE for the case of harmonic oscillator is given by:\n",
    "\n",
    "$\\displaystyle \\frac{dx}{dt} = v$,\n",
    "\n",
    "$\\displaystyle \\frac{dv}{dt} = -\\gamma^2 x$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***HINT:*** Given initial value problem,\n",
    "\n",
    "$\\displaystyle \\frac{du}{dt} = f(t, u),\\qquad u(0)=u_0$,\n",
    "\n",
    "**RK4 algorithm** is defined as follows:\n",
    "\n",
    "$\\displaystyle u^{n + 1} = u^n + \\frac{dt}{6} k_1 + \\frac{dt}{3} \\left( k_2 + k_3 \\right) + \\frac{dt}{6} k_4$,\n",
    "\n",
    "with\n",
    "\n",
    "$k_1 = f(t^n, u^n)$,\n",
    "\n",
    "$\\displaystyle k_2 = f(t^n + \\frac{dt}{2}, u^n + \\frac{dt}{2} k_1)$,\n",
    "\n",
    "$\\displaystyle k_3 = f(t^n + \\frac{dt}{2}, u^n + \\frac{dt}{2} k_2)$,\n",
    "\n",
    "$k_4 = f(t^n + dt, u^n + dt k_3)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Initialise our parameters\n",
    "k = 2.0\n",
    "m = 1.0 # Setting up our constant\n",
    "gamma = np.sqrt(k / m)\n",
    "\n",
    "xi, vi = 0.75, 0.0 # Initialise our Initial conditions for x and v\n",
    "ti, tf = 0.0, 40.0 # Initialise our Initial time conditions \n",
    "dt = 0.01 # Initialise the size of our time step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"\\nHere we've used python definition to create our RK4 algorithm. This function take three arguments (u, f, dt) where 'u' is our varible, 'f' our function\\nand 'dt' the size of our time step.\\nThen we return the RK4 approximation.\\nAll of those terms are describe in the course's notebooks (their form are also given above).\\nPay attention to define k1, k2, ... in the right order because inside of this 'kn' we use 'kn-1'  \\n\""
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Compute RK4 algorithm\n",
    "def rk4(u, f, dt):\n",
    "    # The RHS doesn't depend on time explicitly.\n",
    "    k1 = f(u)\n",
    "    k2 = f(u + dt / 2 * k1)\n",
    "    k3 = f(u + dt / 2 * k2)\n",
    "    k4 = f(u + dt * k3)\n",
    "    \n",
    "    return u + dt / 6 * (k1 + 2 * k2 + 2 * k3 + k4)\n",
    "\n",
    "'''\n",
    "Here we've used python definition to create our RK4 algorithm. This function take three arguments (u, f, dt) where 'u' is our varible, 'f' our function\n",
    "and 'dt' the size of our time step.\n",
    "Then we return the RK4 approximation.\n",
    "All of those terms are describe in the course's notebooks (their form are also given above).\n",
    "Pay attention to define k1, k2, ... in the right order because inside of this 'kn' we use 'kn-1'  \n",
    "'''"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Quick remainder : We can write our equation as follow (see chapter 2.2. RungeKutta):\n",
    "\n",
    "\\begin{equation}\n",
    "    \\frac{d}{dt}\n",
    "    \\begin{pmatrix}\n",
    "        x \\\\\n",
    "        v\n",
    "    \\end{pmatrix}\n",
    "    =\n",
    "    \\begin{pmatrix}\n",
    "        0 & 1 \\\\\n",
    "        -\\gamma^2 & 0 \n",
    "    \\end{pmatrix}\n",
    "    \\begin{pmatrix}\n",
    "        x \\\\\n",
    "        v\n",
    "    \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Where $L = \n",
    "\\begin{pmatrix}\n",
    "    0 & 1 \\\\\n",
    "    -\\gamma^2 & 0\n",
    "\\end{pmatrix}\n",
    "$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"\\nWe've used a for loop that compute the RK4 algorithm for each time step.\\n\""
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Implement the equation above\n",
    "nt = int((tf - ti) / dt) # Computing the number of time step\n",
    "t = np.arange(nt + 1) * dt # Computing a time array\n",
    "L = np.array([[0.0, 1.0], [-gamma ** 2, 0.0]]) # Computing the matrix L\n",
    "\n",
