{ "metadata": { "name": "", "signature": "sha256:fd50607367babf086e266dadfbc364e3a180a80d3eacacbc3edad6e0c29d44bb" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# The Runge-Kutta Method, Higher-Order ODEs and Multistep Methods\n", "\n", "By Thomas P Ogden (<t.p.ogden@durham.ac.uk>)\n", "\n", "_Here we introduce the classical Runge-Kutta method, go beyond first-order ODEs, and take a first look at multistep methods._" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%matplotlib inline" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "In Part 1 we looked at the Explicit Euler method, and checked that it had a global order of accuracy $\\mathcal{O}(h^1)$. Now we'll again take an ODE $y' = f(t,y(t))$ with an initial condition $y(t_0) = y_0$ and look at more advanced numerical methods for solving the problem." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Runge-Kutta Method\n", "\n", "For the Explicit Euler method we truncated the Taylor expansion after the linear term. It's clear that including more terms before truncation will give us a higher-order approximation. The classical Runge-Kutta method does just this, up to an order of accuracy $\\mathcal{O}(h^4)$ \u2014 3 orders higher than Explicit Euler! We're not going to derive the approximation here but you can look it up if you're interested. The finite difference step is given by\n", "\n", "$$\n", "y_{n+1} = y_n + \\tfrac{h}{6} \\left( k_1 + 2k_2 + 2k_3 + k_4 \\right),\n", "$$\n", "\n", "where \n", "\n", "$$\n", "\\begin{align}\n", "k_1 &= f(t_n, y_n), \\\\\n", "k_2 &= f(t_n + \\tfrac{h}{2}, y_n + \\tfrac{h}{2}k_1), \\\\\n", "k_3 &= f(t_n + \\tfrac{h}{2}, y_n + \\tfrac{h}{2}k_2), \\\\\n", "k_4 &= f(t_n + h, y_n + h k_3).\n", "\\end{align}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Implementing the Method in Python\n", "\n", "Just like we did with the Explicit Euler method, we'll define a function to implement the Runge-Kutta method for a first-order ODE system." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def ode_int_rk(func, y_0, t, args={}):\n", " \"\"\" Classical Runge-Kutta (RK4) approximation to a first-order ODE system\n", " with initial conditions.\n", "\n", " Args:\n", " func: (callable) The first-order ODE system to be approximated.\n", " y_0: (array) The initial condition.\n", " t: (array) A sequence of time points for which to solve for y.\n", " args: (dict) Extra arguments to pass to function.\n", "\n", " Out:\n", " y: (array) the approximated solution of the system at each time in t,\n", " with the initial value y_0 in the first row.\n", " \"\"\"\n", "\n", " # Initialise the approximation array\n", " y = np.zeros([len(t), len(y_0)])\n", " y[0] = y_0\n", "\n", " # Loop through the time steps, approximating this step from the prev step\n", " for i, t_i in enumerate(t[:-1]):\n", "\n", " h = t[i+1] - t_i # size of the step\n", "\n", " k_1 = func(t_i, y[i], args)\n", " k_2 = func(t_i+h/2., y[i]+h/2.*k_1, args)\n", " k_3 = func(t_i+h/2., y[i]+h/2.*k_2, args)\n", " k_4 = func(t_i+h, y[i]+h*k_3, args)\n", "\n", " y[i+1] = y[i] + h/6.*(k_1 + 2.*k_2 + 2.*k_3 + k_4) # RK4 step\n", "\n", " return y" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Checking Accuracy\n", "\n", "We'll define our test function `exp` just as we did in Part 1." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from exp import exp #\u00a0same function we defined in Part 1." ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "And just as we did for the Explicit Euler, we'll use this test function to check the order of accuracy of the method by looking over a wide range of stepsizes." ] }, { "cell_type": "code", "collapsed": false, "input": [ "y_0 = np.array([1.]) # Initial condition\n", "\n", "solve_args = {}\n", "solve_args['a'] = 1.\n", "\n", "t_max = 5.\n", "\n", "# Range of stepsizes\n", "max_N = 12\n", "N = 2**np.arange(2, max_N) #\u00a0N = 2, 4, 8, ..., 2^max_N\n", "\n", "order_check = 4 # for visual check of the order of accuracy\n", "\n", "y_end = np.zeros(len(N)) # array to fill with the final values\n", "stepsize = np.zeros(len(N)) # array to fill with the stepsizes\n", "\n", "for i, N_i in enumerate(N): # loop over different numbers of steps\n", "\n", " t = np.linspace(0., t_max, N_i+1)\n", " y_end[i] = ode_int_rk(exp, y_0, t, solve_args)[-1]\n", " \n", " stepsize[i] = t_max/N_i\n", " \n", "plt.loglog(stepsize, abs(y_end - np.exp(solve_args['a']*t_max)), \n", " 'b-o', label='Global error')\n", "plt.loglog(stepsize, stepsize**order_check,'k--', label=r'$h^4$')\n", "plt.xlabel(r'$h$')\n", "plt.legend(loc=2)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 5, "text": [ "<matplotlib.legend.Legend at 0x3851310>" ] }, { "metadata": {}, "output_type": "display_data", "png": 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vH5566ils27aNPbCI/ouXB4lkcPgw8N13dU3uU6lMFo3p7+/P2RXRVWw20/rt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\nLSGEIigoqCAqKiqjoTWJVqu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"text": [ "<matplotlib.figure.Figure at 0x24fe5d0>" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "So we've confirmed that the Runge-Kutta method is $\\mathcal{O}(h^4)$ \u2013 so that the global error gets smaller much quicker with increased number of steps than the Explicit Euler." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example 2: A Forced and Damped Pendulum\n", "\n", "The next problem we're going to look at is a second-order ODE, but crucially _any higher-order ODE can be rewritten as a set of coupled first-order equations_, such that we can apply the solvers we've already designed. We'll see how to do that with this example.\n", "\n", "### The Physical Problem\n", "\n", "Take an idealised pendulum: a weightless string of length $\\ell$, fixed at one end with a mass $m$ at the other. The pendulum is free to swing in a plane subject to gravity, friction proportional to its velocity $v$, and may be driven by an external periodic force $F_d \\cos {\\omega_d t}$.\n", "\n", "We start with Newton's second law applied to the horizontal displacement $x$ for the unforced and undamped pendulum,\n", "\n", "$$m\\ddot{x} = -mg \\sin{\\theta}$$\n", "\n", "where $g$ is the local acceleration due to gravity and $\\theta{(t)}$ is the angle of displacement of the string from vertical at time $t$. We now apply a frictional force proportional to the translational velocity and rearrange to obtain a second-order homogeneous differential equation,\n", "\n", "$$ m \\ddot{x} + k \\ell \\dot{x} + mg \\sin{\\theta} = 0 $$\n", "\n", "where $k$ is the coefficient of friction.