{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Algebraic Manipulation\n",
    "\n",
    "\n",
    "Here is a small sample of algebraic manipulation functions in `Symata`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "using Symata; isymata()   # load Symata and enter symata mode"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Common subexpression elimination"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\left( x + - \\ y \\right)  \\  \\left( - \\ y + z \\right)  +  \\left(  \\left( x + - \\ y \\right)  \\  \\left( - \\ y + z \\right)  \\right) ^{\\frac{1}{2}} $$"
      ],
      "text/plain": [
       "L\"$$  \\left( x + - \\ y \\right)  \\  \\left( - \\ y + z \\right)  +  \\left(  \\left( x + - \\ y \\right)  \\  \\left( - \\ y + z \\right)  \\right) ^{\\frac{1}{2}} $$\""
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ex  = (x-y)*(z-y) + Sqrt((x-y)*(z-y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Cse(expr)` recursivley replaces subexpressions that occur more than once in `expr` with names. The transformed expression is returned with a list of rules that can be used to recover `expr`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ [[x1 + x1^{\\frac{1}{2}}],[x1 \\Rightarrow  \\left(  \\left( x + x0 \\right)  \\  \\left( x0 + z \\right)  \\right) ,x0 \\Rightarrow  \\left( - \\ y \\right) ]] $$"
      ],
      "text/plain": [
       "L\"$$ [[x1 + x1^{\\frac{1}{2}}],[x1 \\Rightarrow  \\left(  \\left( x + x0 \\right)  \\  \\left( x0 + z \\right)  \\right) ,x0 \\Rightarrow  \\left( - \\ y \\right) ]] $$\""
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Cse(ex)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Applying in order the replacement rules in the second list to the expression in the first list results in the original expression.\n",
    "\n",
    "We will use `Splat`, which works like this,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ f \\!  \\left( a,b,c,d \\right)  $$"
      ],
      "text/plain": [
       "L\"$$ f \\!  \\left( a,b,c,d \\right)  $$\""
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(a,b,Splat([c,d]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and `Fold`, which works like this,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ f \\!  \\left(  \\left( f \\!  \\left(  \\left( f \\!  \\left( x,a \\right)  \\right) ,b \\right)  \\right) ,c \\right)  $$"
      ],
      "text/plain": [
       "L\"$$ f \\!  \\left(  \\left( f \\!  \\left(  \\left( f \\!  \\left( x,a \\right)  \\right) ,b \\right)  \\right) ,c \\right)  $$\""
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fold(f, [x,a,b,c])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Apply the replacement rules like this,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\left( x + - \\ y \\right)  \\  \\left( - \\ y + z \\right)  +  \\left(  \\left( x + - \\ y \\right)  \\  \\left( - \\ y + z \\right)  \\right) ^{\\frac{1}{2}} $$"
      ],
      "text/plain": [
       "L\"$$  \\left( x + - \\ y \\right)  \\  \\left( - \\ y + z \\right)  +  \\left(  \\left( x + - \\ y \\right)  \\  \\left( - \\ y + z \\right)  \\right) ^{\\frac{1}{2}} $$\""
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Fold(ReplaceAll, Splat(Cse(ex)))[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "ClearAll(ex)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### `Together`  and `Apart`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Together` rewrites rational expressions as a single fraction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\frac{x \\ y + x \\ z + y \\ z}{x \\ y \\ z} $$"
      ],
      "text/plain": [
       "L\"$$ \\frac{x \\ y + x \\ z + y \\ z}{x \\ y \\ z} $$\""
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Together(1/x + 1/y + 1/z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\frac{x + y}{x \\ y^{2}} $$"
      ],
      "text/plain": [
       "L\"$$ \\frac{x + y}{x \\ y^{2}} $$\""