    "def RHS(u): # Computing the right hand side\n",
    "    return L @ u\n",
    "\n",
    "'''\n",
    "Here we've computed the right hand side of equation (1) above, So L * (x v). To do so I define L as a 2 by 2 numpy array then we use python definition\n",
    "to compute the right hand side (RHS). Our function RHS(u) take one argument, the variable 'u' (which, here, is the matrix (x v)) and return L @ u which is the product of two matrix.\n",
    "Note that we've used '@' and not '*'. Basically, '@' is the numpy version of '*', it is faster and better. \n",
    "remark : we can also use np.dot(x,y) that takes two arguments and multiply them together. \n",
    "'''\n",
    "\n",
    "u = np.empty((nt + 1, 2)) # Creating our containers to store the solutions\n",
    "u[0] = xi, vi # Adding the initial conditions\n",
    "\n",
    "'''\n",
    "We've created a containers to store our solutions. Note that now our containers is a matrix (nt + 1 x 2) that contains the nt + 1 solutions for 'x' and 'v'.\n",
    "Then we add our initial condition 'xi' and 'vi'\n",
    "'''\n",
    "\n",
    "for i in range(nt): # Using RK4 algortihm at each time step\n",
    "    u[i + 1] = rk4(u[i], RHS, dt)\n",
    "\n",
    "'''\n",
    "We've used a for loop that compute the RK4 algorithm for each time step.\n",
    "'''"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The exact solution of our equation is\n",
    "\\begin{equation}\n",
    "    x(t) = c_1 \\cos{\\gamma t} + c_2 \\sin{\\gamma t}\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "    v(t) = -c_1 \\gamma \\sin{\\gamma t} + c_2 \\gamma \\cos{\\gamma t}\n",
    "\\end{equation}\n",
    "\n",
    "with constants $c_1$ and $c_2$ defined by initial conditions $x_0$ and $v_0$ as follows,\n",
    "\\begin{equation}\n",
    "    c_1 = x_0, \\quad c_2 = v_0 / \\gamma\n",
    "\\end{equation}\n",
    "\n",
    "Let's implemented it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute the exact solution\n",
    "c1 = xi # Computing our two constante\n",
    "c2 = vi / gamma\n",
    "\n",
    "def xexact(t): #Computing exact solution for x\n",
    "    return c1 * np.cos(gamma * t) + c2 * np.sin(gamma * t)\n",
    "\n",
    "def vexact(t): #Computing exact solution for v\n",
    "    return -c1 * gamma * np.sin(gamma * t) + c2 * gamma * np.cos(gamma * t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x136b3a4bdf0>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#Plot everything\n",
    "fig, ax = plt.subplots(1, figsize=(6, 5))\n",
    "\n",
    "ax.plot(t, xexact(t), label=\"x(t)\")\n",
    "ax.plot(t, vexact(t), label=\"v(t)\")\n",
    "\n",
    "ax.plot(t, u[:, 0], \"*\", label=\"Approximated x(t)\")\n",
    "ax.plot(t, u[:, 1], \"*\", label=\"Approximated v(t)\")\n",
    "\n",
    "'''\n",
    "Here y[:, 0] means that we are only taking the first column (the 'x' values) and y[:, 1] the second column (the 'v' values)\n",
    "'''\n",
    "\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "\"\\nWe've used the function np.allclose to know if our RK4 approximation of the solution is close enough from the exact solution \\n\""
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(np.allclose(xexact(t), u[:, 0]))\n",
    "\n",
    "'''\n",
    "We've used the function np.allclose to know if our RK4 approximation of the solution is close enough from the exact solution \n",
    "'''"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.9.13 ('base')",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.13"
  },
  "orig_nbformat": 4,
  "vscode": {
   "interpreter": {
    "hash": "a3c7dce546ec049fa41b8595c88c700f11bfc546eadf7e2bff35ee410712721f"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}