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   "source": [
    "# Lecture 5: Matplotlib\n",
    "\n",
    "Today, we will learn \n",
    "* More advanced slicing technique on arrays.\n",
    "* how to use matplotlib's `pyplot` submodule to draw the graph of a function. `pyplot` is a collection of command style functions that make matplotlib work like MATLAB. Each `pyplot` function makes some change to a figure: e.g., creates a figure, creates a plotting area in a figure, plots some lines in a plotting area, decorates the plot with labels, etc.\n",
    "\n",
    "Using `pyplot` is also a first example of the application of vectorization.\n",
    "\n",
    "Reference:\n",
    "[Pyplot tutorial](https://matplotlib.org/tutorials/introductory/pyplot.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr = np.arange(12)+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr[:3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr[3:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr = arr.reshape(3,-1)\n",
    "print(arr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr[1:,:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "arr[:,0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dir(arr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More advanced tricks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.array(range(20)).reshape((4,-1))\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[1:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[1::]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[::-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[::2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[0:2,1:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Setting Values with Slicing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[:,1] = np.zeros(4)\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X[0:2,1:] += 10\n",
    "print(X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### In-class exercise 1\n",
    "Make the following 5 by 5 array, without using `FOR` or `WHILE` loops:\n",
    "```python\n",
    "array([[0, 0, 0, 0, 0],\n",
    "       [0, 1, 2, 3, 0],\n",
    "       [0, 4, 5, 6, 0],\n",
    "       [0, 7, 8, 9, 0],\n",
    "       [0, 0, 0, 0, 0]])\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "from math import pi # so that we can use pi directly instead of math.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Say if we want to plot $y = \\cos(x)$ and $y = \\sin(x)$, first we produce a bunch of equi-distant points using `linspace` function in `numpy`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xs = np.linspace(0, 2*pi, 200)\n",
    "# it generates an array of 200 points between 0 and 2*pi\n",
    "cosxs = np.cos(xs) # evaluates cos at these 200 points\n",
    "sinxs = np.sin(xs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Vectorization remark: this cos is from math module\n",
    "# it cannot be directly applied to xs\n",
    "math.cos(xs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plt is the pyplot function in the matplotlib module\n",
    "# to connect the all the dots\n",
    "# %matplotlib qt\n",
    "plt.plot(xs, cosxs)  # first one is blue\n",
    "plt.plot(xs, sinxs)  # second one is orange"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# optional\n",
    "# we could even annotate the graph we have\n",
    "plt.plot(xs, cosxs, color = 'blue') \n",
    "plt.plot(xs, sinxs, color = 'red') \n",
    "plt.grid(True)\n",
    "plt.axis('auto')\n",
    "plt.annotate('local maximum of $sin(x)$', xy=(pi/2, 1), xytext=(3, 1.5), \n",
    "             arrowprops = dict(facecolor='black',shrink = 0.1))\n",
    "plt.show() # suppress the output, similar to `drawnow` in MATLAB"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## In-class exercise\n",
    "\n",
    "Estimate $\\pi$ by adding the first $100$ terms in the formula: \n",
    "$$\\frac{\\pi}{4} = 1 - \\frac13 + \\frac15 - \\frac17+ \\cdots,$$ \n",
    "which comes from the Taylor expansion of $\\arctan(x)$.\n",
    "\n",
    "Compare the answer to the built-in `math.pi` (you want to  import `math` module) by the following:\n",
    "* Creating a numpy array `estpi` to store the first $100$ approximation along the way. For example, `estpi[0]` should be $4\\cdot 1$, `estpi[1]` should be $4\\cdot \\left(1 - \\frac13\\right)$, `estpi[2]` should be $4\\cdot \\left(1 - \\frac13 + \\frac15\\right)$, etc.\n",
    "* Create another index array of shape `(100,)` to represent the indices, you can use `linspace` or `arange`.\n",
    "* Create a numpy array `actpi` to store the actual $\\pi$ value that has the same shape with `estpi`.\n",
    "* Use `plt` in `pyplot` to plot `estpi` and `actpi` against the index array. Suggestion: use different `color`, `linewidth`, `linestyle`.\n",
    "\n",
    "Reference: [matplotlib.pyplot.plot](https://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html)"
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