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Together(1/(x*y) + 1/y^2) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\frac{x \\  \\left( 1 + y \\right)  +  \\left( 1 + x \\right)  \\ y}{ \\left( 1 + x \\right)  \\  \\left( 1 + y \\right) } $$"
      ],
      "text/plain": [
       "L\"$$ \\frac{x \\  \\left( 1 + y \\right)  +  \\left( 1 + x \\right)  \\ y}{ \\left( 1 + x \\right)  \\  \\left( 1 + y \\right) } $$\""
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Together(1/(1 + 1/x) + 1/(1 + 1/y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default, `Together` only works at the topmost level."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\mathbb{e} ^{x^{-1} + y^{-1}} $$"
      ],
      "text/plain": [
       "L\"$$  \\mathbb{e} ^{x^{-1} + y^{-1}} $$\""
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Together(Exp(1/x + 1/y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Together` is applied at all levels if the option `Deep` is true."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\mathbb{e} ^{\\frac{x + y}{x \\ y}} $$"
      ],
      "text/plain": [
       "L\"$$  \\mathbb{e} ^{\\frac{x + y}{x \\ y}} $$\""
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Together(Exp(1/x + 1/y), Deep => True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Apart` gives the partial fraction decomposition of a rational expression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\frac{y}{1 + x} + \\frac{- \\ y}{2 + x} $$"
      ],
      "text/plain": [
       "L\"$$ \\frac{y}{1 + x} + \\frac{- \\ y}{2 + x} $$\""
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Apart(y/(x + 2)/(x + 1), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the denominator has non-rational roots, the option ``Full => True`` must be given."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\frac{\\frac{- \\ y}{3} + \\frac{ \\left( -2 \\right)  \\  \\left( \\frac{-1}{2} +  \\left( \\frac{-1}{2} \\ \\mathbb{i} \\right)  \\ 3^{\\frac{1}{2}} \\right)  \\ y}{3}}{\\frac{1}{2} +  \\left( \\frac{1}{2} \\ \\mathbb{i} \\right)  \\ 3^{\\frac{1}{2}} + x} + \\frac{\\frac{- \\ y}{3} + \\frac{ \\left( -2 \\right)  \\  \\left( \\frac{-1}{2} +  \\left( \\frac{1}{2} \\ \\mathbb{i} \\right)  \\ 3^{\\frac{1}{2}} \\right)  \\ y}{3}}{\\frac{1}{2} +  \\left( \\frac{-1}{2} \\ \\mathbb{i} \\right)  \\ 3^{\\frac{1}{2}} + x} $$"
      ],
      "text/plain": [
       "L\"$$ \\frac{\\frac{- \\ y}{3} + \\frac{ \\left( -2 \\right)  \\  \\left( \\frac{-1}{2} +  \\left( \\frac{-1}{2} \\ \\mathbb{i} \\right)  \\ 3^{\\frac{1}{2}} \\right)  \\ y}{3}}{\\frac{1}{2} +  \\left( \\frac{1}{2} \\ \\mathbb{i} \\right)  \\ 3^{\\frac{1}{2}} + x} + \\frac{\\frac{- \\ y}{3} + \\frac{ \\left( -2 \\right)  \\  \\left( \\frac{-1}{2} +  \\left( \\frac{1}{2} \\ \\mathbb{i} \\right)  \\ 3^{\\frac{1}{2}} \\right)  \\ y}{3}}{\\frac{1}{2} +  \\left( \\frac{-1}{2} \\ \\mathbb{i} \\right)  \\ 3^{\\frac{1}{2}} + x} $$\""
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Apart(y/(x^2 + x + 1), x, Full=>True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### `Collect`\n",
    "\n",
    "Collect coefficients of powers of `x`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ c +  \\left( a + - \\ b \\right)  \\ x +  \\left( a + b \\right)  \\ x^{2} $$"
      ],
      "text/plain": [
       "L\"$$ c +  \\left( a + - \\ b \\right)  \\ x +  \\left( a + b \\right)  \\ x^{2} $$\""
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Collect(a*x^2 + b*x^2 + a*x - b*x + c, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Collect coefficients of an expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$  \\left( 2 \\ a + b \\right)  \\ x \\ \\text{Log} \\!  \\left( x \\right)  $$"
      ],
      "text/plain": [
       "L\"$$  \\left( 2 \\ a + b \\right)  \\ x \\ \\text{Log} \\!  \\left( x \\right)  $$\""
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Collect(a*x*Log(x) + (b+a)*(x*Log(x)), x*Log(x))"
   ]
  }
 ],
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