.. _tutorial-comprehensions:
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Tutorial: Comprehensions, Iterators, and Iterables
==================================================
.. MODULEAUTHOR:: Florent Hivert and
Nicolas M. ThiĆ©ry
.. linkall
List comprehensions
===================
*List comprehensions* are a very handy way to construct lists in
Python. You can use either of the following idioms::
[ for in ]
[ for in if ]
For example, here are some lists of squares::
sage: [ i^2 for i in [1, 3, 7] ]
[1, 9, 49]
sage: [ i^2 for i in range(1,10) ]
[1, 4, 9, 16, 25, 36, 49, 64, 81]
sage: [ i^2 for i in range(1,10) if i % 2 == 1]
[1, 9, 25, 49, 81]
And a variant on the latter::
sage: [i^2 if i % 2 == 1 else 2 for i in range(10)]
[2, 1, 2, 9, 2, 25, 2, 49, 2, 81]
.. TOPIC:: Exercises
#. Construct the list of the squares of the prime integers between 1 and 10::
sage: # edit here
#. Construct the list of the perfect squares less than 100 (hint: use
:func:`srange` to get a list of Sage integers together with the
method ``i.sqrtrem()``)::
sage: # edit here
One can use more than one iterable in a list comprehension::
sage: [ (i,j) for i in range(1,6) for j in range(1,i) ]
[(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4)]
.. warning:: Mind the order of the nested loop in the previous expression.
If instead one wants to build a list of lists, one can use nested lists as in::
sage: [ [ binomial(n, i) for i in range(n+1) ] for n in range(10) ]
[[1],
[1, 1],
[1, 2, 1],
[1, 3, 3, 1],
[1, 4, 6, 4, 1],
[1, 5, 10, 10, 5, 1],
[1, 6, 15, 20, 15, 6, 1],
[1, 7, 21, 35, 35, 21, 7, 1],
[1, 8, 28, 56, 70, 56, 28, 8, 1],
[1, 9, 36, 84, 126, 126, 84, 36, 9, 1]]
.. TOPIC:: Exercises
#. Compute the list of pairs `(i,j)` of non negative integers such
that ``i`` is at most `5`, ``j`` is at most ``8``, and ``i`` and
``j`` are co-prime::
sage: # edit here
#. Compute the same list for `i < j < 10`::
sage: # edit here
Iterators
=========
Definition
----------
To build a comprehension, Python actually uses an *iterator*. This is
a device which runs through a bunch of objects, returning one at each
call to the ``next`` method. Iterators are built using parentheses::
sage: it = (binomial(8, i) for i in range(9))
sage: it.next()
1
::
sage: it.next()
8
sage: it.next()
28
sage: it.next()
56
You can get the list of the results that are not yet *consumed*::
sage: list(it)
[70, 56, 28, 8, 1]
Asking for more elements triggers a ``StopIteration`` exception::
sage: it.next()
Traceback (most recent call last):
...
StopIteration
An iterator can be used as argument for a function. The two following
idioms give the same results; however, the second idiom is much more
memory efficient (for large examples) as it does not expand any list
in memory::
sage: sum( [ binomial(8, i) for i in range(9) ] )
256
sage: sum( binomial(8, i) for i in xrange(9) )
256
.. TOPIC:: Exercises
#. Compute the sum of `\binom{10}{i}` for all even `i`::
sage: # edit here
#. Compute the sum of the gcd's of all co-prime numbers `i, j` for `i2 )
True
It is well know that if ``2^p-1`` is prime then ``p`` is prime::
sage: def mersenne(p): return 2^p -1
sage: [ is_prime(p) for p in range(20) if is_prime(mersenne(p)) ]
[True, True, True, True, True, True, True]
The converse is not true::
sage: all( is_prime(mersenne(p)) for p in range(1000) if is_prime(p) )
False
Using a list would be much slower here::
sage: %time all( is_prime(mersenne(p)) for p in range(1000) if is_prime(p) ) # not tested
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.00 s
False
sage: %time all( [ is_prime(mersenne(p)) for p in range(1000) if is_prime(p)] ) # not tested
CPU times: user 0.72 s, sys: 0.00 s, total: 0.73 s
Wall time: 0.73 s
False
You can get the counterexample using :func:`exists`. It takes two
arguments: an iterator and a function which tests the property that
should hold::
sage: exists( (p for p in range(1000) if is_prime(p)), lambda p: not is_prime(mersenne(p)) )
(True, 11)
An alternative way to achieve this is::
sage: counter_examples = (p for p in range(1000) if is_prime(p) and not is_prime(mersenne(p)))
sage: counter_examples.next()
11
.. TOPIC:: Exercises
#. Build the list `\{i^3 \mid -10*
[0 1]
[1 0]
[0 1]
[1 1]
[1 1]
[0 1]
[1 1]
[1 0]
[1 0]
[1 1]
sage: for p in Partitions(3): print p
[3]
[2, 1]
[1, 1, 1]
.. skip
Beware of infinite loops::
sage: for p in Partitions(): print p # not tested
.. skip
::
sage: for p in Primes(): print p # not tested
Infinite loops can nevertheless be very useful::
sage: exists( Primes(), lambda p: not is_prime(mersenne(p)) )
(True, 11)
sage: counter_examples = (p for p in Primes() if not is_prime(mersenne(p)))
sage: counter_examples.next()
11*