demo.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "using TypedPolynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x, y, z)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Creating variables\n",
    "@polyvar x y z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x * x = x^2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "TypedPolynomials.Monomial{(x,),1}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Basic arithemtic:\n",
    "@show x * x\n",
    "typeof(x * x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x ^ 3 = x^3\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "TypedPolynomials.Monomial{(x,),1}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@show x^3\n",
    "typeof(x^3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x * y = xy\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "TypedPolynomials.Monomial{(x, y),2}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@show x * y\n",
    "typeof(x * y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x + z = x + z\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "TypedPolynomials.Polynomial{Int64,TypedPolynomials.Term{Int64,TypedPolynomials.Monomial{(x, z),2}},Array{TypedPolynomials.Term{Int64,TypedPolynomials.Monomial{(x, z),2}},1}}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@show x + z\n",
    "typeof(x + z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "xy + 2x - 5y - 10"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = (x - 5) * (y + 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Array{TypedPolynomials.Term{Int64,TypedPolynomials.Monomial{(x, y),2}},1}:\n",
       " xy \n",
       " 2x \n",
       " -5y\n",
       " -10"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "terms(p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-16"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Substitution\n",
    "subs(p, x=>1, y=>2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4y - 8"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Partial substitution\n",
    "subs(p, x=>1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "yz - 5y + 2z - 10"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Variable substitution\n",
    "subs(p, x=>z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Int64"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Julia can infer that this substition produces a value:\n",
    "typeof(subs(p, x=>1, y=>2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TypedPolynomials.Polynomial{Int64,TypedPolynomials.Term{Int64,TypedPolynomials.Monomial{(y,),1}},Array{TypedPolynomials.Term{Int64,TypedPolynomials.Monomial{(y,),1}},1}}"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ...but this one produces a polynomial:\n",
    "typeof(subs(p, x=>1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-16"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Polynomials also provide function-like syntax as a\n",
    "# shorthand for subs():\n",
    "p(x=>1, y=>2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "BenchmarkTools.Trial: \n",
       "  memory estimate:  0 bytes\n",
       "  allocs estimate:  0\n",
       "  --------------\n",
       "  minimum time:     20.928 ns (0.00% GC)\n",
       "  median time:      21.222 ns (0.00% GC)\n",
       "  mean time:        22.839 ns (0.00% GC)\n",
       "  maximum time:     92.882 ns (0.00% GC)\n",
       "  --------------\n",
       "  samples:          10000\n",
       "  evals/sample:     997"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Complete substitution of a polynomial should not allocate\n",
    "# any memory at all:\n",
    "using BenchmarkTools\n",
    "@benchmark subs($p, $x=>1, $y=>2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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