{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ad2fb634-5165-4ed5-8399-53ae0c994ffd",
   "metadata": {
    "tags": []
   },
   "source": [
    "`help(r)` will show the help on R in the module `sage.interfaces.r` object."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75ebc6ef-80e4-451f-a63c-3ab660405084",
   "metadata": {},
   "source": [
    "# 1. R is not a computer algebra system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "08fd56c6-5e4b-4e1d-bffe-b5cec983e361",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1] 10"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r.choose(5, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1e3167bc-8087-4d93-973c-e45ef5661375",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1] 2.937234e+25"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r.choose(100,30)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "613e81ad-bdf0-49c3-af50-872032730488",
   "metadata": {},
   "source": [
    "We see that the output is not an exact number, but a floating-point approximation."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c7e3d94f-b8f7-47b7-8a92-6a53a0ceab96",
   "metadata": {},
   "source": [
    "# 2. Monte Carlo integration"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b7b700e9-3eb6-4977-9bee-5a438512e9a6",
   "metadata": {},
   "source": [
    "We generate one million random numbers uniformly distributed in the interval `[0,1]`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "228ef19a-2d1d-4d04-b2f8-58ec53556870",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "u = r.runif(1000000, min=0, max=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "869e4561-7404-499f-be12-1716a2fc0f84",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "v = r.sqrt(1-u^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89ee23a9-11fd-4eab-8925-5d91bce29969",
   "metadata": {},
   "source": [
    "Observe that `r` did vector arithmetic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c4182ccc-8e2a-4c86-861b-dd29cd9d7bea",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1] 3.141743"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "estpi = 4*r.mean(v)\n",
    "estpi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "edaeb36e-5f0e-43bf-af33-429b23531de8",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<class 'sage.interfaces.r.RElement'>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(estpi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "e2d6109d-a0cd-42f5-a348-6ec4941ed4fd",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.14174254344342"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "estpi.sage()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "32b7bdda-2b79-4dbe-9125-97a708cb743a",
   "metadata": {},
   "source": [
    "Let us consider another example."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01a6e56e-8469-4a6a-bc2d-d48aed053d60",
   "metadata": {},
   "source": [
    "We estimate $\\displaystyle \\int_a^b f(x) dx$\n",
    "by $\\displaystyle (b-a) \\frac{1}{n} \\sum_{x \\in [a,b]} f(x)$.\n",
    "\n",
    "Applied to $\\displaystyle \\int_0^{\\pi/2} 4 \\sin(2 x) e^{-x^2} dx$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0a7a2345-862f-4d3b-96bb-ddd2ea49702c",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.57079632679490"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "halfpi = float(pi)/2.0\n",
    "halfpi"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "a086e3f5-676c-4947-b131-ac15ff1096a5",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "u = r.runif(1000000, min=0, max=halfpi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "c79b24c3-ae28-4173-b471-9b2931d1397f",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1] 2.189133"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "halfpi*r.mean(4*sin(2*u)*exp(-u^2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "b0013801-ba96-4ee0-a7d9-3ba0bbc701fb",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "from rpy2.robjects import r as R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "ac7fb4e0-f369-4dfa-b062-1c75f85313d2",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<rpy2.robjects.functions.SignatureTranslatedFunction object at 0x7f6d38265f40> [RTYPES.CLOSXP]\n",
       "R classes: ('function',)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R('f <-function(x) { 4*sin(2*x)*exp(-x^2) }')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "5a46537f-b58b-48ed-a344-da8a6ec947e9",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "y = R('z = f(3)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "1c4fdb55-c231-4b4b-96cd-7b8a8e1f21fa",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <span>FloatVector with 1 elements.</span>\n",
       "        <table>\n",
       "        <tbody>\n",
       "          <tr>\n",
       "          \n",
       "            <td>\n",
       "            -0.000138\n",
       "            </td>\n",
       "          \n",
       "          </tr>\n",
       "        </tbody>\n",
       "        </table>\n",
       "        "
      ],
      "text/plain": [
       "<rpy2.robjects.vectors.FloatVector object at 0x7f6d3d588880> [RTYPES.REALSXP]\n",
       "R classes: ('numeric',)\n",
       "[-0.000138]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R('z')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "ee23f353-b25b-4b26-875c-6d2d8492c2c1",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "v = R('val = integrate(f, lower=0, upper=pi/2)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "1710f32c-15ce-4c0b-8e4f-16fc6cb803f2",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <span>ListVector with 5 elements.</span>\n",
       "        <table>\n",
       "        <tbody>\n",
       "        \n",
       "          <tr>\n",
       "            <th>\n",
       "            value\n",
       "            </th>\n",
       "            <td>\n",
       "            <rpy2.rinterface.FloatSexpVector object at 0x7f6d380c1f40> [RTYPES.REALSXP]\n",
       "            </td>\n",
       "          </tr>\n",
       "        \n",
       "          <tr>\n",
       "            <th>\n",
