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   "source": [
    "# Lecture 4\n",
    "\n",
    "In this lecture, we will to continue to learn NumPy `array`, and we will use flow control loops to manunipulate arrays."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Review of NumPy array\n",
    "`ndarray` is an arbitrary dimension set of variables of the *same type* (cf. `list` can be concatenate different types). In default, one entry of an array is a 64-bit float (a real number represented by 64 binary digit).\n",
    "* Functions: `shape`, `reshape`, `arange`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# always run this cell block first\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr = np.array(range(10))\n",
    "print(arr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr[-1] # [] is used for array indexing, () is used as input of functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## For loops\n",
    "\n",
    "Syntax:\n",
    "> for INDEX IN A SET:<br><br>\n",
    ">       STUFF THAT WILL BE REPEATED IN EVERY ITERATION"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## While loops\n",
    "\n",
    "Very clever way to repeat actions. \n",
    "\n",
    "Syntax:\n",
    "> while CONDITION FOR DOING THE NEXT ITERATION OF THE LOOP:\n",
    "<br><br>\n",
    ">       STUFF THAT WILL BE REPEATED IN EVERY ITERATION"
   ]
  },
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   "cell_type": "code",
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   "source": [
    "# generate lst which is a list of integers from 0 to 9 using while\n",
    "# code here\n",
    "n = 0 # counter\n",
    "lst = [] # initialize the list\n",
    "while n < 10:\n",
    "    lst.append(n) # the code to be executed if n < 10\n",
    "    n = n + 1 # increase the counter by 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example:  Checking for primeness\n",
    "We want to write a function called `isprime` that checks if a number is prime. Returns `True` if number is prime, `False` otherwise. "
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "def isprime(n):\n",
    "    # insert code here\n",
    "    # we check all the divisor d < n whether n % d is 0\n",
    "    # recall n % d is the modulo operation, \n",
    "    # which return the remainder of the division using divisor d"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing the sum of 1 to 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# while loop version: generating an array\n",
    "arr = np.empty([100], dtype = 'int')\n",
    "k = 1\n",
    "while k <= 100:\n",
    "    arr[k-1] = k\n",
    "    k += 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# for loop version"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Remark: Break and continue keywords\n",
    "\n",
    "These are keywords:\n",
    "* When the `break` was called, it stopped the loop without doing the print statement underneath. \n",
    "* `continue` skips the rest of the current looping of the loop but continues to loop as usual afterwards"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Good exercises (try by yourself):\n",
    "\n",
    "* Using while loops, implement the integer division function `//` as `division(a,b)`. Make sure you are doing the negative number examples correctly. \n",
    "* Write a loop to print the first 100 prime numbers. \n",
    "* Make a list of all the prime numbers up to 1000 (using append to add elements)\n",
    "* Write a function that will return the maximum element of a list\n",
    "* Write a function that will return the sum of the elements of a list\n",
    "* Write a function that takes two lists, and checks to see if they have at least one common element\n",
    "* Estimate pi by adding the first 1000 terms in the formula: \n",
    "$$\\frac{\\pi}{4} = 1 - \\frac13 + \\frac15 - \\frac17+ \\cdots,$$ \n",
    "which comes from the Taylor expansion of $\\arctan(x)$.\n",
    "Write a code snippet to compare the answer to the built-in `math.pi`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vectorization:\n",
    "\n",
    "Instead of iterating along the indices, we can use built-in routine to make our execution more efficient."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr = np.array(range(16))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr = arr.reshape(4,4) + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "arr.reshape(2,2,-1)    # if you put -1, it figures out what the shape should be"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
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   "outputs": [],
   "source": [
    "arr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": true
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   "outputs": [],
   "source": [
    "np.reshape(arr, (2,2,-1)) # -1 means unspecified"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
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   "outputs": [],
   "source": [
    "np.sum(arr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.mean(arr)"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.apply_along_axis()"
   ]
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.apply_along_axis(np.sum, 0, arr)"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "scrolled": false
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   "outputs": [],
   "source": [
    "np.apply_along_axis(np.sum, 1, arr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.sum(arr, axis=1)"
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