08_quadrature.ipynb

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    "<table>\n",
    " <tr align=left><td><img align=left src=\"./images/CC-BY.png\">\n",
    " <td>Text provided under a Creative Commons Attribution license, CC-BY. All code is made available under the FSF-approved MIT license. (c) Kyle T. Mandli</td>\n",
    "</table>"
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   "source": [
    "from __future__ import print_function\n",
    "\n",
    "%matplotlib inline\n",
    "import numpy\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings"
   ]
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    "# Numerical Quadrature\n",
    "\n",
    "**Goal:** Accurately Evaluate definite integrals\n",
    "\n",
    "$$ \n",
    "\\int^b_a f(x) dx\n",
    "$$\n",
    "\n",
    "using a finite number of samples"
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    "### Why?\n",
    "\n",
    "1. Many integrals do not have closed form solutions\n",
    "\n",
    "$$ \n",
    "   \\int^b_a \\sqrt{1 + \\cos^2 x} dx\n",
    "$$"
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    "Solution to ordinary differential equations\n",
    "   \n",
    "   $$\\frac{\\text{d}^2 u}{\\text{d}t^2} = f\\left(u, \\frac{\\text{d} u}{\\text{d}t}, t \\right)$$\n",
    "   \n",
    "   Defining $v = \\frac{\\text{d} u}{\\text{d}t}$ then leads to\n",
    "\n",
    "   $$\\begin{bmatrix}\n",
    "   \\frac{\\text{d} v}{\\text{d}t} \\\\ \\frac{\\text{d} u}{\\text{d}t} \\end{bmatrix} = \\begin{bmatrix} f(u, v, t) \\\\ v \\end{bmatrix}$$\n",
    "   \n",
    "   which can be solved by integration\n",
    "   \n",
    "   $$\\begin{bmatrix}\n",
    "   v \\\\ u \\end{bmatrix} = \\begin{bmatrix} v(t_0) + \\int^t_{t_0} f(u, v, \\hat{t}) d\\hat{t} \\\\ u(t_0) + \\int^t_{t_0} v d\\hat{t} \\end{bmatrix}$$  "
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   "source": [
    "Solving partial differential equations\n",
    "$$\n",
    "    u_t = \\nabla^2 u\n",
    "$$"
   ]
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    "### Basics of Quadrature\n",
    "\n",
    "We want to approximate an integral $I$ with some approximation $I_N$ such that\n",
    "$$\n",
    "    I = \\int^b_a f(x) dx \\approx I_N = \\sum^{N}_{i=1} w_i f(x_i)\n",
    "$$\n",
    "where the $x_i$ are the *quadrature points* or *nodes* and the $w_i$ are the *weights*.  Usually a particular quadrature rule specifies the points $x_i$ resulting in a particular set of weights $w_i$.\n",
    "\n",
    "Convergence requires that\n",
    "$$\n",
    "    \\lim_{N \\rightarrow \\infty} I_N = I.\n",
    "$$"
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    "## Riemann Sums\n",
    "\n",
    "Given $f(x)$ and a partition of the interval $[a,b]$ with $\\{x_i\\}^N_{i=0}$ and $a = x_0 < x_1 < \\ldots < x_N = b$ and $x^*_i \\in [x_i, x_{i+1}]$ we define the Riemann integral as\n",
    "\n",
    "$$\\int^b_a f(x) dx = \\lim_{N\\rightarrow \\infty} \\sum^{N-1}_{i=0} f(x_i^*) (x_{i+1} - x_i)$$\n",
    "\n",
    "This is a general definition and leads to a number of quadrature approaches that depend on how we pick $x_i^* \\in [x_i, x_{i+1}]$."
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    "### Midpoint Rule\n",
    "\n",
    "Choose $x_i^*$ such that\n",
    "\n",
    "$$x_i^* = \\frac{x_{i+1} + x_i}{2}$$"
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    "so that\n",
    "\n",
    "$$I[f] = \\int^b_a f(x) dx \\approx \\sum^{N-1}_{i=0} f\\left(\\frac{x_{i+1} + x_i}{2} \\right ) (x_{i+1} - x_i) = I_N[f]$$\n",
    "\n",
    "over $\\Delta x_i = x_{i+1} - x_i$ "
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    "#### Example:  Integrate using midpoint rule\n",
    "\n",
    "Calculate and illustrate the midpoint rule.  Note that we are computing the cummulative integral here:\n",
    "\n",
    "$$\n",
    "    \\int^x_0 \\sin(\\hat{x}) d\\hat{x} = \\left . -\\cos \\hat{x} \\right|^x_0 = 1 - \\cos x\n",
    "$$"
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    "# Note that this calculates the cummulative integral from 0.0\n",
    "\n",
    "f = lambda x: numpy.sin(x)\n",
    "I = lambda x: 1.0 - numpy.cos(x)\n",
    "x = numpy.linspace(0.0, 2.0 * numpy.pi, 100)\n",
    "\n",
    "num_partitions = 40\n",
    "x_hat = numpy.linspace(0.0, 2.0 * numpy.pi, num_partitions + 1)\n",
    "x_star = 0.5 * (x_hat[1:] + x_hat[:-1])\n",
    "delta_x = x_hat[1] - x_hat[0]"
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    "fig = plt.figure(figsize=(8,6))\n",
    "fig.subplots_adjust(hspace=.5)\n",
    "axes = fig.add_subplot(2, 1, 1)\n",
    "\n",
    "axes.plot(x, numpy.zeros(x.shape), 'k--')\n",
    "axes.plot(x, f(x), 'b')\n",
    "\n",
    "for i in range(num_partitions):\n",
    "    axes.plot([x_hat[i], x_hat[i]], [0.0, f(x_star[i])], 'k--')\n",
    "    axes.plot([x_hat[i + 1], x_hat[i + 1]], [0.0, f(x_star[i])], 'k--')\n",
    "    axes.plot([x_hat[i], x_hat[i + 1]], [f(x_star[i]), f(x_star[i])], 'k--')\n",
    "    \n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"$f(x)$\")\n",
    "axes.set_title(\"Partition and $f(x)$\")\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((-1.1, 1.1))\n",
    "\n",
    "I_hat = numpy.zeros(x_star.shape)\n",
    "I_hat[0] = f(x_star[0]) * delta_x\n",
    "for i in range(1, num_partitions):\n",
    "    I_hat[i] = I_hat[i - 1] + f(x_star[i]) * delta_x\n",
    "    \n",
    "axes = fig.add_subplot(2, 1, 2)\n",
    "\n",
    "axes.plot(x, I(x), 'r')\n",
    "# Offset due to indexing above\n",
    "axes.plot(x_star + delta_x / 2.0, I_hat, 'ko')\n",
    "\n",
    "err = numpy.abs(I(x_star) - I_hat)\n",
    "\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"$f(x)$\")\n",
    "axes.set_title(\"Integral and Approximated Integral, Max Err = {}\".format(err.max()))\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((-0.1, 2.5))\n",
    "\n",
    "plt.show()"
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    "### Newton-Cotes Quadrature\n",
    "\n",
    "Using $N+1$ equally spaced points, evaluate $f(x)$ at these points and exactly integrate the interpolating polynomial:\n",
    "\n",
    "$$I_N[f] = \\int^b_a P_N(x) dx$$"
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   "source": [
    "#### Trapezoidal Rule\n",
    "\n",
    "Use $N = 1$ polynomial to derive the trapezoidal rule."
   ]
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    "Trapezoidal rule uses $N = 1$ order polynomials between each point (i.e. piece-wise defined linear polynomials).  Using the Linear Lagrange basis we can write the interpolating polynomial as \n",
    "\n",
    "$$\n",
    "    P_1(x) = f(x_i)\\frac{x_{i+1}-x}{\\Delta x} + f(x_{i+1})\\frac{x - x_i}{\\Delta x}\n",
    "$$\n",
    "\n",
    "where $\\Delta x = x_{i+1}-x_i$ is the width of the interval."
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    "Integrating this polynomial we have\n",
    "\n",
    "\\begin{aligned}\n",
    "    I_N[f] &= \\int^{x_{i+1}}_{x_i} f(x_i)\\ell_0(x) + f(x_{i+1})\\ell_1(x) dx  \\\\\n",
    "    &=\\frac{1}{\\Delta x}  \\left. \\left[ f(x_i)\\left(x_{i+1}x - \\frac{x^2}{2}\\right) + f(x_{i+1})\\left(\\frac{x^2}{2}-xx_i\\right) \\right] \\right|_{x_i}^{x_{i+1}} \\\\\n",
    "    &= \\frac{1}{\\Delta x}  \\left[ f(x_i)\\left(\\frac{x_{i+1}^2}{2} -x_ix_{i+1} + x_i^2\\right) + f(x_{i+1})\\left(\\frac{x_{i+1}^2}{2} -x_ix_{i+1} + x_i^2\\right) \\right]\\\\\n",
    "    &= \\frac{(x_{i+1} - x_i)^2}{2\\Delta x}\\left[ f(x_i) + f(x_{i+1})\\right]\\\\\n",
    "    & = \\frac{\\Delta x}{2}\\left[ f(x_i) + f(x_{i+1}) \\right] \\\\\n",
    "\\end{aligned}"
   ]
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   "source": [
    "We can also simplify the sum over all the intervals by noting that all but the end points will have total contribution of $\\Delta x$ to the entire sum such that\n",
    "\n",
    "$$\n",
    "    I_N[f] = \\frac{\\Delta x}{2} (f(x_0) + f(x_N) ) + \\sum^{N-1}_{j=1} \\Delta x f(x_j)\n",
    "$$\n",
    "\n",
    "This is known as the composite or extended trapezoidal rule."
   ]
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    "# Note that this calculates the cummulative integral from 0.0\n",
    "\n",
    "f = lambda x: numpy.sin(x)\n",
    "I = lambda x: 1.0 - numpy.cos(x)\n",
    "x = numpy.linspace(0.0, 2.0 * numpy.pi, 100)\n",
    "\n",
    "num_partitions = 20\n",
    "x_hat = numpy.linspace(0.0, 2.0 * numpy.pi, num_partitions + 1)\n",
    "delta_x = x_hat[1] - x_hat[0]"
   ]
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    "fig = plt.figure()\n",
    "fig.subplots_adjust(hspace=.5)\n",
    "axes = fig.add_subplot(2, 1, 1)\n",
    "\n",
    "axes.plot(x, numpy.zeros(x.shape), 'k--')\n",
    "axes.plot(x, f(x), 'b')\n",
    "\n",
    "for i in range(num_partitions):\n",
    "    axes.plot([x_hat[i], x_hat[i]], [0.0, f(x_hat[i])], 'k--')\n",
    "    axes.plot([x_hat[i + 1], x_hat[i + 1]], [0.0, f(x_hat[i+1])], 'k--')\n",
    "    axes.plot([x_hat[i], x_hat[i + 1]], [f(x_hat[i]), f(x_hat[i+1])], 'k--')\n",
    "    \n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"$f(x)$\")\n",
    "axes.set_title(\"Partition and $f(x)$\")\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((-1.1, 1.1))\n",
    "\n",
    "I_hat = numpy.zeros(x_hat.shape)\n",
    "I_hat[0] = (f(x_hat[1]) + f(x_hat[0])) * delta_x / 2.0\n",
    "for i in range(1, num_partitions):\n",
    "    I_hat[i] = I_hat[i - 1] + (f(x_hat[i + 1]) + f(x_hat[i])) * delta_x / 2.0\n",
    "    \n",
    "axes = fig.add_subplot(2, 1, 2)\n",
    "\n",
    "axes.plot(x, I(x), 'r')\n",
    "# Offset due to indexing above\n",
    "axes.plot(x_hat + delta_x, I_hat, 'ko')\n",
    "\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"$f(x)$\")\n",
    "axes.set_title(\"Integral and Approximated Integral\")\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((-0.1, 2.5))\n",
    "\n",
    "plt.show()"
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   "source": [
    "#### Simpson's Rule\n",
    "\n",
    "Simpson's rule uses $N = 2$ order polynomials between each point (i.e. piece-wise defined quadratic polynomials).  \n",
    "\n",
    "The polynomial has the form\n",
    "\n",
    "$$P_2(x) = \\frac{2 f(x_i)}{\\Delta x^2} \\left (x - \\frac{\\Delta x}{2} \\right ) (x - \\Delta x) - \\frac{4 f\\left(x_i + \\frac{\\Delta x}{2}\\right)}{\\Delta x^2}  x (x - \\Delta x) + \\frac{2 f(x_{i+1})}{\\Delta x^2} x \\left (x - \\frac{\\Delta x}{2} \\right )$$"
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    "Integrating this polynomial we have\n",
    "\n",
    "$$\n",
    "    I_N[f] = \\int^{x_{i+1}}_{x_i} P_2(x) dx = \\frac{\\Delta x}{6} f(x_i) + \\frac{2 \\Delta x}{3} f\\left(x_i + \\frac{\\Delta x}{2} \\right ) + \\frac{\\Delta x}{6} f(x_{i+1})\n",
    "$$"
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    "We can also show this by using the method of undetermined coefficients.  \n",
    "\n",
    "Use the general form of the quadrature rule and determine weights $w_j$ by using functions we know the solution to.  These functions can be any representation of polynomials up to order $N=2$ however the monomials $1$, $x$, $x^2$ are the easiest in this case.\n",
    "\n",
    "$$\n",
    "    I_{\\Delta x}[f] = w_0 f(0) + w_1 f(\\Delta x / 2) + w_2 f(\\Delta x)\n",
    "$$"
   ]
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    "$$\\begin{aligned}\n",
    "    &\\text{if}~f = 1:  &I[f] =  \\int^{\\Delta x}_{0} 1 dx = \\Delta x  \\\\\n",
    "    &\\text{if}~f = x:  &I[f] =  \\int^{\\Delta x}_{0} x dx = \\frac{\\Delta x^2}{2} \\\\\n",
    "    &\\text{if}~f = x^2:  &I[f] =  \\int^{\\Delta x}_{0} x^2 dx = \\frac{\\Delta x^3}{3}\n",
    "\\end{aligned}$$\n",
    "\n",
    "What are the corresponding systems of equations?"
   ]
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   "source": [
    "$$\n",
    "    I[f] = w_0 f(0) + w_1 f(\\Delta x / 2) + w_2 f(\\Delta x)\n",
    "$$"
   ]
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   "source": [
    "$$\\begin{aligned}\n",
    "    &\\text{if}~f = 1:  &I[f] =  \\int^{\\Delta x}_{0} 1 dx = \\Delta x & & I_N[1] &= w_0 + w_1 + w_2 \\\\\n",
    "    &\\text{if}~f = x:  &I[f] =  \\int^{\\Delta x}_{0} x dx = \\frac{\\Delta x^2}{2} & & I_N[x] &= w_1 \\frac{\\Delta x}{2} + w_2\\Delta x\\\\\n",
    "    &\\text{if}~f = x^2:  &I[f] =  \\int^{\\Delta x}_{0} x^2 dx = \\frac{\\Delta x^3}{3} & & I_N[x^2] &= \\frac{\\Delta x^2}{4} w_1 + w_2\\Delta x^2\\\\\n",
    "\\end{aligned}$$"
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    "We then have the system of equations:\n",
    "$$\\begin{aligned}\n",
    "    w_0 &+& w_1 &+& w_2 &=\\Delta x \\\\\n",
    "        &\\quad& \\frac{\\Delta x}{2} w_1 &+& \\Delta x w_2  &= \\frac{\\Delta x^2}{2} \\\\\n",
    "        &\\quad& \\frac{\\Delta x^2}{4} w_1 &+& \\Delta x^2 w_2 &=\\frac{\\Delta x^3}{6} \\\\\n",
    "\\end{aligned}$$"
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    "or\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "    1 & 1 & 1 \\\\\n",
    "    0 & \\Delta x / 2 & \\Delta x \\\\\n",
    "    0 & \\Delta x^2 / 4 & \\Delta x^2 \\\\\n",
    "\\end{bmatrix} \\begin{bmatrix}\n",
    "    w_0 \\\\ w_1 \\\\ w_2\n",
    "\\end{bmatrix} = \\begin{bmatrix} \n",
    "    \\Delta x \\\\ \\Delta x^2 / 2 \\\\ \\Delta x^3 / 3\n",
    "\\end{bmatrix} \\Rightarrow \\begin{bmatrix}\n",
    "    1 & 1 & 1 \\\\\n",
    "    0 & 1 / 2 & 1 \\\\\n",
    "    0 & 1 / 4 & 1 \\\\\n",
    "\\end{bmatrix} \\begin{bmatrix}\n",
    "    w_0 \\\\ w_1 \\\\ w_2\n",
    "\\end{bmatrix} = \\begin{bmatrix} \n",
    "    \\Delta x \\\\ \\Delta x / 2 \\\\ \\Delta x / 3\n",
    "\\end{bmatrix}\\Rightarrow \n",
    "\\begin{bmatrix}\n",
    "    1 & 1 & 1 \\\\\n",
    "    0 & 1 / 2 & 1 \\\\\n",
    "    0 & 0 & -1 \\\\\n",
    "\\end{bmatrix} \\begin{bmatrix}\n",
    "    w_0 \\\\ w_1 \\\\ w_2\n",
    "\\end{bmatrix} = \\begin{bmatrix} \n",
    "    \\Delta x \\\\ \\Delta x / 2 \\\\ -\\Delta x / 6\n",
    "\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Leading to \n",
    "\n",
    "$$ w_2 = \\frac{\\Delta x}{6} \\quad w_1 = \\frac{2}{3} \\Delta x \\quad w_0 = \\frac{\\Delta x}{6}$$\n",
    "\n",
    "Another way to write Simpson's rule is to use intervals of three points (similar to one of the ways we did this last time).  The formulation here effectively has a $\\Delta x$ half of what the intervals show but is easier to program."
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    "# Note that this calculates the cummulative integral from 0.0\n",
    "\n",
    "f = lambda x: numpy.sin(x)\n",
    "I = lambda x: 1.0 - numpy.cos(x)\n",
    "x = numpy.linspace(0.0, 2.0 * numpy.pi, 100)\n",
    "\n",
    "num_partitions = 10\n",
    "x_hat = numpy.linspace(0.0, 2.0 * numpy.pi, num_partitions + 1)\n",
    "delta_x = x_hat[1] - x_hat[0]"
   ]
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   },
   "outputs": [],
   "source": [
    "fig = plt.figure()\n",
    "fig.subplots_adjust(hspace=.5)\n",
    "axes = fig.add_subplot(2, 1, 1)\n",
    "\n",
    "axes.plot(x, numpy.zeros(x.shape), 'k--')\n",
    "axes.plot(x, f(x), 'b')\n",
    "\n",
    "for i in range(num_partitions):\n",
    "    axes.plot([x_hat[i], x_hat[i]], [0.0, f(x_hat[i])], 'k--')\n",
    "    axes.plot([x_hat[i + 1], x_hat[i + 1]], [0.0, f(x_hat[i + 1])], 'k--')\n",
    "    coeff = numpy.polyfit((x_hat[i], x_hat[i] + delta_x / 2.0, x_hat[i + 1]), \n",
    "                          (f(x_hat[i]), f(x_hat[i] + delta_x / 2.0), f(x_hat[i+1])), 2)\n",
    "    x_star = numpy.linspace(x_hat[i], x_hat[i+1], 10)\n",
    "    axes.plot(x_star, numpy.polyval(coeff, x_star), 'k--')\n",
    "    \n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"$f(x)$\")\n",
    "axes.set_title(\"Partition and $f(x)$\")\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((-1.1, 1.1))\n",
    "\n",
    "I_hat = numpy.zeros(x_hat.shape)\n",
    "I_hat[0] = delta_x * (1.0 / 6.0 * (f(x_hat[0]) + f(x_hat[1])) + 2.0 / 3.0 * f(x_hat[0] + delta_x / 2.0))\n",
    "for i in range(1, num_partitions):\n",
    "    I_hat[i] = I_hat[i - 1] + delta_x * (1.0 / 6.0 * (f(x_hat[i]) + f(x_hat[i+1])) + 2.0 / 3.0 * f(x_hat[i] + delta_x / 2.0))\n",
    "    \n",
    "axes = fig.add_subplot(2, 1, 2)\n",
    "\n",
    "axes.plot(x, I(x), 'r')\n",
    "# Offset due to indexing above\n",
    "axes.plot(x_hat + delta_x, I_hat, 'ko')\n",
    "\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"$f(x)$\")\n",
    "axes.set_title(\"Integral and Approximated Integral\")\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((-0.1, 2.5))\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Error Analysis\n",
    "\n",
    "From before we have a particular quadrature scheme $I_N$ which we can also write as\n",
    "\n",
    "$$\n",
    "    I_N[f] = \\sum^{N-1}_{i=0} w_i f(x_i).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Define the error $E[f]$ such that\n",
    "\n",
    "$$I[f] = I_N[f] + E_N[f]$$\n",
    "\n",
    "The degree of $I_N[f]$ is the integer $N$ such that $E_N[p_i] = 0 \\quad \\forall i \\leq n$ and $\\exists p_{n+1}$ such that $E[p_{n+1}] \\neq 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Quadrature Accuracy\n",
    "\n",
    "We can also use our polynomial analysis from before to analyze the errors made using both of the aforementioned methods.  From Lagrange's theorem we have the remainder term as before which we can use to look at the error\n",
    "\n",
    "$$R_N(x) = (x - x_0)(x - x_1) \\cdots (x- x_N) \\frac{f^{(N+1)}(c)}{(N+1)!}$$\n",
    "\n",
    "and integrate it to find the form and magnitude of the error on a single interval.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "To find the total error we must sum the error over all the intervals:\n",
    "\n",
    "$$I[f] = \\sum_{i=0}^N \\int^{x_{i+1}}_{x_i} P_N(x) dx + \\sum_{i=0}^N \\int^{x_{i+1}}_{x_i} R_N(x) dx = I_N[f] + E_N[f]$$\n",
    "\n",
    "as we defined before."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Trapezoidal error\n",
    "\n",
    "With $N=1$ we have\n",
    "\n",
    "$$R_1(x) = (x - x_i) (x - x_{i+1}) \\frac{f''(c)}{2}$$\n",
    "\n",
    "Integrating this leads to\n",
    "\n",
    "$$\\int^{x_{i+1}}_{x_i} (x - x_i) (x - x_{i+1}) \\frac{f''(c)}{2} dx = \\frac{\\Delta x^3}{12} f''(c)$$\n",
    "\n",
