<span id="tutorial-programming-python"></span><h1>Tutorial: Programming in Python and Sage</h1>


</pre><p>This tutorial is an introduction to basic programming in Python/Sage. The
reader is supposed to know the elementary notions of programming but is not
supposed to be familiar with the Python language. The memo given here are far
from being exhaustive. In case you need a more complete tutorial, you can have
a look at the <a class="reference external" href="http://docs.python.org/release/2.6.4/tutorial/index.html">Python Tutorial</a>. Also Python's
<a class="reference external" href="http://docs.python.org/release/2.6.4/">documentation</a> and in particular the
<a class="reference external" href="http://docs.python.org/release/2.6.4/library">standard library</a> can be
useful.</p>
<p>A <a href="#id1"><span class="problematic" id="id2">:ref:`more advanced tutorial &lt;tutorial-objects-and-classes&gt;`</span></a>
presents the notions of objects and classes in Python.</p>

<p class="system-message-title">System Message: ERROR/3 (<tt class="docutils">tutorial-programming-python.rst</tt>, line 20); <em><a href="#id2">backlink</a></em></p>
Unknown interpreted text role &quot;ref&quot;.
<p>Here is a further list of resources for people wanting to learn Python:</p>
<ul class="simple">
<li><a class="reference external" href="http://www.korokithakis.net/tutorials/python">Learn Python in 10 minutes</a> ou en français
<a class="reference external" href="http://mat.oxyg3n.org/index.php?post/2009/07/26/Python-en-10-minutes">Python en 10 minutes</a></li>
<li><a class="reference external" href="http://diveintopython.org/">Dive into Python</a>
is a Python book for experienced programmers. Also available in French,
<a class="reference external" href="http://diveintopython.adrahon.org/">Plongez au coeur de Python</a>, and
<a class="reference external" href="http://diveintopython.org/#languages">other languages</a>.</li>
<li><a class="reference external" href="http://www.ibm.com/developerworks/views/opensource/libraryview.jsp?search_by=Discover+Python+Part|">Discover Python</a>
is a series of articles published in IBM's <a class="reference external" href="http://www.ibm.com/developerworks/">developerWorks</a> technical resource center.</li>
</ul>

<h2>Data structures</h2>
<p>In Python, <em>typing is dynamic</em>; there is no such thing as declaring
variables. The function <tt class="docutils literal"><span class="pre">type</span></tt> returns the type of an object <tt class="docutils literal"><span class="pre">obj</span></tt>. To
convert an object to a type <tt class="docutils literal"><span class="pre">typ</span></tt> just write <tt class="docutils literal"><span class="pre">typ(obj)</span></tt> as in
<tt class="docutils literal"><span class="pre">int(&quot;123&quot;)</span></tt>. The command <tt class="docutils literal"><span class="pre">isinstance(ex,</span> <span class="pre">typ)</span></tt> returns whether the
expression <tt class="docutils literal"><span class="pre">ex</span></tt> is of type <tt class="docutils literal"><span class="pre">typ</span></tt>. Specifically, any value is <em>an instance
of a class</em> and there is no difference between classes and types.</p>
<p>The symbol <tt class="docutils literal"><span class="pre">=</span></tt> denotes the affectation to a variable; it should not
be confused with <tt class="docutils literal"><span class="pre">==</span></tt> which denotes mathematical
equality. Inequality is <tt class="docutils literal"><span class="pre">!=</span></tt>.</p>
<p>The <em>standard types</em> are <tt class="docutils literal"><span class="pre">bool</span></tt>, <tt class="docutils literal"><span class="pre">int</span></tt>, <tt class="docutils literal"><span class="pre">list</span></tt>, <tt class="docutils literal"><span class="pre">tuple</span></tt>, <tt class="docutils literal"><span class="pre">set</span></tt>,
<tt class="docutils literal"><span class="pre">dict</span></tt>, <tt class="docutils literal"><span class="pre">str</span></tt>.</p>
<ul>
<li><p class="first">The type <tt class="docutils literal"><span class="pre">bool</span></tt> (<em>Booleans</em>) takes two values: <tt class="docutils literal"><span class="pre">True</span></tt> and <tt class="docutils literal"><span class="pre">False</span></tt>. The
boolean operator are denoted by their names <tt class="docutils literal"><span class="pre">or</span></tt>, <tt class="docutils literal"><span class="pre">and</span></tt>, <tt class="docutils literal"><span class="pre">not</span></tt>.</p>
</li>
<li><p class="first">Sage uses <em>exact arithmetic</em> on the integers. For performance reason, it
uses its own type named <tt class="docutils literal"><span class="pre">Integer</span></tt> (whereas python uses the types <tt class="docutils literal"><span class="pre">int</span></tt>
and <tt class="docutils literal"><span class="pre">long</span></tt>).</p>
</li>
<li><p class="first">A <em>list</em> is a data structure which groups values. It is constructed using
brackets as in <tt class="docutils literal"><span class="pre">[1,</span> <span class="pre">3,</span> <span class="pre">4]</span></tt>. The <tt class="docutils literal"><span class="pre">range</span></tt> function creates integer
lists. One can also create lists using <em>list comprehension</em>:</p>
<pre class="literal-block">
[ &lt;expr&gt; for &lt;name&gt; in &lt;iterable&gt; (if &lt;condition&gt;) ]

