ExercisesSolutions.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suggestions for lab exercises."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Variables and assignment"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Remember that $n! = n \\times (n - 1) \\times \\dots \\times 2 \\times 1$. Compute $15!$, assigning the result to a sensible variable name."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1307674368000\n"
     ]
    }
   ],
   "source": [
    "fifteen_factorial = 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1\n",
    "print(fifteen_factorial)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Using the `math` module, check your result for $15$ factorial. You should explore the help for the `math` library and its functions, using eg tab-completion, the spyder inspector, or online sources."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1307674368000\n",
      "Result correct? True\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "print(math.factorial(15))\n",
    "print(\"Result correct?\", math.factorial(15) == fifteen_factorial)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "[Stirling's approximation](http://mathworld.wolfram.com/StirlingsApproximation.html) gives that, for large enough $n$, \n",
    "\n",
    "\\begin{equation}\n",
    "  n! \\simeq \\sqrt{2 \\pi} n^{n + 1/2} e^{-n}.\n",
    "\\end{equation}\n",
    "\n",
    "Using functions and constants from the `math` library, compare the results of $n!$ and Stirling's approximation for $n = 5, 10, 15, 20$. In what sense does the approximation improve?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "120 118.01916795759007\n",
      "3628800 3598695.6187410355\n",
      "1307674368000 1300430722199.4658\n",
      "2432902008176640000 2.422786846761133e+18\n",
      "Absolute differences:\n",
      "1.9808320424099293\n",
      "30104.38125896454\n",
      "7243645800.53418\n",
      "1.0115161415506944e+16\n",
      "Relative differences:\n",
      "0.016506933686749412\n",
      "0.00829596044393864\n",
      "0.00553933454519939\n",
      "0.004157652622880542\n"
     ]
    }
   ],
   "source": [
    "print(math.factorial(5), math.sqrt(2*math.pi)*5**(5+0.5)*math.exp(-5))\n",
    "print(math.factorial(10), math.sqrt(2*math.pi)*10**(10+0.5)*math.exp(-10))\n",
    "print(math.factorial(15), math.sqrt(2*math.pi)*15**(15+0.5)*math.exp(-15))\n",
    "print(math.factorial(20), math.sqrt(2*math.pi)*20**(20+0.5)*math.exp(-20))\n",
    "print(\"Absolute differences:\")\n",
    "print(math.factorial(5) - math.sqrt(2*math.pi)*5**(5+0.5)*math.exp(-5))\n",
    "print(math.factorial(10) - math.sqrt(2*math.pi)*10**(10+0.5)*math.exp(-10))\n",
    "print(math.factorial(15) - math.sqrt(2*math.pi)*15**(15+0.5)*math.exp(-15))\n",
    "print(math.factorial(20) - math.sqrt(2*math.pi)*20**(20+0.5)*math.exp(-20))\n",
    "print(\"Relative differences:\")\n",
    "print((math.factorial(5) - math.sqrt(2*math.pi)*5**(5+0.5)*math.exp(-5)) / math.factorial(5))\n",
    "print((math.factorial(10) - math.sqrt(2*math.pi)*10**(10+0.5)*math.exp(-10)) / math.factorial(10))\n",
    "print((math.factorial(15) - math.sqrt(2*math.pi)*15**(15+0.5)*math.exp(-15)) / math.factorial(15))\n",
    "print((math.factorial(20) - math.sqrt(2*math.pi)*20**(20+0.5)*math.exp(-20)) / math.factorial(20))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the relative error decreases, whilst the absolute error grows (significantly)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Basic functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Write a function to calculate the volume of a cuboid with edge lengths $a, b, c$. Test your code on sample values such as\n",
    "\n",
    "1. $a=1, b=1, c=1$ (result should be $1$);\n",
    "2. $a=1, b=2, c=3.5$ (result should be $7.0$);\n",
    "3. $a=0, b=1, c=1$ (result should be $0$);\n",
    "4. $a=2, b=-1, c=1$ (what do you think the result should be?)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def cuboid_volume(a, b, c):\n",
    "    \"\"\"\n",
    "    Compute the volume of a cuboid with edge lengths a, b, c.\n",
    "    Volume is abc. Only makes sense if all are non-negative.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    a : float\n",
    "        Edge length 1\n",
    "    b : float\n",
    "        Edge length 2\n",
    "    c : float\n",
    "        Edge length 3\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    volume : float\n",
    "        The volume a*b*c\n",
    "    \"\"\"\n",
    "    \n",
    "    if (a < 0.0) or (b < 0.0) or (c < 0.0):\n",
    "        print(\"Negative edge length makes no sense!\")\n",
    "        return 0\n",
    "    \n",
    "    return a*b*c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "7.0\n",
      "0\n",
      "Negative edge length makes no sense!\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "print(cuboid_volume(1,1,1))\n",
    "print(cuboid_volume(1,2,3.5))\n",
    "print(cuboid_volume(0,1,1))\n",
    "print(cuboid_volume(2,-1,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In later cases, after having covered exceptions, I would suggest raising a `NotImplementedError` for negative edge lengths."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Write a function to compute the time (in seconds) taken for an object to fall from a height $H$ (in metres) to the ground, using the formula\n",
    "\\begin{equation}\n",
    "  h(t) = \\frac{1}{2} g t^2.\n",
    "\\end{equation}\n",
    "Use the value of the acceleration due to gravity $g$ from `scipy.constants.g`. Test your code on sample values such as\n",
    "\n",
    "1. $H = 1$m (result should be $\\approx 0.452$s);\n",
    "2. $H = 10$m (result should be $\\approx 1.428$s);\n",
    "3. $H = 0$m (result should be $0$s);\n",
    "4. $H = -1$m (what do you think the result should be?)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def fall_time(H):\n",
    "    \"\"\"\n",
    "    Give the time in seconds for an object to fall to the ground\n",
    "    from H metres.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    H : float\n",
    "        Starting height (metres)\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    T : float\n",
    "        Fall time (seconds)\n",
    "    \"\"\"\n",
    "    \n",
    "    from math import sqrt\n",
    "    from scipy.constants import g\n",
    "    \n",
    "    if (H < 0):\n",
    "        print(\"Negative height makes no sense!\")\n",
    "        return 0\n",
    "    \n",
    "    return sqrt(2.0*H/g)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.45160075575178754\n",
      "1.4280869812290344\n",
      "0.0\n",
      "Negative height makes no sense!\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "print(fall_time(1))\n",
    "print(fall_time(10))\n",
    "print(fall_time(0))\n",
    "print(fall_time(-1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Write a function that computes the area of a triangle with edge lengths $a, b, c$. You may use the formula\n",
    "\\begin{equation}\n",
    "  A = \\sqrt{s (s - a) (s - b) (s - c)}, \\qquad s = \\frac{a + b + c}{2}.\n",
    "\\end{equation}\n",
    "\n",
    "Construct your own test cases to cover a range of possibilities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def triangle_area(a, b, c):\n",
    "    \"\"\"\n",
    "    Compute the area of a triangle with edge lengths a, b, c.\n",
    "    Area is sqrt(s (s-a) (s-b) (s-c)). \n",
    "    s is (a+b+c)/2.\n",
    "    Only makes sense if all are non-negative.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    a : float\n",
    "        Edge length 1\n",
    "    b : float\n",
    "        Edge length 2\n",
    "    c : float\n",
    "        Edge length 3\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    area : float\n",
    "        The triangle area.\n",
    "    \"\"\"\n",
    "    \n",
    "    from math import sqrt\n",
    "    \n",
    "    if (a < 0.0) or (b < 0.0) or (c < 0.0):\n",
    "        print(\"Negative edge length makes no sense!\")\n",
    "        return 0\n",
    "    \n",
    "    s = 0.5 * (a + b + c)\n",
    "    return sqrt(s * (s-a) * (s-b) * (s-c))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.4330127018922193\n",
      "6.0\n",
      "0.0\n",
      "Negative edge length makes no sense!\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "print(triangle_area(1,1,1)) # Equilateral; answer sqrt(3)/4 ~ 0.433\n",
    "print(triangle_area(3,4,5)) # Right triangle; answer 6\n",
    "print(triangle_area(1,1,0)) # Not a triangle; answer 0\n",
    "print(triangle_area(-1,1,1)) # Not a triangle; exception or 0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Floating point numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Computers cannot, in principle, represent real numbers perfectly. This can lead to problems of accuracy. For example, if\n",
    "\n",
    "\\begin{equation}\n",
    "  x = 1, \\qquad y = 1 + 10^{-14} \\sqrt{3}\n",
    "\\end{equation}\n",
    "\n",
    "then it *should* be true that\n",
    "\n",
    "\\begin{equation}\n",
    "  10^{14} (y - x) = \\sqrt{3}.\n",
    "\\end{equation}\n",
    "\n",
    "Check how accurately this equation holds in Python and see what this implies about the accuracy of subtracting two numbers that are close together."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The calculation gives 1.7319479184152442\n",
      "The result should be 1.7320508075688772\n"
     ]
    }
   ],
   "source": [
    "from math import sqrt\n",
    "\n",
    "x = 1.0\n",
    "y = 1.0 + 1e-14 * sqrt(3.0)\n",
    "print(\"The calculation gives {}\".format(1e14*(y-x)))\n",
    "print(\"The result should be {}\".format(sqrt(3.0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the first three digits are correct. This isn't too surprising: we expect 16 digits of accuracy for a floating point number, but $x$ and $y$ are identical for the first 14 digits."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "The standard quadratic formula gives the solutions to\n",
    "\n",
    "\\begin{equation}\n",
    "  a x^2 + b x + c = 0\n",
    "\\end{equation}\n",
    "\n",
    "as\n",
    "\n",
    "\\begin{equation}\n",
    "  x = \\frac{-b \\pm \\sqrt{b^2 - 4 a c}}{2 a}.\n",
    "\\end{equation}\n",
    "\n",
    "Show that, if $a = 10^{-n} = c$ and $b = 10^n$ then\n",
    "\n",
    "\\begin{equation}\n",
    "  x = \\frac{10^{2 n}}{2} \\left( -1 \\pm \\sqrt{1 - 4 \\times 10^{-4n}} \\right).\n",
    "\\end{equation}\n",
    "\n",
    "Using the expansion (from Taylor's theorem)\n",
    "\n",
    "\\begin{equation}\n",
    "  \\sqrt{1 - 10^{-4 n}} \\simeq 1 - 2 \\times 10^{-4 n} + \\dots, \\qquad n \\gg 1,\n",
    "\\end{equation}\n",
    "\n",
    "show that\n",
    "\n",
    "\\begin{equation}\n",
    "  x \\simeq -10^{2 n} + 10^{-2 n} \\quad \\text{and} \\quad -10^{-2n}, \\qquad n \\gg 1.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is pen-and-paper work; each step should be re-arranging."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "By multiplying and dividing by $-b \\mp \\sqrt{b^2 - 4 a c}$, check that we can also write the solutions to the quadratic equation as\n",
    "\n",
    "\\begin{equation}\n",
    "  x = \\frac{2 c}{-b \\mp \\sqrt{b^2 - 4 a c}}.\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution\n",
    "\n",
    "Using the difference of two squares we get\n",
    "\n",
    "\\begin{equation}\n",
    "  x = \\frac{b^2 - \\left( b^2 - 4 a c \\right)}{2a \\left( -b \\mp \\sqrt{b^2 - 4 a c} \\right)}\n",
    "\\end{equation}\n",
    "\n",
    "which re-arranges to give the required solution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4\n",
    "\n",
    "Using Python, calculate both solutions to the quadratic equation\n",
    "\n",
    "\\begin{equation}\n",
    "  10^{-n} x^2 + 10^n x + 10^{-n} = 0\n",
    "\\end{equation}\n",
    "\n",
    "for $n = 3$ and $n = 4$ using both formulas. What do you see? How has floating point accuracy caused problems here?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For n=3, first formula, solutions are -9.999894245993346e-07 and -999999.999999.\n",
      "For n=3, second formula, solutions are -1000010.5755125057 and -1.000000000001e-06.\n",
      "For n=4, first formula, solutions are -9.094947017729282e-09 and -100000000.0.\n",
      "For n=4, second formula, solutions are -109951162.7776 and -1e-08.\n"
     ]
    }
   ],
   "source": [
    "a = 1e-3\n",
    "b = 1e3\n",
    "c = a\n",
    "formula1_n3_plus = (-b + sqrt(b**2 - 4.0*a*c))/(2.0*a)\n",
    "formula1_n3_minus = (-b - sqrt(b**2 - 4.0*a*c))/(2.0*a)\n",
    "formula2_n3_plus = (2.0*c)/(-b + sqrt(b**2 - 4.0*a*c))\n",
    "formula2_n3_minus = (2.0*c)/(-b - sqrt(b**2 - 4.0*a*c))\n",
    "print(\"For n=3, first formula, solutions are {} and {}.\".format(formula1_n3_plus, \n",
    "                                                                formula1_n3_minus))\n",
    "print(\"For n=3, second formula, solutions are {} and {}.\".format(formula2_n3_plus, \n",
    "                                                                 formula2_n3_minus))\n",
    "\n",
    "a = 1e-4\n",
    "b = 1e4\n",
    "c = a\n",
    "formula1_n4_plus = (-b + sqrt(b**2 - 4.0*a*c))/(2.0*a)\n",
    "formula1_n4_minus = (-b - sqrt(b**2 - 4.0*a*c))/(2.0*a)\n",
    "formula2_n4_plus = (2.0*c)/(-b + sqrt(b**2 - 4.0*a*c))\n",
    "formula2_n4_minus = (2.0*c)/(-b - sqrt(b**2 - 4.0*a*c))\n",
    "print(\"For n=4, first formula, solutions are {} and {}.\".format(formula1_n4_plus, \n",
    "                                                                formula1_n4_minus))\n",
    "print(\"For n=4, second formula, solutions are {} and {}.\".format(formula2_n4_plus, \n",
    "                                                                 formula2_n4_minus))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a difference in the fifth significant figure in both solutions in the first case, which gets to the third (arguably the second) significant figure in the second case. Comparing to the limiting solutions above, we see that the *larger* root is definitely more accurately captured with the first formula than the second (as the result should be bigger than $10^{-2n}$).\n",
    "\n",
    "In the second case we have divided by a very small number to get the big number, which loses accuracy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 5\n",
    "\n",
    "The standard definition of the derivative of a function is\n",
    "\n",
    "\\begin{equation}\n",
    "  \\left. \\frac{\\text{d} f}{\\text{d} x} \\right|_{x=X} = \\lim_{\\delta \\to 0} \\frac{f(X + \\delta) - f(X)}{\\delta}.\n",
    "\\end{equation}\n",
    "\n",
    "We can *approximate* this by computing the result for a *finite* value of $\\delta$:\n",
    "\n",
    "\\begin{equation}\n",
    "  g(x, \\delta) = \\frac{f(x + \\delta) - f(x)}{\\delta}.\n",
    "\\end{equation}\n",
    "\n",
    "Write a function that takes as inputs a function of one variable, $f(x)$, a location $X$, and a step length $\\delta$, and returns the approximation to the derivative given by $g$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def g(f, X, delta):\n",
    "    \"\"\"\n",
    "    Approximate the derivative of a given function at a point.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    f : function\n",
    "        Function to be differentiated\n",
    "    X : real\n",
    "        Point at which the derivative is evaluated\n",
    "    delta : real\n",
    "        Step length\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    g : real\n",
    "        Approximation to the derivative\n",
    "    \"\"\"\n",
    "    \n",
    "    return (f(X+delta) - f(X)) / delta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 6\n",
    "\n",
    "The function $f_1(x) = e^x$ has derivative with the exact value $1$ at $x=0$. Compute the approximate derivative using your function above, for $\\delta = 10^{-2 n}$ with $n = 1, \\dots, 7$. You should see the results initially improve, then get worse. Why is this?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "For n=1, the approx derivative is 1.005016708416795.\n",
