Relationship between elementary differentials and modified equation coefficients.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f55ac2e6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TruncatedBSeries{RootedTree{Int64, Vector{Int64}}, Sym} with 18 entries:\n",
       "  RootedTree{Int64}: Int64[]         => a_∅\n",
       "  RootedTree{Int64}: [1]             => a_τ\n",
       "  RootedTree{Int64}: [1, 2]          => a_[τ]\n",
       "  RootedTree{Int64}: [1, 2, 3]       => a_[[τ]]\n",
       "  RootedTree{Int64}: [1, 2, 2]       => a_[τ²]\n",
       "  RootedTree{Int64}: [1, 2, 3, 4]    => a_[[[τ]]]\n",
       "  RootedTree{Int64}: [1, 2, 3, 3]    => a_[[τ²]]\n",
       "  RootedTree{Int64}: [1, 2, 3, 2]    => a_[[τ]τ]\n",
       "  RootedTree{Int64}: [1, 2, 2, 2]    => a_[τ³]\n",
       "  RootedTree{Int64}: [1, 2, 3, 4, 5] => a_[[[[τ]]]]\n",
       "  RootedTree{Int64}: [1, 2, 3, 4, 4] => a_[[[τ²]]]\n",
       "  RootedTree{Int64}: [1, 2, 3, 4, 3] => a_[[[τ]τ]]\n",
       "  RootedTree{Int64}: [1, 2, 3, 4, 2] => a_[[[τ]]τ]\n",
       "  RootedTree{Int64}: [1, 2, 3, 3, 3] => a_[[τ³]]\n",
       "  RootedTree{Int64}: [1, 2, 3, 3, 2] => a_[[τ²]τ]\n",
       "  RootedTree{Int64}: [1, 2, 3, 2, 3] => a_[[τ][τ]]\n",
       "  RootedTree{Int64}: [1, 2, 3, 2, 2] => a_[[τ]τ²]\n",
       "  RootedTree{Int64}: [1, 2, 2, 2, 2] => a_[τ⁴]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using BSeries, SymPy\n",
    "\n",
    "series = bseries(5) do t, series\n",
    "    return symbols(\"a_$(butcher_representation(t))\", real=true)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "cdb62320",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TruncatedBSeries{RootedTree{Int64, Vector{Int64}}, Sym} with 18 entries:\n",
       "  RootedTree{Int64}: Int64[]         => 0\n",
       "  RootedTree{Int64}: [1]             => a_τ\n",
       "  RootedTree{Int64}: [1, 2]          => a_[τ] - a_τ^2/2\n",
       "  RootedTree{Int64}: [1, 2, 3]       => a_[[τ]] - a_τ^3/6 - a_τ*(a_[τ]/2 - a_τ^…\n",
       "  RootedTree{Int64}: [1, 2, 2]       => a_[τ²] - a_τ^3/3 - a_τ*(a_[τ] - a_τ^2/2)\n",
       "  RootedTree{Int64}: [1, 2, 3, 4]    => a_[[[τ]]] - a_τ^4/24 - a_τ^2*(a_[τ]/6 -…\n",
       "  RootedTree{Int64}: [1, 2, 3, 3]    => a_[[τ²]] - a_τ^4/12 - 2*a_τ^2*(a_[τ] - …\n",
       "  RootedTree{Int64}: [1, 2, 3, 2]    => a_[[τ]τ] - a_τ^4/8 - 5*a_τ^2*(a_[τ] - a…\n",
       "  RootedTree{Int64}: [1, 2, 2, 2]    => a_[τ³] - a_τ^4/4 - a_τ^2*(a_[τ] - a_τ^2…\n",
       "  RootedTree{Int64}: [1, 2, 3, 4, 5] => a_[[[[τ]]]] - a_τ^5/120 - a_τ^3*(a_[τ]/…\n",
       "  RootedTree{Int64}: [1, 2, 3, 4, 4] => a_[[[τ²]]] - a_τ^5/60 - a_τ^3*(a_[τ] - …\n",
       "  RootedTree{Int64}: [1, 2, 3, 4, 3] => a_[[[τ]τ]] - a_τ^5/40 - a_τ^3*(a_[τ] - …\n",
       "  RootedTree{Int64}: [1, 2, 3, 4, 2] => a_[[[τ]]τ] - a_τ^5/30 - 5*a_τ^3*(a_[τ] …\n",
       "  RootedTree{Int64}: [1, 2, 3, 3, 3] => a_[[τ³]] - a_τ^5/20 - a_τ^3*(a_[τ] - a_…\n",
       "  RootedTree{Int64}: [1, 2, 3, 3, 2] => a_[[τ²]τ] - a_τ^5/15 - 7*a_τ^3*(a_[τ] -…\n",
       "  RootedTree{Int64}: [1, 2, 3, 2, 3] => a_[[τ][τ]] - a_τ^5/20 - a_τ^3*(a_[τ]/8 …\n",
       "  RootedTree{Int64}: [1, 2, 3, 2, 2] => a_[[τ]τ²] - a_τ^5/10 - 3*a_τ^3*(a_[τ] -…\n",
       "  RootedTree{Int64}: [1, 2, 2, 2, 2] => a_[τ⁴] - a_τ^5/5 - a_τ^3*(a_[τ] - a_τ^2…"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "modeqn = modified_equation(series)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "bbc7952d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$a_{[[τ]]} - a_{[τ]} a_{τ} + \\frac{a_{τ}^{3}}{3}$"
      ],
      "text/plain": [
       "                         3\n",
       "                      a_τ \n",
       "a_[[τ]] - a_[τ]⋅a_τ + ────\n",
       "                       3  "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(modeqn[RootedTree([1,2,3])])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "c59e533f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$- a_{[τ]} a_{τ} + a_{[τ²]} + \\frac{a_{τ}^{3}}{6}$"
      ],
      "text/plain": [
       "                         3\n",
       "                      a_τ \n",
       "-a_[τ]⋅a_τ + a_[τ²] + ────\n",
       "                       6  "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(modeqn[RootedTree([1,2,2])])"
   ]
  }
 ],
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