Suico_JohnSamuel_module[1].ipynb

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   "source": [
    "# **AP155 Lab Assignment** \n",
    "## Module 1: Python Programming \n",
    "\n",
    "_Instructions_: Answer each problem as completely as you can. Discuss **all** your answers as clearly and concisely as possible. \n",
    "\n",
    "_Scoring Criteria_: 50% - *correctness of code*; 50% - *discussion of the code and results*. Maximum score is **100 points** (Problems 1 to 5 are worth 15, 20, 20, 20, and 25 points, respectively).\n",
    "\n",
    "_Due Date_: **5PM of Sept 24 (Thu)**\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "PeHZ9kP8JaVR"
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   "source": [
    "### Student Information \n",
    "\n",
    "_Full Name (Last Name, First Name)_: Suico, John Samuel \\\n",
    "_Student No._: 18-05897 \\\n",
    "_Section_: WFU-2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "Hc183co-J4eI"
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   "source": [
    "### Submission Information\n",
    "\n",
    "_Date and Time Submitted (MM/DD/YYYY HH:MM AM/PM; most recent upload)_: 9/23/2020 2:30 PM\n",
    "\n",
    "**HONOR PLEDGE** I affirm that I have upheld the highest principles of honesty and integrity in my academic work and that this lab assignment is my own work.\n",
    "\n",
    "**Sign here with your full name:** JOHN SAMUEL SUICO"
   ]
  },
  {
   "cell_type": "markdown",
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    "colab_type": "text",
    "id": "8FQfscyfgI-l"
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   "source": [
    "### Grading Information (c/o Lab Instructor)\n",
    "\n",
    "TOTAL SCORE: **XX**/100\n",
    "\n",
    "Score breakdown:\n",
    "* Problem 1 - XX/15\n",
    "* Problem 2 - XX/20\n",
    "* Problem 3 - XX/20\n",
    "* Problem 4 - XX/20\n",
    "* Problem 5 - XX/25\n",
    "\n",
    "_Date and Time Scored (MM/DD/YYYY HH:MM AM/PM):_ "
   ]
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    "id": "-Vy-oAzUWgHp"
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   "source": [
    "### PROBLEM 1 (15 pts.; 50/50 for code & discussion) \n",
    "**Fibonacci sequence**\n",
    "\n",
    "1. Write a function that returns the nth Fibonacci number. *(10 pts.)*\n",
    "\n",
    "2. Print the nth Fibonacci number, for $n = 1, 10, 10^3, 10^6, 10^9$. *(5 pts.)*"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Here I've written a function called `fibo` which takes an argument $n$ and returns the $n$th number from the Fibonacci Sequence. <br><br> \n",
    "\n",
    "    The Fibonacci sequence is defined such that the $n$th number is the sum of the preceding two numbers $n-1$ and $n-2$, starting from $0$ and $1$. I've defined two separate variables to function as the starting points for the sequence: `a_0` with the value $0$ and `a_1` with the value $1$. <br><br> \n",
    "\n",
    "    To continue the sequence, I update the value of `a_1` while <b>simultaneously</b> updating the value of `a_0` to the <b>previous</b> value of `a_1` (I had previously solved the problem by creating a dummy variable to temporarily 'hold on' to the <b>previous</b> value of `a_1`; this approach is simplified by Python). After looping $n$ times, the value of `a_0`, which corresponds to the value of the $n$th number in the Fibonacci sequence, is returned."
   ]
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   "source": [
    "def fibo(n):\n",
    "    a_0,a_1 = 0,1\n",
    "    for i in range(n):\n",
    "        a_1,a_0 = (a_1 + a_0),a_1\n",
    "    return a_0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Here I've written two functions called `ordinalIndicator` which returns the appropriate ordinal indicator for a number, and `output` which prints the $n$th number of the Fibonacci Sequence. <br><br>\n",
    "    \n",
    "    The function `ordinalIndicator` takes an (assumingly) int or float type argument $n$ and converts the said number into a string. I make use of Python's indexing of strings to pinpoint the number in the argument's <b>ones place</b> and subsequently return the appropriate ordinal indicator. I only wrote this argument to improve the readability of function `output`. Note that this function only works for integers.<br><br>\n",
    "    \n",
    "    The function `output` takes an argument $n$ and prints the string \"This is the $n$<i>(ordinal indicator)<sub>[1]</sub></i> number in the Fibonacci Sequence: <i>($n$th number in the Fibonacci Sequence)<sub>[2]</sub>\"</i>\n",
    "    \n",
    "    <sub>[1] The ordinal indicator is determined by passing the argument $n$ to the function `ordinalIndicator`. </sub> <br>\n",
    "    <sub>[2] The $n$th number in the Fibonacci Sequence is determined by passing the argument $n$ to the function `fibo`.</sub>\n",
    "    \n",
    "    I subsequently call the `output` function 4 times with the following arguments, respectively: $1$, $10$, $1000$, $1000000$."
