Exercises on numpy array
========================

Squared numbers
---------------

Create a list (using a loop) and an array (without loop) containing the square of integer number from 0 to N-1. Compare execution speed with :math:`N=10^6`.


Calculation of pi
-----------------

Without any loop, calculates :math:`\pi` using the formula :

.. math::

   \pi = \sqrt{12}\sum_{k=0}^{\infty} \frac{(-1)^k}{3^k(2k+1)}

Take care of the integer division in the 2.7 version of Python.


Allan variance
--------------

The allan variance of a set of measurements :math:`y_k`, where each point corresponds to a frequency measured during :math:`\tau` is defined as :

.. math::
    \sigma_y^2(\tau) = \frac 12 \left<\left(y_{k+1} - y_k\right)^2\right>_k

1. Write a function that calculates the Allan variance **without any loop**

Using a data set where the duration of each measurement is :math:`\tau_0`, it is possible to calculate the Allan variance for duration :math:`\tau = n\tau_0` that are mutiple of :math:`\tau_0`. In this case, the new value is calculated by taking the average value of :math:`n` consecutive measurements. There is no overlap between the values (there is :math:`n` times less points compared to the initial set). 

2. Write a function ``average_frequency(data, n)`` that calculate the mean value of ``data`` with packets of size ``n``.

It is possible to make this function without any loop. Let us start with an array of size 10 and take :math:`n=2`. The shape of the array is initially: ::

    +--+--+--+--+--+--+--+--+--+--+
    |x0|x1|x2|x3|x4|x5|x6|x7|x8|x9|
    +--+--+--+--+--+--+--+--+--+--+


Using the rechape method (``x.reshape((5,2))``), the array will look like: ::

    +--+--+
    |x0|x1|
    +--+--+
    |x2|x3|
    +--+--+
    |x4|x5|
    +--+--+
    |x6|x7|
    +--+--+
    |x8|x9|
    +--+--+

The method ``mean(axis=1)`` will perform the average line by line.

3. Write the function ``AllanVariance(data, n)`` that calculate the Allan variance for a measurement duration of :math:`n\tau_0`.


Mandelbrot set
--------------

The Mandelbrot set is defined as the set of points :math:`c` of the complex plane such that the following sequence  :

.. math::

    z_0=0

    z_{n+1}=z_n^2+c

is bounded. 

One can show that if there is a value of :math:`n` such that :math:`|z_n|>2` the the sequence is divergent. To calculate the Mandelbrot set, for each value of :math:`c` one calculate 100 iterations. The picture is plotted by giving to each point the smallest value of :math:`n < 100` such that :math:`|z_n|>2` (and 0 if this value doesn't exist). 

Calculate and plot the Mandelbrot set for :math:`c` such that :math:`-2.13 < \operatorname{Re}(c) < 0.77` and :math:`-1.13 < \operatorname{Im}(c) < 1.13`. One can then zoom on the edge of the set.

Note : to plot an image, use the imshow function of pylab. 

The total computation time using numpy efficiently is less than 10 seconds for a 1024x1024 matrix. 

.. figure:: mandel_couleur.png
   :width: 50%
   :align: center