07-Lecture-Challenge-Solutions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# AM10IM - Introduction to Matlab"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lecture Challenge 7 Solutions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Challenge 1__ : Can you write a piece of Matlab code that will __find the 10001st prime number__?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are __many__ ways to do this. One way is to use the Matlab function 'Nthprime(10001)'! and it's one line. This is my solution as I wasn't aware Nthprime(N) exists."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Elapsed time is 284.267 seconds.\n",
      "j =  104743\n"
     ]
    }
   ],
   "source": [
    "%{\n",
    "Challenge 2:\n",
    "We shall find the nth prime:\n",
    "\n",
    "The while loop:\n",
    "    Iterates j=j+1 so will check every number until i=10001\n",
    "    i = i+isprime(j) will add one to i every time j is a prime, once\n",
    "    it reaches 10001 we stop. It will add zero if j is not a prime\n",
    "    The final j will be the 10001st prime\n",
    "%}\n",
    "clear all\n",
    "n = 10001;   % The prime we want\n",
    "i = 0;     % initialising\n",
    "j = 0;     % initialising\n",
    "tic        % simply setting a timer so that we know how long it takes\n",
    "while i<n\n",
    "  j = j+1;\n",
    "  i = i + isprime(j);\n",
    "end\n",
    "toc       % stops the timer and displays to the screen\n",
    "j         % simply displays the final j, the 10001st prime"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another solution is to use if statments in the while loop."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__Challenge 2__: Can you plot the prime counting function $P(m)$ for the first 100 integers? The prime counting function $P\\left(m\\right)$ counts the number of prime numbers less than or equal to the integer $m$. You should be plotting a vector that looks like:\n",
    "\n",
    "\\begin{align}\n",
    "\\begin{pmatrix}\n",
    "1 & 0 \\\\\n",
    "2 & 1 \\\\\n",
    "3 & 2 \\\\\n",
    "4 & 2 \\\\\n",
    "5 & 3 \\\\\n",
    "\\vdots & \\vdots\n",
    "\\end{pmatrix}\n",
    "\\end{align}\n",
    "As there are zero primes less than or equal to 1; one prime less than or equal to two; two primes (3 and 2) that are less than or equal to 3; 2 primes that are less than or equal to 4; 3 primes (2, 3, 5) less than or equal to 5 etc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Elapsed time is 0.0496359 seconds.\n"
     ]
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%{\n",
    "Challenge 2:\n",
    "Shows the prime counting function P for the first n integers:\n",
    "The prime counting function evaluated at 'i' gives the number of primes\n",
    "less than or equal to 'i'.\n",
    "\n",
    "The while loop:\n",
    "    - iterates j until stated in 'n'\n",
    "    - Requires j=j+1 so that it will terminate.\n",
    "    - i = i+isprime(j) will add one to the number of primes if i is prime\n",
    "    - s = [s; j i] i is the number of primes less than or equal to j\n",
    "                   This vector s will be two columns of size n, the first\n",
    "                   column is 1,..,n the second will be the number primes\n",
    "                   less than or equal to i.\n",
    "\n",
    "n - the integer we go up to, the challenge asks for n = 100\n",
    "j - will iterate the while loop, it will run from 1,..,n\n",
    "i - will keep count of the number of primes less than or equal to j.\n",
    "s - will store j and i in a 2xn vector\n",
    "x - x axis for the plot\n",
    "y - y axis for the plot\n",
    "%}\n",
    "n = 100;   % The highest integer\n",
    "i = 0;     % initialising\n",
    "j = 0;     % initialising\n",
    "s = [];    % initialising 's' as the empty array\n",
    "tic        % simply setting a timer so that we know how long it takes\n",
    "while j<n\n",
    "  j = j+1;\n",
    "  i = i + isprime(j);\n",
    "  s = [s; j i];\n",
    "end\n",
    "toc       % stops the timer\n",
    "x = s(:,1);\n",
    "y = s(:,2);\n",
    "plot(x,y,'*')\n",
    "title('Prime Counting Function between 1 and 100')\n",
    "xlabel('x')\n",
    "ylabel('P(x)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Octave",
   "language": "octave",
   "name": "octave"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "GNU Octave",
     "url": "https://www.gnu.org/software/octave/support.html"
    },
    {
     "text": "Octave Kernel",
     "url": "https://github.com/Calysto/octave_kernel"
    },
    {
     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "octave",
   "version": "4.0.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}