   ]
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     "text": [
      "This is the 1st number in the Fibonacci Sequence: 1\n",
      "This is the 10th number in the Fibonacci Sequence: 55\n",
      "This is the 1000th number in the Fibonacci Sequence: 43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875\n",
      "This is the 1000000th number in the Fibonacci Sequence: 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     ]
    }
   ],
   "source": [
    "def ordinalIndicator(n):\n",
    "    convert = str(n)\n",
    "    onesPlace = int(convert[len(convert)-1])\n",
    "    \n",
    "    if onesPlace == 0 or onesPlace>3:\n",
    "        return \"th\"\n",
    "    elif onesPlace == 1:\n",
    "        return \"st\"\n",
    "    elif onesPlace == 2:\n",
    "        return \"nd\"\n",
    "    elif onesPlace == 3:\n",
    "        return \"rd\"\n",
    "    \n",
    "def output(n): \n",
    "    print(\"This is the \",n,ordinalIndicator(n),\" number in the Fibonacci Sequence: \",fibo(n),sep=\"\")\n",
    "\n",
    "output(1)\n",
    "output(10)\n",
    "output(10**3)\n",
    "output(10**6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "LxWuqiCFMsEh"
   },
   "source": [
    "### PROBLEM 2 (20 pts.; 50/50 for code & discussion) \n",
    "**Curve plotting**\n",
    "\n",
    "Make a plot of the so-called deltoid curve, which is defined parametrically by the equations $$x = 2 \\cos\\theta+ \\cos2\\theta, y = 2 \\sin\\theta - \\sin 2\\theta,$$\n",
    "where $0 \\le \\theta < 2\\pi$. \n",
    "\n",
    "1. Take a set of values of $\\theta$ between zero and 2$\\pi$ and calculate $x$\n",
    "and $y$ for each from the equations above. *(10 pts.)*\n",
    "\n",
    "2. Make a plot $y$ as a function of $x$. *(10 pts.)*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Here I've written a class called `deltoidCurve` with the instance variables `theta`, `x`, and `y`. `theta` is initialized using the `linspace` function from the `numpy` module: `theta` is initialized as an array with 60 evenly spaced values on the interval 0 to 2pi. `x` and `y`  are then initialized as arrays using the formulas above in such a way that each index value of `x` and `y` is mapped to an index in `theta`. <br><br> I initialize the variable `createPoints` as an instance of the class `deltoidCurve`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "BygJGx61QCXS"
   },
   "outputs": [],
   "source": [
    "from numpy import cos, sin, pi, linspace\n",
    "\n",
    "class deltoidCurve:\n",
    "    def __init__(self):\n",
    "        self.theta = linspace(0,2*pi,60)\n",
    "        self.x = 2*cos(self.theta) + cos(2*self.theta)\n",
    "        self.y = 2*sin(self.theta) - sin(2*self.theta)\n",
    "\n",
    "createPoints = deltoidCurve()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Here I've called the function `plot` from the `pylab` module, with the arguments `createPoints.x` and `createPoints.y`.\n",
    "    \n",
    "    This creates a graph using the `createPoints.x` array as data points for the x-axis and and the `createPoints.y` array as data points for the y-axis."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x19547b9fec8>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from pylab import plot\n",
    "\n",
    "plot(createPoints.x,createPoints.y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "B65Kd0lNQDhM"
   },
   "source": [
    "### PROBLEM 3 (20 pts.; 50/50 for code & discussion) \n",
    "**3D plotting with matplotlib**\n",
    "\n",
    "To create a 3D plot with matplotlib, you must import the $\\tt mplot3d$ toolkit, in addition to the usual import commands: \n",
    "```\n",
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits import mplot3d\n",
    "```\n",
    "\n",
    "Now, you can create a 3D axis by passing the projection keyword as follows: \n",
    "```\n",
    "fig = plt.figure()\n",
    "ax = plt.axes(projection='3d')\n",
    "```\n",
    "\n",
    "Then, you can add points using the $\\tt scatter3D$ function:\n",
    "```\n",
    "ax.scatter3D(xdata, ydata, zdata, c=zdata, cmap='Greens');\n",
    "```\n",
