numerical integration.ipynb

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    "# NUMERICAL INTEGRATION"
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    "# What is numerical Integration\n",
    "\n",
    "### <b>Numerical</b> Integration is a way to approximate the value of a definite integral\n",
    "\n",
    "### Unlike Analytic solutions which are exact, closed form solutions"
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    "# Numerical Integration\n",
    "There are many options for numerically computing definite integrals:\n",
    "- Trapezoid Rule\n",
    "\n",
    "- Simpsons Rule\n",
    "\n",
    "- Gaussian Quadrature\n",
    "\n",
    "- etc."
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    "## Let's look at the Trapezoid Rule\n",
    "\n",
    "The Trapezoid rule is a way of approximating a definite integral by breaking it up into trapezoids. \n",
    "\n",
    "Let's say we want to compute the integral\n",
    "\n",
    "$$ \\int_0^{\\pi} \\sin(x)dx $$"
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    "We can easily compute it analytically\n",
    "\n",
    "$$ \\int_0^{\\pi} \\sin(x)dx = -\\cos(x) \\big|_0^{\\pi} $$\n",
    "\n",
    "$$ =-(\\cos(\\pi) - \\cos(0)) $$\n",
    "\n",
    "$$ = -(-1 - 1)$$\n",
    "\n",
    "$$ 2 $$\n",
    "\n",
    "But let's try computing using the trapezoid rule."
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    "Below is a visualization showing $\\sin(x)$ and a trapezoid approximation."
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    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "plt.rcParams['figure.figsize'] = (13, 8)\n",
    "plt.rcParams.update({'font.size': 19})\n",
    "\n",
    "def f(x):\n",
    "    return np.sin(x)\n",
    "\n",
    "def trap_plot(n_points):\n",
    "    x = np.linspace(0, np.pi, 1000)          # continuous\n",
    "    x_i = np.linspace(0, np.pi, n_points)    # discrete\n",
    "\n",
    "    plt.plot(x, f(x), label=r'$\\sin(x)$')\n",
    "    plt.plot(x_i, f(x_i), '-o', label=r'$Trap$ ({})'.format(n_points))\n",
    "    plt.fill(x_i, f(x_i), color='C1', alpha=0.15)\n",
    "    plt.vlines(x_i, 0, f(x_i), color='C1', linestyle=':')\n",
    "    plt.xticks(x_i, [r'$x_{}$'.format(n) for n in range(n_points)])\n",
    "    plt.yticks([0, 1], ['$0$', '$1$'])\n",
    "    plt.legend(loc='best')\n",
    "    plt.ylim(0, 1.05)\n",
    "    plt.show()"
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    "trap_plot(9)"
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    "## Area of a Trapezoid\n",
    "\n",
    "The area of a trapezoid is the average height times the base\n",
    "\n",
    "<img src=\"trap2.png\" width=400>\n",
    "\n",
    "For the example shown, the area is \n",
    "\n",
    "$$\n",
    "Area = \\overbrace{\\frac{f(x_{k-1}) + f(x_k)}{2}}^{ave \\, height} \\,\\, \\overbrace{\\Delta x}^{base}\n",
    "$$"
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    "## trapezoid rule equation\n",
    "\n",
    "the equation for the trapezoid rule is \n",
    "\n",
    "$$\n",
    "\\int_a^b f(x)dx \\approx \\frac{f(x_0) + f(x_1)}{2}\\Delta x + \\frac{f(x_1) + f(x_2)}{2}\\Delta x \\cdots \\frac{f(x_{N-1}) + f(x_N)}{2}\\Delta x\n",
    "$$"
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    "Now we can write this as a summation\n",
    "$$\n",
    "\\approx \\sum_{k=1}^N \\frac{f(x_{k-1}) + f(x_k)}{2} \\Delta x\n",
    "$$\n",
    "\n",
    "And we can write a function to compute this"
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    "$$\n",
    "\\approx \\sum_{k=1}^N \\frac{f(x_{k-1}) + f(x_k)}{2} \\Delta x\n",
    "$$"
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   "source": [
    "def trap(f, x):\n",
    "    \"\"\"\n",
    "    computes the integral of f using trapezoid rule\n",
    "    \"\"\"\n",
    "    area = 0\n",
    "    N = len(x)\n",
    "    dx = x[1] - x[0]\n",
    "    \n",
    "    for k in range(1, N):\n",
    "        area += (f(x[k - 1]) + f(x[k])) * dx / 2\n",
    "        \n",
    "    return area"
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   "source": [
    "x = np.linspace(0, np.pi, 20)\n",
    "trap(f, x)"
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    "## Error vs. number of Trapezoids\n",
    "\n",
    "As we increase the number of trapezoids, the approximation gets better ($error \\rightarrow 0$).\n",
    "$$error = (actual - approximation)^2$$"
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    "plt.rcParams['figure.figsize'] = (11, 6)\n",
    "def plot_error(n_points):\n",
    "    for n in range(2, n_points):\n",
    "        x = np.linspace(0, np.pi, n)\n",
    "        plt.plot(n - 1, (trap(f, x) - 2) ** 2, 'bo')\n",
    "\n",
    "        plt.axhline(0, color='black', lw=1)\n",
    "        plt.xlabel('# of trapezoids')\n",
    "        plt.ylabel('error')\n",
    "    plt.show()"
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