\n", "\n", "We want to consider angular displacement, so substitute $\\dot{x} = \\ell \\dot{\\theta}$ and $\\ddot{x} = \\ell \\ddot{\\theta}$, and now apply the driving force \n", "\n", "$$ m \\ell \\ddot{\\theta} + k \\ell \\dot{\\theta} + mg \\sin{\\theta} = F_d \\cos {\\omega_d t} $$\n", "\n", "The equation is nonlinear (due to the sine function), so finding an analytic solution is going to be difficult. Numerical methods are the next line of attack, so we'll try our Runge-Kutta method on the problem. First we must rewrite the second-order equation as a set of coupled first-order equations. Let $y_0 = \\theta$, $y_1 = \\dot{\\theta}$ and $y_2 = \\ddot{\\theta}$. Then \n", "\n", "$$ \n", "\\begin{align*}\n", "y_0' &= y_1 = \\dot{\\theta} \\\\\n", "y_1' &= y_2 = \\ddot{\\theta} = -\\frac{k}{m}\\dot{\\theta} - \\frac{g}{\\ell} \\sin{\\theta} + \\frac{F_d}{m \\ell} cos{\\omega_d t}\n", "\\end{align*}\n", "$$\n", " \n", "is the system we want. We make a final tidying of the parameters by letting $\\alpha = g/\\ell$, $\\beta = k/m$ and $\\gamma = F/m\\ell$\n", "\n", "$$\n", "\\begin{align*}\n", "y_0' &= y_1 \\\\\n", "y_1' &= -\\alpha \\sin{y_0} -\\beta y_1 + \\gamma \\cos{\\omega t}\n", "\\end{align*}\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Setting up the Problem in Python\n", "\n", "We write this pair of euqations up as a Python function to be passed to our ODE integrator:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def pendulum(t, y, args):\n", " \"\"\" A damped and forced pendulum, described as set of two first-order ODEs.\n", "\n", " Args:\n", " t: Time\n", " y: Pendulum system vector [angle, angular velocity] \n", " args['alpha']: gravity_acc/length_of_pendulum\n", " args['beta']: friction_constant/mass_pendulum\n", " args['gamma']: driving_force/mass_pendulum/length_pendulum\n", " args['omega']: driving_freq\n", "\n", " Returns:\n", " dydt: ODE vector\n", " \"\"\"\n", "\n", " dydt = np.zeros(2)\n", "\n", " dydt[0] = y[1]\n", " dydt[1] = (-args['alpha']*np.sin(y[0]) - args['beta']*y[1] + \n", " args['gamma']*np.cos(args['omega']*t))\n", "\n", " return dydt" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we're ready to select some parameters. First we'll remove friction ($k = 0$) and apply no driving force ($F_d = 0$), under which conditions we expect to see simple harmonic motion of the pendulum back and forth." ] }, { "cell_type": "code", "collapsed": false, "input": [ "### Parameters\n", "\n", "gravity_acc = 10. # [m /s2]\n", "\n", "length_pendulum = 1. # [m]\n", "mass_pendulum = 1. # [kg]\n", "friction_constant = 0. # [kg /m /s]\n", "\n", "driving_force = 0. # [N]\n", "driving_freq = 0. # [2\u03c0 /s]\n", "\n", "N = 200\n", "\n", "t = np.linspace(0., 10., N+1) # [s] an array of time steps" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We're going to pass the arguments to the solver as a dict, so let's make that now." ] }, { "cell_type": "code", "collapsed": false, "input": [ "solve_args = {}\n", "solve_args['alpha'] = gravity_acc/length_pendulum\n", "solve_args['beta'] = friction_constant/mass_pendulum\n", "solve_args['gamma'] = driving_force/mass_pendulum/length_pendulum\n", "solve_args['omega'] = driving_freq" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we need some initial conditions. We'll pick $\\theta_0 = \\tfrac{\\pi}{8}$ so that we can check our numerical result with the **small angle approximation**, \n", "\n", "$$\\sin \\theta \\approx \\theta.$$\n", "\n", "With this approximation, the undamped and unforced solution can be found analytically: $y = y_0 \\cos (\\sqrt{\\alpha} t)$ (you can check this by substituting it into the original ODE)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "initial_ang = np.pi/8 # [rad]\n", "initial_ang_vel = 0. # [rad /s]\n", "\n", "initial_cond = np.array([initial_ang, initial_ang_vel])" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solving the Problem\n", "\n", "Now we can solve the ODE system using the Runge-Kutta method we made above and plot the angle $y_0$ over time." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from scipy.integrate import odeint\n", "\n", "# y = odeint(pendulum, initial_cond, t, args=(solve_args,))\n", "\n", "# Solve Pendulum ODE with RK4\n", "y = ode_int_rk(pendulum, initial_cond, t, solve_args)\n", "\n", "y_small_ang = initial_ang*np.cos(np.sqrt(solve_args['alpha'])*t)\n", "\n", "plt.plot(t, y[:,0], c='b', label='Angle')\n", "plt.plot(t, y_small_ang, 'r--', label='Small angle approx.')\n", "plt.xlabel('Time (s)')\n", "plt.ylabel('Angle (rad)')\n", "plt.legend(loc=2)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 11, "text": [ "<matplotlib.legend.Legend at 0x4101950>" ] }, { "metadata": {}, "output_type": "display_data", "png": 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4zXbtpKbMzrYuMGxqP7gMlngTt4Jz48aNMmRlQUUFFcY4Tpywf/h2u3ZQUiIt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"text": [ "<matplotlib.figure.Figure at 0x3a9a250>" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "So we see that the pendulum oscillates as expected and the frequency matches the known small angle result reasonably well for a few swings. (Subject to the error in the small angle approximation. Try making `initial_ang` bigger or smaller to see where the approximation works well.)\n", "\n", "Other perspectives we might be interested in are the the pendulum's trajectory in phase space, which we plot in a **phase diagram** and its spectral properties, via a **power spectrum**. We'll plot those now." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def plot_pendulum(t,y):\n", " \"\"\" Plot Angle, Phase Diagram, FFT. \"\"\"\n", " \n", " fig = plt.figure()\n", "\n", " # Plot Angle\n", " ax_1 = fig.add_subplot(211)\n", " ax_1.plot(t, y[:,0], c='b')\n", " ax_1.set_xlabel('Time (s)')\n", " ax_1.set_ylabel('Angle (rad)')\n", " \n", " # Plot Phase Diagram\n", " ax_2 = fig.add_subplot(223)\n", " ax_2.plot(y[:,0], y[:,1], c='g')\n", " ax_2.set_xlabel('Angle (rad)')\n", " ax_2.set_ylabel('Angular Velocity (rad /s)')\n", " \n", " # Fourier Transform\n", " f_fft = np.fft.fftfreq(len(t), t[1]-t[0])\n", " y_fft = np.fft.fft(y[:,0])/np.sqrt(2*len(t))\n", " \n", " # Plot Power Spectrum\n", " ax_3 = fig.add_subplot(224)\n", " ax_3.plot(f_fft[:N/2]*2*np.pi, abs(y_fft[:N/2]), c='r')\n", " ax_3.set_xlim([0, 30])\n", " ax_3.set_xlabel('Ang Freq ($2 \\pi$ Hz)')\n", " ax_3.set_ylabel('Power')" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 161 }, { "cell_type": "code", "collapsed": false, "input": [ "plot_pendulum(t,y)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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da7XUH4s1SNAju9IzJeB0nTp14OrqCgCoUaMG2rdvj1u3bllVb+XK1ImXFKUXE0PrbWIp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\n1QtTvp+it1zCvPnXL/+CdIeUP69aVYUmFhqj/ZkWjCl3es2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"text": [ "<matplotlib.figure.Figure at 0x1109048d0>" ] } ], "prompt_number": 162 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The phase diagram (green, bottom left) shows each possible physical state for the system. We see the undamped, unforced pendulum follows a regular orbit. The power spectrum (red, bottom right) is found by taking a discrete Fourier transform using NumPy's `fftpack` module. We see a single narrow peak at $\\omega \\approx \\sqrt{\\alpha}$, the pendulum's natural frequency." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Adding Friction\n", "\n", "To better model a physical pendulum we might add a friction constant $k$, and see what happens when we solve the system with this included." ] }, { "cell_type": "code", "collapsed": false, "input": [ "friction_constant = .5 # [kg /m /s]\n", "solve_args['beta'] = friction_constant/mass_pendulum\n", "\n", "# Solve Pendulum ODE with RK4\n", "y = ode_int_rk(pendulum, initial_cond, t, solve_args)\n", "\n", "plot_pendulum(t,y)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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GN/1pi7feYmLU338PTJzIgimEQC5nAT61ajGR5MqOkxNbu7x1iwWmHDrEUkJy\nctjDRYMGzBEuXAiMHMmdHafy8fogZtGiRWq3odTpeXp6wtPTEwkJCZBIJLC2ti7635w5c5R2cPz4\n8TL/d/HiRVGqJ2iCRMJGe1u3Anl5ukld+PNP9nPkSO33pW2WLweCg1kdrcLjqig//sjWrXbsYDf8\nN4V27dhryRIWLJSZydIQeCQih6MclUp9JiQk4Ny5czh58iS2bt2KrVu3CtK5u7s7oqOjERcXh/z8\nfAQFBWHQoEGCtK0NBg5k00inTummv5072fRVy5a66U+bNG3KSo7s2sVC8CvKzZts+m74cGDUqIq3\nZ6hIJIC5OXd4HI6qKHV6X3zxBfr374/Q0FBcuXKl6CUEJiYm2LRpE7y9veHm5gY/P7+iun0DBgwo\nmhYdNWoUunTpgvv378PGxgabtRkKWA69e7Mpo/37td/XgwfAlSssarOyMH8+iyacOhUoZcZbZdLT\ngWHDWGX7tWsNL6qVw+GIh9I8PUdHR1y7dg1VxZAi0QBt5Om9yogRLIw8MREwUmmcrBnLlgELFgBx\ncUCzZtrrR9fcvQu4uLAk/7/+YkEa6qBQMFWRo0fZiLtbN+3YyeFw9B+t5Om1b98ecXFxmtpU6fD2\nZqMUAYJMy4QI+OMPVh29Mjk8AGjThuWNhYSwaFh1IGJSbIcOAWvWcIfH4XDUR2kgS1paGtq1a4eO\nHTuibt26AJh3PXjwoNaN00c8Pdn6yb59zClpg8hI4O+/mbpIZWTqVKbHuWoVYGOjmvMjYtutWcMc\nX2VRXeFwOLpF6fRmaTkREokEPXr00JZNFULb05sA8O67bM1NW7JgM2cygevkZJZvVRmRy1lU6p49\nzIktX172VGdGBnNygYHArFnMWfJ1PA6Ho5XpzcK8iMKXsbExgoKCNDayMjB0KBATA1y7JnzbeXls\nanPIkMrr8ADm4IKCmDzWypVsqjI8vPg2BQXM0bm6sm2XLOEOj8PhVAyVQjEiIyPxySefoFmzZvjq\nq6+KIiyFIjw8HK6urmjfvj1+/vnnEv+Pj49Hr1690LZtW/Ts2RMBAQGC9q8uQ4eyKc7t24Vv+6+/\nWDmeiROFb1vfMDZmTmzLFuCff5hqS/PmzOH37cumPkePZtuFhbGUB+7wOBxORShzejMqKgqBgYEI\nCgpCgwYNMHz4cCxfvhz//POPoAbI5XI4ODjgxIkTsLa2RocOHRAYGFjMsaakpCAlJQVSqRRPnjyB\nk5MTTp2pQy2DAAAgAElEQVQ6Varz1cX0JsBuzIXFJ9WNQCwPT0/gxg3mBIRsV995+RLYtImN9m7e\nZKWB7OxYDp6np3YjZTkcjmGiyf2+zECWNm3aYODAgTh69Cia/qsb9dNPP1XMwlK4fPky7O3tYWtr\nCwDw8fHBgQMHijk0S0tLWFpaAgAsLCzQoUMHJCUlCT7iVAdfX1am5dQpNioRgocPWVTjV1+9WQ4P\nYIoi06ezF4fD4WiLMp3e3r17ERgYiLfeegvvvvsuhg8frpURlLqi0zExMbh9+zY8yiknrQ3B6dcZ\nOJApYezYIZzTW7eOjWjef1+Y9jgcDqcyIYTgtNLozaysLBw4cACBgYE4deoUxo0bB29vb/Tr10/l\nTsoTnS4oKEBISAjWr18PANi+fTsuXbpU6tpeVlYWevbsia+++gqDBw8u/YB0NL0JAJMmsQCLpCTm\nACtCTg7QpAnQpw+T6uJwOBxO+Qg6vVlIrVq1MGbMGIwZMwbPnj3D7t278d1336nl9IQQnS4oKMDQ\noUPh6+tbpsPTNR98wNahtm4FZsyoWFs7dwLPn1e8HQ6Hw+GUjdKRnraRyWRwcHDAyZMnYWVlhY4d\nO5YIZCEijB8/HhYWFkrXFXU50gOATp1YHtmdO5pHFioUrGyMiQkLYuERihwOh6McreTpaRtVRKfP\nnTuH7du3IzQ0FC4uLnBxcUFISIjIljNmzGDqIqGhmrexdy/TpFywgDs8DofD0Saij/SERtcjvdxc\nlk/WuTOgiTIbERNgfvmSjRbftKhNDofD0RSDHOkZOtWqMdmw4GBWCkhdDh5kU5oLFnCHx+FwONqG\nj/QEIDMTaNECkEqBcmJ2SpCXxypgA8Dt27wQKIfD4agDH+mJhJkZG6mdOKHe2t6KFUB0NPDzz9zh\ncTgcji7gIz2ByM0FWrVi+XpXrjCFkfKIiwMcHZnE1u7dOjGRw+FwKhV8pCci1aqxckC3bwOffFL+\ntjk5wLBhbA1PC8puHA6HwykD0Z2esgoLubm56NSpE6RSKTw8PLBy5UoRrFSNd94BPvoI+N//ylZV\nIWJKLpGRrITQv7KmHA6Hw9EBok5vqlJhAQBycnJQo0YN5OXlwc3NDfv374e9vX2pbYo1vVlIXh7Q\nsyerwLB6dXGFlefPgSlTWOHUZcuAzz4TzUwOh8MxeLQiQ6ZNVKmwAAA1atQAwLQ3ZTIZqlatqmtT\nVaZqVRbQMno0S2XYvh0YMABISWFVGR4/Bn74AZg3T2xLORwO581DVKenaoUFhUIBFxcX3L59G6tW\nrSq2T2noospCedSsyVRW1qxhVRi+/ppFeHbqxN7v2FGn5nA4HE6lQCdVFiqKUBUWACAuLg6enp7Y\nsWMHXFxcSt1G7OnN0nj2jBVF5cnnHA6HIxx6Ob0pRIWFQmxtbeHp6YnTp0+X6fT0kXr1xLaAw+Fw\nOIDI0Zvu7u6Ijo5GXFwc8vPzERQUhEGDBhXb5smTJ0hPTwcAPH36FEeOHEG7QhkTDofD4XDUQNQ1\nvVcrLMhkMkyZMqVYhYWNGzciLS0N48ePh1wuh6WlJT766CP06dNHTLM5HA6HY6BwRRYOh8PhGCRc\nkYXD4XA4nHLgTo/D4XA4bwzc6XE4HA7njYE7vTeciiZ6vmnw86Ue/HypBz9f2kd0p6dMcLoQuVwO\nFxcXeHl56dC6yg//kqkHP1/qwc+XevDzpX1ETVmQy+Xw8/MrJjjdt2/fEtqbALB69Wo4OjoiMzNT\nBEs5HA6HUxkQdaT3quC0qalpkeD06yQkJODw4cOYPHkyT0fgcDgcjuaQiOzatYsmT55c9Pe2bdto\nxowZJbYbNmwYRUZGUlhYGA0cOLDcNu3s7AgAf/EXf/EXf1Xyl52dndp+R+vTm+UJTkskEqX7//XX\nX2jYsCFcXFxUmu+OiYnRxEwOh8PhvAHoveD0+fPncfDgQRw+fBi5ubnIyMjAuHHjsHXrVq3ZzOFw\nOJzKiagyZDKZDA4ODjh58iSsrKzQsWPHUiunF3L69Gn8+OOPCA4O1rGlHA6Hw6kMiBrI8qrgtJub\nG/z8/IoJTpc2LarKlCiHw+FwOKVR6QSnORwOh8MpC9GT0zkcDofD0RWVxumpquzCYQFDvXr1Qtu2\nbdGzZ08EBASIbZJBwFWBVCc7Oxvjx4+Hi4sLHB0dcfHiRbFN0mvWr1+PLl26wM3NDXPmzBHbHL3E\nz88PjRo1KlFEXO17v9pJDnqITCYjOzs7io2Npfz8fHJ2dqY7d+6IbZbekpycTNeuXSMiorS0NGrU\nqBE/XyqwYsUKGj16NHl5eYltit4zbtw42rhxIxERFRQUUHp6usgW6S9Pnz4lW1tbysrKIrlcTv37\n96eQkBCxzdI7wsPDKTIykpycnIre0+TeXylGeqoqu3AYlpaWkEqlAAALCwt06NABSUlJIlul33BV\nINV58eIFzpw5Az8/PwAsYK127doiW6W/VK9eHUSEFy9e4OXLl8jJyUHdunXFNkvv6N69e4nzosm9\nv1I4vcTERNjY2BT93aRJEyQmJopokeEQExOD27dvw8PDQ2xT9Jq5c+di+fLlMDKqFF8ZrRIbG4sG\nDRpgwoQJcHJywpQpU/Dy5UuxzdJbqlevjnXr1sHW1haWlpbo2rUrOnbsKLZZBoEm9/5K8Q3maQya\nkZWVBR8fH6xcuRI1a9YU2xy95VVVID7KU45MJsOVK1cwdOhQXLlyBXl5edi1a5fYZuktaWlpmDZt\nGu7cuYO4uDhcuHABhw4dEtssg0CTe3+lcHrW1tZKlV04xSkoKMDQoUPh6+uLwYMHi22OXlOoCtS8\neXOMGjUKoaGhGDdunNhm6S1NmjRB/fr14eXlherVq2PUqFE4cuSI2GbpLZcvX4aHhwfs7e1Rv359\nDB8+HOHh4WKbZRBocu+vFE7P3d0d0dHRiIuLQ35+PoKCgjBo0CCxzdJbiAiTJk1C27ZteaSYCixd\nuhTx8fGIjY3Fzp070bt3by6DVw6Wlpawt7fHpUuXoFAocOjQIfTt21dss/SW7t27IyIiAs+ePUNe\nXh6OHDmCfv36iW2WQaDJvb9SOL3ylF04JTl37hy2b9+O0NBQuLi4wMXFBSEhIWKbZTDw6XTlbNmy\nBbNnz0arVq2QmJgIHx8fsU3SW8zNzfHll1/C29sb3bp1g7OzM3r16iW2WXrHqFGj0KVLF9y/fx82\nNjbYvHmzRvd+rsjC4XA4nDeGSjHS43A4HA5HFbjT43A4HM4bQ7n19B4+fIgNGzYgMjISUVFRkEgk\naNWqFVxdXTFp0iTY2dnpyk4Oh8PhcCpMmSO9wYMHY/bs2Wjbti2WLl2Ka9eu4erVq/j222/h6OiI\nOXPmCBbqXpam2qtwbU0OR7XvAddx5HDKoSx9ssTERKVaaKpsowqlaaq9CtfW5HBU+x5wHUcOp3zK\nHOlZWVkBYGrpcrkcAPD48WOcP3++xDYVpTRNtVfh2pocjmrfA67jyOGUT7lregBzSGfPnoVMJkOn\nTp3QunVrtG7dGqtWrdKFfQBK11e7dOlSqdva29vjwYMHujKNwxGc6dOn45dffinxvirfg1d1HKtW\nrYpZs2YV6Thu3LgR+fn5mDZtGgD+XeEYPnZ2doiJiVFrH6XRmwqFAjVq1MDWrVvh5+eHkJCQYqM9\nXaBOMvCDBw9ARIK/Fi5cyNvl7Wq9XQClOjxVvwfl6Thu3LgRvr6+Wv+uiHHe9LFffqzaf2ny0KZ0\npFe/fn2cPHkSW7ZsQVBQEADoXDGda2ty3iS8vLwQHBwMf39/JCcn4+7du0hMTMSsWbMQGhoKJycn\ndO/eHTY2NmjSpAmuXr2K6dOnIz09HdWrV4eLiwvs7e0BoEjH0cLCAtbW1jAzMwMA/PHHH2IeIocj\nGkpHeitWrMC2bdswefJktGjRAg8ePNC5RA7X1uS8qVy6dAmHDh3Cpk2b8MUXXyAzMxPBwcGIiorC\nli1b0L9/f0ycOBG7d+/GvXv34OvrizNnzpTQcbx27VoxeabFixeLeFQcjniU6fQK0xSkUikCAgIw\ndepUAGwOdc2aNYIaUZqmGgAMGDAAKSkpeqGt2bNnT94ub1fr7b6KRCLBoEGDYGZmhs6dOyM/Px87\nduzAe++9h7///hvt2rWDsbExHj16hLZt28LJyQlbtmyBjY1NCR3HmJgY2NraFrXt7u6udftLQxfn\nTV/65ceqn5Spvblz506EhITg+vXrkEql6N+/P/r166f3kWASiQRlHNIbCRHh2ctniM+IR0JGAhIy\nEpCdnw05ySFXyCEnORSkKPq98KeJkQka1GiAhjUbFntZ1LCAqbGp2IdVaZFIJBg4cCCCg4OxaNEi\n1KpVCx9//DEAwMzMDJmZmQBQ9L933nkHXl5eiI2NLbfdefPmoU2bNpg0aVKxvvh3hWPIaHINl7mm\n5+PjAx8fHxARrl27hpCQELz33nuQyWR4++238e677/LqvnpCgbwAt1Jv4WrSVTx68ajIwcW/YD9f\nylRbgzWSGMFYYgxjI2MUyAsgJ3mp29WrXg8NazaEZS1LOFo4wtnSGVJLKZwaOqGGaQ0hD+2NRpUv\ns4ODAwBgz549Rd/P6OhoODo6FtuuZcuWiIuLK2r30aNHgtvL4RgCSgNZJBIJXF1d4erqigULFuDF\nixc4fvw41q9fz52eSKRkpeBC/AVcTLiICwkXEJEUUeTYjCXGsDKzgk1tG7g2dsUgh0GwMbdBE/Mm\nsKltA2sza5hVNStyboU/JZAUiw4kIqTnpiM1O7X0V04qEjMSsf3WdqyNWAuAOc1W9VvBuZEznBsx\nRyi1lKKxWWNRzpOhUvg5SCTFP5PXozclEglMTU2xf/9+fPjhh/j8