    "giving us a form for the error.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "If we sum up across all the intervals the total error is\n",
    "\n",
    "$$E_N[f] = -\\frac{\\Delta x^3}{12} \\sum_{i=0}^{N} f''(c_i)$$\n",
    "\n",
    "or more illustrative\n",
    "\n",
    "$$E_N[f] = -\\frac{1}{2} \\Delta x^2 (b - a) \\left [ \\frac{1}{N} \\sum^{N-1}_{i=0} f''(c_i) \\right ]$$\n",
    "\n",
    "where the expression in the brackets is the mean value of the second derivative over the interval $[a,b]$.  This also shows that the trapezoidal rule converges quadratically as $\\Delta x \\rightarrow 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Simpson's Rule Error\n",
    "\n",
    "Similarly here we have $N = 2$ and \n",
    "\n",
    "$$R_2(x) = (x - x_i) \\left(x - x_i - \\frac{\\Delta x}{2} \\right) (x - x_{i+1}) \\frac{f'''(c)}{3!}$$\n",
    "\n",
    "Integrating and summing the error contributions we find\n",
    "\n",
    "$$E_N[f] = -\\frac{1}{180} (b - a) \\Delta x^4 f^{(4)}(c)$$\n",
    "\n",
    "Interestingly we have gained two orders of accuracy by increasing the polynomial order by only 1!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Review:  Newton-Cotes Formulas and errors for a single interval \n",
    "\n",
    "given $x\\in[x_i, x_{i+1}]$ and $\\Delta x = x_{i+1} - x_i$, all of the   Newton-Cotes quadrature formula's can be derived by integrating the interpolating polynomial through $N$ equi-spaced points across the interval.  The error term is simply the integral of the Lagrange remainder term $R_N(x)$\n",
    "\n",
    "* Mid-point: 1-point quadrature rule\n",
    "$$\n",
    "    I_M[f] = \\Delta x\\, f(x_i + \\Delta x/2) + O(\\Delta x^3f'')\n",
    "$$\n",
    "* Trapezoidal Rule: 2-point quadrature rule\n",
    "$$\n",
    "    I_T[f] = \\Delta x\\, \\left[\\frac{1}{2}f(x_i) +\\frac{1}{2}f(x_{i+1})\\right] + O(\\Delta x^3f'')\n",
    "$$\n",
    "* Simpson's Rule: 3-point quadrature rule\n",
    "$$\n",
    "    I_S[f] = \\Delta x\\, \\left[ \\frac{1}{6}f(x_i) +  \\frac{2}{3}f(x_i +\\Delta x/2) +\\frac{1}{6}f(x_{i+1})\\right] + O(\\Delta x^5f^{iv})\n",
    "$$\n",
    "\n",
    "Note: error term converges as N increases ($\\Delta x$ gets smaller)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1080x360 with 3 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "func = lambda x: 1 - x**2 + numpy.sin(5*x)\n",
    "x0 = 0.\n",
    "x1 = .5\n",
    "x_buf = .2\n",
    "x = numpy.linspace(x0 - x_buf, x1+ x_buf, 100)\n",
    "x_interval = numpy.linspace(x0, x1, 100)\n",
    "\n",
    "x_samp = numpy.array([ 0.5*(x0 + x1) ])\n",
    "f_samp = func(x_samp)\n",
    "\n",
    "fig = plt.figure(figsize=(15,5))\n",
    "axes = fig.add_subplot(1,3,1)\n",
    "axes.plot(x,func(x),linewidth=3)\n",
    "axes.plot(x_samp, f_samp, 'ro')\n",
    "axes.plot(x_interval, f_samp*numpy.ones(x_interval.shape), 'k--')\n",
    "axes.plot(x0*numpy.ones(2), [0., f_samp],'k--')\n",
    "axes.plot(x1*numpy.ones(2), [0., f_samp],'k--')\n",
    "axes.plot(x0*numpy.ones(2), [0., func(x0)],'b')\n",
    "axes.plot(x1*numpy.ones(2), [0., func(x1)],'b')\n",
    "axes.plot(x, numpy.zeros(x.shape),'k')\n",
    "axes.text(x0-.02,-.2,'$x_i$',fontsize=16)\n",
    "axes.text(x1-.02,-.2,'$x_{i+1}$',fontsize=16)\n",
    "axes.set_title('Midpoint Rule',fontsize=16)\n",
    "axes.set_xlabel('x', fontsize=16)\n",
    "axes.set_ylabel('f(x)', fontsize=16)\n",
    "axes.set_ylim((-0.25, 2.))\n",
    "axes.fill_between(x_interval,0.,func(x_interval))\n",
    "axes.fill_between(x_interval, 0, f_samp*numpy.ones(x_interval.shape), hatch='/', alpha=0)\n",
    "axes.grid()\n",
    "\n",
    "axes = fig.add_subplot(1,3,2)\n",
    "x_samp = numpy.array([x0, x1])\n",
    "f_samp = func(x_samp)\n",
    "axes.plot(x,func(x),linewidth=3)\n",
    "axes.plot(x_samp, f_samp, 'ro')\n",
    "axes.plot(x_samp, f_samp, 'k--')\n",
    "axes.plot(x0*numpy.ones(2), [0., f_samp[0]],'k--')\n",
    "axes.plot(x1*numpy.ones(2), [0., f_samp[1]],'k--')\n",
    "axes.plot(x0*numpy.ones(2), [0., func(x0)],'b')\n",
    "axes.plot(x1*numpy.ones(2), [0., func(x1)],'b')\n",
    "axes.plot(x, numpy.zeros(x.shape),'k')\n",
    "axes.text(x0-.02,-.2,'$x_i$',fontsize=16)\n",
    "axes.text(x1-.02,-.2,'$x_{i+1}$',fontsize=16)\n",
    "axes.set_title('Trapezoidal Rule',fontsize=16)\n",
    "axes.set_xlabel('x', fontsize=16)\n",
    "axes.set_ylabel('f(x)', fontsize=16)\n",
    "axes.set_ylim((-0.25, 2.))\n",
    "axes.fill_between(x_interval,0.,func(x_interval))\n",
    "axes.fill_between(x_samp,0.,f_samp, hatch='/', alpha=0)\n",
    "\n",
    "\n",
    "axes.grid()\n",
    "\n",
    "axes = fig.add_subplot(1,3,3)\n",
    "x_samp = numpy.array([x0, (x0 + x1)/2., x1])\n",
    "f_samp = func(x_samp)\n",
    "p = numpy.polyfit(x_samp, f_samp, 2)\n",
    "f_p2 = numpy.polyval(p, x_interval)\n",
    "axes.plot(x,func(x),linewidth=3)\n",
    "axes.plot(x_samp, f_samp, 'ro')\n",
    "axes.plot(x_interval, f_p2, 'k--')\n",
    "axes.plot(x0*numpy.ones(2), [0., f_samp[0]],'k--')\n",
    "axes.plot(x1*numpy.ones(2), [0., f_samp[-1]],'k--')\n",
    "axes.plot(x0*numpy.ones(2), [0., func(x0)],'b')\n",
    "axes.plot(x1*numpy.ones(2), [0., func(x1)],'b')\n",
    "axes.plot(x, numpy.zeros(x.shape),'k')\n",
    "axes.text(x0-.02,-.2,'$x_i$',fontsize=16)\n",
    "axes.text(x1-.02,-.2,'$x_{i+1}$',fontsize=16)\n",
    "axes.set_title('Simpsons Rule',fontsize=16)\n",
    "axes.set_xlabel('x', fontsize=16)\n",
    "axes.set_ylabel('f(x)', fontsize=16)\n",
    "axes.set_ylim((-0.25, 2.))\n",
    "axes.fill_between(x_interval,0.,func(x_interval))\n",
    "axes.fill_between(x_interval,0.,f_p2, hatch='/', alpha=0)\n",
    "\n",
    "\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "##### Example 1:\n",
    "\n",
    "If $f(x) = \\sin \\pi x$ look at the relative accuracy of midpoint, trapezoidal and simpson's rules for a single interval $x\\in[0,1]$.\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    \\text{Exact:}  &I[f] = \\int^1_0 \\sin \\pi x = \\left . \\frac{-\\cos \\pi x}{\\pi} \\right |^1_0 = \\frac{2}{\\pi} \\approx 0.636619772 \\\\\n",
    "    \\text{Midpoint:}  &I_M[f] = \\Delta x f(1/2) = \\sin (\\pi / 2) = 1 \\\\\n",
    "    \\text{Trapezoid:}  &I_T[f] = \\frac{\\Delta x}{2} (\\sin(0) + \\sin(\\pi)) = 0 \\\\\n",
    "    \\text{Simpson's:}  &I_S[f] = \\frac{\\Delta x}{6} \\sin(0) + \\frac{2 \\Delta x}{3} \\sin(\\pi / 2) + \\frac{\\Delta x}{6} \\sin(\\pi) = \\frac{2 \\Delta x}{3} = \\frac{2}{3}\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Cumulative (global) Error for extended Newton-Cotes formulas\n",
    "\n",
    "For $N$ panels of any given N-C quadrature formula over the interval $x\\in[a,b]$, the approximate integral and error formulas are\n",
    "\n",
    "\\begin{align}\n",
    "I[f] &= \\sum_{i=0}^N I_i + E_N\\\\\n",
    "\\end{align}\n",
    "where \n",
    "$$\n",
    "    E_N = (b-a)O(\\Delta x^p f^{m}(c))\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "* Extended Mid-point: \n",
    "$$\n",
    "    E_{N_M} = (b-a) O(\\Delta x^2 f''')\n",
    "$$\n",
    "* Extended Trapezoidal Rule: \n",
    "$$\n",
    "    E_{N_M} = (b-a) O(\\Delta x^2 f''')\n",
    "$$\n",
    "* Extended Simpson: \n",
    "$$\n",
    "    E_{N_M} = (b-a) O(\\Delta x^4 f^{(v)})\n",
    "$$\n",
    "\n",
    "Where $\\Delta x = \\frac{b -a }{N}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute the error as a function of delta_x for each method\n",
    "f = lambda x: numpy.sin(numpy.pi * x)\n",
    "\n",
    "num_partitions = numpy.array([ 2**n for n in range(0, 16)])\n",
    "delta_x = numpy.empty(len(num_partitions))\n",
    "error_mid = numpy.empty(len(num_partitions))\n",
    "error_trap = numpy.empty(len(num_partitions))\n",
    "error_simpson = numpy.empty(len(num_partitions))\n",
    "\n",
    "I_true = 2.0/numpy.pi\n",
    "\n",
    "for (j, N) in enumerate(num_partitions):\n",
    "    x_hat = numpy.linspace(0.0, 1.0, N + 1)\n",
    "    delta_x[j] = x_hat[1] - x_hat[0]\n",
    "\n",
    "    # Compute Midpoint\n",
    "    x_star = 0.5 * (x_hat[1:] + x_hat[:-1])\n",
    "    I_hat = 0.0\n",
    "    for i in range(0, N):\n",
    "        I_hat += f(x_star[i]) * delta_x[j]\n",
    "    error_mid[j] = numpy.abs(I_hat - I_true)/I_true\n",
    "    \n",
    "    # Compute trapezoid\n",
    "    I_hat = 0.0\n",
    "    for i in range(1, N):\n",
    "        I_hat += (f(x_hat[i + 1]) + f(x_hat[i])) * delta_x[j] / 2.0\n",
    "    error_trap[j] = numpy.abs(I_hat - I_true)/I_true\n",
    "    \n",
    "    # Compute simpson's    \n",
    "    I_hat = 0.0\n",
    "    for i in range(0, N):\n",
    "        I_hat += delta_x[j] * (1.0 / 6.0 * (f(x_hat[i]) + f(x_hat[i+1])) + 2.0 / 3.0 * f(x_hat[i] + delta_x[j] / 2.0))\n",
    "    error_simpson[j] = numpy.abs(I_hat - I_true)/I_true\n",
    "\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "order_C = lambda delta_x, error, order: numpy.exp(numpy.log(error) - order * numpy.log(delta_x))\n",
    "axes.loglog(delta_x, error_mid, 'ro', label=\"Midpoint\")\n",
    "axes.loglog(delta_x, error_trap, 'bo', label=\"Trapezoid\")\n",
    "axes.loglog(delta_x, error_simpson, 'go', label=\"Simpson's\")\n",
    "axes.loglog(delta_x, order_C(delta_x[0], error_trap[0], 2.0) * delta_x**2.0, 'b--', label=\"2nd Order\")\n",
    "axes.loglog(delta_x, order_C(delta_x[0], error_simpson[0], 4.0) * delta_x**4.0, 'g--', label=\"4th Order\")\n",
    "axes.legend(loc='best')\n",
    "axes.set_xlabel('$\\Delta x$', fontsize=16)\n",
    "axes.set_ylabel('Relative Error', fontsize=16)\n",
    "axes.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "### Recursive Improvement of Accuracy\n",
    "\n",
    "Say we ran the trapezoidal rule with step size $2 \\Delta x$, we then will have\n",
    "\n",
    "$$\\begin{aligned}\n",
    "\t\t\t\\int^{x_2}_{x_0} f(x) dx  &= \\frac{2 \\Delta x}{2} (f_0 + f_2) =  h (f_0 + f_2) \\Rightarrow \\\\\n",
    "\t\t\t\\int^b_a f(x)dx &\\approx I_{2\\Delta x}[f] = \\sum^{N/2-1}_{j=0} \\Delta x (f_{2j} + f_{2j+2}) \\\\\n",
    "\t\t\t&= \\Delta x (f_{0} + f_{2})  + \\Delta x (f_{2} + f_{4})  + \\cdots + \\Delta x (f_{N-2} + f_{N}) \\\\\n",
    "\t\t\t&= \\Delta x\\left ( f_0 + f_N +  2 \\sum^{N/2-1}_{j=1} f_{2j} \\right )\n",
    "  \\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "Now compare the two rules for $\\Delta x$ and $2 \\Delta x$:\n",
    "\n",
    "$$\\begin{align}I_{\\Delta x}[f] &= \\frac{\\Delta x}{2} \\left (f_0 + f_N + 2 \\sum^{N-1}_{j=1} f_j \\right)\\\\ I_{2 \\Delta x}[f] &= \\Delta x \\left ( f_0 + f_N +  2 \\sum^{N/2-1}_{j=1} f_{2j} \\right )\\end{align}$$\n",
    "\n",
    "$$I_{\\Delta x}[f] = \\frac{1}{2} I_{2\\Delta x} + \\Delta x(f_1 + f_3 + \\cdots + f_{N-1})$$\n",
    "\n",
    "Here we see we can actually reuse the work we did to calculate $Q_{2 \\Delta x}[f]$ to refine the integral."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Arbitrary Intervals (Affine Transforms)\n",
    "\n",
    "A lot of quadrature rules are defined on specific intervals, e.g $x\\in[0,1]$ or $x\\in[-1,1]$.  However any two intervals can always be mapped onto each other through an  *affine transform* or *affine map* which is a __linear  transformation__.\n",
    "\n",
    "$$\n",
    "    x = \\alpha\\xi + \\beta\n",
    "$$\n",
    "where $\\alpha$ is a scaling and $\\beta$ is a shift.  \n",
    "\n",
    "Given $\\alpha$ and $\\beta$,  the inverse map is simply\n",
    "\n",
    "$$\n",
    "    \\xi = \\frac{x - \\beta}{\\alpha}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example:  \n",
    "\n",
    "Map $ x \\in [a,b]\\rightarrow\\xi \\in [-1,1]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(5,5))\n",
    "axes = fig.add_subplot(1,1,1)\n",
    "axes.plot(x,func(x),linewidth=3)\n",
    "axes.fill_between(x_interval,0.,func(x_interval))\n",
    "axes.plot(x, numpy.zeros(x.shape),'k')\n",
    "axes.text(x0-.02,-.2,'$a$',fontsize=16)\n",
    "axes.text((x0+x1)/2.-.02,-.2,'$x$',fontsize=16)\n",
    "axes.text(x1-.02,-.2,'$b$',fontsize=16)\n",
    "axes.set_xlabel('x', fontsize=16)\n",
    "axes.set_ylabel('f(x)', fontsize=16)\n",
    "axes.text(x0-.06,-.4,'$-1$',fontsize=16)\n",
    "axes.text((x0+x1)/2.-.02,-.4,'$\\\\xi$',fontsize=16)\n",
    "axes.text(x1-.02,-.4,'$1$',fontsize=16)\n",
    "axes.set_ylim((-0.5, 2.))\n",
    "axes.grid()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$x =  \\frac{b - a}{2}\\xi + \\frac{a+b}{2}$$\n",
    "\n",
    "$$\\xi = \\left( x - \\frac{a + b}{2}\\right) \\frac{2}{b-a}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Given an Affine mapping,  it is easy to transform any integral from one coordinate system to another\n",
    "\n",
    "$$\n",
    "    I[f] = \\int^b_a f(x) dx = \\int^1_{-1} f(x(\\xi)) \\frac{dx}{d\\xi} d\\xi = \\frac{b - a}{2} \\int^1_{-1} f(x(\\xi)) d\\xi\\\\\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "and the $N$ point quadrature rule for a single interval is\n",
    "$$\n",
    "    I_N[f]_i = \\frac{\\Delta x_i}{2} \\sum_{n=1}^{N} w_n f(x(\\xi_n))\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "and the cumulative integral over $M$ panels\n",
    "$$\n",
    "    I_N[f] = \\sum_{i=1}^M I_N[f]_i\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Example:  Newton-Cotes Rules\n",
    "\n",
    "We can rewrite our previous quadrature rules so that they are given on the interval $\\xi \\in [-1, 1]$ instead of $x \\in [x_i, x_{i+1}]$.  Recall that a general quadrature rule can be written as\n",
    "$$\n",
    "    \\sum^N_{i=1} w_i f(\\xi_i)\n",
    "$$\n",
    "where $w_i$ are the weights and $\\xi_i$ are the points specified to evaluate the function at. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    " For Newton-Cotes rules we know that the points $\\xi_i$ are uniformly distributed on $[-1, 1]$ but we still need to define the weights.  For a 2-Point trapezoid rule we can do this to find that\n",
    "$$\n",
    "    \\int^1_{-1} f(x) dx \\approx f(-1) + f(1)\n",
    "$$\n",
    "so the quadrature points are at  $\\xi_0 = [-1, 1]$ with weights $w = [1, 1]$  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Note that if we map this using our affine transform we would get back the original trapezoid rule:\n",
    "$$\n",
    "    I_2[f] = \\frac{dx}{d\\xi}\\sum_{n=1}^{2} w_n f(x(\\xi_n))  = \\frac{\\Delta x}{2}(f(a) + f(b)) \n",
    "$$\n",
    "\n",
    "Note: $dx = \\Delta x$, and $d\\xi = 2$ (\\[-1, 1\\])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Similarly for Simpson's rule we have\n",
    "$$\n",
    "    \\xi = [-1, 0, 1] \\quad \\text{and} \\quad w = \\left[\\frac{1}{3}, \\frac{4}{3}, \\frac{1}{3} \\right].\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def ncintegration(f, a, b, N):\n",
    "    \"\"\" Approximate \\int_a^b f(x)dx using  Newton-Cotes quadrature rules with N<=3 quadrature points\"\"\"\n",
    "        \n",
    "    assert (N > 0)\n",
    "    assert (N < 4)\n",
    "    # Build up a nested list of integration points and weights for the interval [-1, 1]:\n",
    "    xNC = []\n",
    "    wNC = []\n",
    "    # 1 point Newton-Cotes (Mid-point) \n",
    "    xNC.append(numpy.array([0.]))\n",
    "    wNC.append(numpy.array([2.]))\n",
    "    # 2 point Newton-Cotes quadrature (Trapezoidal)\n",
    "    xNC.append(numpy.array([ -1., 1. ]))\n",
    "    wNC.append(numpy.array([ 1. , 1.]))\n",
    "    # 3 point Newton-Cotes quadrature (Simpson's rule)\n",
    "    xNC.append(numpy.array([-1., 0., 1.]))\n",
    "    wNC.append(numpy.array([1., 4., 1.])/3.)\n",
    "    \n",
    "    # make affine map between x and xi\n",
    "    xi_map = lambda xi: (b - a)/2. * xi + (a + b) / 2.\n",
    "    \n",
    "    # Calculate I[f] for each NC method\n",
    "    I = (b - a)/2. * numpy.sum(wNC[N-1]*f(xi_map(xNC[N-1])))\n",
    "    return I\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7071067811865475\n"
     ]
    }
   ],
   "source": [
    "# Note that this calculates the cumulative integral from 0.0\n",
    "\n",
    "f = lambda x: numpy.sin(x)\n",
    "I = lambda x: - numpy.cos(x)\n",
    "\n",
    "a = numpy.pi/4.\n",
    "b = 3*numpy.pi/4.\n",
    "b = numpy.pi/2\n",
    "I_true = I(b) - I(a)\n",
    "print(I_true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = numpy.linspace(0.0, 2.0 * numpy.pi, 100)\n",
    "x_interval = numpy.linspace(a, b, 100)\n",
    "fig = plt.figure(figsize=(5,5))\n",
    "axes = fig.add_subplot(1,1,1)\n",
    "axes.plot(x,f(x),linewidth=3)\n",
    "axes.fill_between(x_interval,0.,f(x_interval))\n",
    "axes.plot(x, numpy.zeros(x.shape),'k')\n",
    "\n",
    "axes.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N = 1, I_N = 0.7256132880348577, err = 0.026172152977030898\n",
      "N = 2, I_N = 0.670379265333622, err = 0.051940551031480015\n",
      "N = 3, I_N = 0.7072019471344457, err = 0.00013458497419382203\n"
     ]
    }
   ],
   "source": [
    "# Calculate Newton-Cotes Quadrature and relative errors for 1,2, and 3 point NC quadrature schemes\n",
    "for N in range(1,4):\n",
    "    I = ncintegration(f,a,b,N)\n",
    "    err = numpy.abs(I - I_true)/numpy.abs(I_true)\n",
    "    print('N = {}, I_N = {}, err = {}'.format(N, I, err))\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Optimal Quadrature Methods\n",
    "\n",
    "Newton-Cotes formulas assume a fixed, regularly spaced set of quadrature points and calculate quadrature rules based on integrating the interpolating polynomial.  \n",
    "\n",
    "Can we do better?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Actually, if we allow both the points and weights to be chosen in an optimal way, we can generate optimal quadrature rules with $N$ points that can exactly integrate polynomials up to order $2N-1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Generalized Gaussian Quadrature\n",
    "\n",
    "Given $g(x) \\in P_N(x)$ with roots $\\{x_i\\}^N_{i=1}$ with the property\n",
    "$$\n",
    "    \\int^1_{-1} w(x) x^i g(x) dx = 0 \\quad \\forall i < N,\n",
    "$$\n",
    "\n",
    "implies that  $g(x)$ is __*orthogonal* to all monomials__ $x^i$ ($i <N$) with respect to the weight function $w(x)$.\n",
    "\n",
    "Note: see Wikipedia page on Gaussian Quadrature"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Recall something similar:\n",
    "$$\n",
    "    <x, y> = \\sum^N_{i=1} x_i y_i = ||x|| \\cdot ||y|| \\cos \\theta.\n",
    "$$\n",