      "For n=2, the approx derivative is 1.000050001667141.\n",
      "For n=3, the approx derivative is 1.0000004999621837.\n",
      "For n=4, the approx derivative is 0.999999993922529.\n",
      "For n=5, the approx derivative is 1.000000082740371.\n",
      "For n=6, the approx derivative is 1.000088900582341.\n",
      "For n=7, the approx derivative is 0.9992007221626409.\n"
     ]
    }
   ],
   "source": [
    "from math import exp\n",
    "for n in range(1, 8):\n",
    "    print(\"For n={}, the approx derivative is {}.\".format(n, g(exp, 0.0, 10**(-2.0*n))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have a combination of floating point inaccuracies: in the numerator we have two terms that are nearly equal, leading to a very small number. We then divide two very small numbers. This is inherently inaccurate.\n",
    "\n",
    "This does not mean that you can't calculate derivatives to high accuracy, but alternative approaches are definitely recommended."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Prime numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Write a function that tests if a number is prime. Test it by writing out all prime numbers less than 50."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a \"simple\" solution, but not efficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def isprime(n):\n",
    "    \"\"\"\n",
    "    Checks to see if an integer is prime.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    n : integer\n",
    "        Number to check\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    isprime : Boolean\n",
    "        If n is prime\n",
    "    \"\"\"\n",
    "    \n",
    "    # No number less than 2 can be prime\n",
    "    if n < 2:\n",
    "        return False\n",
    "    \n",
    "    # We only need to check for divisors up to sqrt(n)\n",
    "    for m in range(2, int(n**0.5)+1):\n",
    "        if n%m == 0:\n",
    "            return False\n",
    "    \n",
    "    # If we've got this far, there are no divisors.\n",
    "    return True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function says that 2 is prime.\n",
      "Function says that 3 is prime.\n",
      "Function says that 5 is prime.\n",
      "Function says that 7 is prime.\n",
      "Function says that 11 is prime.\n",
      "Function says that 13 is prime.\n",
      "Function says that 17 is prime.\n",
      "Function says that 19 is prime.\n",
      "Function says that 23 is prime.\n",
      "Function says that 29 is prime.\n",
      "Function says that 31 is prime.\n",
      "Function says that 37 is prime.\n",
      "Function says that 41 is prime.\n",
      "Function says that 43 is prime.\n",
      "Function says that 47 is prime.\n"
     ]
    }
   ],
   "source": [
    "for n in range(50):\n",
    "    if isprime(n):\n",
    "        print(\"Function says that {} is prime.\".format(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "500 years ago some believed that the number $2^n - 1$ was prime for *all* primes $n$. Use your function to find the first prime $n$ for which this is not true."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could do this many ways. This \"elegant\" solution says:\n",
    "\n",
    "* Start from the smallest possible $n$ (2).\n",
    "* Check if $n$ is prime. If not, add one to $n$.\n",
    "* If $n$ is prime, check if $2^n-1$ is prime. If it is, add one to $n$.\n",
    "* If both those logical checks fail, we have found the $n$ we want."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The first n such that 2^n-1 is not prime is 11.\n"
     ]
    }
   ],
   "source": [
    "n = 2\n",
    "while (not isprime(n)) or (isprime(2**n-1)):\n",
    "    n += 1\n",
    "print(\"The first n such that 2^n-1 is not prime is {}.\".format(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "The *Mersenne* primes are those that have the form $2^n-1$, where $n$ is prime. Use your previous solutions to generate all the $n < 40$ that give Mersenne primes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n=2 is such that 2^n-1 is prime.\n",
      "n=3 is such that 2^n-1 is prime.\n",
      "n=5 is such that 2^n-1 is prime.\n",
      "n=7 is such that 2^n-1 is prime.\n",
      "n=13 is such that 2^n-1 is prime.\n",
      "n=17 is such that 2^n-1 is prime.\n",
      "n=19 is such that 2^n-1 is prime.\n",
      "n=31 is such that 2^n-1 is prime.\n"
     ]
    }
   ],
   "source": [
    "for n in range(2, 41):\n",
    "    if isprime(n) and isprime(2**n-1):\n",
    "        print(\"n={} is such that 2^n-1 is prime.\".format(n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4\n",
    "\n",
    "Write a function to compute all prime factors of an integer $n$, including their multiplicities. Test it by printing the prime factors (without multiplicities) of $n = 17, \\dots, 20$ and the multiplicities (without factors) of $n = 48$.\n",
    "\n",
    "##### Note \n",
    "\n",
    "One effective solution is to return a *dictionary*, where the keys are the factors and the values are the multiplicities."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This solution uses the trick of immediately dividing $n$ by any divisor: this means we never have to check the divisor for being prime."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def prime_factors(n):\n",
    "    \"\"\"\n",
    "    Generate all the prime factors of n.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    n : integer\n",
    "        Number to be checked\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    factors : dict\n",
    "        Prime factors (keys) and multiplicities (values)\n",
    "    \"\"\"\n",
    "    \n",
    "    factors = {}\n",
    "    \n",
    "    m = 2\n",
    "    while m <= n:\n",
    "        if n%m == 0:\n",
    "            factors[m] = 1\n",
    "            n //= m\n",
    "            while n%m == 0:\n",
    "                factors[m] += 1\n",
    "                n //= m\n",
    "        m += 1\n",
    "        \n",
    "    return factors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Prime factors of 17 are dict_keys([17]).\n",
      "Prime factors of 18 are dict_keys([2, 3]).\n",
      "Prime factors of 19 are dict_keys([19]).\n",
      "Prime factors of 20 are dict_keys([2, 5]).\n",
      "Multiplicities of prime factors of 48 are dict_values([4, 1]).\n"
     ]
    }
   ],
   "source": [
    "for n in range(17, 21):\n",
    "    print(\"Prime factors of {} are {}.\".format(n, prime_factors(n).keys()))\n",
    "print(\"Multiplicities of prime factors of 48 are {}.\".format(prime_factors(48).values()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 5\n",
    "\n",
    "Write a function to generate all the integer divisors, including 1, but not including $n$ itself, of an integer $n$. Test it on $n = 16, \\dots, 20$.\n",
    "\n",
    "##### Note\n",
    "\n",
    "You could use the prime factorization from the previous exercise, or you could do it directly."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we will do it directly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def divisors(n):\n",
    "    \"\"\"\n",
    "    Generate all integer divisors of n.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    n : integer\n",
    "        Number to be checked\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    divs : list\n",
    "        All integer divisors, including 1.\n",
    "    \"\"\"\n",
    "    \n",
    "    divs = [1]\n",
    "    m = 2\n",
    "    while m <= n/2:\n",
    "        if n%m == 0:\n",
    "            divs.append(m)\n",
    "        m += 1\n",
    "        \n",
    "    return divs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The divisors of 16 are [1, 2, 4, 8].\n",
      "The divisors of 17 are [1].\n",
      "The divisors of 18 are [1, 2, 3, 6, 9].\n",
      "The divisors of 19 are [1].\n",
      "The divisors of 20 are [1, 2, 4, 5, 10].\n"
     ]
    }
   ],
   "source": [
    "for n in range(16, 21):\n",
    "    print(\"The divisors of {} are {}.\".format(n, divisors(n)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 6\n",
    "\n",
    "A *perfect* number $n$ is one where the divisors sum to $n$. For example, 6 has divisors 1, 2, and 3, which sum to 6. Use your previous solution to find all perfect numbers $n < 10,000$ (there are only four!)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can do this much more efficiently than the code below using packages such as `numpy`, but this is a \"bare python\" solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def isperfect(n):\n",
    "    \"\"\"\n",
    "    Check if a number is perfect.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    n : integer\n",
    "        Number to check\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    isperfect : Boolean\n",
    "        Whether it is perfect or not.\n",
    "    \"\"\"\n",
    "    \n",
    "    divs = divisors(n)\n",
    "    sum_divs = 0\n",
    "    for d in divs:\n",
    "        sum_divs += d\n",
    "    \n",
    "    return n == sum_divs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6 is perfect.\n",
      "Divisors are [1, 2, 3].\n",
      "Prime factors dict_keys([2, 3]) (multiplicities dict_values([1, 1])).\n",
      "28 is perfect.\n",
      "Divisors are [1, 2, 4, 7, 14].\n",
      "Prime factors dict_keys([2, 7]) (multiplicities dict_values([2, 1])).\n",
      "496 is perfect.\n",
      "Divisors are [1, 2, 4, 8, 16, 31, 62, 124, 248].\n",
      "Prime factors dict_keys([2, 31]) (multiplicities dict_values([4, 1])).\n",
      "8128 is perfect.\n",
      "Divisors are [1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064].\n",
      "Prime factors dict_keys([2, 127]) (multiplicities dict_values([6, 1])).\n"
     ]
    }
   ],
   "source": [
    "for n in range(2,10000):\n",
    "    if (isperfect(n)):\n",
    "        factors = prime_factors(n)\n",
    "        print(\"{} is perfect.\\n\"\n",
    "              \"Divisors are {}.\\n\"\n",
    "              \"Prime factors {} (multiplicities {}).\".format(\n",
    "            n, divisors(n), factors.keys(), factors.values()))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 7\n",
    "\n",
    "Using your previous functions, check that all perfect numbers $n < 10,000$ can be written as $2^{k-1} \\times (2^k - 1)$, where $2^k-1$ is a Mersenne prime."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In fact we did this above already:\n",
    "\n",
    "* $6 = 2^{2-1} \\times (2^2 - 1)$. 2 is the first number on our Mersenne list.\n",
    "* $28 = 2^{3-1} \\times (2^3 - 1)$. 3 is the second number on our Mersenne list.\n",
    "* $496 = 2^{5-1} \\times (2^5 - 1)$. 5 is the third number on our Mersenne list.\n",
    "* $8128 = 2^{7-1} \\times (2^7 - 1)$. 7 is the fourth number on our Mersenne list."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 8 (bonus)\n",
    "\n",
    "Investigate the `timeit` function in python or IPython. Use this to measure how long your function takes to check that, if $k$ on the Mersenne list then $n = 2^{k-1} \\times (2^k - 1)$ is a perfect number, using your functions. Stop increasing $k$ when the time takes too long!\n",
    "\n",
    "##### Note\n",
    "\n",
    "You could waste considerable time on this, and on optimizing the functions above to work efficiently. It is *not* worth it, other than to show how rapidly the computation time can grow!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "100000 loops, best of 3: 7.65 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit isperfect(2**(3-1)*(2**3-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10000 loops, best of 3: 83.9 µs per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit isperfect(2**(5-1)*(2**5-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000 loops, best of 3: 1.6 ms per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit isperfect(2**(7-1)*(2**7-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 loops, best of 3: 6.39 s per loop\n"
     ]
    }
   ],
   "source": [
    "%timeit isperfect(2**(13-1)*(2**13-1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's worth thinking about the operation counts of the various functions implemented here. The implementations are inefficient, but even in the best case you see how the number of operations (and hence computing time required) rapidly increases."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Logistic map"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Partly taken from Newman's book, p 120."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The logistic map builds a sequence of numbers $\\{ x_n \\}$ using the relation\n",
    "\n",
    "\\begin{equation}\n",
    "  x_{n+1} = r x_n \\left( 1 - x_n \\right),\n",
    "\\end{equation}\n",
    "\n",
    "where $0 \\le x_0 \\le 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Write a program that calculates the first $N$ members of the sequence, given as input $x_0$ and $r$ (and, of course, $N$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def logistic(x0, r, N = 1000):\n",
    "    sequence = [x0]\n",
    "    xn = x0\n",
    "    for n in range(N):\n",
    "        xnew = r*xn*(1.0-xn)\n",
    "        sequence.append(xnew)\n",
    "        xn = xnew\n",
    "    return sequence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Fix $x_0=0.5$. Calculate the first 2,000 members of the sequence for $r=1.5$ and $r=3.5$ Plot the last 100 members of the sequence in both cases.\n",
    "\n",
    "What does this suggest about the long-term behaviour of the sequence?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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78nPNpPQheXrRkbu4XJWfqxap+PloWHXBvGgQSLUI66qFScOJiYmlkmUofhwt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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105b6d048>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy\n",
    "from matplotlib import pyplot\n",
    "%matplotlib inline\n",
    "\n",
    "x0 = 0.5\n",
    "N = 2000\n",
    "sequence1 = logistic(x0, 1.5, N)\n",
    "sequence2 = logistic(x0, 3.5, N)\n",
    "pyplot.plot(sequence1[-100:], 'b-', label = r'$r=1.5$')\n",
    "pyplot.plot(sequence2[-100:], 'k-', label = r'$r=3.5$')\n",
    "pyplot.xlabel(r'$n$')\n",
    "pyplot.ylabel(r'$x$')\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This suggests that, for $r=1.5$, the sequence has settled down to a fixed point. In the $r=3.5$ case it seems to be moving between four points repeatedly."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Fix $x_0 = 0.5$. For each value of $r$ between $1$ and $4$, in steps of $0.01$, calculate the first 2,000 members of the sequence. Plot the last 1,000 members of the sequence on a plot where the $x$-axis is the value of $r$ and the $y$-axis is the values in the sequence. Do not plot lines - just plot markers (e.g., use the `'k.'` plotting style)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x106922278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy\n",
    "from matplotlib import pyplot\n",
    "%matplotlib inline\n",
    "\n",
    "# This is the \"best\" way of doing it, but numpy hasn't been introduced yet\n",
    "# r_values = numpy.arange(1.0, 4.0, 0.01) \n",
    "r_values = []\n",
    "for i in range(302):\n",
    "    r_values.append(1.0 + 0.01 * i)\n",
    "x0 = 0.5\n",
    "N = 2000\n",
    "for r in r_values:\n",
    "    sequence = logistic(x0, r, N)\n",
    "    pyplot.plot(r*numpy.ones_like(sequence[1000:]), sequence[1000:], 'k.')\n",
    "pyplot.xlabel(r'$r$')\n",
    "pyplot.ylabel(r'$x$')\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4\n",
    "\n",
    "For iterative maps such as the logistic map, one of three things can occur:\n",
    "\n",
    "1. The sequence settles down to a *fixed point*.\n",
    "2. The sequence rotates through a finite number of values. This is called a *limit cycle*.\n",
    "3. The sequence generates an infinite number of values. This is called *deterministic chaos*.\n",
