{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Understanding gduals and floats\n",
    "(by Dario Izzo)\n",
    "\n",
    "Arithmetic operators +,-,+,/ and \\** work on a gdual as well as mathematical functions (from pyaudi). But not everything must be a gdual. You can operate between gduals and floats too, and between gdual having different truncation order."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Importing stuff"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pyaudi import gdual_double as gdual\n",
    "from pyaudi import sin, cos, tan"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Arithmetic between floats and gduals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\[ 3.3+{dx}+\\mathcal{O}\\left(4\\right) \\]"
      ],
      "text/plain": [
       "3.3+dx"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = gdual(1.2, \"x\", 3) # a gdual\n",
    "y = 2.1                # a float\n",
    "f = x + y              # this will be a gdual\n",
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "NOTE: the gdual f defined above only contains derivative infos w.r.t the variable \"x\". You cannot make derivatives w.r.t \"y\" as y was not defined as a gdual "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The result is a float: 4.183307552861899\n",
      "The result is a gdual (contains only derivatives w.r.t. x): 4.18331+5.06537*dx+1.54332*dx**2-0.0437224*dx**3\n",
      "The result is a gdual (contains derivatives w.r.t. x and y): -0.324612*dy+0.104137*dx*dy**2-0.0437224*dx**3+4.18331-0.250974*dx*dy-0.0345762*dy**3+5.06537*dx-0.0827871*dx**2*dy+0.10332*dy**2+1.54332*dx**2\n"
     ]
    }
   ],
   "source": [
    "def f(x,y):\n",
    "    return x*x+2*tan(x/(y+1))-sin(x)\n",
    "\n",
    "xf = 2\n",
    "yf = 3\n",
    "x = gdual(2., \"x\", 3)\n",
    "y = gdual(3., \"y\", 3)\n",
    "print(\"The result is a float: {}\".format(f(xf,yf)))\n",
    "print(\"The result is a gdual (contains only derivatives w.r.t. x): {}\".format(f(x,yf)))\n",
    "print(\"The result is a gdual (contains derivatives w.r.t. x and y): {}\".format(f(x,y)))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Promoting a gdual\n",
    "\n",
    "You can also perform arithmetic operations with gduals defined with a different differentiation order. In this case the resulting gdual will be \"promoted\" to the higher order"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1.9+dy+dx\n",
      "Order: 4\n",
      "Degree: 1\n"
     ]
    }
   ],
   "source": [
    "x = gdual(0.1, \"x\", 2)\n",
    "y = gdual(-2, \"y\", 4)\n",
    "f = x+y\n",
    "print(f)\n",
    "print(\"Order: {}\".format(f.order)) # The order is the maximum order terms will be retained during the\n",
    "# computations using the truncated Taylor Polynomial algebra computations\n",
    "print(\"Degree: {}\".format(f.degree)) # The degree is the actual degree of the polynomial\n",
    "#which will be less or equal than the order"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "NOTE: be careful when using gdual promotion. You may \"lose\" information implicitly if the order equals the degree. "
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}