    "If $<x, y> = 0$ then the vectors $x$ and $y$ are orthogonal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Given the above $g(x)$ there then exists a set of weights $\\{w_i\\}$ such that\n",
    "\n",
    "$$\\int^1_{-1} w(x) P_j(x) dx = \\sum^N_{i=1} w_i P_j(x_i) \\quad \\forall j \\leq 2 N - 1$$\n",
    "\n",
    "In other words, given a polynomial basis function and weight and orthogonality to all polynomials of order $i < N$ we can exactly integrate polynomials of order $2 N - 1$.  For a proof and considerably more detail see https://en.wikipedia.org/wiki/Gaussian_quadrature\n",
    "\n",
    "Choosing the correct weighting function and basis leads to a number of useful quadrature approaches:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Gauss-Legendre Quadrature\n",
    "\n",
    "General Gauss-Legendre quadrature uses $w(x) = 1$ and $g(x) = \\ell_N(x)$ which are the Legendre Polynomials, which can be shown to have weights\n",
    "\n",
    "$$w_i = \\frac{2}{(1-x_i^2)(\\ell'_n(x_i))^2}$$\n",
    "\n",
    "and $x_i$ is the $i$th root of $\\ell_N$.  \n",
    "\n",
    "Note: $\\ell_N(x)$ here denotes Legendre polynomial, not Lagrange"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "The first few Legendre Polynomials are \n",
    "\\begin{align}\n",
    "    \\ell_1(x) &=x \\\\\n",
    "    \\ell_2(x) &= \\frac{1}{2}(3x^2 -1)\\\\\n",
    "    \\ell_3(x) &= \\frac{1}{2}x(5x^2 - 3)\\\\\n",
    "       \\vdots\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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QmqZpmmMVZMI3k2HZQ9C0D8zYDO2Hn3dIan4qy44s4+bWN5C7/zRK3M51orYXncgBTz/jz8zkfCZ2moiI8MneTxwblKZpmuY4p2NhztWw93u49v9g4vfgH3nBYYv3LcaszEz0HEouRq0wIMxk11B1Isfo7AaQkZxHuE84N7W5ie+OfEdqfqpjA9M0TdPsy2KBTe/AwuFGs/p9P8PgJ8Dlwl7oWUVZfHnwS4a3GE7AgTPkeYcBEKhr5Pbn6iF4+biTmZIPwL2d76WopIjP93/u4Mg0TdM0u8lOgs9uhdX/gA7XG73Sm/ev9PCvDn5FbnEuk7pMIn97LIUhLXBzd8EnwLPSc+qCTuRWAWHeZCbnAdAqoBXXtbiOLw58QU5RjoMj0zRN0+rckTUwZxCc3AI3vA1jPgHvynufF5gLWLxvMYOaDKJDcAfyYrdTGNoa/1BvxMV+Q89AJ/JzjESef+7nyV0nk12czVeHvnJgVJqmaVqdMhfBry8YNXFTCEyLgj6ToJpx4N8f+Z60gjQmd5mM+cwZzElJ5HkEE2jn5+OgE/k5gWEmctILMReVANC5UWcGRg5k8b7FFJYUOjg6TdM0zebSjsHHI2Hzf43kPW0dhHWs9jSzxcyivYvoHtqdPuF9yNu+HYWQU+Ru9x7roBP5OaW//NLn5GDUylPzU/nhyA+OCkvTNE2rC7uXGr3SU4/A2E+N5nT3miXhlcdXEp8Tz+QukxER8mJjKQpuiqUEAkJ1IneYgFCjOaRsIu8X0Y+uIV35eM/HeolTTdO0hqAo15gn/ZvJEN4J7t8EnW6u8ekWZeGj3R/RNrAtQ5oNASA/djslnY1Ocbpp3YFKhwtkWDu8AYgIk7tM5nTOaVYdt+vqqpqmaZqtJeyCuUPgj//B1U/AvSuMOdMvwvpT6zmScYRJXSbhIi6Y09MpPHyY4hadAHTTuiN5mtzx8nU/r8MbwDXNr6FtYFsW7F6ARVkcFJ2maZp2yZSCbXONudILs+HuH2DY/4HrxS0AqpRi/u75NPFtwqhWowDIi4kBoCCouUOGnoFO5OcJCPUmMyXvvG0u4sLkrpM5knGEdafWOSgyTdM07ZLkpRmrlf38FLS+Bu7fDK2HXFJRWxO2sjt1N5O6TMLNxfgSkBcdjXh5kat8CAiz/9Az0In8PIFhpgtq5AAjW46kqW9T5u+aj7G0uqZpmub0jm+C2YPg8K8w4t8w4UvwCbnk4ubvnk+Ydxi3tL3l3La86Bi8e/QgM7XgXF8re9OJvIyAMG9y0gsptg5BK+Xm4sbkrpPZe3YvW85scVB0mqZpWo2UmGHdv+GTG42e6FNWw8AHqh0bXpUdyTuITozmns734OHqYVwmM5PCAwfw6tOHzNR8hzwfB53Iz1Pa2zAr5cJa+U1tbiLMFMa83fPsHZamaZpWU5mnjQS+/jXoNg6mr4fGPWpd7Lxd8wjyDOL29ref25YXux2UQnXsjcWsHNJjHXQiP8+5seQVNK97uHpwX+f7iE2KZXvSdnuHpmmaplXnwHKYcxUk7oLR82D0nD+Xt6yF/Wf3szF+I3d1uguT+5/JOi86GvHwoDCkFeCYMeSgE/l5SpeeKzsErazb2t9GsFewrpVrmqY5k+ICWP6E0aktsAVM3wDdx9ms+Pm75+Pr7sv4DuPP254XHY13t25kpRcD9l++tJRO5GV4ervh7ed+3qQwZXm7eTOx00Q2x29mb+peO0enaZqmXSDloDGsLHo+DHwIJv8KjdrYrPijGUdZfWI1d3S4A38P/3PbS3JyKNi3D1O/vmSm5BtDzwI9bHbdi6ETeTkBoX+uglaR8VeMx8/Dj7m75toxKk3TNO08SsH2T2HeUMhOgAlfw4hXwc22yXTe7nl4uXkxsdPE87bnx8aCxYKpb18yk/OMoWe16ExXGzqRlxMQZiKjgmfkpXw9fJnYcSLrTq3jYNpBO0amaZqmAVCQCUsnwbKHoWkfmLEZ2g+3+WVOZJ3g52M/M+6KcQR5BZ23Ly86Gtzd8e7Rg4zkfIc1q4NO5BcIDDORm1FIUUHlc6tP6DgBX3dfXSvXNE2zt9MxxmIn+36Aa/8PJn4P/pF1cqn5u+bj7uLOPZ3vuWBfbnQ03l27ojw8yUrNPzfNtyPoRF5OUIS1w1tS5c3rAZ4BTOg4gV9P/Mrh9MP2Ck3TNO3yZbHAprdh4QijWX3SShj8BLi41snlTmWf4qejPzGm/RhCvM+fRMaSm0vBnr2Y+vYlKyUfS4kiKMKnTuKoCZ3Iyyl9M9ITK0/kABM7TsTkZmL+rvn2CEvTNO3ylZ0En90Kq1+EDtfDjI3QrF+dXvKj3R/hKq7c1+W+C/bl/bEDSkow9e17LlfoRO5EAkKNuXKrqpEDBHoFMr7DeFYeX8nRzKN2ik7TNO0yc2Q1zBkEJ7fCjf+FMZ+Ad2CdXvJMzhl+iPuBW9vdSpgp7IL9edHR4OqKqWePc7kiMEI/I3caru4u+Id4kZ6QW+2xd3e6Gy83LxbsWmCHyDRN0y4j5iL45f/gs9vAJxSmRUHve2s1zWpNLdyzEIDJXSdXuD8vOhqvLp1x8fEhPSEXU4AHnt4Xt5KaLdkkkYvISBE5KCJHROTpCva/LSI7rK9DIpJRZl9JmX3LbBFPbQVF+JBeTY0coJF3I8a0H8OKYys4mXXSDpFpmqZdBtKOGs/Cf3sX+kyCqWshrINdLp2Um8S3h7/llra3EOETccF+S34++bt349O3LwDpSXkObVYHGyRyEXEFPgBGAZ2AO0SkU9ljlFKPKqV6KKV6AO8B35bZnV+6Tyl1U23jsYWgCBMZyXlYSqpff/y+Lvfh5uKme7BrmqbZwu6lMGcwpMXB2E/hhreNhU/sZMHuBSilmNJ1SoX782K3Q3Expv4DUEqRnph3rpO0o9iiRt4POKKUOqqUKgK+AG6u4vg7gCU2uG6dCYowYTErslILqj02xDuEsVeMZfnR5bpWrmmadqmKcrniwLvwzWQI7wQzNkGnqlKJ7SXmJvLN4W+4ue3NNPFtUuExeVu3gLs7pt69yMsqoijf3CASeRPgVJmfT1u3XUBEWgCtgLVlNnuJSIyIbBWRWyo6z97O9VyvQfM6wKQuk3B3cde1ck3TtEuRsAvmDiEicS0MfhLuXQGBze0eRmltfFq3aZUek7t1G97du+FiMjlFj3UAez+dHw8sVUqVXfC7hVIqXkRaA2tFZLdSKq78iSIyDZgGEB4eTlRUlM2CysnJOa+8kiIFQOzm3ZxIq1nHioGmgfwY9yPd87sT5n5hL0d7KX8v9VlDuZeGch+g78UZ1ev7UIom8ctpE/cxxe7+bG//DIUu/WHjJruHkm5O55v4b+jv259DMYc4xKELjpHcXEL37iX3+r9yPCqKtMNGrjhwbCdxSefnCru+L0qpWr2AgcCqMj8/AzxTybF/AFdWUdYi4Pbqrtm7d29lS+vWrbtg20dPblSrP9lX4zJS8lJUn8V91DMbnrFhZBevonuprxrKvTSU+1BK34szqrf3kXtWqc/HK/UPf6X+N1apnFSH3svLW15WPT7toeKz4ys9JvOXX9S+Kzqo3OhopZRS6784qObOjFIWi+WCY219L0CMqiQn2qJpPRpoJyKtRMQDo9Z9Qe9zEekABAFbymwLEhFP699DgEHAPhvEVGvBESYyEqsfglYqxDuEcVeMY/mx5RzLPFaHkWmaptVzxzfB7EHGGPER/4Y7vgCfRg4LJyEngW8Of8PotqNp7Nu40uPytm5DvL3x7tYNgIzEXIIiTA5bLKVUrRO5UsoMPASsAvYDXyml9orISyJSthf6eOAL6zeLUh2BGBHZCawDXlNKOUUiD4zwIT0xj/PDrdq9Xe7Fw8WDebv0euWapmkXKDHDun/DJzcaPdGnrIaBD9hlbHhVFuw25gKZ2nVqlcflbt2KqXdvxMNYYS09Mc+hE8GUsskzcqXUCmBFuW0vlPv5xQrO+w3oaosYbC0o3ERhnpn87GJM/jVbFi/EO4TxHcbz6b5PmdZtGq0CWtVxlJqmafVE5mn4Ziqc/A26T4C/zgJPX0dHRUJOAt8e+Zbb2t1GpG/li68UJydTFBdH4K2jASgqMJOTXkhQuGM7uoGe2a1SQZHGt6z0i2heB7i38714unoye+fsughL0zSt/jmw3GhKT9wFo+fB6NlOkcQB5u6aiyCVjhsvlbdtGwCm/gOAPxfWKs0VjqQTeSVqunhKeY28GzGhwwRWHlupV0bTNO3yVlwAy5+ALyZAUEuYvgG6j3N0VOecyjrF90e+Z0z7MRXO4lZW7patuAQE4NXRmGHu3NAzXSN3Xr6Bnrh5ul50jRyMWrnJ3cSHOz6sg8g0TdPqgZSDMP9aiJ4PAx+Cyb9CozaOjuo8c3bNwc3FrdrauFKK3K1b8OnXD3E1lk1NT8xFXIQAB65DXkon8kqIixAUbiLjImvkYKyMdnenu1l9cjX7zjpF3z1N0zT7UApiP4G5QyAnCe5cCiNeBbea9TWyl6OZR/np6E+Mv2I8oabQKo8tPnUK85kETAP6n9uWkZhHQKg3rm6OT6OOj8CJBYabLrppvdTEThPx9/Dngx0f2DgqTdM0J1WQCUsnwY8zjfXC798M7f7i6KgqNHvHbDxdPZnUdVK1x+Zu3QqAz8CB57alJ+URGO745+OgE3mVgiNNZKcVUFxYUv3B5fh5+HFfl/vYcHoDO1N21kF0mqZpTuR0DMy5Gvb9AMNegInfg1/Vz50d5WDaQVYeX8ldHe8i2Cu42uPztm7FLTQUj1bGSCRLiYWMpDyCnaCjG+hEXqVAayeGjBrOuV7ehA4TCPYK5oM/dK1c07QGymKBTW8by44qBZNWwtWPg4vzppcPd3yIn7sf93S+p9pjlcVC7tZtmAYOODfxS1ZqAZYSdS5HOJrz/qadQOmKNulJF9/hDcDkbmJSl0lsSdhCTGKMLUPTNE1zvOwk+OxWWP0idLgBZmw0mtSd2N6ze1l7ai0TO08kwDOg2uMLDx+mJC0NnwHnN6sDDl/1rJRO5FUIDDMhAukJl1YjBxh3xTjCvMN474/3LmqWOE3TNKd2ZDXMGQQnt8KN/4Uxi8A70NFRVeu97e8R4BnAxI4Ta3R87iZjARefQVee25aeYFTudCKvB1zdXfAP8b7kDm8AXm5eTO8+ne3J29kUb/8VfTRN02zKXAS/PA+f3QY+oTAtCnrf6/BpVmsiOjGazWc2M6XLFHw9ajYhTe7mzXi2a4t7ePi5belJeZj8PfA0uddVqBdFJ/JqBEWYyLjEpvVSo9uOpqlvU979410symKjyDRN0+ws7ajxLPy396DPZJi6FsI6ODqqGlFK8e72dwnzDmN8h/E1OseSn09eTCw+g646b3vpYinOQifyagRF+JCRlI/FcunN4u6u7jzY80EOpB3glxO/2DA6TdM0O9n1NcwZDGlxMHYx3PCWsfBJPbExfiM7UnYwvft0vNy8anROXkwMqqgIn0GDzm1TSpGemHdu9k9noBN5NYIiTZSYLWSl5NeqnFEtR9E2sC0f/PEBZovZRtFpmqbVscIc+P4B+HYKhHeGGZuh003Vn+dELMrCu9vfpZlfM0a3G13j83I3bUY8PDD16X1uW15WEYV5ZqeYY72UTuTVaNTEeI5y9kxOrcpxdXFlZs+ZHM86zrK4C5Zr1zRNcz4Ju2DeENjxOQx+Cu5dDoHNHB3VRVt1fBUH0w/yYI8HcXep+XPt3N82Y+rTBxfvP1sezsYbuaBRY+dY9AV0Iq9WUKQPCJyNr91zcoChzYbSLaQbH+74kMKSQhtEp2maVgeUgq2zYcEwKMqFe36Ea58DV5usfG1XxZZi3v/jfdoFtWNUq1E1Py8xkcLDR85rVoc/c0FwE920Xm+4e7gSEOJNWi1r5AAiwsxeM0nKS+LLA1/aIDpN0zQbyz0LS+6AlU9Dm2uNpvRWVzs6qkv2w5EfOJl9kod7PIyL1Dzl5W7+DQCfq87v6JZ2JgeTvwfevs4zd7xO5DUQ3NjHJjVygP6R/RkYOZD5u+eTXZRtkzI1TdNs4thGY2x43BoY+Trc8QX4NHJ0VJcs35zP7B2z6R7anaHNhl7UubmbN+MaGoJn+3bnbT8bn0twY+epjYNO5Cy6mGcAACAASURBVDXSqIkvmcl5mIsvfs71ijzS+xEyCjNYtHeRTcrTNE2rlRIzrH0VPrkRPHxgymoYMKNejA2vyuf7Pyc5P5lHez96bnrVmlAlJeT+9hu+Vw467zyLRZGekHuu75Sz0Im8Bho18UWp2s3wVlanRp0Y1XIUi/ctJiUvxSZlapqmXZLM0/DJDbDhDegxAaath8jujo6q1jILM/lo90cMaTqE3uG9qz+hjIJ9+ynJyLigWT0rJR9zsYVGTvR8HHQir5HSZpTa9lwv66GeD1FcUszcXXNtVqamadpF2f8jzB4Eibvh1vlwy4fg6Vy1zUu1YPcCcopzmNlr5kWfm7t5MwA+Vw48b3tpDgh2oh7roBN5jQSGeePiJqTZ6Dk5QHP/5tze/naWHlrK8czjNitX0zStWsX5sPxx+PIuCG5lLHbSbayjo7KZhJwEPt//OTe2uZH2Qe0v+vzcTZvw6tQJt0bn9w9IO5MLAsGRukZe77i4uhAU4WPTGjnA9O7T8XD14L0/3rNpuZqmaZVKPgDzh0H0ArjyYZj0CwS3dnRUNvXhzg8BeKjHQxd9bklOLnk7dlww7AyMMeT+Id64e7rWOkZb0om8hho18TG+jdlQiHcI93S+h19O/MKe1D02LVvTNO08SkHsIpg3FHKS4M5vYPgr4OY8w6hs4XD6YZbFLWN8h/FE+kZe9Pl527aC2VxhIk87k0sjJ+uxDjqR11ijxr7kpBdSkFts03Lv7XwvwV7BvBX7ll7mVNO0upGfAV/fCz/+DZr3h/s3Q7vrHB1Vnfjv9v9icjMxtevUSzo/J2o9Lj4+mHr3Om+7ubiEjOR8p+uxDjqR11hphzdb18p93H2Y3m060YnRbIzfaNOyNU3TOBUNc6+GAz/BdS/CXd+BX4Sjo6oT0YnRrD+9nsldJxPodfFroyulyNmwAZ9BgxD386dyTU/IQ1mU040hB53Ia6z0W5gtZngrb8wVY2jh34K3Yt7SC6pommYbFgtsfMtYdhTgvpVw1aPg0jA/9i3KwpsxbxLhE8FdHe+6pDIKDx7EnJSE75AhF+wr/ezXNfJ6zDfIEw9vN5vN8FaWu4s7j/R6hLjMOL4/8r3Ny9c07TKTnQifjYY1/zRWKpu+EZr1dXRUdernYz+z7+w+ZvacWeNlSsvLiVoPgO/gC6ekPRufi4ubEBDmfEu36kReQyJCo8a277lealjzYfQM68n7f7xPXrFtJp7RNO0ydHi1MTb85Da46T24/WPwvvhm5vqksKSQd7e/S8fgjlzf+vpLLidnwwa8OnfGLTT0gn1nz+QSFOGDq6vzpU2bRCQiI0XkoIgcEZGnK9h/r4ikiMgO62tKmX33iMhh6+seW8RTV4IbGz3X66JTmojwWO/HOFtwVk/dqmnaxTMXwS/Pw/9uA99wmBYFve6u99Os1sSS/Us4k3uGx/o8dlELo5RlTk8nf8eOCpvVwWhad8Ye62CDRC4irsAHwCigE3CHiHSq4NAvlVI9rK8F1nODgX8A/YF+wD9EJKi2MdWVRk18Kcwzk5tRVCfl9wjrwV9a/IVFexfpqVs1Tau5s3GwcDj89h70nQJT10BYB0dHZReZhZnM2z2Pq5pcxYDIAZdcTu6mzWCx4Dtk8AX7CvOKyUkvdMrn42CbGnk/4IhS6qhSqgj4Ari5hueOAH5VSqUppdKBX4GRNoipTtTFVK3lPdLrEYotxXyw44M6u4amaQ1HWFIUzB0Macdg3Gdw/X/A3fme49aVubvmklucy2O9H6tVOTkbNuAaHIxX164X7DtrHa3kjD3WwTaJvAlwqszPp63byrtNRHaJyFIRaXaR5zqFRtb5dc/G110ib+7fnPFXjOe7I99xKP1QnV1H07R6rjAHvrufTvvfhoiuMGMTdLzR0VHZ1YmsEyw5sIRb2t5Cu6B21Z9QCVVSQu7GjfhefRVSQa/+tHjn7bEO4Gan6/wILFFKFYrIdOAT4NqLKUBEpgHTAMLDw4mKirJZcDk5OTUuz80L9m2PI8vjmM2uX16Xki54iRfP/vIsD4Y9eFHL713MvTi7hnIvDeU+QN+Ls/DNjqPTvjfxzk/gcORoElpORO2IA+IcHVqtXOx7Mj95Pq7Kld75vWv1XrofPUpwRgYnQkM5VEE5Z2IsuLhDzM4tNf48tue/L1sk8nigWZmfm1q3naOUOlvmxwXAG2XOHVru3KiKLqKUmgfMA+jTp48aOnRoRYddkqioKGpaXtauHeRnFzF0aD+bXb8i6fvSeT36dVzbuTK46YXPbCpzMffi7BrKvTSU+wB9Lw6nFGybAxtfAFMI3PsTZ46b6999VOJi3pPoxGh2ndjFzJ4zuanbTbW6bvKOHZx1daXf1Km4BgRcsP/bmFg8m8E119R8OVR7/vuyRdN6NNBORFqJiAcwHlhW9gARKTvh7U3AfuvfVwHDRSTI2sltuHWb02rU2If0hDwslrqdTnVch3G09G/JmzFvUmyx7bSwmqbVQ7lnYcl4WPk0tBlmNKW3vKr68xogi7IwK3oWkT6RTOw0sdbl5azfgHfPHhUmcaWUMce6k61BXlatE7lSygw8hJGA9wNfKaX2ishLIlL6NWmmiOwVkZ3ATOBe67lpwMsYXwaigZes25xWcGNfSswWMpPrdqy3u4s7j/V+jGOZx/j64Nd1ei1N05zcsY0wZxDErYVRb8AdS8CnUfXnNVDL4paxP20/j/R65JInfylVnJRE4f79+A6ueNhZbkYRhXlmp1uDvCybPCNXSq0AVpTb9kKZvz8DPFPJuQuBhbaIwx5CmhlvZuqpHIIi6vYb2tBmQ+kf0Z/ZO2dzfevrCfC88NuipmkNWIkZ1r8GG96ERm1hwlcQ2c3RUTlUXnEe725/l24h3RjValStyzs3m1sl48dTT2UDf372OyPnm6LGyQVH+uDiJqSczK7za4kIT/R9whgnuWtenV9P0zQnknEKFl0PG2ZBjwnGBC+XeRIH+Hjvx6Tkp/Bk3ycvqiNwZbLXrsG9aVM821fc6z3lVDYIhDR13kRur17rDYarmwuNGvsab64ddAjuwOh2o/n8wOeMaT+GlgEt7XJdZ2UpKqIkLQ3z2bOUpKWhzGbE1RVcXBE3V1z9/XGLiMA1KMgm/8k1zSH2LYNlDxkLn9y6ALqNcXRETiEhJ4FFexYxsuVIeoT1qHV5ltxc8rZsJeiO8ZV+XqSczCYwzISHl/OmS+eNzImFNvMlbkcKSim7JIuHez7MquOrmBUziw+GXR4TxSilcE1KJvOn5RTs20fB/n0UHjhISXp6jc4XDw/cwsPxaNkSr86d8OrcGe8uXXCLiNAJXnNexfmw6jmI+Qga94TbF0Jwa0dH5TTein0Lhar15C+lcjZtRhUV4XvtsEqPSTmVTWRr536sqRP5JQhp5se+zQnkpBfiF1y7jhY1up53CDO6zeA/sf9h4+mNXN30wpV5GgJVXExebCzZa9eSsy6KkFOnOAOIuzue7dvjO+xaPJo0wTW4EW4hjXANCjbWDC4xoywWVLGZkswMzIlJFCclYk5IpDAujrO//QYlJQC4hYfjO/hqfAYPxmfglbj6Om9PVO0yk3wAlk6C5L1w5cNw7Qvg5uHoqJxGbFIsK4+v5P7u9xPpG1n9CTWQs3YNrgEBmHr3qnB/QU4xOWmFhAzxs8n16opO5JcgtLnxpqaczLZLIge4s+OdLD28lDei32BA4wG4u7hXf1I9URgXR/qSL8hctgxLVhbi6YnPgAGcvfoquo8bh2fr1kbCvkSWggIKDxwgf89e8n7/nawVP5Px9VJwd8enb18CbrkZv+uuw8VksuFdaVoNKQXbP4GfnwYPH7jzG2h3naOjciollhJe//11InwiuK/LfTYpU5nN5EStx3foEMSt4lRY+gi19DPfWelEfgkaNfVFxHiTW/e4cLm7uuDu6s6TfZ7kobUPsWT/Eu7ufLddrltXVEkJ2avXkL5kCXlbtyLu7viNGIH/qJH4DByIi8lEXFQUXldcUetruXh54d2jB949ehB8151Gzf+PP8hZv57sVb9w5qm/42Iy4TdyJIG3jsa7d2/d/K7ZR34G/Pg32Pc9tB4Ko+eBX7ijo3I63x35jv1p+5k1eBbebraZRz4vdjslmZlVN6tbOzWHNtOJvMFx93AlMMKHVDv0XC9rcNPBDGoyiDk753B96+tp5F3/xpEqpchZt46Ut9+m8PAR3BpHEvroowTefhtujexzP+Lujk+/fvj060fY44+THxtLxvffk71yFZnffotXp04E33cf/iNH1KolQNOqdCraaErPPgPXvQhX/g0qmOf7cpdVlMV7f7xHr7BejGg5wmblZq9ZjXh44HvVoEqPST2VjW+wJ16+zv05oP/VXKLQ5r6knKq7xVMqIiI81ecp8s35vPfHe3a9ti3kxcZy4s67OP3Ag6iiYpq8/RZtf/2VkOnT7JbEyxMXF0x9+9L41Vdpt2kjES/9E0tBAWeefJIjfxnO2Y8WYsmr28l/tMuMxQIb/wMLR4AAk1bBVY/qJF6JuTvnkl6QztP9nrZZS5lSipw1a43WP5/K+8mknMpx+to46ER+yUKb+ZGbUUheVt2sTV6Z1oGtGd9hPN8e/pa9Z/fa9dqXypyeTvwTT3LizrsoPn2aiH/+k9Y//Yj/qFHG0DEn4eLtTdDYsbT+6UeazpmNR4sWJM+axZHhI0j77H9Yiuz7XmsNUHYiLL4F1rwEnW42pllt2sfRUTmtuIw4Pt//Obe2u5WOjTrarNzCQ4cojo/Hd1jla3cVFZjJSM5z+ufjoBP5JSv9lpZqp/HkZd3f436CvIL417Z/YVEWu1+/ppRSZK1cydHrbyBr1SpCHnqINqtWEjRurFM3WYuLC35Dh9Lik0W0+Px/eLZsSdIrr3B05Cgyvv0OZe0Br2kX5fCvMHsQnPodbnrPGFrm5dzDmhxJKcW/f/833u7ezOw106ZlZ69ZAyL4XXNNpcekns4B5fzPx0En8ktWOl2fvSaGKcvfw59Hez/KrpRdLItbVv0JDmBOTSV+5t+If+RR3Bs3ptU3Swl96EFcvG3TUcVeTL160XzxpzRbsADXoCASnn2W42PGkvfHH44OTasvzEXG2PD/3Q6+4TB9PfS6G3SHyir9euJXtiVs4+GeDxPsFWzTsnPWrMW7WzfcQivvrPzn1Kw6kTdYniZ3/EO8SDlp3+fkpW5qcxPdQ7vzduzbZBVlOSSGyuRt/4Njt95Gzvr1hD3xOC2/WIJX+/aODuuSiQi+Vw2i5dKvafzmm5hTUzlxxwTOPPMs5tRUR4enObOzcfDRX2DL+9B3KkxdC6G1H4nR0OUV5zErZhZXBF3BmPa2ndWuODGRgr178R1WeW91MHqse/u54xPo/GP5da/1Wght5ueQGjmAi7jwbP9nGf/TeGbvmM3f+/3dIXGUpZQi/fPPSXrtddwjI2n59Vc2GT7mLESEgBuux++aoaTOmcPZRZ+Q/euvhD3xOIFjxyINpLNSYb6ZrJR8cjIKyU0vICe9kKJC43FCwmkLG5IO4e7him+Qp/XlhX+IF54m531c4hC7voKfHgUXNxj3P+h4g6MjqjcW7F5AYm4ir1/9Om4utk1T2WvWAOBXxfNx+LOjW30YiqoTeS2ENPcj7o8UCvPNeHrb/1fZqVEnxrQfw5IDSxjdbjTtgxxX67Xk55P44otk/rAM36FDafz6axWu7dsQuPj4EPb44wTceiuJL71E4ov/JOvnlUS+8jIezZo5OryLlp6YS8KRTBKPZpJ4LIv0xFxQf+53cRHcvYxOiWYz5MYnUlxYgqVEnVdOYLiJiFb+hLcOILJtAMGRPvXiQ9DmCnNgxROwcwk0vxJumw8BTR0dVb1xMuski/Yu4obWN9ArvOIZ12oje9UveLRpg2ebNpUeU1JsIf1MLi261I8hvjqR10LZDm9N2gc5JIaHez7MqhOr+Pe2f7NwhGNWgzWnp3Nq2nQK9uwh5OGHCLn//gZTO62KZ6tWNF+4kIylS0l+/Q2O3nQzYY8+StBddzr1/SulSDuTy5HtycTFJpOeaAyv8/JxJ7y1P+37hhEU6YNvkBe+QZ6Y/DwQFyMhR0VFMXToYJRFkZ9TTI61xp6WkEvSsSxO7D3Lga2JAASEedO2VxhteoUR0sz38kjqZ3YYY8PTj8GQv8Pgp8BVf8zWlFKK16Nfx93F3WbzqZdlTk0lLyaGkBnTqzzu7JkcLBZVLzq6gU7ktVI6LCH1VI7DEnmgVyAze87k5a0vs/zYcnyx71J7xWfOcHLKVIrj42n6/nv4VfPcqaEREYLGjMH3qqtIePFFkv71L7LXrqXx66/hHu5cM3QVFZg5sCWRPetPk56Yhwg0bhdIlyFNad4pmIAw7xonW3ERTP4emPw9CGvBuRkOlVJkpRZwan8acduT2f7LSWJXniAg1JsuQ5rQcVBjh7Re1TmlYOts+PUF8AmFe36Ellc5Oqp6Z3f+bjakbOCJPk8QarL9rJnZq9eAxYLfiJFVHnduRrfmzrt0aVkN8H+U/Zj8PTAFeNhlbfKq3NbuNr4/8j1vRr/JU6FP2e26hXFxnJw8BUtODs0XzMfUt6/dru1s3CMjaTZnDpnffkviq//i2E03E/HKy/j/5S+ODo2s1Hx2R51m3+YEivLNhLX0Z8iEK2jdIxSTv2078ogIAaHeBIQ2ocvgJuTnFHFsRyoHtiSweekRfv/xGB2ujKTb0KYEhjeQue1zU+H7B+DwKmg/Cm75EEy27WV9OcgrzmNp2lLaBrZlQscJdXKN7F9W4dGyZaVrj5dKPZWDh5cr/o3qxygbnchrKbS54zq8lXJ1ceW5Ac8xYfkEfsr4ib/y1zq/Zv6uXZyaNh3c3Gjx2WK8OnSo82s6OxEh8LbbMPXuTfwTTxL/8Exyx46FQVc6JJ7cjEJ+X36M/ZsTAGjTK5Tu1zYjwo5LMnr7etDpqsZ0uqoxySey2LX2NHs3xLM76jQd+kfQ76bWdlt4qE4c2wDfTIX8NBj1BvSbpoeVXaK5u+aSXpLOOwPeqZNFoczp6eRu+51GU6ZU2/KUciqbkGZ+5x4pOTudyGsptJkfJ/emYS4qwc3DcbOUdW7UmXFXjOOLA1+w9+xeOjfqXGfXyt+zl5OTJuMaFETzjxbg0bx5nV2rPvJo2ZKWn/+PlPfe4+yCjwjetInC9u3xbNXKLtcvzDfzx6oT7FxzCotF0WVwE3oOb+7whBnWwp/r7uvEwFvbsOPXk+yOiudwTDJdr2lK75Et8PKpR73eS8wQ9W9jqtVGbeHOryGym6OjqrfiMuL4dO+nDPAZUCcd3ACyV6+GkhL8Rwyv8jhLiYXU0zl0ubpJncRRF5y3R049EdrMD2VRpMY7Zjx5WQ/3fBhfF19e2fIKJZa6mX2s8PBhTk2Zgqu/Py0+/UQn8UqIhwdhjz9Os/nzcc3M5PjtY8j65Zc6vaZSiv2/neGz57cQu/IErXqEMuHF/gwe397hSbwsnwBPBt3ejjtfGkC7vmHsWH2Sxc9vYc/60yiLqr4AR8s4CYv+ChvfhJ53GhO86CR+yZRSvLL1FUzuJm4KuqnOrpO96hfcmzfHs2PVU72mJ+VRUmwhpJ48HwedyGsttIXR4S35uGOb1wH8PPy4NfhW9pzdwzeHv7F5+UUnT3Jy0mRwd6P5xwtxj4y0+TUaGt+rBnH2uWfxaNOG+Jl/I+mNWSiz2ebXyUzJY9l/d7D20wMERZoY+2xfhk/uTECo8z6H9gv2Ytg9nRj/fD9Cm/uxfskhvntruzH8zVnt+wHmXAVJ++C2j+DmD4w1xLVL9tPRn4hJiuHR3o/i51o3vcRLMjLI3boV/xHDq21WTz5uTLAV1sK/TmKpCzqR15JvkCemAA+SjmU6OhQAept60y+iH+9sf4fUfNvNOlacmMjJe+9DFRXR/KOP8GjRwmZlN3SW4GBafLaYoAkTSFu4kJOTJmNOT7dN2SUWtv9ygi9e+p2k41kMmXAFox/rVS8WeijVqIkvNz/Sg2vv7kDamVy+eOV3YlYco6TEidYRKM43Jnf56m4IbgMzNkDX2x0dVb2XWZjJmzFv0i2kG7e2u7XOrpO9dh2YzfgNr34Z1MRjWXh4uxFUjzpj6kReSyJCRKsAko45xzSpIsJzA54j35zPrOhZNimzJCuLk5OnUJKZSbMFC+r1dKuO4uLhQcQL/0fj118jf8cOjo8dR+Hhw7UqMye9gO/f/oMt38bRtGMwE/4xgC6Dm9SbDjpliQgdr2zMHf/oT+vuoWxbdoxvZ20nKzXf0aFB8n6Yfy3ELIQrZxrLjga3dnRUDcI7298hszCT5wc8j4vUXTrKWrUS9yZN8OpSfd+hpGNZhLesPx3dQCdymwhv5U9mSj75Oc6xzGXrgNZM6TqFFcdWsDl+c63KUmYz8Y8+RtGJEzT98EO8u3axUZSXp4Cbb6bF4k+xFORzfPwdZK9bd0nlHN+VypevRJNyKofr7u3IX+/vim+Qp42jtT+fAE9GTO3CiKldyEjM5ctXo4nbnuyYYJSCmI9h3jWQmwJ3fQPDXwY35597uz7YnrSdpYeWclfHu2y6RGl5JVlZ5P62Bb8RI6ptVi8uLCEtPofwVvVrVkqdyG0gvKXxLMVZauUAU7pOoaV/S17e+jL55kuv1SS99jq5mzcT+eI/8Onfz4YRXr68u3en1ddf49GyJacfeJCzH32EUjXr5FVitrBp6WGWf7gL32BPxj3blysGRDa4WdPa9g5j3PP9CAw3sXLeHtZ/fhBzsR2Xj83PgK/vgZ8egeYDYMZmaHud/a7fwBWXFPPSlpeI9InkgR4P1Om1ctatg+LianurAySfyEIpo3JWn+hEbgOhLfwQgaTjzpPIPV09eWHgC8TnxDN359xLKiN9yRLSP/uM4HvvJfB2/TzQltwjImjx2WL8Ro4gedabJP7zn9V2gsvPLuKHd/5g5+pTdB3alNue6t1wJlWpgH+IN7c+0Ysef2nOng3xfPfmdnIzCuv+wie3wZyr4cByuO6fcNe34Odcs/TVdx/v/Zi4zDieH/A8Jve6/TecuWIFbo0j8eratdpjSytjpZWz+kInchvw8HIjuLGvU9XIAfpG9OXmNjfzyd5POJR+6KLOzf3tNxJfeRXfIUMIe/KJOorw8ubi7U2T//yHRlOnkPHFl5x+8CEsuRX32D4bn8PXr8WQfCKb4ZM7M3h8e9zcHTdvgb24urkw6La2jJrRlbTEPOvvoI7+n1lKYMOb8PEoY1KXSavgqkfAiefNr49OZp1k7s65DG8xnMFNB9fptcxpaeRu2kzA9dfXaP2DpONZ+Id44e1Xvx6f6H+hNhLeyp/k41lONw72iT5P4Ofhx0tbXsKiatYLuDg+ntOPPoZn69Y0/s+biGvDTxiOIi4uhD3+OBEvvkjOxo2cmHg3xcnnPxM+viuVb96IpcRsYfTjvWjX9/KrHbbuEcptT/ZCXOC7N7dzJNbGz82zEmDxLbD2Zeh8C8zYCE372PYaGkopXtr6Eh6uHnZZejl71SpjEpgbaraEbNLRzHr3fBxslMhFZKSIHBSRIyLydAX7HxORfSKyS0TWiEiLMvtKRGSH9bXMFvE4QngrfwrzzGQk5zk6lPMEegXyZN8n2Zmyky8Pflnt8aq4mPjHHoeSEpp+8D6uvvVnUoT6LGj8OJp++AGFx49zYvwdFB47BsDONadYPnsXgeEmxjzdt941+dlSSFM/xjzdl5Bmfqyav4eYFcdr3LegSodWwZxBcDoGbnrfGB/uVf8+zOuDZXHL2Jawjb/1+hthprA6v17m8uV4tmuLZw1G2uSkF5CbWVTvno+DDRK5iLgCHwCjgE7AHSLSqdxhfwB9lFLdgKXAG2X25SulelhfdTetTx0rffOdrXkd4IbWN3Bl4yt5J/YdEnISqjw2+e13yN+5k8hXXtGzttmZ39ChtPj0UywFBRy/8y42zt/Kpq8P07pHKKOf6NUgeqXXlsnfg1se7Un7/uFsW3aUTV8dvuRWMLEUw8pn4fOx4BcJ09ZDr4l6rvQ6kpqfyhvRb9AzrCdjrxhb59crPnOG/JhY/K+/oUadQUs/uyMu0xp5P+CIUuqoUqoI+AK4uewBSql1SqnSqupWoKkNrutUgiJ8cPdydcpELiK8MPAFFIqXt75caS0me9060hYuJGjCHfiPrH7iBM32vLt0ptnixRxofiu7YvNo396NEVO74O7Aefydjau7C9fd04nuw5qxa91pVi/ad/GTx5yNo9f2v8PWD4yFTqasgVA9P0Jdeu3318g35/PilS/W6ZjxUlkrVgDgf8P1NTo+8VgWLm5CSNP61wppi99mE+BUmZ9PW7dVZjLwc5mfvUQkRkS2isgtNojHIVxchPCW/iQ6yQxv5TXxbcLMnjPZGL+RFcdWXLC/OCGBhKefwbNjR8L+XvfPrrSKmYtLiFqdy+nAnrTJ2kbTjx8ip47naK+PxEUYdHtbBtzSmkO/J7Hiw90UF9VweNrOL2DuYLwKkmH85/DXWeDuPHPRN0TrTq5j1fFVzOg+g9YB9plMJ/On5Xj36IFH05rVG5OOZRLazA9X9/rXdUxq+4xJRG4HRiqlplh/ngj0V0o9VMGxdwEPAUOUUoXWbU2UUvEi0hpYCwxTSsVVcO40YBpAeHh47y+++KJWcZeVk5ODrw2eBSftspC6HzreJri4OaZ5rqp7sSgLbye+Tao5lWcbP/vnvMYlJQT95y3c4uNJe/ZZSsLr/tlVTdjqfXG0mt6Hxaw4uVGRmwQRPYWQpnkEfjgb96NHyZp4FwVXOmY51LKc8T1Ji1MkxChMjaD5EMHVveL/e67mPNodnktEUhQZAZ2IaTEDt+D6P9WwM74nZeVb8nn1zKv4uPjwVORTuErlrUu2uhfXM2cIeellssaNI/+aodUeryyK/d8ogtpAZC/bJHJbvy/XXHNNYNa9oAAAIABJREFUrFKq4h6YSqlavYCBwKoyPz8DPFPBcdcB+4GwKspaBNxe3TV79+6tbGndunU2KefojmT1/vQ1Kv5wuk3KuxTV3cvhtMOqx6c91FPrnzq3LWXuPLXvig4qY9myOo7u4tjqfXG0mtxHcaFZ/fDOdvX+jDVq3+b4c9tL8vLUiUmT1b4rOqiziz+rwyhrxlnfk0PRieqD+9eqb2bFqML84gsPiN+u1H97KPVioFJr/6WUudhp7+ViOft9vPjbi6rbJ93UnpQ91R5rq3tJevttta9jJ1WcklKj45NPZKn3p69RB39PsMn1lbL9+wLEqEpyoi2+ekQD7USklYh4AOOB83qfi0hPYC5wk1Iqucz2IBHxtP49BBgE7LNBTA5ROmwh6ajzPScv1TaoLVO7TmXFsRVsOL2BgkOHSH3vPfxGjKjxEA3NtsxFJayYvYtTB9K5dmJHOl7Z+Nw+F29vms7+EN9hw0h65RVS5813YKTOq12fcP6fvfMOj6Jq+/A9u+m99xB6gID0Ir13AZGigKJ0ERFeBUEUFaSICKJI702aKEiRHgi9BUgInYT03vtm93x/BPwQKSnbEva+Li6WnZnn/IbdnWfmnKd0GlaL2Afp7F18jfzcR8V1hICzv8KqTqDIhaF/QbupIDfSreBXhIuxF9l5Zyfv1nwXP6eX1zlXB0II0vftx/L11zFycirSMY+LeblWLHuBbqCGNXIhRAGF0+UHKXzi3i6EuCFJ0gxJkh5Hof8AWAE7nkozqwlckiTpGnAcmCuEKLOO3MLGBGtHM+LC9HOd/DEj64ykql1VZgZ8Q+SUz5FZW+P29fRyV+azLPBfJ/7f1rAyExO8flqITY8eJCxYQPxPP6kn7aqc8R9nnhwHWwbCwS+gWif48DRUbKlrma8M2Ypsvjr9Fd7W3nxU/yOtjZt7/TqKiIhiPZjEhaZhbm2MjVPZjJVQy22pEGI/sP+p96Y/8fqZRYqFEGeAl9fNK0O4VrIh9r5+O3JjuTHftfiOHdMGogiJwXPRIowcHHQt65VDWaDiwLKgR068xjOd+GMkY2M85n2PZG5G0rLlUFCA86efGm6+nqJao8JiOYdX32DfjIu84XAao24/QJORhrQyLbPoyiKiM6NZ23Ut5kbmWhs37a+9SCYmWHcqem38wo5nNmX291T2wvP0HLdKtmSm5GmnJnQpqJJkRP9TgtM1Ja75ae9HZqAQlUpwZG0I4SHJtBtc41/T6c9DkstxnzEDu3feJmnVauK/n2d4Mn8aZQHV0lbQ0WYB0blVOWi1DWWjEQYnrmUuxl5ky60tDKo5iIauDbU2rsjPJ33vXqw6tC9yMau8bAUpsdllsqLbYwyOXM08Lgyjr2loUFi9LXrqVIzt7DnWrwrfnP2GjPwMXct6ZRBCcOK329y7HE/zvlWp1fLlTvwxkkyG2/Tp2A8ZQvK6dcTNmWNw5o9JeVhYJz1gPtWb+9BmQGXC7uRzbMNNvSudXJ7JVmTz9Zmv8bLyYnz98VodO8PfH2VqKnZ9+xb5mPiwwmtfWazo9hhDxIeacfa2Rm4kI+ZeGlXq60ca19Mkb9xEXshNPH/5mS/quzN4/2DmX5rPt82/1bW0V4Jzux8QEhBNg64+1O9c/Op5kiThOu0LJLmM5PUbQKnC9ctpZXZaUC3c+BP2jAehKiyxWqcftYHcPBnndz/A1MKYVgOqvdr/R1ril8BfiMiIYE2XNRrvbPY0aX/8iZGLC5bFSNWMvpeKJBkcuYEnkBvLcK1kQ/TdVF1LeSaK2FgSFi/Gql07bDp1ojbwgd8HrA5eTSefTrT0NAQDaZLAw+Fc+fshfq08aNa75IUxJEnCZcoUkBuRvGYNPHbur5qjys+Gg1Ph8jrwbFjoxB0q/bO5YVcf8rIUXD0SgZmlMU16Vnq+LQOl5nLcZTbf3Mw7Nd6hsVtjrY5dkJhI5smTOA77oFiNnqLvpuJcwRoTs7LrDg1T6xrAo7odiREZ5OW8uL+0LoibMxdUKlynffHPex/W+5AqtlX4+vTXpOXp75JAWefuxTjO/H6Pqg1daP2Ob6mdriRJuEz6DIcPPiBl0ybi5859tabZ40JgZftCJ97ik8K2ow7/dtSSJNH8rarUaO7Oxb2hhJyO1o3WV4BsRTZfnvoSTytPJjSYoPXx0/7aC0oltn2KXiC0QKEkLjQdj2p2GlSmeQyOXAN4VLNDCPQuej0z4BQZBw/iNGbMv8oWmspNmdVqFsm5ycy5MEeHCssv0XdTOLI+BI9qdnR8vxYymXqenCVJwmXyJOzfe5fk9RuI/2F++XfmQsClNbCyHWQnwZBd0GkGyI2fubskSbQd7EuFWg74b75N+I0kLQt+NZh/aT5RmVF81/I7rU+pCyFI27ULs7qvYVqlSpGPiw/LQFmgMjhyA//FrZItMpmkV9Prqrw8Yr+biUnFijgM++A/2/0c/RhVdxT7HuzjUJihtrc6yUsX7F8ahK2TOd3G1FF7LWdJknCdOhX7QYNIXrOGhAULy68zz0mB7e/B3ong07wwN7xqh5ceJpfL6DKqNg4elvy9IpiECENwpzoJiAxgx50dvO/3vlaj1B+TGxJC3t272L35ZrGOe3yNdq9qcOQGnsLYVI5LRWui76boWso/JK1aheJhOG7Tv0JmYvLMfUbUGYGfox8zz80kMSdRywrLJ9np+Tw8IZDJJXqOq4uZ5bOfGkuLJEm4fvUldm8PJGnlShIX/6qRcXRK+HlY1gpu7y98Ah/8O1gVPaDUxMyInh/VxdTCiH2Lr6HIKqc3O1omLS+Nr898TVW7qlot/PIvDX/8iWRigk23bsU6LvpuCo6elhr7XWoLgyPXEB7V7IgPyyh6RyYNkh8RQdLyFdh07/bCaE5jmTGzW84mW5HNt2e+Lb9PdVqiIF/Jvl+vUZALPT6qi42TZvP1JUnCbfp0bPv2JfHXX0lcWU7KuaqUcHJ+YWqZTA7DDhWuicuKf/mysjel57i6KPKUPDwp/r+Uq4ESM+v8LFJyU5jdcjamclOtj6/Kzyf9r7+w7tgBuW3Rc8GVShUxD9LxqGavQXXaweDINYR7VTtUKkHcA92vk8cvWAByeZHak1a2q8yEhhPwj/Tnz3t/akFd+UQIwbENN4kPz8Dr9cIWt9pAkslwnzkDm+7dSfhxAckbN2llXI2RHgMb+8CxmeDXB0afBK/STd06elrRdVQd8tLh8JoQVIYc8xLzd9jfHAg9wJi6Y6jpWFMnGjKP+6NMS8O2mNPqieGZFOQpy/z6OBgcucZwr2qHJEGUjtfJswMDyTjwN47DhmHs6lqkYwbXHEwTtybMvTCXiPSIlx9g4D9cPhDG3UvxvN6nCjZe2k0Jk+RyPL6fi1XHDsTNmkXKjh1aHV9t3DkIy1pA5CXotbgwtcxMPdW3vGs54FZfIux6Iud3/6drsoEiEJsVy8yzM6njVIfhdYbrTEfarl0YOTsXK3ccIOrR0qd71bJb0e0xBkeuIUzNjXDytiZGh45cCEH8vB+QOzvh+IwAt+chk2TMajkLuUzOlFNTKFAZph+Lw/3AeM7vCaV6U9cSFXxRB5KxMZ4LFmDZqhWx078uTM0pKxTkwd9fwJYBYO0Oo/yhwbtqL7PqUA38Wnty5WA4t8/FqNV2eUclVHx56ksUKgVzW83FSKabHGxFdDSZAQHYvtW3WLnjADF3U7FztcDSVvvLAerG4Mg1iEdVO2JD01EqVDoZP+PQYXICA3EePx6ZpWWxjnWzdGN6s+lcT7jOyuvlZK1VCyREZHBkbQiulWxoN6SGTgu0yExM8PrlZywaNyZ6yhQyjh7VmZYik3QfVneCc79C45Ew4ig4+2pkKEmSaDWwGp7V7Ti26RaxerAMVlbYGLKR87HnmdJkChVsdHOzCpC6cycIgX3//sU6TqUSRN9LKxfT6mBw5BrFo5odSoWK+Ifa708u8vOJ//FHTKtVK1bd4SfpWqkrPSv3ZPn15VxLuKZmheWP7PR89i+5jpmlMd3G1MHIuHhPCJpAZmaG15IlmPn5ETVhIllnzuha0vO5thWWt4bUcHh7C/SYD8aabSspl8voOqoOVnam7F8WRGZKrkbHKw/cTr7NoiuLaO/dnjerFm9dWp2IggJSd/6OZauWGHt6FuvY5OhM8nMKDI7cwMtxr1a49qKLdfKUrVtRhIfjMnlSsaecnuSLpl/gauHK1ICpZCmy1KiwfKFSqji0KpicTAXdxtTRq+k6uZUlFVYsx6RiRSI+Gkd2YKCuJf2bvAzYNQr+GA3udWHMaajRQ2vDm1kZ02NsXQrylPy9IlhnM2hlgdyCXKYETMHW1JZvmn+j0xmnTH9/CuLjsR84sNjHRt0pvCYbHLmBl2JuZYKDh6XW18mVaWkk/roEy+bNsWxZutrp1ibWzG41m6jMKOacN1R9ex5ndt0n6k4q7Qb74uKjf80X5HZ2eK9ehZGLMxGjRpN786auJRUSHVj4FB60A9p+AUP/AtviPV2pAwcPSzq8X5O40HRObruj9fHLCgsvL+Re6j2+a/Ed9ma6TdtK2boNI1dXrNq0KfaxMXdTsXYww9pBszM+2sLgyDWMR1U7Yu6noVJq7y4/ac1alOnphU/jarhjbujakBF1RrD7/m72P9ivBoXlizsXYrl2NII67bzwbeauaznPxdjFBZ81a5BZWhI+chT5YWG6EyMEnP0VVnUqDG4buhfafl6YJ64jqtR3oUFXH0JORXMjIEpnOvQV/wh/ttzawpCaQ2jh2UKnWvIjI8k6fRq7fv2QjIoXaCeEIPpearl5GgeDI9c4HtXtUOQpSYjI1Mp4UmYmyRs3YtOtG2Y1aqjN7od1P6Secz1mnptJRIYhJe0xiZEZHN94C/eqtrToV1XXcl6KsacnFdasBqWS8GHDUcTFaV9EVmJhRPrBL6BaZxhzCirq1jE8pmmvylSo5cDJbXeIDTUEvz0mLiuOr05/RU2HmkxsOFHXckjdvgMkCbv+/Yp9bEpsNjkZCjyqGxy5gSLy+K4v6o52yrVaHjyEyM3FaZx6SyUayYz4vvX3SEhMOTkFhUqhVvtlkdwsBQeWBWFqYUSXkbWRy8vGz8m0cmW8V65EmZZG+LDhFKRosZTwgxOwtEXh393nw9ubwcJBe+O/BJlMotNwP6zsTPl7eTDZ6fm6lqRzlColU09NJU+Zx7zW8zCRP7vEs7YQCgWpu3Zh1aYNxm5uxT7+cX11jzJeX/1JysaVpwxjaWuKvbslkTeTNT5WQWIiFv7+2PTsgWnlkve6fh4eVh580/wbride59fAcljLuxgIleDouhAyU/LoOlq/gtuKgnltP7yWLEEREUHEqNEoMzUcyKhUwNEZsKE3mNnAyKPQZKTac8PVweOsg9wsBYdW33jlK7+tClrFxdiLfNH0CyraVtS1HDKOHkOZmIj928UPcgOICEnGysEUWxfNlkzWJgZHrgUq1HQg+m6axuuuJ61cBUolzmPHamyMzhU70696P9YEr+Fs9FmNjaPvXDn0kLCgJFr0q4Zb5bJZGcqyaRM8f1pIbkgIkePGocrX0NNnajis7Q4BP0L9IYUFXtzqaGYsNeHkZU2bd3yJup3Chb8e6FqOzgiMD2TptaV0r9Sd3lV661oOACnbtmLk4V6iQF6lUkXkrWQq1HTQacS9ujE4ci3g7eeAskCl0bamirh4UrZuJbdpU0wqVtTYOACTG0+msm1lpgZMfSW7pEXdTuH87gdUbeRCnbbaj7BWJ9bt2+MxexbZ584R/elnCKWabzZv/AlLW0L8zcISq70Xg0nxihPpiprN3anZwp3LBx4SFvTqfc9Tc1OZfHIy7pbufNXsK71wfHl375J99hz2A98uUVptXGg6+blKvGs5akCd7jA4ci3gUc0OuZGMiBDNTa8nrViBUCrJ6l68Nn4lwdzInPlt5pOlyOLzk5+jVOm+w5u2yErL4+DqG9i5Wui8cpu6sO3dG9epU8g4fJjYb9TU9S4/G/76BHYMBaeqMCYA6hQ/MEnXtB5YHSdvK46sDSE9KUfXcrSGSqiYdnoaSTlJzG87HysTK11LAiB54yYkU1PsBhSvkttjIkKSkSTwqlH2O549icGRawFjEznuVW0J15AjV8TEkLp9O3ZvvonS2VkjYzxNVfuqTGs2jQuxF1h2fZlWxtQ1hUVfbqDILaDLqNqYmOmmvrQmcBg6FMfRo0ndsYOEhT+VzlhcCKxsD5fXFbYbHXYQHCqpRae2MTKR03VUbYSAg69QsZj1N9ZzMvIknzX6DD9HP13LAUCZmkranj3Y9noDI/uSOeLwkGRcKtqU+f7jT2Nw5FqiQi1HUmKyyEhWfwnIpFWrEULgNGa02m2/iD5V+9C7Sm+WX1vOmWg9Lv2pJs7/FUr03VTaDq6Bo4d+PKGoE+cJn2A3YABJK1aQtHZd8Q0IARdXw8p2kJ0IQ3ZBpxkgL9sXTVtnCzoMrUn8wwxO77qnazkaJzA+kEVXFtHZpzPv1HhH13L+IWXHDkRuLvZD3i3R8bmZCuIfplOhlv5kSagLgyPXEhX8Cr88EWqOXi9ITib199+x7dWr2PWG1cG0ZtOoYleFqQFTic+O1/r42iL8RhJX/n5IrZYe+DYtfspLWUCSJNy+no51ly7Ef/89abt3F/3gnBTY/h7s+x/4NIcPz0DVDpoTq2Uq13Ombgdvgo5Hcj+w/H7PU3JT+OzEZ/9kqOjL0pEoKCBly29YNGuGmW/1EtmIuJUMAir4la/1cTA4cq3h4GGJha2J2tfJUzZtQuTl4Th8mFrtFpXH6+U5BTlMPjm5XLY8zUzJ4/DaEBw9LWk1oJqu5WgUSS7H44d5WDRrRvQX08jw93/5QeHnYFkruL2/8Al88O9g5aJxrdrm9Ter4FLRhmMbbpGWUP7Wy1VCxdRTU0nJTWF+m/lYm1jrWtI/ZBw5QkFMDA7vlexpHArXx03MjXDx0Z/zUhcGR64lJEmiQk0HIm4mqy0vVZmZRfLmLVh37IBplSpqsVkSqthV4atmX3E57jKLrizSmQ5NoFKqOLzmBgUKFV1G1sbIRPcdzTSNzMQEr8WLMatRg6gJE8m+8pwmKyolnPihMLVMJodhhwrXxGXl87IiN5LRZUThevGhVcEoC8rXevnya8s5HXWaKU2mUMuxlq7l/IvkDRsx9vIqUV11KCzLGh6SjHcNe2RlpHBTcVDLGUmS1FWSpNuSJN2TJGnKM7abSpK07dH285IkVXxi29RH79+WJKmLOvToK95+DuRlF5DwMEMt9lJ37ECVlobjiBFqsVca3qjyBgN9B7LuxjoOPzysazlq4+K+sMJ18XeqY+9WNtKm1IHcyhLvFcsxdnUlYswYcu881UgkPaawuMvx78DvTRgdAF4NdSNWi9g4mdPhvcL18rO77utajtoIiAxg6bWl9KrSi/7VSxYRrilygm+Qc+UK9kMGl7iTY0pMNlmpeXiXw/VxUIMjlyRJDvwKdANqAe9IkvT07dxwIEUIURVYCHz/6NhawNuAH9AVWPLIXrnEu4YDSBAeklRqW6r8fJLXrsWiaVPM69ZVg7rS83njz3nN+TW+PPUlD9LKfhGNiJvJXDoQRo3m7nrdDEVTGDk64r16NTIzMyJGjEQR9aiRyJ2DsKwFRF2G3kvgrVWF1dpeESrXd6ZOOy+uHYvgwdUEXcspNZEZkUwJmEJ1++p82exLvVkXf0zKxg3ILCywe+utEtt4fM01OPLn0wS4J4R4IITIB7YCT5cA6g2sf/R6J9BBKvy29Aa2CiHyhBChwL1H9sol5tYmOHtbq2WdPP2vvyiIj8dx5Eg1KFMPxnJjfmzzI2ZGZkw8PpFsRbauJZWY7PR8Dq8Nwd7VgtYDSxZcUx4w8fLEe+VKVLm5hA8fTuVrSwsbnlh7wKgTUH+wXpZZ1TQt+lbFuYI1xzbc1EgmirbILcjlf/7/QyBY2HYh5kb6VbZUERtL2v4D2Pbti9y65GvbESHJ2LlaYOOoX+enLtThyD2BJ9thRT5675n7CCEKgDTAsYjHlisq1HIgNjSdvJySB4UJpZKkVasxrVUTyxbN1aiu9LhZujGv9TzC0sOYfma6eoqLaJnHddTzcwroMrI2xqbldpKoSJj5Vsd7zhcoIh6i3HYJZd0RMOIIOL+6NzhyYxmdh/uhUgoOr7mh1TbF6kIIwezzs7mZfJM5LefgbeOta0n/IXndelCpcHj//RLbKFAoibqbWi7Tzh5TZipaSJI0ChgF4Orqin9RommLSGZmplrtvYisfIFQCQ7+HoCNV8meZEwDA7ELDSV1xAjCT5z41zZtnsuL6Gnbkz1hezBLNaOjbccS2dDVuSTeFMSFCNwbSQTdvQR3S2dPXz6TkuIae4zqd5bj1sqM6BOWBK1/QKpVABiX7fxwdXwuLvUFUefS2LnEH5c6ugmiKul5nMw4yR/Jf9DVtivivsD/fvFtqJsnz0XKysLpt9/Ia9SQM/fuwr2S/RAzYwVKhSBVGYW/f7Qa1b5kXG3+7oUQpfoDvA4cfOLfU4GpT+1zEHj90WsjIBGQnt73yf1e9Kdhw4ZCnRw/flyt9l5EgUIpln/iL45tCCmxjdC33xF3O3YSqoKC/2zT5rm8CJVKJT71/1TUWVdHBEQGlMiGLs4l5kGqWPLhMXFg+XWhUqnUYlNfPpNik5suxO8jhfjaRog13YRIjRRnZ80WIb41RMSECc/8/pUl1PW5HFl3Qywec1RE3kpWi73iUpLzuBhzUdRbX098dOQjoVQp1S+qhDx5LvG//ipCfGuInFu3S2UzYPsdseSjYyI/V7vfV3X/7oFL4jk+UR23kBeBapIkVZIkyYTC4LU9T+2zBxj66HU/4NgjYXuAtx9FtVcCqgEX1KBJb5EbyfDxcyT0emKJ0tBygoLJCQzEoRQRnNpAkiRmNJ9BNftqTD45mfD0cF1Leil5OQUcXn0DSzvTclNHvcREXSnMDQ/aAW2/gKF/ga0nuc1fx2XSZ2Qc+Ju4WbPL5NKJumk1sDp2LhYcXnODnEz9718ekxnDpyc+xcvaizmt5iCT9C8dS5WdTcqGjVi1bVviAjBQ+KD64GoC3jUcyvUSWak/QVG45j2Owqfpm8B2IcQNSZJmSJLU69FuqwFHSZLuAf8Dpjw69gawHQgB/gY+EkKU+w4cles5k5OhIPZBWrGPTX4UwWnbt68GlKkXC2MLFrVbhFySM/7YeLIUGu55XQqEEJzYfIuM5Dw6j/DD1KJsTxuXGJUKziyG1Z1BmQ/v74O2nxfmiT/CcfhwHIYNI2XLFhKXLNGhWP3AxMyIziP8yMlScGz9Tb2+uckpyOGT45+Qr8zn5/Y/61XRlydJ/X0XytRUHEeVLpg3MTKTjKRcKtfTTg8KXaGWWzEhxH4hRHUhRBUhxKxH700XQux59DpXCNFfCFFVCNFECPHgiWNnPTrOVwhxQB169B2f2o7IjKRip64o4uNJP/B3qSM4tYmXtRfz28wnLD2MqQFTUQn9DAq6dTaGu5fiadqrUpntL15qMhNgS384NA2qd4ExpwrLrT4Dl88+xbZPHxJ/WUzKb79pWaj+4extTYu3qhIWlESQf6Su5TwTIQTfnv2WW8m3mNtqLpVs9bORjVAoSFq7BvOGDbFo0KBUth5cTUCSoOJrTmpSp5/o35zKK4CJuRFevg6EXk0o1t176tZtUFCAw5DBGlSnfpq6N+WzRp9xPOI4iwMX61rOf0iJzeLk1jt4+tpTv7OPruXohgf+hbnhoQHQfT4M3AQWz4/ylWQy3GfOwKptW2JnzCT9wCtxD/5C6rT1omIdR07/fo/ESPUUfVInq4NXs+/BPsbVH0cb75JVSNMG6fv3UxAdg+PI0he6Cr2agFsVWyxsTNSgTH8xOHIdUbmeE+mJuSRFZRZpf1V+PinbtmHVujUmFStqVpwGGFxzMG9Ve4uVQSv56/5fupbzD0qFikOrb2BkLKfTB7WQyV6xdXGlAo58Cxv6gJkdjDwGTUYWKTdcMjbGc+ECzBs0IGry52SePq0FwfqLJEm0H1oTM0vjwna3efqzSnj04VEWXVlEt0rdGFlHf2pP/AeViqRVqzCtVq3E5Vgfk5aQTVJUVrmfVgeDI9cZleo6gwQPAos2vZ6+bz/KpCTsS9E0QJdIksS0ptNo5NqIr898zdX4q7qWBMDZP+6TGJFJ+6E1sbQz1bUc7ZLyENZ2g1MLoMG7MOo4uNUulgmZuTneS5dgWqkSkR+PJ+f6dQ2JLRuYW5nQ6YNapMRlc2r7nZcfoAVuJt1k6qmpvOb0GjOaz9DrIE7TwEDy7t7DcdSoUut8EJgIYHDkBjSHhY0J7lVseXA18aX7CiFI3rgBk6pVsGyuXwVgioOx3JiFbRfiZunGJ8c/ITpTezmdzyIsKJFrxyKo086LSuV8De0/3PijMCo94Tb0WwO9fgGTktWSl9vY4L1qJUYODkSMGk3e/fJTg7wkeNVwoEEXH0JOx3D3UpxOtSRkJ/DxsY+xNbVlUftFmBmZ6VTPixBKJVZ792FStQo23buV2t6Dqwk4eVth41Q+q7k9icGR65DK9ZxJisp8aUvEnEuXyAu5icOQd/X6broo2JnZsbjDYhRKBR8f+1hnkexZaXkcXX8TR08rmvfVXec4rZOfDXvGw473wakajAmA2iWvYf0YYxcXKqxZDUZGhA8fgSJatzdpuqbJG5VwrWSD/+bbpCfqpuVpbkEunxz/hPT8dH5p/wtO5vp9s5p+4G+MYmJwHjeu1Km1WWl5xIamvRJP42Bw5Drl8ZfsZdHryZs2I7O1xbZ3rxfuV1aobFuZ+W3mcz/1Pp+e+FTrPcyFSnBkbQgFeUo6j/DDyLj85pf+i7gQWNkOrqyHFhNg2N9gX1Ft5k0qVKDCqpWosrIIHz6CguTS9xQoq8jlMjoN80MI3ZRwVQkVUwOmEpwYzJxWc6haXi8fAAAgAElEQVThUEOr4xcXUVBA4uLFKDw9se7cudT2Qq8lgng1ptXB4Mh1io2TOY5eVoS+wJEXJCSQcfQodn36IDMvP1NEzT2bM63ZNE5HnWbW+Vlazb0NPBxO5K0UWg6ohoP7K9CaVAi4uLrQiWcnw7t/QKdvQa7+XHmzGjXwXroERXQ0ESNGoswsWjBnecTW2Zy2g32JfZDOxX1hWh17waUFHAk/wqTGk+hQoYNWxy4JaXv3kh8WRlbPnkhq6GcfejUBG2dzHDxegd83BkeucyrXcybmQRrZ6c+uCJW66w8oKMBu4EAtK9M8/av3Z3jt4ey8s5M1wWu0MmZcaDrndz+gSn1narX00MqYOiUnBba/C/v+Bz4t4MPTUKW9Roe0aNQIz0U/kXvnDpEfjkWVW3a7g5WW6o3dqNHMjUsHwoi6k6KVMX+79RvrQ9YzqMYghtQcopUxS4NQKEhcshTTWjXJq1f6lsx5OQVE3k6hcj3nMr8UWVQMjlzHVK7nDAJCr/33qVyoVKTu2IFFkyaYVtbP4g2lZXyD8XSr2I2frvzE36F/a3Ss/NwCDq25gYWtCW1fhRKsD8/C0pZw+wB0mgmDd4KVi1aGtm7bFo+5c8m+dImoif9DKBRaGVcfafV2dWydzTmyNoTcLM3+P/hH+DP3wlzaerVlcuPJZeI7nrZnD4rwcJzHfayWlrgPgxJRKcUrM60OBkeucxw9LbFxMuPe5fj/bMs6fQZFZCR2AwfoQJl2kEkyZracSQOXBnxx6gsuxV7S2FgnfrtNRmIOnYb7YWZZjkuwqpRwYh6s6144fT78ELQYD2qYsiwOtj174Db9KzKPHyfmyy8RKv2s6qdpTMyM6Dzcj+z0fI5t0FwJ1+DEYCafnExNh5p83/p75DL9j/0Q+fkkLlmKWZ06WLVrqxab9y7HY2FjglslG7XYKwsYHLmOkSSJ6k3ciLydQmZK3r+2pWzbitzeHutOnXSkTjuYyk1Z1G4RXtZejD82njsp6s+/vXUuhjvn42jUoxIeVe3Ubl9vSI+GDb3h+KzCaPTRJ8Gzoc7k2L/zDs4TPiFt9x7ivtNuLIQ+4eJjQ7M+VQi9lkjwiSi12w9LC2PskbE4mDmwuMNiLIwt1D6GJkjZuhVFVBTO48erZfYgN1PBw+AkqjV2RXqFijsZHLke4NvUDQTcuRD7z3uKuHgyj/tj2/dNZCblu7wgFKalLe+4HHMjcz48/KFac8xT47I58dsdPKrZ0ah7RbXZ1TtuH4ClLSDqMvReAn1Xgpnun0ocR4/+p8lKwk+LdC1HZ9Tr4E0FPwdO77xHYqT6ggDTCtIYc2QMkiSxvNNyvU8ze4wyLY3EX5dg2bw5li1bqMXm3UtxqJQC32ZuarFXVjA4cj3AztUCt8o23D4f+88TS+rvO0GpxH5A+Z1Wfxp3K3eWdlpKTkEOow+PJlNZ+oudUqHi4Kpg5EYSnYaV0xKsBXlw4HP47W2w9Sx8Cq8/WC3rjepAkiRcJn2G3YABJC1fTuKKlbqWpBMkmUSHobUwsTDi0KpgtZRwzcjPYEn8EpJzk1nSYQk+NmWnV0Di8hUo09NxmTxJbWv5t8/H4uhpiZOXlVrslRUMjlxP8G3mTnJ0FokRmQilktQdO7Fs/jomPmXnh6kOqttX5+f2PxOdGc2y+GVkK7JLZe/sn49KsL5bEyt7/a1qVWIS78GqjnB+GTQdAyOOFhZ60TMkScLt6+nY9OxJwoIFJG/ZomtJOsHCxoRO79ciJTabUzvulspWnjKP8cfGE6uI5ae2P+Hn5KcmlZonPzKSlI0bsX3zTcxqqCfHPTUum7jQdHybupeJID91YnDkekLVhi7IjCRun4slMyCAgpgY7AaUv5SzotDIrRHz2swjPD+cCccnkK98dmreywgLSuTa0QjqtPEsfxGsQsDVLbC8NaRFwNu/QbfvwUh/68VLcjkec2Zj1b49cTNmkvrHn7qWpBO8aznQoEsFQk5Fl7iEq0KlYNKJSVyKu8QQxyE09yxbpZsTFiwEuRznT8arzebt87FIElRv4qo2m2UFgyPXE8wsjalUx4k7F2NJ3rYTuZMT1h00m++rz3So0IFBjoM4G3OWyScnF7v6W2bKEyVY+1XVkEodkZcBu0bBnx+CR3348AzU6K5rVUXiccc0y+avEzNt2ivb/rRJr8qFJVw33XppieanUQkV009P53jEcaY2mUpjq8YaUqkZcq5dI33/fhyHfYCxq3qcrlAJbp+Lxbumw6vX/AiDI9crqjd1IydDQXhQPLa9eyEZl+MUqSLQzKoZnzf+nKPhR/n6zNeoRNHSl1RKFYfX3KAgX0mXkeWsBGt0YGGzk+Cd0G4aDN0DNmWrsI3M1BSvxYsxr1+fqEmTyTh2TNeStI5cLqPzcD+QJA6tCkZZULTvthCC2edns/fBXj6u/zGDag7SsFL1IoQgbt4PyJ2ccBg2XG12o++lkpGcS/Wmr1aQ22MMjlyP8KntiImRkliXxtj17atrOXrBkFpDGFt3LHvu7+H7C98XKX3p4v4wou+m0maQL/Zu5aREo0oFZ36BVZ1AmQ/v74M2k6EM5Ao/C5mFBd7Ll2FWsyZRn0wg89Sr18vcxsmc9u/WIP5hBmf/LFrHuJ8Df2bb7W184PeBfvcVfw4ZBw6Qc/kyzh9/jNxKfb/N2+diMTaVl78ltCJicOR6hEwu4Z4aRKJzXXCvoGs5esOYumN4t9a7bLm1hUVXFr3QmUfeSubS/jBqNHOjRjN3LarUIJkJsKU/HPoSqneBMafAp2ytiT4LuZUVFVauwKRKFSLHjSPr/AVdS9I6VRq4UKeNJ9eORBB6/cUtjVdeX8mqoFX0r96fiQ0nlrmALmVmJnFz5mJWqxZ2/Urfce8xBflK7l2Jp0oDZ4xNy+aNbWkxOHI9IjckBOfbB1FJRty/8uKOaK8SkiQxqdEk+lXvx+rg1Sy5tuSZ+2Wn53N4TQj2rha0eru6llVqiAf+sKwFhAZA9/kwcBNYOOhaldqQ29lRYc1qjL08iRgzhuxLmqvsp68071cVJ28rjq4PISP52XXp1wav5efAn+lZuSfTmk4rc04cIPGXXyhITMTtm69L3ab0SUKvJaLIVeJbXm7cS4DBkesRaX/8iU1eLHbOZtw882r3c34aSZL4qtlXvFn1TZZdW8aya8v+tV2lEhxZe4O8nAI6j6iNiZmRjpSqCaUCjnwDG/qAmR2MPAZNRupNbrg6MXJwwGfdOozd3QkfNZrsK1d0LUmrGBnL6TKiNqoCweHVN1A+1fJ0w40NLLi8gG4Vu/Fdi+/KROnVp8m9dYvkTZuxGzgA89deU6vtm2eisbI3xbNaOa7Y+BIMjlxPUOXnk/7XX9h07EDttt7EPkgn/mG6rmXpFTJJxjfNv6FXlV78evVXVgWt+mfb5QNhRNxModWAamW/GERKGKztBqcWQoP3YNRxcKuta1UaxcjJiQrr1mLs4kLEyFHkXL2qa0laxc7VgrZDfIm5n8b5Px/88/7mm5v54dIPdPLpxOxWs8ukExcqFbHfzkBuY4PLhAlqtZ0ck0XEzRT8Wnm8UiVZn8bgyPWETH9/lGlp2L75JjWau2NsKuf68Uhdy9I7ZJKMGc1n0L1SdxZdWcSa4DVE3Ezmwt5QfJu6lf3WpMG7CqPSE+5Av7XQ62cwKScBey/B2MWFCuvXIXdyJHzESHKuXdO1JK1SvbEbtVt7Eng4nNBrCfx26zfmXphLhwod+L719xjJyuYsU9off5ATGIjLpEnI7dT71BzkH4ncSIZfK0+12i1rGBy5npC26w+MXFywbN4cU3Mjarzuzt1Lcc/tU/4qI5fJmdVyFt0qdmPFmTXsWX4JB3dL2gzyLZNrhwDkZ8Oe8bDzA3D2hTEBUPvVy1wwdnXFZ/165Pb2hA8fQXZgoK4laZWW/avhXMGa/Wuu8cuJZbT1bssPrX/AWFY2U1ELUlKI/2E+5g0bYtunt1pt52UruHUulmqNXTC3Lv/9KF6EwZHrAQUJCWQGBBTmjj8KAqnT1hNVgeBGgPo7JZUHjGRGfNd8FgMiJqLIV5LY8ipGJmX06xx3A1a0hSsboOX/4IMDYP9qleZ9EmM3N3w2bkDu6EDE8BGv1Jq53FhGWpsQ8gpyeSvsE+Y1n4+xvGw6cYC4WbNRZmbiNn06kprb6N48E0NBnpLX2nmr1W5ZpIxe+coXaX/tBaUS2zff/Oc9ezdLKvg5EnwiqsjFIl41Lu15iGmCPdkt7rMiYjELrywsW20yhYCLq2BFO8hNhff+hI5fF/YQf8UxdnPDZ8MGjFxcCB8x8pWJZl92bRk/35tPWoubmKXYcWFXmK4llZj0w4dJ37sXpzFjMPNVbxaJSiUI8o/EvaotzhWs1Wq7LGJw5HpA2p9/Ylb3NUwrV/7X+6+19yI7PZ/7V+J1pEx/uX8lnsDD4dRu7cmUQWMZ6DuQtcFrmXthbpErwOmU7GTYNgT2fQqVWsGY01C5ra5V6RXGrq5U2LAeYzc3wkeOIuvceV1L0hhCCBZdWcSvV3+lV5VeTH/7f9TvVIHgk1HcOheja3nFpiAlhdhvvsW0Zk2cRo9Su/2HQYmkJ+YansYfYXDkOib39h3y7tzBtlev/2yrUNMBO1cLrh0zBL09SXJ0FkfX38S1kg0t+1dDJsmY1nTaP0Vjvjr9VbFrs2uVh2cLA9ruHITO38GgHWD1alakehnGLi74rF+HsacHEaNHk3nihK4lqR2lSsl3575jVdAq+lXvx8wWM5HL5DTrUxlPXzv8N98mITxD1zKLRdzMmSjT0/GYO0cjpaavHYvEyt6UyvXKRu91TVMqRy5JkoMkSYclSbr76G/7Z+xTT5Kks5Ik3ZAk6bokSQOf2LZOkqRQSZKuPvpTrzR6yiLp+/aBXI5N167/2SbJJF5r50V8WDqxoWk6UKd/5OUUcGB5EEamcrqOqoPcuPAr/LhozLh649hzfw8T/SeSp8zTsdqnUCnhxDxY171w+nz4IWj+Mah57bC8YeTsjM/GjZhWqULEuI9JP3hI15LUhkKpYGrAVLbf2c7w2sOZ3mw6Mqnw+yCTy+gyojbmVsYcWBZETmbZCHxN//tv0vcfwPmjsZj5+qrdflJUJlG3U6jT1guZ3PDbgdI/kU8BjgohqgFHH/37abKB94QQfkBX4CdJkp7MQZgkhKj36M8rlTwqhCB9714smzfHyNHxmfv4NnPDxEzOdcNTOUIlOLI2hPSEHLqOrI2V/b+7HEmSxOi6o/mi6Rf4R/gz9shYMvMzdaT2KdKjYUNvOD4Lar8Fo0+CZwNdqyozGNnbU2H9Osxr1yZq4kTSdu/WtaRSk1OQw/jj4zkQdoCJDScyoeGE/2RdmFub0G1MHbLT8zm06gYqpX4vGxUkJRH77QzM/PxwHDFCI2NcPx6JkbGMWi3KeKqpGimtI+8NrH/0ej3Q5+kdhBB3hBB3H72OBuIBwzwikBN4FUV0NLY9ezx3HxMzI2q18uTepThSYrO0qE7/uHQgjLDribToXxWPF1RxeqfGO8xpNYfLcZcZdnAYiTkvrmGtaRwTL8DSFhB1Bfoshb4rwcxGp5rKInJrayqsWolFkyZEfz6F5C1bdC2pxKTmpjLq0ChOR53m69e/ZljtYc/d18XHhjaDqhN5K4Vzux88dz9dI1QqoqdORZWZifuc2UhG6s97z0jO5da5GHybuWFmZQgKfUxpHbmrEOJxJEYs8MLmspIkNQFMgCdb/cx6NOW+UJKkV6qRbPq+fUimplh16PjC/ep3qoDcRM7FvaFaUqZ/PLiawIW9oVRv6kqdtl4v3b9n5Z783P5nwtLDGLJ/CKFpOvi/K8iDA59TJ3gW2HrB6BNQb1C5LLOqLWSWlngvX4ZVu3bEzZhJwuJfy1amAhCZEcm7B94lJCmEH9v+SL/q/V56TM3mHvi19iTwUDh3LsRqQWXxSV6zhqyTAbhMnYJZdc30Ori0r/B33LBbRY3YL6tIL/sRSJJ0BHhWk9dpwHohhN0T+6YIIf6zTv5omzvgDwwVQpx74r1YCp37CuC+EGLGc44fBYwCcHV1bbh169YXn1kxyMzMxMpKy2U9lUqcp0whv1p10ka9vB1h3HUViSFQpauEmd3zHYFOzkVDPD6X3BRB6FGBqQ1UbC8hMyq6I3yY95Bl8ctQoWK082gqm1V++UFqwDw7kloh87HODOWBS1ciaoxAlNGiHk+iN98vpRKbTZsxP3uW7DZtyBg4oNixBro4l/C8cJbFL0OJklHOo6hiVqXIx6qUgof+gpwkqNhBwsKx8HegD5+J8f372P+4gLx69UgbOaLEN6svOpe8DMG9/QKHauDeQP/XxtX9ubRr1+6yEKLRMzcKIUr8B7gNuD967Q7cfs5+NsAVoN8LbLUF9hZl3IYNGwp1cvz4cbXaKwoZJwNEiG8NkX74cJH2z8nMFysmnBD7llx74X66OBdNcfz4cZGZmivWTTkl1n5+SmSm5pbITnhauOixq4douLGhOBJ2RM0qn0KlEuLKJiG+cxdibkUhbh0od5+JvqBSqUTs9/NEiG8NETlxolDl5RXreG2fS0BkgGi8qbHovKOzuJ9yv0Q2sjPyxIZpp8XqSQEiPSlHCKH7z0SRnCzutGkr7nbsJArS00tl60XncmhNsFg27niJrwPaRt2fC3BJPMcnlva2Zg8w9NHrocB/IlAkSTIB/gA2CCF2PrXN/dHfEoXr68Gl1FNmSN+7F5m1NZatWxdpfzNLY+p19Cb0WiJxYa9GMxWVUnBgWRC5mQp6jH0NS9uSrbx423izsdtGfB18meg/kTXBazQzHZubDrtGwu6xhYFsH54G3/9mIxhQD5Ik4Tp5Ei6TPiN9/wHCR49GmaGfaVpbbm7ho6Mf4WPjw6bum6hsV7KZIXMrE3qMrYsyX8m+JdfJz9VtmqUQgpipX1CQlITnggXIrTVTnCUpOpM7F+Ko086rxNeB8kxpHflcoJMkSXeBjo/+jSRJjSRJetyaagDQGnj/GWlmmyVJCgKCACfgu1LqKROocnPJOHwY6y6dkZkUvUZw3fbemFkac2GP/ga8qAshBNEXBHGh6XT8oFapqzfZm9mzuvNqOlfszMLLC5l+ZjoKpUJNaoGoy7C8NQT/Du2mwXu7wcYQVasNHIcPx33OHLIvXuLhoMEoYvSngEqBqoDvzn3HnAtzaO3VmvVd1+NsUbpYXwcPSzqPqE1yVCZH1oboNEYgadUqMv39cZ00CfM6muvQd/GvUIxN5TTo/OqWLn4RpQorFEIkAR2e8f4lYMSj15uATc85vn1pxi+rZPqfQJWdjW3PnsU6zsTciPpdKnB2132i76a+MHK7rHPhr1DSHkLTXpWo0sBFLTbNjMyY13oelWwrsezaMiIyIljYdiH2Zs8M6ygaKhWcXQxHvwUrN3h/P/i8rha9BoqO3Zt9MHZ1IXL8J4QNfBvv5cswq1lTp5rS89P5zP8zzsac5X2/95nQYILa2pD61HakRb9qnNpxF8dcoJ1azBaLjKNHSViwEJvu3bB/d4jGxkkIz+B+YAKNelQ0RKo/B/2PGCiHpO/bi5GzMxaNGxf72DptvTC3MeHc7vtlLlq3qASfjOLS/jDsKqs/OlUmyfio3kfMbTWXoIQgBu0bxJ2UOyUzlhkPW/rD4a+getfCjmUGJ64zLJs3x2fzZpDLeTh4CJknT+pMS2haKEP2D+Fi7EW+bf4tnzb6VO29xF9r78Vr7bxIug2Bh8PVavtl5N66RdSkyZjVro377Nka7Tp4bvcDTC2MqNfBUI71eRgcuZZRZmaSeeIk1t26/tPprDgYm8hp0qMiMffSuHMhTgMKdcuDqwmc/O02PnUc8WgkaewC0aNyD9Z0XUOeMo8h+4fwd9jfxTNw/1hhbnhoAPT4EQZuAgsHjWg1UHTMfKtTcetWjH18iBjzIckbNmj9hvd4+HHe2fcOqbmprOi8gr7VNNOOVpIkWvavho03nPn9HncuaictrSAhgYgPxyK3tsZr8WJkZmYaG+tBYALhN5Jo0NUHUwvD0/jzMDhyLZN53B+Rn49N124ltlGrlSeulWw4teNumSnbWBRi7qdxaPUNnH1s6DKiNpJMs/nWdZ3rsq3nNnztfZl0YhILLi94eY12pQKOfAMb+4K5PYw6Do1Lnm5jQP0Yu7pQcdNGrNq3I272HGK++gqRr/nfiUqoWHJ1CeOPj8fHxodtPbfR2K34s27FQZJJeDaT8Khmx9F1N4m4lazR8VR5eUSMG4cyNRWvJb9i7KqeZa9nkZ9TwMmtt3H0tKKu4Wn8hRgcuZbJOHQQIxcXzOvVLbENmUyi3ZAa5GcXcGbnPTWq0x1J0ZnsW3INK3tTen70Gsam6p2GfB7OFs6s6bLmn+5pHx75kOTc51wMU8JgbTc4tRAavAej/MHVTys6DRQPmaUlXj//jOOHY0jb+TsPPxhGQVKSxsZLy0tj/LHxLL22lF5VerG+63rcrdw1Nt6TyOQS3T+sg52rBQeWBWmswYooKCB68ufkXruOx/dzMffT7Hf/3J/3yUrPp92QGsgNNdVfiOF/R4uosrLIPBmAdefOSKVslOHoaUX9zhW4dS5W43fhmiYlNovdP11FbiTjjY/rYW5d9Eh+dWAsN+bLZl8yo/kMrsRdof+e/lyOu/zvnYJ3FXYsS7gN/ddBr5/BxEKrOg0UD0kmw+WTT/D4cT65wcGE9u9PTlCQ2scJSghiwF8DOB19mqlNpvJdi+8wM9LcdPOzMLUw5o2P62JmYczuRYEkRqq3x4BQqYj58isyDh7E5fPPsencWa32nyb2QRpBJ6N4ra0XrpUM5YxfhsGRa5HMEycQeXnYdO2iFnuNulfE1tkc/823KchXqsWmtkmNz2b3wkAQgj4T62PrbK4zLW9We5MtPbZgbmzO8IPDWRW0ClVeBuz5GHZ+AM6+MOYU+L2pM40Gio9tjx74bN6MhMTDQYNJ2bpNLevmQgg2hmzkvb/fA2BD1w0MqjlIo4FfL8LK3ozeE+tjbCJn90+BJEWrx5kLIYj7bhZpf/6J07hxOH7wvlrsPg9lgYrjm25hZWdK097aqcRY1jE4ci2SfvAQcmcnzOvXV4s9IxM5bQf7kp6Qw8V9YWqxqU3SE3PYvTAQpVLQe0J97N0sdS0JXwdftvbYSiefTiy6soixm1qSeHUztPwffHAA7A15rGUR89p+VPx9JxbNmhH7zTfETJkKpVg3T81NZcLxCcy7OI9Wnq3Y/sZ26jjXUaPikmHrbE7vCfWRySV2/3RVLY2WEhYsJGXLFhw++ACnj8aqQeWLCTwUTnJ0Fq3f8cXETP2NV8ojBkeuJVTZ2WSePIlNp04lilZ/Hl41HKjxuhuBh8OJupOiNruaJj0xhz8XBqLIU9J7Qj0cPfWgfvcjrIwtmWdWjS+T07koV/JW1RqcrNGusIe4gTKLkb093suX4fTxONL27MFh7vfk3b1bbDvnYs7x1p63OBl1ks8afcaidouwNbXVgOKSYedqQZ+J9UEIdi8MJDUuu0R2hBAk/PorSStXYvf2QFwmT9L4bEN2kuDi/lCqNHCm0mtOGh2rPGFw5Foi82QAIicH687qmVZ/klYDqmPrbM7BlcHkZ+l/bnlSVCa7frhMfk4BvT6ph5OXZso6lojsZNg2BOnAJAY6N2ZrpzU4Wbnz0dGPmHVuFrkFubpWaKAUSDIZzh99hPeKFcgz0gnt15+U334r0lR7vjKf+RfnM/LQSCyMLdjcfTND/YbqbCr9Rdi7WdJ7Qn1UKsGu+ZeJf1i8ss5CpSJuzhwSf1mMbZ8+uE2frvHzzErLI+KUwNLWlDaDfDU6VnnD4Mi1RMahg8gdHLBo/OzmNaXBxNyI7h/WQalQEXFK6PV6ecy9VP748QoAb37aABcfPQpkeXi2MKDtzkHoPAsGbaeaZxO29NjCu7XeZevtrQzcO5AbiTd0rdRAKbFq1ZKkL7/EonFjYr+dQeRH4yhIef6M1u3k2wzeP5j1IesZ6DuQ7W9sp5ZjLS0qLj6Onlb0/awhRsZy/lwQWOSgWKFQED1lCikbNuIw9D3cZ88qdXDuy1AWqDi4IhhlPnT/sA7mVtoNeC3rGBy5FlDl5pLhfwJrNU+rP4m9myUdh/mRmwLHN9/Sy6pvYdcT2b3oKubWJvSd3FB/ptNVSvD/HtZ1L5w+H3EYmo/7py2mqdyUyY0ns7zTcjIVmQzeP5hFVxaRryw/OfyvIipbW7xXLMd16hSyAgII7dWbjGPH/7WPQqVg6bWlvL3vbeKz4/ml/S982exLzI10F5RZHOxcLXhrckOsHc3Yu/ga9y7Hv3B/VU4OEePGkb7nL5wnTMBlyhSNO3GAU9vvEnM/Dc8mkn7N0JURDI5cC2QGBCCys7HpotmUjUqvOeFcW+LO+TiuH4vU6FjFQQhB8Mko9i8LwtHDkr6fNcDGUU8uhGlRsL4X+M+GOv0Ly6x6PDsYsblHc/7o/Qe9qvRiVdAqBvw1gKAE9aczGdAekkyGw9ChVNy+Dbm9PZFjxxL12SQKUlK4lXyLQfsGseTqEjr7dGZ379209W6ra8nFxtLOlDc/bYCrjw0HVwVz9Uj4M2/0FbGxPBz6PlkBp3D79lucxozWyrLBjYAogk9GUb9zBWx99G+ZoixgcORaIOPgIeR2diWqrV5cnP2gUl0nTu+8y+3z2inZ+CIK8pUc23CTE1tu413Tgd4T62s9T/y53D4Ay1pAdCD0WQZ9V4Dpi58GbExsmNFiBks7LiVTkcmQA0OYd3Ee2YqSBRQZ0A/Matak0s4dOI0bR/rffxPcpQMLfxhAQlY8P7X7ie9bf4+dWdltUmRmacwbn9Sjcl1nTu+8x6HVN/7VAjXr/AVC3+pH/r17eP28CPuBA7Si635gPCe33sG7lgPN+lTRypjlEYMj1zCq/Hwyjx/HqmCY/xkAACAASURBVGMHJGPNRz1LkkTHD2rhUd2OI2tDCPLX3ZN5emIOu+Zf4dbZWBr1qEjPj17Tj3QSRS7snwy/vQ223jD6JNR7p1gmWnq25I/ef/BWtbfYGLKRXn/24mj4UQ0JNqANJBMTbvV5jXkfOhNpkcOEPxSsOVKFVqry4WCMTeR0HV2bZn0qc/9yPL/Pu0xKXBZJa9YSPmwYcltbKu7YjnXHjlrRc/NMDAdXBONcwZrOw/2Qabgkc3nG4Mg1TPa5c6iysrDp1ElrY5qYGdFzXF0qvubEya13uLQ/TOtr5qHXEtg+5yJpCTn0GPsaTd+orPHa6UUi8S6s7ggXlkOzsTDiCDhVLZEpaxNrpr8+nY3dNmJrasuE4xP4+NjHRGVGqVm0AU0TlxXHp/6fMvboWJI8rXDdtBaXKZ+juBrEgzd6Ef/jAlRZpc/J1jWSJNGwa0XeGF+P7NRctn99iqC1R7Bq156K27dhWkU7Ny3XjkZwbMNNvGrY03tCfcwsDamdpUEPHo/KNxlHjyFZWGDRrJlWxzUyLrz7PrbhJuf3PCAvW0HzvlU17kyz0vII2HaX+1ficfSyouuo2ti56EEpUyHg6hbY/xkYmcE728C3q1pM13Opx9aeW9kcspkl15bQ649eDPUbyog6I7Aw1oNzN/BccgtyWXdjHWuC16BUKRlXbxwf1P4AE7kJvN8M2x49iP9xAUkrV5K2Zw/O4z/GtndvJKOye+kUQmAbeo5GlxZz3bs/N/yGk1XTEXuFEZoOPxVCcHFvKBf3hVG5vjOdh/khNzY8T5aWsvttLAMIlYrMY8ewatkSmamp1seXy2V0HFoLUzMjrh6JIC4snfbv1sTOVf3ORQjBzTMxnPn9HgX5Kpr2rkz9zhX0o9lBbjrsnQjBO6Fiq8K1cBsPtQ5hLDPm/drv07VSV3668hMrg1ay+95uJjScgKXQfcU6A/9GCMGVrCvM/nM2MVkxdPLpxP8a/g8va69/7Wfk7IzH3DnYDRxA3Jy5xEz7kqRVq3Ee/zHWXbpoJaJbnSji44mbOZOMw0ewqVOHt75qzu0Icy7secCWb8/zep8q1G7tqZEb/rSEHPw33yLyVgo1mrvTbrAvMn24PpQDDI5cg+QGB1OQkIB1h/Y60yDJJFq9XR1nH2tO77zH1pkXaNyzIvU6qcfJCpUg9Hoilw+EEf8wA49qdrQd7KsX5VYBiLoMO4dBagS0+xJa/Q9kmuus5mbpxtxWc3nb922+v/A9X5z6Am8Tb8yizHjd43W9LB7yqnEx9iI/XfmJ64nX/6+9O4+PqjofP/452feQhYRAkIAEQiIhJMgiZV8riqgoIMhqwVa/PxdUbK1tbati8VurtV8VsZXFFhUXVESUJexL2JewGJAlrCGBkD2TzPn9MRc6YvaZzBKe9+uVV2buPffOc3Jm8sy999xz6BjWkRd/9mKt040GdO1K3IeLKVy1ipzXX+f0E0/i26kTkTNmEDxkcKPdVmov5pIS8hYsJHfePHRZGVFPzSR88mSUlxddE6BdSnPW/vsQ6xYfIXPjGVKHteHm1Ci7XLc2mzV7V59i6xfHUB6KfuM6kNSncb4s3KgkkTeigpWrwNOToH79nBqHUopOt7XkpqQI1i8+wpbPj5G14wJpw+NomxzZoFNblZVmsjLOs2PFSS6dLSIk0o+BExNI6BnjGh9Qsxk2/x1W/RGCY2DK13CT4y5vpESl8MGID1h2bBmvbn6VGStn0L1Fdx5LfYzk5skOi0P8V2ZuJm/sfIONZzYSFRDFuPBxzBoxC886frFTShE8eDBBAwZwZdkycv7xD04//jjesbGET5xIs3vvwSPQRb7AGrTJxOVPPuXiP/5BRU4OQQMGEPX00/i2a/ujcqHN/bnz/6XwfcZ5MpYd59t5BwiNOkbqsDZ07NECT6+G/Y84sS+XHd+c4MLxK7TpHEG/cR0JDnfszHA3Aknkjahg9SoCunXDs5lr3LYSGOrL8BmdObYrh/UfH2HFu/vxC/SmQ/doEm6LITI2qMYjxtJCEyczczmxP5eTB/IoLTIR0SqQIdMSaZ8a5TqnyQovwGcPw9FV0GmkZcpR/zCHh+GhPLjz5jvxO+nHhRYXmLt3LuO/Hk//2P7M6DKDWyJvcXhMN6JDeYeYu3cu3534jlDfUJ7q9hRjOo5hy4YtdU7i1pSnJ6EjRxIyYgQFq1aR96/3Of/SS+S8+Sahd91F6Ki78EtMdOrZl8orV8j/fCl5HyzCdOIk/l270upvrxGQllbtNkopOnRvQftu0RzblcPOFSdYs/AQmz7JonViOHG3RHBTUkSNt49qrck7U8ShzWc5vPUcJQUmAkJ9GDItkfhu0XJGqpFIIm8k5SdOUJ51lLD7HXM/Zn2069qcuC6RZB/M4+Cms+xff5q9a7Lx8vUkNNKPkEh/Qpr7o82akgITJQXlFF8p59LZIrQG/2Bv2nSOoH1aFG1uiXCtD2fWKksSL7sCd7wGaVPAyfF5K2/GdxrPqPajWJS5iAWZCxi3bBy9W/ZmRpcZdI2yz2x44sf25exj7t65pGenE+QdxIzkGUxKmkSwj31GDlOenoQMHUrI0KGU7N5N3oIFXF68mEsLF+IbH0/oqFGE/Hw43i3t2x+jOlpryg4d4tK//0P+V1+hS0rw65JM9KxZBA0YUOfPqYeHon1aFDenNif74CWObD/Pif25ZG2/AArCogMICPUhINgH/2AfPDwVV3JLyc8p4UpOCaaySjw8FXHJkXTqFcNNSeGu8yW/iZJE3kgKVq0GIGjgICdHUjUPD8VNSZZv2KVFJo7tyiHvTBH5F0u4fKGEk5l5eHiqax/W0Ob+li8At0QS1SbYNU6fW6s0weo/w8a/QfMEmLgUol1rLOxA70BmdJnBhMQJLD60mAWZC5i4fCKpUalMTJpI/9j+DTpCFP9l1mY2nN7AgswFbD27lVDfUB5NeZRxncYR4tN44/r7p6TQKiWFyvx8rixfTv5nn3NhzhwuzJmDb3w8gX37ENS3HwFdU1A+9hsQSZtMeB86xPnNmylYk47p5EmUnx8hd4wgbOw4/G9JavC+lVK0TgyndWI42qzJOVXAif25XDxVSElBORdOFlBSYKLSZCYk0o+Q5v60im9GWEwgN3dt7joDP90AJJE3koLVq/Dt2BGf2FbODqVWfoHeJP7MMUcNjSLvB/hkmqVjW9pkGPYy+LjubV+B3oFM6zyNBzo9wJIjS1iUuYjH1zxO6+DWjO80nrvb3y23rdVTSUUJXx79koWZCzl+5ThRAVE8kfYEYzqOIdDbcdetPUNDCRs7lrCxYyk/fpyCNekUrltL3oKF5L33T5S3N74JCfglJeJ/yy34tm+PV0wMXpGRtXaYMxcXYzp7ltLMg5QePEhpZial+/cTXljIJR8fAnr2IGLKZEJuvx3PUPtOq6o8FFFtQlxrkiNxjSTyRlBx6RIlO3cR+fAMZ4fS9O3/BL58HFBw3/uQdLezI6ozfy9/Hkx8kHEJ41h1chULMxcye9ts/r7r74xoO4LRHUbTKaKTs8N0ad9f+p4lR5bw5bEvKSgvIDEikdl9ZjM0bijeHs4dZMQnLo6IKZOJmDKZysIiirdspnjXLkoPZHJl2ddcXvzhfwt7euLVvDmeYWGWW9q8PFEenuiKCirz8qjIy0OXlFwrrry98e3QgZDbb+d4WBg9pv/C5TraCceRRN4ICtekg9nssqfVm4TyIlg+C3YthNjucO88CGvj7KgaxMvDi2FxwxgWN4w9OXv46PBHLD26lI+OfERSRBL3xN/DsLhhhPra9yjLXRWUF7DyxEo++f4T9uTswdvDm8FtBjOm4xhSo1Jdq8+GwTMokODBg68Nf6rNZkwnT1J2/DgV585jOn+OinPnqbx8GW2uhIpKtLkSD48AfNrG4RUegVdkBF7Nm+ObkIBvu3bXhnw+nJ4uSfwGJ4m8ERSsXoVXixb4JbnWNdom49x+WDLFMtxqn5nQ/9eW6UebgC7Nu9CleReeufUZvjr2FUuOLOFPW/7Ey9tepk+rPoxoN4J+sf3w87qxbuEpryxnffZ6lv2wjLWn1lJuLicuJI6nuj3FyJtHEubn+LsSbKE8PPCJi8MnLs7ZoYgmQBK5nZlLSynasJFm99zjkkcGbk1ryJgHK54D/2Yw8XNo19/ZUTWKUN9QxncazwMJD5CZl8nXx75m+Q/LWXNqDQFeAfRu1ZuBNw2kT6s+TfZIvaC8gA2nN7Dm5BrWn15PoamQcL9w7ut4H7e3vZ3OkZ3lMyYEksjtrmjLFnRpKUEDnTeaW5NUnAdf/A8c+grih8KotyAw0tlRNTqlFEkRSSRFJPFk2pNknM9gxfEVpJ9K57sT3+GlvEiLTqNny570iulFQniC2/Z8N2szh/MOs/XsVjad2UTG+QwqzBWE+4UzNG4oQ9sMpUdMD7w85N+WENZs+kQopcKBD4E44Dhwv9b6UhXlKoF9xtOTWuuRxvK2wGIgAtgBPKi1LrclJmcrWrcO5e9PQPfGn3v8hnF8I3z6C8tAL0NftMxa5mZjXNuDp4cnPWN60jOmJ8/3fJ79F/ez5tQa1mav5fWdr/M6rxPiE0L3Ft1JiUqhS/MuJEYkWiYAcUGmShOH8g6xJ2cPuy7sIuNcBpfKLP8+2oW248FODzLgpgEkRya77ZcTIRzB1q+2zwKrtNazlVLPGs9nVVGuRGudUsXyV4DXtNaLlVJvA9OAt2yMyWm01hSmryWwVy887Hiv6A1LV0L6bFj7CoTFwUPfQUsZPAUso8YlN08muXkyj6U+xsWSi2w9u5UtZ7ew7ew2Vp5cCVgmc+kU3omO4R3pENaB+LB44sPiG/We6qoUlheSdTmLI5eOcOTSEQ7nHSYzN5Nys+V7e3RANH1i+9Azpic9YnoQFRDl0PiEcGe2JvK7gP7G4/lAOlUn8p9QlotbA4EHrLb/A26cyMuzsjCdOUPEDLntzGb5p0nZ/TzkH4DkMTDif8HXPiNyNUWR/pGMaDeCEe1GAJBTnMPenL3sydnD3ot7+eb4N3x85ONr5cP9wokNiqVVcCtig2KJDogm3D+cCL8IIvwjCPYJxt/LHz9Pv2qvQ2utKasso7iimMLyQnJLc8ktySWvNI8LxRfILswmu8Dyk1uae227IO8g4sPiGZswli7Nu5DcPJkWgS0a9w8kRBNmayKP1lqfNR6fA6KrKeenlNoOVACztdafYzmdfllrXWGUyQZcf/SUGhSuWwdAUN8+To7EzR36Gpb+iuCyEhj1NqSMc3ZEbqd5QHMGtRnEoDaWWyC11pwvPs+RS0fIupzFySsnyS7MZm/OXr49/i2VurLK/SgU/l7+eBt3BZhMJrwXe2OqNFFSUYJGV7mdh/IgJjCG2KBY+rfuT2xwLPHNLGcDYgJjpJOaEHaktK76g3itgFIrgaq+Lj8HzNdaN7Mqe0lr/ZP7QJRSrbTWp5VS7YDVwCAgH9iitW5vlGkNLNdaVzmThFJqOjAdIDo6Om3x4sV1qV+dFBYWEhQUZPN+wv73r6jiYvKe/60domoYe9XFGTwqy2l37H1iTy+jIOhmtsf9ChXZ3tlh2czV26RSV1JYWUiBuYCCSstPqS6lzFxGuS6nzFxGJZZEbzKZ8Pb2xhNPfDx88FW++Cgf/Dz8CPEMIcgziBAPy29P5drXtV29XeqqqdQDpC41GTBgwA6tdbcqV2qtG/wDHAZijMcxwOE6bPM+MBpQwEXAy1jeC1hRl9dNS0vT9rRmzRqb91GRn68zE5P0+f/9q+0B2cAedXGKC4e1/r/eWv8+ROvlz2ptKnXfulynqdRDa6mLK2oq9dBa6lITYLuuJifa2vX3C2CS8XgSsPT6AkqpMKWUr/E4EugNZBqBrTGSerXbu4uiTZugspKgfn2dHYp70Rp2LYK5/aDgDDzwEQx/Gbx8nR2ZEEK4BVsT+WxgiFLqe2Cw8RylVDel1DyjTCdgu1JqD5bEPVtrnWmsmwU8qZTKwnLN/D0b43GawvS1eISG4t+li7NDcR+l+fDJQ7D0EWiVBg9vhA7DnB2VEEK4FZs6u2mtc7Fc775++XbgIePxJqBzNdsfA7rbEoMr0GYzhevXE9S7N8pLBquok+wd8MlUuHwKBv4WfvYkyL3CQghRb5J17KD0wAEqc3MJ6t/P2aG4PrMZNr0Bq/8EwTEwZTnc1MPZUQkhhNuSRG4HhelrQSkC+8htZzUqvACfzYCjq6HTSBj5Bvi712QXQgjnMZlMZGdnU1pa6uxQahUaGsrBgwfrvZ2fnx+xsbF4e9d9IihJ5HZQuG4d/snJeIVJUqpW1ipLEi8rgDteg7QpIPcSCyHqITs7m+DgYOLi4lx+LIKCggKCg+s3iJXWmtzcXLKzs2nbtm2dt7vxBqy2s4qLFyndt09Oq1enohy+fR4W3QMBkfCLNdBtqiRxIUS9lZaWEhER4fJJvKGUUkRERNT7jIMckduoaONGAAL7yG1nP5H3A3wyDU7vsCTvYS+Bt7+zoxJCuLGmmsSvakj95IjcRoUbNuIZHo5fYidnh+Ja9i2Bd/pCbhbcv8ByOl2SuBDCzSmlmDlz5rXnr776Kn/4wx/qvP38+fOJj48nPj6e+fPn2yUmSeQ20GYzRZs2EXjbbagbcFrNKpUXWe4L/2QaNE+AhzdA4l3OjkoIIezC19eXTz/9lIsXL9Z727y8PF544QW2bt3Ktm3beOGFF7h06Sczf9ebZB8blB0+TGVuLoG9ezs7FNdwbh/M7Q+7PoA+T1luLWt2k7OjEkIIu/Hy8mL69Om89tpr9d52xYoVDBkyhPDwcMLCwhgyZAjffPON7THZvIcbWOGGDQAE9r7NyZE4mdaw7V349reW28kmLoV20vlPCNF4XvjyAJlnrth1n4ktQ/j9nUm1lnvkkUdITk7mmWee+dHyDz74gDlz5gBgNpvxMM7Utm/fniVLlnD69Glat259rXxsbCynT5+2OW5J5DYo2rgJ3w4d8I6KcnYozlOcB0sfhcPLIH4ojHoLAiOdHZUQQjSakJAQJk6cyBtvvIG//3/7/owfP57x48cDDbv9rKEkkTeQubiYkh07CJswwdmhOM/xjfDpLywDvQx7CXr+Sm4rE0I4RF2OnBvT448/TmpqKlOmTLm2rLYj8latWpGenn6tfHZ2Nv3797c5FknkDVSckYE2mQj82Q14fdxcCevmwNpXICwOHvoOWnZ1dlRCCOEw4eHh3H///bz33ntMnToVqP2IfNiwYfzmN7+51sHt22+/5eWXX7Y5FknkDVS4cSPK15eAtDRnh+JY+dnw6XQ4sRGSx8KIV8HXMaePhBDClcycOZM333yzzuXDw8N5/vnnufXWWwH43e9+R3h4uM1xSCJvoKINGwm49VY8/PycHYrjHFpmubWs0gR3vwNdxjo7IiGEcKjCwsJrj6OjoykuLq7X9lOnTr12BG8vksgbwHTmDOXHjtHsvvucHYpjmEotPdIz3oWYLjD6XxBxs7OjEkIIgSTyBinatAmAoBvh+njOYVgyFc7vh56PwODfg5evs6MSQghhkETeAIUbNuIVFYVP+/bODqXxaA27FsLyWZahVR/4GDoMdXZUQgghriOJvJ50ZSVFmzcTPHBg0x28vzQfvnoC9n8CbfvC3XMhJMbZUQkhhKiCJPJ6Kt2/H3N+ftO97Sx7ByyZYumdPvC38LMnwcPT2VEJIYSohiTyeiravBmAwF69nByJnZnNsOkNWP0nCG5pGSf9ph7OjkoIIUQtZNKUeirashXfhAS87HDvn8soOA8f3Asrfw8JI+Dh9ZLEhRCiCrZOYzp8+HCaNWvGHXfcYbeYJJHXg7m0lJKdOwns0YSSXNZKeLs3nNhkmTP8vvng38zZUQkhhEuyZRpTgKeffpqFCxfaNSZJ5PVQsns3urycgF49nR2K7SrKLfeGL7oXAiJhejp0mypjpQshRA1smcYUYNCgQXafTEWukddD0eYt4OlJQLdbnR2KbfKOwZJpcGanJXkPe8lyi5kQQriL5c/CuX323WeLzvDz2bUWa+g0po1FEnk9FG3ZjH/nzngGBTo7lIbbtwS+fBw8PCyn0ZNGOTsiIYRwKzKNqZuqLCigdN9+ImZMd3YoDVNeBF8/A7sXQesecO88aHaTs6MSQoiGqcORc2NqyDSmjUUSeR0VZ2wHs5nAHm54ffzsXsswq7lZ0Pdp6PcseErTCyFEQzVkGtPGIp3d6qh46xaUry/+XVOcHUrdaQ1b34F5g6C8ECZ9YRnkRZK4EELYbObMmfXuvd6nTx/uu+8+Vq1aRWxsLCtWrLA5Dpv+oyulwoEPgTjgOHC/1vrSdWUGANbd+xKAsVrrz5VS7wP9gHxj3WSt9W5bYmosRZu34J/aFQ9fN5kwpDjPMuXo4a+hw3C46/8gMMLZUQkhhFuzdRrT9evX2zskm4/InwVWaa3jgVXG8x/RWq/RWqdorVOAgUAx8K1VkaevrnfVJF6Rm0vZkSME9nST0dyOb4C3elvuER8+G8YtliQuhBBNlK3nWO8C+huP5wPpwKwayo8Glmut6/cVxsmKt24FILCniw8EU1lB3A//hrUfQ1hbmPYdtHSjSwFCCCHqzdYj8mit9Vnj8TkgupbyY4H/XLfsRaXUXqXUa0oplzxvXbR5Cx5BQfglJTk7lOrlZ8P8O4k78SEkj4EZayWJCyHEDUBprWsuoNRKoEUVq54D5mutm1mVvaS1DqtmPzHAXqCl1tpktewc4APMBY5qrf9YzfbTgekA0dHRaYsXL66lanVXWFhIUFBQtesjnv8dlTEtuPyrX9ntNe0p4uJWEg69gdIV7LtpCvlthjs7JLuorV3cRVOpB0hdXFFTqQfUXpfQ0FDat2/vwIgarrKyEk/Phs0cmZWVRX5+/o+WDRgwYIfWuluVG2itG/wDHAZijMcxwOEayj4GzK1hfX/gq7q8blpamranNWvWVLuuPDtbZ3ZM0Lnz59v1Ne2ivETrr2Zq/fsQrd/uo/XFrBrr4m6aSl2aSj20lrq4oqZSD61rr0tmZqZjArGDK1euNHjbquoJbNfV5ERbT61/AUwyHk8CltZQdhzXnVY3jshRSilgFLDfxnjsrmiL5fp4gKvdP55zGN4dCBnvQq9HYdpKiLjZ2VEJIYRwMFsT+WxgiFLqe2Cw8RylVDel1LyrhZRScUBrYO1123+glNoH7AMigT/bGI/dFW/bhmdYGL7xLnI6R2vYuQDe6QeF52H8Ehj2Inj5ODsyIYRo8myZxnT37t306tWLpKQkkpOT+fDDD+0Sk0291rXWucCgKpZvBx6yen4caFVFuYG2vL4jFGdkENCtG8rDBcbOKc23jJN+4FNo2w/umQvBVXVfEEII0RiuTmP661//msjIyHptGxAQwIIFC4iPj+fMmTOkpaUxbNgwmjWzbepoF8hOrst0+jSmM2cIuNUFZjs7lQFv/wwyl8Kg38GDn0sSF0IIB7NlGtMOHToQHx8PQMuWLYmKiiInJ8f2mGzeQxNWlJEBQEB3JyZysxk2vQ6r/wzBLWHqN9C6u/PiEUIIF/DKtlc4lHfIrvtMCE9gVveahkKxsMc0ptu2baO8vJybb7a9b5Mk8hoUb8vAIzQU3w4dnBNAwXn4bAYcWwOJo+DO18HftlMwQgghbGPrNKZnz57lwQcfZP78+deSvS0kkdfAqdfHs1bCZw9DWaElgadOAqUcH4cQQriguhw5N6aGTmN65coVRowYwYsvvkjPnva5G0oSeTVM585hOnWKsPEPOPaFK8ph9R9h098hKhEmfQVRCY6NQQghRI0aMo1peXk5d999NxMnTmT06NF2i0U6u1Wj+Or1cUd2dMs7Bv8cZkni3abCL1ZLEhdCCBdV32lMP/roI9atW8f7779PSkoKKSkp7N5t+1xhckRejeJtGXgEB+OX4KBEum+J5dYyDw+4fyEkjnTM6wohhKgzW6YxnTBhAhMmTLB7TJLIq1GckUFAaiqqgWPl1ll5EXz9DOxeBK17wr3vQrObGvc1hRBCNBmSyKtgunCB8uPHaXaf/a5hVOnsXlgyFXKzoO/T0O9Z8JQmEUIIUXeSNapQsn070IjXx7WGre/Ad89DQARM+gLa9m2c1xJCCNGkSSKvQlFGBh4BAfglJjbCznNh6SNwZDl0GA53/R8ERtj/dYQQQtwQJJFXoTgjA/+0NJSXnf88P6yHT38BxbkwfDb0eFjuDRdCCGETuf3sOhW5uZRnHbXvafXKClj9Isy/E7wD4KGV0POXksSFEELYTBL5dYozrl4f72afHeZnw/w7YN1foMs4mLEOYrrYZ99CCCEcypZpTE+cOEFqaiopKSkkJSXx9ttv2yUmSeTXKd6+HeXvj39Sku07O/glvNUbzu2De96Fu98C3yDb9yuEEMIprk5jWp+BYK6KiYlh8+bN7N69m61btzJ79mzOnDljc0ySyK9TvHMH/l26oHx8Gr4TUwksmwkfToCwOMtRePL9dotRCCGEc9gyjamPjw++vr4AlJWVYTab7ROTXfbSRFQWFlF26DCRDz/c8J3kHIaPp8CFA9DrURj0e/Cy4UuBEEKInzj30kuUHbTvNKa+nRJo8Zvf1FrOlmlMT506xYgRI8jKymLOnDm0bNnS5rglkVsp2bMbzGb8U1Prv7HWsHMBLJ8FPoEwfgnED7F/kEIIIZzKlmlMW7duzd69ezlz5gyjRo1i9OjRREdH2xSPJHIrJTt2gocH/in17IxWmg9fPgYHPoO2/eCeuRDconGCFEIIUacj58bU0GlMr2rZsiW33HIL69evt3kmNEnkVop37sQ3oSOeQfXokJa93TLMan625TR6b2PiEyGEEE1WQ6Yxzc7OJiIiAn9/fy5dusSGDRt44oknbI5FMo5Bm0yU7NlDQGpa3TYwm2H9Xy3TjmoNU7+BPk9KEhdCiBtEfacxHUexRwAADJlJREFUPXjwID169KBLly7069ePp556is6dO9schxyRG0oPHUKXlBCQVofr4wXn4bPpcCwdku6GO/4G/s0aPUYhhBDOZcs0pkOGDGHv3r12j0kSuaF4xw6A2ju6fb8SPpthmX70zjcgdaKM0CaEEMJpJJEbSnbsxDs2Fu/qeg9WlMOqF2DzmxCVBKP/CVEJjg1SCCGEuI4kcgCtKd61i6Det1W9PvcofDINzuyCWx+CoX8Gb/+qywohhBAOJIkc8MzJofLiRfyr6ui292P46nHw8IQxi6DTnY4PUAghhKiGJHLAO+sowI87upUVwtdPw55/Q+uecO88aNbaSREKIYQQVZNEDvgczcIjNBSfdu0sC87uhSVTLKfU+z4D/WaBp/yphBBCuB656RnLEXlA164opWDL2zBvkKVX+qQvYeBzksSFEEIAtk1jetWVK1eIjY3l0UcftUtMNiVypdR9SqkDSimzUqraCbyVUsOVUoeVUllKqWetlrdVSm01ln+olHL47CIVeXl4nT+Pf+cE+M84+GYW3DwQHt4Ibfs4OhwhhBAuzJZpTK96/vnn6du3r91isvWIfD9wD7CuugJKKU/gH8DPgURgnFIq0Vj9CvCa1ro9cAmYZmM89VaycycAAcffgqOrYPgrMG4xBEY4OhQhhBAuzpZpTAF27NjB+fPnGTp0qP1ismVjrfVBsJxqqEF3IEtrfcwouxi4Syl1EBgIPGCUmw/8AXjLlpjqq3jpOygPjV+rABjzIcTUc8IUIYQQDrf+oyNcPFVYe8F6iGwdRJ/7O9RarqHTmJrNZmbOnMmiRYtYuXKl3eJ2xMXfVsApq+fZQA8gArista6wWt6qup0opaYD08EyLF56erpdgoupNOEZH8LGW16g8vAlOGyf/TpLYWGh3f42ztZU6tJU6gFSF1fUVOoBtdclNDSUgoICAEzlJiorK+36+qZy07X910QpxZgxY5gzZw7+/v6UlZVRUFDAyJEjGTlyJACVlZV4enpe26agoIB33nmHQYMGERoaSmlpKeXl5VW+Xmlpab3atNZErpRaCVQ1J+dzWuuldX4lG2mt5wJzAbp166b79+9vnx3360f62rXYbX9Olp6eLnVxMU2lHiB1cUVNpR5Qe10OHjx4bUaxgROSHBTVTwUHBzNr1qxr05j6+voSHBxc6xH5rl27WL9+Pe+99x6FhYWUl5cTHh7O7Nmzf7R/Pz8/unbtWud4ak3kWuvB9algFU4D1jdgxxrLcoFmSikv46j86nLHknHShRBC1FNDpjH94IMPrj1+//332b59+0+SeEM44vazDCDe6KHuA4wFvtBaa2ANcHVG9UmAw47whRBCCFvUdxrTxmLTNXKl1N3A34HmwDKl1G6t9TClVEtgntb6dq11hVLqUWAF4An8U2t9wNjFLGCxUurPwC7gPVviEUIIIRqTLdOYWps8eTKTJ0+2S0y29lr/DPisiuVngNutnn8NfF1FuWNYerULIYQQogFkZDchhBDCjUkiF0IIIdyYJHIhhBBuw9JPuulqSP0kkQshhHALfn5+5ObmNtlkrrUmNzcXPz+/em0n03oJIYRwC7GxsWRnZ5OTk+PsUGpVWlpa74QMli8rsbGx9dpGErkQQgi34O3tTdu2bZ0dRp2kp6fXa3Q2W8ipdSGEEMKNSSIXQggh3JgkciGEEMKNKXfs/aeUygFO2HGXkYDzB8y1D6mL62kq9QCpiytqKvUAqUtN2mitm1e1wi0Tub0ppbZrrbs5Ow57kLq4nqZSD5C6uKKmUg+QujSUnFoXQggh3JgkciGEEMKNSSK3mOvsAOxI6uJ6mko9QOriippKPUDq0iByjVwIIYRwY3JELoQQQrixGyaRK6XuU0odUEqZlVLV9iRUSg1XSh1WSmUppZ61Wt5WKbXVWP6hUsrHMZFXGWO4Uuo7pdT3xu+wKsoMUErttvopVUqNMta9r5T6wWpdiuNrUbd6GOUqrWL9wmq5u7VJilJqs/E+3KuUGmO1zultUt1732q9r/F3zjL+7nFW635tLD+slBrmyLivV4d6PKmUyjTaYJVSqo3Vuirfa85Sh7pMVkrlWMX8kNW6Scb78Xul1CTHRv5TdajLa1b1OKKUumy1zmXaRSn1T6XUBaXU/mrWK6XUG0Y99yqlUq3WNU6baK1viB+gE9ARSAe6VVPGEzgKtAN8gD1AorHuI2Cs8fht4JdOrMtfgGeNx88Cr9RSPhzIAwKM5+8Do12gTepUD6CwmuVu1SZAByDeeNwSOAs0c4U2qem9b1XmV8DbxuOxwIfG40SjvC/Q1tiPpwvXY4DVZ+GXV+tR03vNhesyGXizim3DgWPG7zDjcZgr1+W68v8D/NNF26UvkArsr2b97cByQAE9ga2N3SY3zBG51vqg1vpwLcW6A1la62Na63JgMXCXUkoBA4ElRrn5wKjGi7ZWdxkx1DWW0cByrXVxo0ZVf/WtxzXu2CZa6yNa6++Nx2eAC0CVAzw4QZXv/evKWNdxCTDIaIe7gMVa6zKt9Q9AlrE/Z6i1HlrrNVafhS1A/aaacpy6tEl1hgHfaa3ztNaXgO+A4Y0UZ13Uty7jgP84JLJ60lqvw3JgVJ27gAXaYgvQTCkVQyO2yQ2TyOuoFXDK6nm2sSwCuKy1rrhuubNEa63PGo/PAdG1lB/LTz8ULxqnfV5TSvnaPcK6qWs9/JRS25VSW65eHsDN20Qp1R3LkclRq8XObJPq3vtVljH+7vlY2qEu2zpKfWOZhuXo6aqq3mvOUte63Gu8b5YopVrXc1tHqXM8xqWOtsBqq8Wu1C61qa6ujdYmTWoaU6XUSqBFFaue01ovdXQ8tqipLtZPtNZaKVXtrQfGN8HOwAqrxb/Gkmx8sNwiMQv4o60xV/P69qhHG631aaVUO2C1UmofliTiUHZuk4XAJK212VjssDYRFkqpCUA3oJ/V4p+817TWR6veg0v4EviP1rpMKTUDyxmTgU6OyVZjgSVa60qrZe7WLg7VpBK51nqwjbs4DbS2eh5rLMvFcnrEyzgSubq80dRUF6XUeaVUjNb6rJEULtSwq/uBz7TWJqt9Xz1yLFNK/Qt4yi5BV8Ee9dBanzZ+H1NKpQNdgU9wwzZRSoUAy7B8udxitW+HtUk1qnvvV1UmWynlBYRi+WzUZVtHqVMsSqnBWL6A9dNal11dXs17zVkJo9a6aK1zrZ7Ow9JX4+q2/a/bNt3uEdZdfd4jY4FHrBe4WLvUprq6NlqbyKn1H8sA4pWlN7QPljfUF9rSU2ENlmvNAJMAZx7hf2HEUJdYfnKtyUg0V68zjwKq7H3pALXWQykVdvU0s1IqEugNZLpjmxjvqc+wXD9bct06Z7dJle/968pY13E0sNpohy+AscrSq70tEA9sc1Dc16u1HkqprsA7wEit9QWr5VW+1xwW+U/VpS4xVk9HAgeNxyuAoUadwoCh/PisnKPV5f2FUioBS0ewzVbLXK1davMFMNHovd4TyDe+qDdem9irJ5+r/wB3Y7kmUQacB1YYy1sCX1uVux04guXb3nNWy9th+eeUBXwM+DqxLhHAKuB7YCUQbizvBsyzKheH5Vugx3Xbrwb2YUkWi4AgV60HcJsR6x7j9zR3bRNgAmACdlv9pLhKm1T13sdyen+k8djP+DtnGX/3dlbbPmdsdxj4ubPaoY71WGn8D7jaBl/U9l5z4bq8DBwwYl4DJFhtO9VoqyxgiqvXxXj+B2D2ddu5VLtgOTA6a3yWs7H0s3gYeNhYr4B/GPXch9VdUo3VJjKymxBCCOHG5NS6EEII4cYkkQshhBBuTBK5EEII4cYkkQshhBBuTBK5EEII4cYkkQshhBBuTBK5EEII4cYkkQshhBBuTBK5EKJaSqlApdQhpdQ2pZS31fKhSimzUuqRmrYXQjQ+GdlNCFEjY2zyLcBrWutnlVLRWIbL3Kq1ruv82EKIRiKJXAhRK6XUE8CrwDAsM7N1BrporS86NTAhhCRyIUTtjFnZlmGZ69oHGKK1XuXcqIQQINfIhRB1oC3f+BcCvsAeSeJCuA5J5EKIWimlWgCvAzuBLkqpx5wckhDCIIlcCFEj47T6fKAMGAz8DXhFKZXs1MCEEIBcIxdC1EIpNRP4CzBQa71WKeWDpRe7L9BNa13i1ACFuMHJEbkQolpKqVTgJeBlrfVaAK11OTAOiAP+6rzohBAgR+RCCCGEW5MjciGEEMKNSSIXQggh3JgkciGEEMKNSSIXQggh3JgkciGEEMKNSSIXQggh3JgkciGEEMKNSSIXQggh3JgkciGEEMKN/X8AZcRYhZiuuwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from scipy.special import legendre\n",
