.. _tutorial-comprehensions: ================================================== 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 1] [1 0] [0 1] [1 0] [1 1] [1 1] [0 1] [1 1] [1 0] 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