    "\n",
    "Using just your plot, or new plots from this data, work out approximate values of $r$ for which there is a transition from fixed points to limit cycles, from limit cycles of a given number of values to more values, and the transition to chaos."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first transition is at $r \\approx 3$, the next at $r \\approx 3.45$, the next at $r \\approx 3.55$. The transition to chaos appears to happen before $r=4$, but it's not obvious exactly where."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mandelbrot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Mandelbrot set is also generated from a sequence, $\\{ z_n \\}$, using the relation\n",
    "\n",
    "\\begin{equation}\n",
    "  z_{n+1} = z_n^2 + c, \\qquad z_0 = 0.\n",
    "\\end{equation}\n",
    "\n",
    "The members of the sequence, and the constant $c$, are all complex. The point in the complex plane at $c$ is in the Mandelbrot set only if the $|z_n| < 2$ for all members of the sequence. In reality, checking the first 100 iterations is sufficient.\n",
    "\n",
    "Note: the python notation for a complex number $x + \\text{i} y$ is `x + yj`: that is, `j` is used to indicate $\\sqrt{-1}$. If you know the values of `x` and `y` then `x + yj` constructs a complex number; if they are stored in variables you can use `complex(x, y)`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Write a function that checks if the point $c$ is in the Mandelbrot set."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def in_Mandelbrot(c, n_iterations = 100):\n",
    "    z0 = 0.0 + 0j\n",
    "    in_set = True\n",
    "    n = 0\n",
    "    zn = z0\n",
    "    while in_set and (n < n_iterations):\n",
    "        n += 1\n",
    "        znew = zn**2 + c\n",
    "        in_set = abs(znew) < 2.0\n",
    "        zn = znew\n",
    "    return in_set"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Check the points $c=0$ and $c=\\pm 2 \\pm 2 \\text{i}$ and ensure they do what you expect. (What *should* you expect?)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Is 0.0 in the Mandelbrot set? True.\n",
      "Is (2+2j) in the Mandelbrot set? False.\n",
      "Is (2-2j) in the Mandelbrot set? False.\n",
      "Is (-2+2j) in the Mandelbrot set? False.\n",
      "Is (-2-2j) in the Mandelbrot set? False.\n"
     ]
    }
   ],
   "source": [
    "c_values = [0.0, 2+2j, 2-2j, -2+2j, -2-2j]\n",
    "for c in c_values:\n",
    "    print(\"Is {} in the Mandelbrot set? {}.\".format(c, in_Mandelbrot(c)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Write a function that, given $N$\n",
    "\n",
    "1. generates an $N \\times N$ grid spanning $c = x + \\text{i} y$, for $-2 \\le x \\le 2$ and $-2 \\le y \\le 2$;\n",
    "2. returns an $N\\times N$ array containing one if the associated grid point is in the Mandelbrot set, and zero otherwise."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy\n",
    "\n",
    "def grid_Mandelbrot(N):\n",
    "    x = numpy.linspace(-2.0, 2.0, N)\n",
    "    X, Y = numpy.meshgrid(x, x)\n",
    "    C = X + 1j*Y\n",
    "    grid = numpy.zeros((N, N), int)\n",
    "    for nx in range(N):\n",
    "        for ny in range(N):\n",
    "            grid[nx, ny] = int(in_Mandelbrot(C[nx, ny]))\n",
    "    return grid"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4\n",
    "\n",
    "Using the function `imshow` from `matplotlib`, plot the resulting array for a $100 \\times 100$ array to make sure you see the expected shape."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x10749ba20>"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1072fcfd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib import pyplot\n",
    "%matplotlib inline\n",
    "\n",
    "pyplot.imshow(grid_Mandelbrot(100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 5\n",
    "\n",
    "Modify your functions so that, instead of returning whether a point is inside the set or not, it returns the logarithm of the number of iterations it takes. Plot the result using `imshow` again."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from math import log\n",
    "\n",
    "def log_Mandelbrot(c, n_iterations = 100):\n",
    "    z0 = 0.0 + 0j\n",
    "    in_set = True\n",
    "    n = 0\n",
    "    zn = z0\n",
    "    while in_set and (n < n_iterations):\n",
    "        n += 1\n",
    "        znew = zn**2 + c\n",
    "        in_set = abs(znew) < 2.0\n",
    "        zn = znew\n",
    "    return log(n)\n",
    "\n",
    "def log_grid_Mandelbrot(N):\n",
    "    x = numpy.linspace(-2.0, 2.0, N)\n",
    "    X, Y = numpy.meshgrid(x, x)\n",
    "    C = X + 1j*Y\n",
    "    grid = numpy.zeros((N, N), int)\n",
    "    for nx in range(N):\n",
    "        for ny in range(N):\n",
    "            grid[nx, ny] = log_Mandelbrot(C[nx, ny])\n",
    "    return grid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x1076dd3c8>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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2yPVyjbhTTcCOnPHDmEqOUc+YyeUG2N4a6rJhGu0+1Wj6jK729LWalWzWy9bm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\naPoMLXqNps/Qotdo+gwteo2mz/j/ATDB0rbGcD/MAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10746ae48>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib import pyplot\n",
    "%matplotlib inline\n",
    "\n",
    "pyplot.imshow(log_grid_Mandelbrot(100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 6\n",
    "\n",
    "Try some higher resolution plots, and try plotting only a section to see the structure. **Note** this is not a good way to get high accuracy close up images!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a simple example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x107793cc0>"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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OH29Yg85kYhd/pPgtE531cHFVBhc2RBM3u4Bl8efQms1obW0NRGeHWOno4quP\noLF+4LLjpbIUPv1d32Mbd4j5srNr3+NaHd19rvbywe9JPUEJKVyJT+I2Qr3nCWexNPDZO9AQtwo5\nWDzpzc+qWR54nIbLTdwL8CP9Zdi9dSON92mG6+Pd8f/bRLTf3OWx50T/tDox0pudCCOMKu9G1Wki\n5B++QvukEbzE7kHXzU6sWdWOU2UDlaUiwCzzJJDqj7OfBr3RQFhlGa2OTuMKJmqo96atTQSflJWF\n8c3Xm8dcV+6NueTemAuAh2c9jz72CQABARVotP2FTAYiuqSQ2bZIxlpPbww63bhClqfMELQ5OWPU\n6rA2GzGbxXDa2w+ee1Wc3/EifP57CE13ZNGLIkqqaARRQJXefhzKWMULez8CQA5yYdPqTgJ9zRgd\n9WQPItvU5uRM5z8ko2trQd3ejtWWSsqo1fLO9mdge9/yeZEx5EXG4NNYz4t7PsStrQXpviGo0SAi\nJ794b+Tfi705+JmYsnQ5P++nxdmFL1ds7PdEaXDz4Dc7bC74HX2vyYuM4Va0Hunjs1iPPs07g7Qd\nUFfN2vMnaHJxw62thUNfyKTvcMYvVkPnMAk3BkKrl9j+t264NtaD1cqJtAyuxCeRdPUaCccPkHle\nzWN/5cxzP27hl09vp8bTm8eO7mN15iky58wbMj3YYLS0uGG1qjh2dD03bTevPWls8OKd330fgG89\n/1tCQu+h0Qy/m+5EWgYn0jIAiCgvZv3ZY7y77ekx92NKlg//Tv5bQMQStPwii5YmEeTThVWlQmVb\nB/5i9RauxCeNuP6Y4rs8/+XHWG1u6teefolNpw9z8+eFXNmwFOvfD53oYSw4GA385Tuvobetj3VN\nS6+cFSOA6cCKTeIl0ROeDfB/dr1Cjaf3hLbd+/t5d9vT3A6NZFXmaVzbWrvFP0fDqx+9hW9DHbtX\nbeJqXBKqt64x+78OsPb/9eWtHc/xk9+/zm92PE+NpzePHttPo6v7gNmAhsJqFV/Sr9/4U2prx7fF\ndzQ88+w7zIoqQJLGuHnIau3+3+/i6pX57PvyMfHmp9L0WT7szeLVfX05Bp2O13d+h+e/+gTvMY6h\naz29+e2jzwIia+yn67ezTv4avRecsUenh+HrzyHn8vSKPTh3FC6fEUujO16c6t7AmZQF/UZQI+V3\nj+5CZZUx6PSAmPpseUFo6xu0Ov7Prle6BTsPLlmDdSQhp72oq/XlvXfFmnx7++TGon/x+VNs2ryH\nOYlZo753nxsjAAAgAElEQVRWbzTyw09+x++37qRuKI/1AEzJiCDnQByezY1cjZtLcWAI8XfzWXnp\nDPXunny08TEqvf3wbahD9/19pIS3UfliBpfm9JM/GBC90YB7awvVXn3Xiz1amrCo1LQ423f5yKup\ngacPfkFAXTWSLLP/E9vmmApx40039A4iNkOnh2d/CA0BPuxevYUS/4lT8VC1GAhc+BryO1uoS4ns\nvoHtRfrNa6TmZrFn1eZ+f/fRcCsvgdMnV2Eya6mt8Rv+ggnC3aORpcuOkTovc/jCvVDJMv511dR4\nevcRRp22IwK/+hp8G+qQZJk2RycuJyRT5hfYx+FR7eXD6iV64lrKaRuFkotBp6faq/8/2v2OLHsQ\nWFPJpjNHuzXl938iVgNcPXoCmaYbhk5oaxOK0WoN+NXXsun0YY4sXD5hsfhWBy3l//Yocnyo2JE0\nAXg3NbLu/HGMGi27V28RCURGSObFxRTejaapyYOqypFlBppImho9OH9uKRaLmvT550d8nVWSxuww\nnBJDcHz+UhxszqJ6d08aXd0HvFFDI6HIKXFA/ffpQEdhOzd/cY+btvc5l0WQUNv4FKgmHKNBCLrk\n5wi/QcWJMmYVn0X9uGVivmutCnnLxP4NnTvaCK6u4MiC5VgHCKoZjMyLi7mcuWhS/QAjoa7Wl0uZ\niwCGNQZ+9bUszL6MSaPh8KIVA8qkD8eUGIKs+6LEBiM7JoFKH7/uZA7TCelaFc1v5HLjqhDZvHhy\n5kindbRB5inx+8IVoNVDSOU9zNVh09boDkWFjz8n0zJodXLmckLysOUtZg2ZmYuxmNVcv5ZGff3E\nOkvHSm2NHwX5ccMaAotKRafeAbNazf05h4qLI7hdMLCKc2+m3Fk4FKNZLZgswipLMWp0mO9VEJ6b\nhesSWLrWFlk4QwxBb/KyeuL0A2qrCDuVTYnBFXltxFR3bUQE1Fahslr5ZtGKYcu2tLhReDcak0nL\n8aPrMJun9b8/AC3NbuTfimd2bO6gZeo8vAb8/KUl4Vw4t5RbecM/eKf/NzGNCKypYmXmGdqcnKnU\nODLblkCmsGDwLbfTna7txNufhQgpn7CT+dRbA/FMXDdsKu3pQEBtNU6dHUP2taHBi8YGL8rLgzl2\nZMMk9m78VFcHcOzo+iENwf2UloZhMuo4f24pd24Pn9gVFEMwKtafO0bAjUJUarBeh99/MPw1MwF3\nTzEiKLghRGSDQypYd+wArz0z/bfrXosbPMinudkdk1FH5sXF3fPtmYjFrKG+zgdPr7oh4wtkWaKh\n3ps9n++koWF0y4eKIRglJw6IG8fFXchlm6bp6sBo2PQkzJotVhF0DpCbo8I02VrgdsRo1AESe3c/\nQVHhzPN53E99vTe//c2f8IM/+QXOLq0DGgNZlmhtceWtX79q+/yjQzEEo2T942LnnCSBXi808mY6\nn70jNg8tXS8CvBweCePt7TuGvW46YjDo+dXrf47RoMNkmpilyqnAYNDzxv/8hO+8/D94e9f2O19X\n68vbv/3BmIwAKIZg1PR+UI5SKHbaYjLCsq0i9uFX/wIBceV83/wuBq2Ot7fvYueh3ZxMWzxp0uBj\noaQ4gn1fPdr9ZBwgZmbGYzDo+eiDF9n+6KeEhIoNe7k3EzlxfC0WswaDYeyBWoohGCGPHttP0AOc\noDHnspjmNNRCW4uR9spa1I4qdsqfkv1aObU/S4fpt4pLdlYq2VkptLc7T2k04GTRUO/F1we24ugk\nBDRamt3t8rmnxBB8uUcMO1NSLxMaVjQVXRiSiPLibnXlLhLu3OLKAUOfLcUPUoal6l6fy8VdxO9b\nrVZu/dc9wkMg6u4FLs/ScC9wfPve7cm1q2lkXsyYFtGAk0lFhf1Xc6bEEFy/JpRU2tud8b8TQ2Bg\nGbFxN4e5avIw6PQ0VMm4fJHdnWfxrFnMoyvLekRPH1TcPESiGLNZRErOSYP8bwqJblOjfsTK3eDw\nKe3fxQsZtLc7U5Af99AZgYliSqcGBflxFOTHERp2j7ZWFxwcO0iYkz38hRNMhY8/7bMSmN+eDbKI\nxDObhAZgY/2DbQh8A3r0/jQaobGYc0WIq+jbjWinaJmko8OxW8zj7OkVtLa6DnOFwmiYFj6CkuJw\nSorDcXdvRK83EDnrNirV1EboOLsIiS6VCqLiITxK7NgDsbtQo4Wqsint4oQQNgsS5wmZs7BYFXdC\nI/n6gkTaE3B38/whdSMnivY2Z27enMvB/dsmve2HhWmVJrSpyYPPPt1FWWnolId/1tfA2SNQUgSf\nvAXNDeK4u6eYPw8RxzLjqa8ReyfMag3H0zJoffsxjnzvyUk3Aq2trlRWBilGYBKYVoYARDDIu29/\nj4qKYNraXKbMIARFq1myy4m3/+CEVi91q/osXi127504OCXdmnBMJqH798jToDMZefnz9wiqrsS5\nox2NefI2UxgMes6fW8pbb76qGIFJYEqESfjp8G2q1BYkYMsjX5CUfGXiO3YfkiyjslrRdhj4yXtv\n4GgxggRffghXz83cvQXDIUliOuQb2JNj0mLLYbZn5SauTtJQaN+Xj3HtahqyPO2eVTOb6SRMMhKs\nFpHH7fChTXR2OrBg4dlJbV+25eSzOjvy+rde4tt7PsSrqYE1jwj/wIXjk9qdSUOWhdhpTQX89z+I\nY8+9asXLl25psbkFN1l3/gQgpMF+s+N5jHYMSd67+0nychMUIzCJTFtD0EV7uzMXzi3FYtaweMnJ\nSW3bs7mRHYe/BMC1rZWvPxfS6k0TJEc+nbBYhMDKUy+LRKJL14s0YwAq2UqboxMX5qax5eQhJOw3\nqtz9+U5uF8RiNNpXzkxhaKa9IQDhRJxs8YiA2mrWnztGeIXIgvD152JnXlfG34cBlUrkLbRabBut\nmurJuH4Rs1qDW2sL4RUlfLF6C6YxKOLcj8mk5eD+bRTkx48rVFZhbMyYsVdZ2fD528bKsqvn8blP\nMdm1vZWY4ruYTXDkS7h24eEwAiERIvNxRq8ERDFzerIl+dfVEFxdgUGvpzgghJtRsf3ks0dLa6sr\nRw5vJCc7RTECU8SMGBEAVFcFcP7cUjQaM3OTro67PqfODlJuibRTGdcu4tnUyPmkdKq9fPCvqyH+\nbk8+MlkGO45+pze2HIpz00GtAlWv/5A7eUIFOSQCGlzd7eI4bGr05NKlhVy6uHjcdSmMnRljCEBs\ntTx7ZrldDIFreyubTh8GoCBsFvGF+d3JOxdmX2b+DdGGRityMVpMkH35wR8VtLVAbaXII7jGtmpX\nEDaLwNoqGmracHIBIsC1vY3E2z2qOQVhURhGqVDc2ODF1avpnDuz3H4fQGFMzChDAGIuWV4WSmBQ\nGZI09jU8o0ZLsU3eau/KjWw/dgCvpgbCK0pEyu372LBDDAq60jW2NUN9/23hMx6/QIiz6X9aJRXl\nfgHsXbmRjOuZBLuV497aAq3NBNRW8fiRfVR5+xJcXcEvn3mZGt3o/DiFhVGcObXS/h9CYdTMOEPQ\n2ODF++99mz/9yc/R6QxjrqdPbj9E7sPFWZfQDZGeaKNNq6OzAy6dhsN7xtz8tMTBUYRNn/4Gtu2C\nuk4tb7/0DEadjv1LhdMg4/pFFl8X6UtrPbz4aOPjvPLZuyLWYBR0djrQ0eFk98+gMDambUDRUOj1\nnfzoz/4NB4cOO/VK8Oix/czLzeqXiqt3LkYQcmXH9/dN1fYgsGITrNwsgoqqK+BXr+sx3XsVXO2r\nwCLLEiePr+XUyVV2rVdhBAwSUDRjVg16YzTqeeO1P7f7kuLBJWs4m9x3ZcKg0/Hfu17pk0tu8WpY\n96hdm54WnDsqYgYAfPzhT/6uZ6OVPTl8aDMXzo8uKanCxDLjpgYgnihtbS7dGWvtwdaThwitLMO1\nvZW863A3Hxa9JHIxNrh58MGmHTx6dB933ynnVpZIG/agkTRfJDwBEUPg7CpGB/am0+AwZm09hYlh\nRhqCLg7u38a69fvxD6gYvvAQbD15iMTbuTh3CPmngFBojgpg96rV3bnkFuRcwau5kSv14O0PIU5i\nh96DRFFBz6qIuydseMrEzq93Y1WpOLpg2Zjz6ilMf2a0ISgqjKKtbfzZjYsDQ/pl0W1ycaOwlxJP\ncWAIs0qLSJrfjt6hR7j0QTIGtVXiBWL1wGqwUvifd1iyFtbIMifSMsad9CTz4mLKSsLs0FsFezKj\nDYG9GEkuxqyYBFJzs4ieXQdAZSk4u0FaBlye3P1Qk4reASQVGLU6LGr1uOsryI+bdglHFYZxFkqS\n9DtJkqokScrudcxLkqRvJEnKlyTpkCRJHr3O/a0kSQWSJOVJkrR+Ijs+FdwJjaTO3RMQS4haLSx8\nQB3fnR2Qex2WbxBqTU0urvjX1ZCal01o5QMozfSQM5y37W1g433H/gb4Rpbl2cAR23skSUoAngIS\nbNe8LknShK9KVFYG0dLiNmH1q6xWIsvuEVVaRIWvP7c6PLl7S+gRBIRAaeGENT2lNDfCvo+FViPA\n0qsXePzIVzx+5KsBA64UZjZDTg1kWT4lSVLEfYe3AV2pV98FjiOMwXbgQ1mWTUCRJEm3gQXA0Dmd\nx8nRwxtwdmolOfXyhNSvsZjZeOYo/vU1qC0W9rwPh85NSFPTCrVGKBX1NuUtTi4Y9HpanZzHVGdj\ngxcmZbVgWjIWH4G/LMs2lxJVQJcrOYi+N30pMP3T6Q6DUavjjZ3f5scf/Brfhjo0GnGTzMQU6AOh\n1YmlQrNJaBCo1WJ/hU8AvPyXQpbN2AkaHRyfv4TL8clj9hV89MELin9gmjIuZ6Esy7I0VHrWB3DP\n3vrHhQPt9DdT3RP78OKfgo8f7PsEsjLFrsNNT4iRgKET/uefhDF46mXYqDuC3mjg1Dxlp+CDxlgM\nQZUkSQGyLFdKkhQIVNuOlwG90+CE2I7159hPe36PWAmRK8fQjYljzp08oksK2bNyU79zWq3Q+5/p\n6PTw4o/BP1iMcNY9Css2wN08OLQbHnlGhFA//yPx8+QB6Owws0R7Ea3ZzNEFy/rV+eixA+SHz+Lm\nrNgp+EQKA1J4HIqOD1tsLP/Se4EXgJ/bfu7udfwDSZJ+gZgSxAAXB6xh1U/H0OzgnD+3FJXawtyk\na3apr8Q/GKfODl7Y+1H3MXdbpM25o3A9c/Br124TgTm9duhOK4LCIG2JCCX28RdGAMDVXbz0DhBm\nyyQuSaIMCKESFzdw7mjHta21X72PHttP/N18igOCSbmVg1NnR79wbYUpIHJl3wftiX8csNiQhkCS\npA8RjkEfSZJKgP8F/G/gE0mSvgsUATsBZFm+KUnSJ8BNwAz8UJ6kHU01Nf40NHgNX3CENLu40ujq\nTnRJ3yWBs0fg8mmRKHQwAsOgrsZuXbE7Ti4wJ1X4BjS99EavnoPiOyKpS+AWX75IXtB9bu2FEwSG\n9r/5exNWUcrVuCTuBYWQfOsGcwtugixzNmXBkNcpTA+GWzV4ZpBTawc6KMvyz4CfjbdT04E6dy8O\nL1yOxmJh2dXzqC0WXN0heSEU5ov5c1ySEPK4cKLvtfHJ0Fgnyk036qrFiKZrT8G5pHTaHJ3QVOXj\n2VSJkys0O7tyJT6p+xq1xYJTZzsJd/P7ZIR26WhjUZZYrcmaPYfrs+fQ4ObB3ZBwLCoVrc7jj/pU\nmBwegNnuxFDv7sGJ9CU4GA0svp5JbqaF8GjhTAsIEQE34dGQew3Sl/Zc5+Yh8geW35s+hiAiRjj8\nyouFEGlDrRgBJC2AzMR51Hh6k+LqTtjqUpqgX7h1ZmIqAF7NjQTVVOLbUMv8G1dxa2th6dXzXJud\nSJW3L1aVWE0oCgqjKEgJI55JKIZgCPRGIzH37qKSrZzYDxufEDf67ERxvqJEpAfb+nT/a739IdYm\n6WfoFH6DqSAkQkQHtrVCzmXR/8AQKLgJiek95a7FJnItNnHQekKryrv9JOEVpfg01lPt5YNJreHA\nsrWsP3sMp852mlyU5KQzkQfGELS1utLc5IGbe+OQ5Zw6O3AwdFJvCxUeDL3RQHRJITsPCV+ojz/o\nbXvzW5xcaHVyRu9uZHNow4DXz00XLxAiH6//8+g+j71IXgiBoeDoLPpz+yYc+gJ+8HcSVV6+mEcY\nE7D6wkmiSoq6P3txYAhnk+fz+NF9WCWJL1dsmOBPojCRPDCG4PKlhVitKrY88sWQ5RLu3iLhzi1+\n/8hTQ5YLrq5k+/GeBIdPvSx+GnQ6zqQu4EzKQmKK7/KtPR/T2S5uNEmCTr0DFpUKrdmEzmTCYhZp\n1Xvj4Ahmc0/4rr1Qa0TdvTm+XxzrMkpqjeirUavlg807MGp1qC2WYYOEOvUOmDUaMhNTOTa/Zy70\n1mPfsu+HUJgSHhhDYG8Kg8N4e/sz/PDj3/U5fjBjDVcSkrvf11XB2/8Nf/aPYm3+03XbuBMSwapL\nZ1hx6QxFBfD+r/rWveNFuHFF5EqwJxExsOv7/Y/3lhOMiIEXfgQqo5E/ff/XIMF7W3YOm+n403Xb\nePTYfvt2WGHa8FAagojyYn7y3hvd79/Z9nS/qYIsSX0Sd7yz7Rnq3T1oc3TCapPtKQoK452/+AGt\nL8EvPcWIoNXJReRMtJWRZeGg0zvAd38ijIWLG9y0hTwkpkF0Aux+b2yfZd2jYjkQxHLgYA/2s8nz\nOZ+UTkR5CY8f+QoAtdXCO9ueoTgwZNh2Hju6j4S7+VhUKlLzsikKCuXzNVvH1mmFaccDZQjy8+PR\nHzKwdv3AT6603Ov4NtTx7rZn+uQ1VI8gtfH6c8fYu3IjDW7du64xaTSYPD3AE+73TFyOT0ZjMTO/\n/TyPPQ8HPgUPL9A5iPNL10PqImEUnJzhO38ORiN8/BsR1efhBacPQf6Noft15QzcyobYRFiybuAy\nJ9MWczFxHk0ubt0ZnYxaLR9u2kFxYDCmEYRKurS3cT4pDZf2NublZlHrMfK4DaNRxx8/eZamJo/h\nCytMCQ+UIWhrdaGmxq/f8eWXz1ISEEKpXxCV3n6U+QVS5e3LJ+u3AyLTkX4AGXMHQ2f370E1lWw4\ne5Rj85dRGDz80liTqxv17p44OInVA919ATzevlBVCjWVItIvLEps+nnseYiKg4snxLnhqKsRL98B\nVMQOLllDs7MLpf5BNLn03aptlVSUBARj0owsi/HJtAwaXd3QmUxYVSpiC2+z89AeTBoNXy7fgHkI\nYyLLKkpLwjCZ7JcxWcG+PFCGYDDqPLxpd3TEtbUFB6ORMr9AOnV6sm3KRBqLGadOcdNHF9/tF1HY\nRWRZMddiG0dkCKKL7zLndh4gnHXxKeL4qdRFtDk5k3g7F1f3cvS9nHtqNSTYyhUVQEPdyD9jaREc\n+lyMMLryFuaHR1Hj2VfpudbDm4NL1mBWq0e8YgD0+cxlfoFElJfQ5OLK4qxLQr1IpeLSnNRRjRQU\npg8PnCFobPQkOyu1T1q0G1GxRJcUElZZ1i9YRiBxIyqWRld39EYDru2t3AmJ6FMi9VYOjp0dRJcU\nUuPlQ4ktPdpghNkUkc8mz4e6DqSPc5FfTuFsygJanZxpdnYhJKAcAENLEwm9ci1evwBNA69KDkpV\nuYhXmN8re1hqXjYXE1NpdHXvPtbg5s6ZcYb9Vnn5cjZ5PnmRMd35Hu7PBaEws3jgDMFg+RG9mhqo\n8vblRlRcv2sCaqupsI2tq718aXF2IbPLA2cjpvgujp0dzC24SY2n97CGoHc9DndqiPpFAbkLV2J1\nEsIc2TEJ3SOSmOK7fQzB6W96RERHg4OT2E3YxbIr57gbEt7HENiDkoDgbhHTA0t7os0jyouxqFR9\n/CgGgwO3chOwWsevd6gwcTxwhgAGzo94MXHeoOUPLF3T/XtOdH9D0UW1lw+dOn2/+fZA9K7H0VdL\n+l8Eka+TuN8t6dLehq+dkig21kHWRYiO7znmW19LhY8/bY4Tn14s4W4+d0Ii+hiC5iZ39u55YsLb\nVhgfD6QhaGzw4g/vfYdvf/cNvL3rUKksY65LkmXc2lpQW618uWIDt0MjR11Hg5sH727rH4fsYDCQ\nmpfF+nPHsVigtUkct4yxuyGR8PgLfY+tuHyOFmcXcnpbhwli/9K+e9HMZi0tLUrI8UzggTQEAEaD\nnjdf/zO+94P/xse3mqGFlAbHwdDJqx/9Fp3JiCTLSLKMbKf0P4uzMlmVeRoQuwJ/9S/i+AhWMwdE\nlsW1XeEPVknFp+se4c4YjNd4kWWJosJZfPTBC8MXVphyZmQS1NHg5NTGjp0fEB4+erlhn4Y6vrXv\nU36/7WmMGi3bThwkOzq+e24/XnQmI/NvXGXjmaNYrdBu2/L/9n+OTdNArYHQSCE/BiL8t9Q/yC75\nCEZLTnYKBw88QmeH4/CFFSaPBykJ6mhob3dm397Hyb81+qGxWrbi3NFOm6MTrU7OfJ2xyq5PV6NW\nh8GWZVSlEkt/Lm6gGuN9azH3GBOAbScOEjQWr+M4uXJ5AceOrFeMwAzigZ0a9Ka+3puTx9diMmmZ\nkzgyTf7AmkoWZ13i0/XbMdn0vOrc7b9Gfjs0kvc3P4FHSxNbTo1NEXVuOnT5Qjs64LN3YPuz4Fdf\ni95oACCp4KZQDbJh1GjZvXrLiKIKR0tri5sSRTjDeCgMAUBlZSAXzi+luDgSX98q0ucPnm4hrKKU\nuMICCsKiyA+PmtB+Nbq6ozOZSCroG0u8YIUQOMnPgYIhwoyTFoCXjwhPnpsudjoaO4UA6fINsCjr\nEvF38wmuriC4uoJqLx8uzE3DolJjVdk