</pre><p>For example:</p>

{{{id=0|
[ i^2 for i in range(10) if i % 2 == 0 ]
///
[0, 4, 16, 36, 64]
}}}

</li>
<li><p class="first">A <em>tuple</em> is very similar to a list; it is constructed using
parentheses. The empty tuple is obtained by <tt class="docutils literal"><span class="pre">()</span></tt> or by the
constructor <tt class="docutils literal"><span class="pre">tuple()</span></tt>. If there is only one element, one has to
write <tt class="docutils literal"><span class="pre">(a,)</span></tt>. A tuple is <em>immutable</em> (one cannot change it) but it
is <em>hashable</em> (see below). One can also create tuples using
comprehensions:</p>

{{{id=1|
tuple(i^2 for i in range(10) if i % 2 == 0)
///
(0, 4, 16, 36, 64)
}}}

</li>
<li><p class="first">A <em>set</em> is a data structure which contains values without
multiplicities or order. One creates it from a list (or any
iterable) with the constructor <tt class="docutils literal"><span class="pre">set()</span></tt>. The elements of a set must
be hashable:</p>

{{{id=2|
set([2,2,1,4,5])
///
set([1, 2, 4, 5])
}}}

</li>
<li><p class="first">A <em>dictionary</em> is an association table, which associate values to
keys. Keys must be hashable. One creates dictionaries using the
constructor <tt class="docutils literal"><span class="pre">dict</span></tt>, or using the syntax:</p>
<pre class="literal-block">
{&nbsp;key1 : value1, key2 : value2 ...}&nbsp;

</pre><p>For example:</p>

{{{id=3|
age = {'toto' : 8, 'mom' : 27}; age
///
{'toto': 8, 'mom': 27}
}}}

</li>
<li><p class="first">Quotes (simple <tt class="docutils literal"><span class="pre">'</span> <span class="pre">'</span></tt> or double <tt class="docutils literal"><span class="pre">&quot;</span> <span class="pre">&quot;</span></tt>) encloses <em>character
strings</em>. One can concatenate them using <tt class="docutils literal"><span class="pre">+</span></tt>.</p>
</li>
<li><p class="first">For lists, tuples, strings, and dictionaries, the <em>indexing
operator</em> is written <tt class="docutils literal"><span class="pre">l[i]</span></tt>. For lists, tuples, and strings one
can also uses <em>slices</em> as <tt class="docutils literal"><span class="pre">l[:]</span></tt>, <tt class="docutils literal"><span class="pre">l[:b]</span></tt>, <tt class="docutils literal"><span class="pre">l[a:]</span></tt>, or
<tt class="docutils literal"><span class="pre">l[a:b]</span></tt>. Negative indices start from the end.</p>
</li>
<li><p class="first">The <tt class="docutils literal"><span class="pre">len</span></tt> function returns the number of elements of a list, a
tuple, a set, a string, or a dictionary. One writes <tt class="docutils literal"><span class="pre">x</span> <span class="pre">in</span> <span class="pre">C</span></tt> to
tests whether <tt class="docutils literal"><span class="pre">x</span></tt> is in <tt class="docutils literal"><span class="pre">C</span></tt>.</p>
</li>
<li><p class="first">Finally there is a special value called <tt class="docutils literal"><span class="pre">None</span></tt> to denote the
absence of a value.</p>
</li>
</ul>


<h2>Control structures</h2>
<p>In Python, there is no keyword for the beginning and the end of an
instructions block. Blocks are delimited thanks to indentation. Most
of the time a new block is introduced by <tt class="docutils literal"><span class="pre">:</span></tt>. Python has the
following control structures:</p>
<ul>
<li><p class="first">Conditional instruction:</p>
<pre class="literal-block">
if &lt;condition&gt;:
    &lt;instruction sequence&gt;
[elif &lt;condition&gt;:
    &lt;instruction sequence&gt;]*
[else:
    &lt;instruction sequence&gt;]

</pre></li>
<li><p class="first">Inside expression exclusively, one can write:</p>
<pre class="literal-block">
&lt;value&gt; if &lt;condition&gt; else &lt;value&gt;

</pre></li>
<li><p class="first">Iterative instructions:</p>
<pre class="literal-block">
for &lt;name&gt; in &lt;iterable&gt;:
    &lt;instruction sequence&gt;
[else:
    &lt;instruction sequence&gt;]

</pre><pre class="literal-block">
while &lt;condition&gt;:
    &lt;instruction sequence&gt;
[else:
    &lt;instruction sequence&gt;]

</pre><p>The <tt class="docutils literal"><span class="pre">else</span></tt> bloc is executed at the end of the loop if the loop is
ended normally, that is neither by a <tt class="docutils literal"><span class="pre">break</span></tt> nor an exception.</p>
</li>
<li><p class="first">In a loop, <tt class="docutils literal"><span class="pre">continue</span></tt> jumps to the next iteration.</p>
</li>
<li><p class="first">An iterable is an object which can be iterated through. Iterable
types include lists, tuples, dictionaries, and strings.</p>
</li>
<li><p class="first">An error (also called exception) is raised by:</p>
<pre class="literal-block">
raise &lt;ErrorType&gt; [, error message]

</pre><p>Usual errors includes <tt class="docutils literal"><span class="pre">ValueError</span></tt> and <tt class="docutils literal"><span class="pre">TypeError</span></tt>.</p>
</li>
</ul>


<h2>Functions</h2>

<p class="first admonition-title">Note</p>
<p>Python functions vs. mathematical functions</p>
<p class="last">In what follows, we deal with <em>functions</em> is the sense of
<em>programming languages</em>. Mathematical functions are handled by
sage in a different way. In particular it doesn't make sense to do
mathematical manipulation such as additions or derivations on
Python functions.</p>