889RpUoVzJ07t4TTk0qlOHHi\nBABW2Xzs2LFaPgoORz+pdKWFKuOUTWJGIvbd24ez/5zFxYSLePSCPaWbGpnCtbErOjfpDI8mHuho\n3RFNazeFsZGxzmwjIsSlx+F6ynXceHwDNx7fwPWU64hLjyvapo1FGwxoOQCeLT3RrWk3Pj1aDtq8\nfjt37oyjR4/C3NxcuL5Onwa8vIDYWKB+fQGs5HBUR5NrmDs9PSUtOw277+zGzts7cebRGRAITWs3\nhUcTD3hYe6CzTWdILaWoZlJNbFNL5UXuC9x8fBNXkq4gJCYEpx+dRr48H+ZVzdHPrh887T3Rv2V/\nWNayFNtUvUKb1++mTZuQl5dXlJwuSF/r1gEffghERgIuLgJYyeGoDnd6MGynl56bjn1392Hn7Z04\n+fAk5CRHG4s2GOU0CiOdRqJV/VZim6gxWflZOPnwJA5FH8Lh6MNIzEwEALhbucO7tTf8XPy4A4Ty\n6zc8PBxz5syBTCbDlClTMHPmzGL/j4qKKlal/OHDh/jmm28wa9YstftSie+/Bz77DAgNBXi1b46O\nEdTpXb16tajB0pQgXF1dNbNSyxia0yMihMaG4ufLP+NIzBHky/PRom4L+LT1gY+TD5waOqmlSGMI\nEBFuPr6JQ9GH8Nf9v3Ah4QJMjEwwtM1QTHOfhreavVXpjllVyrt+5XI5HBwccOLECVhbW6NDhw4I\nDAwsM31HoVDA2toaly9fLiZfpkpfKrNgAbBsGbB3L+DtXbG2OBw1ETR68+OPP4ZEIkF+fj4uXLiA\npk2bQiKR4NGjR+jSpQvOnj1bYYPfZIgIJx6ewKLTi3Au/hwa1WyE6R2mY5TTKLhbuVfqm75EIoGz\npTOcLZ2xoPsC3H96H79G/IrN1zcj6HYQHBs4Ypr7NIxtPxa1q9UW21y94VXBaQBFgtNlOb0TJ07A\nzs6uVIcnGBkZ7Gd6uvb64HAEpEynFxYWBoB9sRYvXow+ffoAAEJDQ/H777/rxLjKCBHh2INjWHR6\nES4kXEAT8yb4pf8vmOQ6SW/X57RNq/qt8NM7P2FJ7yUI+jsIayPWYuaRmfjsxGcY3W405nWZZ9BT\nu0KhjvA6wHJtR48eXW6b/v7+Rb/37NlT/SRj7vQ4OiQsLKzIN2kMKcHW1pZyc3OL/s7NzaXmzZsr\n2000VDgkUVAoFHTo/iHqtL4TwR9k85MNrbuyjnILcpXv/AZyJfEK+e33o2pLqpHJYhOacWgGpWal\nim2W1inv+t29ezdNnjy56O9t27bRjBkzSt02Ly+PLCwsKDW17HMmyHdl0CAigOjrryveFoejJppc\nw0q1Nzt16oQxY8Zg79692LNnD3x9fdGpU6eKedo3jAfPHqBHQA8M+GMAUrJS8NvA3xAzKwYfuH+A\nqiZVxTZPL3G3csfGwRvxaM4jTHGdgnUR62D/sz2+P/s9cmW5YpsnCuoIrx85cgRubm5o0KCBdo0q\nHOk9f67dfjgcgVAavZmbm4vDhw/jyJEjkEgk6N+/Pzw9PVG1qn7erPUpkIWIsCFyA+YenQsTIxP8\n8PYPmCCdgCrGVcQ2zeC4m3YXn574FMH3g9G0dlMs67MMPk4+MJIofW4zKMq7fmUyGRwcHHDy5ElY\nWVmhY8eOZQay+Pj4oH///hg/frxGfamMmxtLVxg7Fti6tWJtcThqwlMWoD9O73HWY0wJnoLg+8Ho\n3bw3AgYHwKa2FgMK3hBCY0Mx79g8XEu5Bncrd6z3Wg+ppVRsswRD2fV7+vTpYikLhakIAwYMwMaN\nG2FpaYns7Gw0a9YMsbGxRaWENOlLJVq2BGJiWIL6wYMVa4vDUROtOL0HDx7g008/xZ07d4rq6Ekk\nEjx8+FBzS19DWe4RANja2sLc3BzGxsYwNTXF5cuXS21LH5zewaiDmHxwMjLyMvBd3+8wq9OsSjci\nERMFKbDj5g58euJTPHv5DD+98xOmuU+rFBGvurx+BemrYUMgLQ3o3h0IDxfGMA5HRQRNWShk8eLF\nGDFiBJYsWYJ9+/Zh/fr1goZAy+Vy+Pn5Fcs96tu3b4kpG4lEgrCwMNSrV0+wvoUmV5aLGYdnYOO1\njZBaSnHK+xTaNmwrtlkqIVfIkSfPg1zxb9WFf6svEBHqVKujV9JhRhIjjHUei3ft38X4/eMx/fB0\nnIo7hQ1eG3iKg67h0ZscA0PpSM/FxQXXrl2Ds7MzIiIiALBaXDdu3BDEgAsXLmDRokUICQkBAHz3\n3XcAgM8++6zYds2bN0dERATqK9H3E2uk97LgJYYEDcHxB8fxaddPsajXIr1bu3uR+wJXkq7g5uOb\nRWWGCl9JmUllVlUAWGWFRjUboWHNhmhUqxGszazRvlF7SC2lcGzgKNqxKkiBFedXYEHoAtiY2yBo\nWBA6WHcQxRYhMKiRXl4eUO3fNBsbG+Cff4QxjMNREa2M9KpXrw65XI4ePXpg6dKlaN68OWrVqqWx\nka+jau6RRCJB7969YWRkhA8//BBTpkwps80K5x6pSU5BDgbvHIyTD09i46CNmOgyUav9qUr002ic\nfnQaFxMu4mLCRdxJuwMCu0BqmNYoqr7Qu3lvWJtZo3a12kXlhYwkRkXC1c9ePsPjrMd4nP0Yqdmp\nuJ5yHcFRwUWVHUyNTOHYwBFSSyl6NOsBz5aeaFSrkU6O0UhihE+6foJuTbvBZ48Pum7qih/e/gGz\nO802iOlOQfKOxKJwlGdqykd6HINB6UjvypUraN26NV6+fIm1a9ciMTERM2fORPv27QUxYM+ePQgJ\nCcH69esBANu3b8elS5fw888/F9suOTkZjRs3xt27d+Hp6YmtW7eie/fuJQ9IxyO9nIIceAV64VTs\nKWwevBnjpWVHy+mC2OexCLodhKDbQbiech0AG6UVClV7NPGAa2NX1Kter0JOQa6QI+ZZDK6nXGev\nx9cRmRyJ1OxUACzlYGDLgRjQagBcG7vqZE3z2ctn8DvghwNRB/Bx54+x/O3lBuH4XsWgRnoPHgD2\n9kCLFsDDh0BBAWCi9DmawxEMwUd6crkcf/75J5YvXw4zM7NiIyihUDX3qHFjVpOtTZs28Pb2xuXL\nl0t1erokOz8bAwMHIvxROLYM2YKxzuLUKHtZ8BKbrm3CtpvbcCmRjZI9mnhg5Tsr4dnSEy3rtRT8\n5m9sZAwHCwc4WDhgpNNIACxF48bjGzh0/xD+iv4Li04vgv9pf9jVtcM092mY6DIR9aprb022XvV6\n2DdyH2YdmYUVF1bAWGKM7/p+Z3COz2AoHOk1a8ac3osXvLwQR/9Rlr3esWNHysjIUDvrXVUKCgqo\nRYsWFBsbS3l5eeTs7Ex37twptk12dnaRDampqeTg4EBHjx4ttT0VDkkQsvKy6K3Nb5HRIiPacXOH\nTvp8nez8bFpxfgU1Wt6I4A+S/iql789+T7HPY0Wx53VSs1Ip4FoAddvUjeAPqrakGk3YP4EuJ1zW\nar8KhYKm/TWN4A/6/MTnpFAotNqfkCi7fk+fPk0uLi7Url07WrNmTanbZGVl0bhx40gqlVKbNm3o\nwoULGvWllLAwpsYyYQL7GRNTsfY4HDXR5BpWusfcuXOpR48e9PPPP9Pu3btp9+7dtGfPHo0MLIuw\nsDCSSqXk5OREq1evLnrf09OTkpOT6eHDh+Ts7EzOzs7Uu3dv+vXXX8tsS1dOb9KBSWS0yIgCbwXq\npL9XycnPoeXnllPD5Q0J/qDeW3pTWGyYzu1QhxspN+iD4A+o5rc1Cf4gzx2edDv1ttb6kyvk9P7B\n9wn+oC9Pfmkwjq+861cmk5GdnR3FxsZSfn5+qQ+IRETjxo2jjRs3EhF7qExPT1e7L5U4cOA/CTKA\nKCKiYu1xOGqiyTWsdE1vwoQJAFBiimjz5s2CjzqFQBdrIiExIei/oz8+6/oZlvVdptW+XicyORK+\ne31x98ldvN3ibXzd42t0a9pNpzZUhIy8DPwa8SuWnlmKrPwsvO/2Pvx7+qNhzYaC96UgBaYGT8WG\naxuwuOdifNXjK8H7EJryrl9VIp1fvHgBFxcXlfJoK/xd2b6dKbFs2ABMngycOAH8K0zP4egCrURv\nBgQEaGpPpeRF7gtMCZ4CxwaOWNhzoc76lSlk+OHcD1gYthANazbEUd+j6GfXT2f9C4V5VXPM7zof\nfi5+WBS2COsi1mH7ze1Y0W8FJrtOFnT9zUhihN+8fkOePA9fh7GHg17NDbfQqSqRzrGxsWjQoAEm\nTJiAiIgIdO7cGWvWrEH16tVLbbNCkc6vrukBPIKTo3W0WmVh/vz5dP/+/TKHiFFRUTR//ny1h5ba\nppxDEgS//X5ktMiILiVc0mo/r5KYkUhdNnYh+ING7hpJT3OeaqUfhUJBadlpdDPlJh2NOUoH7x0s\neh2JPkKRSZGUnJlMMrlMsD7vpt2lPlv6EPxBo/eMpoxc4dePs/OzqeWaltR0ZVN6kftC8PaFpLzr\nV5UqC1euXCGJREIHDx6knJwcGjt2LG3ZskXtvlRi6VI2rRkVxX5u2FCx9jgcNdHkGi5zpNevXz/M\nnz8fycnJaNWqFWxtbUFEiIuLw/3799G4ceMi3b83hSPRR7Dp+iZ81vUzdLTuqJM+EzMS0WtLLyRn\nJWPHezswymmUYKOhuPQ4hMaG4kriFVxJuoK/U/9GnjxP6X7GEha56drYFW6N3dDFpgvcrdw1Skto\nbdEax8Yew7Izy/B12NeISIrAruG70L6RMCkxAMtJ3Oq9FV03dcWckDnYNHiTYG3rElUinZs0aYL6\n9evDy8sLADBq1Chs3boV48aNE96gjAyWo9fo35xMXmmBYwAoXdNLSkrCzZs3ERMTAwBo2bIl2rVr\nBysrK50YqC7aWtPLys9C619ao3a12oh8P1InJYESMhLQa0svPM56jBDfEHSx6VLhNuNfxCPw70Ds\nurMLEUlMYce8qjncrdzhYukCG3MbWJlZwbKWJWqY1ijaL0+eh5SsFCRnJiMxMxG3Um/hatJVJGcl\nAwAsa1nCq5UXhjkOQ98WfTVygKfjTmPUnlF4kfcCR8YcwVvN3qrw8b7Kl6Ff4tsz3+KAzwEMchgk\naNtCIUSVhc6dO2PVqlXo0KEDZs2aBRcXF0yaNEmtvlRi+nQgKAhITWXO7/PPgSVLNG+Pw1ETja5h\nIYea+oC2DmnD1Q0Ef9DpuNNaaf91El4kkN1qOzJfZk4X4ksPOVeHqCdRNHH/RDJZbELwB3X4vQP9\ncPYHupN6h+QKucbtJmUk0bYb22j4n8Op1tJaBH9QyzUtac3FNRpNVSZnJlPrX1pTzW9r0rl/zmls\nV2nkyfJI+quUGi5vSGnZaYK2LRTKrl9lkc5EbOmhU6dOZGdnR0OGDKGsrCyN+lKKry9Rixbs93r1\niKZPr1h7HI6aaHIN89JCKtJ9c3ekZafh7vS7Wk92VpACfbf2xZWkKzgx9gQ6NdG8aG9WfhbmH5+P\nXyN+RVWTqpjiOgVzPOagRd0WAlrMyJPlYc/dPfj58s+4mHARDWs2xLe9v8VE6cQiSTNVSMpMQo+A\nHkjNTsXxsccFnUq+kXID0t+k+KbXN/jyrS8Fa1coDEqRZdAgID4euHYNsLMDOndmEZ0cjo7Q5Brm\n9W5U4MGzBzj7z1mMdx6vE3WP36/+jlNxp/BTv58q5PDOPDqD9uva49eIXzGz40w8mvMIa/qv0YrD\nA4CqJlUxut1oXJh0ARcmXUDLei0xJXgKOqzvgLtpd1Vux8rMCqHjQlG/en0M3jkYT3OeCmajs6Uz\n3m7xNn67+htkCplg7b6RZGQA5ubs9zp1ePQmxyBQOtJ7+vSp0soG+oQ2npQXnlqIb8K/wT9z/0ET\n85ISaULyKP0RnNY5waOJB475HtPYye66vQuj945Gs9rNEDAkQKVcvlxZLk48PIGz/5zFtZRrePDs\nAVKzU5GVnwVjI2PUMK0BKzMr2NaxhXtjd3S26Yxetr1Q3bT0cHgiwp+3/8TMIzORU5CDjYM2FkmW\nqcKNlBtwX++OEW1HYMd7O1TeTxn77+2Hd5A39o3chyGthwjWrhAY1EjP1RWwtgaCg1l+Xl4ecPas\ncAZyOErQShHZli1bQiqVYuLEiejfv7/e6xgKfdNQkAItVrdAq/qtcGzsMcHaLQvvIG+ceHgCf0/7\nG9HCjyMAACAASURBVM3qNNOojR03d2Dc/nHo3KQzDo0+pLTG3J20O/jx/I/48/afyC7IhqmRKZwa\nOsHBwgGNajZCrSq1oCAFsvOzkZiZiOhn0bidehtykqOmaU0Mbj0YszvNLnMaMjEjESN2j8D5+PNY\n+c5KzPGYo/KxfHP6G3wd9jX2jNiD99q8p9Z5KAuZQoYWq1vAwcIBx8ceF6RNoTAop2dvD3h4sCnN\noUOBqCjg77+FM5DDUYJWAlnkcjkdPXqURo4cSS1atKDPPvuMoqKi1F481BUqHJJanIo9RfCHTvQ1\nkzKSyGiREX1+4nON27iRcoNMF5tSz4CelJmXWe626S/TaWrwVII/qPqS6jT5wGQKiQ6h3IJcpf1k\n52dTSHQITQ2eSrWX1Sb4g97Z9g5FPSn92siX5dN7Qe8R/EEbIzeqfDz5snxy/c2VrFdYU4G8QOX9\nlLHk9BKCP+he2j3B2hQCoa9frfZlYUE0bRr73c+PyNq64kZxOGqgyTWs1h4nT56kxo0bk7m5Ob31\n1lt07pyw0XVCIPRNY1HYIoI/KCuv9Ag4Ifnx3I8Ef9DdtLsa7Z9bkEvt17WnRssbUWpWarnbPnj2\ngOxW25HRIiOaGzK3RDTj05yntOPmDpp1eBYN/GMgvbX5Leq9pTeN3jOavjn9