31XHmxcXU1vpRXjb83gWF6cVDYwgAystCKC8Lwcu7FrXa\nQuq8gVVILWo1NV7eZMdMvKcdhDDq3IK+aqexc0WocVsztDQJPYSc+3K4JKbBopVCLakrGtrRGeZl\niFiEy4kpJNUXEVt0m3tBoZxMy6DWw5OrcUnYmyuXF5B5IYP6eu/hCytMOx4qQ9BFfZ0PJ0+sQa0W\n63SxsTfRO/TsKyjzC6TML3BC+xBWWYpRo6PSp//eiNgkoSgM4B8C7t5CfdhkFBuMunD3FOHLXapg\nzc6u3duJ3efCN4uWEXykCc/mRu6ERPTREbAXFoua3JuJnDyxhlZlqXDG8lAaAoCWZjf27haBLlu3\nfYa7exM+vtW4ujZPaLuhlWXozCaWXL1Am5Mz12ITCazp69Bbuw0qfP1paW8jPrnH+7dmW48hCI3s\nkUrzsiUPqvL2HXRrsFdzI74Ndf32HowHo1FHUWEU+758XNlQNMN5aA1Bb77auwOAVasPERd/A71D\nJy4uLXZtQ5JlvJoaWHPxJKGVZRi0egx6PSFV5WhNJsxqNY29IvK+WraetNzrfRKOqjXg7Sdu/u3P\nirRkIFYK2rUDRzw2ubhR6+nN7KLbaMwmDi5ZM6LIyKHo7HSgrdWV+npvPvnouXHVpTA9UAxBL44d\nXc+xo+tJSrnCxk1fIkkyWm3/7cljQWcy8t3d7/O7R59l/dlj3IyK5VpsImqrldii26y+cJLXnnmp\nzzVpudf7vPf2hVf+H3jt/xciKF1cPgMng2Zj+XF/QYLdqzejsZhZfvkcGdcuojFb+HTDdowj3H58\nPxaLmuysVL4+8MiYrleYnjzwAUVjQa22oNGY8PGt5tvf/dXwF4wQvdGAUatDazZjUauwqNSk5mWz\n9eTXNLh69DMEjx3d1z8FuQwGg1A66jIGZjOYJTV3Z0fz4cbH+7W79eQh2hydOJc8n4iyYpZfOcev\ndzw/ps9w5vRKTp1chVmZCsxMBgkoUgzBEKg1Zvz9K3juhd/abWRwPw6GThJv57Eo61I/Q+Da1oqj\noZM5d/KIKinkdOoint3/xz5lDi9cTq4t16BBq6PJtf+w37WtFatKRZujEzqTEZf2durdR7/Of/zo\nOq5cXkB7+9jSoitMAwYxBMrUYAgsZg2VFcF8/OFzbNv+2bBS6WMhrLKM+TeuDniuxdmFFmcXOnXJ\n3IqIHjBUuMXZdRCNhb71dGHU6qh31426n8eObCDreqpiBB5QFEMwDFariqLCKL4+uAUHh05SUi8T\nGlZkt/rFzkYLRxYuH7RMs4srzS6u+NdPXVLFioogWlrG52RUmL4ohmCE3MqbA4i9CwutKiIi7tql\n3jK/QE6mZZA7a/awZVsdnTmevqQ7F2Nvgqsr8Kuv5WrcXLv0qzcXlUChBx7FEIySgnwhOCJJ8ph2\nNN5PhY8/FT4DKI/eh3trM7Pv3aHDwREZiZwrEDYLIsvuobZaCC8vwamzw66GoGuF4OzpFbS2KsFC\nDzKKIRgDBflxuLi02MUQDIdPYz3ejfUE1VSy+uKp7uNduRhTbuWQciuHRlc3u4cOm81aDh3cgtGo\nt2u9CtMPxRCMkY4OJxoavPD0rJ/QdiJL73U7E2s8vfG1SRt35WJscnWjXe9IUXAYubNm2z16UOHh\nQDEEYyQvdw6dnY7sePIDHB3bJ6ydzMRUMhNTUVssBNdUsGv/ZwA8+UonKtnKF/OXcSU+ieRbObyw\n9yNqPL35rR3yEVqtKtrbnAH77lBUmJ4ohmAc3CuK5P3ff4eXvvfahLcVWV7M+rPH+LcXXwXg1Y9+\ni0+vxAdZs+cIXUI7yajV1fnw1ps/wmJR1IcfBh54haKJRJYlLJMk030vMJQ/bH0Si0qNRaXm7W3P\ncC8otKcvkoRFrcaistOfVJYUI/AQoRiCcdLY4MWHf/j2hO++M2k0NDu7ojOZeGHvR+w8tBv/uhqW\nXz7XPwxZQWGUKIZgnJhMWkpLQ5HlyfkqLWo1l+ak4tnchIOhk7sh4ZT6T6x2gsKDj+IjmGFYVCpu\nRMWiMxlYfvkcpf5BVHcJEigojBFlRGAHLBYNly4uomMSVXuvxiXZPZWZwsOLYgjsgNms4eiRDbS1\nuQxf2E7EFN+l1tMbp852Em/nElxdMWltKzx4KIZghuBg6CSwtgqVbCWkqpxtxw9yKyIatdXK4uuZ\nJOXfJKyyjMBeeQwCa6twMBjG1J5WayIouKQ7d6TCg41iCGYIAXXVbDp9GN+GOr695wPUVgtmtYYT\nafTm5ZAAAA8gSURBVBn8ZsfzlPoH8uShPWw6fRj31mYkWebJQ3sIqS4fU3senvU8+9zbaLUmO38S\nhemI4iy0I7JVhSxLY86zOBT3gkLZv3Qdr374FlaVil9t39UnlLgrzbpffS1/8vHv+MW3vo9sBck6\nvnyNKpUyIngYUEYEduS9d1+iuDhiQuqOKCvm23s+wKDT8d+7XqHO3XPAcrUeXrz5xAv84JO3+fL/\na2D+G3tIvJ07YNnh0OkM/ODV/8TLq274wgozGsUQ2JH2dmcs5okZZGksFjr1Dry39Sl2HPkK99aB\nZdc9Wpp47MhXfLb2EZa95MbNHcsJqywj/ca1UbcpSTLOzq3KqOAhQDEE05CtJw/x7P4/ElVa1H2s\n3Nef3as2UeIfxIm0DNocnQa8ts3RmZNpGZT6B3Fm12YKEuPRmYwsuX6RhdmX+5XfevJQHwfjQGzc\nvBf/AGVV4kFG8RFMQ4oDQ1iZeZoVl86ScOcWQHfOQmDIfIwGna77/N2QcACuxSZS5hdIjWePtqHW\nbGbthRMk5d/Ao6WJE2kZlAQED1hnROQdVq76hjOnVlJaGmaXz6gwvVAMwTQkKyYBjdnM4qxMFuRc\nocLHf1xhxIXB4RQGh3e/d+5oZ2H2ZRZmX+baSQuz4m6zVK3mbPJ87gWGDlhHzOw8LBY1F84voWSC\n/CAKU8eQUwNJkn4nSVKVJEnZvY79VJKkUkmSrtpem3qd+1tJkgokScqTJGn9RHb8QedKfBLlviKV\nUZlfoF3Vh9QWCx4tTWTFJFDZoifPLxKDVses0nuEVpYNel1c/A0WLT5NSGix3fqiMD0YbkTwNvBL\n4Pe9jsnAL2RZ/kXvgpIkJQBPAQlAMHBYkqTZsiw/VJ6mysogfP2qxp1DMbCmik69A3dCIjDo9ATW\nVo1I23AwvJobUVss1Hh60+zi2p0j8TFpHxfmplHuG0BKXjahlWWDThEAYuNu0t7uTGmJMkV4kBhy\nRCDL8imgYYBTAy1Kbwc+lGXZJMtyEXAbWDDuHs4wjh7ewN3bMeOuZ/25Y1R5+fLO9mco9Q9iQfaV\ncdU3+94dkvJvdL+XZBnvxnr2rtjYPfK4FjeXsykP3Z9MgbGvGvxIkqTrkiT9VpKkrpQ5QUBprzKl\niJGBwijRmUxY1GqsNpGRnOg49qzaNMxVQ3N+blqf3AlduRg9W5oGLC/JMjrTxGR3Uph+jMUQvAFE\nAilABfAfQ5SdGbnNphnf2f0+mXNSyY6Jn7A2DDr9kIFJPo11/PiD3yjG4CFh1KsGsixXd/0uSdJb\nwJe2t2VAb5dziO1Yf479tOf3iJUQuXK03Xig0ZmMbD35NQeWruWmLa/hRGDQDS5TrpJl9EaDIl06\n0yk8DkX/t717DY6qvOM4/v0HsgnBXJRrQIaEi1hQ7uIFFNARZeqo9AKo7Vi1rVPbvulM6/RN1TeO\n05m20xmdVlucInKRqSPFqY7BSwI0gQgEEo2YBDYkwSSSkOuyySbZpy/OhlzIbjbJ7tlL/p+ZHXbP\nnj3neTib357Lc54nd9jZRhwEIpJpjOltXbIF6L2icBDYIyJ/xjokWAgUDrmQjS+OdLXjzudLVlIz\nY1bE1t+Ums7ezd+na6JeYY5p2RsG/tDmvTTkbAG3sojsBdYDU0WkGngB2CAiy7F2+53AswDGmFIR\n2Q+UAt3Ac8buoZbjyDfTZ9I6OTSjC50uWjXg2n9qWisbNh4K+JlkTydLzp2lKnM23gl9R5Bfn13M\nmdMrQ1IuFT0CBoEx5rEhJr8ZYP6XgZfHWqjxYPH5r+lISuZ8v4Y+E3u6WVd0nOKblvg9dg9Gd/dE\nCvLvudoLcXnZzdTX9TVISpnsuvp8ze35pKS4rllGT8IEmtIyMIMODurrZlFTPfea+VVs0/0+m91a\n/hVVmTdaVwYS+roLn9TZwdKyUlJd7Xx221raU0be21FrSzoV5Yvo6k7kf0c20O3nBqgrrskcPbwR\ngATxWsO3ZTmZMrVvtGXXpBQKlq5medkXlM5bhDspmQuV86j9Ri8ExSMNAptlfVPFpE43Zc1ptDRB\nxsUKZsyGSR1u5tZW8d69D434uLyxcSqXG6dSWzubw7n3jeizh/Os+dfcns+q244xZUrD1fcSjJcF\nVU4q5mTjTkqmuHgF5eXhO3mpIkeDwGbvr3+AJw/uw/uKk6ICuHEZ3LM1icsrs9n34JYRL6+lJYPj\nBes4dXJsDYEKj99FT88E1q7LIy29GcDqAWm1/zsdVfzQ25BDLDm5g4mJ3X7fn9ThZoK35+rrr87A\nP3Jn8v76B5jsvoIE2fTC7U7B5bqOTz9+YMwh0Ovkidv57NNNdHQkA1bDpsc/eJf0tlY6OpLp7tLf\njXilWzbEHv3eO8xfUOb3/acP7GHG5Uv0H5so+2IVj+98lQ5HEn/58S/oTHQMu55dO39KY8M0vN7Q\nZvkXJcvo6Ehm22NvWb0hPfFzvAkTOLBnG+cqbgrpulT00CAIsYQEb8A+C3c9tJWth/4DVHPHRrgu\nFSorDE2p6ex8eDueif6HTmtsnMret58CoK0tLSxjExojOJ3zefWvv8WR1MlTT/+d/e/8iOqquRij\nzYvilQaBzTYVfMaMxkus2wTFty7jxC1LueKC/XMSAw5YcvHiHD787yM0N4/+smKwursSaW6+noQE\nL7t3PU19fSbd3eEd21FFlgaBzYpvWsLXWQsAqJ8yjUbfcGXuAJ+pdM7nSN691NXa29LQ603QHonG\nCQ2CELpz7WGmTLkUcJ5A3YwN5fy5hRwruJsLF7LHUjSlAtKrBiG0bPlJ0jOaQ7Y8KwTWcf7cgpAt\nU6mh6B5BlKq6kE3+0fVUVs6LdFHUOKB7BFHq6JENGgLKNhoEUaitLU3P0qvQ6+nx+5YeGtgoweu9\n2v1YILvfeoaGhmk2lEiNC8ZYj0v+T2RrENgkyePhuf1v8tZDW2nMuCHSxVHjSUkJ5OSA13+H4hoE\nIeBweNj+xL/IyBiqw2dLV2Ii+x7cQktqmt95PB4H+3b/xJZGQ2qcOHECjhyB9vaAs2kQhICIl5kz\na5k40f/NRl6RYcclMCaBurpMv/0IKBW03Fyoq4OGBmgZuqfq/vQbFyXa21LJzb2fnh7dJGqM8vLg\n1CloDX6QHf3WRQm3O4XTp1ZHuhgqVl25Aid9o10XFoLr2u7nAtEgCLM0Vxuzvq3jbPbYRz9S6hqN\njVBTY/36f/LJqBejQTBGDoeHuVlORIY+I5vqaufmynINAhV6TU3WycCCgjEvSoNgjNIzmti6fVff\n6/ZWOh1JdPgGD7k4PZOL00c/pLlS12hpga4u6xCgcOihQ0ZKg2AMEhK8OBIHDgn23cOHKJ2/iNOL\nbolQqVTc8vi+awcOgNMZ0kVrEIzB3CwnP9z29oBp/77/YXomaMttFWKdnfDaa1YYeEI/HqUGwSis\nXnOM1auPkejw4HB0DnjPkzjwHoHqqiwK8u8ecPgwWHVVFgcP/CAsZVVx4PJl2LMH2tqspsJhoEEw\nQnfceZRVtx3j+usvB5yvpHgFJcXLuXJlMpcbp7LH19cgwMb7csjM7Bsf1uNx0NSkzY7VIOXlcOyY\ntQfQ0DD8/GOgQTBC06bVDxsCp4tW8XnhXQOGGTt/buBVg9TUvsYebW3++ypU41BREVRVWX/81dW2\nrFKDIMSKT6+8JgQGGxwKSgFw/LjVEKiszGoebCMNghGqvDBvQDfiyZPcLF5ijQxf+uVS8vPvoeHS\n9EgVT8Uatxu+/NJ6HsTNQeGiQTBCJWdWUHJmxdXX6enNJCZ2AfBxzmZaW3U3XwXJ5bJC4IMPIl0S\nxITpLKTfFYoYXrR3nUpFlfZ261FVFYEQeAkzxEg1ukeglF06OqzuwvLzrUcU0SBQyi7vvgvnz4M3\n+vaINQiUCqfOTtixw7o3oL09YAeikaRBoFQ41NT09RPY0BCwv8BooEGgVCiVllqP3pOBMUKDQKlQ\nOHnS6iSkutq21oChpEGg1Gj19FgB4PVaHYSE+X6AcNIgUGqk2tvhwgXrBGBODnT77706VmgQKBWs\nlharb8CaGvjoo0iXJqQ0CJQajstl/eoXFFi3BcchDQKl/Om95NfbECiOaRAoNZTOTnj9datTELc7\n0qUJOw0CpXqdPQuHD1vPvV6ru3Cbb8qLFA0CpY4ft3oFbm62vUOQaKFBoMavvDzrUqDTGdNtAEJB\ng0CNL2McIzBeRSYInLmQvSEiq7aF1i+61NVBfb31fNgxAiuBrPCXKcpEJggqc2PrizRSWr/oUFtr\nnfE/c8Z6BKUSDQKlYp0x1oAgxsCHH8bUHYCRpEGg4ovHA2+8YbUDUEGLTOelSqmIGarzUtuDQCkV\nfXTYXqWUBoFSKgJBICIPishZESkXkeftXn+oiUiliBSLSJGIFPqm3SAih0SkTERyRCQj0uUMloi8\nKSL1IlLSb5rf+ojI733b8qyIbIpMqYPnp34vikiNbxsWicjmfu/FVP1GzRhj2wOYAFRgXahNBE4D\n37GzDGGokxO4YdC0PwK/8z1/Hngl0uUcQX3uBlYAJcPVB1js24aJvm1aASREug6jqN8LwG+GmDfm\n6jfah917BGuACmNMpTGmC9gHPGJzGcJh8FnYh4Gdvuc7gUftLc7oGWOOAE2DJvurzyPAXmNMlzGm\nEusPZY0d5RwtP/WDa7chxGD9RsvuIJgN9O/itcY3LZYZ4GMROSEiP/NNm2GM8bVppR6YEZmihYy/\n+szC2oa9Ynl7/lpEzojIjn6HPvFUv4DsDoJ4vFa51hizAtgM/FJE7u7/prH2MeOm3kHUJxbr+jcg\nG1gO1AJ/CjBvLNZvWHYHwUVgTr/XcxiYuDHHGFPr+/cS8B7WrmO9iMwEEJFM4NvIlTAk/NVn8Pa8\n0TctphhjvjU+wD/p2/2Pi/oFw+4gOAEsFJEsEXEA24CDNpchZEQkRURSfc8nA5uAEqw6Pemb7Ung\nQGRKGDL+6nMQ2C4iDhHJBhYChREo35j4wq3XFqxtCHFSv2DYeq+BMaZbRH4FfIR1BWGHMeYrO8sQ\nYjOA90QErP/L3caYHBE5AewXkWewbmfbGrkijoyI7AXWA1NFpBr4A/AKQ9THGFMqIvuBUqAbeM73\nqxq1hqjfC8AGEVmOtdvvBJ6F2KzfaGkTY6WUtixUSmkQKKXQIFBKoUGglEKDQCmFBoFSCg0CpRQa\nBEop4P/qEhJJQ1GZygAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1076f54a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pyplot.imshow(log_grid_Mandelbrot(1000)[600:800,400:600])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Equivalence classes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An *equivalence class* is a relation that groups objects in a set into related subsets. For example, if we think of the integers modulo $7$, then $1$ is in the same equivalence class as $8$ (and $15$, and $22$, and so on), and $3$ is in the same equivalence class as $10$. We use the tilde $3 \\sim 10$ to denote two objects within the same equivalence class.\n",
    "\n",
    "Here, we are going to define the positive integers programmatically from equivalent sequences."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Define a python class `Eqint`. This should be\n",
    "\n",
    "1. Initialized by a sequence;\n",
    "2. Store the sequence;\n",
    "3. Define its representation (via the `__repr__` function) to be the integer length of the sequence;\n",
    "4. Redefine equality (via the `__eq__` function) so that two `eqint`s are equal if their sequences have same length."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Eqint(object):\n",
    "    \n",
    "    def __init__(self, sequence):\n",
    "        self.sequence = sequence\n",
    "        \n",
    "    def __repr__(self):\n",
    "        return str(len(self.sequence))\n",
    "    \n",
    "    def __eq__(self, other):\n",
    "        return len(self.sequence)==len(other.sequence)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Define a `zero` object from the empty list, and three `one` objects, from a single object list, tuple, and string. For example\n",
    "\n",
    "```python\n",
    "one_list = Eqint([1])\n",
    "one_tuple = Eqint((1,))\n",
    "one_string = Eqint('1')\n",
    "```\n",
    "\n",
    "Check that none of the `one` objects equal the zero object, but all equal the other `one` objects. Print each object to check that the representation gives the integer length."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Is zero equivalent to one? False, False, False\n",
      "Is one equivalent to one? True, True, True.\n",
      "0\n",
      "1\n",
      "1\n",
      "1\n"
     ]
    }
   ],
   "source": [
    "zero = Eqint([])\n",
    "one_list = Eqint([1])\n",
    "one_tuple = Eqint((1,))\n",
    "one_string = Eqint('1')\n",
    "\n",
    "print(\"Is zero equivalent to one? {}, {}, {}\".format(zero == one_list, \n",
    "                                                     zero == one_tuple,\n",
    "                                                     zero == one_string))\n",
    "print(\"Is one equivalent to one? {}, {}, {}.\".format(one_list == one_tuple,\n",
    "                                                     one_list == one_string,\n",
    "                                                     one_tuple == one_string))\n",
    "print(zero)\n",
    "print(one_list)\n",
    "print(one_tuple)\n",
    "print(one_string)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Redefine the class by including an `__add__` method that combines the two sequences. That is, if `a` and `b` are `Eqint`s then `a+b` should return an `Eqint` defined from combining `a` and `b`s sequences.\n",
    "\n",
    "##### Note\n",
    "\n",
    "Adding two different *types* of sequences (eg, a list to a tuple) does not work, so it is better to either iterate over the sequences, or to convert to a uniform type before adding."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "class Eqint(object):\n",
    "    \n",
    "    def __init__(self, sequence):\n",
    "        self.sequence = sequence\n",
    "        \n",
    "    def __repr__(self):\n",
    "        return str(len(self.sequence))\n",
    "    \n",
    "    def __eq__(self, other):\n",
    "        return len(self.sequence)==len(other.sequence)\n",
    "    \n",
    "    def __add__(a, b):\n",
    "        return Eqint(tuple(a.sequence) + tuple(b.sequence))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4\n",
    "\n",
    "Check your addition function by adding together all your previous `Eqint` objects (which will need re-defining, as the class has been redefined). Print the resulting object to check you get `3`, and also print its internal sequence."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The sum is 3.\n",
      "The internal sequence is (1, 1, '1').\n"
     ]
    }
   ],
   "source": [
    "zero = Eqint([])\n",
    "one_list = Eqint([1])\n",
    "one_tuple = Eqint((1,))\n",
    "one_string = Eqint('1')\n",
    "\n",
    "sum_eqint = zero + one_list + one_tuple + one_string\n",
    "print(\"The sum is {}.\".format(sum_eqint))\n",
    "print(\"The internal sequence is {}.\".format(sum_eqint.sequence))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 5\n",
    "\n",
    "We will sketch a construction of the positive integers from *nothing*.\n",
    "\n",
    "1. Define an empty list `positive_integers`.\n",
    "2. Define an `Eqint` called `zero` from the empty list. Append it to `positive_integers`.\n",
    "3. Define an `Eqint` called `next_integer` from the `Eqint` defined by *a copy of* `positive_integers` (ie, use `Eqint(list(positive_integers))`. Append it to `positive_integers`.\n",
    "4. Repeat step 3 as often as needed.\n",
    "\n",
    "Use this procedure to define the `Eqint` equivalent to $10$. Print it, and its internal sequence, to check."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The 'final' Eqint is 10\n",
      "Its sequence is [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]\n",
      "That is, it contains all Eqints with length less than 10.\n"
     ]
    }
   ],
   "source": [
    "positive_integers = []\n",
    "zero = Eqint([])\n",
    "positive_integers.append(zero)\n",
    "\n",
    "N = 10\n",
    "for n in range(1,N+1):\n",
    "    positive_integers.append(Eqint(list(positive_integers)))\n",
    "    \n",
    "print(\"The 'final' Eqint is {}\".format(positive_integers[-1]))\n",
    "print(\"Its sequence is {}\".format(positive_integers[-1].sequence))\n",
    "print(\"That is, it contains all Eqints with length less than 10.\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Rational numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of working with floating point numbers, which are not \"exact\", we could work with the rational numbers $\\mathbb{Q}$. A rational number $q \\in \\mathbb{Q}$ is defined by the *numerator* $n$ and *denominator* $d$ as $q = \\frac{n}{d}$, where $n$ and $d$ are *coprime* (ie, have no common divisor other than $1$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Find a python function that finds the greatest common divisor (`gcd`) of two numbers. Use this to write a function `normal_form` that takes a numerator and divisor and returns the coprime $n$ and $d$. Test this function on $q = \\frac{3}{2}$, $q = \\frac{15}{3}$, and $q = \\frac{20}{42}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def normal_form(numerator, denominator):\n",
    "    from fractions import gcd\n",
    "    \n",
    "    factor = gcd(numerator, denominator)\n",
    "    return numerator//factor, denominator//factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3, 2)\n",
      "(5, 1)\n",
      "(10, 21)\n"
     ]
    }
   ],
   "source": [
    "print(normal_form(3, 2))\n",
    "print(normal_form(15, 3))\n",
    "print(normal_form(20, 42))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Define a class `Rational` that uses the `normal_form` function to store the rational number in the appropriate form. Define a `__repr__` function that prints a string that *looks like* $\\frac{n}{d}$ (**hint**: use `len(str(number))` to find the number of digits of an integer). Test it on the cases above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Rational(object):\n",
    "    \"\"\"\n",
    "    A rational number.\n",
    "    \"\"\"\n",
    "    \n",
    "    def __init__(self, numerator, denominator):\n",
    "        \n",
    "        n, d = normal_form(numerator, denominator)\n",
    "        \n",
    "        self.numerator = n\n",
    "        self.denominator = d\n",
    "        \n",
    "        return None\n",
    "    \n",
    "    def __repr__(self):\n",
    "        \n",
    "        max_length = max(len(str(self.numerator)), len(str(self.denominator)))\n",
    "        \n",
    "        if self.denominator == 1:\n",