<p>One defines a function using the keyword <tt class="docutils literal"><span class="pre">def</span></tt> as</p>
<pre class="literal-block">
def &lt;name&gt;(&lt;argument list&gt;):
     &lt;instruction sequence&gt;

</pre><p>The result of the function is given by the instruction
<tt class="docutils literal"><span class="pre">return</span></tt>. Very short functions can be created anonymously using
<tt class="docutils literal"><span class="pre">lambda</span></tt> (remark that there is no return here):</p>
<pre class="literal-block">
lambda &lt;arguments&gt;: &lt;expression&gt;

</pre><p class="first admonition-title">Note</p>
<p>(functional programming)</p>
<p class="last">Functions are objects as any other objects. One can assign them to
variables or return them.</p>




<h1>Exercises</h1>

<h2>Lists</h2>

<h3>Creating Lists I: [Square brackets]</h3>
<p><strong>Example:</strong></p>

{{{id=4|
L = [3, Permutation([5,1,4,2,3]), 17, 17, 3, 51]
L
///
[3, [5, 1, 4, 2, 3], 17, 17, 3, 51]
}}}

<p><strong>Exercise:</strong> Create the list $[63, 12, -10, 'a', 12]$, assign it to
the variable <tt class="docutils literal"><span class="pre">L</span></tt>, and print the list.</p>

{{{id=5|

///
}}}

<p><strong>Exercise:</strong> Create the empty list (you will often need to do this).</p>

{{{id=6|

///
}}}

<h3>Creating Lists II: range</h3>
<p>The <em>range</em> function provides an easy way to construct a list of
integers. Here is the documentation of the <em>range</em> function:</p>
<pre class="literal-block">
range([start,] stop[, step]) -&gt; list of integers

Return a list containing an arithmetic progression of integers.
range(i, j) returns [i, i+1, i+2, ..., j-1]; start (!) defaults to 0.
When step is given, it specifies the increment (or decrement). For
example, range(4) returns [0, 1, 2, 3].  The end point is omitted!
These are exactly the valid indices for a list of 4 elements.

</pre><p><strong>Exercise:</strong> Use <em>range</em> to construct the list $[1,2,ldots,50]$.</p>

{{{id=7|

///
}}}

<p><strong>Exercise:</strong> Use <em>range</em> to construct the list of <em>even</em> numbers
between 1 and 100 (including 100).</p>

{{{id=8|

///
}}}

<p><strong>Exercise:</strong> The <em>step</em> argument for the <em>range</em> command can be negative. Use
<em>range</em> to construct the list $[10, 7, 4, 1, -2]$.</p>

{{{id=9|

///
}}}

<h3>Creating Lists III: list comprehensions</h3>
<p><em>List comprehensions</em> provide a concise way to create lists from other lists
(or other data types).</p>
<p><strong>Example</strong> We already know how to create the list $[1, 2, dots, 16]$:</p>

{{{id=10|
range(1,17)
///
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
}}}

<p>Using a <em>list comprehension</em>, we can now create the list
$[1^2, 2^2, 3^2, dots, 16^2]$ as follows:</p>

{{{id=11|
[i^2 for i in range(1,17)]
///
[1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256]
}}}


{{{id=12|
sum([i^2 for i in range(1,17)])
///
1496
}}}

<p><strong>Exercice:</strong></p>
<p>The sum of the squares of the first ten natural numbers is</p>

<p class="system-message-title">System Message: ERROR/3 (<tt class="docutils">tutorial-programming-python.rst</tt>, line 301)</p>
<p>Unknown directive type &quot;math&quot;.</p>
<pre class="literal-block">
.. math:: (1^2 + 2^2 + ... + 10^2) = 385


</pre><p>The square of the sum of the first ten natural numbers is</p>

<p class="system-message-title">System Message: ERROR/3 (<tt class="docutils">tutorial-programming-python.rst</tt>, line 305)</p>
<p>Unknown directive type &quot;math&quot;.</p>
<pre class="literal-block">
.. math:: (1 + 2 + ... + 10)^2 = 55^2 = 3025


</pre><p>Hence the difference between the sum of the squares of the first ten natural
numbers and the square of the sum is</p>

<p class="system-message-title">System Message: ERROR/3 (<tt class="docutils">tutorial-programming-python.rst</tt>, line 310)</p>
<p>Unknown directive type &quot;math&quot;.</p>
<pre class="literal-block">
.. math:: 3025 - 385 = 2640


</pre><p>Find the difference between the sum of the squares of the first one hundred
natural numbers and the square of the sum.</p>

{{{id=13|

///
}}}


{{{id=14|

///
}}}


{{{id=15|

///
}}}

<h4>Filtering lists with a list comprehension</h4>
<p>A list can be <em>filtered</em> using a list comprehension.</p>
<p><strong>Example:</strong> To create a list of the squares of the prime numbers between 1
and 100, we use a list comprehension as follows.</p>

{{{id=16|
[p^2 for p in [1,2,..,100] if is_prime(p)]
///
[4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409]
}}}

<p><strong>Exercise:</strong> Use a <em>list comprehension</em> to list all the natural numbers below
20 that are multiples of 3 or 5. Hint:</p>
<ul class="simple">
<li>To get the remainder of 7 divided by 3 use <em>7%3</em>.</li>
<li>To test for equality use two equal signs (<em>==</em>); for example, <em>3 == 7</em>.</li>
</ul>

