{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Demystifying the Beigel-Reingold-Spielman construction" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from math import *\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "import numpy as np\n", "\n", "plt.rcParams['font.size'] = '16'\n", "color_palette = sns.color_palette()" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "### A helper to plot the graph of a function\n", "def plot_func(f, xs=None, label=None, title=None):\n", " if xs is None:\n", " xs = np.arange(-100,100,0.5)\n", " if label is None:\n", " label=\"(unknown)\"\n", " \n", " ys = np.array(list(map(f, xs)))\n", " \n", " plt.figure(figsize=(12,8))\n", " plt.grid()\n", " plt.axvline(x=0, color=\"black\", linewidth=\"1\")\n", " plt.axhline(y=0, color=\"black\", linewidth=\"1\")\n", "\n", " if title is not None:\n", " plt.title(title)\n", " \n", " plt.plot(xs, ys, label=label, linewidth=2)\n", " plt.legend()\n", " plt.show() " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# The hyperbolic tangent\n", "\n", "$$\\tanh(x) = \\frac{e^x - e^{-x}}{e^x + e^{-x}}$$\n", "\n", "Often, this function is used as a smooth approximation for the sign function. " ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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