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===== A bit of Python =====

Python is available on any Linux/Unix machine including department machines and Macs.

You can download and install python on your own computer by following
instructions at [[http://www.python.org]]

You can use the Python interpreter interactively by typing //python// at a terminal window.
Ipython is a nicer front end to python that is invoked with
  ipython
To quit, type control-d

To run python code in a file //code.py//, either type
  run code.py
in //ipython//, or type
  python code.py
at the unix command line.

When in //ipython//, you may type python statements or expressions
that are evaluated, or //ipython// commands.  See the
[[http://showmedo.com/videotutorials/video?name=1000010&fromSeriesID=100|Video
tutorial on using ipython]], in five parts by Jeff Rush, for help
getting started with //ipython//.

Documentation is immediately available for many things.  For example:
<code>
> ipython
asa:~$ ipython 
Python 2.7.3 (v2.7.3:70274d53c1dd, Apr  9 2012, 20:52:43) 
Type "copyright", "credits" or "license" for more information.

IPython 0.13.2 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.

In [1]: list?
Type:		type
Base Class:	<type 'type'>
String Form:	<type 'list'>
Namespace:	Python builtin
Docstring:
    list() -> new list
    list(sequence) -> new list initialized from sequence's items

In [2]: help(list)
Help on class list in module __builtin__:

class list(object)
 |  list() -> new list
 |  list(sequence) -> new list initialized from sequence's items
 |  
 |  Methods defined here:
   .
   .
   .
 |  append(...)
 |      L.append(object) -- append object to end
 |  
   .
   .
   .
 |  
 |  sort(...)
 |      L.sort(cmp=None, key=None, reverse=False) -- stable sort *IN PLACE*;
 |      cmp(x, y) -> -1, 0, 1
</code>

What is the value of $(100\cdot 2 - 12^2) / 7 \cdot 5 + 2\;\;\;$?
<code python>
  In [301]: (100*2 - 12**2) / 7*5 + 2
  Out[301]: 42
</code>

In order to compute something like $\sin(\pi/2)$ we first need to //import// the //math// module:
<code python>
In [303]: import math
In [304]: math.sin(math.pi/2)
1.0
</code>

How do I find out what other mathematical functions are available?
  help("math")


===== Plotting =====

Let's get on to that all important step of visualizing data.  We will be using the [[http://matplotlib.org |matplotlib]] Python package for that.  Let's start by plotting the function $f(x) = x^2$.

First, let's generate the numbers. Well, there are tons of ways to do so.  First, using a''for'' loop.
<code python>
In [3]: f = []

In [4]: for i in range(10) :
   ...:     f.append(i**2)
   ...:     

In [5]: f
Out[5]: [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
</code>

To plot the data, first import the ''pyplot'' module.

<code python>
In [6]: import matplotlib.pyplot as plt

In [7]: plt.plot(range(10), f)
Out[7]: [<matplotlib.lines.Line2D at 0x10549b590>]

</code>
In order to actually see the plot you need to do:

<code python>
In [8]: plt.show()
</code>
As an alternative, you can put matplotlib in interactive mode before plotting using the command ''plt.ion()''.

Python has some nifty syntax for generating lists.  Watch this!  A [[http://www.secnetix.de/olli/Python/list_comprehensions.hawk|list comprehension]]!!

<code python>
In [9]: f = [i**2 for i in range(10)]

In [10]: f
Out[10]: [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
</code>

There's an alternative way of doing this using ''numpy'':

<code python>

In [11]: import numpy as np

In [12]: f = np.arange(10)**2

In [13]: f
Out[13]: array([ 0,  1,  4,  9, 16, 25, 36, 49, 64, 81])
</code>

Note that plotting functions to accept either lists or ''numpy'' arrays, so a fast way of doing our plot is
<code python>
In [14]: plt.plot(np.arange(10), np.arange(10)**2)
</code>

For a smoother plot:
<code python>
In [14]: x = np.arange(10, 0.1)

In [15]: plt.plot(x, x**2, 'ob')
Out[15]: [<matplotlib.lines.Line2D at 0x1054162d0>]
</code>
/* {{ Notes:plot2.png?400  }}*/


We can add a second plot to the same axes by calling //plot// again:
<code python>
In [16]: plt.plot(x, x, 'dr')
Out[16]: [<matplotlib.lines.Line2D object at 0x3608990>]
</code>

/*{{ Notes:plot3.png?400  }}*/


===== Matrices in Python =====

Can I work with vectors and matrices in python?