{{{id=17|

///
}}}

<h4>Nested list comprehensions</h4>
<p>List comprehensions can be nested!</p>
<p><strong>Examples:</strong></p>

{{{id=18|
[(x,y) for x in range(5) for y in range(3)]
///
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2), (4, 0), (4, 1), (4, 2)]
}}}


{{{id=19|
[[i^j for j in range(1,4)] for i in range(6)]
///
[[0, 0, 0], [1, 1, 1], [2, 4, 8], [3, 9, 27], [4, 16, 64], [5, 25, 125]]
}}}


{{{id=20|
matrix([[i^j for j in range(1,4)] for i in range(6)])
///
[  0   0   0][  1   1   1][  2   4   8][  3   9  27][  4  16  64][  5  25 125]
}}}

<p><strong>Exercise:</strong></p>
<p>A <em>Pythagorean triple</em> is a triple $(x,y,z)$ of <em>positive</em> integers satisfying
$x^2+y^2=z^2$. The Pythagorean triples whose components are at most $10$ are:</p>

<p class="system-message-title">System Message: ERROR/3 (<tt class="docutils">tutorial-programming-python.rst</tt>, line 398)</p>
<p>Unknown directive type &quot;math&quot;.</p>
<pre class="literal-block">
.. math:: [(3, 4, 5), (4, 3, 5), (6, 8, 10), (8, 6, 10)]\,.


</pre><p>Using a filtered list comprehension, construct the list of Pythagorean triples
whose components are at most $50$.</p>

{{{id=21|

///
}}}


{{{id=22|

///
}}}

<h3>Accessing individual elements of lists</h3>
<p>To access an element of the list, use the syntax <tt class="docutils literal"><span class="pre">L[i]</span></tt>, where $i$ is the
index of the item.</p>
<p><strong>Exercise:</strong></p>
<blockquote>
<ol class="arabic">
<li><p class="first">Construct the list <tt class="docutils literal"><span class="pre">L</span> <span class="pre">=</span> <span class="pre">[1,2,3,4,3,5,6]</span></tt>. What is <tt class="docutils literal"><span class="pre">L[3]</span></tt>?</p>

{{{id=23|

///
}}}

</li>
<li><p class="first">What is <tt class="docutils literal"><span class="pre">L[1]</span></tt>?</p>

{{{id=24|

///
}}}

</li>
<li><p class="first">What is the index of the first element of <tt class="docutils literal"><span class="pre">L</span></tt>?</p>

{{{id=25|

///
}}}

</li>
<li><p class="first">What is <tt class="docutils literal"><span class="pre">L[-1]</span></tt>? What is <tt class="docutils literal"><span class="pre">L[-2]</span></tt>?</p>

{{{id=26|

///
}}}

</li>
<li><p class="first">What is <tt class="docutils literal"><span class="pre">L.index(2)</span></tt>? What is <tt class="docutils literal"><span class="pre">L.index(3)</span></tt>?</p>

{{{id=27|

///
}}}

</li>
</ol>
</blockquote>


<h3>Modifying lists: changing an element in a list</h3>
<p>To change the item in position <tt class="docutils literal"><span class="pre">i</span></tt> of a list <tt class="docutils literal"><span class="pre">L</span></tt>:</p>

{{{id=28|
L = [&quot;a&quot;, 4, 1, 8]
L
///
['a', 4, 1, 8]
}}}

<!-- link -->

{{{id=29|
L[2] = 0
L
///
['a', 4, 0, 8]
}}}

<h3>Modifying lists: append and extend</h3>
<p>To <em>append</em> an object to a list:</p>

{{{id=30|
L = [&quot;a&quot;, 4, 1, 8]
L
///
['a', 4, 1, 8]
}}}

<!-- link -->

{{{id=31|
L.append(17)
L
///
['a', 4, 1, 8, 17]
}}}

<p>To <em>extend</em> a list by another list:</p>

{{{id=32|
L1 = [1,2,3]
L2 = [7,8,9,0]
print L1
///
[1, 2, 3]
}}}

{{{id=33|
print L2
///
[7, 8, 9, 0]
}}}

<!-- link -->

{{{id=34|
L1.extend(L2)
L1
///
[1, 2, 3, 7, 8, 9, 0]
}}}

<h3>Modifying lists: reverse, sort, ...</h3>

{{{id=35|
L = [4,2,5,1,3]
L
///
[4, 2, 5, 1, 3]
}}}

<!-- link -->

{{{id=36|
L.reverse()
L
///
[3, 1, 5, 2, 4]
}}}

<!-- link -->

{{{id=37|
L.sort()
L
///
[1, 2, 3, 4, 5]
}}}


{{{id=38|
L = [3,1,6,4]
sorted(L)
///
[1, 3, 4, 6]
}}}

<!-- link -->

{{{id=39|
L
///
[3, 1, 6, 4]
}}}

<h3>Concatenating Lists</h3>
<p>To concatenate two lists, add them with the operator <tt class="docutils literal"><span class="pre">+</span></tt>. This is not a commutative operation....</p>

{{{id=40|
L1 = [1,2,3]
L2 = [7,8,9,0]
L1 + L2
///
[1, 2, 3, 7, 8, 9, 0]
}}}

<h3>Slicing Lists</h3>
<p>You can slice a list using the syntax <tt class="docutils literal"><span class="pre">L[start</span> <span class="pre">:</span> <span class="pre">stop</span> <span class="pre">:</span> <span class="pre">step]</span></tt>. This will
return a sublist of <tt class="docutils literal"><span class="pre">L</span></tt>.</p>
<p><strong>Exercise:</strong> Below are some examples of slicing lists. Try to guess what the output will be before evaluating the cell.</p>