    "            frac = str(self.numerator)\n",
    "        else:\n",
    "            numerator = str(self.numerator)+'\\n'\n",
    "            bar = max_length*'-'+'\\n'\n",
    "            denominator = str(self.denominator)\n",
    "            frac = numerator+bar+denominator\n",
    "        \n",
    "        return frac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n",
      "-\n",
      "2\n",
      "5\n",
      "10\n",
      "--\n",
      "21\n"
     ]
    }
   ],
   "source": [
    "q1 = Rational(3, 2)\n",
    "print(q1)\n",
    "q2 = Rational(15, 3)\n",
    "print(q2)\n",
    "q3 = Rational(20, 42)\n",
    "print(q3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Overload the `__add__` function so that you can add two rational numbers. Test it on $\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{6} = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Rational(object):\n",
    "    \"\"\"\n",
    "    A rational number.\n",
    "    \"\"\"\n",
    "    \n",
    "    def __init__(self, numerator, denominator):\n",
    "        \n",
    "        n, d = normal_form(numerator, denominator)\n",
    "        \n",
    "        self.numerator = n\n",
    "        self.denominator = d\n",
    "        \n",
    "        return None\n",
    "    \n",
    "    def __add__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.denominator + b.numerator * a.denominator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __repr__(self):\n",
    "        \n",
    "        max_length = max(len(str(self.numerator)), len(str(self.denominator)))\n",
    "        \n",
    "        if self.denominator == 1:\n",
    "            frac = str(self.numerator)\n",
    "        else:\n",
    "            numerator = str(self.numerator)+'\\n'\n",
    "            bar = max_length*'-'+'\\n'\n",
    "            denominator = str(self.denominator)\n",
    "            frac = numerator+bar+denominator\n",
    "        \n",
    "        return frac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "print(Rational(1,2) + Rational(1,3) + Rational(1,6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4\n",
    "\n",
    "Overload the `__mul__` function so that you can multiply two rational numbers. Test it on $\\frac{1}{3} \\times \\frac{15}{2} \\times \\frac{2}{5} = 1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Rational(object):\n",
    "    \"\"\"\n",
    "    A rational number.\n",
    "    \"\"\"\n",
    "    \n",
    "    def __init__(self, numerator, denominator):\n",
    "        \n",
    "        n, d = normal_form(numerator, denominator)\n",
    "        \n",
    "        self.numerator = n\n",
    "        self.denominator = d\n",
    "        \n",
    "        return None\n",
    "    \n",
    "    def __add__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.denominator + b.numerator * a.denominator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __mul__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.numerator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __repr__(self):\n",
    "        \n",
    "        max_length = max(len(str(self.numerator)), len(str(self.denominator)))\n",
    "        \n",
    "        if self.denominator == 1:\n",
    "            frac = str(self.numerator)\n",
    "        else:\n",
    "            numerator = str(self.numerator)+'\\n'\n",
    "            bar = max_length*'-'+'\\n'\n",
    "            denominator = str(self.denominator)\n",
    "            frac = numerator+bar+denominator\n",
    "        \n",
    "        return frac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n"
     ]
    }
   ],
   "source": [
    "print(Rational(1,3)*Rational(15,2)*Rational(2,5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 5\n",
    "\n",
    "Overload the [`__rmul__`](https://docs.python.org/2/reference/datamodel.html?highlight=rmul#object.__rmul__) function so that you can multiply a rational by an *integer*. Check that $\\frac{1}{2} \\times 2 = 1$ and $\\frac{1}{2} + (-1) \\times \\frac{1}{2} = 0$. Also overload the `__sub__` function (using previous functions!) so that you can subtract rational numbers and check that $\\frac{1}{2} - \\frac{1}{2} = 0$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Rational(object):\n",
    "    \"\"\"\n",
    "    A rational number.\n",
    "    \"\"\"\n",
    "    \n",
    "    def __init__(self, numerator, denominator):\n",
    "        \n",
    "        n, d = normal_form(numerator, denominator)\n",
    "        \n",
    "        self.numerator = n\n",
    "        self.denominator = d\n",
    "        \n",
    "        return None\n",
    "    \n",
    "    def __add__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.denominator + b.numerator * a.denominator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __mul__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.numerator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __rmul__(self, other):\n",
    "        \n",
    "        numerator = self.numerator * other\n",
    "        return Rational(numerator, self.denominator)\n",
    "    \n",
    "    def __sub__(a, b):\n",
    "        \n",
    "        return a + (-1)*b\n",
    "    \n",
    "    def __repr__(self):\n",
    "        \n",
    "        max_length = max(len(str(self.numerator)), len(str(self.denominator)))\n",
    "        \n",
    "        if self.denominator == 1:\n",
    "            frac = str(self.numerator)\n",
    "        else:\n",
    "            numerator = str(self.numerator)+'\\n'\n",
    "            bar = max_length*'-'+'\\n'\n",
    "            denominator = str(self.denominator)\n",
    "            frac = numerator+bar+denominator\n",
    "        \n",
    "        return frac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "0\n",
      "0\n"
     ]
    }
   ],
   "source": [
    "half = Rational(1,2)\n",
    "print(2*half)\n",
    "print(half+(-1)*half)\n",
    "print(half-half)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 6\n",
    "\n",
    "Overload the `__float__` function so that `float(q)` returns the floating point approximation to the rational number `q`. Test this on $\\frac{1}{2}, \\frac{1}{3}$, and $\\frac{1}{11}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Rational(object):\n",
    "    \"\"\"\n",
    "    A rational number.\n",
    "    \"\"\"\n",
    "    \n",
    "    def __init__(self, numerator, denominator):\n",
    "        \n",
    "        n, d = normal_form(numerator, denominator)\n",
    "        \n",
    "        self.numerator = n\n",
    "        self.denominator = d\n",
    "        \n",
    "        return None\n",
    "    \n",
    "    def __add__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.denominator + b.numerator * a.denominator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __mul__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.numerator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __rmul__(self, other):\n",
    "        \n",
    "        numerator = self.numerator * other\n",
    "        return Rational(numerator, self.denominator)\n",
    "    \n",
    "    def __sub__(a, b):\n",
    "        \n",
    "        return a + (-1)*b\n",
    "    \n",
    "    def __float__(a):\n",
    "        \n",
    "        return float(a.numerator) / float(a.denominator)\n",
    "    \n",
    "    def __repr__(self):\n",
    "        \n",
    "        max_length = max(len(str(self.numerator)), len(str(self.denominator)))\n",
    "        \n",
    "        if self.denominator == 1:\n",
    "            frac = str(self.numerator)\n",
    "        else:\n",
    "            numerator = str(self.numerator)+'\\n'\n",
    "            bar = max_length*'-'+'\\n'\n",
    "            denominator = str(self.denominator)\n",
    "            frac = numerator+bar+denominator\n",
    "        \n",
    "        return frac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.5\n",
      "0.3333333333333333\n",
      "0.09090909090909091\n"
     ]
    }
   ],
   "source": [
    "print(float(Rational(1,2)))\n",
    "print(float(Rational(1,3)))\n",
    "print(float(Rational(1,11)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 7\n",
    "\n",
    "Overload the `__lt__` function to compare two rational numbers. Create a list of rational numbers where the denominator is $n = 2, \\dots, 11$ and the numerator is the floored integer $n/2$, ie `n//2`. Use the `sorted` function on that list (which relies on the `__lt__` function)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Rational(object):\n",
    "    \"\"\"\n",
    "    A rational number.\n",
    "    \"\"\"\n",
    "    \n",
    "    def __init__(self, numerator, denominator):\n",
    "        \n",
    "        n, d = normal_form(numerator, denominator)\n",
    "        \n",
    "        self.numerator = n\n",
    "        self.denominator = d\n",
    "        \n",
    "        return None\n",
    "    \n",
    "    def __add__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.denominator + b.numerator * a.denominator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __mul__(a, b):\n",
    "        \n",
    "        numerator = a.numerator * b.numerator\n",
    "        denominator = a.denominator * b.denominator\n",
    "        return Rational(numerator, denominator)\n",
    "    \n",
    "    def __rmul__(self, other):\n",
    "        \n",
    "        numerator = self.numerator * other\n",
    "        return Rational(numerator, self.denominator)\n",
    "    \n",
    "    def __sub__(a, b):\n",
    "        \n",
    "        return a + (-1)*b\n",
    "    \n",
    "    def __float__(a):\n",
    "        \n",
    "        return float(a.numerator) / float(a.denominator)\n",
    "    \n",
    "    def __lt__(a, b):\n",
    "        \n",
    "        return a.numerator * b.denominator < a.denominator * b.numerator\n",
    "    \n",
    "    def __repr__(self):\n",
    "        \n",
    "        max_length = max(len(str(self.numerator)), len(str(self.denominator)))\n",
    "        \n",
    "        if self.denominator == 1:\n",
    "            frac = str(self.numerator)\n",
    "        else:\n",
    "            numerator = '\\n'+str(self.numerator)+'\\n'\n",
    "            bar = max_length*'-'+'\\n'\n",
    "            denominator = str(self.denominator)\n",
    "            frac = numerator+bar+denominator\n",
    "        \n",
    "        return frac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[\n",
      "1\n",
      "-\n",
      "3, \n",
      "2\n",
      "-\n",
      "5, \n",
      "3\n",
      "-\n",
      "7, \n",
      "4\n",
      "-\n",
      "9, \n",
      "5\n",
      "--\n",
      "11, \n",
      "1\n",
      "-\n",
      "2, \n",
      "1\n",
      "-\n",
      "2, \n",
      "1\n",
      "-\n",
      "2, \n",
      "1\n",
      "-\n",
      "2, \n",
      "1\n",
      "-\n",
      "2]\n"
     ]
    }
   ],
   "source": [
    "q_list = [Rational(n//2, n) for n in range(2, 12)]\n",
    "print(sorted(q_list))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 8\n",
    "\n",
    "The [Wallis formula for $\\pi$](http://mathworld.wolfram.com/WallisFormula.html) is\n",
    "\n",
    "\\begin{equation}\n",
    "  \\pi = 2 \\prod_{n=1}^{\\infty} \\frac{ (2 n)^2 }{(2 n - 1) (2 n + 1)}.\n",
    "\\end{equation}\n",
    "\n",
    "We can define a partial product $\\pi_N$ as\n",
    "\n",
    "\\begin{equation}\n",
    "  \\pi_N = 2 \\prod_{n=1}^{N} \\frac{ (2 n)^2 }{(2 n - 1) (2 n + 1)},\n",
    "\\end{equation}\n",
    "\n",
    "each of which are rational numbers.\n",
    "\n",
    "Construct a list of the first 20 rational number approximations to $\\pi$ and print them out. Print the sorted list to show that the approximations are always increasing. Then convert them to floating point numbers, construct a `numpy` array, and subtract this array from $\\pi$ to see how accurate they are."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def wallis_rational(N):\n",
    "    \"\"\"\n",
    "    The partial product approximation to pi using the first N terms of Wallis' formula.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    N : int\n",
    "        Number of terms in product\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    partial : Rational\n",
    "        A rational number approximation to pi\n",
    "    \"\"\"\n",
    "    \n",
    "    partial = Rational(2,1)\n",
    "    for n in range(1, N+1):\n",
    "        partial = partial * Rational((2*n)**2, (2*n-1)*(2*n+1))\n",
    "    return partial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[\n",
      "8\n",
      "-\n",
      "3, \n",
      "128\n",
      "---\n",
      "45, \n",
      "512\n",
      "---\n",
      "175, \n",
      "32768\n",
      "-----\n",
      "11025, \n",
      "131072\n",
      "------\n",
      "43659, \n",
      "2097152\n",
      "-------\n",
      "693693, \n",
      "8388608\n",
      "-------\n",
      "2760615, \n",
      "2147483648\n",
      "----------\n",
      "703956825, \n",
      "8589934592\n",
      "----------\n",
      "2807136475, \n",
      "137438953472\n",
      "------------\n",
      "44801898141, \n",
      "549755813888\n",
      "------------\n",
      "178837328943, \n",
      "35184372088832\n",
      "--------------\n",
      "11425718238025, \n",
      "140737488355328\n",
      "---------------\n",
      "45635265151875, \n",
      "2251799813685248\n",
      "----------------\n",
      "729232910488125, \n",
      "9007199254740992\n",
      "----------------\n",
      "2913690606794775, \n",
      "9223372036854775808\n",
      "-------------------\n",
      "2980705490751054825, \n",
      "36893488147419103232\n",
      "--------------------\n",
      "11912508103174630875, \n",
      "590295810358705651712\n",
      "---------------------\n",
      "190453061649520333125, \n",
      "2361183241434822606848\n",
      "----------------------\n",
      "761284675790187924375, \n",
      "151115727451828646838272\n",
      "------------------------\n",
      "48691767863540419643025]\n",
      "[\n",
      "8\n",
      "-\n",
      "3, \n",
      "128\n",
      "---\n",
      "45, \n",
      "512\n",
      "---\n",
      "175, \n",
      "32768\n",
      "-----\n",
      "11025, \n",
      "131072\n",
      "------\n",
      "43659, \n",
      "2097152\n",
      "-------\n",
      "693693, \n",
      "8388608\n",
      "-------\n",
      "2760615, \n",
      "2147483648\n",
      "----------\n",
      "703956825, \n",
      "8589934592\n",
      "----------\n",
      "2807136475, \n",
      "137438953472\n",
      "------------\n",
      "44801898141, \n",
      "549755813888\n",
      "------------\n",
      "178837328943, \n",
      "35184372088832\n",
      "--------------\n",
      "11425718238025, \n",
      "140737488355328\n",
      "---------------\n",
      "45635265151875, \n",
      "2251799813685248\n",
      "----------------\n",
      "729232910488125, \n",
      "9007199254740992\n",
      "----------------\n",
      "2913690606794775, \n",
      "9223372036854775808\n",
      "-------------------\n",
      "2980705490751054825, \n",
      "36893488147419103232\n",
      "--------------------\n",
      "11912508103174630875, \n",
      "590295810358705651712\n",
      "---------------------\n",
      "190453061649520333125, \n",
      "2361183241434822606848\n",
      "----------------------\n",
      "761284675790187924375, \n",
      "151115727451828646838272\n",
      "------------------------\n",
      "48691767863540419643025]\n"
     ]
    }
   ],
   "source": [
    "pi_list = [wallis_rational(n) for n in range(1, 21)]\n",
    "print(pi_list)\n",
    "print(sorted(pi_list))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.47492599  0.29714821  0.21587837  0.16943846  0.1394167   0.11842246\n",
      "  0.10291902  0.09100266  0.08155811  0.07388885  0.06753749  0.06219131\n",
      "  0.05762923  0.05369058  0.05025576  0.04723393  0.04455483  0.0421633\n",
      "  0.04001539  0.03807569]\n"
     ]
    }
   ],
   "source": [
    "import numpy\n",