DYXHhRclqWfkZtCP\n534k82XmVOWbKrQpclOZ9r299W0yXWxKN1Nuqnxc++/uJ/iD9t/dr/I+ykjMSCT4g1acXyFYm0Kg\n7PpVRXC6WbNm1K5dO5JKpdShQweN+1JKlSpEn37Kfv/oI6IaNSrWHoejJppcw0plyJ48eYIdO3Zg\n69ataNSoEX755Rd4eXnhypUreP/99/F3JZ/OSMxIRIMaDVCzSk2t97X15lZ0tO6I1hatNdo/4HoA\nbj6+iQM+B9CgZoMyt0vJSkH3zd2RK8vF6Qmni633Pc56jC9Cv8C2m9uQL89HTdOasK9njzrV6iBX\nlovz8efxx60/AADWZtaY6zEXMzrOwMddPoZve1+M2TsGfgf98CTnCT7p+kmxfquaVMUfQ/9Am/+1\nwaSDk3Bh0gWVojoHtBoAazNr/Hb1NwxuPVijc/M6VmZWaFCjgVoBNmIjl8vh5+eHEydOwNraGh06\ndEDfvn1L5OlJJBKEhYWhXj3tlXFCXh6Qnw/U/nfqvE4dICeHvVelivb65XAqiFKn16VLF/j6+uLA\ngQPF1B8K36/sJGYmwtrcWuv9xKXH4ebjm1j97mqN29hwbQPaNWwHr1Ze5W438cBEPH/5HBcmXYCz\npXPR+xcTLsIr0AsZeRmY5DIJE6QTSlVaef7yOY49OIbfI3/HvOPzEHAjAAd8DqBF3RYI8Q2B715f\nzD8xH44NHDGg1YBi+1rUsMCKfiswfv94HHtwDP1b9ld6XCZGJvBz8cOS8CVIzU4VTJy6tUVr3Ht6\nT5C2dMHly5dhb28PW1tbAICPjw8OHDhQwukB0P664IsX7Gdh9Gbduv+936DsBy4OR2yUpiwsWbIE\nX3/9dTGHt2vXLgDF1d0rQnh4OFxdXdG+ffsSFdPV2UYbJGYmwtpM+07vnxf/AAAcGzhqtP/9p/cR\nkRSBSS6Tyg02upRwCSExIfim1zfFHF5SZhIG/jEQtavWxvWp17F2wFp0tO5YqrJK3ep1MdJpJE6O\nO4ngUcFIzEhE7y298TTnKUyMTLBlyBY4NnDEjCMzSk0L8HHyQf3q9bH5uuqVOro37Q4C4U7aHZX3\nUUZri9YGNdIrTXA6MTGxxHYSiQS9e/eGi4sL1q9fX26b/v7+RS+1hHwLxaZfTVkAeNoCR6uEhYUV\nu2Y1Qtn8p4uLS4n3pFKp2vOoZaFKjTBV64gRCb+m1+CHBvT+wfcFbbM0Am8FEvyhcY25A/cOEPyh\ntEDr3JC5VG1JtRJqKR+FfEQmi000Cuy4nHCZjBcZ09yQuUXv7bq9i+APOvHgRKn7TNg/gSx/tFS5\nj7jncQR/0O8Rv6ttX1n8dP4ngj90ps5SUFBArVu3Lneb8q5fVQSniYiSkpKIiOjOnTtka2tL4eHh\navellKtXmcj0/n/XWYOD2d+XtVsgmMN5FU2u4TJHekeOHMHMmTORkJCAWbNmYebMmZg5cyZ8fHwE\n1d18dcrG1NS0aMpG3W20QZ4sD2k5aTqZ3kzKTALA1po04VH6IwBQmuZw78k9tLZoDbOqZsXeD3sU\nhl62veBg4aB23x2sO8CzpScORx8ueq9Pc1ZX7cbjG6Xu06JOC6RkpSBXlqtSHza1bVDFuApinsWo\nbV95bQJM41QXmJiYwNHREdeuXSvxv0GBg7Dq4qpy91dFcBoAGjduDABo06YNvL29cfny5QpaXgqF\nI71X1/QAPtLj6D1lrulZWVnBzc0NBw4cgJubW9Eagbm5OXr1Eq4mmSo1wlTZ5lUqVCPsFZ6+ZEog\nDWpof43iRS5bI6lVpZZG+8tJDgCQoPw8SmMj4zLXewiarwM1q90M956UXB9TkKLc/cqy5XVkChkK\n5AXFRLArSqHSiy4+30KePXsGd3d3SKVSWFlZ4enTp3jy9AlirGLw3Kb8KgXu7u6Ijo5GXFwcrKys\nEBQUhMDAwGLb5OTkQC6Xw8zMDGlpaTh8+DDWrFkj/IG8vqbHnR7HQCjT6Tk7O8PZ2RljxoyBqamp\n1gxQJdld3YR4jed6X6NBjQYwMTLRyUiged3mAFhAi309e7X3L5QWi02PLTdy066uHU4+PIms/Kxi\nDrZ70+74NeJXxL+ILxoBqcPPnsXXWUNiWHVv18aupW4f/SwaVmZWZaq5vE5yZjIIJKgiTnJWMiSQ\noFGtRoK1qYyFC0sWHk7JTsGoiFEY7zUeZ7eVrWhiYmKCTZs2wdvbGzKZDFOmTCkKYhkwYAA2btyI\nly9fwtvbGwBQv359zJ07F/36aaHYcFlrery8EEfPKdPpDR8+HLt27YKra8mblkQiwc2bNwUxQJUp\nG1WndYTG1NgULeq2wP1n97Xel0N9Nq1478k9jZxeq/qtAACRyZHlCjQPbTMUqy+tRuCtQExxm1L0\n/hyPOfj96u8YtWcUjo09VqER1ZOcJ/j0xKdoY9EGPW17lvh/TkEOgu8Ho7+98sjNQmLTYwFA0Knm\npMwkNKjJHmx0Rc+ePZGfn4+LFy/irbfeQk5ODkJjQoEIoHmd5kr379GjR6nTo4cOHSr6/fr164La\nXCo8kIVjoJS5prd6NQudDw4OLvE6ePCgYAa8OmWTn5+PoKAgDBo0SO1ttEWr+q0Q9SRK6/0UrqWV\nNkWo0v71HdC2QVsEXA8od7tuTbvBo4kHFoQuQFp2WtH7tnVssWXIFpyPP49um7rhduptjeyIehJV\nVCFhq/fWUqM/111Zh/TcdEx1m6pyu3vv7kUV4yroZK25APfrJGclo3GtxoK1pwqbNm2Ch4cHJk6c\nCABISEjAJ5NZPmPhaN8gKJzeLFzTq1kTMDHhTo+j95Tp9AqDVYgIjRo1gq2tLWxtbdGokbBTQa9O\n2bi5ucHPz6/YlE1KSkq522gbh/oOiH4WrXRtqqLUq14PVmZWOPuPZoK9EokEk10n41LiJVxNulru\ndr8N/A0ZeRkYtHMQsvOzi/43vO1wBI8KxqMXjyD9TQq/A36ISIpQad3t4fOH+Pjox3D+1RnJmckI\n8Q2Bu5V7ie1upNzAF6FfwKuVl8pFYvNkedhxaweGtB6CutXrqrSPMogIf6dqrm+qKUuXLsWZM2dg\n/u8IqVWrVnj65CmMJEawMVd/Wlk0MjJYEnrVf4spSyS80gLHIFA6rzNs2DBcuHCh6G8jIyMMHz4c\nV65cEcwIVaZsytpG27Sq3wq5slzEv4jX+g3St50vVlxYgYSMBI3WrsY7j8eys8sw9a+puDj5YpnT\ndu0btUfg0ED8v73zjmvq/P74J2wQcSFDUEFQZAiJiqLWhbMojq+i4Cpu21rraK2/rx2g1dqvddRR\nV90DrXuvWhA3KsuBCCoqCLJUlqzk/P64JTISMkgCyPN+ve4ruTf3nnOem+fm5Fnn+Bz0Qd/dfXHI\n55C423BQm0GInRmLwJBAbI3Yiu2R29HcpDm6tegGl6YusDaxhqGuIYpFxVx+wfQY3Ei8gejX0dDm\naWOs61j82vdXWBhbVNAbkRyBAXsGoLFhY2wdslXusdq99/Yi830mJvInKnxPpHH95XUkvE1AQM8A\nlcmUBz09PRgZfeg6TktLQ15uHqxNrKGrrb6xc5VTOq1QCczpMWoBMhenZ2VlQa9UWCFdXV28qUOD\n1SVjbbEZ6u/inN5xOkQkwpa7lS8olkYjw0b4w+sP3E2+i1+v/lrpuf9x/A8O+hzEvdf30H5ze+y/\nv1/cojM1MsVar7VImpuE7UO3o7N1Z1x9cRXfB38P/+P+GH1oNMYeGYvZ52fjwIMDMDUyxfJ+y/Hs\n62fYOWxnBYdHRNgesR29dvaCgY4BQvxDKp1sU5qkrCTMuzAPXZt3Rb9W/ZS6L5LYHb0bRrpGKsvc\nIC+DBg3CN998g7y8POzatQu+vr5o6N5QrvG8GgVzeozaiqyFfL1796b169dTYWEhFRQU0Lp166hX\nr14KLwjUFHIUSSEy8zJJb7EezTozS6VypfHpnk/J8jdLKiwuVFrG6IOjiRfAo33R+2Se+yD1AQk2\nCggBoO7butOZx2dIKBJKPDe3MJfiM+Lp/uv7FJMWQ2m5aSQSiSqVf/X5Veq7qy8hANRjew96/va5\n3OUQiUTktdeLDH82pMfpj+W+Thb5RfnUcFlDGnt4rMpkyotQKKRNmzbRiBEjaMSIEbR582ayWG5B\n/sf8iagWZVkYPJiofOCKvn2JunSpmlEMhgIoU4dlXhEXF0edOnWi+vXrU/369cnDw4Pi4uKUMlAT\nqONHw++QHzX4pYFGMi2cij1V5ej/uYW51HN7T9IO1KYjD4/IPL9YWEyb7mwiy98sCQEgh7UOFBgS\nSFEpUTKdmiReZb2iP8L+oK5buxICQE1+bUJrb62V6kwlIRKJ6KszXxECQL/f/F1hGyqjJFrMubhz\nKpUrD3///Tfl5eWJ96NTosuUUVb9lSfLAhEXxYjP59PgwYOlnlOlZ6VHD6KePcseGzmSyNFReZkM\nhoIoU4dlJpEtIScnBwBgbKzc4mlNoY4knKHPQ9FzR09sHbIVkwSTVCq7PESEofuH4sKTCwifHq50\nLM7sgmz039Mft5Nu49e+v2Jul7kyx9AKhYU49PAQNtzZgGsvroFAMK9njo7NOqK9ZXvYNLSBVX0r\nmBqZQounBR6Ph+yCbKTkpCAxKxGRryNx99VdPEx7CAKhTZM2+NL9S0wWTFYoS4WIRJh5ZiY23NmA\nuR5z8Vv/31SWvPh90XvwN/FRKCxE3FdxGl2uAAATJkzAzZs30ahRI/To0QOP6j3C+aLzeLXwFUyN\nTCutv0KhEA4ODmWyLAQFBUmc1LVy5UrcvXsX2dnZUmdbV+lZEQiA5s2B0rKnTgVOnQKSk5WTyWAo\niFoyp799+xaBgYEIDQ0FwK0z+vHHH9GgQeXZuKsLdTg9IoLLBhfU062HsKlqCOlUjpScFLj84QLb\nRra4Pum60hMcsgqyMPH4RByJOYKhDkOxY9gONDRoKNe1r3Ne4+Tjk7j64iruvLojdmSVUeIgu1h3\nwbC2w+DU1ElhZ/Xm/RvMOD0Dfz34C991+w6/9PlFZQ4PAL67+B3+d/1/uDDuAvrZqW6MUFFevXqF\nfQf2YX7gfCAbEAm52cGV1d8bN24gMDAQ585xC/+XLVsGoGLg98TERPj7+2PhwoVYuXIlTp48KVFe\nlZ6VVq2Abt2A3bs/HPv2W2DdOuD9e+VkMhgKokwdlvk3d+LEibC0tMTGjRtBRNi5cycmTpyII0eO\nKG1obYPH42FGhxmYdW4W7r66iw7NOqhVn4WxBTYO3gifgz5YemUpfupVMYqHPJjom+CQzyH8fut3\nfHvxWziud8RSz6X4jP+ZxPVzpTE3NseU9lMwpf0UAEB+cT6Ss5ORlJ2EzPeZEJEIIhLBWM8YlsaW\nsDC2ELdUlOXCkwuYdHwSXue+xrI+yzC/23yVOrywpDD8duM3TBFMqTaHt3v3bly9ehXR0dHI188H\nuRPWzpAva4i84fjmzJmD5cuXI6tkAXklKB2yT9JElkaNgPx8bjMwkE8Og6EAISEhimUDkYSs/s+W\nLVtSUVGReL+oqIhatmypcD+qppCjSErx5v0bMvzZkKaemKoW+ZIYd2QcIQC0NXxrlWXdTrpNHn96\nEAJAgo0CuvT0klLjdeogLiOOPjv6GSEA5LjOke4k3VG5jvyifHJe70zWK63p7fu3KpcvL40bNyZ3\nd3fatm0bdVjWgVqvaV3me6is/sqTZeHkyZP0xRdfEBFRcHCwesb0RCIiXV2iBQvKHl+/nsu0kJKi\nnFwGQ0GUqcMylyw0adIEhw8fBnGTXnD06FGYmppWzdPWQhoaNISfix/23tsrzn2nbrZ4b8EAuwGY\ncmIKdkburJKsjs064vqk69j3n31Iz0tHn119wN/Ex5a7W8osUNckMWkxGHdkHBzWOeDAgwOY33U+\n7k5TfUtaRCLMOD0DD9IeYPPgzWhgUH1d8+np6di2bRuepj3F3T13kb8hH+PHj5frWnnC8V2/fh0n\nTpyAra0t/Pz88M8//2DChAkqLQMKCoCiIslLFgC2bIFRs5HlFSMjI2n48OFkbW1N1tbW9J///Iei\noqKU8MmaQY4iKc3TzKdkvNSY+u7qq7FWUl5hHvXd1Zd4ATzaHbVbZTK33N1CbhvcCAGgBr80oCnH\np9DxR8fVPkM18V0irb6xWjyz02iJEX1z/htKzk5Wiz6hSEgTj00kBIACggPUokMR3r17R6dPn6YO\nPh2I15xHrexb0fjx48WfV1Z/i4qKqFWrVvTs2TMqKCioNK8kEVFISIh6WnopKVyLbv36ssdPn+aO\n37ypnFwGQ0GUqcNyX1FYWEiFhcqvHdMU6nR6REQbb28kBID+CPtDrXpKk1uYS7139CatQC3acHuD\nyhyuSCSiq8+v0pjDY6j+0vqEAJD+Yn36dM+ntOzKMroQf6HKCVaTspLoyMMjtODiAuq2tRvxAniE\nAJDbBjdaErpErQlci4XF5H/MnxAA+in4J7XpUYR27drRxCkTqZ5vPfLe6F3hc1n1NyQkhPh8Prm4\nuNDvv39YyuHl5UXJyckVzvX2rqhDXl1SefyYc267y/0Ju3aNO35O80tBGHUTZeqw1NmbK1asEL8v\nPZmAiMDj8TB37lz1NT+rgDpmb5aGiDBw70BcfXEV0TOiYdfYTm26SpNbmIuRB0fiXPw5+Lr4YtPg\nTTDRN5F9oZwUCgtx5fkVnHp8CqfiTpVJ1mptYg1HU0dYGFvAwtgC5vXM0bReU2jxtCAiEdf1DcKb\n92+QlJ2EpOwkvMp+hSeZT5CUnQQA0NXShZuFG4a0GQIfZx+0NW2rMtslIRQJMeXkFOyI3IGAngFK\nTwZSB2MOj8FfEX8hxD8En9h/UuYzdddflei6exfo2BE4fhwoHfj94UPA2RkICgJ8fVVnKIMhBZXO\n3szOzlbpzLmPBR6Ph61DtsLlDxdMPD4RwZ8FQ1tLW+166+nVw+kxp7Hs6jL8EPwD7ry6g79G/gWB\npUAl8vW09dCnVR/0adUHqwauQkZeBqJeRyEiOQIRKRGIy4zD44zHSMlJQYGwQKocAx0DWNW3QrP6\nzdDbtjc6WnZEZ+vO4FvwYaCjmRl9r7JfYdLxSTj/5HyNc3i/Hv4VQTODYJBjgMHrBsPOzg47d+6E\ni4tLdZsmP+UTyJbAxvQYtQHVNTRrBpoq0o6IHVWOnKIsoQmhZLXCivQW69Gam2uoWFisMd0ikYje\nvn9L8RnxFJcRR/EZ8fQk8wk9zXxKGXkZ1T4j9NCDQ9T418Zk+LMhbby9sVptKc/Ldy9J21qb7KfY\n0/uC91RUVET79u2jjh07is/R5COptK6jR7luzPDwssdzc7njv/xSdeMYDDlQpg7LFYZsxowZxOfz\niYgoKiqKFi9erLh1UpAnrFLLli2pXbt2xOfzyd3dvVJ5mvrREIlENCRoCOkv1qfI5EiN6CxNWm4a\nDdo7iBAAct3gShfiL2jchprEu/x34mUPHTd3pEdpj6rbpDIIRULqt6sf8RryKDYtVny8uLi4zBKg\nWuH0du7knFt8fNnjIhGRnh7Rd99V3TgGQw6UqcMylyyMHz8e3t7e4v127dohKChIJa1MoVCISZMm\n4ciRI7h79y62bt2KmJiYCufxeDyEhIQgIiICYWHqj4giDyV56ZrWa4p+u/vhYdpDjeo3NTLFSb+T\nODDygDjkmNdeL9xPva9RO6obIsLxR8fhttENu6N34/vu3+P6pOvipLw1hfVh63Hx6UW4urpi3aJ1\nCA8Px927dzFv3jzw+fzqNk8xymdNL4Hl1GPUAmQ6vXfv3sHLy0u8LxQKUVAgfUxHEcLCwmBvbw8b\nGxvo6urC19cXx48fl3guaWhwXxEsjC3wz4R/oKOlA8+dnhrJsF4aHo+HUc6jEPNlDH7r9xtuJN6A\n20Y3TDs5DU/fPNWoLZqGiHDs0TF02NwBww4Mg66WLq5MvILFnotrXF66mLQYzP97Pga1HoTgY8Gw\nt7fH//3f/+G///0v7OzssGPHDrllhYaGon379nB1dcXatRUjueTn56Nz587g8/nw8PDAqlWrVFiS\nf5E2pgcwp8eo8cgMQ2ZpaYm7d7lM3AUFBdiwYQPs7e1VolzesEo8Hg+enp7Q0tLCF198galTp1Yq\nV+nQSkrQuklrXJpwCb129oLnLk9c9r8M+8aquT/yoq+jj3ld58Gf74/FoYux/vZ6/Bn+JwbaD8Tn\nHT+HV2svjUy20QQiEuH4o+NYFLoIkSmRsGtkhx1Dd2Cs61iNB4+Wh5zCHIw7Og5GWkYYrTsav/76\nKwYMGICZM2dCS0sLISEhWL16tVyySnpGSgec7tu3b5mA0wYGBggODoaRkREKCgrQoUMHeHt7q+yZ\nBcC19PT1P2RNLw1zeowajsxfic2bN2PixImIiYlB48aNwefzsXfvXrkV9OvXDykpKRWOL1myRO7Z\nodeuXYOlpSViYmLg5eWFtm3bonv37lLPL+30NIFjU0dcmnAJvXf2hudOzvHZNtJ8UtAmRk2weuBq\nfNP1G2y5uwVbwrdgyP4haNGgBaZ3mI7JgskwNzbXuF2qIDk7GQcfHsTWiK2Ifh0N+8b22DlsJ8a0\nG1MjnR0AvMt/h0/3foqolCgMeT4E+6/uh6enJxYtWgRvb2/MnTu3wp+ywMBAqfJK94wAEPeMlM+y\nUJKZPScnB8XFxdCX5JyqgqS4myUwp8eo4Uj9tXBycsKYMWPg5+eH0NBQ5ObmQigUwkRaZZfCxYsX\npX528+ZNmWGVAK61CQCOjo4YPnw4wsLCKnV61YGLmQv+Hv835/j+bfG1aNCiWmyxNrFGYO9AfN/j\nexyPPY4NdzZg4T8LERASgN62veFl74VBbQZpvEWqKJnvM3H44WHsf7AfIQkhEJEIbuZu2DVsF/za\n+dVYZwcAGXkZGLBnAKJfR+PAyANYOn4pbt68CV1dXUyePBlDhw5VeK2rvD0jIpEIAoEADx48wOrV\nq8tcUx6lekXevZPu9KysgMhIgIgb42MwVIhaA05HRETQd999R61atSJ3d3dauXIlJSUlKT/NRgLy\nhFXKzc2lrKwsIiJKTU0lBwcHOn/+vFSZlRRJI9xJukMNfmlArX5vpdJs31UlJi2Gvjn/DTmsdSAE\ngBAAarO2Dc0+O5suxF+g/KL86jaRioXFFJkcSRtub6BBeweRziIdQgCo9ZrW9OM/P9KD1AfVbaJc\npGSnkMsfLqS/WJ9OPz5NRCSe/VxC+f0SKqu/8gScLs2zZ8/I0dGRwssvLZBDV6UMGkTUvr3kz9at\n42Z2vnihnGwGQwGUqcNyJZG9efMm9u/fjyNHjsDOzg5+fn6YNm1a1bztv1y+fBmzZ89GcXExpk6d\nilmzZgEABg0ahK1bt+L9+/cYPnw4AC749ahRozB9+nSp8jQZ0UIatxJvwWufF4qERdjivQWjXUZX\nqz3leZL5BGfizuBM/BkEPwtGgbAA+tr6cDFzAd+CDzdzN7hZuMHN3E2twZnTctNwM/EmbibexI3E\nG7j96jZyCrlkxc1NmmO082j4tfODwEJQawIlJGYlos+uPkjMSsQJ3xPo06oPAEBbW1vc7QgA79+/\nh6GhIQCuzpakAaqs/t68eRMBAQHifHq//PILtLS08N1330m155tvvoG1tTVmz55d4TOln5UePQBt\nbSA4uOJnt24BHh7AkSPAv88tg6Eu1JJEtgQiQkhICObMmYOHDx+isLBQKSPVTU1wegDw4t0L+B7y\nxY3EG/i84+dYOWClxiKSKEJuYS6CE4IR/CwYUa+jEPU6Cul56eLPbRrawLmpMyyMLWBWz6zCZmpk\nCh54EJIQQpGwwmvG+wwkZiXi5buXeJnFbSX7aXlpAABtnjb4Fnx4WHugi3UXeFh7oFWjVrXG0ZXw\n7M0z9NnVB+l56Tg79iy6teimsIzK6m9xcTEcHBxw6dIlNGvWDJ06daqQOT09PR06Ojpo2LAhMjIy\n0KNHD6xZswZ9+vRRSFel8PlAy5ZcGLLyvH/PdX3Onw8sWaK4bAZDAdTi9MLCwrB//34cOnRInK5k\n5MiRNTa9UE1xegBQJCzCwn8WYvn15RBYCPCXz181fhyNiJCck4zIlEhEpUQh8nUkYtNjkZqbirS8\nNBSLipWW3UC/AZo3aI7mJs1hbWKN1o1bw8PaAx2adYCRrpFsATWYxxmP0WdXH+QW5uL8uPNwt3JX\nSo6s+iurZyQtLQ2fffYZhEIhLCwsMGrUKEyePFkpXVKxtQU++aRs1vTSCASAmRlw/rzishkMBVCp\n0/vvf/+LAwcOoFGjRvDz88Po0aMlTjKpadQkp1fCydiT+OzYZygWFWPrkK3wcfapbpOUQkQixmj7\nBQAAH/lJREFUvM1/yznA3DSk5qYiPS8dBII2TxvaWtoVXhsaNBQ7ufr69au7CCqHiLAzaie+Pvc1\nDHQMcHH8Rbiauyotr1YEnG7SBPDzA9atk/z51Klc92Z6OpvMwlArKg04ra+vj3PnzqF169ZVNqyu\n4+3gjcgZkRh9aDRGHRqFL59/iRX9V0BfR8VTydWMFk8LjQ0bo7FhY7VnSagNpOSkYNrJaTj5+CS6\nt+iOHcN2oFWjVtVtlnohqnzJAsBlYPjzTyAhgWsVMhg1CKkRWX766Sfm8FRIiwYtcNn/MuZ1mYf1\nt9fDbaMbTsaerHGtUoZ8HLh/AM5/OOPi04tY2X8lQvxDPn6HBwC5uUBxsWynBwC3b2vGJgZDAWSG\nIWOoDj1tPfzW/zecHXsWADBk/xD0290PUSlR1WwZQ17S89Ix+tBo+B72hX1je0RMj8CcLnOgxasj\nj9K9e9xr20pa+i4ugJ4ecOeOZmxiMBSg0ieViMosHmeohoH2A3Hv83tY++laRKREQLBJgCknpiA5\nO7m6TWNUwonYE3D5wwVHY45iiecSXJt0re5185YEfO/USfo5+vqAqytzeowaicy/p6WDTTNUh662\nLmZ2mon4r+Ixt8tc7IrahdZrW+Pn0J+RV5RX3eYxSvH87XNMODoBQ/cPhWV9S9yZdgf/7f7fGh0R\nRm2EhXFRV5o1q/w8d3cuw7pIpBm7GAw5qdTp8Xg8dOnSRWrmA0bVaWTYCL/1/w0xX8ZggP0A/BD8\nAxzWOWBP9B6IiP1gVCdxGXGYdHwS7NfaY//9/fi++/e4NeVWlWZnVhVZWRZevnyJ3r17w9nZGb16\n9VIog4Nc3L5deSuvhI4duQkv8fGq1c9gVBVZIVvatm1LPB6PTE1NycXFhVxcXKhdu3YKh37RFHIU\nqUZzOeEyddjUgRAAsl1tS/+7+j9Kz02vbrPqFPde3yO/Q36kFahFBj8b0Kwzs+jlu5ca0V1Z/S0u\nLiY7Ozt69uwZFRYWSgzbl5ycTBEREURElJaWRubm5hXOkUeXRDIyuBBjS5fKPjcqijt3717FdDAY\nCqDM773M/pmzZ8+q2+8yStGjZQ+ETQ3D4YeHse72Osz/ez5+DPkRvi6++NL9S3Rs1rG6Tfxoufvq\nLpZcWYKjj47CWM8Y33T5BnO7zK0xmSnkybJgYWEBCwsLAICpqSnc3d3x6tWrCpkYlKJkjE6elp6T\nE2BgwF0zZkzVdTMYKkKm0yt5wFJTU5Gfn69uexjg1sP5OPvAx9kH917fwx+3/8Du6N3YEbkDnaw6\n4YuOX2C0y+gaGdasNnL95XX8HPozzsafRUODhvixx4+Y1XkWmhg1qW7TyiBvloUS4uPj8eDBA3h4\neEg9R6EsCyWTWDrK8cdLR4eLzMKWLTBUiCqyLMgMQ3bp0iX4+/sjMzMThoaGyMzMhJOTE+7fv18l\nxeqiJkZkUQXv8t9hd/RurL+9Ho/SH6GJYRNMFkzGjI4zqiV3X23nxbsXOPTwEA48OICwpDCYGpli\nrsdcfOH+hVqDbMuisvp7+PBhnDt3Dlu2bAEA7NmzB7du3ZI4tpeTk4NevXrhhx9+wNChQxXWJZGh\nQ4HYWODRI/nOnzUL2LqVG9vT/jiSGDNqFsr83sucvblixQpcu3YN9vb2SE1NxZ49e9CjRw+ljSzP\npEmTYG5ujnbt2kk9R9bgfV2ggUEDzOw0Ew+/eIhLEy6hp01PrLixAnZr7ODxpwd+Dv0ZkSmRH6XD\nVxWJWYlYfXM1um7tiparW2LehXkoFhVj1YBVSPg6Af/X/f+q1eHJwsrKSq78k0VFRRgxYgTGjRsn\n1eEpDBGXQUGers0SOnYE8vLkd5IMhiaQNegnEAiIiKhLly7ivHZt27ZVePBQGqGhoRQeHk4uLi4S\nP5dn8L40chTpo+Hlu5f08+WfqfOWzsQL4BECQFYrrGj6yel0MvYk5RbmVreJ1U5SVhL9fvN36ra1\nmziPIH8jn5aGLqW4jLjqNq8CldVfefJPikQiGj9+PM2ZM6dKuirw4gU3MWXtWvmvefCAu2b7dvmv\nYTAUQJnfe5ljeo0bN0Z2dja8vLwwcuRIWFlZqWZQ/F+6d++OhIQEqZ/LM3hfV7E2scbCHguxsMdC\nvM55jbPxZ3Hq8SnsvbcXm+5ugoGOATxtPTG49WAMajOo2jK5axIRiRCTFoPghGAcfHgQV55fAYHQ\nzqwdFvdeDB8nHziYOlS3mUqho6ODbdu2Yfjw4eIsCyXPQUmWhfj4eOzZsweurq4QCAQAuLx7AwcO\nrJrykrE5RVp6Dg6AuTmwbRvw2Wcs+DSjRiDT6R07dgyGhob4/vvvERISgqSkJAwbNkwTtgFQfPAe\nUHBw/iPB3Ngc/nx/+PP9USgsROjzUJx6fAonH5/EmbgzwBnAuakzOll1At+CD4GFAG4WbjDRrySG\nYi0gOTsZt5JuISwpDLeSbuF20m1kF2YDAJyaOiGgVwB8nHzg2LRm/klSdGC+Z8+eiIiIqHD89OnT\nALjZmyJ1LAgPCwN0dQE3N/mv0dYGAgKAzz8Hjh1jSWUZNQK5k8gqS79+/ZCSklLh+NKlS+Ht7Q0A\nSEhIgLe3N+6VxPUrhSKD98DHO5FFWYgIsRmxOP34NC4+vYiIlAik5qaKP7drZCd2ggJLAfgWfFga\nW9bIBK65hbkITw7HraRb3JZ4Cy+zuDEuHS0duJm7obNVZ3Sy6oQuzbugTZM21Wyx4tTY1EKenkB2\ntuKzMYuLOUdZWAg8eMDF5GQwVIRKUwsZGxtL/eHj8XjIysqSS8HFixcVMqg88g7eMyTD4/HQ1rQt\n2pq2xbyu80BESMlJQURKBCJTIhGREoGI5Agcjjksvsasnhn4Fny0btwaFsYWMK9nDnNj8zKvhrqG\nKrVTKBLide5rJGUlITErEUnZSUjKSuJes7ljTzKfQEhCAIBtQ1t0a9FN7OQEFgKV28T4F5GIW283\nfrzi1+roAMuXA4MGAX/8AcyerXr7GAwFUHtLTx4qa+kVFxfDwcEBly5dQrNmzdCpUycEBQVJHdNj\nLT3lyCrI4jKllzjClAgkvE3A2/y3Es830Tcp4wQtjC1gqGMIAkFEIvFGVG6/1OfvCt5xDi4rCSk5\nKWKHVoKOlg6s6lvBysQKVvWt4NDEAZ2tOSdnVs9ME7dF49TIll5MDLfYfMcObmxOUYiA/v25WJzx\n8UDjxorLYDAkoNLM6SW8ePFC4vEWLVQzKcLPzw+XL19GRkYGzMzMsGjRIkycOFE8MG9hYYHLly9j\