{{{id=41|
L = range(20)
L
///
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
}}}

<!-- link -->

{{{id=42|
L[3:15]
///
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
}}}

<!-- link -->

{{{id=43|
L[3:15:2]
///
[3, 5, 7, 9, 11, 13]
}}}

<!-- link -->

{{{id=44|
L[15:3:-1]
///
[15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4]
}}}

<!-- link -->

{{{id=45|
L[:4]
///
[0, 1, 2, 3]
}}}

<!-- link -->

{{{id=46|
L[:]
///
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]
}}}

<!-- link -->

{{{id=47|
L[::-1]
///
[19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0]
}}}

<p><strong>Advanced exercise:</strong> The following function combines a loop with the some of
the list operations above. What does the function do?</p>

{{{id=48|
def f(number_of_iterations):
       L = [1]
       for n in range(2, number_of_iterations):
           L = [sum(L[:i]) for i in range(n-1, -1, -1)]
       return numerical_approx(2*L[0]*len(L)/sum(L), digits=50)
///
}}}

<!-- link -->

{{{id=49|
f(10)
///
3.1413810483870967741935483870967741935483870967742
}}}

<h2>Tuples</h2>
<p>A <em>tuple</em> is an <em>immutable</em> list. That is, it cannot be changed once it is
created. The syntax for creating a tuple is to the use parentheses.</p>

{{{id=50|
t = (3, 5, [3,1], (17,[2,3],17), 4)
t
///
(3, 5, [3, 1], (17, [2, 3], 17), 4)
}}}

<p>We can create a tuple from a list, or vice-versa.</p>
<!-- link -->

{{{id=51|
tuple(range(5))
///
(0, 1, 2, 3, 4)
}}}

<!-- link -->

{{{id=52|
list(t)
///
[3, 5, [3, 1], (17, [2, 3], 17), 4]
}}}

<p>Tuples behave like lists in many respects:</p>
<table border="1" class="docutils">
<colgroup>
<col width="30%">
<col width="35%">
<col width="35%">
</colgroup>
<thead valign="bottom">
<tr><th class="head">Operation</th>
<th class="head">Syntax for lists</th>
<th class="head">Syntax for tuples</th>
</tr>
</thead>
<tbody valign="top">
<tr><td>Accessing a letter</td>
<td><tt class="docutils literal"><span class="pre">list[3]</span></tt></td>
<td><tt class="docutils literal"><span class="pre">tuple[3]</span></tt></td>
</tr>
<tr><td>Concatenation</td>
<td><tt class="docutils literal"><span class="pre">list1</span> <span class="pre">+</span> <span class="pre">list2</span></tt></td>
<td><tt class="docutils literal"><span class="pre">tuple1</span> <span class="pre">+</span> <span class="pre">tuple2</span></tt></td>
</tr>
<tr><td>Slicing</td>
<td><tt class="docutils literal"><span class="pre">list[3:17:2]</span></tt></td>
<td><tt class="docutils literal"><span class="pre">tuple[3:17:2]</span></tt></td>
</tr>
<tr><td>A reversed copy</td>
<td><tt class="docutils literal"><span class="pre">list[::-1]</span></tt></td>
<td><tt class="docutils literal"><span class="pre">tuple[::-1]</span></tt></td>
</tr>
<tr><td>Length</td>
<td><tt class="docutils literal"><span class="pre">len(list)</span></tt></td>
<td><tt class="docutils literal"><span class="pre">len(tuple)</span></tt></td>
</tr>
</tbody>
</table>
<p>Trying to modify a tuple will fail.</p>

{{{id=53|
t = (5, 'a', 6/5)
t
///
(5, 'a', 6/5)
}}}

<!-- link -->

{{{id=54|
t[1] = 'b'
///
Traceback (most recent call last):
TypeError: 'tuple' object does not support item assignment
}}}

<h2>Generators</h2>
<p>&quot;Tuple-comprehension&quot; does not exist.  The syntax produces something called a
generator.  A generator allows you to process a sequence of items one at a
time.  Each item is created when it is needed, and then forgotten.  This can
be very efficient if we only need to use each item once.</p>

{{{id=55|
(i^2 for i in range(5))
///
<generator object <genexpr> at 0x...>
}}}


{{{id=56|
g = (i^2 for i in range(5))
g[0]
///
Traceback (most recent call last):
TypeError: 'generator' object is unsubscriptable
}}}

<!-- link -->

{{{id=57|
[x for x in g]
///
[0, 1, 4, 9, 16]
}}}

<p><tt class="docutils literal"><span class="pre">g</span></tt> is now empty.</p>
<!-- link -->

{{{id=58|
[x for x in g]
///
[]
}}}

<p>A nice 'pythonic' trick is to use generators as the argument to functions.  We
do <em>not</em> need double parentheses for this.</p>

{{{id=59|
sum( i^2 for i in srange(100001) )
///
333338333350000
}}}

<h2>Dictionaries</h2>
<p>A <em>dictionary</em> is another built-in data type. Unlike lists, which are indexed by a range of numbers, dictionaries are &quot;indexed&quot; by <em>keys</em>, which can be any immutable object. Strings and numbers can always be keys (because they are immutable). Dictionaries are sometimes called &quot;associative arrays&quot; in other programming languages.</p>
<p>There are several ways to define dictionaries. One method is to use braces, {}, with comma-separated entries given in the form <em>key:value</em>.</p>