    "where xdata, ydata, and zdata are arrays containing the x, y, and z values, respectively.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "tEgx1p4hTXdW"
   },
   "source": [
    "Make a plot similar to the one in the figure below (yours does not have to be exactly the same dimensions, but should follow the same shape):\n",
    "\n",
    "![fig_spiral.png](data:image/png;base64,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)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "3eB9x6rsTwlk"
   },
   "source": [
    "The equation for a circular helix with radius $R$ and pitch $2\\pi a$ is\n",
    "\n",
    "$$ x(t) = R \\cos(t),\\ y(t) = R \\sin(t),\\ z(t)=at. $$\n",
    "\n",
    "1. Plot the circular helix as a gray curve. *(10 pts.)*\n",
    "\n",
    "2. Plot individual random points that lie on the curve (these can be exactly on the curve-- easier to implement-- or slightly off the curve, as in the figure). To do this, you can use the numpy function $\\tt np.random.random(N)$ to generate N random points between 0 to 1. *(10 pts.)* \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Here I've written a class called `circularHelix` with the instance variables `R`, `a`, `t`, `x`, `y`, and `z`. <br> `R` and `a` are initialized with arbitrarily chosen values. `t` is initialized using the `linspace` function from the `numpy` module as an array with 100 evenly spaced values from 0 to 4pi. `x`, `y`, and `z` are initialized as arrays using the formulas above in such a way that each index value of `x`, `y`, and `z` is mapped to an index value in `t`. <br><br> \n",
    "\n",
    "    I've also written the functions `drawCurve` and `plotRandomonCurve` under the class `circularHelix`. <br><br>\n",
    "\n",
    "    The function `drawCurve` creates a plot of the circular helix as a gray curve. Calling the function creates an instance of the `figure` and `axes` classes from the module `matplotlib.pyplot`: `fig` and `ax`, respectively. The class instance variables `x`, `y`, and `z`, and the string `gray` are then passed as arguments for the function `plot3D` to plot the circular helix in gray. Also, the `drawCurve` function has a parameter `doMore` with the default argument `False`. If ever `doMore` is `True`, it returns the variable `ax` (which is an instance of the class `axes`).<br><br>\n",
    "\n",
    "    The function `plotRandomonCurve` creates random erratic points that lie on the circular helix plot. The function also has a parameter `doMore` with the default value `True`. When called, the function begins by calling `drawCurve` with the argument `doMore` so that it returns `ax` which is assigned to the variable `ax`<sub>[1]</sub>. The function then initializes the variable `t_rand` as an array of 100 random values from 0 to 4pi with the help of the `random.random()` function from the `numpy` module. It then initialize the arrays `x_rand`, `y_rand`, and `z_rand` using the formulas above<sub>[2]</sub> in such a way that each index value of `x_rand`, `y_rand`, and `z_rand` is mapped to an index value in `t_rand`. Finally, the `scatter3D` function is called using the `axes` class instance `ax`, and the data sets `x_rand`, `y_rand`, and `z_rand` are plotted. <br>\n",
    "    \n",
    "    <sub>[1] the variable names `doMore` and `ax` are the same as in the function `drawCurve` for coherence</sub> <br>\n",
    "    <sub>[2] the formulas are modified to add random values from 0 to 1.5, except for `z_rand`; also note that the class instance variables `R` and `a` are unchanged</sub><br><br>\n",
    "    To plot the circular helix as a gray curve, I instantiate the class `circularHelix` to call the function `drawCurve`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "7Z7Ty041Ri9r"
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from mpl_toolkits import mplot3d\n",
    "%matplotlib inline\n",
    "from numpy import cos,sin,pi,linspace, random\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "class circularHelix:\n",
    "    def __init__(self):\n",
    "        self.R,self.a = 10,3\n",
    "        self.t = linspace(0,4*pi,100)\n",
    "        self.x = self.R*cos(self.t)\n",
    "        self.y = self.R*sin(self.t)\n",
    "        self.z = self.a*self.t\n",
    "    \n",
