Attachment 'chap2-list_and_for.rst'

Chapter 2: Lists and for loops

Authors:	Vincent Delecroix
License:	CC BY-SA 3.0

In the first tutorial, you learnt how to call Sage functions and solve basic problems. To do more advanced computations you will need to learn a bit of programming. The aim of this tutorial is to introduce you to some basic Python programming: list that is one of the builtin Python data structures and for loops.

For reference you can have a look at [pydoc-datastructures] and [pydoc-controlflow].

Manipulating lists

In Python lists are constructed using square brackets ([ and ]) such as:

sage: l = [1, "two", -5.7, Primes()]

As you can notice, the data contained in a list is not necessarily homogeneous. To access elements of a list, one also uses square brackets:

sage: l[0]    # the first element

sage: l[2]    # the third elements

sage: l[-1]   # the last element

sage: l[4]    # this is out of range

An important function is len that returns the length of the list:

sage: len(l)   # number of elements in l

Given a list l and a non-negative integer n the operation l * n duplicates n times the list l:

sage: l * 3

Given two lists l1 and l2 the operation l1 + l2 is the concatenation of l1 and l2:

sage: [1, 2, 3] + [4, 5, 6]

Exercise 2.1

Construct the list that contains 1 time 1, 2 times 2, 3 times 3, ..., 10 times 10 (i.e. [1, 2, 2, 3, 3, 3, ..., 10]

A for loop is a control flow that will allow to repeat some instructions for all elements of an iterable such as list.:

sage: l = [1, "hello", 3]
sage: for x in l:
....:     print(x)

You can have more than one instructions inside a for loop:

sage: l = [1, 2, 3, 4, 5]
sage: for x in l:
....:     y = 3 * x^ - 2
....:     print(y)

And it is possible to nest them:

sage: l1 = [1, 2, 3]
sage: l2 = [4, 7, 13]
sage: for x in l1:
....:     for y in l2:
....:         print(x, y, x^y)

Often you might want to perform a for loop for integers in a given range. To do so there are the functions range (from Python) and srange (from Sage).:

sage: for i in range(10):
....:     print(i)

sage: for i in srange(10):
....:     print(i)

The difference between range and srange is that range produces Python integer while srange produces Sage integers. You can compare:

sage: for i in range(10):
....:     print(i.is_prime())

sage: for i in srange(10):
....:     print(i.is_prime())

For both range and srange there are three possible syntaxes

range(stop): integers from 0 to stop − 1 included
range(start, stop): integers from start to stop − 1
range(start, stop, step): integers from start to stop and with a difference of step between each consecutive terms

Exercise 2.2

Write a for loop which prints the factorization for each integer from 1 to 100.

Exercise 2.3

Write a for loop which prints the primitive of the functions sin(x), cos(x), tan(x), log(x), exp(x), sinh(x) and cosh(x).

Write a for loop which prints the derivates of the functions f_n(x) = xⁿ for all values of n from 1 to 20.

Exercise 2.4

Redo exercise 2.1 using a for loop

Repeat it with 100 instead of 10

Exercise 2.5

Write some code that displays the following figure using a loop

sage: print('*')
....: print('**')
....: print('***')
....: print('****')
....: print('*****')
....: print('******')

Write some code that displays the following figure using a loop:

sage: print('*')
....: print('**')
....: print('***')
....: print('****')
....: print('*****')
....: print('******')
....: print('*****')
....: print('****')
....: print('***')
....: print('**')
....: print('*')

Write some code that displays the following figure using a loop:

sage: print('   *')
....: print('  ***')
....: print(' *****')
....: print('*******')
....: print(' *****')
....: print('  ***')
....: print('   *')

Exercise 2.6

What is the value of

²⁰⎲⎳_k = 1k^k

Solve Eulerproblem48 < https : ⁄ ⁄ projecteuler.net ⁄ problem = 48 > .

Note that Python tuple (constructed with parenthesis) and Python string (constructed with quotes ' or ") behave the same with respect to many operations:

sage: t1 = (0, "hello", -2/3)

sage: t1[0]

sage: t1[-1]

sage: t1 * 3

sage: s1 = "I am in"
sage: s2 = "Saint-Flour"
sage: s1 + " " + s2

Exercise 2.7

Here is a list describing the size of 64 files:

sage: L = ["12K", "12K", "12K", "12K", "12K", "12K", "12K", "12K", "16K",
....: "16K", "16K", "20K", "20K", "20K", "24K", "24K", "24K", "24K", "24K",
....: "24K", "24K", "24K", "24K", "28K", "32K", "40K", "4K", "4K", "4K", "4K",
....: "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K",
....: "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K", "4K",
....: "4K", "4K", "4K", "4K", "8K", "8K", "8K", "8K", "8K", "8K"]

Compute the total size of the files, their average and the median

(hint: to convert a string s like "12" into an integer simply do ZZ(s))

How many files are there of each size? (you might want to look the available method on list object)

Construct the string that is the concatenation of each file size separated by a space

Can you construct the same string by increasing size?

(hint: you might want to sort the list first... use tab-completion to find the appropriate list method to do this)

And by decreasing size?