{{{id=60|
d = {3:17, &quot;key&quot;:[4,1,5,2,3], (3,1,2):&quot;goo&quot;, 3/2 : 17}
d
///
{3/2: 17, 3: 17, (3, 1, 2): 'goo', 'key': [4, 1, 5, 2, 3]}
}}}

<p>Dictionaries behave as lists and tuples for several important operations.</p>
<table border="1" class="docutils">
<colgroup>
<col width="28%">
<col width="32%">
<col width="40%">
</colgroup>
<thead valign="bottom">
<tr><th class="head">Operation</th>
<th class="head">Syntax for lists</th>
<th class="head">Syntax for dictionaries</th>
</tr>
</thead>
<tbody valign="top">
<tr><td>Accessing elements</td>
<td><tt class="docutils literal"><span class="pre">list[3]</span></tt></td>
<td><tt class="docutils literal"><span class="pre">D[&quot;key&quot;]</span></tt></td>
</tr>
<tr><td>Length</td>
<td><tt class="docutils literal"><span class="pre">len(list)</span></tt></td>
<td><tt class="docutils literal"><span class="pre">len(D)</span></tt></td>
</tr>
<tr><td>Modifying</td>
<td><tt class="docutils literal"><span class="pre">L[3]</span> <span class="pre">=</span> <span class="pre">17</span></tt></td>
<td><tt class="docutils literal"><span class="pre">D[&quot;key&quot;]</span> <span class="pre">=</span> <span class="pre">17</span></tt></td>
</tr>
<tr><td>Deleting items</td>
<td><tt class="docutils literal"><span class="pre">del</span> <span class="pre">L[3]</span></tt></td>
<td><tt class="docutils literal"><span class="pre">del</span> <span class="pre">D[&quot;key&quot;]</span></tt></td>
</tr>
</tbody>
</table>
<!-- link -->

{{{id=61|
d[10]='a'
d
///
{3/2: 17, 10: 'a', 3: 17, (3, 1, 2): 'goo', 'key': [4, 1, 5, 2, 3]}
}}}

<p>A dictionary can have the same value multiple times, but each key can only
appear once and must be immutable.</p>

{{{id=62|
d = {3: 14, 4: 14}
d
///
{3: 14, 4: 14}
}}}


{{{id=63|
d = {3: 13, 3: 14}
d
///
{3: 14}
}}}


{{{id=64|
d = {[1,2,3] : 12}
///
Traceback (most recent call last):
TypeError: unhashable type: 'list'
}}}

<p>Another way to add items to a dictionary is with the <tt class="docutils literal"><span class="pre">update()</span></tt> method:</p>

{{{id=65|
d = {}
d
///
{}
}}}

<!-- link -->

{{{id=66|
d.update( {10 : 'newvalue', 20: 'newervalue', 3: 14, 'a':[1,2,3]} )
d
///
{'a': [1, 2, 3], 10: 'newvalue', 3: 14, 20: 'newervalue'}
}}}

<p>We can iterate through the <em>keys</em>, or <em>values</em>, or both, of a dictionary.</p>

{{{id=67|
d = {10 : 'newvalue', 20: 'newervalue', 3: 14, 'a':[1,2,3]}
///
}}}

<!-- link -->

{{{id=68|
[key for key in d]
///
['a', 10, 3, 20]
}}}

<!-- link -->

{{{id=69|
[key for key in d.iterkeys()]
///
['a', 10, 3, 20]
}}}

<!-- link -->

{{{id=70|
[value for value in d.itervalues()]
///
[[1, 2, 3], 'newvalue', 14, 'newervalue']
}}}

<!-- link -->

{{{id=71|
[(key, value) for key, value in d.iteritems()]
///
[('a', [1, 2, 3]), (10, 'newvalue'), (3, 14), (20, 'newervalue')]
}}}

<p><strong>Exercise:</strong> Consider the following directed graph.</p>
<img alt="media/graph0.png" src="media/graph0.png">
<p>Create a dictionary whose keys are the vertices of the above directed graph,
and whose values are the lists of the vertices that it points to. For
instance, the vertex 1 points to the vertices 2 and 3, so the dictionary will
look like:</p>
<pre class="literal-block">
d = {&nbsp; ..., 1:[2,3], ... }&nbsp;

</pre>
{{{id=72|

///
}}}

<p>Then try</p>
<!-- skip -->

{{{id=73|
g = DiGraph(d)
g.plot()
///
}}}

<h2>Using Sage types: The srange command</h2>
<p><strong>Example:</strong> Construct a $3 times 3$ matrix whose $(i,j)$ entry is the
rational number $frac{i}{j}$. The integer generated by <tt class="docutils literal"><span class="pre">range</span></tt> are
python <tt class="docutils literal"><span class="pre">int</span></tt>. As a consequence, dividing them does euclidian division.</p>

{{{id=74|
matrix([[ i/j for j in range(1,4)] for i in range(1,4)])
///
[1 0 0][2 1 0][3 1 1]
}}}

<p>Whereas Dividing sage <tt class="docutils literal"><span class="pre">Integer</span></tt> gives a fraction</p>

{{{id=75|
matrix([[ i/j for j in srange(1,4)] for i in srange(1,4)])
///
[  1 1/2 1/3][  2   1 2/3][  3 3/2   1]
}}}

<h2>Modifying lists has consequences!</h2>
<p>Try to predict the results of the following commands.</p>

{{{id=76|
a = [1, 2, 3]
L = [a, a, a]
L
///
[[1, 2, 3], [1, 2, 3], [1, 2, 3]]
}}}