n9uzZ4sH7WbNmSZXHnJ5qKSguQGpuKlJyUvA69zVe57wu81r6eH5xPrR4WuKNx+OV3QevzGf19erD\n2sRa7NSs6luV2W9ar2ndSdnzLzXS6e3cCfj7c92TTk7KKYuK4harz54NrFypnAwGoxxqcXouLi7i\nbs43b97g1atXbHE6g6EmZNXf0NDQMn8Av/rqqwrnTJo0CadPn4aZmZnE3hN5dYmZORPYtQt486Zq\ni8wnTwZ27wYuXQK6d1deDoPxL2pxeuU5e/YsTpw4gQ0bNiikSFMwp8eozVRWf4VCIRwcHPD333/D\nysoK7u7uErv6r1y5AmNjY0yYMKHqTi83F7C3B/h8oKpxeFNTgZ49gYQE4NAhbpyPwagCaonIUp6B\nAwfi8uXLil7GYDCqSOk1q7q6uuI1q+Xp3r07GjVqpBqlK1cCKSnADz9UXZaZGRAaCjg7A8OGAUFB\nVZfJYCiIzHV6K1asEL8vKCjA1atX0Z11TTAYGkeZNauyqHRNa2oq8L//Af/5D9C1a5X0iGnaFPjn\nH2DIEGDsWM6hfv01oFW3xm4ZyqGKgNMynV52drZ4TM/AwAALFixAly5dqqSUwWAojjrWTpZ2ehVY\ntAh4/x5YulS1Sk1MuK5SX19g7lxg717g99+Bbt1Uq4fx0VH+j1lgYKDCMmQ6vUofCgaDoTE0umY1\nLg7YtAmYNo0LJ6ZqDA25KC1BQcD8+cAnn3BOMDAQaFP7ggowag8yJ7J4e3uXGSzk8XgwMTGBu7s7\npk+fDgODmpXTjU1kYdRmKqu/iqxZrWztqzy64OPDtcaePOHiZ6qT3FyuG/V//wPy84FevYCpU7lu\n1Rr2+8KoWahlIouzszOEQiFGjhyJESNGQCQSwcTEBNevX8fUqVOVNpbBYChG6YDTHTp0wKRJk8oE\nnC4J9+fn54euXbvi8ePHaN68ObZv366Yon/+4WZXfvut+h0eANSrx7Xwnj3julKfP+fG+5o1A8aN\nA/btAzIy1G8Ho04gs6UnEAhw7do1GBkZAQDy8vLQrVs3XL9+HW3atCnT3VITYC09Rm2m2henX7jA\nBYa2tuYiqBgba8SWMohEnOPdtYtrbaancxNdOnTgxv26dOG2UpN6GHUTlcbeLOHNmzeIjY0Vpyl5\n/Pgx3rx5A0NDQ5iamipnKYPBqHkcPgz4+XFRV86frx6HB3AOrm9fbhMKOed7+jQQHMyNM65ezZ3X\ntCnQrh3g6sq9tm3LjQc2acLSGDGkIrOld+bMGSxYsEA8c4yI8Msvv6BXr17YsmULZtewALKspceo\nzVRLS08k4nLeTZ8OeHhwDqZhQ43YoDBFRUB0NHDjBhAZyb2/f5+bZVpCo0ZA69aAjQ3QsiW3NW/O\ndZdaWnJdtjoy/+8zagFqjciSmJgIHo8HKysrpYzTFMzpMWozGnd68+YBf/0FvHzJBYU+coQbY6tN\nCIXceGBsLPD4MbfFxXFjgy9ecGmNSsPjAaam3GL5kq1JE+5YkyZcQOxGjT5sDRpwm5ERa0HWMNTm\n9BITE3Ht2jUUFBSIj02YMEFxCzUAc3qM2ozGnZ6uLjBgADB6NDBq1MeX704kAl6/BhITgeRk4NUr\nbktN/bClpXETZTIzuYwQ0tDR4Zxf/frcZmLCvRobf9jq1Su7GRl92AwNJW8GBtxWlbimdRS1OL2F\nCxfixIkT6Nq1K/RKPRDSkrhWN+r60QgJCVFLBnYml8ktjcadXmYm15rREOq6byrRKxRyQbUzM4G3\nb7n3b94A795x++/ecVt2Nre9ewfk5HzYsrO55RdFRZxOADI0lkVHh3N++voftpJ9Pb0Px/T0JG+6\nuoCeHkJevUIve3vxPnR1Odm6umW3kmM6OhXfS9q0tSu+/ruF3LjB3d9SxzTRKlbLRJajR48iIiIC\n+vr6ShtWGfJEhLexsYGJiQm0tbWhq6uLsLAwtdhSGbXtx5PJrZ1yZSFPlgV5zhGjQYcH1HCnp63N\ndXFWdYJeURGQl4eQRYvQ6/PPgby8D9v79x+2/HxuK9kvKOC2kuMl+wUFXBdtQQEn480bbr+o6MPx\nkveFhQjJz0cvkahqZVCQEEhw8DxeWScoadPSqvheS6vse0mflewrgUyn5+rqioSEBDioIyoDgIkT\nJ+Krr76qtLuUx+MhJCQEjVnySUYdRigUYtKkSWWyLPTt27fM4nR5zmGoGV3dD92g9vaa1x8QAPz0\nE1BczDnDoqKy70vvl38VCrn35fdL3pf+vGRfKATOnAH69Cl7TJ5NJCr7KhRyXcySPiv9WvJeCWQ6\nvbS0NLRr1w6dOnUSR27n8Xg4ceKEUgrL0717dyQkJMg8j43TMeo6pbMsABBnWSjt0OQ5h1EH4PE+\ndGNqgrdvge++04yu0ijThUoyCA4OrrCFhITIukwhnj17Ri4uLlI/t7W1JVdXV+Lz+bR58+ZKZdnZ\n2REAtrGtVm52dnZS6/bBgwdpypQp4v3du3fTzJkzFT6HPSts+1i2yp4Xachs6ZXvB79y5QqCgoLQ\ns2dPWZcCAPr16ycOj1SapUuXwtvbWy4Z165dg6WlJWJiYuDl5YW2bdtKTW8UHx8vl0wGo7YhT5YF\nRTIxsGeFUReRa4VmeHg4goKC8Ndff8HW1hYjRoyQW8HFixeVNq4ES0tLAICjoyOGDx+OsLAwltOP\nUeeQJ8uCRjMxMBi1EKlOLzY2FkFBQThw4ACaNm0KHx8fEFGVE/gpSl5eHoRCIerXr4+0tDScOXMG\na9as0agNDEZNoGPHjoiLi0NCQgKaNWuGAwcOIKhc9nF5zmEw6jTS+j15PB55e3vT8+fPxcdsbGwU\n7j+Vha+vL1laWpKenh5ZW1vTtm3biIjIy8uLkpOT6enTp+Tm5kZubm7k6elJGzduVLkNDEZtISQk\nhPh8Prm4uNDvv/8uPl7yvFR2DoPBIJK6OP3YsWMICgrCrVu3MHDgQPj4+GDy5MlyzbRkMBgMBqMm\nIjMiS05ODo4fP46goCAEBwdjwoQJGD58OPr3768pGxkMBoPBUAkyk8gaGxtj7NixOHXqFF6+fAmB\nQIBly5Zpwja5CQ0NRfv27eHq6lppeDShUAiBQCD3rFFZcvPz89G5c2fw+Xx4eHhg1apVKpH78uVL\n9O7dG87OzujVqxd27NihErkAFwHH3Nwc7dq1q7Isee+7umxURK667qm66kAJitZZVepWNTY2NnB1\ndYVAIECnTp3Upkda/VFnuaXpVGeZK6vT6iprZTrVWdbKnjOFy1q9vatVp7i4mOzs7OjZs2dUWFhI\nbm5u9PDhQ4nnrlixgsaMGUPe3t4qk5ubm0tERPn5+eTs7ExxcXFVlpucnEwRERFERJSWlkbm5uZS\ny6SovaGhoRQeHl7pukh5ZCly39Vho6Jy1XlP1VEHSlCkzsqDMt+bqrCxsaGMjAy165FUf9Rdbml1\nVp1lllSnY2Ji1FpWaTqJ1P/9SnrOlCmrzJZeTad0BApdXV1xBIryJCYm4syZM5gyZYpc0V3klVuS\nUT4nJwfFxcUyY5TKI9fCwgJ8Ph8AYGpqCnd3d7x69Uol9nbv3l0cWacqsuTVpy4bFZWrznuqjjoA\nKF5n5UGZ702VqKoclSGp/qi73JXVWXWVWVKdTkpKUmtZpeksQZ3fr6TnTJmy1nqnl5SUhObNm4v3\nra2ty3wJJcyZMwfLly+HlpZ8RZZXrkgkgpubG8zNzTFz5swy11RFbgnx8fF48OABPDw8VCq3qrKU\n0adKG6siV9X3VF11QNE6Kw/q+g7kgcfjwdPTEwKBAFu2bNGIzhKqq9yaKnPpOq2pspZ/jtRdVknP\nmTJlrRXpg6VFdVmyZIlcEShOnToFMzMzCASCMusMqyoXALS0tBAVFYWEhAR4eXmhW7dumD9/fpXl\nAtw/Gl9fX6xatQr16tVTib3yoOrIH1W5RtVyy99TVciVVAcEAkGV5Eqrs1VFXd+BPCgSWUnVVFe5\nNVHm8nVaE2WV9Bypu6ySnjNlylornF5lUV1u3rwpMwLF9evXceLECZw5cwb5+fnIysrChAkTqiy3\nNDY2NvDy8sLly5dVIreoqAgjRozAuHHjMHToUABVvw/yoq7IH+qKFiKvXEn3VJX2lq4DlTk9eeRK\nq7O7du2SaXdlVGfEluqMrFRd5VZ3mSXVaXWXVdpzpKnvt/Rz5uHhoXhZ1TbqqCGKioqoVatW9OzZ\nMyooKJA5kBkSEkKDBw9Widy0tDR68+YNERGlp6eTk5MT/f3331WWKxKJaPz48TRnzhyZdioitwRZ\nAb7lkaXofVe1jYrKVdc9VVcdKI28dVYelPneVEFubi5lZWUREVFqaio5ODjQ+fPn1aavfP3RRLnL\n61R3maXVaXWWVZpOdZdV2nNWXFyscFlrvdMjki9KRelz5Z0JJ0tudHQ0CQQCcnV1pf79+9Off/6p\nErlXrlwhHo9Hbm5uxOfzic/n09mzZ6ssl0h6BBxlZCkT+UOVNioiV133VF11oPy5qpq9WZludaLJ\nyErS6o86y12iU1dXV6xT3WWurE6rq6zSdKq7rJU9Z4qWVebidAaDwWAwPhZq/exNBoPBYDDkhTk9\nBoPBYNQZmNNjMBgMRp2BOT0Gg8Fg1BmY02MwGAxGnYE5PQaDwWDUGZjT+0g4duwYtLS0EBsbWyU5\nO3bswFdffaXQNfHx8fDz86uSXn9/fxw+fBgAMHToUCQnJ1dJHoPBYEiCOb2PhKCgIAwePBhBQUFV\nkqNMLLu1a9di8uTJFY4XFxcrpLdE97hx47Bx40aF7WAwGAxZMKf3EZCTk4Nbt25h3bp1OHDggPh4\nSEgI+vTpA19fXzg5OWHhwoXiz86cOQMbGxsIBAIsWLBAnKS0dKyCtLQ0DBs2DE5OTnBzc0NYWFgF\n3UVFRTh79iz69u0LAAgICMC0adPQrVs3+Pv74/nz5+jRowfat2+PkSNHIjw8XHztzJkzYWNjgwED\nBuDt27di3UOGDMG+fftUe5MYHyWq6uEojba2NgQCgXh78eKFymSXRigU4pNPPgERobCwEL/++itW\nrFiB5cuXKyTH2Ni4zH5lvTX5+fkai3daU2FO7yPg+PHjGDhwIFq0aIGmTZuWcSxXrlxBYGAgIiIi\ncOLECSQmJgIAFixYgKNHj+LatWu4d++exBbe119/jYkTJ+Lhw4fYu3cvAgMDK5zz5MkTWFhYlDkW\nGhqKY8eOYc+ePTAzM8PFixcRHh6Ob7/9FtOmTQMAhIeHIzw8HNHR0Vi1ahXOnTsntkFfXx+GhoYS\nM0owGKVRVQ9HaYyMjBARESHeWrRoUeZz4sI3VlnPiRMn0KtXL/B4PBw8eBBjx47FvHnzEBYWhvv3\n78stp/yzW1lvjYGBAVxdXREcHKy03bUd5vQ+AoKCguDj4wMA8PHxKfMD0KlTJzg4OEBfXx9du3bF\ntWvXkJiYCB6PB4FAACMjI4waNUriQ3zx4kUEBARAIBBg/PjxePjwIfLz88ucExcXBxsbG/E+j8fD\n4MGD0bRpU/H+jz/+iI4dO2LGjBl49OgRAODs2bMYOXIkTExM4OTkhM6dO5eRa2dnp9J/74yPD0k9\nHAkJCXBycsKXX34JJycnzJgxA0VFReJrFi9ejBYtWqB79+6YPHkyVqxYIZeuhIQEODo6Ytq0aXB1\ndcXLly+xZ88euLi4oHXr1vj888/F5y5duhQtWrTAJ598UqmOLVu2YMyYMQC45+jkyZMAgFatWiEu\nLk6pewJ86K3ZtGmTuLVqa2sLT09PAMCYMWM0ns+wJlErUgsxpJOZmYng4GDcv38fPB4PQqEQPB5P\n3EVSOpuznp4eCgoKKvwTlPavlYhw/PjxCv90S8Pj8SpcX5JiBAD279+P9PR0XL16Fbm5uTA3N5d4\nnSSbVJk8lfHxIamHo3Hjxnj06BHWrl2LtWvXwsvLCzdu3ECPHj2QmpqKffv24c6dOygsLESXLl3g\n7OxcQe779+/FqaFatWolnmAVGxuLZcuWYfPmzYiJicHmzZtx9+5d6OrqYty4cbh16xbs7OywY8cO\nhIWFobCwEF27dpWoAwCio6Ph4OAAgOt5EYlEAICoqCjMmjULSUlJ+P7773H//n1oa2ujSZMmGDJk\nCKZPny7VXoD7TRg6dCimT5+O6dOno7i4GJ6enpg3bx4AoG3btmV6g+oa7FellnPo0CFMmDABCQkJ\nePbsGV68eAFbW1tcuXJF6jVWVlYgIkRGRiIvLw+HDh2S2CUyYMAArF27FgUFBQCAyMjICue0bt0a\nCQkJUnUlJSWhZcuW0NfXx5YtW8QP9qeffoqjR48iKysLMTExuHnzZpnrnj59ijZt2shzCxh1FEk9\nHDweD1ZWVujTpw+0tLTQs2dP3LhxAwBw4cIF9O/fH2ZmZrC2tkbfvn0l/uEzNDQUd22WODwAaNKk\niTh/3KVLl/DkyRN4eHigQ4cOiIiIQHBwMC5cuIABAwbAwsICLVq0kKojKysL2tra0NbWBsB1OxoZ\nGSEkJASenp6wsrLC8+fPsX37dsyZMwdff/01Tp8+XcHhlbc3IiICixYtKqNz1qxZ6NOnDwYNGiQu\nR0ZGBoRCobK3vlbDnF4tZ//+/Rg+fHiZYyNGjBD/AEjr3//ll18wbNgwdOvWDS1atECDBg0AlJ1F\nuWbNGrx48QKOjo5wdnbG5s2bK8ixtbWtMPZWWudnn32Gq1evol27digsLBQPupd0u7i6umL27Nn4\n9NNPxdcUFhYiLy9P3CpkMMpT0sMxefJk2NraYvny5Th48CCICA0bNhSfp6enJ+6Sl7eHQxrlx677\n9+8vdjQxMTFYsGCB3LIk9ZBkZGTg+vXrmD9/PgCga9euiI2NRYMGDRRawlNa7o4dO/Dy5Uv89NNP\nEm2ok6gm2xGjtpGTk0NERHl5eTRq1ChavXq10rJmzZpFFy5cUJVpdODAAfrxxx9VJo/x8bFp0yaa\nMWNGmWM9e/aky5cvl0nk+ttvv1FAQAAREb1+/ZocHR0pNTWVEhMTqXnz5rRixYoKso2NjSscK58g\n9uHDh9SsWTNxwtKMjAx6/vw5paenU5s2bSglJYVevHhB1tbWEnUQETVr1oyKioqIiEvOumbNGhIK\nhVRUVCRORPzDDz9QZmYmTZs2jd69eydRTnl7t2/fTjNnzqQ7d+6Qi4uLOPlqCenp6eTg4CBRVl2A\ntfTqKFu2bIFAIECbNm2gp6cnsdtEXr766its27ZNZbbt3bsXM2bMUJk8xseHtB6OZcuWSZ3NaGZm\nBl9fX3To0AG+vr5wd3cX93BIOr+y446Ojli1ahWGDx8OZ2dn9O/fHykpKWjSpAn8/f3h7u4OPz8/\n9O/fX2qL0tXVVTxZa+PGjfjhhx9gbm4OMzMzcasyJSUFjRo1Qps2bZCUlCTTrtL769evR2ZmJnr3\n7g2BQCCeOR0TE4P27dtLlFUXYElkGQxGnSE3Nxf16tVDamoqunbtikOHDoHP56tNX2BgIIyNjcWT\nSEpz9OhR3LlzB0uWLFGbfkl88cUX8PHxQe/evTWqt6bAWnoMBqPOMG3aNPD5fHTo0AFTp05Vq8Mr\nQVrLcejQoQgJCVHJmj95yc/PR3R0dJ11eABr6TEYDAajDsFaegwGg8GoMzCnx2AwGIw6A3N6DAaD\nwagzMKfHYDAYjDoDc3oMBoPBqDMwp8dgMBiMOgNzegwGg8GoM/w/KhfJA7GbGjgAAAAASUVORK5C\nYII=\n", "text": [ "<matplotlib.figure.Figure at 0x110bdd7d0>" ] } ], "prompt_number": 163 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We see that over time the friction attenuates the pendulum's swing. The phase space trajectory no longer orbits consistently but spirals in, and the spectral peak has been widened out to include a spread of lower frequency components." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###\u00a0Adding a Driving Force" ] }, { "cell_type": "code", "collapsed": false, "input": [ "driving_force = 10. # [N]\n", "driving_freq = 10. # [2\u03c0 /s]\n", "\n", "solve_args['gamma'] = driving_force/mass_pendulum/length_pendulum\n", "solve_args['omega'] = driving_freq\n", "\n", "# Solve Pendulum ODE with RK4\n", "y = ode_int_rk(pendulum, initial_cond, t, solve_args)\n", "\n", "plot_pendulum(t,y)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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CqRnxxYvalcPXl3x4dOElRRZffUVvp1LHc1WmhH/4gTpbdV1opPfP//6nXPm5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\nb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"text": [ "<matplotlib.figure.Figure at 0x110d60510>" ] } ], "prompt_number": 164 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The driven system is of course more interesting! The phase space trajectory looks chaotic, but the pendulum's behaviour is clear in the power spectrum, where we see we now have an additional peak at $\\omega_F$.\n", "\n", "Anyway, there's plenty interesting to explore in the damped forced pendulum system, but as we're just interested in the integration methods here, we'll move on. The takeaway is: by converting our second-order ODE into a system of first-order ODEs, we are able to solve it using the methods we've already written. And crucially, this can be done for any higher-order ODE." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## The Two-Step Adams-Bashforth Method\n", "\n", "Remember that with these different methods, we're always looking to do the same thing: to choose a numerical approximation to the integral in\n", "\n", "$$ y(t_{n+1}) = y(t_n) + \\int_{t_n}^{t_{n+1}} \\! f(t,y(t)) \\, \\mathrm{d}t. $$\n", "\n", "So far, we've looked at **single-step** methods where we only use one known point to estimate the next one. But one way to improve the accuracy of our method is to use more known points. Such approximations are called **multistep methods**, and we'll look at an example now. \n", "\n", "Where the Expliit Euler method takes the slope $f$ to be a constant on the interval $[t_n, t_{n+1}]$, the idea behind _Adams-Bashforth_ methods is to approxmiate $f$ by a [Lagrange interpolating polynomial](lagrange):\n", "\n", "$$ P(t) = \\sum_{j=1}^{m}{P_j(t)} $$\n", "\n", "where\n", "\n", "$$ P_j(t) = y_j \\prod_{\\substack{k=1 \\\\ k \\ne j}}^{m}{ \\frac{t - t_k}{t_j - t_k} }. $$\n", "\n", "Here $P(t)$ is the polynomial of degree $\\le (m-1)$ that passes through the $m$ points $(t_1, y_1 = f(t_1))$, $(t_2, y_2 = f(t_2))$ $\\dots$ $(t_m, y_m = f(t_m))$. We'll take the linear $(m = 2)$ interpolant on the point $t_{n}$ and an earlier point $t_{n-1}$, so we have\n", "\n", "$$ P(t) = f(t_n, y_n)\\frac{t - t_{n-1}}{t_n - t_{n-1}} + f(t_{n-1}, y_{n-1})\\frac{t - t_{n}}{t_{n-1} - t_n}. $$\n", "\n", "Now if we put this approximating polynomial into the integral of, we find\n", "\n", "\\begin{align}\n", "\\int_{t_n}^{t_{n+1}} \\! f(t,y(t)) \\, \\mathrm{d}t \\approx \\int_{t_n}^{t_{n+1}} \\! P(t) \\, \\mathrm{d}t &= \\int_{t_n}^{t_{n+1}} \\! \\left[ f(t_n, y_n)\\frac{t - t_{n-1}}{t_n - t_{n-1}} + f(t_{n-1}, y_{n-1})\\frac{t - t_{n}}{t_{n-1} - t_n} \\right] \\mathrm{d}t \\\\\n", "&= \\frac{(t_n - t_{n+1})}{2(t_{n-1}-t_n)} \\left[ f(t_n,y_n)(t_n + t_{n+1} - 2t_{n-1}) - f(t_{n-1},y_{n-1})(t_n - t_{n+1}) \\right]\n", "\\end{align}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Step Sizes\n", "\n", "If we let $h_1 := t_n - t_{n-1}$ and $h_2 := t_{n+1} - t_n$ then\n", "\n", "$$ \\int_{t_n}^{t_{n+1}} \\! P(t) \\, \\mathrm{d}t = \\frac{h_2}{2 h_1} \\left[ (2 h_1 + h_2) f(t_n,y_n) - h_2 f(t_{n-1},y_{n-1}) \\right]. $$\n", "\n", "Putting this back into $\\color{grey}{[1]}$, we get\n", "\n", "$$ y(t_{n+1}) \\approx y(t_{n}) + \\frac{h_2}{2 h_1} \\left[ (2 h_1 + h_2) f(t_n,y_n) - h_2 f(t_{n-1},y_{n-1}) \\right] $$\n", "\n", "and our sequence of approximation points $y_n$ is calculated as\n", "\n", "$$ y_{n+1} = y_n + \\frac{h_2}{2 h_1} \\left[ (2 h_1 + h_2) f(t_n,y_n) - h_2 f(t_{n-1},y_{n-1}) \\right] $$\n", "\n", "for $n = 1, 2, \\dots N$.\n", "\n", "**If the steps are of equal size**, i.e. $h := h_1 = h_2$ we find\n", "\n", "$$ y_{n+1} = y_n + \\frac{3}{2} h f(t_n,y_n) - \\frac{1}{2} h f(t_{n-1}, t_{n-1}) $$\n", "\n", "which is the [standard two-step Adams-Bashforth method][multistep].\n", "\n", "\n", "\n", "[lagrange]: http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html\n", "[multistep]: http://en.wikipedia.org/wiki/Linear_multistep_method#Families_of_multistep_methods" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Accuracy\n", "\n", "Replacing $f(t,y(t))$ with the interpolant $P(t)$ [incurs a global error][wiki_order_cite] of order $\\mathcal{O}(h^m)$, so in the case of the two-step method we have $\\mathcal{O}(h^2)$. \n", "\n", "Note that if you follow the same derivation with $m = 1$ you get the Euler method \u2014 so the Euler method is also in fact the one-step Adams-Bashforth method. \n", "\n", "[wiki_order_cite]: http://en.wikipedia.org/wiki/Linear_multistep_method#CITEREFIserles1996" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Next\n", "\n", "In [Part 3][part3] we'll look at an example of a **stiff** problem, which resists solution by the Explicit Euler method, and so discover another measure of a numerical method to go with its accuracy: the numerical **stability**. We'll then introduce **implicit methods**, which allow us to solve stiff problems.\n", "\n", "[part3]: ./8_Stiff-Problems-Implicit-Methods-and-Computational-Cost.ipynb" ] } ], "metadata": {} } ] }