    "    def drawCurve(self,doMore=False):        \n",
    "        fig = plt.figure()\n",
    "        ax = plt.axes(projection = '3d')\n",
    "        ax.plot3D(self.x,self.y,self.z, 'gray')\n",
    "        if doMore == True:\n",
    "            return ax\n",
    "    \n",
    "    def plotRandomonCurve(self,doMore=True):\n",
    "        ax = self.drawCurve(doMore)\n",
    "        \n",
    "        #generates erratic data points\n",
    "        t_rand = 4*pi * random.random(100)\n",
    "        x_rand = self.R*cos(t_rand) + 1.5*random.random(100)\n",
    "        y_rand = self.R*sin(t_rand) + 1.5*random.random(100)\n",
    "        z_rand = self.a*t_rand\n",
    "        \n",
    "        ax.scatter3D(x_rand,y_rand,z_rand,cmap='Greens')\n",
    "    \n",
    "circularHelix().drawCurve()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. To plot random points on the circular helix, I instantiate the class `circularHelix` to call the function `plotRandomonCurve`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "circularHelix().plotRandomonCurve()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "uq6NtvYBOEMI"
   },
   "source": [
    "### PROBLEM 4 (20 pts.; 50/50 for code & discussion)\n",
    "\n",
    "**Determine underflow and overflow limits**\n",
    "\n",
    "1. Write a program that determines the underflow and overflow limits (within a\n",
    "factor of 2) for Python on your computer. Here is a sample pseudocode:\n",
    "\n",
    "```\n",
    "under = 1.\n",
    "over = 1.\n",
    "begin do N times\n",
    "    under = under / 2.\n",
    "    over = over * 2.\n",
    "    write out: loop number, under, over\n",
    "end do \n",
    "```\n",
    "\n",
    "You may need to increase N if your initial choice does not lead to underflow and overflow. *(10 pts.)*\n",
    "\n",
    "2. Check where under- and overflow occur for double-precision floating-point numbers (floats). Give your answer in decimal. *(5 pts.)*\n",
    "\n",
    "3. Check where under- and overflow occur for double-precision floating-point numbers. *(5 pts.)*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Here I've written a class<sub>[1]</sub> `checkLimit` which, when initialized, checks the overflow  and underflow limits of Python for my computer. The class uses two `while` loops to calculate the underflow and overflow limits, each loop having their own separate iterator variables`uf_counter` and `of_counter`, respectively. The float type variables `uf_limit` and `of_limit` are used to save each iteration of `uf_counter` and `of_counter`, respectively, before updating each iterator variable's values by a factor of 2<sub>[2]</sub>. As such, when `uf_counter` and  `of_counter` hit their respective limits <b>0.0</b> and <b>inf</b>, the values for the underflow and overflow limits are retained. The class also initializes two variables `uf_loop` and `of_loop` which are used to count the number of loops until underflow and overflow occurs, respectively. <br><br> <sub>[1] I wrote a class because I didn't want to make two separate functions to calculate the underflow and overflow limits.</sub><br>\n",
    "<sub>[2] As in the sample code, both iterator variables are initialized with the value `1.0`, with each iteration multiplying and dividing `of_counter` and `uf_counter` by a factor of 2, respectively</sub>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "hMqWOsbIMoUh"
   },
   "outputs": [],
   "source": [
    "class checkLimit:   \n",
    "    def __init__(self):\n",
    "        #iterator variable\n",
    "        self.uf_counter,self.of_counter = 1.0,1.0\n",
    "        #saves limits\n",
    "        self.uf_limit,self.of_limit = 0.0,0.0\n",
    "        #checks for loop number:\n",
    "        self.uf_loop, self.of_loop = 0,0\n",
    "        \n",
    "    \n",
    "        while self.of_counter != float('inf'):\n",
    "            self.of_loop += 1\n",
    "            self.of_limit = self.of_counter\n",
    "            self.of_counter *= 2\n",
    "        \n",
    "        while self.uf_counter != 0:\n",
    "            self.uf_loop += 1\n",
    "            self.uf_limit = self.uf_counter\n",
    "            self.uf_counter /= 2\n",