<!-- link -->

{{{id=77|
a.append(4)
L
///
[[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]
}}}

<p>Now try these:</p>

{{{id=78|
a = [1, 2, 3]
L = [a, a, a]
L
///
[[1, 2, 3], [1, 2, 3], [1, 2, 3]]
}}}

<!-- link -->

{{{id=79|
a = [1, 2, 3, 4]
L
///
[[1, 2, 3], [1, 2, 3], [1, 2, 3]]
}}}

<!-- link -->

{{{id=80|
L[0].append(4)
L
///
[[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]
}}}

<p>You can use the command <em>deepcopy</em> to avoid these issues.</p>

{{{id=81|
a = [1,2,3]
L = [deepcopy(a), deepcopy(a)]
L
///
[[1, 2, 3], [1, 2, 3]]
}}}

<!-- link -->

{{{id=82|
a.append(4)
L
///
[[1, 2, 3], [1, 2, 3]]
}}}

<p>The same issues occur with dictionaries.</p>

{{{id=83|
d = {1:'a', 2:'b', 3:'c'}
dd = d
d.update( { 4:'d' } )
dd
///
{1: 'a', 2: 'b', 3: 'c', 4: 'd'}
}}}

<h1>Loops and Functions</h1>
<p>For more verbose explanation of what's going on here, a good place to look is
at the following section of the Python tutorial:
<a class="reference external" href="http://docs.python.org/tutorial/controlflow.html">http://docs.python.org/tutorial/controlflow.html</a></p>

<h2>While Loops</h2>
<p>While loops tend not to be used nearly as much as for loops in Python code.</p>

{{{id=84|
i = 0
while i &lt; 10:
       print i
       i += 1
///
0123456789
}}}


{{{id=85|
i = 0
while i &lt; 10:
       if i % 2 == 1:
           i += 1
           continue
       print i
       i += 1
///
02468
}}}

<p>Note that the truth value expression in the <em>while</em> loop is evaluated using
bool().</p>

{{{id=86|
bool(True)
///
True
}}}


{{{id=87|
bool('a')
///
True
}}}


{{{id=88|
bool(1)
///
True
}}}


{{{id=89|
bool(0)
///
False
}}}

<!-- skip -->

{{{id=90|
i = 4
while i:
       print i
       i -= 1
///
4321
}}}

<h2>For Loops</h2>
<p>Here is a basic for loop iterating over all of the elements in the list <tt class="docutils literal"><span class="pre">l</span></tt>:</p>

{{{id=91|
l = ['a', 'b', 'c']
for letter in l:
       print letter
///
abc
}}}

<p>The <em>range</em> function is very useful when you want to generate arithmetic
progressions to loop over. Note that the end point is never included:</p>
<!-- skip -->

{{{id=92|
range?
///
}}}


{{{id=93|
range(4)
///
[0, 1, 2, 3]
}}}


{{{id=94|
range(1, 5)
///
[1, 2, 3, 4]
}}}


{{{id=95|
range(1, 11, 2)
///
[1, 3, 5, 7, 9]
}}}


{{{id=96|
range(10, 0, -1)
///
[10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
}}}


{{{id=97|
for i in range(4):
       print i, i*i
///
0 01 12 43 9
}}}

<p>You can use the <em>continue</em> keyword to immediately go to the next item in the
loop:</p>

{{{id=98|
for i in range(10):
       if i % 2 == 0:
           continue
       print i
///
13579
}}}

<p>If you want to break out of the loop, use the <em>break</em> keyword:</p>

{{{id=99|
for i in range(10):
       if i % 2 == 0:
           continue
       if i == 7:
           break
       print i
///
135
}}}

<p>If you need to keep track of both the position in the list and its value, one (not so elegant) way would be to do the following:</p>

{{{id=100|
l = ['a', 'b', 'c']
   for i in range(len(l)):
       print i, l[i]
///
}}}

<p>It's cleaner to use <em>enumerate</em> which provides the index as well as the value:</p>

{{{id=101|
l = ['a', 'b', 'c']
   for i, letter in enumerate(l):
       print i, letter
///
}}}

<p>You could make something similary to the <em>enumerate</em> function by using <em>zip</em>
to zip two lists together:</p>

{{{id=102|
l = ['a', 'b', 'c']
   for i, letter in zip(range(len(l)), l):
       print i, letter
///
}}}

<p>For loops work using Python's iterator protocol.&nbsp; This allows all sorts of different objects to be looped over.&nbsp; For example,</p>

{{{id=103|
for i in GF(5):
       print i, i*i
///
0 01 12 43 44 1
}}}

<p>How does it work?</p>

{{{id=104|
it = iter(GF(5)); it
///
<generator object __iter__ at 0x...>
}}}

{{{id=105|
it.next()
///
0
}}}

{{{id=106|
it.next()
///
1
}}}

{{{id=107|
it.next()
///
2
}}}

{{{id=108|
it.next()
///
3
}}}

{{{id=109|
it.next()
///
4
}}}

{{{id=110|
it.next()
///
Traceback (most recent call last):
StopIteration
}}}

<p>..skip</p>

{{{id=111|
R = GF(5)
   R.__iter__??
///
}}}

<p><em>yield</em> provides a very convient way to produce iterators.&nbsp; We'll see more
about it in a bit.</p>