    "print(numpy.pi-numpy.array(list(map(float, pi_list))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The shortest published Mathematical paper"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A [candidate for the shortest mathematical paper ever](http://www.ams.org/journals/bull/1966-72-06/S0002-9904-1966-11654-3/S0002-9904-1966-11654-3.pdf) shows the following result:\n",
    "\n",
    "\\begin{equation}\n",
    "  27^5 + 84^5 + 110^5 + 133^5 = 144^5.\n",
    "\\end{equation}\n",
    "\n",
    "This is interesting as\n",
    "\n",
    "> This is a counterexample to a conjecture by Euler ... that at least $n$ $n$th powers are required to sum to an $n$th power, $n > 2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Using python, check the equation above is true."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Does the LHS 61917364224 equal the RHS 61917364224? True\n"
     ]
    }
   ],
   "source": [
    "lhs = 27**5 + 84**5 + 110**5 + 133**5\n",
    "rhs = 144**5\n",
    "\n",
    "print(\"Does the LHS {} equal the RHS {}? {}\".format(lhs, rhs, lhs==rhs))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "The more interesting statement in the paper is that\n",
    "\n",
    "\\begin{equation}\n",
    "  27^5 + 84^5 + 110^5 + 133^5 = 144^5.\n",
    "\\end{equation}\n",
    "\n",
    "> [is] the smallest instance in which four fifth powers sum to a fifth power.\n",
    "\n",
    "Interpreting \"the smallest instance\" to mean the solution where the right hand side term (the largest integer) is the smallest, we want to use python to check this statement.\n",
    "\n",
    "You may find the `combinations` function from the `itertools` package useful."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy\n",
    "import itertools"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `combinations` function returns all the combinations (ignoring order) of `r` elements from a given list. For example, take a list of length 6, `[1, 2, 3, 4, 5, 6]` and compute all the combinations of length 4:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(1, 2, 3, 4), (1, 2, 3, 5), (1, 2, 3, 6), (1, 2, 4, 5), (1, 2, 4, 6), (1, 2, 5, 6), (1, 3, 4, 5), (1, 3, 4, 6), (1, 3, 5, 6), (1, 4, 5, 6), (2, 3, 4, 5), (2, 3, 4, 6), (2, 3, 5, 6), (2, 4, 5, 6), (3, 4, 5, 6)]\n"
     ]
    }
   ],
   "source": [
    "input_list = numpy.arange(1, 7)\n",
    "combinations = list(itertools.combinations(input_list, 4))\n",
    "print(combinations)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can already see that the number of terms to consider is large.\n",
    "\n",
    "Note that we have used the `list` function to explicitly get a list of the combinations. The `combinations` function returns a *generator*, which can be used in a loop as if it were a list, without storing all elements of the list."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How fast does the number of combinations grow? The standard formula says that for a list of length $n$ there are\n",
    "\n",
    "\\begin{equation}\n",
    "  \\begin{pmatrix} n \\\\ k \\end{pmatrix} = \\frac{n!}{k! (n-k)!}\n",
    "\\end{equation}\n",
    "\n",
    "combinations of length $k$. For $k=4$ as needed here we will have $n (n-1) (n-2) (n-3) / 24$ combinations. For $n=144$ we therefore have"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Number of combinations of 4 objects from 144 is 17178876.0\n"
     ]
    }
   ],
   "source": [
    "n_combinations = 144*143*142*141/24\n",
    "print(\"Number of combinations of 4 objects from 144 is {}\".format(n_combinations))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2a\n",
    "\n",
    "Show, by getting python to compute the number of combinations $N = \\begin{pmatrix} n \\\\ 4 \\end{pmatrix}$ that $N$ grows roughly as $n^4$. To do this, plot the number of combinations and $n^4$ on a log-log scale. Restrict to $n \\le 50$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from matplotlib import pyplot\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n = numpy.arange(5, 51)\n",
    "N = numpy.zeros_like(n)\n",
    "for i, n_c in enumerate(n):\n",
    "    combinations = list(itertools.combinations(numpy.arange(1,n_c+1), 4))\n",
    "    N[i] = len(combinations)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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4PUfSK865nCSZ2YOS5kv6Z0k/NrPPSspUq0YAAOCP8pz2xo3ktBEufme4p0jqKrvuljTX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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10727d2b0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pyplot.figure(figsize=(12,6))\n",
    "pyplot.loglog(n, N, linestyle='None', marker='x', color='k', label='Combinations')\n",
    "pyplot.loglog(n, n**4, color='b', label=r'$n^4$')\n",
    "pyplot.xlabel(r'$n$')\n",
    "pyplot.ylabel(r'$N$')\n",
    "pyplot.legend(loc='upper left')\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With 17 million combinations to work with, we'll need to be a little careful how we compute."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One thing we could try is to loop through each possible \"smallest instance\" (the term on the right hand side) in increasing order. We then check all possible combinations of left hand sides.\n",
    "\n",
    "This is computationally *very expensive* as we repeat a lot of calculations. We repeatedly recalculate combinations (a bad idea). We repeatedly recalculate the powers of the same number.\n",
    "\n",
    "Instead, let us try creating the list of all combinations of powers once."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2b\n",
    "\n",
    "1. Construct a `numpy` array containing all integers in $1, \\dots, 144$ to the fifth power. \n",
    "2. Construct a list of all combinations of four elements from this array.\n",
    "3. Construct a list of sums of all these combinations.\n",
    "4. Loop over one list and check if the entry appears in the other list (ie, use the `in` keyword)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "nmax=145\n",
    "range_to_power = numpy.arange(1, nmax)**5.0\n",
    "lhs_combinations = list(itertools.combinations(range_to_power, 4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then calculate the sums:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "lhs_sums = []\n",
    "for lhs_terms in lhs_combinations:\n",
    "    lhs_sums.append(numpy.sum(numpy.array(lhs_terms)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, loop through the sums and check to see if it matches any possible term on the RHS:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The LHS terms are [ 27  84 110 133].\n",
      "The RHS term is 144.\n"
     ]
    }
   ],
   "source": [
    "for i, lhs in enumerate(lhs_sums):\n",
    "    if lhs in range_to_power:\n",
    "        rhs_primitive = int(lhs**(0.2))\n",
    "        lhs_primitive = (numpy.array(lhs_combinations[i])**(0.2)).astype(int)\n",
    "        print(\"The LHS terms are {}.\".format(lhs_primitive))\n",
    "        print(\"The RHS term is {}.\".format(rhs_primitive))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lorenz attractor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Lorenz system is a set of ordinary differential equations which can be written\n",
    "\n",
    "\\begin{equation}\n",
    "  \\frac{\\text{d} \\vec{v}}{\\text{d} \\vec{t}} = \\vec{f}(\\vec{v})\n",
    "\\end{equation}\n",
    "\n",
    "where the variables in the state vector $\\vec{v}$ are\n",
    "\n",
    "\\begin{equation}\n",
    "  \\vec{v} = \\begin{pmatrix} x(t) \\\\ y(t) \\\\ z(t) \\end{pmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "and the function defining the ODE is\n",
    "\n",
    "\\begin{equation}\n",
    "  \\vec{f} = \\begin{pmatrix} \\sigma \\left( y(t) - x(t) \\right) \\\\ x(t) \\left( \\rho - z(t) \\right) - y(t) \\\\ x(t) y(t) - \\beta z(t) \\end{pmatrix}.\n",
    "\\end{equation}\n",
    "\n",
    "The parameters $\\sigma, \\rho, \\beta$ are all real numbers."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Write a function `dvdt(v, t, params)` that returns $\\vec{f}$ given $\\vec{v}, t$ and the parameters $\\sigma, \\rho, \\beta$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def dvdt(v, t, sigma, rho, beta):\n",
    "    \"\"\"\n",
    "    Define the Lorenz system.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    v : list\n",
    "        State vector\n",
    "    t : float\n",
    "        Time\n",
    "    sigma : float\n",
    "        Parameter\n",
    "    rho : float\n",
    "        Parameter\n",
    "    beta : float\n",
    "        Parameter\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    dvdt : list\n",
    "        RHS defining the Lorenz system\n",
    "    \"\"\"\n",
    "    \n",
    "    x, y, z = v\n",
    "    \n",
    "    return [sigma*(y-x), x*(rho-z)-y, x*y-beta*z]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Fix $\\sigma=10, \\beta=8/3$. Set initial data to be $\\vec{v}(0) = \\vec{1}$. Using `scipy`, specifically the `odeint` function of `scipy.integrate`, solve the Lorenz system up to $t=100$ for $\\rho=13, 14, 15$ and $28$.\n",
    "\n",
    "Plot your results in 3d, plotting $x, y, z$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy\n",
    "from scipy.integrate import odeint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "v0 = [1.0, 1.0, 1.0]\n",
    "sigma = 10.0\n",
    "beta = 8.0/3.0\n",
    "t_values = numpy.linspace(0.0, 100.0, 5000)\n",
    "rho_values = [13.0, 14.0, 15.0, 28.0]\n",
    "v_values = []\n",
    "for rho in rho_values:\n",
    "    params = (sigma, rho, beta)\n",
    "    v = odeint(dvdt, v0, t_values, args=params)\n",
    "    v_values.append(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib import pyplot\n",
    "from mpl_toolkits.mplot3d.axes3d import Axes3D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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11Vfz/vvv06lTJ1asWAHA3XffzbRp02Ilrx588EHOOuusjMbs4NBCXmNNGQmK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7Js5FGUyil+qRjFRCAx0+s9k4ZTedxTgbFUQDFRPoLK3KkoSo2kYXJ9qTb93vVVNt5Vrs8LFmsr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\naO+J/5eUQGIi9O2r0revk379VPr1U+ncOTKnWtgK1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N68eZSWlmrvxdpm1ytnVVyw4c72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\nCFr/BzelCt0mXMciAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1097a77b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = pyplot.figure(figsize=(12,6))\n",
    "for i, v in enumerate(v_values):\n",
    "    ax = fig.add_subplot(2,2,i+1,projection='3d')\n",
    "    ax.plot(v[:,0], v[:,1], v[:,2])\n",
    "    ax.set_xlabel(r'$x$')\n",
    "    ax.set_ylabel(r'$y$')\n",
    "    ax.set_zlabel(r'$z$')\n",
    "    ax.set_title(r\"$\\rho={}$\".format(rho_values[i]))\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Fix $\\rho = 28$. Solve the Lorenz system twice, up to $t=40$, using the two different initial conditions $\\vec{v}(0) = \\vec{1}$ and $\\vec{v}(0) = \\vec{1} + \\vec{10^{-5}}$.\n",
    "\n",
    "Show four plots. Each plot should show the two solutions on the same axes, plotting $x, y$ and $z$. Each plot should show $10$ units of time, ie the first shows $t \\in [0, 10]$, the second shows $t \\in [10, 20]$, and so on."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t_values = numpy.linspace(0.0, 40.0, 4000)\n",
    "rho = 28.0\n",
    "params = (sigma, rho, beta)\n",
    "v_values = []\n",
    "v0_values = [[1.0,1.0,1.0],\n",
    "             [1.0+1e-5,1.0+1e-5,1.0+1e-5]]\n",
    "for v0 in v0_values:\n",
    "    v = odeint(dvdt, v0, t_values, args=params)\n",
    "    v_values.append(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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PLF7IOxbxRcyUsar0NHq9XoxGY1yhTDV0n4k0AOVLQpblpCGw1j4AycJOIqwk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\noUOHDh06dOjQ0Wzx/wHIEUgjyLnlGwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1897b61d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = pyplot.figure(figsize=(12,6))\n",
    "line_colours = 'by'\n",
    "for tstart in range(4):\n",
    "    ax = fig.add_subplot(2,2,tstart+1,projection='3d')\n",
    "    for i, v in enumerate(v_values):\n",
    "        ax.plot(v[tstart*1000:(tstart+1)*1000,0], \n",
    "                v[tstart*1000:(tstart+1)*1000,1], \n",
    "                v[tstart*1000:(tstart+1)*1000,2], \n",
    "                color=line_colours[i])\n",
    "    ax.set_xlabel(r'$x$')\n",
    "    ax.set_ylabel(r'$y$')\n",
    "    ax.set_zlabel(r'$z$')\n",
    "    ax.set_title(r\"$t \\in [{},{}]$\".format(tstart*10, (tstart+1)*10))\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This shows the *sensitive dependence on initial conditions* that is characteristic of chaotic behaviour."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Systematic ODE solving with sympy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are interested in the solution of\n",
    "\n",
    "\\begin{equation}\n",
    "  \\frac{\\text{d} y}{\\text{d} t} = e^{-t} - y^n, \\qquad y(0) = 1,\n",
    "\\end{equation}\n",
    "\n",
    "where $n > 1$ is an integer. The \"minor\" change from the above examples mean that `sympy` can only give the solution as a power series."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Compute the general solution as a power series for $n = 2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y, t = sympy.symbols('y, t')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/latex": [
       "$$y{\\left (t \\right )} = \\frac{t^{2}}{2} \\left(C_{1} - 1\\right) + \\frac{t^{3}}{6} \\left(2 C_{1} + 1\\right) + \\frac{t^{4}}{24} \\left(C_{1} \\left(5 C_{1} + 1\\right) + C_{1} + 3\\right) + \\frac{t^{5}}{120} \\left(3 C_{1} \\left(- 6 C_{1} + 5\\right) - 8 C_{1} - 9\\right) + C_{1} + C_{1} t + \\mathcal{O}\\left(t^{6}\\right)$$"
      ],
      "text/plain": [
       "        2             3               4                             5         \n",
       "       t ⋅(C₁ - 1)   t ⋅(2⋅C₁ + 1)   t ⋅(C₁⋅(5⋅C₁ + 1) + C₁ + 3)   t ⋅(3⋅C₁⋅(-\n",
       "y(t) = ─────────── + ───────────── + ─────────────────────────── + ───────────\n",
       "            2              6                      24                          \n",
       "\n",
       "                                         \n",
       "6⋅C₁ + 5) - 8⋅C₁ - 9)                ⎛ 6⎞\n",
       "───────────────────── + C₁ + C₁⋅t + O⎝t ⎠\n",
       "   120                                   "
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.dsolve(sympy.diff(y(t), t) + y(t)**2 - sympy.exp(-t), y(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Investigate the help for the `dsolve` function to straightforwardly impose the initial condition $y(0) = 1$ using the `ics` argument. Using this, compute the specific solutions that satisfy the ODE for $n = 2, \\dots, 10$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y(t) == 1 - t**2/2 + t**3/2 - 7*t**4/24 + 3*t**5/40 + O(t**6)\n",
      "y(t) == 1 - t**2/2 + 2*t**3/3 - 13*t**4/24 + 11*t**5/60 + O(t**6)\n",
      "y(t) == 1 - t**2/2 + 5*t**3/6 - 7*t**4/8 + 49*t**5/120 + O(t**6)\n",
      "y(t) == 1 - t**2/2 + t**3 - 31*t**4/24 + 4*t**5/5 + O(t**6)\n",
      "y(t) == 1 - t**2/2 + 7*t**3/6 - 43*t**4/24 + 169*t**5/120 + O(t**6)\n",
      "y(t) == 1 - t**2/2 + 4*t**3/3 - 19*t**4/8 + 137*t**5/60 + O(t**6)\n",
      "y(t) == 1 - t**2/2 + 3*t**3/2 - 73*t**4/24 + 139*t**5/40 + O(t**6)\n",
      "y(t) == 1 - t**2/2 + 5*t**3/3 - 91*t**4/24 + 151*t**5/30 + O(t**6)\n",
      "y(t) == 1 - t**2/2 + 11*t**3/6 - 37*t**4/8 + 841*t**5/120 + O(t**6)\n"
     ]
    }
   ],
   "source": [
    "for n in range(2, 11):\n",
    "    ode_solution = sympy.dsolve(sympy.diff(y(t), t) + y(t)**n - sympy.exp(-t), y(t), \n",
    "                                ics = {y(0) : 1})\n",
