{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "7cff2fc3-de18-46c0-8f17-0614c7b681fb",
   "metadata": {},
   "source": [
    "## For loop"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2ea21b76-0048-4aed-98fa-f84483264ff4",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# for loop using `in` and a vector\n",
    "a = [1, 2, 3]\n",
    "for x in a\n",
    "    println(\"iteration: $(sin(x))\")   # interpolation of the value of sin(x)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e18bfeb7-f490-49c7-8537-cb8ca20ae0c4",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# for loop using `in` and a range\n",
    "inds = 1:5\n",
    "for x in inds\n",
    "    println(\"iteration: $(sin(x))\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "caac0593-6acd-4692-946a-437c09efcadc",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# for loop using unicode sysmbol `∈`\n",
    "inds = 1:5\n",
    "for x ∈ inds\n",
    "    println(\"iteration: $(sin(x))\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "59d1ceab-faf3-4f64-a656-8c5d728ca9fc",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# for loop using `=` and interpolation in the body of the loop\n",
    "inds = 1:5\n",
    "for x = inds\n",
    "    println(\"iteration: $(sin(x))\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9c868b28-eefc-432b-8818-8bbd04afbf5d",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "# for loop using a range and interpolation \n",
    "ran = range(0.0, 2.0, 6)\n",
    "for x = ran\n",
    "    println(\"iteration: $(log(x + 1.0))\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9f0b5d62-8f1f-4070-bcfc-eb00ed8bf3f5",
   "metadata": {},
   "source": [
    "## An example of a function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3ca1d206-0850-4b24-af84-6126d6e3c388",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "function fun(x, y)\n",
    "    println(\"x = $x, y = $y\")\n",
    "    return sqrt(x^2 + y^2)   # `return` here is optional\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "71dafdd4-6976-4da6-8d06-45f09af988c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "fun(3.0, 4.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "38999779-6430-47d2-9127-10a7d81ee74c",
   "metadata": {},
   "source": [
    "## Chebyshev polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "494581dd-b64e-4934-8ec4-9405d80db307",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "T(x, n) = cos(n*acos(x))  # Generic Chebyshev Polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8f854358-d51d-44c4-b44b-32e7c977baae",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "T(0.5, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ce406713-3966-4ef2-87d0-2c593fb87977",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "T(0.0, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6676cdfb-f118-4657-888d-1a73be59648e",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "T(1.0, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f93f6f8e-daa8-47a8-acae-b49e0fa2dc94",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "t2(x) = 2*x^2 - 1.0 # Chebyshev polynomial T_2(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "021e33a6-23fc-4efe-b728-f8c1b050d293",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "t2(1.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2ccaa1ae-34f2-445b-b021-224b95bdec91",
   "metadata": {},
   "source": [
    "## Benchmarking"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bbb3a46d-1901-4826-9fa1-ad7e359982fe",
   "metadata": {},
   "outputs": [],
   "source": [
    "# ] add BenchmarkTools"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9c3d7c2f-4b9b-43e1-8ee2-aeb9feca86b3",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "using BenchmarkTools"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e6b44be5-fc87-42fc-9b45-a29fd3f55b8f",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "x = 2/3\n",
    "n = 2\n",
    "@btime T($x, $n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3a25bcb5-184b-48f5-9566-3fa2715ecf8b",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "@btime t2($x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e26e9b2-a773-427e-a6de-654c73ac4e6f",
   "metadata": {},
   "source": [
    "### First draft:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "350e13d0-9224-4d13-bf43-75ef7ebfae37",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "function mychebyshev(x, n)\n",
    "    tp = 1.0\n",
    "    tc = x\n",
    "    tn = 0.0\n",
    "    for i in 1:(n-1)\n",
    "        tn = 2*x*tc - tp\n",
    "        tp = tc\n",
    "        tc = tn\n",
    "    end\n",
    "    return tn\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fe0ab782-0e7e-465d-a857-40137b2b2de1",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "mychebyshev(0.0, 9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3f5c9cc8-7a1a-4056-a3a6-ca56a57b8398",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "using PyPlot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "311dd08b-3d91-430c-9051-70c4b649a80c",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "x = range(-1.0, 1.0, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c51c5cb1-644b-4bef-b54a-d23b31bcba66",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "y = mychebyshev.(x, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "000852d5-feb7-4325-9a52-bb6f40f92c02",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "plot(x, y)\n",
    "grid(true)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c2accc4c-cc56-4283-95e6-e9069c518a78",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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