SymPy: Reducing Symbolic Sums Over Constants

PY-ReduceSum.ipynb

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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Reducing Symbolic Summations with SymPy\n",
    "\n",
    "This gist shows how to use [SymPy 1.5](https://www.sympy.org) to transform symbolic sums over products of constant terms (with respect to the summation index) and indexed terms like so:\n",
    "\n",
    "$$\n",
    "\\sum^{m-1}_{i=0} (a \\cdot A_i + b \\cdot B_i + c)\n",
    "\\rightarrow \n",
    "a\\sum^{m-1}_{i=0}A_i + b\\sum^{m-1}_{0}B_i + m \\cdot c\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get started with the example above we need to setup a SymPy session for symbolic summation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import IndexedBase, Sum, expand, simplify, factor_terms,Eq\n",
    "from sympy.abc import *\n",
    "\n",
    "A = IndexedBase('A')\n",
    "B = IndexedBase('B')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's work on the example above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{i=0}^{m - 1} \\left(a {A}_{i} + b {B}_{i} + c\\right)$"
      ],
      "text/plain": [
       "Sum(a*A[i] + b*B[i] + c, (i, 0, m - 1))"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = Sum(a*A[i] + b*B[i] + c, (i,0,m-1))\n",
    "s"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First we pull the sum apart by expanding it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{i=0}^{m - 1} c + \\sum_{i=0}^{m - 1} a {A}_{i} + \\sum_{i=0}^{m - 1} b {B}_{i}$"
      ],
      "text/plain": [
       "Sum(c, (i, 0, m - 1)) + Sum(a*A[i], (i, 0, m - 1)) + Sum(b*B[i], (i, 0, m - 1))"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1 = expand(s)\n",
    "s1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To evaluate the sum over the constant $c$, we factor out all constants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\sum_{i=0}^{m - 1} {A}_{i} + b \\sum_{i=0}^{m - 1} {B}_{i} + c \\sum_{i=0}^{m - 1} 1$"
      ],
      "text/plain": [
       "a*Sum(A[i], (i, 0, m - 1)) + b*Sum(B[i], (i, 0, m - 1)) + c*Sum(1, (i, 0, m - 1))"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s2 = factor_terms(s1)\n",
    "s2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get rid of the odd $\\sum^{i}_{m-1} 1$ we need one final step:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{i=0}^{m - 1} \\left(a {A}_{i} + b {B}_{i} + c\\right) = a \\sum_{i=0}^{m - 1} {A}_{i} + b \\sum_{i=0}^{m - 1} {B}_{i} + c m$"
      ],
      "text/plain": [
       "Eq(Sum(a*A[i] + b*B[i] + c, (i, 0, m - 1)), a*Sum(A[i], (i, 0, m - 1)) + b*Sum(B[i], (i, 0, m - 1)) + c*m)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s3 = s2.doit()\n",
    "Eq(s,s3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try a more complex example now:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{i=0}^{m - 1} \\left(a {A}_{i} + b {B}_{i} + c\\right)^{2}$"
      ],
      "text/plain": [
       "Sum((a*A[i] + b*B[i] + c)**2, (i, 0, m - 1))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = Sum((a*A[i] + b*B[i] + c)**2, (i,0,m-1))\n",
    "e"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "for this we get:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{i=0}^{m - 1} \\left(a {A}_{i} + b {B}_{i} + c\\right)^{2} = a^{2} \\sum_{i=0}^{m - 1} {A}_{i}^{2} + 2 a b \\sum_{i=0}^{m - 1} {A}_{i} {B}_{i} + 2 a c \\sum_{i=0}^{m - 1} {A}_{i} + b^{2} \\sum_{i=0}^{m - 1} {B}_{i}^{2} + 2 b c \\sum_{i=0}^{m - 1} {B}_{i} + c^{2} m$"
      ],
      "text/plain": [
       "Eq(Sum((a*A[i] + b*B[i] + c)**2, (i, 0, m - 1)), a**2*Sum(A[i]**2, (i, 0, m - 1)) + 2*a*b*Sum(A[i]*B[i], (i, 0, m - 1)) + 2*a*c*Sum(A[i], (i, 0, m - 1)) + b**2*Sum(B[i]**2, (i, 0, m - 1)) + 2*b*c*Sum(B[i], (i, 0, m - 1)) + c**2*m)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1 = factor_terms(expand(e)).doit()\n",
    "Eq(e,e1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# About this Jupyter Notebook\n",
    "\n",
    "This _Gist_ was created using:\n",
    "* the [Jupyter Lab](https://jupyter.org/) computational notebook.\n",
    "* the _python3_ kernel\n",
    "* [SymPy 1.5](https://www.sympy.org) - for symbolic math."
   ]
  }
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