    "x = numpy.linspace(-1,1,100)\n",
    "\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1,1,1)\n",
    "for N in range(5):\n",
    "    Pn = legendre(N)\n",
    "    axes.plot(x, Pn(x), label='N={}'.format(N))\n",
    "\n",
    "axes.grid()\n",
    "axes.set_title('Legendre Polynomials', fontsize=18)\n",
    "axes.set_xlabel('x',fontsize=16)\n",
    "axes.legend(loc='best')\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Which lead to the quadrature points and weights...\n",
    "\n",
    "<table width=\"80%\">\n",
    "    <tr align=\"center\"><th>$$N$$</th> <th align=\"center\">$$x_i$$</th> <th align=\"center\"> $$w_i$$ </th></tr>\n",
    "    <tr align=\"center\"><td>$$1$$</td>           <td> $$0$$ </td> <td> $$2$$ </td> </tr>\n",
    "    <tr align=\"center\"><td>$$2$$</td>           <td> $$\\pm \\sqrt{\\frac{1}{3}}$$ </td> <td> $$1$$ </td> </tr>\n",
    "    <tr align=\"center\"><td rowspan=2>$$3$$</td> <td> $$0$$ </td> <td> $$8/9$$ </td> </tr>\n",
    "    <tr align=\"center\">                     <td> $$\\pm \\sqrt{\\frac{3}{5}}$$ </td> <td> $$5/9$$</td> </tr>\n",
    "    <tr align=\"center\"><td rowspan=2>$$4$$</td> <td> $$\\pm \\sqrt{\\frac{3}{7} - \\frac{2}{7} \\sqrt{\\frac{6}{5}}}$$</td> <td> $$\\frac{18 + \\sqrt{30}}{36}$$ </td> </tr>\n",
    "    <tr align=\"center\">                     <td> $$\\pm \\sqrt{\\frac{3}{7} + \\frac{2}{7} \\sqrt{\\frac{6}{5}}}$$</td> <td>$$\\frac{18 - \\sqrt{30}}{36}$$ </td> </tr>\n",
    "    <tr align=\"center\"><td rowspan=3>$$5$$</td> <td> $$0$$ </td> <td> $$\\frac{128}{225}$$ </td> </tr>\n",
    "    <tr align=\"center\">                     <td> $$\\pm \\frac{1}{3} \\sqrt{5 - 2 \\sqrt{\\frac{10}{7}}}$$</td> <td> $$\\frac{322 + 13\\sqrt{70}}{900}$$</td> </tr>\n",
    "    <tr align=\"center\">                     <td> $$\\pm \\frac{1}{3} \\sqrt{5 + 2 \\sqrt{\\frac{10}{7}}}$$</td> <td> $$\\frac{322 - 13\\sqrt{70}}{900}$$</td> </tr>\n",
    "</table>\n",
    "\n",
    "Note: weights always add up to 2 for \\[-1,1\\] for each N."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "##### Example 2:  2-Point Gauss-Legendre Quadrature\n",
    "\n",
    "Let $N=2$ on $x \\in [-1,1]$\n",
    "\n",
    "$$I_2[f] = w_0 f(x_0) + w_1 f(x_1)$$\n",
    "\n",
    "From above table we are told that $x_0 = -\\sqrt{\\frac{1}{3}}$, $x_1 = \\sqrt{\\frac{1}{3}}$ and $w_0 = w_1 = 1$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "We can actually solve for the 2 weights and 2 locations using the method of *undetermined coefficients*\n",
    "\n",
    "I.e.  Assume \n",
    "$$\n",
    "    I_2[f] = w_0 f(x_0) + w_1 f(x_1)\n",
    "$$\n",
    "\n",
    "and solve for $w_0, w_1, x_0, x_1$ that exactly integrate the first 4 monomials, $1, x, x^2, x^3$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "    \n",
    "$$\\begin{aligned}\n",
    "    &\\text{if}~f = 1:  &I[f] =  \\int^{1}_{-1} 1 dx = 2 & & I_2[1] &= w_0 + w_1\\\\\n",
    "    &\\text{if}~f = x:  &I[f] =  \\int^{1}_{-1} x dx = 0 & & I_2[x] &= w_0 x_0 + w_1 x_1\\\\\n",
    "    &\\text{if}~f = x^2:  &I[f] =  \\int^{1}_{-1} x^2 dx = \\frac{2}{3} & & I_2[x^2] &= w_0 x_0^2 + w_1 x_1^2\\\\\n",
    "    &\\text{if}~f = x^3:  &I[f] =  \\int^{1}_{-1} x^3 dx = 0 & & I_2[x^3] &= w_0 x_0^3 + w_1 x_1^3\\\\\n",
    "\\end{aligned}$$\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    &w_0 + w_1 = 2\\\\\n",
    "    &w_0 x_0 + w_1 x_1 = 0\\\\\n",
    "    &w_0 x_0^2 + w_1 x_1^2 = \\frac{2}{3}\\\\\n",
    "    &w_0 x_0^3 + w_1 x_1^3 = 0\\\\\n",
    "\\end{aligned}$$\n",
    "\n",
    "Note that we need to solve for 4 unknowns $x_0$, $x_1$, $w_0$, and $w_1$.  Solving these non-linear equations leads to\n",
    "\n",
    "$$x_0 = -\\sqrt{\\frac{1}{3}}, x_1 = \\sqrt{\\frac{1}{3}}  \\quad \\text{and} \\quad w_0 = w_1 = 1 $$\n",
    "\n",
    "Note: Integration is linear transformation\n",
    "\n",
    "Note: Quadrature points for GL are never at endpoints of interval"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "##### Example: Gauss-Legendre Quadrature single interval\n",
    "\n",
    "let $f(x) = \\cos(x)$:\n",
    "\n",
    "$$\n",
    "    I[f] = \\int_0^{\\pi/2} f(x) dx = \\left.\\sin(x)\\right|_0^{\\pi/2} = 1.\n",
    "$$\n",
    "\n",
    "so the affine map from $x\\in[0,\\pi/2]\\rightarrow\\xi\\in[-1,1]$ is\n",
    "\n",
    "\\begin{align}\n",
    "x &= \\frac{\\pi}{4}\\xi + \\frac{\\pi}{4} \\\\\n",
    "&= \\frac{\\pi}{4}\\left(\\xi + 1\\right)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 360x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = lambda x: numpy.cos(x)\n",
    "I = 1.\n",
    "\n",
    "x = numpy.linspace(-1.,2, 100)\n",
    "x_interval = numpy.linspace(0., numpy.pi/2, 100)\n",
    "fig = plt.figure(figsize=(5,5))\n",
    "axes = fig.add_subplot(1,1,1)\n",
    "axes.plot(x,f(x),linewidth=3)\n",
    "axes.fill_between(x_interval,0.,f(x_interval))\n",
    "axes.plot(x, numpy.zeros(x.shape),'k')\n",
    "\n",
    "axes.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "def glintegration(f, a, b, N):\n",
    "    \"\"\" Approximate \\int_a^b f(x)dx using  Gauss-Legendre quadrature rules with N<=4 quadrature points\"\"\"\n",
    "        \n",
    "    # Build up a nested list of gauss points and weights:\n",
    "    xGl = []\n",
    "    wGl = []\n",
    "    # 1 point Gauss-legendre quadrature\n",
    "    xGl.append(numpy.array([0.]))\n",
    "    wGl.append(numpy.array([2.]))\n",
    "    # 2 point Gauss-legendre quadrature\n",
    "    xGl.append(numpy.array([ -1./numpy.sqrt(3.), 1./numpy.sqrt(3)]))\n",
    "    wGl.append(numpy.array([ 1. , 1.]))\n",
    "    # 3 point Gauss-legendre quadrature\n",
    "    xGl.append(numpy.array([-numpy.sqrt(3. /5.), 0., numpy.sqrt(3. / 5.)]))\n",
    "    wGl.append(numpy.array([5., 8., 5.])/9.)\n",
    "    # 4 point Gauss-Legendre quadrature\n",
    "    xGl.append(numpy.array([-numpy.sqrt(3.0 / 7.0 - 2.0 / 7.0 * numpy.sqrt(6.0 / 5.0)), \n",
    "           numpy.sqrt(3.0 / 7.0 - 2.0 / 7.0 * numpy.sqrt(6.0 / 5.0)),\n",
    "          -numpy.sqrt(3.0 / 7.0 + 2.0 / 7.0 * numpy.sqrt(6.0 / 5.0)),\n",
    "           numpy.sqrt(3.0 / 7.0 + 2.0 / 7.0 * numpy.sqrt(6.0 / 5.0))]))                 \n",
    "    wGl.append(numpy.array([(18.0 + numpy.sqrt(30.0)) / 36.0, (18.0 + numpy.sqrt(30.0)) / 36.0,\n",
    "         (18.0 - numpy.sqrt(30.0)) / 36.0, (18.0 - numpy.sqrt(30.0)) / 36.0]))\n",
    "    \n",
    "    # make affine map between x and xi\n",
    "    xi_map = lambda xi: (b - a)/2. * xi + (a + b) / 2.\n",
    "    \n",
    "    I = (b - a)/2. * numpy.sum(wGl[N-1]*f(xi_map(xGl[N-1])))\n",
    "    return I\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "hide_input": false,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N = 1, I = 1.1107207345395915, err = 0.11072073453959153\n",
      "N = 2, I = 0.9984726134041148, err = 0.0015273865958852495\n",
      "N = 3, I = 1.0000081215554983, err = 8.121555498341593e-06\n",
      "N = 4, I = 0.9999999771971155, err = 2.2802884469541596e-08\n"
     ]
    }
   ],
   "source": [
    "a = 0.\n",
    "b = numpy.pi/2\n",
    "I_true = 1.\n",
    "err = numpy.zeros(4)\n",
    "for N in range(1,5):\n",
    "    I = glintegration(f, a, b, N)\n",
    "    err[N-1] = numpy.abs(I_true - I)\n",
    "    print('N = {}, I = {}, err = {}'.format(N,I, err[N-1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N = 1, I = 1.1107207345395915, err = 0.11072073453959153\n",
      "N = 2, I = 0.7853981633974483, err = 0.21460183660255172\n",
      "N = 3, I = 1.0022798774922104, err = 0.0022798774922103693\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "errNC = numpy.zeros(4)\n",
    "for N in range(1,4):\n",
    "    I = ncintegration(f, a, b, N)\n",
    "    errNC[N-1] = numpy.abs(I_true - I)\n",
    "    print('N = {}, I = {}, err = {}'.format(N,I, errNC[N-1]))\n",
    "    \n",
    "fig = plt.figure(figsize=(8, 6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "axes.semilogy(range(1,5),errNC,'bs', markersize=10, label='Newton-Cotes')\n",
    "axes.semilogy(range(1,5),err,'ro', markersize=10, label='Gauss-Legendre')\n",
    "axes.set_ylabel('Error')\n",
    "axes.set_xlabel('quadrature points (N)')\n",
    "axes.legend(loc='best')\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "I_true = 1279.4999999999998\n",
      "N = 1, I = 964.21875\n",
      "N = 2, I = 1262.625\n",
      "N = 3, I = 1279.5\n",
      "N = 4, I = 1279.5000000000002\n"
     ]
    }
   ],
   "source": [
    "# Cubic f(x): Exact interpolation at/after 2N-1 (N=2)\n",
    "f = lambda x: - 4*x**3 - 3.*x**2 + 2.*x +300.\n",
    "int_f = lambda x:  - x**4 - x**3. + x**2 + x*300.\n",
    "\n",
    "# 4th order f(x): N=3\n",
    "f = lambda x: x**5 - 4*x**3 - 3.*x**2 + 2.*x +300.\n",
    "int_f = lambda x: x**6/6. - x**4 - x**3. + x**2 + x*300.\n",
    "\n",
    "a = 1.\n",
    "b = 4.\n",
    "I_true = int_f(b) - int_f(a)\n",
    "print('I_true = {}'.format(I_true))\n",
    "for N in range(1,5):\n",
    "    I = glintegration(f, a, b, N)\n",
    "    print('N = {}, I = {}'.format(N,I))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "hide_input": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = numpy.linspace(a - 1, b + 1, 100)\n",
    "x_interval = numpy.linspace(a, b, 100)\n",
    "fig = plt.figure(figsize=(8,6))\n",
    "axes = fig.add_subplot(1,1,1)\n",
    "axes.plot(x,f(x),linewidth=3)\n",
    "axes.fill_between(x_interval,0.,f(x_interval))\n",
    "axes.plot(x, numpy.zeros(x.shape),'k')\n",
    "\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([6.97744637e-06, 2.52446381e-05, 4.78241289e-05, 6.60913206e-05,\n",
       "       7.30687670e-05, 6.60913206e-05, 4.78241289e-05, 2.52446381e-05,\n",
       "       6.97744637e-06, 1.66533454e-16])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute Gauss-Legendre based quadrature and affine transforms\n",
    "f = lambda x: numpy.sin(x)\n",
    "I = lambda x: 1.0 - numpy.cos(x)\n",
    "x = numpy.linspace(0.0, 2.0 * numpy.pi, 100)\n",
    "\n",
    "num_partitions = 10\n",
    "x_hat = numpy.linspace(0.0, 2.0 * numpy.pi, num_partitions + 1)\n",
    "delta_x = x_hat[1] - x_hat[0]\n",
    "\n",
    "xi_map = lambda a,b,xi : (b - a) / 2.0 * xi + (a + b) / 2.0\n",
    "xi_0 = -numpy.sqrt(1.0 / 3.0)\n",
    "xi_1 =  numpy.sqrt(1.0 / 3.0)\n",
    "I_hat = numpy.zeros(x_hat.shape)\n",
    "I_hat[0] = (f(xi_map(x_hat[0], x_hat[1], xi_0)) + f(xi_map(x_hat[0], x_hat[1], xi_1))) * delta_x / 2.0\n",
    "for i in range(1, num_partitions):\n",
    "    I_hat[i] = I_hat[i - 1] + (f(xi_map(x_hat[i], x_hat[i+1], xi_0)) + f(xi_map(x_hat[i], x_hat[i+1], xi_1))) * delta_x / 2.0\n",
    "    \n",
    "error = numpy.abs(I_hat - I(x_hat + delta_x))\n",
    "error[:-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "    \n",
    "fig = plt.figure()\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "axes.plot(x, I(x), 'r')\n",
    "# Offset due to indexing above\n",
    "axes.plot(x_hat + delta_x, I_hat, 'ko')\n",
    "\n",
    "axes.set_xlabel(\"x\")\n",
    "axes.set_ylabel(\"$f(x)$\")\n",
    "axes.set_title(\"Integral and Approximated Integral\")\n",
    "axes.set_xlim((0.0, 2.0 * numpy.pi))\n",
    "axes.set_ylim((-0.1, 2.5))\n",
    "axes.grid()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "##### Example 3:\n",
    "\n",
    "If $f(x) = e^x$ look at the relative accuracy of midpoint, trapezoidal, Simpson's and 2-point Gauss-Legendre quadrature for a single interval $x \\in [-1,1]$.\n",
    "\n",
    "$$\\begin{aligned}\n",
    "    \\text{Exact:} &I[f] = \\int^1_{-1} e^x = \\left . e^x \\right |^1_{-1} = e - \\frac{1}{e} \\approx 2.350402387 \\\\\n",
    "    \\text{Midpoint:} &I_2[f] = 2 e^0 = 2 \\\\\n",
    "    \\text{Trapezoid:} &I_2[f] = \\frac{2}{2} (e^{-1} + e^1) = e + \\frac{1}{e} = 3.08616127 \\\\\n",
    "    \\text{Simpson's:} &I_2[f] = \\frac{2}{6} e^{-1} + \\frac{4}{3} e^0 + \\frac{2}{6} e^1 = \\frac{4}{3} + \\frac{1}{3} (e^{-1} + e^1) \\approx 2.362053757 \\\\\n",
    "    \\text{Gauss-Legendre:} &I_2[f] = e^{-\\sqrt{\\frac{1}{3}}} + e^{\\sqrt{\\frac{1}{3}}} \\approx 2.342696088\n",
    "\\end{aligned}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "hide_input": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8IMRK/e/pNRXaXHJyMvv27WPhwoV069aN+vXr07ZtW6ZMmcKgQYMA9Rb6hAkTNNe4uLgwe/ZsRowYgbm5OXXr1mX16tUkJyczaNAgzMzMaNiwIeHh4Zpr9u3bh6IobN68mRYtWmBkZETr1q05fvy45pw7d+4wfPhw7OzsMDIyws3NjdB8mX0uX75M//79MTc3x9zcnBdffJGrV69qjoeEhODl5cWvv/6Ku7s75ubmvPDCCyQlJZX4+ffu3YuPjw9mZmZYWlrSrl07zpw5o5XvrVQz5U0ci09OQ6BOHJvz6xF2rd6h7tkNarDu0UMNukZG0KiROknsjz/U47VqqTO1x42DL76A7dvVpVedO6vHO3eGL79Ub5n3768G/nzrpvO80NKJBS82w8nKGAVwsjJmwYvN5GxuSSvkCDy/02tg00TIevAX950r6nsA74EV0qSZmRlmZmZs3LiRTp06YVR4skoJQkNDmTt3LjNnzuSrr77itddew9/fn0GDBjF37lwWLFjAsGHDuHz5coE6p0yZwtKlS3FycuLDDz+kT58+REdHY2Jiwnvvvcdff/3F5s2bqVOnDrGxsdx8sFQlNzeXfv36YWxszO7duwGYMGECL7zwAkePHkV5cGswLi6O1atX89tvv3Hv3j0GDRrEzJkz+frrr4t8huzsbPr168eoUaNYtWoVWVlZnDhxAt38zwIl6VHkZb3KyeGTsAvopqaweMeX1E++Tv3kBGrfS1bPmztXfT5tagp37kD79jBkyMOkGs2aqec1a6ZmzNICufxKqjBCiGrzat26tSjJuXPnSjz2yD5tKsQHFkVfnzZ98rpLsW7dOlGrVi1haGgofHx8xOTJk8WhQ4c0x7t06SLGjx+veV+/fn0xaNAgzfuUlBQBiDfffFNTFhsbKwBx9OhRIYQQW7ZsEYD4+eefC1xnaWkp/vOf/wghhOjbt694/fXXi+1jeHi40NHREbGxsZqy6OhooSiK2LFjhxBCiA8++EAYGhqK5ORkzTlz584V7u7uxdZ569YtAYiIiIgyv0dl0crPX6rasrOFuH1b/To3V4gZM4QYMECIFi2EMDcXAoQYM0a4TNssXKduEHFW9uJAvWbiF+8A8dFzr4rxz08TIiamcj9DFbR79+7K7oJUBuCYKCYmyhF4fneuPl65lgwYMIDevXuzb98+Dh48yPbt21m8eDHz5s3j3XffLfYab29vzddmZmaYmJjQLG/0ANR5sOtSYmJiges65FsfamZmRrNmzTh37hwA48aN46WXXuL48eP06NGDvn370qVLF0DNt+3o6IiLi4vmejc3NxwdHTl37hzdu3cHoH79+lhaWmrOcXR0LNKHPNbW1owYMYKePXvSrVs3unXrxksvvUS9evXK/J5J1c8jPZfOyXk4G/v77+HEiYcTyGJj1fXRmzc/TFupKOrIuXNn9b/t2+MYcZ/45DS6jPm2QNVOVsbqkixJekbIZ+D5WTo/XrkWGRkZ0aNHD2bNmsWBAwcYNWoUISEhZGZmFnt+4VnpiqIUKMu7pZ2bm/vIfQgMDOTSpUtMmTKFpKQkevfuzeuvv17mdUq+nZ6K61dpffj+++85fPgwzz33HBs3bsTDw4OwsLBH7rNUPRT3XHrjoh85N2E6jBoFfn7qXtz510KvWKHO9k5IUJ8xv/02jBjx8Pj58+pryxb47DMIDgYfH6b29MBYv+BjGDlxTHoWyRF4ft1mFXwGDqBvrJY/ZU2aNCE7O5v09HSt1nvo0CHc3NwAuHfvHmfOnOHVV1/VHLe1tWX48OEMHz6cwMBABg8ezFdffYWnpyfXrl0jLi5OMwqPiYnh2rVrNGnS5In61Lx5c5o3b860adMIDAzkhx9+oGfPnk9Up1QFnDsHhw9DdDSmmw7wy82rWKWl4jfmPwD0OxlGk8g96h7d7u7QtSt4eT28fvt2NUVl4a1A85RQLvftlmoKGcDzy5uotnO2etvc0lkN3hU0gQ3g1q1bvPzyy4wcORJvb2/Mzc05duwYH3/8Md26dcPCwkKr7c2dO5fatWvj6OjI7NmzMTAwYMiQIQDMmjWLVq1a0bRpU7Kzs1m/fj1ubm4YGhrSvXt3vL29GTp0KEuXLgXgzTffpFWrVvj7l289a2xsLF9//TXPP/88Tk5OxMTEcPr0acaNG6e1zyuVzyPd7r5xA06dKppU49AhdUb2ypXwySegq0sj89pcsnLgTJ0G6OVkk62rx+xub/Burwmc/fSl4jthalru/suJY1JNIAN4Yd4DKzRgF2ZmZoaPjw9Lly4lKiqKjIwMnJycGDJkCO+9957W21u4cCGTJ0/mwoULNG3alM2bN2P64BeloaEhM2fOJDY2FiMjI3x8fNi0aROg3grfsGEDEydOpGvXrgB0796dzz//vMAt9MdhYmLCxYsXefnll0lKSqJOnToMHTqUadOmaefDSuWSd7s79/59Gt+Op96F6/z9x3+JtcrA9c51NW+0uzusWQMTH6zSMDZWyxo0gNRUNYBPnAhjxkC9egxZvI/45LQC7dwytVKfS0uSVC6KyLfVX1XXpk0bcezYsWKPRUZG4unp+ZR7VH1s3bqV3r17c/PmTWxtbSu7O1onf/6qx9rARAg1o1WhvNGjzH3YaV6fgIsH+ea3eZrT7xibY+nVGL7+Glq2VNdUx8aqgbu4rFeF+jWjmA1N5JroyhdROJ2oVOUoinJcCNGmcLkcgUvSM6K4zFcz153C5NoVAozuPQzSPXuqu4mdPAmtWz+sQFHA2RndtvXBvD4nnBoz/vlpXKrlwGUre1KMzIhdmG+Pbmdn9fUI5HNpSdI+GcAlqQopV+7o9HSIjeV/oesZfPUS0TbO7HFrjWVaCkeXDccgN/vhuQYGYGenBvCGDSE09OFWoA+yXp1duAuS00gyrcUWz86aS5/0drd8Li1J2iUDeA3RuXNnqtPjkpqoxNzRwAsuJg9H0Obm8K9/qbfAPTzUCWRC8MmDen7xDmCPW2vuGJnxdfsBXLOozYJ3+qvPp52cHq6zNjdXl14VMrWnR7G3u+UyLEmqWmQAl6QK8Ngj6dxcvl+zn2aX4zDKymCvm3pr+9M1c/Bd/BekpTw8189PDeCKAi++qC61cnfnjf8lc0yvFreNH6xcUBQWPzdcHTk/xkoBebtbkqoHGcAlSctKGkkrWVn0s85WNybJS4rx3nvqntsxMWx4kPUqzspBs1Y6xtqJWyaWDBvqXzDrVZ6FCzVf9m4Sz/71f4EWRs7ydrckVX0ygEuSln0SdkETvF/66w/6nY2gfnICjvNvQm6umv3q3j3Q0QE9PfXWds+eLIrK5rShDXG1HB/W1eU1nKyMGTal7BG0HDlLUs0iA7gkadm1fOudLdNSMM+8z0nHxmyw8uPN0b3UUXTefISQEM25DU7G890TPnuWI2dJqjlkAJckLXO0MtZsWvJdu/58164/oM7ifnNEySNpOYKWJOlxyAAuSVr2JLO45QhakqRHJbORSU/NiBEj6NOnT2V3o8K90NKJBS82w8nKGAV15C13HJMkSdsqPYArimKqKMoxRVGe/d/sJbhx4waTJk2iYcOGGBkZYWdnR8eOHfn8889JTU0t9VpFUVi3bt0jtZOVlcW0adPw9vbG1NQUBwcHhgwZwuXLl7XxMcq0dOlSfv7558e6xsXFhUWLFlVQjyrOCy2d+N90f2IX9uZ/0/1l8JYkSevKHcAVRVmhKEqioihnCpX3UhTlgqIoUYqiTH+EqqYBa8rbj+ouLi6OVq1asX37dubMmcOJEyc4fPgw7777Ljt37mTjxo1aa+v+/fucOHGCmTNncuLECTZs2MCVK1fo1asX2dnZZVfwhCwtLbGysqrwdiRJkmoEIUS5XsBzQCvgTL4yXSAacAMMgD+BJkAzYHOhlx3QAxgEjAD6lNVm69atRUnOnTtX4rHHsTl6s+ixtodotrKZ6LG2h9gcvVkr9ZakV69ewtnZWaSmphZ7PDc3t9TrAbF27doy27l7926x5WfPnhWAOH36dInXxsbGCkCsWrVK+Pr6CkNDQ+Hh4SHCwsIKnLdnzx7Rrl07YWhoKOzs7MRbb70lMjIyNMdfe+010bt3b837Ll26iHHjxokZM2YIGxsbUbt2bTF58mSRk5OjOQ4UeJVEWz9/Sappdu/eXdldkEry52ohPm0qWjvoCFFMTCz3CFwIsRf4p1BxOyBKCBEjhMgEfgX6CSH+EkL0KfRKBPwAH2AI8IaiKJV6S39LzBZCDoSQcC8BgSDhXgIhB0LYErOlQtq7desWYWFhjB8/XpPSs7Dypup8VHfv3gWgVq1aZZ77zjvvMHHiRE6dOkWPHj3o168f8fHxAMTHxxMYGEjLli05efIk3333Hb/88gszZswotc5Vq1ahp6fHgQMHWLZsGaGhoaxevRqA9evX4+zszKxZs0hISCAhIeEJP60kSVI1cXoNbJoId66UeIq2Z6E7Aflbuwq0L+lkIcRMAEVRRgBJQojcwucoijIaGA1Qp04dIiIiiq3L0tKSlJSUYo89qiXHlpCek16gLD0nnSXHlvBc7eeeqO7i/PnnnwghqFevXoG+N27cmDt37gDwyiuvEBoaWmo9aWlpZX72nJycIudkZmby1ltvERgYWOr3L+85/MiRIwkMDARg7ty5bNu2jdDQUGbNmkVoaCj29vZ89NFH6Ojo4OzsTEhICMHBwbzzzjuYmJiQlZVFdna2pp2cnBw8PDyYOnUqAA4ODjz33HNs376dPn36oK+vj46ODgYGBpo/cErqY3p6eon/NiRJKllqaqr8f6cK8jn4LvFZ97hJkbCoUSWWkQkhVpZy7BvgG1DzgZeUtzYyMhJzc/Mn6kdiWmKJ5U9ad3HygpKxsXGB+vfv309OTg6jR48mJycHc3Nzxo4dW2ACWP7JbYWvL05KSkqBc7Kzsxk1ahQpKSls2bJFc6y4dszMzADw8/MrUEeHDh2Ijo7G3Nyc6OhoOnbsiKWlpeZ49+7dyczM5MaNG3h7e6Ovr4+enp6mDl1dXVq2bFmgzrp163L79m1NmaIoGBoalvn5jIyMaNmyZannSJJUlMwHXnXcSb/D7rjdhEeHMyojlhglF3N9Q7INDYo9X9sBPB6om++984OyasHe1J6Ee0Vv09qb2ldIew0aNEBRFM6fP1+g3NXVFQATExNN2ezZs5kyZYpW2s3Ozmbw4MH89ddfREREYGNjUyHt5CntMYC+vn6Rc3NzS/6LU5Ik6VmRk5vD6RunaemgDj5GbhzJ+sj1mBmY0dTQlGxTUwxNzLlC8bfRtf3M+SjQUFEUV0VRDFAnqGlvGnUFC24VjJGuUYEyI10jglsVTbmoDTY2NgQEBLBs2bIyl4vZ2dnRoEEDzau8srKyeOWVVzh9+jS7d+/G3r7gHyeltXPo0CHN10IIjhw5gqenJwCenp4cOnSoQPDdv38/BgYGuLu7l7u/BgYG5OTklH2iJElSNXAp