Lab 4

Note: This lab has more problems than we expect everyone to finish during the lab time. If you do not have time to finish all problems, you can continue working on them later (at home, if you have a computer with python set up, in the CSIT lab rooms outside teaching hours, or on one of the computers available across campus), or return to them in a later lab.

Objectives#

The purpose of this week’s lab is to:

understand the indexing of (1-dimensional) sequences;
do some computations over sequences that require iterating through them using loops; and
practice reading and debugging code.

Exercise 0: Reading and debugging code#

The following are attempts to define a function that takes three (numeric) arguments and checks if any one of them is equal to the sum of the other two. For example, any_one_is_sum(1, 3, 2) should return True (because 3 == 1 + 2), while any_one_is_sum(0, 1, 2) should return False.

(a) All of the functions below are incorrect. For each of them, find examples of arguments that cause it to return the wrong answer.

Function 1#

def any_one_is_sum(a,b,c):
    sum_c=a+b
    sum_b=a+c
    sum_a=b+c
    if sum_c == a+b:
        if sum_b == c+a:
            if sum_a == b+c:
                return True
    else:
        return False

Function 2#

def any_one_is_sum(a,b,c):
    if b + c == a:
        print(True)
    if c + b == a:
        print(True)
    else:
        print(False)
    return False

Function 3#

def any_one_is_sum(a, b, c):
    if a+b==c and a+c==b:
        return True
   else:
        return False

(b) For each of the three functions above, can you work out how they are intended to work? That is, what was the idea of the programmer who wrote them? What comments would be useful to add to explain the thinking? Is it possible to fix them by making only a small change to each function?

Exercise 1: Debugging loops#

Below are two attempted solutions to the problems of summing the odd and even digits in a number (from Lab 3), respectively. Both of them, however, have some problems: For some arguments, they may return the wrong answer, or not return at all because the loop never ends.

(a) For each of the two functions, find arguments that cause it to return an incorrect answer, and arguments that cause it to get stuck in an infinite loop (either or both may be possible). Arguments to the function must (of course!) be non-negative integers.

Hint: Add print calls inside the loop to see what is happening. Print the variables that appear in the loop condition, so you can see if they are changing or not (if they are not, then the loop is stuck).

def sum_odd_digits(number):
    dsum = 0
    # only count odd digits
    while number % 2 != 0:
        # add the last digit to the sum
        digit = number % 10
        dsum = dsum + digit
        # divide by 10 (rounded down) to remove the last digit
        number = number // 10
    return dsum

def sum_even_digits(number):
    m = 1 # the position of the next digit
    dsum = 0 # the sum
    while number % (10 ** m) != 0:
        # get the m:th digit
        digit = (number % (10 ** m)) // (10 ** (m - 1))
        # only add it if even:
        if digit % 2 == 0:
            dsum = dsum + digit
        m = m + 1
    return dsum

(b) Like in the previous problem, can you work out how the functions are intended to work? That is, what was the idea of the programmer who wrote them? What comments would be useful to add to explain the thinking? Is it possible to fix the errors you uncovered in your testing by making only a small change to each function?

(c) (advanced) Here is a more complex function:

def mystery(m):
    assert type(m) is int and m >= 0, "m must be a non-negative integer"
    while m > 1:
        i = 2
        while i < m:
            while m % i == 0:
                m = m // i
            i = i + 1
    return i

(It is called mystery because it’s not very obvious what it does.)

The mystery function is meant to work on any non-negative integer input, but some arguments will cause it to get stuck in an infinite loop. Can you find examples of arguments that will cause this?

Learning to read and debug code is a very important skill (and it will also show you the value of good naming and commenting!). There are more debugging exercises towards the end of the lab.

Sequence types#

We have already seen a number of times that all values in python have a type, such as int, float, str, etc. To determine the type of a value we can use the function type(_some expression_). python has three built-in sequence types: lists (type list), strings (type str) and tuples (type tuple). These sequence types are used to represent different kinds of ordered collections.

To write a list literal, write its elements, separated by commas, in a pair of square brackets:

In [1]: my_list = [1, 2, 3, 4, 5, 6]

In [2]: type(my_list)
Out [2]: ...

The elements that you write can be expressions. These are evaluated, and the resulting values become the elements of the list:

In [3]: my_list = [2, 2 + 1, 2 * 2, 2 + 3]

In [4]: my_list
Out [4]: ...

The NumPy array type#

The NumPy library provides a type for represening n-dimensional arrays of values (usually numbers, but also other types, such as Booleans), and functions for doing calculations with arrays.

NumPy is not part of python’s standard library, but it is installed on the CSIT lab computers, and other computers across campus. (If you want to be able to use your own python setup, you have to ensure that you have NumPy installed. The Anaconda distribution includes NumPy, SciPy and matplotlib by default, which is one reason why we recommend it. Read the guide to setting up python for more information.)