    "    print(ode_solution)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Using the `removeO` command, plot each of these solutions for $t \\in [0, 1]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x18b0bbcf8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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GISKSLiZNgjPPhJEj4fzzk/7xShgiIulk8eKwi1///nDTTUkdQaWEISKSbr75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\niIhITJQwREQkJv8PHuxO6Ytj2+UAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18b047ac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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6feopOFe3sGQkzfQWkZSZOTMkiwED4IYb4o6mTNPy5iKS/lasgC5dwszwQYMg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\negAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18a0a14e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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5NXEiHHxw3NFIIYuh2GBVKj4oUhN9+8If/xhWg69cGXc0UsjypdhgVZEmDDMr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\n7QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18b27dd30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x18b31bdd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x18b2a0390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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vXrq9Q4dQoypb9qijSpNT2c8fckhINuW3l01OeXm6QEH4d7BjR/gDY/Pm8N/x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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18b1a13c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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RObLGUa3YzBoBjwPfAYqA98xshruvqtZ0nruPiKoOERFJXpQ9hf7AJ+6+zt1L\ngD8C36+hnUVYg4iI1EGUodAB2FDl+cby16py4BwzW2Zms8ysV4T1iIhILSIbPiLs8GuzBChw92Iz\nGwZMA7pHWJOIiBxBlKFQBBRUeV5A6C0kuPvuKo9nm9kTZna8u2+rvjIzu7fK07nuPree6xURafCi\nDIXFQDcz6wR8DvwIuLpqAzNrC3zp7m5m/QGrKRDcXfMOIiJpEFkouHupmf0UeB1oBExy91VmNrr8\n/YnAFcAYMysFioGRUdUjIiK1M/dkhv5FRKQh0BnNIiKSoFAQEZEEhYKIiCQoFEREJEGhICIiCQoF\nERFJUCiIiEiCQkFERBL+P9r5BNn5tP1KAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x188d79d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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wOdAeaAp8BBxb8Zq7u7/22ms+cOBAd3d/7733vHfv3p6Ptm/f7h07dvSFCxf6N9984927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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18afbc5c0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "for n in range(2, 11):\n",
    "    ode_solution = sympy.dsolve(sympy.diff(y(t), t) + y(t)**n - sympy.exp(-t), y(t), \n",
    "                                ics = {y(0) : 1})\n",
    "    sympy.plot(ode_solution.rhs.removeO(), (t, 0, 1));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Twin primes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A *twin prime* is a pair $(p_1, p_2)$ such that both $p_1$ and $p_2$ are prime and $p_2 = p_1 + 2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Write a generator that returns twin primes. You can use the generators above, and may want to look at the [itertools](https://docs.python.org/3/library/itertools.html) module together with [its recipes](https://docs.python.org/3/library/itertools.html#itertools-recipes), particularly the `pairwise` recipe."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution\n",
    "\n",
    "Note: we need to first pull in the generators introduced in that notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def all_primes(N):\n",
    "    \"\"\"\n",
    "    Return all primes less than or equal to N.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    \n",
    "    N : int\n",
    "        Maximum number\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    \n",
    "    prime : generator\n",
    "        Prime numbers\n",
    "    \"\"\"\n",
    "    \n",
    "    primes = []\n",
    "    for n in range(2, N+1):\n",
    "        is_n_prime = True\n",
    "        for p in primes:\n",
    "            if n%p == 0:\n",
    "                is_n_prime = False\n",
    "                break\n",
    "        if is_n_prime:\n",
    "            primes.append(n)\n",
    "            yield n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can generate pairs using the pairwise recipe:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from itertools import tee\n",
    "\n",
    "def pair_primes(N):\n",
    "    \"Generate consecutive prime pairs, using the itertools recipe\"\n",
    "    a, b = tee(all_primes(N))\n",
    "    next(b, None)\n",
    "    return zip(a, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could examine the results of the two primes directly. But an efficient solution is to use python's [filter function](https://docs.python.org/3/library/functions.html#filter). To do this, first define a function checking if the pair are *twin* primes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def check_twin(pair):\n",
    "    \"\"\"\n",
    "    Take in a pair of integers, check if they differ by 2.\n",
    "    \"\"\"\n",
    "    p1, p2 = pair\n",
    "    return p2-p1 == 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then use the `filter` function to define another generator:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def twin_primes(N):\n",
    "    \"\"\"\n",
    "    Return all twin primes\n",
    "    \"\"\"\n",
    "    return filter(check_twin, pair_primes(N))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now check by finding the twin primes with $N<20$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3, 5)\n",
      "(5, 7)\n",
      "(11, 13)\n",
      "(17, 19)\n"
     ]
    }
   ],
   "source": [
    "for tp in twin_primes(20):\n",
    "    print(tp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Find how many twin primes there are with $p_2 < 1000$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again there are many solutions, but the itertools recipes has the `quantify` pattern. Looking ahead to exercise 3 we'll define:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def pi_N(N):\n",
    "    \"\"\"\n",
    "    Use the quantify pattern from itertools to count the number of twin primes.\n",
    "    \"\"\"\n",
    "    return sum(map(check_twin, pair_primes(N)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAOBAMAAADd6iHDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nMqsI8sYEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjklEQVQIHWNgEFIyYWA4vuGqAwNjAIN/AsP6\n/58YGNi/MPA3MBx6LsDAwLmSYf8BhgcMEABUAGVyNzEwPEuPBQozakQzMFgzyE8AKdECkcwNICbX\naiDB+pGBUYCB+QvfAQamnwz8v4FMrgMMrL8ZuAwYOD6yMjBwHGBgS2DwLwDangW0bmqoJQODqF40\nAwCxgyFSEPpuIAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$35$$"
      ],
      "text/plain": [
       "35"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi_N(1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Let $\\pi_N$ be the number of twin primes such that $p_2 < N$. Plot how $\\pi_N / N$ varies with $N$ for $N=2^k$ and $k = 4, 5, \\dots 16$. (You should use a logarithmic scale where appropriate!)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We've now done all the hard work and can use the solutions above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy\n",
    "from matplotlib import pyplot\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "N = numpy.array([2**k for k in range(4, 17)])\n",
    "twin_prime_fraction = numpy.array(list(map(pi_N, N))) / N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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lhRzy6gXs4e7PtU1JradDXsm74YYQLL//vVZ5FClX0ZcALkUKlOS5wzHHhDvpJ0+OXY2I\ntAUFShYKlLbx3nshUGbMCPepiEh5iboeilSW7t3D0sGnnw4ffBC7GhEpNepQJG8/+AEsWwa/+pVW\neRQpJ+pQpOgmT4bly+FnP4tdiYiUEnUoUpBXX4VDDoGnnoIhQ2JXIyJJUIciUQwcCNdcA+PHw1/+\nErsaESkF6lCkYO4wbhx06wa33BK7GhFpLV02nIUCpXgaV3mcMAEuugg6d45dkYgUSoe8JKqaGnjy\nSWhogP79YepU2LgxdlUiEoMCRVqtb98wLctjj0F9PQwYANOmwaZNsSsTkWJSoEhivvAFePBBeOCB\n8Bg0CP793+Hjj2NXJiLFoHMo0maefhouvRRWr4af/AROPFE3QoqUMp2Uz0KBUjrc4be/hcsuC88n\nTYLRoxUsIqVIgZKFAqX0uMOsWXD55fC5z4Vgqa2NXZWINKVAyUKBUro+/hjuuQfq6sLJ/KuughEj\nYlclIqDLhiVlqqrgtNPCZcbf+AZ8/eswdiwsXBi7MhFpLQWKRNGhA5x9NixdGtZWOfroMI3Lq6/G\nrkxECqVAkaiqq8N69cuWwdChYcLJM8+Et96KXZmI5EuBIiWhc2f40Y9Cx9KjBxxwAJx/PrzzTuzK\nRCRXChQpKTU14QqwxYuhY0fYd1+4+GKtECmSBgoUKUndu8P114eT9R9+GKbLv+mmcPmxiJQmXTYs\nqbBsGZxwAhx6aAiW9u1jVyRSfnQfShYKlPK0dm241LhDB5g5E7p0iV2RSHnRfShSMT77WXjkEejZ\nEw47DFasiF2RiDSlQJFU6dABbr8dvvlNOPhg3RApUkp0yEtSa+ZM+N73whT5X/lK7GpE0k+HvKRi\njRsXJpz81rdC1yIicalDkdRbuhTGjAnzgk2eDO30zySRgugqrywUKJVn1aowyWTv3jB9epjSRUTy\no0NeIkDXrvC734WFu448MgSMiBSXAkXKRnV1WGvlsMPCFWBLl8auSKSyKFCkrLRrB1OmhPm/Dj0U\nfv/72BWJVA4FipSls84K51JOOCFcXiwibU8n5aWsLVwIxx4L554LP/xhOMciItnpKq8sFCjS1IoV\nIVSGD4fbbgt324vIp6XyKi8zG21mDWa21MwmtrDNTZn3XzKzYU1ef9PMFprZfDObW7yqJa169YL/\n+Z+wWNcxx4RJJkUkeUUPFDOrAm4BRgODgfFmtk+zbcYA/d19AHA2cFuTtx2odfdh7j6iSGVLynXp\nAg8+CP37h5P1y5fHrkik/MToUEYAy9z9TXffBMwExjbb5nhgOoC7zwFqzGy3Ju/rSLjkrX17uPXW\nMFXLwQfDiy/GrkikvMQIlF5A038fvp15LddtHHjCzOaZ2VltVqWUJTO46CK48cYwoeQjj8SuSKR8\nxFj3Ltez5S11IV9095Vm1g143Mwa3P2Z5hvV1dV98t+1tbXU1tbmW6eUsa9/PZxbOeEEuPxyOO+8\n2BWJFF99fT319fWJfV7Rr/Iys1FAnbuPzjy/BNji7lc32eZnQL27z8w8bwAOd/d3m33WFcB6d7+u\n2eu6ykty8vrrYWLJY46Ba6/VxJJS2dJ4ldc8YICZ7WlmHYGTgIeabfMQcDp8EkBr3P1dM9vZzLpk\nXu8EHA0sKl7pUm769oXnnoMXXgjLC//5z7ErEkmvogeKu28GzgceBV4B7nP3xWZ2jpmdk9nmN8Dr\nZrYMmAY0HpDYHXjGzBYAc4CH3f2xYv8MUl4+9zl49FHYeWcYNQr++79BDa5I/nRjo0iGe1iw6/LL\nQ8hcdRUcfnjsqkSKR3fKZ6FAkdb4+OMwa3FdHfTrB5MmwQjd8SQVII3nUERKWlUVnHYaNDTA3/1d\nuCJs7NgwL5iItEyBItKCDh3g7LPDuipHHAFHHw3jx8Orr8auTKQ0KVBEdqC6Gi68EJYtg6FD4ZBD\n4Mwz4a23YlcmUloUKCI56twZfvSj0LH06AEHHADf/W6YdFJEFCgieaupCSfqFy8Oh8X23TesEPnB\nB7ErE4lLgSJSoO7d4frrYdEiWLcO9t47XBn24YexKxOJQ4Ei0kq9eoWFu+bOhTfeCFPkX3ON7rqX\nyqNAEUlI375hHfunn4bnnw/BcvPN8NFHsSsTKQ4FikjC9tkH7r8/TI3/29/CwIFwxx2weXPsykTa\nlu6UF2ljzz0Hl14a1ra/8ko46STNaiylSVOvZKFAkVLjDk8+GYJlzRo46qgwEeXIkWF6F9MapFIC\nFChZKFCkVLnD7Nmha5k9G+bMCSfvR44Mj1GjwrxhNTWxK5VKpEDJQoEiabJyZQiWOXNCyLzwQrhy\nrLGDGTky3KHfoUPsSqXcKVCyUKBImm3eDK+8srWDmT07TPMybNjWLmbkSOjdW4fKJFkKlCwUKFJu\n1q6FefO2DZn27bftYg48MEwPI1IoBUoWChQpd+7w5ptbw2XOnDC9fv/+W0Nm2DAYPBh22il2tZIW\nCpQsFChSiT76CF56aev5mAUL4LXXQsjst9+2j912i12tlCIFShYKFJFg40b43/8NQdP0sdNOnw6Z\nvffWif9Kp0DJQoEi0jJ3WL780yHz9tswaBDsv/+2QbPLLrErlmJRoGShQBHJ3/r18PLL24bMwoUh\nUJp3M/36haWSpbwoULJQoIgkY8uWMIPySy+FczKNQbNqFQwZEk78H3ggDB8e1oXRIbN0U6BkoUAR\naVtr1oTu5cUXw42YL7wQ7pXZd9+tAaOQSR8FShYKFJHiW78e5s/fGjAKmfRRoGShQBEpDQqZdFGg\nZKFAESldCpnSpUDJQoEiki7bC5lhw8LkmEOGhEfXrrGrLV8KlCwUKCLpt359uLJswYJwOXPjo7p6\n24AZOjRMMaN5zFpPgZKFAkWkPLmHGzAbw2XRovBnQwPsvvu2QTNkSLj7v2PH2FWnhwIlCwWKSGXZ\nvDnMW9a0k1m0KBw269dvayfTGDR77aVlmLNRoGShQBERCHOZNTRs7WQag+ZPfwqHyRoDZtCgsKhZ\nz56w666VGzYKlCwUKCKyPWvXhkkzGwPm1VfDypkrV4ZzN7vvHsJle4+amvJb4EyBkoUCRUQKtXEj\nvPPO1oBp6fHXv+44dHr2hC5dYv9EuVOgZKFAEZG2tn791uBpKYBWrAiHz7YXOD16hD933jn2T6RA\nyUqBIiKlwB3WrdsaLtvrfKqrd9zt9OjRtitwKlCyUKCISJq4w+rVW8OlpeD5v/8Lh9C2Fzr77Vd4\n6ChQslCgiEg52rIFPvgge9g0htCsWdC7d2Gfr0DJQoEiIpK/1gZKhV5tLSIiSVOgiIhIIhQoIiKS\nCAWKiIgkQoEiIiKJUKCIiEgiFCgiIpIIBYqIiCQiSqCY2WgzazCzpWY2sYVtbsq8/5KZDctnXxER\nKb6iB4qZVQG3AKOBwcB4M9un2TZjgP7uPgA4G7gt130lefX19bFLKCsaz+RoLEtLjA5lBLDM3d90\n903ATGBss22OB6YDuPscoMbMds9xX0mY/tImS+OZHI1laYkRKL2A5U2ev515LZdteuawb1G05hc5\nn313tG1L7+fzevPXYvwlLcZ4FjqW23svl7HTeOb2XiX/buaybRr+rscIlFxnbSzpxTX1S5asNP4P\nMNvrCpTc3leg5L9tGv6uF322YTMbBdS5++jM80uALe5+dZNtfgbUu/vMzPMG4HBgrx3tm3ldUw2L\niBSgNbMNt0+ykBzNAwaY2Z7ASuAkYHyzbR4CzgdmZgJojbu/a2Yf5LBvqwZEREQKU/RAcffNZnY+\n8ChQBdzh7ovN7JzM+9Pc/TdmNsbMlgEbgAnb27fYP4OIiHxaWS6wJSIixac75UVEJBEKFBERSURF\nBIqZ7WVmPzez+2PXUg7MbKyZ3W5mM83sqNj1pJmZDTKz28zsP83szNj1lAMz62Rmz5vZMbFrSTsz\nqzWzZzK/o4fvaPuKCBR3f8Pdvx27jnLh7g+6+9nAdwhX2kmB3L3B3c8FxgFfiV1PmbgYuC92EWVi\nC7AO2IlwI/l2pTZQzOxOM3vXzBY1e12TRxagwPG8jDC3mjSR71ia2XHAI4SphKSZfMYz0zG/Arwf\no9Y0yPP38xl3HwP8ELhyhx/u7ql8AIcCw4BFTV6rApYBewIdgAXAPk3evz923aX6yGc8CbMYXA0c\nGbvuUnwU8ruZ2ebB2LWX4iPP381JwFTCrQW/JnMlqx6FjWeT9zvm8v/PGDc2JsLdn8nc4NjUJ5NH\nApjZTGCsmb0LTAb2N7OJ3uzOeslvPIEvA0cCnzGz/u4+rYillrw8fze7AycC1cBTRSwzNfIZT3e/\nLPP8W8D7nvm/oWyV5+/nIMKh2Brg5h19dmoDpQXZJpUc6e5/Ihzvl/y0NJ7fJYdfLtlGS2P5NPB0\nnJJSLet4Nj5x9+lFryjdWvr9/Cnwq1w/JLXnUFqgf40kS+OZHI1lsjSeyUpkPMstUFYAfZo870MO\nVyZIizSeydFYJkvjmaxExrPcAuWTiSfNrCPhktaHIteUZhrP5Ggsk6XxTFYi45naQDGze4HngIFm\nttzMJrj7ZsIsxY8SLh28zzV5ZE40nsnRWCZL45msthxPTQ4pIiKJSG2HIiIipUWBIiIiiVCgiIhI\nIhQoIiKSCAWKiIgkQoEiIiKJUKCIiEgiFCgiIpIIBYqIiCRCgSJSBGb2fTNbb2Y9Ms8PMbMXzOzU\n2LWJJEVTr4gUgZkdDnwVeNfdp2Ze+6a7/2fcykSSow5FpDi6AzcC4wHMrAvwYdSKRBJWbis2ipQq\nc/d3zGxtZlnV3YAXYhclkiR1KCLFNQM4BdjN3d+PXYxIkhQoIm3MzHYHVmaePgCcCFi8ikTahgJF\npO0dBLwI4O7rgJeBblErEmkDOoci0obM7EtAHbAT8MvMy/8BfBCrJpG2osuGRUQkETrkJSIiiVCg\niIhIIhQoIiKSCAWKiIgkQoEiIiKJUKCIiEgiFCgiIpIIBYqIiCTi/wFF29HtAvI9RgAAAABJRU5E\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18b1b9588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pyplot.semilogx(N, twin_prime_fraction)\n",
    "pyplot.xlabel(r\"$N$\")\n",
    "pyplot.ylabel(r\"$\\pi_N / N$\")\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For those that have checked Wikipedia, you'll see [Brun's theorem](https://en.wikipedia.org/wiki/Twin_prime#Brun.27s_theorem) which suggests a specific scaling, that $\\pi_N$ is bounded by $C N / \\log(N)^2$. Checking this numerically on this data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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//3l2qw2XkkTaAQa8BNS5+2XJhJR+ixeHLU779IkdiYhk2YABYarx/PnhQ+lj\nj8WOqHClJJG57v5X4EXgbDO7JaGYUu/558OezqutFjsSEcm6rl3Dtr5XXRW6uk49FRYtih1V/kpJ\nIuea2Z7u/oK7X+zuJyYWVcqp3ImIJO073wmtks8+g+22g4ceih1RfkrZT+SN3GysNkfTe0WkHDp3\nDosShw2Dk0+GH/84VAlPM+0nUgS1RESknA48MLRK2reHbbeFceNiR9Q87SdSIPdQAfS552Cjjcp+\nORFp4x55JLRK+vaFq68OO2UmKQ1lT9qUt94KFTpV7kREKmHffcOH1i5dQqvk7rtjR/RVSiIFWjEe\nYkXnbRGRwqyxRmiFjBoVVroffTS8807sqAIlkQJpkaGIxLLXXqEGV01N+Dv0t7+FLvaYlEQKpHIn\nIhLT6qvD0KGhDtell8bfMz5qEjGzm81svpk918IxtWY21cxmmFl9BcNrkloiIpIGu+0WqgGfcUbc\nOKLOzjKzvYGPgdvcfbsmXl8XmAAc6O6vm1lXd3+vmXOVfXbWkiWw7rrw4YdarS4i1SHTs7Pc/Qlg\nQQuHHAuMcffXc8c3mUAq5fnnYYstlEBERFZI+5hIT6CLmT1qZpPN7PiYwWiRoYjIV7WPHUArOgA7\nAfsBnYCJZvYvd5/b1MF1dXX//bm2tpba2tpEg1G5ExHJuvr6eurr6xM7X/QV62ZWA4xtZkxkELC6\nu9flHg8Hxrv76CaOLfuYyP77wznnQP/+Zb2MiEjFZHpMJA/3AX3NrJ2ZdQJ2B56PEYi7WiIiIo1F\n7c4ys5FAP6Crmc0DhhC6sHD3Ye4+28zGA9OB5cBN7h4libz9dviuelkiIl+KmkTc/Zg8jvk98PsK\nhNMilTsREVlZ2ruzUkMzs0REVqYkkieNh4iIrExJJE8qdyIisrLoU3yTUs4pvivKnSxYAB07luUS\nIiJRVPsU31SYNQs231wJRESkMSWRPGhQXUSkaUoiedCguohI05RE8qCWiIhI05REWqFyJyIizVMS\nacX8+bB8OWy8cexIRETSR0mkFSp3IiLSPCWRVmg8RESkeUoirdB4iIhI85REWqFyJyIizVPZkxao\n3ImIVDuVPSmj2bNhs82UQEREmqMk0oJp0zSoLiLSEiWRFmg8RESkZUoiLdD0XhGRlimJtEDTe0VE\nWqYk0oz58+GLL2CTTWJHIiKSXkoizVC5ExGR1imJNEPjISIirYueRMzsZjObb2bPNfN6rZktNLOp\nua+LKhEa+VjfAAAGbElEQVSXxkNERFoXPYkAtwD9WznmMXffMfd1aSWC0vReEZHWRU8i7v4EsKCV\nwyo6MvH55zBnDmyzTSWvKiKSPdGTSB4c2NPMppnZg2a2dbkvOHs21NTA6quX+0oiItnWPnYAeZgC\ndHf3T83sIOBeoFdTB9bV1f3359raWmpra4u6oMqdiEi1qq+vp76+PrHzpaKKr5nVAGPdfbs8jn0F\n2NndP2j0fGJVfM89Fzp3hsGDEzmdiEhqVX0VXzPrZhZWa5jZboTE90Erv1YStURERPITvTvLzEYC\n/YCuZjYPGAJ0AHD3YcAA4Kdm9gXwKfD9csekmVkiIvlJRXdWEpLqzpo/H/r0gfff12p1Eal+Vd+d\nVWkrWiFKICIirVMSaUTlTkRE8qck0ojKnYiI5E9JpBG1RERE8qeB9QY+/xzWWScMqnfqlFBgIiIp\npoH1BL3wAvTooQQiIpIvJZEGtMhQRKQwSiINaJGhiEhhlEQaUEtERKQwSiINqCUiIlIYJZGcd96B\nxYuhe/fYkYiIZIeSSI7KnYiIFE5JJEeLDEVECqckkqNyJyIihdOK9ZwZM2D99aFbtwSDEhFJuVJX\nrCuJiIi0YSp7IiIi0SiJiIhI0ZRERESkaEoiIiJSNCUREREpmpKIiIgULXoSMbObzWy+mT3XynG7\nmtkXZnZkpWITEZGWRU8iwC1A/5YOMLN2wFBgPKDqVmVWX18fO4SqovuZLN3PdImeRNz9CWBBK4ed\nDowG3i1/RKL/pMnS/UyW7me6RE8irTGzTYDDgOtzT0Vbll7Kmzff323tuJZeb+q1Up4rt2KvWcjv\nxbifWbqXhf5usfezkOfbyv1M+r3Z1POV+L+e+iQCXA2cn6tpYkTszsriG0tJpOVjlUSURJqSxf/r\nTT1fif/rqaidZWY1wFh3366J117my8TRFfgU+LG739/ouPj/EBGRDCqldlb7JAMpB3fffMXPZnYL\nIdnc38RxGnAXEamw6EnEzEYC/YCuZjYPGAJ0AHD3YTFjExGRlqWiO0tERLIpCwPrIiKSUkoiIiJS\ntKpMIma2mZkNN7NRsWOpBmZ2mJndaGZ/M7MDYseTdWa2lZldb2Z3mdlJsePJOjNbw8z+bWaHxI4l\n68ys1syeyL0/++XzO1WZRNz9FXc/OXYc1cLd73P3nwD/AxwdO56sc/fZ7v5T4PvAgbHjqQLnAXfG\nDqJKLAcWAasBr+fzC5lJIs0VajSz/mY228zmmtmgWPFlTZH38yLgz5WLMjsKvZ9m9m1gHPC3Ssea\ndoXcy1zL+HlUEqlZBb43n3D3g4HzgV/ldQF3z8QXsDewI/Bcg+faAS8CNYRpwc8CfRq8Pip23Gn9\nKuR+EhZ7DgX2ix13Wr+KeX/mjrkvduxp+yrwvXkp8AfgH8C95Gac6qu4+9ng9VXz/fsZfZ1Ivtz9\nidzK9oZ2A15091cBzOxvwGFmNh+4HNjBzAa5+9BKxpoFhdxPYH9gP2BtM9vStX5nJQW+PzcAjgQ6\nAo9WMMxMKOReuvtFucc/At713F9A+VKB782tCF2s6wLX5HP+zCSRZmwCzGvw+HVgd3f/gNB/L4Vp\n7n6eTp5vKPmK5u7nY8BjcULKrCbv5YoH7v6XikeUbc29N38D3FPIiTIzJtIMfepIlu5nsnQ/k6N7\nmazE7mfWk8gbQPcGj7uT54wCaZLuZ7J0P5Oje5msxO5n1pPIZKCnmdWY2aqE6acrFWeUvOl+Jkv3\nMzm6l8lK7H5mJonkCjU+BfQys3lmdqK7fwGcRpiZ8Txwp7vPihlnVuh+Jkv3Mzm6l8kq9/1UAUYR\nESlaZloiIiKSPkoiIiJSNCUREREpmpKIiIgUTUlERESKpiQiIiJFUxIREZGiKYmIiEjRlERERKRo\nSiIiZWJmZ5vZx2a2Ue7xXmb2jJkdFzs2kaSo7IlImZhZP+AgYL67/yH33FHuflfcyESSo5aISPls\nAPwROAbAzNYCPooakUjCsr6zoUiambu/ZWYLc9uOdgOeiR2USJLUEhEpvzuAHwDd3P3d2MGIJElJ\nRKQMzGxD4M3cwzHAkYDFi0ikPJRERMpjV2AKgLsvAmYA60eNSKQMNCYikjAz2xeoA1YDRueevh14\nP1ZMIuWiKb4iIlI0dWeJiEjRlERERKRoSiIiIlI0JRERESmakoiIiBRNSURERIqmJCIiIkVTEhER\nkaL9P9go3QsBZ5TCAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x18b1b34e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pyplot.semilogx(N, twin_prime_fraction * numpy.log(N)**2)\n",
