{
 "cells": [
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   "execution_count": null,
   "id": "5baec308-f765-4d81-81c0-ada21cfdcc31",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\"\"\"\n",
    "    myfib(n)\n",
    "\n",
    "Calculate Fibonacci number F_n using the recurrence relation\n",
    "F_1 = 1, F_2 = 1, and F_n = F_n-1 + F_n-2 for n > 2.\n",
    "\"\"\"\n",
    "function myfib(n)\n",
    "\n",
    "    fp = 1\n",
    "    if n == 1\n",
    "        return fp\n",
    "    end\n",
    "\n",
    "    fc = 1\n",
    "    if n == 2\n",
    "        return fc\n",
    "    end\n",
    "\n",
    "    for i = 3:n\n",
    "        fn = fc + fp\n",
    "        fp = fc\n",
    "        fc = fn\n",
    "    end\n",
    "\n",
    "    return fc\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "49884960-941b-41ed-b01d-78cc3a65ce1f",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "?myfib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eb821ac9-b2d9-49da-8c09-c38643e11678",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "myfib.(1:10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ddee6fd-38a0-407f-bfbe-579d1711fdfb",
   "metadata": {},
   "source": [
    "\n",
    "A **recursive version** of the same function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0304fe59-42b4-49e6-8c7d-f64de53b7f43",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "function myfib1(n)\n",
    "    \n",
    "    if n == 1\n",
    "        return 1\n",
    "    elseif n == 2\n",
    "        return 1\n",
    "    end\n",
    "\n",
    "    return myfib1(n-1) + myfib1(n-2)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6c61e693-89b3-4989-bb7e-cdede1cce12e",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "myfib1.(1:11)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f70b4a49-7b3d-4f34-a273-6db4b9e7f796",
   "metadata": {},
   "source": [
    "Elegant but very slow"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6ee1dc3f-2f89-47e1-b48d-31262a105a87",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "@time myfib(51); # faster than 10^{-6} sec."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bdd20dea-2bbc-4f8d-8368-403368e6a481",
   "metadata": {},
   "outputs": [],
   "source": [
    "@time myfib1(51); # 79 sec."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b47301f-d942-4d0f-b13d-a20de295cc7c",
   "metadata": {},
   "source": [
    "A version of the function that uses a different style of code in the main loop: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b7c7a506-6273-496e-9601-147aa6d8aa9b",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\"\"\"\n",
    "    myfib2(n)\n",
    "\n",
    "Calculate Fibonacci number F_n using the recurrence relation\n",
    "F_1 = 1, F_2 = 1, and F_n = F_n-1 + F_n-2 for n > 2.\n",
    "\"\"\"\n",
    "function myfib2(n)\n",
    "\n",
    "    fp = 1\n",
    "    if n == 1\n",
    "        return fp\n",
    "    end\n",
    "\n",
    "    fc = 1\n",
    "    if n == 2\n",
    "        return fc\n",
    "    end\n",
    "\n",
    "    for i = 3:n\n",
    "        fc, fp = fc + fp, fc\n",
    "    end\n",
    "\n",
    "    return fc\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8a498381-528f-40d2-9e46-38e920894dce",
   "metadata": {},
   "outputs": [],
   "source": [
    "myfib2.(1:11)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "707a277b-d813-4378-9667-678b398a5c12",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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