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   "source": [
    "# Understanding Taylor models\n",
    "(by Sean Cowan)\n",
    "\n",
    "Analogously to the arithmetic operators of gduals, the interface with Taylor models in pyaudi is similar, if not identical."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "858713f3-d65f-439c-8b3f-8fc4c132d0a7",
   "metadata": {},
   "source": [
    "## Importing stuff"
   ]
  },
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   "id": "8b879243-90e2-423a-ad3d-0cd5d21b603a",
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   "source": [
    "from pyaudi import gdual_double as gdual, taylor_model, int_d"
   ]
  },
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   "source": [
    "## Defining function\n",
    "\n",
    "We will try to approximate a function — shown below, as one could also do with gduals, and show how Taylor models differ from gduals.\n",
    "\n",
    "$$\n",
    "f(x) = \\frac{1}{x} + x\n",
    "$$\n",
    "\n",
    "Now, to approximate a function using Taylor expansions, we need to define the expansion order and expansion point:"
   ]
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  {
   "cell_type": "markdown",
   "id": "f10c1e6f-8740-43db-880f-4fe98127236a",
   "metadata": {},
   "source": [
    "## Taylor polynomial (with gduals)"
   ]
  },
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   "id": "584da1af-f21c-43b4-b86a-46c08b675ab8",
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.5+0.125*dx**2+0.03125*dx**4+0.75*dx-0.0625*dx**3\n"
     ]
    }
   ],
   "source": [
    "order = 4\n",
    "exp_points = {\"x\": 2.0}\n",
    "\n",
    "x = gdual(exp_points[\"x\"], \"x\", order)\n",
    "fx = 1 / x + x\n",
    "print(fx)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "049504ba-c789-4068-acd9-48ede4bec944",
   "metadata": {},
   "source": [
    "## Taylor model\n",
    "The result is the expected fourth order Taylor polynomial of function $f(x)$. If however, we would like to do the same computation with Taylor models by encompassing the higher order terms into a remainder interval, we also need to define the domain of validity and initial remainder bound. The same steps are performed, yet now there is a non-zero remainder bound."
   ]
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "Taylor Polynomial:\n",
       "2.5+0.125*dx**2+0.03125*dx**4+0.75*dx-0.0625*dx**3\n",
       "Remainder Bound: [-2.12558e-07, 2.12558e-07]\n",
       "Expansion Point: {x: 2}\n",
       "Domain: {x:[1.9, 2.1]}"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rem_bound = int_d(0.0, 0.0)\n",
    "domain = {\"x\": int_d(1.9, 2.1)}\n",
    "\n",
    "x = gdual(exp_points[\"x\"], \"x\", order)\n",
    "tm_x = taylor_model(x, rem_bound, exp_points, domain)\n",
    "\n",
    "tm_fx = 1 / tm_x + tm_x\n",
    "tm_fx"
   ]
  },
  {
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   "id": "09648bf9-297d-4175-b016-ce3230fa70eb",
   "metadata": {},
   "source": [
    "This remainder bound means that, given a Taylor polynomial valid within the domain of $[1.9, 2.1]$, one would need to add the lower and upper bound as a constant to that Taylor polynomial to ensure that, whatever $x$ value is chosen, the function value of $f(x)$ will be within that range. This differs fundamentally from the information that the gdual provides, since it will provide no mathematical guarantee on the distance from the actual function."
   ]
  }
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