<h3>Exercices</h3>
<p>For each of the following sets, compute the list of its elements and their
sum. Use two different ways, if possible: with a loop and using lists
comprehension:</p>
<blockquote>
<ol class="arabic simple">
<li><tt class="docutils literal"><span class="pre">n</span></tt> premiers termes de la série harmonique</li>
</ol>

<p class="system-message-title">System Message: WARNING/2 (<tt class="docutils">tutorial-programming-python.rst</tt>, line 1268)</p>
Enumerated list ends without a blank line; unexpected unindent.
<blockquote>

<p class="system-message-title">System Message: ERROR/3 (<tt class="docutils">tutorial-programming-python.rst</tt>, line 1268)</p>
<p>Unknown directive type &quot;math&quot;.</p>
<pre class="literal-block">
.. math:: \sum_{&nbsp;i=1}&nbsp;^n \frac{&nbsp;1}&nbsp;{&nbsp;i}&nbsp;


</pre></blockquote>
<ol class="arabic simple">
<li>les entiers impair entre <tt class="docutils literal"><span class="pre">1</span></tt> et <tt class="docutils literal"><span class="pre">n</span></tt>;</li>
<li>les <tt class="docutils literal"><span class="pre">n</span></tt> premiers entiers impairs;</li>
<li>les entiers entre <tt class="docutils literal"><span class="pre">1</span></tt> et <tt class="docutils literal"><span class="pre">n</span></tt> et qui ne sont divisibles ni par <tt class="docutils literal"><span class="pre">2</span></tt>
ni par <tt class="docutils literal"><span class="pre">3</span></tt> ni par <tt class="docutils literal"><span class="pre">5</span></tt>;</li>
<li>les <tt class="docutils literal"><span class="pre">n</span></tt> premiers entiers qui ne sont
divisibles ni par <tt class="docutils literal"><span class="pre">2</span></tt> ni par <tt class="docutils literal"><span class="pre">3</span></tt> ni par <tt class="docutils literal"><span class="pre">5</span></tt>.</li>
</ol>
</blockquote>



<h2>Functions</h2>
<p>Functions are defined using the <em>def</em> statement, and values are returned using
the <em>return</em> keyword:</p>

{{{id=112|
def f(x):
       return x*x
///
}}}

<!-- link -->

{{{id=113|
f(2)
///
4
}}}

<!-- link -->

{{{id=114|
def fib(n):
       if n &lt;= 1:
           return 1
       else:
           return fib(n-1) + fib(n-2)
///
}}}

<!-- link -->

{{{id=115|
[fib(i) for i in range(10)]
///
[1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
}}}

<p>Functions are first class objects like any other.&nbsp; For example, they can be
passed in as arguments to other functions:</p>
<pre class="literal-block">
.. link

</pre>
{{{id=116|
f
///
<function f at 0x...>
}}}

<!-- link -->

{{{id=117|
def compose(f, x, n):
       for i in range(n):
           x = f(x)
       return x
///
}}}

<!-- link -->

{{{id=118|
compose(f, 2, 3)
///
256
}}}

<!-- link -->

{{{id=119|
def add_one(x):
       return x + 1
///
}}}

<!-- link -->

{{{id=120|
compose(add_one, 2, 3)
///
5
}}}

<p>You can give default values for arguments in functions:</p>

{{{id=121|
def add_n(x, n=1):
       return x + n
///
}}}

<!-- link -->

{{{id=122|
add_n(4)
///
5
}}}

<!-- link -->

{{{id=123|
add_n(4, n=100)
///
104
}}}

<!-- link -->

{{{id=124|
add_n(4, 1000)
///
1004
}}}

<p>You can return multiple things from a function:</p>

{{{id=125|
def g(x):
       return x, x*x
///
}}}

<!-- link -->

{{{id=126|
g(2)
///
(2, 4)
}}}

<!-- link -->

{{{id=127|
type(g)
///
<type 'function'>
}}}

<!-- link -->

{{{id=128|
a,b = g(100)
///
}}}

<!-- link -->

{{{id=129|
a
///
100
}}}

<!-- link -->

{{{id=130|
b
///
10000
}}}

<p>You can also take variable number of arguments and keyword arguments in a
function:</p>

{{{id=131|
def h(*args, **kwds):
       print type(args), args
       print type(kwds), kwds
///
}}}

<!-- link -->

{{{id=132|
h(1,2,3,n=4)
///
<type 'tuple'> (1, 2, 3)<type 'dict'> {'n': 4}
}}}

<p>Let's use the <em>yield</em> instruction to make an generator for the Fibonacci
numbers up to n:</p>

{{{id=133|
def fib_gen(n):
       if n &lt; 1:
           return
       a = b = 1
       yield b
       while b &lt; n:
           yield b
           a, b = b, b+a
///
}}}

<!-- link -->

{{{id=134|
for i in fib_gen(50):
       print i
///
112358132134
}}}

<h3>Exercices</h3>
<blockquote>
<ol class="arabic simple">
<li>Write a function <tt class="docutils literal"><span class="pre">is_even</span></tt> returning <tt class="docutils literal"><span class="pre">True</span></tt> if <tt class="docutils literal"><span class="pre">n</span></tt> is even
and <tt class="docutils literal"><span class="pre">False</span></tt> otherwise.</li>
<li>Write a function <tt class="docutils literal"><span class="pre">every_other</span></tt> which takes a list <tt class="docutils literal"><span class="pre">l</span></tt> and returns a
list containing every other element of <tt class="docutils literal"><span class="pre">l</span></tt></li>
<li>Write a function computing the n-th Fibonacci number. Try to improve
performance.</li>
</ol>
</blockquote>