       "            abs.error\n",
       "            </th>\n",
       "            <td>\n",
       "            <rpy2.rinterface.FloatSexpVector object at 0x7f6d380d6b00> [RTYPES.REALSXP]\n",
       "            </td>\n",
       "          </tr>\n",
       "        \n",
       "          <tr>\n",
       "            <th>\n",
       "            subdivisions\n",
       "            </th>\n",
       "            <td>\n",
       "            <rpy2.rinterface.IntSexpVector object at 0x7f6d380d6d80> [RTYPES.INTSXP]\n",
       "            </td>\n",
       "          </tr>\n",
       "        \n",
       "          <tr>\n",
       "            <th>\n",
       "            message\n",
       "            </th>\n",
       "            <td>\n",
       "            <rpy2.rinterface_lib.sexp.StrSexpVector object at 0x7f6d380d5440> [RTYPES.STRSXP]\n",
       "            </td>\n",
       "          </tr>\n",
       "        \n",
       "          <tr>\n",
       "            <th>\n",
       "            call\n",
       "            </th>\n",
       "            <td>\n",
       "            <rpy2.rinterface.LangSexpVector object at 0x7f6d380df9c0> [RTYPES.LANGSXP]\n",
       "            </td>\n",
       "          </tr>\n",
       "        \n",
       "        </tbody>\n",
       "        </table>\n",
       "        "
      ],
      "text/plain": [
       "<rpy2.robjects.vectors.ListVector object at 0x7f6d380a0cc0> [RTYPES.VECSXP]\n",
       "R classes: ('integrate',)\n",
       "[FloatSe..., FloatSe..., IntSexp..., StrSexp..., LangSex...]\n",
       "  value: <class 'rpy2.rinterface.FloatSexpVector'>\n",
       "  <rpy2.rinterface.FloatSexpVector object at 0x7f6d380d7700> [RTYPES.REALSXP]\n",
       "  abs.error: <class 'rpy2.rinterface.FloatSexpVector'>\n",
       "  <rpy2.rinterface.FloatSexpVector object at 0x7f6d380df780> [RTYPES.REALSXP]\n",
       "  subdivisions: <class 'rpy2.rinterface.IntSexpVector'>\n",
       "  <rpy2.rinterface.IntSexpVector object at 0x7f6d380df7c0> [RTYPES.INTSXP]\n",
       "  message: <class 'rpy2.rinterface_lib.sexp.StrSexpVector'>\n",
       "  <rpy2.rinterface_lib.sexp.StrSexpVector object at 0x7f6d380d5440> [RTYPES.STRSXP]\n",
       "  call: <class 'rpy2.rinterface.LangSexpVector'>\n",
       "  <rpy2.rinterface.LangSexpVector object at 0x7f6d382e50c0> [RTYPES.LANGSXP]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R('val')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "90ec6a45-29e4-4b0c-8a31-9e618f6ade6a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <span>FloatVector with 1 elements.</span>\n",
       "        <table>\n",
       "        <tbody>\n",
       "          <tr>\n",
       "          \n",
       "            <td>\n",
       "            2.190752\n",
       "            </td>\n",
       "          \n",
       "          </tr>\n",
       "        </tbody>\n",
       "        </table>\n",
       "        "
      ],
      "text/plain": [
       "<rpy2.robjects.vectors.FloatVector object at 0x7f6d384884c0> [RTYPES.REALSXP]\n",
       "R classes: ('numeric',)\n",
       "[2.190752]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[int(0)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "df917703-67c6-4e0b-8c4f-5cea865b6e15",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.190751642453249"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[int(0)][int(0)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "f619ace2-184a-475d-b201-31e6c673e95f",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <span>FloatVector with 1 elements.</span>\n",
       "        <table>\n",
       "        <tbody>\n",
       "          <tr>\n",
       "          \n",
       "            <td>\n",
       "            0.000000\n",
       "            </td>\n",
       "          \n",
       "          </tr>\n",
       "        </tbody>\n",
       "        </table>\n",
       "        "
      ],
      "text/plain": [
       "<rpy2.robjects.vectors.FloatVector object at 0x7f6d380fc040> [RTYPES.REALSXP]\n",
       "R classes: ('numeric',)\n",
       "[0.000000]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[int(1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "5b167ddf-6f04-41bb-bbe7-557c3df0bd7d",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.4322229146869754e-14"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v[int(1)][int(0)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "d2c0beb8-a55f-4e54-ab88-098c49edfdf3",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <span>ListVector with 2 elements.</span>\n",
       "        <table>\n",
       "        <tbody>\n",
       "        \n",
       "          <tr>\n",
       "            <th>\n",
       "            x\n",
       "            </th>\n",
       "            <td>\n",
       "            <rpy2.rinterface.FloatSexpVector object at 0x7f6d380d6a80> [RTYPES.REALSXP]\n",
       "            </td>\n",
       "          </tr>\n",
       "        \n",
       "          <tr>\n",
       "            <th>\n",
       "            y\n",
       "            </th>\n",
       "            <td>\n",
       "            <rpy2.rinterface.FloatSexpVector object at 0x7f6d380d63c0> [RTYPES.REALSXP]\n",
       "            </td>\n",
       "          </tr>\n",
       "        \n",
       "        </tbody>\n",
       "        </table>\n",
       "        "
      ],
      "text/plain": [
       "<rpy2.robjects.vectors.ListVector object at 0x7f6d3d5ad380> [RTYPES.VECSXP]\n",
       "R classes: ('list',)\n",
       "[FloatSexpVector, FloatSexpVector]\n",
       "  x: <class 'rpy2.rinterface.FloatSexpVector'>\n",
       "  <rpy2.rinterface.FloatSexpVector object at 0x7f6d37add300> [RTYPES.REALSXP]\n",
       "  y: <class 'rpy2.rinterface.FloatSexpVector'>\n",
       "  <rpy2.rinterface.FloatSexpVector object at 0x7f6d380d7780> [RTYPES.REALSXP]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R('plot(sin, -pi, pi)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "1a9e01fe-c2ab-471d-81cd-3549e326ba58",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 9.5",
   "language": "sage",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}