Exercises on graphics (and numpy)
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Measurement of pi (Monte Carlo)
-------------------------------

This exercise can be solved without any loop.

* Using the ``rand`` function, creates two array X and Y with ``N=1000`` random samples from a uniform distribution over ``[-1, 1[``.

* Plot the points

* Plot a circle of radius 1 and plot using a different color the points inside the circle.

* How many points are inside the circle ?

* The number of points is proportional to the area of the circle. Deduce an approximate value of :math:`\pi`. One can use :math:`10^6` points (without plotting the figure...)


Bode plot
---------

The transfer function of a damped oscillator can written in the Fourier space using the formula : 

.. math::

    H(\omega) = \frac{\omega_0^2}{\omega_0^2 - \omega^2 + 2j\zeta\omega\omega_0}


We would like to draw the three following graphs : 

* Bode plot for :math:`\omega_0/2\pi = 1 \mathrm{kHz}` and :math:`\zeta=0.1`

  .. image:: exo1_1.*
     :scale: 80%

* Tranfer amplitude function for :math:`\omega_0/2\pi = 1 \mathrm{kHz}` and different values of :math:`\zeta`.

  .. image:: exo1_2.*
     :scale: 80%

* Bode plot for the product of two tranfer function :math:`\omega_0/2\pi = 1 \mathrm{kHz}`, :math:`\zeta=0.5`, :math:`\omega_1/2\pi = 2 \mathrm{kHz}`, :math:`\zeta=0.5`.

  .. image:: exo1_3.*
     :scale: 80%

You will uses the following functions :

* ``loglog``

* ``semilogx``

* ``xlabel``, ``ylabel`` et ``title``

* ``grid`` 

* ``subplot(ny, nx, n)`` 

* The ``label`` optional parameter and the function ``legend``.

* The ``angle`` function of numpy can be used to calculate the phase of a complex number. For the last graph, one should modify the phase in order for the plot to be continuous (do it without any loop!).

You should also know how to 

* make a string with accents (for french labels!).

* format a string to insert parameters

* For the greek letters, one can use unicode or latex formula.