(hint: have a look at the documentation of the method you used for sorting in the previous question)

Exercise 2.8

Let S₀ be the unit square with vertices (0, 0), (1, 0), (1, 1) and (0, 1). We define the square S_n obtained by joining the middle of each side of S_n − 1.

Draw on the same pictures S₀, S₁, S₂, ... up to S₁₀.

Do the same graphics starting from another quadrilateral which is not regular

Do the same starting from a pentagon (five vertices)

Exercise 2.9

What does the following code?:

sage: x = 1.0
sage: for i in range(10):
....:     x = (x + 2.0 / x) / 2.0

What is the difference with:

sage: x = 1
sage: for i in range(10):
....:     x = (x + 2 / x) / 2

Exercise 2.10

Solve Euler problem 13. To simplify the exercise, the list of numbers has been created in the following cell:

sage: L = [37107287533902102798797998220837590246510135740250,
....: 46376937677490009712648124896970078050417018260538,
....: 74324986199524741059474233309513058123726617309629,
....: 91942213363574161572522430563301811072406154908250,
....: 23067588207539346171171980310421047513778063246676,
....: 89261670696623633820136378418383684178734361726757,
....: 28112879812849979408065481931592621691275889832738,
....: 44274228917432520321923589422876796487670272189318,
....: 47451445736001306439091167216856844588711603153276,
....: 70386486105843025439939619828917593665686757934951,
....: 62176457141856560629502157223196586755079324193331,
....: 64906352462741904929101432445813822663347944758178,
....: 92575867718337217661963751590579239728245598838407,
....: 58203565325359399008402633568948830189458628227828,
....: 80181199384826282014278194139940567587151170094390,
....: 35398664372827112653829987240784473053190104293586,
....: 86515506006295864861532075273371959191420517255829,
....: 71693888707715466499115593487603532921714970056938,
....: 54370070576826684624621495650076471787294438377604,
....: 53282654108756828443191190634694037855217779295145,
....: 36123272525000296071075082563815656710885258350721,
....: 45876576172410976447339110607218265236877223636045,
....: 17423706905851860660448207621209813287860733969412,
....: 81142660418086830619328460811191061556940512689692,
....: 51934325451728388641918047049293215058642563049483,
....: 62467221648435076201727918039944693004732956340691,
....: 15732444386908125794514089057706229429197107928209,
....: 55037687525678773091862540744969844508330393682126,
....: 18336384825330154686196124348767681297534375946515,
....: 80386287592878490201521685554828717201219257766954,
....: 78182833757993103614740356856449095527097864797581,
....: 16726320100436897842553539920931837441497806860984,
....: 48403098129077791799088218795327364475675590848030,
....: 87086987551392711854517078544161852424320693150332,
....: 59959406895756536782107074926966537676326235447210,
....: 69793950679652694742597709739166693763042633987085,
....: 41052684708299085211399427365734116182760315001271,
....: 65378607361501080857009149939512557028198746004375,
....: 35829035317434717326932123578154982629742552737307,
....: 94953759765105305946966067683156574377167401875275,
....: 88902802571733229619176668713819931811048770190271,
....: 25267680276078003013678680992525463401061632866526,
....: 36270218540497705585629946580636237993140746255962,
....: 24074486908231174977792365466257246923322810917141,
....: 91430288197103288597806669760892938638285025333403,
....: 34413065578016127815921815005561868836468420090470,
....: 23053081172816430487623791969842487255036638784583,
....: 11487696932154902810424020138335124462181441773470,
....: 63783299490636259666498587618221225225512486764533,
....: 67720186971698544312419572409913959008952310058822,
....: 95548255300263520781532296796249481641953868218774,
....: 76085327132285723110424803456124867697064507995236,
....: 37774242535411291684276865538926205024910326572967,
....: 23701913275725675285653248258265463092207058596522,
....: 29798860272258331913126375147341994889534765745501,
....: 18495701454879288984856827726077713721403798879715,
....: 38298203783031473527721580348144513491373226651381,
....: 34829543829199918180278916522431027392251122869539,
....: 40957953066405232632538044100059654939159879593635,
....: 29746152185502371307642255121183693803580388584903,
....: 41698116222072977186158236678424689157993532961922,
....: 62467957194401269043877107275048102390895523597457,
....: 23189706772547915061505504953922979530901129967519,
....: 86188088225875314529584099251203829009407770775672,
....: 11306739708304724483816533873502340845647058077308,
....: 82959174767140363198008187129011875491310547126581,
....: 97623331044818386269515456334926366572897563400500,
....: 42846280183517070527831839425882145521227251250327,
....: 55121603546981200581762165212827652751691296897789,
....: 32238195734329339946437501907836945765883352399886,
....: 75506164965184775180738168837861091527357929701337,
....: 62177842752192623401942399639168044983993173312731,
....: 32924185707147349566916674687634660915035914677504,
....: 99518671430235219628894890102423325116913619626622,
....: 73267460800591547471830798392868535206946944540724,
....: 76841822524674417161514036427982273348055556214818,
....: 97142617910342598647204516893989422179826088076852,
....: 87783646182799346313767754307809363333018982642090,
....: 10848802521674670883215120185883543223812876952786,
....: 71329612474782464538636993009049310363619763878039,
....: 62184073572399794223406235393808339651327408011116,
....: 66627891981488087797941876876144230030984490851411,
....: 60661826293682836764744779239180335110989069790714,
....: 85786944089552990653640447425576083659976645795096,
....: 66024396409905389607120198219976047599490197230297,
....: 64913982680032973156037120041377903785566085089252,
....: 16730939319872750275468906903707539413042652315011,
....: 94809377245048795150954100921645863754710598436791,
....: 78639167021187492431995700641917969777599028300699,
....: 15368713711936614952811305876380278410754449733078,
....: 40789923115535562561142322423255033685442488917353,
....: 44889911501440648020369068063960672322193204149535,
....: 41503128880339536053299340368006977710650566631954,
....: 81234880673210146739058568557934581403627822703280,
....: 82616570773948327592232845941706525094512325230608,
....: 22918802058777319719839450180888072429661980811197,
....: 77158542502016545090413245809786882778948721859617,
....: 72107838435069186155435662884062257473692284509516,
....: 20849603980134001723930671666823555245252804609722,
....: 53503534226472524250874054075591789781264330331690]