    "pyplot.xlabel(r\"$N$\")\n",
    "pyplot.ylabel(r\"$\\pi_N \\times \\log(N)^2 / N$\")\n",
    "pyplot.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A basis for the polynomials"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the section on classes we defined a `Monomial` class to represent a polynomial with leading coefficient $1$. As the $N+1$ monomials $1, x, x^2, \\dots, x^N$ form a basis for the vector space of polynomials of order $N$, $\\mathbb{P}^N$, we can use the `Monomial` class to return this basis.\n",
    "\n",
    "## Exercise 1\n",
    "\n",
    "Define a generator that will iterate through this basis of $\\mathbb{P}^N$ and test it on $\\mathbb{P}^3$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again we first take the definition of the crucial class from the notes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Polynomial(object):\n",
    "    \"\"\"Representing a polynomial.\"\"\"\n",
    "    explanation = \"I am a polynomial\"\n",
    "    \n",
    "    def __init__(self, roots, leading_term):\n",
    "        self.roots = roots\n",
    "        self.leading_term = leading_term\n",
    "        self.order = len(roots)\n",
    "        \n",
    "    def __repr__(self):\n",
    "        string = str(self.leading_term)\n",
    "        for root in self.roots:\n",
    "            if root == 0:\n",
    "                string = string + \"x\"\n",
    "            elif root > 0:\n",
    "                string = string + \"(x - {})\".format(root)\n",
    "            else:\n",
    "                string = string + \"(x + {})\".format(-root)\n",
    "        return string\n",
    "    \n",
    "    def __mul__(self, other):\n",
    "        roots = self.roots + other.roots\n",
    "        leading_term = self.leading_term * other.leading_term\n",
    "        return Polynomial(roots, leading_term)\n",
    "    \n",
    "    def explain_to(self, caller):\n",
    "        print(\"Hello, {}. {}.\".format(caller,self.explanation))\n",
    "        print(\"My roots are {}.\".format(self.roots))\n",
    "        return None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Monomial(Polynomial):\n",
    "    \"\"\"Representing a monomial, which is a polynomial with leading term 1.\"\"\"\n",
    "    explanation = \"I am a monomial\"\n",
    "    \n",
    "    def __init__(self, roots):\n",
    "        Polynomial.__init__(self, roots, 1)\n",
    "        \n",
    "    def __repr__(self):\n",
    "        string = \"\"\n",
    "        for root in self.roots:\n",
    "            if root == 0:\n",
    "                string = string + \"x\"\n",
    "            elif root > 0:\n",
    "                string = string + \"(x - {})\".format(root)\n",
    "            else:\n",
    "                string = string + \"(x + {})\".format(-root)\n",
    "        return string"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can define the first basis:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def basis_pN(N):\n",
    "    \"\"\"\n",
    "    A generator for the simplest basis of P^N.\n",
    "    \"\"\"\n",
    "    \n",
    "    for n in range(N+1):\n",
    "        yield Monomial(n*[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then test it on $\\mathbb{P}^N$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "x\n",
      "xx\n",
      "xxx\n"
     ]
    }
   ],
   "source": [
    "for poly in basis_pN(3):\n",
    "    print(poly)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This looks horrible, but is correct. To really make this look good, we need to improve the output. If we use"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "class Monomial(Polynomial):\n",
    "    \"\"\"Representing a monomial, which is a polynomial with leading term 1.\"\"\"\n",
    "    explanation = \"I am a monomial\"\n",
    "    \n",
    "    def __init__(self, roots):\n",
    "        Polynomial.__init__(self, roots, 1)\n",
    "        \n",
    "    def __repr__(self):\n",
    "        if len(self.roots):\n",
    "            string = \"\"\n",
    "            n_zero_roots = len(self.roots) - numpy.count_nonzero(self.roots)\n",
    "            if n_zero_roots == 1:\n",
    "                string = \"x\"\n",
    "            elif n_zero_roots > 1:\n",
    "                string = \"x^{}\".format(n_zero_roots)\n",
    "        else: # Monomial degree 0.\n",
    "            string = \"1\"\n",
    "        for root in self.roots:\n",
    "            if root > 0:\n",
    "                string = string + \"(x - {})\".format(root)\n",
    "            elif root < 0:\n",
    "                string = string + \"(x + {})\".format(-root)\n",
    "        return string"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "then we can deal with the uglier cases, and re-running the test we get"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1\n",
      "x\n",
      "x^2\n",
      "x^3\n"
     ]
    }
   ],
   "source": [
    "for poly in basis_pN(3):\n",
    "    print(poly)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An even better solution would be to use the `numpy.unique` function as in [this stackoverflow answer](http://stackoverflow.com/questions/10741346/numpy-most-efficient-frequency-counts-for-unique-values-in-an-array) (the second one!) to get the frequency of all the roots."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "An alternative basis is given by the monomials\n",
    "\n",
    "\\begin{align}\n",
    "  p_0(x) &= 1, \\\\ p_1(x) &= 1-x, \\\\ p_2(x) &= (1-x)(2-x), \\\\ \\dots & \\quad \\dots, \\\\ p_N(x) &= \\prod_{n=1}^N (n-x).\n",
    "\\end{align}\n",
    "\n",
    "Define a generator that will iterate through this basis of $\\mathbb{P}^N$ and test it on $\\mathbb{P}^4$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def basis_pN_variant(N):\n",
    "    \"\"\"\n",
    "    A generator for the 'sum' basis of P^N.\n",
    "    \"\"\"\n",
    "    \n",
    "    for n in range(N+1):\n",
    "        yield Monomial(range(n+1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x\n",
      "x(x - 1)\n",
      "x(x - 1)(x - 2)\n",
      "x(x - 1)(x - 2)(x - 3)\n",
      "x(x - 1)(x - 2)(x - 3)(x - 4)\n"
     ]
    }
   ],
   "source": [
    "for poly in basis_pN_variant(4):\n",
    "    print(poly)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I am too lazy to work back through the definitions and flip all the signs; it should be clear how to do this!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Use these generators to write another generator that produces a basis of $\\mathbb{P^3} \\times \\mathbb{P^4}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Hopefully by now you'll be aware of how useful `itertools` is!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from itertools import product"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def basis_product():\n",
    "    \"\"\"\n",
    "    Basis of the product space\n",
    "    \"\"\"\n",
    "    yield from product(basis_pN(3), basis_pN_variant(4))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Basis element is (1) X (x).\n",
      "Basis element is (1) X (x(x - 1)).\n",
      "Basis element is (1) X (x(x - 1)(x - 2)).\n",
      "Basis element is (1) X (x(x - 1)(x - 2)(x - 3)).\n",
      "Basis element is (1) X (x(x - 1)(x - 2)(x - 3)(x - 4)).\n",
      "Basis element is (x) X (x).\n",
      "Basis element is (x) X (x(x - 1)).\n",
      "Basis element is (x) X (x(x - 1)(x - 2)).\n",
      "Basis element is (x) X (x(x - 1)(x - 2)(x - 3)).\n",
      "Basis element is (x) X (x(x - 1)(x - 2)(x - 3)(x - 4)).\n",
      "Basis element is (x^2) X (x).\n",
      "Basis element is (x^2) X (x(x - 1)).\n",
      "Basis element is (x^2) X (x(x - 1)(x - 2)).\n",
      "Basis element is (x^2) X (x(x - 1)(x - 2)(x - 3)).\n",
      "Basis element is (x^2) X (x(x - 1)(x - 2)(x - 3)(x - 4)).\n",
      "Basis element is (x^3) X (x).\n",
      "Basis element is (x^3) X (x(x - 1)).\n",
      "Basis element is (x^3) X (x(x - 1)(x - 2)).\n",
      "Basis element is (x^3) X (x(x - 1)(x - 2)(x - 3)).\n",
      "Basis element is (x^3) X (x(x - 1)(x - 2)(x - 3)(x - 4)).\n"
     ]
    }
   ],
   "source": [
    "for p1, p2 in basis_product():\n",
    "    print(\"Basis element is ({}) X ({}).\".format(p1, p2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I've cheated here as I haven't introduced the `yield from` syntax (which returns an iterator from a generator). We could write this out instead as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def basis_product_long_form():\n",
    "    \"\"\"\n",
    "    Basis of the product space (without using yield_from)\n",
    "    \"\"\"\n",
    "    prod = product(basis_pN(3), basis_pN_variant(4))\n",
    "    yield next(prod)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Basis element is (1) X (x).\n",
      "Basis element is (1) X (x(x - 1)).\n",
      "Basis element is (1) X (x(x - 1)(x - 2)).\n",
      "Basis element is (1) X (x(x - 1)(x - 2)(x - 3)).\n",
      "Basis element is (1) X (x(x - 1)(x - 2)(x - 3)(x - 4)).\n",
      "Basis element is (x) X (x).\n",
      "Basis element is (x) X (x(x - 1)).\n",
      "Basis element is (x) X (x(x - 1)(x - 2)).\n",
      "Basis element is (x) X (x(x - 1)(x - 2)(x - 3)).\n",
      "Basis element is (x) X (x(x - 1)(x - 2)(x - 3)(x - 4)).\n",
      "Basis element is (x^2) X (x).\n",
      "Basis element is (x^2) X (x(x - 1)).\n",
      "Basis element is (x^2) X (x(x - 1)(x - 2)).\n",
      "Basis element is (x^2) X (x(x - 1)(x - 2)(x - 3)).\n",
      "Basis element is (x^2) X (x(x - 1)(x - 2)(x - 3)(x - 4)).\n",
      "Basis element is (x^3) X (x).\n",
      "Basis element is (x^3) X (x(x - 1)).\n",
      "Basis element is (x^3) X (x(x - 1)(x - 2)).\n",
      "Basis element is (x^3) X (x(x - 1)(x - 2)(x - 3)).\n",
      "Basis element is (x^3) X (x(x - 1)(x - 2)(x - 3)(x - 4)).\n"
     ]
    }
   ],
   "source": [
    "for p1, p2 in basis_product():\n",
    "    print(\"Basis element is ({}) X ({}).\".format(p1, p2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Anscombe's quartet"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Four separate datasets are given:\n",
    "\n",
    "| x    | y     | x    | y    | x    | y     | x    | y     |\n",
    "|------|-------|------|------|------|-------|------|-------|\n",
    "| 10.0 | 8.04  | 10.0 | 9.14 | 10.0 | 7.46  | 8.0  | 6.58  |\n",
    "| 8.0  | 6.95  | 8.0  | 8.14 | 8.0  | 6.77  | 8.0  | 5.76  |\n",
    "| 13.0 | 7.58  | 13.0 | 8.74 | 13.0 | 12.74 | 8.0  | 7.71  |\n",
    "| 9.0  | 8.81  | 9.0  | 8.77 | 9.0  | 7.11  | 8.0  | 8.84  |\n",
    "| 11.0 | 8.33  | 11.0 | 9.26 | 11.0 | 7.81  | 8.0  | 8.47  |\n",
    "| 14.0 | 9.96  | 14.0 | 8.10 | 14.0 | 8.84  | 8.0  | 7.04  |\n",
    "| 6.0  | 7.24  | 6.0  | 6.13 | 6.0  | 6.08  | 8.0  | 5.25  |\n",
    "| 4.0  | 4.26  | 4.0  | 3.10 | 4.0  | 5.39  | 19.0 | 12.50 |\n",
    "| 12.0 | 10.84 | 12.0 | 9.13 | 12.0 | 8.15  | 8.0  | 5.56  |\n",
    "| 7.0  | 4.82  | 7.0  | 7.26 | 7.0  | 6.42  | 8.0  | 7.91  |\n",
    "| 5.0  | 5.68  | 5.0  | 4.74 | 5.0  | 5.73  | 8.0  | 6.89  |"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "Using standard `numpy` operations, show that each dataset has the same mean and standard deviation, to two decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "set1_x = numpy.array([10.0, 8.0, 13.0, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0])\n",
    "set1_y = numpy.array([8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68])\n",
    "set2_x = numpy.array([10.0, 8.0, 13.0, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0])\n",
    "set2_y = numpy.array([9.14, 8.14, 8.74, 8.77, 9.26, 8.10, 6.13, 3.10, 9.13, 7.26, 4.74])\n",
    "set3_x = numpy.array([10.0, 8.0, 13.0, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0])\n",
    "set3_y = numpy.array([7.46, 6.77, 12.74, 7.11, 7.81, 8.84, 6.08, 5.39, 8.15, 6.42, 5.73])\n",
    "set4_x = numpy.array([8.0, 8.0, 8.0, 8.0, 8.0, 8.0, 8.0, 19.0, 8.0, 8.0, 8.0])\n",
    "set4_y = numpy.array([6.58, 5.76, 7.71, 8.84, 8.47, 7.04, 5.25, 12.50, 5.56, 7.91, 6.89])\n",
    "\n",
    "data_x = set1_x, set2_x, set3_x, set4_x\n",
    "data_y = set1_y, set2_y, set3_y, set4_y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Results for x:\n",
      "Mean: 9.00. Variance 10.00. Standard deviation 3.16.\n",
      "Mean: 9.00. Variance 10.00. Standard deviation 3.16.\n",
      "Mean: 9.00. Variance 10.00. Standard deviation 3.16.\n",
      "Mean: 9.00. Variance 10.00. Standard deviation 3.16.\n",
      "Results for y:\n",
      "Mean: 7.50. Variance 3.75. Standard deviation 1.94.\n",
      "Mean: 7.50. Variance 3.75. Standard deviation 1.94.\n",
      "Mean: 7.50. Variance 3.75. Standard deviation 1.94.\n",
      "Mean: 7.50. Variance 3.75. Standard deviation 1.94.\n"
     ]
    }
   ],
   "source": [
    "print(\"Results for x:\")\n",
    "for x in data_x:\n",
    "    print(\"Mean: {:.2f}. Variance {:.2f}. Standard deviation {:.2f}.\".format(numpy.mean(x),\n",
    "                                                                            numpy.var(x),\n",
    "                                                                            numpy.std(x)))\n",
    "print(\"Results for y:\")\n",
    "for data in data_y:\n",
    "    print(\"Mean: {:.2f}. Variance {:.2f}. Standard deviation {:.2f}.\".format(numpy.mean(data),\n",
    "                                                                             numpy.var(data),\n",
    "                                                                             numpy.std(data)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "Using the standard `scipy` function, compute the linear regression of each data set and show that the slope and correlation coefficient match to two decimal places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Slope: 0.500. Correlation: 0.816.\n",
      "Slope: 0.500. Correlation: 0.816.\n",
      "Slope: 0.500. Correlation: 0.816.\n",
      "Slope: 0.500. Correlation: 0.817.\n"
     ]
    }
   ],
   "source": [
    "from scipy import stats\n",
    "\n",
    "for x, y in zip(data_x, data_y):\n",
    "    slope, intercept, r_value, p_value, std_err = stats.linregress(x, y)\n",
    "    print(\"Slope: {:.2f}. Correlation: {:.2f}.\".format(slope, r_value))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Plot each dataset. Add the best fit line. Then look at the description of [Anscombe's quartet](https://en.wikipedia.org/wiki/Anscombe%27s_quartet), and consider in what order the operations in this exercise *should* have been done."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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w4cBOhyQpw+zbkjQk1RsQQwgTgJ8CP48xXjrMc7xKS1LNqrcbEEfr2/ZsSbWu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\nA00RhL1te9xiAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1101d2940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib import pyplot\n",
    "\n",
    "fit_x = numpy.linspace(2.0, 20.0)\n",
    "fig = pyplot.figure(figsize=(12,6))\n",
    "for i in range(4):\n",
    "    slope, intercept, r_value, p_value, std_err = stats.linregress(data_x[i], data_y[i])\n",
    "    ax = fig.add_subplot(2,2,i+1)\n",
    "    ax.scatter(data_x[i], data_y[i])\n",
    "    ax.plot(fit_x, intercept + slope*fit_x)\n",
    "    ax.set_xlim(2.0, 20.0)\n",
    "    ax.set_xlabel(r'$x$')\n",
    "    ax.set_ylabel(r'$y$')\n",
    "pyplot.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}