{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Examples from Mathematica Stack Exchange\n",
    "\n",
    "This notebook contains examples from [Mathematica Stack Exchange](http://mathematica.stackexchange.com/) applied to Symata.\n",
    "\n",
    "Disclaimer to avoid any possible confusion.\n",
    "\n",
    "Neither Symata nor the Symata language are affiliated in any way with [Mathematica and/or the Wolfram language](http://wolfram.com). Symata is an open source project. Mathematica and Wolfram language are software products developed and licensed by [WRI](http://wolfram.com)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Symata"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "FloatFormat(Short);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [Determining all possible traversals of a tree](http://mathematica.stackexchange.com/questions/5753/determining-all-possible-traversals-of-a-tree)\n",
    "\n",
    "The [following example](http://mathematica.stackexchange.com/questions/5753/determining-all-possible-traversals-of-a-tree/5756#5756) example is from L. Shifrin.\n",
    "\n",
    "`C` is a tree:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = [a,[[a1,[a12,b12,c12]],[b2,[a22,b22,c22]],[c3,[a32,b32,c32,d32]]]];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "trav(tree_List) := Flatten(trav([], tree), 1)\n",
    "trav(accum_List, [x_, y_List]) := Map(yy -> trav([accum, x], yy), y)\n",
    "trav(x_,y_) := Flatten([x,y])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\left[  \\left[ a,a1,a12 \\right] , \\left[ a,a1,b12 \\right] , \\left[ a,a1,c12 \\right] , \\left[ a,b2,a22 \\right] , \\left[ a,b2,b22 \\right] , \\left[ a,b2,c22 \\right] , \\left[ a,c3,a32 \\right] , \\left[ a,c3,b32 \\right] , \\left[ a,c3,c32 \\right] , \\left[ a,c3,d32 \\right]  \\right]  $$"
      ],
      "text/plain": [
       "L\"$$  \\left[  \\left[ a,a1,a12 \\right] , \\left[ a,a1,b12 \\right] , \\left[ a,a1,c12 \\right] , \\left[ a,b2,a22 \\right] , \\left[ a,b2,b22 \\right] , \\left[ a,b2,c22 \\right] , \\left[ a,c3,a32 \\right] , \\left[ a,c3,b32 \\right] , \\left[ a,c3,c32 \\right] , \\left[ a,c3,d32 \\right]  \\right]  $$\""
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trav(C)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [What are the use cases for different scoping constructs?](http://mathematica.stackexchange.com/questions/559/what-are-the-use-cases-for-different-scoping-constructs/)\n",
    "\n",
    " In this example, `Module` creates a *closure*. We want to use big integers, so we use `big\"1\"` for one of the values.\n",
    "\n",
    "[This example](http://mathematica.stackexchange.com/questions/559/what-are-the-use-cases-for-different-scoping-constructs/569#569) is also by L. Shifrin\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "Module([prev, prevprev, this],\n",
    "begin \n",
    "    reset() := (prev = big\"1\"; prevprev = 1); \n",
    "    reset(); \n",
    "    nextFib() := (this = prev + prevprev; prevprev = prev; prev = this)\n",
    "end\n",
    ");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ 113796925398360272257523782552224175572745930353730513145086634176691092536145985470146129334641866902783673042322088625863396052888690096969577173696370562180400527049497109023054114771394568040040412172632376 $$"
      ],
      "text/plain": [
       "L\"$$ 113796925398360272257523782552224175572745930353730513145086634176691092536145985470146129334641866902783673042322088625863396052888690096969577173696370562180400527049497109023054114771394568040040412172632376 $$\""
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reset()\n",
    "\n",
    "a = Table(nextFib(),[1000]);\n",
    "\n",
    "a[-1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Return all free symbols in an expression\n",
    "\n",
    "Below is another example from L. Shifrin. `allsyms` returns all free symbols in `expr`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "ClearAll(a,b)\n",
    "\n",
    "allsyms(expr_) := Cases(expr , s_Symbol => HoldComplete(s),[0,Infinity])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\left[ \\text{HoldComplete} \\!  \\left( a \\right) ,\\text{HoldComplete} \\!  \\left( b \\right) ,\\text{HoldComplete} \\!  \\left( x \\right)  \\right]  $$"
      ],
      "text/plain": [
       "L\"$$  \\left[ \\text{HoldComplete} \\!  \\left( a \\right) ,\\text{HoldComplete} \\!  \\left( b \\right) ,\\text{HoldComplete} \\!  \\left( x \\right)  \\right]  $$\""
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "allsyms(a + b * (1 - x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### [Finding real roots of negative numbers (for example $\\sqrt[3]{-8}$)](http://mathematica.stackexchange.com/questions/3886/finding-real-roots-of-negative-numbers-for-example-sqrt3-8)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `Power` function returns the principal root, not necessarily a real root."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\left[ 2 \\  \\left(  \\left( -1 \\right) ^{\\frac{1}{3}} \\right) ,1. + 1.73205\\mathbb{i} \\right]  $$"
      ],
      "text/plain": [
       "L\"$$  \\left[ 2 \\  \\left(  \\left( -1 \\right) ^{\\frac{1}{3}} \\right) ,1. + 1.73205\\mathbb{i} \\right]  $$\""
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[(-8)^(1/3), (-8.0)^(1/3)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`CubeRoot` and `Surd` give real roots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\left[ -2,-2 \\right]  $$"
      ],
      "text/plain": [
       "L\"$$  \\left[ -2,-2 \\right]  $$\""
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[CubeRoot(-8), Surd(-32,5)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Surd` returns unevaluated if the root is even."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Surd::noneg: Surd is not defined for even roots of negative values.\n",
      "└ @ Symata /home/lapeyre/.julia/dev/Symata/src/wrapout.jl:29\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$ \\text{Surd} \\!  \\left(  \\left( -8 \\right) ,4 \\right)  $$"
      ],
      "text/plain": [
       "L\"$$ \\text{Surd} \\!  \\left(  \\left( -8 \\right) ,4 \\right)  $$\""
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Surd(-8,4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "or complex"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Warning: Surd::preal: The parameter I should be real valued\n",
      "└ @ Symata /home/lapeyre/.julia/dev/Symata/src/wrapout.jl:29\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$ \\text{Surd} \\!  \\left( \\mathbb{i},3 \\right)  $$"
      ],
      "text/plain": [
       "L\"$$ \\text{Surd} \\!  \\left( \\mathbb{i},3 \\right)  $$\""
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Surd(I,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Version and date"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Symata version     0.4.6\n",
      "Julia version      1.6.0-DEV.58\n",
      "Python version     3.8.3\n",
      "SymPy version      1.5.1\n"
     ]
    }
   ],
   "source": [
    "VersionInfo()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2020-05-29T22:44:02.976"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "InputForm(Now())"
   ]
  }
 ],
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