Tricks in SymbolicNumericalIntegration package

Tricks_in_SymbolicNumericalIntegration_package.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tricks in `SymbolicNumericalIntegration` package"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Julia Version 1.8.5\n",
      "Commit 17cfb8e65ea (2023-01-08 06:45 UTC)\n",
      "Platform Info:\n",
      "  OS: Linux (x86_64-linux-gnu)\n",
      "  CPU: 8 × Intel(R) Core(TM) i5-1035G7 CPU @ 1.20GHz\n",
      "  WORD_SIZE: 64\n",
      "  LIBM: libopenlibm\n",
      "  LLVM: libLLVM-13.0.1 (ORCJIT, icelake-client)\n",
      "  Threads: 1 on 8 virtual cores\n",
      "Environment:\n",
      "  JULIA_NUM_THREADS = \n"
     ]
    }
   ],
   "source": [
    "versioninfo()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Symbolics\n",
    "using SymbolicNumericIntegration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{c}\n",
       "x \\\\\n",
       "y \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "2-element Vector{Num}:\n",
       " x\n",
       " y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@variables x y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f1 (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f1(x) = sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\sin\\left( x \\right)\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f1(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-cos(x), 0, 0)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "temp = integrate(f1(x), x)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Failure example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "g1 (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g1(x) = temp[1]"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`g1` doesn't work at all."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       " - \\cos\\left( x \\right)\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g1(pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       " - \\cos\\left( x \\right)\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g1(y)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Successful example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "g2 (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g2(y) = substitute(g1(x), Dict(x=>y))"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`g2` works well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       " - \\cos\\left( x \\right)\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g2(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       " - \\cos\\left( y \\right)\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "-cos(y)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g2(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g2(pi)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrix version"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "f2(x) = cos(x); f3(x) = exp(2x); f4(x) = sinh(x);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "A (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A(x) = [f1(x) f2(x); f3(x) f4(x)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{cc}\n",
       "\\sin\\left( x \\right) & \\cos\\left( x \\right) \\\\\n",
       "e^{2 x} & \\sinh\\left( x \\right) \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "2×2 Matrix{Num}:\n",
       " sin(x)          cos(x)\n",
       "       exp(2x)  sinh(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Matrix{Tuple{SymbolicUtils.Symbolic{Real}, Int64, Int64}}:\n",
       " (-cos(x), 0, 0)         (sin(x), 0, 0)\n",
       " ((1//2)*exp(2x), 0, 0)  (cosh(x), 0, 0)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Note: `integrate(A(x), x)` somehow fails. Element-wise integration works.\n",
    "temp = integrate.(A(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Matrix{SymbolicUtils.Symbolic{Real}}:\n",
       " -cos(x)         sin(x)\n",
       " (1//2)*exp(2x)  cosh(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "temp2 = (e -> e[1]).(temp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "B (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Note: `substitute(temp2, Dict(x => y))` somehow fails. Element-wise substitution works.\n",
    "B(y) = (e -> substitute(e, Dict(x => y))).(temp2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{equation}\n",
       "\\left[\n",
       "\\begin{array}{cc}\n",
       " - \\cos\\left( x \\right) & \\sin\\left( x \\right) \\\\\n",
       "\\frac{1}{2} e^{2 x} & \\cosh\\left( x \\right) \\\\\n",
       "\\end{array}\n",
       "\\right]\n",
       "\\end{equation}\n",
       " $$"
      ],
      "text/plain": [
       "2×2 Matrix{Num}:\n",
       " -cos(x)          sin(x)\n",
       " (1//2)*exp(2x)  cosh(x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "B(x)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 1.8.5",
   "language": "julia",
   "name": "julia-1.8"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "1.8.5"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
 "nbformat_minor": 2
}