+RL/Of4fXl77Mraf2NLqm1ZcvXsVgGm+09gzYg+33rmFRX1vLCxtMdQ3LLGuco/AFUX5BXUSmq2iKFeBD4QQ3ymKMgEIQ52RvkIIcba8bTxtvd16A7D0xFKu37uOvak9wa2CNeUVYfny5fj6+tK6dWtCQkJo3rw5enp6HD9+nD///JOAgIAy64iLi+PUqVMFytzc3LCwsChQlp2dzcsvv8zRo0fZtGkTiqJw/fp1QJ1DYGxsXGo7X375JY0aNaJZs2YsX76cS5cuMW7cOACCgoIIDQ0lKCiI4OBgYmJimD59OhMmTChwJ+Fxubi4sG/fPoYNG4ahoSG2trblrkuSJOlpS81MRQiBuaE5a86u4ZV1rwDgZO5E/8b96eneEysjdXltO6d2muuuZ90tu/LipqZX1dfTWEZWGRISEsTEiROFu7u7MDAwEKampqJNmzZi/vz5JS7/ykOhZVZ5r02bNhU47+7du5rlYMW9vv/++xLbyLvu559/Fh06dBCGhoaiUaNGYuvWrQXOy1tGZmBgoFlGlp6erjle3DKy8ePHF6ij8DkHDx4U3t7ewtDQUC4jk6QKIJeRaVdObo44Fn9MzN87X/it9BP6s/XFF0e+EEIIkZCSID498Kk4m3i2zCXCPdb2EAY7R6oAACAASURBVF4rvYTXSi9h5GIkRDExURF5WZGqgTZt2ohjx44VeywyMlJzO1cqqvAktscRFxeHq6srR48epU2bNlrumXbIn78klY+cxPbkMrIzMNQz5H7WfVyXupJ4T50Q3cK+BQFuAQxuNpgW9i0eq868Zc3pOelEhUSRFptWZDJRlZiFLkmSJEnVRVpWGvsu7yMsKozwmHCcLZzZNnQbJvomjG41Gg9bD3q49aCOWZ1yt5H/kW4UUcWeIwO4JEmSJJVCCKFZTTM5bDJfHP2CjJwMDHUN6Vy/M/9q8C/NuXP852it3d5uvent1hvlZeV4ccdlAJfK5OLiQnV61CJJkvSkEu8lsiN6B+Ex4UTERXA26CxmBmY0tm1MUNsgAtwDeK7+c5jol3+S7pOSAVySJEmSHtgTt4dJYZM4ef0kADbGNvRw70FyejJmBma80fqNSu7hQzKAS5IkSTWOEILzSecJjw4nPCacsa3H0tejL7WMa2FuaM48/3kEuAfQ0r4lujq6ld3dYskALkmSJNUYaVlpvLntTcKjw7lyV93hrKF1Q1Iz1c20vOt4s2fEnsrs4iOTAVySJEl6JmXlZHHo6iHCo8Mx0DXg/S7vY6RnxNFrR2nn1I73nnuPHm49cK3lWtldLRcZwCVJkqRnyuozq/nlzC/sit1FSmYKOooO/Tz6AWq+hVNjTlV4quanQQZwSZIkqdrKy+AVERfB4oDF6Orosv/yfv688SdDmg0hwD0Af1d/zXalUHqCpepEBnDpqRkxYgRJSUls3ry5srsiSVI1Fns7lp9O/0R4dDiHrh4iR+RgZmDG+LbjaWjTkEUBi/gs8LNnJlCXRNvZyKRyuHHjBpMmTaJhw4YYGRlhZ2dHx44d+fzzz8vMUqYoCuvWrXvktt5//30aN26MqakptWrVolu3bhw4cOBJP8IjWbp0aYFc44/CxcWFRYsWVVCPJEmqDvIyeJ1JPAPA3//8TUhECJk5mUzvNJ09I/bwzzv/0NCmIQCGeobPfPAGOQKvdHFxcfj6+mJhYcGcOXPw9vbG2NiYs2fP8u2332JjY8OQIUO01p6HhwdffPEFrq6upKWlsWTJEnr16sXff/9NnTrl3/bvUVhaWlZo/ZIkPRuycrIIiw4jPDqcsOgwLt66CMDcrnPxsvPCz8WPxKmJ2JrU8OyExWU4qaqvp5GNLHnjRnGxq78419hTXOzqL5I3btRKvSXp1auXcHZ2FqmpqcUeLytjDSDWrl1bZjslZTW7c+eOAMT27dtLvDYvG9mqVauEr6+vMDQ0FB4eHiIsLKzAeXnZyAwNDTXZyDIyMjTHi8tGNm7cODFjxgxhY2MjateuLSZPnixycnI0xymUNa0kMhuZJJVPVchGlpfB64/oP4QQQmRmZwqz+WbCeK6xCPw5UCw5uOSRMng9q4BjopiYKEfg+dzZtImE92ch0tMByL52jYT3ZwFg2bev1tu7desWYWFhzJ8/H1NT02LPqcjbQJmZmXzzzTdYWFjQokXZmXLeeecdPv30U7y9vfniiy/o168fUVFRODk5ER8fT2BgIMOHD2flypVER0fz73//Gx0dHRYvXlxinatWrSI4OJgDBw5w6tQphgwZQuvWrRk8eDDr16+nefPmjBw5UpN3XJKkZ0P83Xh2xOwgPDqcHTE7SLqfRDO7Zpwedxp9XX32v74fD1sPjPSMKrurVZZ8Bp5P4pJQTfDOI9LTSVwSWiHtRUVFIYTAw8OjQLmzszNmZmaYmZkxduxYrbe7efNmzMzMMDIyYsmSJezYseORbp+PGzeOgQMH0rhxY5YuXUrdunX58ssvAVi+fDmOjo4sX74cT09P+vTpw8KFC1m2bBn3798vsc4mTZowe/ZsGjVqxMCBA+natSs7d+4EwNraGl1dXczNzbG3t8fe3l473wBJkp66tKw0IuIiNO/fDn+b1ze8zq7YXQQ2COTn/j+zY/gOzfHm9s1l8C6DHIHnk52Q8FjlFWXfvn3k5OQwevRo0h/8QTF27NgCE8DKmtxWmq5du3Lq1CmSkpL4z3/+w8CBAzl48CAODg6lttOhQwfN1zo6OrRv355z584Baj5uHx8fdHQe/k3YqVMnMjMziYqKwtvbu9i+FC53dHQkMTGx3J9NkqSqQQjBX4l/qVuVRoez99JeMnIyiJkYg2stV2Z2nsnMzjNpZtesRkw4qwgygOej5+BA9rVrxZZXhAYNGqAoCufPny9Q7uqq7gpkYvIwy83s2bOZMmWKVto1NTWlQYMGNGjQAB8fHxo2bMi3337L+++/r9V28pT2P6e+vn6Rc3Nzc7XaviRJT8eN1BsY6BpQy7gWa86uYdD/DQKgae2mmgxeDubq71PvOsX/US89OnkLPR+7SW+hGBW8ZaMYGWE36a0Kac/GxoaAgACWLVtW5ojazs5OE3QbNGig1X7k5uaSkZFRZjuHDh3SfC2E4MiRI3h6egLg6enJoUOHCgTf/fv3Y2BggLu7e7n7ZmBgQE5OTrmvlyRJS06vgSVeEGKl/vf0GjKyM9gVu4tpO6bR8uuW2C+256fTPwHQza0bK55fwdVJVzkTdIZPe35Krwa95G1xLZIj8HzyJqolLgklOyEBPQcH7Ca9VSET2PIsX74cX19fWrduTUhICM2bN0dPT4/jx4/z559/EhAQUGYdcXFxnDp1qkCZm5sbFhYWBcru3r3Lxx9/TN++fXFwcODmzZt88cUXXL16lYEDB5bZzpdffkmjRo1o1qwZy5cv59KlS5rJZUFBQYSGhhIUFERwcDAxMTFMnz6dCRMmFLiT8LhcXFzYt28fw4YNw9DQEFvbGr5sRJIqw+k1sGkiIus+dwHLO1dI2/gmdTaNICU7DT0dPXzr+moyeAHYmtjyesvXK7ffzzgZwAux7Nu3QgN2YW5ubpw8eZIFCxbw/vvvc+XKFfT19fH09CQoKIgJEyaUWcfUqVOLlG3atIk+ffoUKNPT0+Ps2bOsWLGCW7duYWNjQ9u2bdm7d2+Jz6jzW7hwIZ9++iknTpygfv36/Pbbbzg7OwPg5OTEtm3bmDp1Ki1atMDKyoohQ4Ywf/78R/xOFG/27NmMGTMGd3d3MjIyUFdUSJL0tNy6f4ud2ycTnvUP4WTTCB3+wBTj7HTeN7Kg8Uur8XPxw9zQvLK7WuMo1ekXYps2bcSxY8eKPRYZGam5nSsVlZKSgrl5+f4Hi4uLw9XVlaNHj9KmTRst90w75M9fksonIiICPz8/zfuc3BxN/uugLUF8dewrBAJTFEyNzDAwNcfTyILg28n0vpcGIcmV1POaQ1GU40KIIr985QhckiSphov6J6rAbPHY4FgsjSzxretLHdM6mJ76gV8tjMjQVQN7AhBiaw0mCr0rt+s1mgzgkiRJNdTOmJ28evhVru1RV9+4WrnyStNXuJ91H0sjS4Z6DwUg4NoOMrLuFLg2XUeHpbUsZQCvRDKAS2VycXGRz54lqRrLyc3h6LWjmlF2cPtgXm76Mo7mjriYujDTfyYB7gG413Ivdtnn9ay7xdZbUrn0dMgALkmS9Iy6n3Wf135/jT9i/iA5PRkFhTaObTTPuD1rezLPax5+bf1Krcfe1J6Ee0U3tLI3lbsjViYZwCVJkp4BqZmpRMRFEBYVhom+CR/1+AgTfRMS7yXyYuMXCXAPoLtbd2xMbB677uBWwYQcCCE95+FW00a6RgS3CtbmR5AekwzgkiRJ1diKkyv48c8fOXDlAFm5WRjrGTOgyQDN8T0j9jxxG73d1CfdS08s5fq969ib2hPcKlhTLlUOGcAlSZKqibwMXhFxEfyn73/Q19XnTOIZ7mTcYZLPJALcA/Ct51shu531dustA3YVIwO4JElSFXYh6QJfH/+a8Ohwzt48C0Ad0zrEJcfR0KYhiwIWoaPIXbFrIhnAJUmSqoj8Gbz8Xf1p5dCKhNQElh9dznP1n2NEixEEuAcUyOAlg3fNJQO49NSMGDGCpKQkNm/eXNldkaQqIzMnk7Vn1xIeoy7xup56HYCPu39MK4dWdKrXiX+m/YOJfvlzCkjPJvmnWxVw48YNJk2aRMOGDTEyMsLOzo6OHTvy+eefl5mlTFEU1q1bV652x4wZg6IoLFq0qFzXP66lS5cWyDX+KFxcXJ5a/ySpQhTK4pVxchU7Y3ay8cJGQB1Bj986nq1/b6WrS1e+7/c9VyddZaqvmuNAT0dPBm+pWHIEXsni4uLw9fXFwsKCOXPm4O3tjbGxMWfPnuXbb7/FxsaGIUOGaL3ddevWceTIERwdHbVed0ksLS2fWluSVCU8yOJ1MSuVbWQTducCERuHkQZ42XnxvMfz6OnocWLMCVysXOTtcOmxyH8thVw8fJ0f3v0fX4zdxQ/v/o+Lh69XaHvjxo1DR0eHY8eOMWjQIJo0aYKrqyt9+vTh999/Z/DgwVpv89KlSwQHB/Pf//4XfX39Ms+Pi4tDURT++9//0qlTJ4yMjGjcuDHh4eEFztu7dy/t27fHyMiIOnXqMGnSJDIzMzXHR4wYUSBDmp+fH0FBQbz77rvY2tpiZ2fHlClTNDnF/fz8uHTpElOnTkVRlGJ3iJKkqujW/Vv8fv53xB8fQlYas8nkLSWDaHL5tzBgk3E9Dow8oDnfrZabDN7SY5P/YvK5ePg6u1edJ/WfDABS/8lg96rzFRbEb926RVhYGOPHj8fU1LTYc7QdtLKzsxk8eDDvvffeY2fveuedd5g4cSKnTp2iR48e9OvXj/j4eADi4+MJDAykZcuWnDx5ku+++45ffvmFGTNmlFrnqlWr0NPT48CBAyxbtozQ0FBWr14NwPr163F2dmbWrFkkJCSQkFB0JyhJqgqycrLYe2kv7+16j3b/aUftT2rTf3V/ou5eZoupCbF16uLh1JD6zo3oaWpNn/t3ZPpN6YnJAJ7PwQ3RZGfmFijLzszl4IboCmkvKioKIQQeHh4Fyp2dnTEzM8PMzIyxY8dqtc0PPvgAW1tbxo0b99jXjhs3joEDB9K4cWOWLl1K3bp1+fLLLwFYvnw5jo6OLF++HE9PT/r06cPChQtZtmwZ9+/fL7HOJk2aMHv2bBo1asTAgQPp2rUrO3fuBMDa2hpdXV3Mzc2xt7fH3l5u2yhVDUIIov6JIvFeIgC/nf+NLiu7sHD/Qgx0DQjxC+HgqINE2tYjxNaauyYm6OsbkqCvR4itNVtqO1fyJ5CeBfIZeD55I+9HLa8o+/btIycnh9GjR5Oerm5dOHbs2AITwMqa3FaciIgIVq5cyalTp0o8p7R2OnTooPlaR0eH9u3bc+7cOUDNx+3j44OOzsO/CTt16kRmZiZRUVF4e3sX217hckdHRxITEx/vg0nSU5Ccnszu2N2ERYcRHh1ObHIsi3osYnLHyQS4B7B+4Hr8Xf2xNHo41yPAuhbpMouXVEFkAM/HzNqw2GBtZm1YIe01aNAARVE4f/58gXJXV1cATEwezjydPXs2U6ZMeaL2IiIiSEhIwMHBQVOWk5PDtGnTCA0N5erVq1ppp7DSHgMUfgavKIrmGbgkVabs3Gxu3ruJg7kDaVlpOCx2ID07HXMDc/xd/ZnScQq9G6ph2MrIiv6e/YvUIbN4SRVJBvB8OvRzZ/eq8wVuo+sZ6NChn3uFtGdjY0NAQADLli3jzTffxMzMrMRz7ezssLOze6L2goKCeOmllwqU9ezZk8GDB/PGG2+U2c6hQ4fw9/cH1FuIR44c0dTn6enJmjVryM3N1YzC9+/fj4GBAe7u5f/+GRgYkJOTU+7rJelxXEq+pBlh74zdiXcdb/aM2IOxvjFLey3F09YTH2cf9HXLnvwJMouXVLFkAM+nUXv1f6qDG6JJ/ScDM2tDOvRz15RXhOXLl+Pr60vr1q0JCQmhefPm6Onpcfz4cf78808CAgLKrCMuLq7IbXE3NzcsLCwKlBUXnPX19bG3ty/yHL44X375JY0aNaJZs2YsX76cS5cuaZ6lBwUFERoaSlBQEMHBwcTExDB9+nQmTJhQ4E7C43JxcWHfvn0MGzYMQ0NDbG1ty12XJBWWlpWGsb4xACM3jOT7U98D4GzhzIuNX+RfDf+lOXd069GPXb/M4iVVJBnAC2nU3r5CA3Zhbm5unDx5kgULFvD+++9z5coV9PX18fT0JCgoiAkTJpRZx9SpU4uUbdq0qcCSLW1YuHAhn376KSdOnKB+/fr89ttvODurk3GcnJzYtm0bU6dOpUWLFlhZWTFkyBDmz5//RG3Onj2bMWPG4O7uTkZGBkIIbXwU6Vl3eg3snA13roKlM3SbBd4DyRW5nEg4QXi0uuvZoauHuDb5GtbG1vRt1JfmdZoT4B5AY9vGWlkBIrN4SRVJqU6/ENu0aSOOHTtW7LHIyMjHXhZVk6SkpGBuXr5lK3Fxcbi6unL06FHatGmj5Z5ph/z5SxoPNk8hKw0AgUDRN2F7u5EMO/kNt9JuAdDSviUB7gFM8plEHbM6ldnjShUREYGfn19ld0MqhaIox4UQRX75yhG4JEnPlPt/hLA3K4Vwsgknm2kYMDwrjQan1/Kvhv+ip3tPurt1r9FBW3o2VGoAVxSlMzD0QT+aCCE6VmZ/JEmqvtKy0nj+1+fZdzeSDAV0UDAxMuFT01pYp2XTO+UmP/b/sbK7KUlaU+4ArijKCqAPkCiE8MpX3gtYCugC3wohFpZUhxBiH7BPUZQXgKPl7YtUsVxcXOSzZ6lKuZF6gz9i/iAsOgwzAzOW916Osb4xJvomBBrbcNbcFH1jM3R0dMgGQkxzwUSRa6+lZ8qTjMBXAssAzZ+0iqLoAl8APYCrwFFFUTaiBvMFha4fKYTI27FjCDDqCfoiSVINsOzIMr47+R2nrqurLmxNbHnJ8+HSyA2DNhDw304Yys1TpBqg3AFcCLFXURSXQsXtgCghRAyAoii/Av2EEAtQR+tFKIpSD7gjhEgpb18kSXq2CCGITIokPDqciLgI1ry8BgNdA66nXsfS0JL5/vMJcA+gpUPLIklA5OYp0rPizqZNJC4JpamhUevijmv7GbgTcCXf+6tA+zKuGQV8X9JBRVFGA6MB6tSpQ0RERLHnWVpakpIi/wYoSU5OzjP9/UlPTy/x34ZUfcSkxrA2fi3H/jlGUmYSAHWN67I2fC1Oxk501+lOd5fukAMpF1PYe3FvkTqsdK24nXO72HL5b6So1NRU+X2pgowOH8F81Sp08mV0LKzSZ6ELIT4o4/g3wDegLiMrablDZGRkuZdJ1QRPsoysOjAyMqJly5aV3Q2pNIXWZmd2fZdDtZwJjw6nd8PedKjbAb3Lehw6e4juDboT4BZAD/ceuFi5PFYz02KmFbt5yrSO0/Bz89PqR3oWyGVkVdPfs+eQXUrwBu0H8Higbr73zg/KJEmqyR6szc7Ius+3ZBF+5wK7NgwjFYGuooutiS0d6nagY92O3Jx6Ez2d8v9qkpunSM+C7EdIn6ztAH4UaKgoiitq4B6EOkFNkqQaKDk9mV2xu8jYPpnBWWnoAR+QgQUwTOgRYGKP/8S/NBm8dBSdIs+0y6O3W28ZsKVqTc/Bgexr10o9p9z/pyiK8gtwEPBQFOWqoiijhBDZwAQgDIgE1gghzpa3DenJmJmZsXLlyqferpeXFyEhIU+9XalqOJFwgg8jPsR3hS+2H9syYM0APr5/jS2mJgQ6O2FXtxENnD3oY2pD//t3C6TflCRJZTfpLRQjo1LPKXcAF0IMFkI4CCH0hRDOQojvHpRvFUI0EkK4CyHmlbf+mmLBggW0bdsWCwsLateuTd++fTlz5sxTaz85OZkpU6bg5uaGgYEBdnZ2DBo0qEiKU0kqSVxyHCtPrdTsFfDZ4c/4cM+HZOdmM6PTDPaO2EuIbWNCbK1J0NdDV0+fBH09Qmyt2VLbuZJ7L0lVk2XfvjjMmY2eo2OJ51T6JLaaLiIigqCgINq2bYsQglmzZtG9e3fOnTuHtbV1hbZ9+/ZtOnZUN79bunQpzZs359q1a8ybN4+2bduya9cu2rZtW+y1mZmZGBgYVEi/cnNzEUKgq6tbIfVLj6CEZCAAqZmp7I7drUm7+fc/fwPQ3qk9nrU9md11NosDFmNjYqOpLsC6FulybbYkPRbLvn2x7NuXs4pyvLjjT/6wSXoiYWFhvP7663h5edGsWTN++uknbt68yf/+9z/NOS4uLsydO5cxY8ZgYWGBs7Mzn3zySYF6oqKi8PPzw8jICA8PDzZv3lxm2zNnzuTy5cvs3LmTvn37Uq9ePXx8fNiwYQMNGzbk9ddf14yqRowYQZ8+ffjoo49wdnbWZCFLTEykX79+GBsbU79+fVasWFGknTt37jB69Gjs7OwwNzenS5cu5E9Ks3LlSszMzNi6dSteXl4YGBgQGRlZru+npAV5yUDuXAEEuXcuc2zjOOIPfwXAjugdPP/r83x/6nsa2jQktGco54LO0di2MQD1LOsVCN4g12ZLUkWQI/AqJiUlhdzcXGrVqlWgfMmSJXz44YdMnTqVbdu2MXHiRDp16kSHDh3Izc2lf//+1KpVi4MHD3L//n2Cg4PJyMgosZ3c3Fx+/fVXhg4dimOhWzQ6OjpMnjyZYcOGcfr0aZo3bw7Anj17sLS0ZPv27QUC+6VLl/jjjz8wMTFh0qRJxMXFaeoSQtC7d28sLS3ZvHkz1tbW/PDDD/j7+3PhwgUcHBwAdR33nDlz+Prrr6ldu7amXKoEO2dzNeseOx4kA9lBDrdyBAt2f8D09mPp7tadna/uxLeuL4Z6ho9Upb2pPQn3is6qtTd9eql7Jam6EAJOn4Zff4XSsto++wG8uPWNAwdCUBDcvw//+lfR4yNGqK+kJHjppaLHx42DV16BK1dg+PCCx55wQ4Tg4GBatGhBhw4dCpQHBARocoO/+eabfPbZZ+zcuZMOHTrwxx9/cO7cOWJjY6lXrx4AoaGhdO7cucR2bt68ye3bt0tMwdmkSRMALly4oAngRkZGrFixAkND9Zf2xYsX2bZtG/v378fX1xeAH374ATc3N009u3fv5tSpU9y8eRNjY2MA5syZw6ZNm/jpp5945513AHWjmWXLltG6dbEbDkkV7H7Wfa6nXsetlhtpd67gTiqZCtgLhd7o0VPo0SNdTc9pbmiOv6v/Y9Uf3Cq42LXZwa2Ctfo5JKk6u3gRfvlFDdznz4OuLrz4YsnnP/sBvBp5++232b9/P/v37y/y/Nfb27vAe0dHRxIT1a3kIyMjcXJy0gRvgPbt26Ojo90nJF5eXprgndeujo4O7dq105TVr1+/wIj++PHj3L9/n9q1axeoKz09nejoaM17PT09WrRoodX+SiUTQnD6xmnCo8MJjwln36V9tHJoxYFRBzC2rMtb6VfZa1WLVCMTEnJysbydTG09m7IrLoFcmy1JxYuLg7p11WD91VcQGgpdusBbb6nBu3btkkfhz34AL21EbGJS+nFb29KP1637xCPuPJMmTeLXX39l9+7dBUawefT19Qu8VxSF3NzccrdXu3ZtrKysOHfuXLHH88obNWqkKTM1NS32XKWUezy5ubnUqVOHffv2FTlmYWGh+drQ0FBOWtOGUiaf/ZP2D9bG6sTIoeuH8suZXwDwsvNifNvxBDYMBGBLy/7siv2NdB3155qgo0OIrQ249n+iCWdybbYkqeLjYe1aWL0aDh2CXbuga1eYMkV9lTLxvIBnP4BXA8HBwaxevZrdu3fTuHHjx77e09OT+Ph4rly5Qt266kZ4R44cKTXA6+joMGjQIH788Uc+/PDDAqPm3NxcFi9eTNOmTTW3z4vTuHFjcnNzOXLkiGY2++XLl7mWb/OBVq1acePGDXR0dIr9w0TSorzJZ1nqre6MO5f534axhJ1ZRXjKZU7fOE3C5ATsTO14tfmr9HTvSQ/3HjiaF/xtsTTpsCZ450nXUViadFjOGJekJ3DtGgwZAnv3qs+5W7SAhQsh79f+owbuPDKAV7Lx48fz008/8fvvv1OrVi2uX78OqJuwmJmZPVId3bt3p3Hjxrz66qssWbKEtLQ0Jk2ahJ5e6T/eefPmsXPnTrp3785HH31EixYtiI+PZ/78+fz999/s2rWr1NG1h4cHvXr1YsyYMXzzzTcYGxvz9ttva5515/XN19eXfv368fHHH9O4cWOuX7/O9u3b6d69e6nP6aXHI/74kJys++ihsIUsXiaNtFzQ/3sTvi5dmNt1rmaXs14NepVYz/V71x+rXJKk4iUnw2+/qV+//jrY2UFuLoSEqNOoPDyerH65jKySLV++nJSUFLp164aDg4PmtWjRokeuQ0dHh99++43c3Fzat2/Pq6++ynvvvVfgeXVxrK2tOXToEL169WLixIm4u7vTt29fjI2NOXLkSIlrwPNbuXIlrq6u+Pv707dvX4YMGYKLi4vmuKIobN26FX9/f9544w08PDwYOHAgFy5cKDL7XXp8SfeTWH1mNSM3jKTu3Uh+JAsAL3T5NwZsEsb8I8zZ/dpuZnSega2JbZl1ljQzXM4Yl6SypabCf/8L/fpBnTowciT8+KN6TE9PHX3PmvXkwRtAyVsOVB20adNG5F8/nF9kZGSJM6qlZz8bWU37+adlpfHcyuc4fu04AoGVkRXds3NpoavLHts6XNfTxT47h+DbyfTWs4FJj76735aYLcXOGA/pGCKfYT+DZDayJ5eRAXnjpVdfhZ9+AicndZQ9aBC0aVP6crCyKIpyXAjRpnC5vIUuSVVJoUlowv99opxbaWaLWxha8FP/nzDWN8bT1pO+jfoS4B5AG8c2hO39kJD8k8/09co1+UzOGJeksmVmwh9/qEu+fv9dnYzWpAlMngyjR0PHjqDlhUBFyAAuSVVFoUloH9+J4svfhxCHOhnRrZYbAzwHaE7/sf+PBS7X5uQzOWNckoqXkAAffAD/93/wzz9gZQUvvwx5C4VKmferdTKAS1Ily87N5mj8UcK3TiAi6x+2YYIRV8wMtwAAIABJREFUCjlAC6HDO8Z1CHhjH+7W7qXWIyefSZL25ebCgQOQlaUu9TIxUSemBQaqt8cDAqCC0kKUSQZwSaokJxJOMH/ffHbG7iQ5PRlFQENdfbrXrs1dI+OHz7Dv3YcygjfI7UolSVuEgOPH1dvjq1fD1avQqRPs2weWluoovIxFPk+FnIUuSRXh9BpY4gUhVrDEi5TjP7DpwiYmbJ1ARFwEAFk5WRyOP8wAzwGsfmk1q2w8sHBuyB1jE4SiPHbKzeBWwRjpFswfLLcrlaTH9+qr0LYtfPYZtGwJP/8MW7c+PF4VgjfIEbgkad+DZ9k5Wff5iEzC7pznwOYRZAMm+iY0rd0UPxc/2jm14/JblzVr7QP+/OyJUm7KyWeS9PguXlRH2evWwY4d6lrtIUPU2+X9+0OhvFJVigzgkqRtO2dDVhq6KPxAFqbAZGFAT1NHOk46r8ngVXiTHG2k3JSTzySpbLduwYoV6i3yEyfUss6dITFRDeCBgZXbv0clb6FLkrbducoWUxMCnB3Rr+eBrXMjOpta0/Xe7VLTb8oNVCSp4iQkwIUL6tcpKfDOO+qt8MWL1cSSe/eCl1fl9vFxyQAuSVq2pbYzIbbWJOjroejqPvKzbPkMW5K0KykJvvkG/P3VjVUeZC/GxUWdmHb4MLz9Njg/2jSTKkcG8Gpq5cqVj7xXekU7duwYiqIQFxdX2V2pEpbWsiK90A4O6rNsq1Kv6+3Wm5COITiYOqCg4GDqIHc/k6RymjABHBxgzBg1icisWWrikDxOTpXXN22RAbwKWbBgAYqiMGHChALlLi4uj7U3elkOHz7M888/j7W1NYaGhjRu3JgPP/yQ9PT0si+WyvQkz7L/v717j++57v84/nhvs83mPOa0mImcCxNTMkRFRV1+jnUlhyI6CF2hkq5LqBRKlC7K1UEnV+ikuCxyHjnkNOeQzMhhhtn2/v3xsWW2MXb4fr/b8367fW/2fX9OL99tn9fe78/70CGsAz90/oGND23kh84/KHmLZMPp005HtG7dIPU2VreuMyvaL7/A1q3OAiIFbbZldWJzEytXruTdd9+lQYMGeXqdefPm0blzZ3r27MnChQsJCgpi+fLlDB06lEWLFrFw4UJ8s5iVIDExMcttOZWX585vGo8tkvfOnYPvv3c6os2bBwkJTo17xw6oXx8GDHB1hHlPNXA3cOLECXr27MmMGTMofcmYhcjISPbt28ewYcMwxmToubxo0SLq1atHYGAgrVq1Ys+ePVleJyEhgT59+tC+fXtmzpxJo0aNqFq1Kt27d2f+/Pn8/PPPTJo0KW1/YwxTpkzh/vvvJzAwkBEjRgDw/fffU6tWLfz9/WnRogUxMTEZrrV8+XJatmxJQEAAlStXZsCAAZw8+VcNNDIykgEDBjB06FDKlSvHLbfcck2fnTvSs2yRvHH+vDN9KcCGDdCpkzP06+9/h6gopzNa/fouDTFfKYG7gUceeYTOnTvTqlWrDNvmzJlDSEgIL7zwAocOHeLQob9qdufOnWPs2LHMmDGDFStWcPz4cfr375/ldRYsWEBcXBzPpPbkuEijRo1o06YNH3/8cbry0aNH0759ezZt2sTAgQPZv38/nTp1om3btqxfv57HH388w/k2bdpEu3btuPfee9mwYQNz5sxh/fr19O7dO91+H374IdZali5dyqxZ6ef19mR6li2Se5KTYfFi51l2xYowfLhT3qSJs5jIoUMwdSq0bAne3q6NNb8V+Cb0yPcjM5R1qduFx5o8RsL5BNp/1D7D9l439aLXTb2IS4ij82edM2wfED6ArvW6sv/Efh7874PptkX1irqq+KZPn87OnTv58MMPM91epkwZvL29KV68OBUqpG+CTUpKYsqUKdxwYWHZoUOH0rt3b6y1GWrqQFpNOatlN+vUqcP06dPTlXXt2pW+ffumvR8xYgRVqlRh8uTJGGOoVasWMTExPP/882n7vPrqq3Tt2pUhQ4aklU2dOpWGDRsSGxtLcHAwANWqVWPChAlZfjaeTOOxRXJu9GiYNg3++AMCA+Hee+FvF9bzMQbatHFtfK5W4BO4O9u+fTsjRozg559/pkjqUjZXwc/PLy15A1SqVInExET+/PNPypQpkysxhoenX4J269atNGvWLN0fCBEREen2Wbt2LTt37uTTTz9NK0tdd37Xrl1pCbxx48a5EqOIeD5rnQ5nX38Nzz/vJOgTJ5xlObt1gw4dnIVE5C8FPoFfrkYcUCTgstvLBpS97PbrSl531TXui61YsYK4uDjq1q2bVpacnMySJUuYNm0ap0+fxs8v64k/fC6ZkDc1qaakpGS6f82aNQHYsmVLps+ct2zZkrZPqsDAwOz9Zy6SkpJC3759GTx4cIZtlS8au3Et5xaRgmXLFqcj2uzZTgc0Hx9nec7ateH1110dnXsr8AncnXXq1ClDDffhhx+mRo0ajBgxIq1Xtq+vL8nJyTm+Xrt27QgKCuLVV1/NkMDXrVvHokWLGD9+/GXPUbt2bb788st0zfQrV65Mt0+jRo3YvHkz119/fY5jFpGCJyUFvLycZ9ht2zpfR0bCsGFw//0QFOTqCD2DOrG5UKlSpahXr166V2BgIGXKlKFevXppCTI0NJSlS5dy8OBB4uLirvl6gYGBTJ8+nW+++YbevXvzyy+/8NtvvzF79mzuvfdebr31Vp588vI9pfv378/evXt56qmn2L59O1988QXTpk1Lt88//vEPVq9eTf/+/fnll1/YuXMnX3/9NY8++ug1xy4inm3/fmfa0iZN4KWXnLIWLeCtt+DgQVi0CPr1U/K+GkrgHuCll15i//79VK9enXLlyuXoXPfddx9LliwhNjaW1q1bU6NGDUaNGkXfvn354YcfrjgWu0qVKsyZM4fvv/+eG2+8kTfeeINxF09vBDRo0IAlS5awd+9eWrZsyY033sjw4cMpX758jmIXEc8zfbqTqKtUgaFDnbKwMOdfPz8YOBAqaIqEa2JSOxd5gvDwcBsdHZ3ptq1bt2bZu1rg1KlTFC9e3NVh5Bl9/0WuTVRUFJGRkbl2vmPHnDHZ99/vvL//fufZdrdu0LUr6Mna1TPGrLXWhl9armfgIiKSIydPOrOhzZ4NCxZAUhLs2eMsGvLhh+o9nlfUhC4iIld0Yv58drRuw9baddjRug0n5s8H4IcfoHx5ePBB2LgRBg+GtWuhalXnOCXvvKMauIiIXNaJ+fM59PwLnEtIZllCS7492J5bBq7hMaBx83vo189pIm/WzOlRLvlDCVxERC7rv88vZd6e51gY35ZTKSUo6XWcev6biX1jIjXuuYfJk10dYeGkBC4iIumkpDgTrNSr57yfuL0HuxKr06bYIu4s/h3NA5dTxCSRdCjjlM2Sf5TARUQEa2H1aqcj2mefQVwcHD4MpUrBhJteI+joJvy8EtMd41OxoouiFVAnNhGRQm/RImdsdrNm8PbbzmQrH3wA/hdWxQ0f0RX/gPTpwvj7Ezz4KRdEK6lUAxcRKWS2bYNPP3UmWPHygpAQqFULXnwROnZ0at0XK3nPPQDEvjGRpEOH8KlYkeDBT6WVi2sogYuIFAJ79jhJe/Zs2LDBWe3rxRfhttvghhvgu+8uf3zJe+5RwnYzakL3UO+//z7FihVzdRgAREdHY4xh7969rg5FRC4SH+/8ay3cfjsMH+6My540CQ4cgBdecG18kjNK4C724osvYoxJ96pwycTAoaGhvPbaa7l2zVWrVnHvvfdSpkwZ/Pz8qFWrFqNHj+bs2bO5dg0RcY3YWOc5dsuWznPt8+ed2vbMmbB3LyxfDk88AZUquTpSySklcDdwww03cOjQobTXpk2b8uxa8+bNo0WLFgQFBbFw4UJiYmIYNWoU7777Lu3atSMxMTHLYy+3Lafy8twihcHy5dCuHVSs6CwQcuQIDBoE584522+77a/Z0aRgUAJ3Az4+PlSoUCHtdfGKY5GRkezbt49hw4al1dAvtmjRorRlSFu1asWePXuyvE5CQgJ9+vShffv2zJw5k0aNGlG1alW6d+/O/Pnz+fnnn5k0aVLa/sYYpkyZwv33309gYCAjRowA4Pvvv6dWrVr4+/vTokULYmJiMlxr+fLltGzZkoCAACpXrsyAAQM4efJkuv/XgAEDGDp0KOXKlcuwPrmIXF58PHz8MWze7Lw/fx527YJnn3WmNN282Wkid5MnbZIHlMDdwO7du6lUqRLVqlWjW7du7N69O23bnDlzCAkJ4YUXXkiroac6d+4cY8eOZcaMGaxYsYLjx4/Tv3//LK+zYMEC4uLieOaZZzJsa9SoEW3atOHjjz9OVz569Gjat2/Ppk2bGDhwIPv376dTp060bduW9evX8/jjj2c436ZNm2jXrh333nsvGzZsYM6cOaxfv57evXun2+/DDz/EWsvSpUuZNWvWVX1mIoXRmTPw5ZfQpQsEB0PPnk4SB6eGvXMnjBkD9es7zeZSsOVbL3RjTBgwEihpre18oSwQeBtIBKKstR/l9nUzWyWvSxd47DFISID27TNu79XLecXFQefOGbcPGOAsi7d/vzOB/8Wioq4uvqZNm/L+++9Tq1YtYmNj+de//kXz5s3ZvHkzQUFBlClTBm9vb4oXL57h2XhSUhJTpkzhhhtuAGDo0KH07t0ba22GmjqQVlPOatnNOnXqMH369HRlXbt2pW/fvmnvR4wYQZUqVZg8eTLGGGrVqkVMTAzPP/982j6vvvoqXbt2ZciQIWllU6dOpWHDhsTGxhIcHAxAtWrVmDBhwtV8XCKFjrVOMk5JcXqL79/vJO8+fZz7UPPmzn5K2IVPthK4MWYGcDcQa62td1H5ncAkwBt4z1o7LqtzWGt3A32MMV9cVHw/8IW1dr4x5lMg1xO4u7vrrrvSvW/WrBlhYWF88MEHPP3005c91s/PLy15A1SqVInExET+/PNPypQpkyvxhYenX4J269atNGvWLN0fCBEREen2Wbt2LTt37uTTTz9NK0tdd37Xrl1pCbxx48a5EqNIQZOU5FQGZs92pjRdtswZrz16NFSp4nRQ89Eg4EIvuz8C7wNvAWntnMYYb2AK0BY4AKwxxszDSeZjLzm+t7U2NpPzhgCpPbaSsx929l2uRhwQcPntZctefvt11119jftKihUrRt26ddmxY8cV9/W55Dc4NammpKRkun/NmjUB2LJlS6bPnLds2ZK2T6rAwMBsxX2xlJQU+vbty+DBgzNsq1y5co7OLVKQ/forTJsGn3/u9CYvVgw6dXKazgMC4OGHXR2huJNsJXBr7RJjTOglxTcDOy/UrDHGzAY6WmvH4tTWs+MAThJfTxbP440xjwCPAJQvX56oLDJmyZIlOXXqVDYv677Onj3L1q1bad68edr/x8fHh4SEhHT/v9QhXxeXJSQkABAfH4+fn1+68yYnJxMREUGZMmUYO3Ysn3zySbrt69evZ9GiRbz00kvpznnmzJl076tXr87cuXM5efJk2h8MP/30U9p1T506Rf369dm4cSPly5fP8P9LSkri1KlTJCcnk5iYmGvfs7Nnz2b5syHirqyFmJjilCt3jjJlEvnf/8oxfXotIiKO8thjsTRrdgw/vxRWr867GOLj4/W746mstdl6AaHArxe974zTbJ76/kHgrcscHwRMA3YBwy+UBQIzgalAzyvF0LhxY5uVLVu2ZLnNnQ0ZMsRGRUXZ3bt325UrV9oOHTrY4sWL271796bt07ZtW9uhQwd74MABe+TIEWuttTNnzrSBgYHpzrV48WILpO1zsZMnT1prrZ0zZ4718fGxDz/8sF23bp3dt2+f/eSTT2zlypVtixYt7Llz59KOAeznn3+e7jz79u2zvr6+9oknnrDbtm2zn3/+uQ0JCbGA3bNnj7XW2g0bNtiiRYvaRx991K5bt87u2LHDzp8/3z7yyCNp52nZsqUdOHBgzj68i3jq918Kn5QUazdutHbECGvDwqwFa8eOdbadPWvthV/VfLN48eL8vaBcNSDaZpIT8+0pirX2KND/krLTQKFuFDpw4ADdu3cnLi6OcuXK0axZM1auXEnViwZsvvTSSzz66KNUr16dc+fOpT1Pvhb33XcfS5YsYcyYMbRu3ZqEhARCQ0Pp27cvzz77LL6+vpc9vkqVKsyZM4enn36ad955h8aNGzNu3DgeeOCBtH0aNGjAkiVLeO6552jZsiXJycmEhYVx3333XXPcIgXB+fPOQiEbNoC3N7RpA8895zSTA/j5OS8RgJhVf7Bi7i6uK1sz0w5DJrvJ4EIT+tf2Qic2Y0wE8KK19o4L74cDWKcJPU+Eh4fb6OjoTLdt3bo1y97V4jS1Fy9e3NVh5Bl9/8Ud7d3rLM25Zw9MneqUPfusM6HK3/7m9CZ3taioKCIzG64jLhWz6g8Wf7SNpMQUxn85gN+ObM8wziAnNfA1QA1jTDXgINAN6JGD84mIeLxDh5ykPXs2rFzplDUssZkNi3sQUKksw7WKl2TDirm7SErMvENyqmxN5GKM+QRYAdxgjDlgjOljrU0CBgELgK3AZ9bazTmMWUTE48TFOfNKgJO8n3rK6Tk+6u9b+bFWBz6q2BlfEkn6/XcOPf8CJ+bPd23A4vbij5274j7ZSuDW2u7W2orW2iLW2hBr7b8vlH9rra1pra1urR2Tw3hFRDzGiRPwwQdw111QoYIz9AvggQecsdvr10PP/YOobHenO86ePUvsGxNdELF4kmJlrtwZQlOpiohchbNnnU5nwcHOjI3btsGwYZA6n1FQEKR2x0i6aOrji2VVLpIqomN1