Of course!  No data analysis tool is worth the bytes it burns if it
doesn't. The ''numpy'' package provides the required magic.
Let's create an array that represents the following matrix:
\[\left ( \begin{array}{cc}
	1 & 2\\
	3 & 4\\
	5 & 6
	\end{array} \right ) \]
by doing
<code>
In [17]: import numpy as np

In [18]: m = np.array([[1,2], [3,4], [5,6]])

In [19]: m
Out[19]: 
array([[1, 2],
       [3, 4],
       [5, 6]])
</code>

Let's construct the matrices
\[a = \left ( \begin{array}{cc}
	2 & 2 & 2\\
	2 & 2 & 2\\
	2 & 2 & 2
	\end{array} \right ) \]
and
\[b = \left ( \begin{array}{cc}
	1 & 2 & 3\\
	4 & 5 & 6\\
	7 & 8 & 9
	\end{array} \right ) \]
<code python>
In [16]: a = np.ones((3,3)) * 2

In [17]: a
Out[17]: 
array([[ 2.,  2.,  2.],
       [ 2.,  2.,  2.],
       [ 2.,  2.,  2.]])

In [18]: b = np.resize(np.arange(9)+1,(3,3))

In [19]: b
Out[19]: 
array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]])
</code>

What is the value of $a * b$?
<code python>
In [20]: a * b
Out[21]: 
array([[ 2,  4,  6],
       [ 8, 10, 12],
       [14, 16, 18]])
</code>
The ''*'' operator does a component-wise multiplication.  Use the
''numpy'' function ''dot'' to do matrix multiplication.

<code python>
In [22]: np.dot(a,b)
Out[22]: 
array([[24, 30, 36],
       [24, 30, 36],
       [24, 30, 36]])
</code>

An array is transposed by
<code python>
In [23]: b.transpose()
Out[23]: 
array([[1, 4, 7],
       [2, 5, 8],
       [3, 6, 9]])

In [24]: b.T
Out[24]: 
array([[1, 4, 7],
       [2, 5, 8],
       [3, 6, 9]])
</code>

Elements and sub-matrices are easily extracted:
<code python>
In [25]: b
Out[25]: 
array([[1, 2, 3],
       [4, 5, 6],
       [7, 8, 9]])

In [26]: b[0,0]
Out[26]: 1

In [27]: b[0,1]
Out[27]: 2

In [28]: b[0:2, 1:3]
Out[28]: 
array([[2, 3],
       [5, 6]])
</code>

Let's multiply the first row of a $a$ by the second column of $b$.
<code python>
In [29]: np.dot(a[0], b[:,1])
Out[29]: 30.0

In [30]: np.dot(a[0],b.T[1])
Out[30]: 30.0
</code>

How do I find the inverse of a matrix?  
<code python>
In [2]: z = np.array([[2,1,1],[1,2,2],[2,3,4]])

In [3]: z
Out[3]: 
array([[2, 1, 1],
       [1, 2, 2],
       [2, 3, 4]])

In [4]: np.linalg.inv(z)
Out[4]: 
array([[ 0.66666667, -0.33333333,  0.        ],
       [ 0.        ,  2.        , -1.        ],
       [-0.33333333, -1.33333333,  1.        ]])

In [5]: np.dot(z, np.linalg.inv(z))
Out[5]: 
array([[ 1.,  0.,  0.],
       [ 0.,  1.,  0.],
       [ 0.,  0.,  1.]])
</code>