Modifying lists

Here are three ways to modify an already existing list l:

l[i] = j: modify the element at position i to become j
l.append(j): append the element j at the end of l
l.pop(): remove the last element of the list l and return it
l.extend(ll): add to l the content of the iterable ll

(there is also l.insert(i, j) that we will not use)

Exercise 2.11

What is the value of the list l at the end of the execution:

sage: l = [1, 2, 3]
sage: l.append(-1)
sage: l[1] = 7

(think about it before executing the lines)

What is the value of the list l at the end of the execution:

sage: l = [1]
sage: l.extend(l)
sage: l.extend(l)

(think about it before executing the lines)

Exercise 2.12

Given the following list of integers:

sage: l = [231, 442, 534, 667, 827, 314, 299, 351, 257, 688, 661, 123,
....: 567, 247, 151, 222, 605, 307]

Modify it so that each number at an even position is replaced by its double

Exercise 2.13

Let the following lists:

sage: t1 = [31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31]
sage: t2 = ['January', 'February', 'March', 'April', 'May', 'June',
....: 'July', 'August', 'September', 'October', 'November', 'December']

Using t1 and t2 create a new list t3 containing all elements of the two lists alternating them in such a way that each Month is followed by the corresponding number of days, that is, ['Janvier',31,'Février',28,'Mars',31, etc...]

Exercise 2.14

Using the recurrence relation satisfied by the binomial numbers

(n + 1k) = (nk) + (nk − 1)

compute the list (200), (201), …, (2020). In order to do that you need to start from the list [1] and design a loop that constructs successively [1, 1], then [1, 2, 1], then [1, 3, 3, 1], etc. (It can be done using only one list.)

Modify your loop to compute the Stirling numbers of the second kind that satisfies

S(n + 1, k) = kS(n, k) + S(n, k − 1)

with initial conditions S(0, 0) = 1 and S(n, 0) = S(0, n) = 1.

List comprehension

List comprehension is a flexible way to build list. To build the list of squares n² from n = 1 to n = 10 one can do:

sage: l = [n^2 for n in srange(1, 11)]
....: print(l)

Exercise 2.15

Construct the same list of squares using a for loop and the method .append().

Exercise 2.16

Construct the list of powers of 5 for all values of exponents in the interval [1, 20]

Exercise 2.17

The Fibonacci sequence is defined by F₀ = 0, F₁ = 1 and for all n ≥ 2, F_n = F_n − 1 + F_n − 2.

Make the list of the first 50 Fibonacci numbers F_n.

Using a for loop, print the values of F²_n − F_n − 1F_n + 1 for n between 1 and 48.

Using a for loop, print the values of F_2n − F²_n − F²_n − 1.

What do you remark? Could you prove it?

Exercise 2.18

What happens when you try to modify a string or a tuple?

Exercise 2.19

Solve the Euler problem 18 and 67 about "maximum sum paths"

Exercise 2.20 (Champernowne constant)

The Champernowne constant is the real number obtained by concatenating all natural numbers in base 10

C = 0.1234567891011121314151617181920212223…

The Champernowne word is the sequence of digits. Using a for loop, construct the begining of the Champernowne word obtained by concatenating the integers from 1 to 100

(hint: use the method n.digits() of Sage integers together with the method l.extend(ll) on lists)

Then solve Euler problem 48

Further comments

To learn more about iterators (that can be thought as "lazy lists") and in particular how to construct them, you can have a look at the Sage thematic tutorial on Comprehensions [sagett-comprehensions].

References

[pydoc-datastructures]	https://docs.python.org/2.7/tutorial/datastructures.html

[pydoc-datastructures]

https://docs.python.org/2.7/tutorial/datastructures.html

[pydoc-controlflow]

https://docs.python.org/2.7/tutorial/controlflow.html

[sagett-comprehensions]

https://doc.sagemath.org/html/en/thematic_tutorials/tutorial-comprehensions.html

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