    "        #to offset the last loop, we deduct 1 from the loop counting variable\n",
    "        self.uf_loop -= 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. and 3. Here, I've used the `print` function while creating an instance of the class `checkLimit` to print both instance variables `of_loop` and `of_limit`. I then do the same for the instance variables `uf_loop` and `uf_limit`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Overflow occurs at loop number 1024 ; Overflow limit is:  8.98846567431158e+307\n",
      "Underflow occrs at loop number 1074 ; Underflow limit is:  5e-324\n"
     ]
    }
   ],
   "source": [
    "print(\"Overflow occurs at loop number\",checkLimit().of_loop,\"; Overflow limit is: \",checkLimit().of_limit)\n",
    "print(\"Underflow occrs at loop number\",checkLimit().uf_loop,\"; Underflow limit is: \",checkLimit().uf_limit)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "colab_type": "text",
    "id": "kBX9EyMLOnXj"
   },
   "source": [
    "### PROBLEM 5 (25 pts.; 50/50 for code & discussion)\n",
    "\n",
    "**Calculating the sum of a simple series**\n",
    "\n",
    "In spite of the power of your trusty computer, calculating the sum of even a simple series may require some thought and care. Consider the two series\n",
    "\n",
    "$$\\begin{equation}\n",
    "S^{\\rm (up)} = \\sum_{n=1}^{N} \\frac{1}{n},\\ S^{\\rm (down)} = \\sum_{n=N}^{1} \\frac{1}{n}.\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "Both series are finite as long as $N$ is finite, and when summed analytically both give the same answer. Nonetheless, because of round-off error, the numerical value of $S^{\\rm (up)}$ will not be precisely that of $S^{\\rm (down)}$.\n",
    "\n",
    "1. Calculate $S^{\\rm (up)}$ and $S^{\\rm (down)}$ as functions of $N$. *(10 pts.)*\n",
    "\n",
    "2. Make a log-log plot of $(S^{\\rm (up)}-S^{\\rm (down)})/(|S^{\\rm (up)}|+|S^{\\rm (down)}|)$ vs. $N$. *(10 pts.)*\n",
    "\n",
    "3. Observe the linear regime on your graph and explain why the downward sum is generally more precise. *(5 pts.)*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. Here, I've written a class called `sumSeries` with the instance variable `N` which is initialized as itself. I've also written multiple functons under the class which I will explain briefly.<br><br>\n",
    "\n",
    "    The function `genericUp` takes an argument $n$ and then returns the sum of the series given above. The calculation is facilitated through a `for` loop with the iterating variable `i` initialized at $1$ with a $+1$ increment. Each iteration adds the value  $\\frac{1}{i}$ to the variable `sumUp`, until `i` equals $n$ (thereby calculating $S^{\\rm (up)}$). The function `genericDown` also returns the sum of the series given above, however, the `for` loop's iterating variable `i` is initialized with the value `n` and instead decrements by `-1`. Each iteration adds the value  $\\frac{1}{i}$ to the variable `sumDown`, until `i` equals $1$ (thereby calculating $S^{\\rm (down)}$). <br><br>\n",
    "    \n",
    "    The functions `specificUp` and `specificDown` also calculate the sum of the series $\\frac{1}{n}$, except that they are specifically used to calculate the sums using the instance variable `N`. The said functions actually only pass the argument `N` to the `genericUp` and `genericDown` functions.<sub>[1]</sub>\n",
    "    \n",
    "    <sub>[1] The `specificUp` and `specificDown` functions are a bit redundant and are only used to answer Problem no. 1</sub> <br><br>\n",
    "    \n",
    "    I first initialize an instance of the `sumSeries` class called `testme`. I then print out the string \"S(up): $S^{\\rm (up)}$\"<sub>[2]</sub> on the first line, and then the string \"S(down): $S^{\\rm (down)}$<sub>[3]</sub>\" on the next line.\n",
    "    \n",
    "    <sub>[2] $S^{\\rm (up)}$ is calculated by calling the class function `specificUp` using the class instance `testme` for $N = 1000$</sub> <br>\n",
    "    <sub>[3] $S^{\\rm (down)}$ is calculated by calling the class function `specificDown` using the class instance `testme` for $N = 1000$</sub>\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "colab": {},
