{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SageMath Software\n",
    "\n",
    "KME Meeting -- February 25, 2019"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Brief history of SageMath\n",
    "\n",
    "- Originally called SAGE: **S**ofware for **A**lgebraic **G**eometry **E**xploration\n",
    "    * It is a complete free and open-source distribution of mathematics software for OS X and Linux. (All under GPL compatible licenses.)\n",
    "    * Now called SageMath.\n",
    "<br>\n",
    "<br>\n",
    "\n",
    "- SageMath is a computer algebra system based on the Python programming language as an open-source alternative designed to compete with the big-name M’s.\n",
    "    * It was created in 2005 by William Stein and is now maintained and expanded with the help of the open-source community.\n",
    "    * **Mission statement:** *Create a viable, free open-source alternative to Magma, Maple, Mathematica, and Matlab.*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Working with numbers in SageMath"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "2 + 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "5 * 6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "10 / 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "11 / 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "N(11 / 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "5 ^ 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "gcd(135, 75)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "lcm(135, 75)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "factor(37238117074)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "factorial(645)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "645!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "N(e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "pi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "N(pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "log(45)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "log(16,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Calculus in SageMath"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f(x) = sin(3*x) + 2*exp(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.diff(x)(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.integrate(x,0,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "integrate(f(x),x,0,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.integrate(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.integrate(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "var('t,y,z')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.integrate(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.taylor(x,0,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "g(x,y) = sin(x-y)*y*cos(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "g.integrate(x).integrate(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "integrate( integrate( x * y, x, 0, y-3), y, 0, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting functions in SageMath"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = plot(f,-2,2)\n",
    "p.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = plot(f,-2,2,frame=True, axes=False, gridlines=True, color='purple', thickness = 2)\n",
    "p.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "implicit_plot(y^2 - x^3 - x^2, (x,-2,2), (y,-2,2),frame=True, axes=False, gridlines=True, color='red')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "q = implicit_plot(y^2 - x^3 - x^2, (x,-2,2), (y,-2,2),frame=True, axes=False, gridlines=True, color='red')\n",
    "(p+q).show(aspect_ratio='automatic')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "L = [(0,0)]\n",
    "for i in range(2000):\n",
    "    last = L[-1]\n",
    "    L.append((last[0]+(random()-.5), last[1]+(random()-.5)))\n",
    "\n",
    "g = line(L, thickness=.4, color='blue')\n",
    "g += points(L, pointsize=10, color='grey')\n",
    "g.show(aspect_ratio=1, figsize=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f(x) = 3*sin(10*x) + 2*x\n",
    "t1 = f.taylor(x,pi/32,1)\n",
    "t2 = f.taylor(x,pi/32,2)\n",
    "pf = f.plot(color='red',ymin=-10,ymax=10,thickness=2)\n",
    "p1 = t1.plot(color='green')\n",
    "p2 = t2.plot(color='purple')\n",
    "pf + p1 + p2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "@interact\n",
    "def my_taylor(degree=slider(0, 20, 1, 1)):\n",
    "    p1 = plot(f, -2, 2, color='red', ymin=-10, ymax=10,thickness=2)\n",
    "    p2 = plot(f.taylor(x, 0, degree), -2, 2, ymin=-10, ymax=10)\n",
    "    show(p1 + p2, figsize=8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "frames = [pf]\n",
    "for k in range(1,20):\n",
    "    frames.append(pf + plot(f.taylor(x,0,k),color='blue',ymin=-10,ymax=10))\n",
    "a = animate(frames)\n",
    "a.show(delay=60,iterations=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 0,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = plot3d(g,(x,-3,3),(y,-3,3), mesh=True)\n",
    "P.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### What else?"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 8.7.beta5",
   "language": "",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.15"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}