fHwvn6I1l4+IyGWcPQvffedMYfrEE+Dv7/QmHzgQunVzepVnNY2pT8WKJP3+e6blIpdTs6kzdfaKubuy3EcJXETkEufPw48/Oh3RvvoKTp2C0FAnaXt7wzffZO88wYOf4tDzL2AvTLwEYPz9CR78VN4ELgVKzaYVqNm0Ar3GxqzNbLua0EVEgORk5wXw0kvQoQPMm+csaLRgAezY4STvq1Hynnuo+M+X8KlUCYzBp1IlKv7zJfVCl1yhGriIFFopKc5Qr9mznU5o778Pd9wBDz0EN98M7drlfGKVkvfco4QtV+3YmWO8uepN/jz7Z5b7KIGLSKETH+/Usj/9FH77zUnSd98NpUs726+/3nmJ5LfD8Yd5fcXrvB39NvGJ8dxf+/4s91UTuoeKiorCGENcXJyrQyEuLg5jjBZEELe2eTN8+63zddGiznjt+vXhP/9xVv764gun1i3iKl9u+ZLQSaG8tuI17q55Nxv6b+DLLl9mub8SuIuFhoZijMnw6tChQ9o+kZGRDBo0KNeuuW3bNrp370758uXx8/OjWrVqDBkyhD//zLqpRsQT7dwJY8Y4ibpePejf31kBzNsbYmLg66+dcdslSrg6Uimsdhzdwa+xvwLQNKQpPev3ZNvAbXzyt09oUL7BZY9VAnexNWvWcOjQobTXunXrMMbQpUuXPLne6tWradKkCadOneKrr75ix44dvPnmm3z33Xc0b96c48ePZ3lsYmJinsSU1+eWwmnUKKhRw1kspFQpePNNWL36ryFfV1i3RyRPbTq8ie5fdqfWlFo88+MzAISUCOG9e9+jRlCNbJ1DCdzFypUrR4UKFdJe3377LSVKlEhL4L169eKnn35iypQpabXzvXv3ph2/YcMGmjZtSkBAAOHh4axbty7La1lr6d27NzVr1mTevHlERERQpUoV7r77bhYuXMi+ffsYOXJk2v6hoaG8+OKL9O7dm1KlStGzZ0/A+aOjcePG+Pv707BhQ1atWpXhWlu2bKFDhw4UL16c4OBgunfvzh9//JG2vVevXtx9992MHz+ekJAQQkJCcvpRSiH2xx/w1ltw663OSl/gdEZ77TXnGffSpTBokDNjmogrrf19LR1nd6TBtAZ8HfM1w5oPY2bHmdd0LiVwN2Kt5d///jcPPPAARYsWBWDSpElERETw8MMPp9XSr7vuurRjhg8fzrhx41i3bh1BQUH07Nkzy+VG169fz+bNmxkyZAheXum/9ZUqVaJnz5588skn6Y5//fXXqVWrFtHR0bz88svEx8fToUMHwsLCiI6OZty4cQwdOjTduQ4dOsRtt91GvXr1WL16NQsXLiQ+Pp6OHTuSkvLX5Pw//fQTGzdu5Pvvv2fRokU5/vykcElIgOnTnSU5K1eGxx+Hkyfh2DFne/PmMGQIXPTrIuISqet3AyzcvZCff/uZ0ZGj+e2p3xh3+zjKFyt/Tect8L3QI9+PzFDWpW4XHmvyGAnnE2j/UfsM23vd1IteN/UiLiGOzp91zrB9QPgAutbryv4T+3nwvw+m2xbVK+qaY/3xxx/Zs2cP/fr1SysrWbIkvr6+BAQEUCGT6sM///lPWrVqBcALL7zArbfeysGDBzOt0cbExABkuexmnTp1eO+99zhy5AjBF9Y5bNmyJc8880zaPu+++y6JiYnMnDmTYsWKUa9ePUaOHMmDD/71OUydOpUbb7yR8ePHp5XNmjWLMmXKEB0dzc0Xegr5+/szY8YM/LQAsmTTiRNw4ADUres8y37qKSd5jxwJXbs65SLuwlrLtzu+ZczSMQxsMpCeDXoy6OZBPNbkMYr75Xx55wKfwD3J9OnTadKkCTfeeGO2j2nQ4K9ODpUqVQIgNjY215qkw8PD073funUrDRo0oFixYmllEamTQF+wdu1alixZkm6fVLt27UpL4PXq1VPylis6fdqZ+Wz2bKcXee3a8MsvEBjoLBpSpUrGqUxPzJ9P7BsTSTp0CJ+KFQnWEp6Sj5JTkpmzdQ4v//wy6/9YT5WSVSjiXQSAQN/AXLtOgU/gl6sRBxQJuOz2sgFlL7v9upLX5ajGfbHY2Fjmzp3LlClTruq4IkWKpH1tLtzFLm6mvljNmjUB5/l0w4YNM2zfsmULpUuXply5cmllgYFX/8OWkpJChw4deO211zJsK1/+r6aiazm3FC6vvOKM1z592nl+3b+/U9NOVbVqxmNOzJ+fbvrS1CU8ASVxyRedP+/MV9u+omZQTWZ2nEnP+j3TEnhu0jNwN/H+++/j5+dH9+7dM2zz9fUlOXWOxxy46aabqF27NhMmTMiQ5H///Xc++ugjunfvnvaHQGZq167Npk2bOH36dFrZypUr0+3TqFEjNm/eTNWqVbn++uvTvYoXz3mzkRRM5887U5b27g2p639Uq+YM81q82Gk6nzjxr1W/shL7xsR0c4+DlvCUvHU26SzvRL/DqXOnAOjXqB+fdv6ULY9toddNvfIkeYMSuFuw1vLee+/RrVu3TJudQ0NDWb16NXv37iUuLi7LGvaVGGOYMWMGMTExdOzYkRUrVrB//36++eYbbr/9dqpWrcq//vWvy56jR48e+Pj40Lt3bzZv3syPP/7ImDHpl4IfOHAgJ06coGvXrqxatYrdu3ezcOFCHnnkEU6dOnVNsUvBlJwMP/0EAwZApUpw553w5ZewaZOz/f/+D6ZNg8jI7M9DriU8Jb/EJ8YzYfkEwiaF0f+b/ny51Zl0pX2N9nSp2wVvr6ucPP8qKYG7gaioKHbs2JGu89rFhg4diq+vL3Xq1KFcuXL89ttv13ytZs2asXr1agIDA+nYsSPVq1dn4MCB3HnnnSxfvpzSqXNJZqFYsWJ8/fXX7Nixg0aNGjF06NB0ndXAeRa/bNkyvLy8uPPOO6lbty4DBw7Ez89Pz7wFa//qKX74MLRqBbNmwe23Oyt/HT7sDAG7Vlkt1aklPCW3JKck88+f/knViVUZ+uNQaperzaK/L+KhGx/K1zhMVkOO3FF4eLiNjo7OdNvWrVuz7F0tcOrUqQLdfK3vv3uzFtavdzqiffYZ1KrlrLENsHCh0yyeW10iLn0GDs4SnloFLHNRUVFERka6OgyPkHA+gYAiAQC0/qA1gb6BjGwxkmYhzfL0usaYtdba8EvLC3wnNhFxrenTnQlVYmKcZvC2bZ3n2qluvz13r5eapNULXXLLgZMHeHXZq8zaOItfB/xK5RKV+bbnt/j7+Ls0LiVwEclVu3c7tezHH3dq1X/+6YzVHjIE7r8fypbN+xi0hKfkhl3HdjHu53F8sOEDUmwKDzR4AIvTau3q5A1K4CKSCw4ccNbTnj3bmW8coGFD51n2sGFw0VxAIh4hLiGOOm/XwWDo16gfw24ZRmipUFeHlY4SuIhcE2udCVR27IALUwzQqJEzdrtLl7/GaF9mVKKIW4n+PZofd/3I8BbDKRtQlpkdZ9IqtBUVi7tnB8gClcCttZcdwywFkyd1xPR0x47Bf//r1LSrVoX33oPrr4dJk+Cuu5zVv0Q8zZJ9SxizdAw/7PqB0v6l6de4H2UDytKjfg9Xh3ZZBWYYWZEiRThz5oyrwxAXOHPmTLoZ6ST3zZ0L99zjzIbWty/s2fNXsjYGnnjiysn7xPz57Gjdhq2167CjdRtOzJ+f94GLXEbM0RhazGxBy/dbsv6P9YxrM469T+2lbEA+dNTIBQWmBh4cHMzBgwepXLkyRYsWVU28ELDWcubMGQ4ePJhuilbJuTNn4PvvoWNH8PJyZkJbvx6efBK6dXOayq/mV0zTm4q7SLEpHI4/TMXiFSkbUJZjZ44x+c7J9G3Ul6JFiro6vKtSYMaBA5w8eZLY2FjOnz+fj1F5hrNnz+Lv7/pek7mtSJEiBAcHU6JECVeH4vESE+GHH5zm8blzIT7emSXtttucrwMCnGR+LXa0bkNS6vyoF/GpVIka/9NSsq5UWMaBJ6UkMfvX2Yz9eSz+Pv5E94vGGOMRj14LxTjwEiVK6EaehaioqEwXMBEB2LwZbr0Vjh+H0qWhe3dn0ZBbbnG2ZzLD71XR9KbiKueSzvHBhg8Yv2w8u//cTb3gegyNGIrFYjBun7wvp0AlcBG5spQUWLbsr45ozzzj9CLv0sVpMr/9dvD1zd1r+lSsmHkNXNObSh77eNPHPPrWLKJUAAAZrUlEQVT1ozSp1ITX273OPTfcg5cpGN2/lMBFConoaPj4Y2eSlYMHoWhRZ3lOgCJF4J138u7awYOfynR60+DBT+XdRaVQOnH2BFPWTKFS8Ur0uqkXPer3oErJKrSu1tqja9uZKRh/hohIBtbC9u1/vX/tNXjrLWjc2EnksbHw+uv5E0vJe+6h4j9fwqdSJTAGn0qVNDe55Kojp48wctFIqkyswsj/jWT5/uUA+Pn40SasTYFL3qAauEiBs3270zw+ezZs2+a8brgBxo+HqVOdZ9yuoOlNJa+8veZthv04jDPnz/C3On9jxK0jaFix4Pf5UQIXKSB+/RUefNAZ7mWM03v8ySchdYRd6sxoIgXB7j93U9KvJEEBQYSWCqVznc48e8uz1C5XeFYlVBO6iIc6eBAmToQvvnDeh4Q4vcVffx3274eoKOcZd6lSV39uTboi7mrLkS08+N8HqflmTSasmABA+xrt+aDTB4UqeYNq4CIe5cgR+PJLp3l8yRLnOXevXtC5s5Ooly7N+TU06Yq4o3WH1jFm6RjmbJ1DQJEAnmz6JINuHuTqsFxKCVzEzZ054/QYB2ds9uLFUKsWjBrlzIp2ww25e73YNyam6y0OYM+eJfaNiUrg4jIvL32ZRbsX8VyL53iy2ZMeM91pXlICF3FDp0/D/PlOTXvhQti711lH++WXnWTeoEHerfKlSVfE1ay1/Lj7R8b+PJa3279N7XK1mXjnRIr7Fqekf0lXh+c2lMBF3Mj27U7Nev58SEiAypXhkUcgKcnZ3qxZ3segSVfEVVJsCvO2z2PM0jFE/x5N5eKV2X9yP7XL1SakRIirw3M7SuAiLpSY6NSwy5RxkrOvr9NE/tBDznSmt9xy7fOPXytNuiKukGJTaPZeM9b8voaw0mG8e/e7/P3Gv+Pn4+fq0NyWErhIPktOdnqIz54Nc+Y4a2z36OEk8GrV4PffwdvbdfGlPueOfWMiSYcO4VOxIsGDn9Lzb8l1icmJfBPzDZ1qdcLLeNGlbheebPokXet1xcdL6elK9AmJ5LPWrZ0e5MWKOXOPd+sG7dr9tT0nyfvE/Pm5kng16YrkpYTzCby37j1eXf4qB04eYGWflTQNacrQ5kNdHZpHUQIXySPWwrp1Tk17wQJYswb8/GDQIHjiCWjf/q/e5blBw7/E3SWcT2Dyqsm8vuJ1jiQcoUWVFrx3z3vcXPlmV4fmkZTARfLI+fPQtq2zlvYdd0BcnNMp7f/+L2+up+Ff4q5SbEraCmBvrHyDRhUbMbLFSFpUbeHiyDybErhIHvH1hblzoW5dp5NaXtPwL3E3h04dYsKKCSzcvZDoR6IJKBLA5sc2awx3LlECF8lDLfKxgqHhX+Iu9h7fyyvLXmHGLzNISkmie/3unDp3itJFSyt556J8S+DGmDBgJFDSWtv5Qllt4EmgLLDIWjs1v+IRKWg0/EvcwZqDa4j4dwTeXt70urEX/7j1H4SVDnN1WAVStkaYGmNmGGNijTG/XlJ+pzFmuzFmpzHm2cudw1q721rb55Kyrdba/kAX4JarDV5E/qI1t8VVfjn0C19u+RKAxpUa82Lki+x+Yjfv3POOknceym4N/H3gLWBWaoExxhuYArQFDgBrjDHzAG9g7CXH97bWxmZ2YmPMvcAA4D9XFblIAZTTYWAa/iV5JWbVH6yYu4v4Y+coVsaPiI7Viau0mzFLx/Dtjm+pVqoa99W+Dy/jxXO3PefqcAsFY63N3o7GhAJfW2vrXXgfAbxorb3jwvvhANbaS5P3pef5IrUJ/ZLyb6y1HTIpfwR4BKB8+fKNZ8+ena14Jb34+HiKFSvm6jDkMvxXrab4Rx/hlZiYVpbi68upnj0521TDbCRvZOfecHxvCr+vAZvsvP/NJ4b/lpzGDt8NlPApwf+F/B+dKneimI/uMXmhVatWa6214ZeW5+QZeGVg/0XvDwBNs9rZGBMEjAEaGmOGW2vHGmMigfsBP+DbzI6z1r4LvAsQHh5uIyMjcxBy4RUVFYU+O/e246V/knRR8gbwSkwkaMECavzjGRdFJQVddu4NH4xYRkryWRLNWfxsUZLMeWK9D9A96XGmDx9LoG9g/gQr6eRbJzZr7VGg/yVlUUBUfsUg4s40DEzcUXJKMlEJ3/FD2Y+plliX7icHE3a+LqNjP8KHIkreLpSTZRIOAtdd9D7kQpmIXIOshntpGJi4QmJyIjN+mUHtKbV5v/QYUkimRuKNadt9KEKxMlpoxJVyksDXADWMMdWMMb5AN2Be7oQl4tlOzJ/PjtZt2Fq7Djtat+HE/PlXPCZ48FMYf/90ZRoGJq4yYtEI+szrQzHfYky+aTovnJxJ+NnWadt9fL2I6FjdhRFKtprQjTGfAJFAWWPMAWCUtfbfxphBwAKcnuczrLWb8yxSEQ9xrXOSaxUwcaWT504ydc1UWldrTZPKTRjYZCBtqrXhzuvvxBhDTIWMvdBrNq3g6rALtWz3QncH4eHhNjo62tVheCR1Yss/O1q3yXxGtEqVqPG/RS6ISCRr836cx7oi65i0ahLHzx5nVMtRvBj5oqvDkosYY3K9F7qIZEKd0cRTjFkyhjGrxnAm+Qwdb+jIyBYjaVK5iavDkmxSAhfJZZqTXNzZ/hP7qVyiMl7Gi2SbTERQBBPvn0j98vVdHZpcpZx0YhORTKgzmrijmKMxPDz3YcImhzF321wAnr/teZ6v/bySt4dSDVwkl6kzmriTjYc38vLSl/ls82f4+fgxIHwA4ZWcx6nGGBdHJzmhBC6SBzQnubiDFJvCfZ/ex5HTR/jHLf9gcMRgggODXR2W5BIlcBGRAsJay+K9i5kWPY0POn1A0SJF+azzZ4SVDqN00dKuDk9ymRK4iIiHs9byzY5vGLN0DCsPrKRCsQpsP7qdmyrcRONKjV0dnuQRJXAREQ92NOEobWa1YcPhDVQtWZW327/Nww0fxt/H/8oHi0dTAhcR8TDnk8+z/o/1NKnchDJFy1C/fH0GNxtMj/o9KOJdxNXhST5RAhcR8RBnk84y45cZvLLsFWJPx/Lb4N8oG1CW/9z3H1eHJi6gBC4i4ubiE+OZFj2NCSsm8Ef8H0SERDCl/RSCiga5OjRxISVwERE3t+/4Pob9OIw21drw8f0fExkaqTHcogQuIuJuDscf5o2VbxCXEMd7975H3eC6bB+0nZpBNV0dmrgRTaUqIuIm9p/YzxPfPUHopFBeWfYKZ5LOkJySDKDkLRmoBi4i4ga+2PIFPb7sgcXyYIMHefbWZ5W05bKUwEVEXOTX2F9JTE6kUcVG3FrlVvqH92dIxBCqlqrq6tDEA6gJXUQkn605uIZOsztRf2p9hi8aDkCFYhWYfNdkJW/JNtXARUTyyYr9KxgVNYofd/9Iaf/SjGo5iieaPuHqsMRDKYGLiOQhay0pNgVvL29WHVzFxsMbGX/7eAaED6C4X3FXhyceTAlcRCQPpNgU/rv1v4xaMJqmRzvQKK4tgWUa8MPdK2jQvJqrw5MCQAlcRCQXnU8+zye/fsLYn8eyLW4bwckheJ8qCkDiMVgxex/+3kWp2bSCiyMVT6cELiKSizp/3pl52+fRoHwDBpwfTe24CLzwTtuelJjCirm7lMAlx9QLXUQkB04nnmbiyon8eeZPAJ64+QnmdZvH+kfXUzfu1nTJO1X8sXP5HaYUQKqBi4hcg+Nnj/PW6reYuHIiR88cpZR/KXrd1Is2YW3S9ilWxi/TZF2sjF9+hioFlGrgIiJXIcWmMHLRSKpOrMrzi5+nWUgzlvdeTq+bemXYN6JjdXx8099mfXy9iOhYPZ+ilYJMNXARkWw4cfYEJf1L4mW82HB4A3dUv4MRLUZwU4Wbsjwm9Tn3irm7iD92jmJl/IjoWF3PvyVXKIGLiFzGrmO7GL9sPB9t+ohfB/xKtdLV+KrbV/h4Ze/2WbNpBSVsyRNK4CIimdgcu5mxP4/lk18/oYhXEfo07IO/jz9AtpO3SF7ST6GIyCWOJhyl0buNKOJVhKebPc3TEU9TsXhFV4clko4SuIgIsGTfEn7Y9QP/av0vggKC+LTzp7So0oKggCBXhyaSKfVCF5FCy1rLgp0LuG3mbbR8vyXvrn2X2NOxAHSq1UnJW9yaauAiUihtj9tOzzk9WXtoLSElQph05yT6NupLQJEAV4cmki1K4CJSaCSlJHHw5EGqlqpKxeIVMcYw/Z7pPNjgQfx8NLmKeBYlcBEp8M4lnWPWhlmMXzYefx9/Ng7YSAm/Eqzpt8bVoYlcMyVwESmwEs4nMH3tdF5d/ioHTx0kvFI4I1uMdHVYIrlCCVxECpyYVX+wYu4u/nfma2aVGkeT0s2Y8cAM2oa1xRjj6vBEcoUSuIgUGHEJcbwwZwwnNhian7qbxrSibFIlah6tT2hcLUx1JW8pODSMTEQ83u+nfufpBU9TdWJVpu2cxH6zCwBvfAg7XzdtDW6RgkQ1cBE3ktr0q4Uvsm/yqskM+3EYySnJ9KjfgyqLIqmYFJphP63BLQWNauAibiJm1R8s/mhbWqKJP3aOxR9tI2bVHy6OzP1sPbKVw/GHAagXXI+Hb3qYmMdjmHXfLGqUuCHTY7QGtxQ0SuAibmLF3F0kJaakK1PTb3q/HPqFzp91pu7bdXl1+asAtK7Wmml3TyOsdBigNbil8FATuoibyKqJV02/sHz/cv615F98t/M7SvqVZESLETzZ9MlM99Ua3FJYKIGLuIliZfwyTdZq+oUpa6aw5vc1vNz6ZR5r8hgl/Utedn+twS2FgZrQRdyEmn4dKTaFudvm0vS9pmw8vBGA19u9zt4n9zK8xfArJm+RwiLfauDGmDBgJFDSWtv5QpkX8E+gBBBtrf0gv+IRcTfu1PSbF73hr3TOpJQkPtv8GWN/Hsuvsb8SVjqMI6ePAFC+WPkcXVukIMpWAjfGzADuBmKttfUuKr8TmAR4A+9Za8dldQ5r7W6gjzHmi4uKOwIhwFHgwNWHL1KwuEPTb2pv+NQOdam94YFrju1K50yxKTSZ3oT1f6yndtna/Oe+/9CtXjd8vPSUTyQr2f3teB94C5iVWmCM8QamAG1xku8aY8w8nGQ+9pLje1trYzM57w3AcmvtOxcS+6KrC19EctvlesNfawLP7JwJiQlMmP82024ejZfxol+jflQoVoFOtTrhZfR0T+RKspXArbVLjDGhlxTfDOy8ULPGGDMb6GitHYtTW8+OA0Diha+TM9vBGPMI8AhA+fLliYqKyuap5WLx8fH67CRb4o+lZFF+7pp/hi4+5xlzmqUB81gc+AWnvI9Tf14w9UrWow514DQsObzkmq4h10b3Bs+Vk/apysD+i94fAJpmtbMxJggYAzQ0xgy/kOjnAG8aY1oAmf7WWmvfBd4FCA8Pt5GRkTkIufCKiopCn51kx74flmXZGz4y8pZrPufRP4+zMPBTogL/yxmveGqfbUKn5IcY1HFgTkOWHNC9wXPl2wMma+1RoP8lZQlAn/yKQUSuLKJj9XTPqyFnveGTUpKI6FidHz7ayIqA76iReCN3xPeguqlDq561citskUInJwn8IHDdRe9DLpSJiAfLrd7w+47v45Vlr7BozyI2DdhEOxrgM/cjkv/00eQqIrkgJwl8DVDDGFMNJ3F3A3rkSlQi4lI56Q2/PW4745aN48ONH2IwPHTjQ8QnxrtFD3uRgiS7w8g+ASKBssaYA8Aoa+2/jTGDgAU4Pc9nWGs351mkIuL2on+P5ubpN+Pv489j4Y8xtPlQrit53ZUPFJGrlt1e6N2zKP8W+DZXIxIRj7LywEp2HdtFzwY9aVyxMa+1e40HGjxAcGCwq0MTKdA0S4KIXDVrLf/b8z/GLB3D4r2LCSsdRrd63fD28ubpiKddHZ5IoaDZEkTkqqw+uJqIf0dw+39uZ1vcNia0m8CG/hvw9vJ2dWgihYpq4CJyRckpycQnxlPSvyTexpvDpw8ztcNUet3UC38ff1eHJ1IoKYGLSJbOJ5/nw40fMm7ZOJpf15wZ986gcaXG7Hx8p2rcIi6mBC4iGZw5f4YZv8zgleWv8NuJ37ipwk3cW/PetO1K3iKupwQuIhmMihrFq8tfpfl1zZnaYSp3XX8XxhhXhyUiF1ECFxGOnTnGm6ve5Paw27mlyi080fQJ2tdoT8uqLZW4RdyUErhIIXY4/jCvr3idt6PfJj4xHmMMt1S5hZASIYSUCHF1eCJyGUrgIoXUqMWjeGX5KyQmJ9KlbhdG3DqC+uXruzosEckmJXCRQmTnsZ2ElgrFx8uH4n7F6VGvB8/e+iw1gmq4OjQRuUpK4CKFwMbDG3l56ct8vuVzPrr/I7rV68bQ5kNdHZaI5IASuEgBturAKsYsHcP8mPkU8y3G0IihtApt5eqwRCQXKIGLFFApNoWHvnqIIwlHGB05mkE3D6JM0TKuDktEcokSuEgBYa3lu53f8faat5ndeTbFfIvxRZcvqFqyKsX9irs6PBHJZVrMRMTDJack8/nmz2n0biM6fNyBTbGb2HlsJwD1guspeYsUUKqBi3iwP8/8SfMZzdkWt42aQTWZce8MHmjwAEW8i7g6NBHJY0rgIh7mbNJZVh9czW1Vb6N00dK0rNqS0ZGj+Vvtv2mOcpFCRAlcxEPEJ8bzTvQ7TFgxgaNnjrJ/8H6CA4OZdvc0V4cmIi6gBC7i5o6fPc6bq95k0qpJHD1zlNbVWjOyxUjKBZRzdWgi4kJK4CJuylqLMYYjp48w+qfR3FXjLka2GEmzkGauDk1E3IASuIibOXDyAK8ue5UjCUf4+G8fUyOoBnue3MN1Ja9zdWgi4kY0jEzETew8tpN+8/oRNimMt6Pfxt/Hn+SUZAAlbxHJQDVwETfwxZYv6PpFV4p4FaFfo34Mu2UYoaVCXR2WiLgxJXARF4n+PZrzyeeJuC6CVqGtGNZ8GE81e4oKxSq4OjQR8QBqQhfJZ0v2LeGOD++gyfQmvBD1AgBBAUGMu32ckreIZJtq4CL55Ke9P/Hc4uf4+befCQ4MZvzt4xkQPsDVYYmIh1ICF8lDKTaFFJuCj5cPW45sYd/xfUy+czJ9G/WlaJGirg5PRDyYmtBF8sjJcyepP7U+09dOB6BPoz7sfGInjzd9XMlbRHJMNXCRPFLCrwStQ1unDQHz9fZ1cUQiUpAogYvkoTfbv+nqEESkgFITuoiIiAdSAhcREfFASuAiIiIeSAlcRETEAymBi4iIeCAlcBEREQ+kBC4iIuKBlMBFREQ8kBK4iIiIB1ICFxER8UBK4CIiIh5ICVxERMQDKYGLiIh4IGOtdXUM2WaMOQHsyOXTlgROuPg813Ls1R5TFoi7ymvIX3Lr5yS/uUPc+RFDXlwjN86Z03Po3uDe8uv3q6q1tlyGUmutx7yAd931nDk5z7Uce7XHANGu/v558isvfvYKS9z5EYO73htyeg7dG9z75erfL09rQp/vxufMyXmu5di8+Cwka576ebtD3PkRg7veG3J6Dt0b3JtLP2uPakKXa2eMibbWhrs6DhFxL7o3eC5Pq4HLtXvX1QGIiFvSvcFDqQYuIiLigVQDFxER8UBK4CIiIh5ICVxERMQDKYELxphOxpjpxphPjTHtXB2PiLgHY0yYMebfxpgvXB2LZKQE7uGMMTOMMbHGmF8vKb/TGLPdGLPTGPPs5c5hrf3KWtsP6A90zct4RSR/5NK9Ybe1tk/eRirXSr3QPZwx5jYgHphlra13ocwbiAHaAgeANUB3wBsYe8kpeltrYy8cNwH4yFq7Lp/CF5E8ksv3hi+stZ3zK3bJHh9XByA5Y61dYowJvaT4ZmCntXY3gDFmNtDRWjsWuPvScxhjDDAO+E7JW6RgyI17g7g3NaEXTJWB/Re9P3ChLCuPA7cDnY0x/fMyMBFxqau6Nxhjgowx04CGxpjheR2cXB3VwAVr7WRgsqvjEBH3Yq09itM3RtyQauAF00Hguoveh1woE5HCTfeGAkQJvGBaA9QwxlQzxvgC3YB5Lo5JRFxP94YCRAncwxljPgFWADcYYw4YY/pYa5OAQcACYCvwmbV2syvjFJH8pXtDwadhZCIiIh5INXAREREPpAQuIiLigZTARUREPJASuIiIiAdSAhcREfFASuAiIiIeSAlcRETEAymBi0gaY8x0Y4w1xrzh6lhE5PI0kYuIAGCMKQr8ARQHjgCVL8zcJSJuSDVwEUnVCSgBvAIEA3e6NhwRuRwlcBFJ9RCwB3gepwb+0MUbjTHXG2POG2NeuqR8qjHmlDEmPP9CFRElcBHBGFMJuB340Fp7HpgN3GOMKZ26j7V2J/Ae8JQxJujCcS8AvYH7rLXR+R+5SOGlBC4iAA8A3sCHF97PAvyArpfs99KF/Z41xvQFRgEPWmsX5legIuJQJzYRwRizGThlrW12UdlW4Li1NuKSfccAQwAf4Elr7ZR8DVZEANXARQq9C8+u6wD/uWTTf4Bmxpial5TvwKmdr1DyFnEdJXAReQg4D3x6SfmHgAX+nlpgjGkDvAOsAG4xxjTIryBFJD01oYsUYsYYX+B3YJm1tmMm2xcDYUAo0BCIwqmZDwZigM3W2g75Fa+I/MXH1QGIiEt1AIKA/caYTpls3w1EAn2BfwE/AI9ba1OMMaOBGcaY26y1S/IrYBFxqAYuUogZY74CMtS8M2GBJcAd1tpzF471Bn4F/rTWNs+7KEUkM0rgIiIiHkid2ERERDyQEriIiIgHUgIXERHxQErgIiIiHkgJXERExAMpgYuIiHggJXAREREPpAQuIiLigZTARUREPND/AwWbdmfrhTC4AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 576x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute the error as a function of delta_x for each method\n",
    "f = lambda x: numpy.sin(numpy.pi * x)\n",
    "I = 2.0 / numpy.pi\n",
    "\n",
    "# num_partitions = range(50, 1000, 50)\n",
    "num_partitions = range(5, 50, 5)\n",
    "delta_x = numpy.empty(len(num_partitions))\n",
    "error_trap = numpy.empty(len(num_partitions))\n",
    "error_simpson = numpy.empty(len(num_partitions))\n",
    "error_2 = numpy.empty(len(num_partitions))\n",
    "error_3 = numpy.empty(len(num_partitions))\n",
    "error_4 = numpy.empty(len(num_partitions))\n",
    "\n",
    "for (j, N) in enumerate(num_partitions):\n",
    "    x_hat = numpy.linspace(0.0, 1.0, N)\n",
    "    delta_x[j] = x_hat[1] - x_hat[0]\n",
    "    \n",
    "    # Compute trapezoid\n",
    "    I_hat = 0.0\n",
    "    for i in range(0, N - 1):\n",
    "        I_hat += (f(x_hat[i + 1]) + f(x_hat[i])) * delta_x[j] / 2.0\n",
    "    error_trap[j] = numpy.abs(I_hat - I)\n",
    "    \n",
    "    # Compute simpson's    \n",
    "    I_hat = 0.0\n",
    "    for i in range(0, N - 1):\n",
    "        I_hat += delta_x[j] * (1.0 / 6.0 * (f(x_hat[i]) + f(x_hat[i+1])) + 2.0 / 3.0 * f(x_hat[i] + delta_x[j] / 2.0))\n",
    "    error_simpson[j] = numpy.abs(I_hat - I)\n",
    "    \n",
    "    # Compute Gauss-Legendre 2-point\n",
    "    xi_map = lambda a,b,xi : (b - a) / 2.0 * xi + (a + b) / 2.0\n",
    "    xi = [-numpy.sqrt(1.0 / 3.0), numpy.sqrt(1.0 / 3.0)]\n",
    "    w = [1.0, 1.0]\n",
    "    I_hat = 0.0\n",
    "    for i in range(0, N - 1):\n",
    "        for k in range(len(xi)):\n",
    "            I_hat += f(xi_map(x_hat[i], x_hat[i+1], xi[k])) * w[k]\n",
    "    I_hat *= delta_x[j] / 2.0\n",
    "    error_2[j] = numpy.abs(I_hat - I)\n",
    "    \n",
    "    # Compute Gauss-Legendre 3-point\n",
    "    xi_map = lambda a,b,xi : (b - a) / 2.0 * xi + (a + b) / 2.0\n",
    "    xi = [-numpy.sqrt(3.0 / 5.0), 0.0, numpy.sqrt(3.0 / 5.0)]\n",
    "    w = [5.0 / 9.0, 8.0 / 9.0, 5.0 / 9.0]\n",
    "    I_hat = 0.0\n",
    "    for i in range(0, N - 1):\n",
    "        for k in range(len(xi)):\n",
    "            I_hat += f(xi_map(x_hat[i], x_hat[i+1], xi[k])) * w[k]\n",
    "    I_hat *= delta_x[j] / 2.0\n",
    "    error_3[j] = numpy.abs(I_hat - I)\n",
    "    \n",
    "    # Compute Gauss-Legendre 4-point\n",
    "    xi_map = lambda a,b,xi : (b - a) / 2.0 * xi + (a + b) / 2.0\n",
    "    xi = [-numpy.sqrt(3.0 / 7.0 - 2.0 / 7.0 * numpy.sqrt(6.0 / 5.0)), \n",
    "           numpy.sqrt(3.0 / 7.0 - 2.0 / 7.0 * numpy.sqrt(6.0 / 5.0)),\n",
    "          -numpy.sqrt(3.0 / 7.0 + 2.0 / 7.0 * numpy.sqrt(6.0 / 5.0)),\n",
    "           numpy.sqrt(3.0 / 7.0 + 2.0 / 7.0 * numpy.sqrt(6.0 / 5.0))]\n",
    "    w = [(18.0 + numpy.sqrt(30.0)) / 36.0, (18.0 + numpy.sqrt(30.0)) / 36.0,\n",
    "         (18.0 - numpy.sqrt(30.0)) / 36.0, (18.0 - numpy.sqrt(30.0)) / 36.0]\n",
    "    I_hat = 0.0\n",
    "    for i in range(0, N - 1):\n",
    "        for k in range(len(xi)):\n",
    "            I_hat += f(xi_map(x_hat[i], x_hat[i+1], xi[k])) * w[k]\n",
    "    I_hat *= delta_x[j] / 2.0\n",
    "    error_4[j] = numpy.abs(I_hat - I)\n",
    "    \n",
    "fig = plt.figure(figsize=(8, 6))\n",
    "axes = fig.add_subplot(1, 1, 1)\n",
    "\n",
    "# axes.plot(delta_x, error)\n",
    "axes.loglog(delta_x, error_trap, 'o', label=\"Trapezoid\")\n",
    "axes.loglog(delta_x, error_simpson, 'o', label=\"Simpson's\")\n",
    "axes.loglog(delta_x, error_2, 'o', label=\"G-L 2-point\")\n",
    "axes.loglog(delta_x, error_3, 'o', label=\"G-L 3-point\")\n",
    "axes.loglog(delta_x, error_4, 'o', label=\"G-L 4-point\")\n",
    "\n",
    "order_C = lambda delta_x, error, order: numpy.exp(numpy.log(error) - order * numpy.log(delta_x))\n",
    "axes.loglog(delta_x, order_C(delta_x[0], error_trap[0], 2.0) * delta_x**2.0, 'r--', label=\"2nd Order\")\n",
    "axes.loglog(delta_x, order_C(delta_x[0], error_simpson[0], 4.0) * delta_x**4.0, 'g--', label=\"4th Order\")\n",
    "axes.loglog(delta_x, order_C(delta_x[1], error_3[1], 5) * delta_x**5, 'b--', label=\"5th Order\")\n",
    "axes.loglog(delta_x, order_C(delta_x[1], error_4[1], 7.0) * delta_x**7.0, 'g--', label=\"7th Order\")\n",
    "\n",
    "axes.legend(loc='best',fontsize=14)\n",
    "axes.set_xlabel('$\\Delta x$', fontsize=16)\n",
    "axes.set_xlim((5e-3, delta_x[0]))\n",
    "axes.grid()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Other Quadrature Families\n",
    "\n",
    " - Clenshaw-Curtis (Gauss-Chebyshev): If $w(x) = \\frac{1}{\\sqrt{1 - x^2}}$ and $g(x)$ are Chebyshev polynomials then we know the roots of the polynomials to be $x_i = \\cos\\left(\\frac{2i-1}{2N} \\pi \\right)$ (the Chebyshev nodes) and we can derive that $w_i = \\frac{\\pi}{N}$.\n",
    " - Gauss-Hermite:  If $w(x) = e^{-x^2}$ and $g(x)$ are Hermite polynomials $H_i(x)$ then\n",
    "   $$w_i = \\frac{2^{N-1} N! \\sqrt{\\pi}}{N^2 (H_{N-1}(x_i))^2}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## SciPy Integration Routines\n",
    "\n",
    "SciPy has a number of integration routines that we have derived here including general purpose integrators that can control error."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "import scipy.integrate as integrate\n",
    "# integrate?"
   ]
  }
 ],
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