    "colab_type": "code",
    "id": "GABjwlXCR97i"
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "S(up):  7.485470860550343\n",
      "S(down):  7.485470860550341\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "from numpy import abs,linspace\n",
    "\n",
    "class sumSeries:\n",
    "    def __init__(self,N):\n",
    "        self.N = N\n",
    "    \n",
    "    def genericUp(self,n):\n",
    "        sumUp = 0\n",
    "        for i in range(1,n+1):\n",
    "             sumUp += 1/i\n",
    "        return sumUp\n",
    "    \n",
    "    def genericDown(self,n):\n",
    "        sumDown = 0\n",
    "        for i in range(n,0,-1):\n",
    "            sumDown += 1/i\n",
    "        return sumDown\n",
    "    \n",
    "    def specificUp(self):\n",
    "        return self.genericUp(self.N)\n",
    "    \n",
    "    def specificDown(self):\n",
    "        return self.genericDown(self.N)\n",
    "    \n",
    "    def makePlot(self):\n",
    "        y = list()\n",
    "        \n",
    "        #calculating x:\n",
    "        x = linspace(1,self.N,self.N)\n",
    "        \n",
    "        #calculating y:\n",
    "        for i in range(1,self.N+1):\n",
    "            y.append( abs( (self.genericUp(i) - self.genericDown(i)) / ( abs((self.genericUp(i)) + abs(self.genericDown(i))) ) ))\n",
    "        \n",
    "        plt.xscale(\"log\")\n",
    "        plt.yscale(\"log\")             \n",
    "        plt.scatter(x,y)\n",
    " \n",
    "testme = sumSeries(1000)\n",
    "print(\"S(up): \",testme.specificUp())\n",
    "print(\"S(down): \",testme.specificDown())\n",
    "#testme.makePlot()\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. Here I've called the function `makePlot` using the class instance `testme`. <br><br>\n",
    "    The function `makePlot` creates a log plot of $|(S^{\\rm (up)}-S^{\\rm (down)})/(|S^{\\rm (up)}|+|S^{\\rm (down)}|)|$ vs. $N$. Firstly, the function initializes the empty list `y` to store the data points for the y-axis of the plot. The data points for the x-axis of the plot are initialized in the array `x` using the `linspace` function from the `numpy` module, inputting all positive integers from 1 to $N$. To calculate the data points for the y-axis of the plot, a `for` loop function is used to `append` each iteration of the formula  $(S^{\\rm (up)}-S^{\\rm (down)})/(|S^{\\rm (up)}|+|S^{\\rm (down)}|)$ to the `y` list.<sub>[1]</sub>. The function then changes the x and y axes of the plot to the log scale, and then plots the data points.\n",
    "    \n",
    "    <sub>[1] The functions `genericUp` and `genericDown` are used extensively in the formula</sub>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "testme.makePlot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. Addition and subtraction using float type variables tends to produce a rounding error. The graph shows that as $N$ increases, the difference between $S^{\\rm (up)}$ and $S^{\\rm (down)}$ increases, thereby implying that the rounding error accumulates as $N$ increases. The downward sum generally means that the computer will be calculating the sum by adding smaller numbers initially, until $N = 1$. This means that the relative difference between each consecutive term $\\frac{1}{i}$ will be smaller than that of the upward sum (i.e. which starts with larger relative differences). The smaller relative differences mean that there is less room for rounding error in the downwards approach (i.e. because of the similarity in their least significant digits). These errors accumulate across each operation, but the errors accumulate much faster in the upwards approach, hence the rising peaks in the graph as $N$ increases."
   ]
  }
 ],
 "metadata": {
  "colab": {
   "authorship_tag": "ABX9TyN5Dr2e2AmvNUdrlNIzM1Ej",
   "name": "lab_assignment_module1.ipynb",
   "provenance": [
    {
     "file_id": "1kLceW-symdPQWu82l-5hm4b5SGUszX3t",
     "timestamp": 1598784193168
    },
    {
     "file_id": "1FJlZnhC4nq-8ag5P6s8R5x9AryccJwSV",
     "timestamp": 1598775074680
    }
   ]
  },
  "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": "3.7.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}