Like any library (module) in python, to use NumPy you must first import it:

In [1]: import numpy

Remember that the names of all functions in the imported module are prefixed with the module name. That is, you have to write numpy.linspace(...) instead of just linspace(...). When you import a module, you can give it a shorthand name. For example, if your write

In [1]: import numpy as np

the functions in the NumPy module will be prefixed with just np instead of numpy. In all the examples below, we will assume you have imported NumPy with the abbreviation np.

Information about the functions that NumPy provides is available through the built-in help system. However, you can also find documentation of NumPy and SciPy on-line; the on-line documentation can be easier to navigate. You can also find some tutorials at numpy.org.

We can create an array from a list:

In [1]: my_list = [1, 2, 3, 4, 5, 6]

In [2]: my_array = np.array(my_list)

In [3]: my_array
Out [3]: ...

In [4]: type(my_array)
Out [4]: ...

or simply:

In [5]: np.array([3, 1.2, -2])
Out [5]: ...

As shown in the lectures, there are also several functions for creating arrays with specific contents:

In [6]: np.zeros(10)
Out [6]: ...

In [7]: np.ones(10)
Out [7]: ...

In [8]: np.linspace(-2, 2, 21)
Out [8]: ...

In [9]: np.arange(4,9)
Out [9]: ...

linspace(from, to, num) returns an array of num floating point numbers evenly spaced between from and to.

arange(from,to) returns an array of consecutive integers, starting with from and ending at to - 1. If you provide just one argument, as in arange(to), the starting number defaults to 0.

Indexing sequences#

Both list and NumPy’s ndarray are called sequence data types. Every element in a sequence has an index (position). The first element is at index 0. The length of a sequence is the number of elements in the sequence. The index of the last element is the length minus one. The built-in function len returns the length of any sequence.

Indexing a sequence selects a single element from the sequence (for example, a character if the sequence is a string). Python also allows indexing sequences from the end, using negative indices. That is, -1 also refers to the last element in the sequence, and -len(seq) refers to the first.

Exercise 2(a)#

This exercise is to play with the list sequence type. Execute the following in the python shell. For each expression, try to work out what the output will be before you evaluate the expression.

In [1]: my_list =  [1, 2, 3, 4, 5, 6]

In [2]: my_list[1]
Out [2]: ...

In [3]: my_list[4]
Out [3]: ...

In [4]: my_list[-1]
Out [4]: ...

In [5]: L = len(my_list)

In [6]: my_list[L - 1]
Out [6]: ...

In [7]: my_list[1 - L]
Out [7]: ...

They should all run without error. Is the result of each expression what you expected?

Exercise 2(b)#

As we saw above, you can turn a list into a NumPy array:

In [1]: my_array =  np.array(my_list)

You can now try the indexing expressions that you did with my_list above on my_array instead. Is there any difference in the result?

Exercise 2(c)#

What does the following statement do?

In [1]: my_array = np.arange(1,7)

Iteration over sequences#

Python has two kinds of loop statements: the while loop, which repeatedly executes a suite as long as a condition is true, and the for loop, which executes a suite once for every element of a sequence. (To be precise, the for loop works not only on sequences but on any type that is iterable. All sequences are iterable, but later in the course we will see examples of types that are iterable but not sequences.)

Both kinds of loop can be used to iterate over a sequence. Which one is most appropriate to implement some function depends on what the function needs to do with the sequence. The for loop is simpler to use, but only allows you to look at one element at a time. The while loop is more complex to use (you must initialise and update an index variable, and specify the loop condition correctly) but allows you greater flexibility; for example, you can skip elements in the sequence (increment the index by more than one) or look at elements in more than one position in each iteration.

In the lectures so far, we have only used while loops, and they are sufficient to solve all the problems in this lab. The syntax and execution of the for loop is described in the text books (Downey: Section “Traversal with a for loop” in Chapter 8; Punch & Enbody: Sections 2.1.4 and 2.2.13).

Exercise 3(a)#

The following function takes one argument, a sequence, and counts the number of elements in it that are negative. It is implemented using a while loop.

def count_negative(sequence):
    count = 0
    index = 0
    while index < len(sequence):
        if sequence[index] < 0:
            count = count + 1
        index = index + 1
    return count

Note that the function will work on any sequence type (e.g., both list and array), as long as it contains only numbers.

Rewrite this function so that it uses a for loop instead.

To test your function, you can use the following inputs:

[-1, 0, -2, 1, -3, 2] (3 negative numbers)
np.linspace(2, -2, 5) (2 negative numbers)
np.arange(5) - 3 (3 negative numbers)
np.sin(np.linspace(0, 4*np.pi, 50)) (25 negative numbers)
np.zeros(10) (0 negative numbers)

You can create more test cases by making variations of these, or using other array-creating functions.

Exercise 3(b)#

Write a function called is_increasing that takes a sequence (of numbers) and returns True iff the elements in the array are in (non-strict) increasing order. This means that every element is less than or equal to the next one after it. For example,

for [1, 5, 9] the function should return True
for [3, 3, 4] the function should return True
for [3, 4, 2] the function should return False

Is it best to use a for loop or a while loop for this problem? (Note: Downey describes different solutions to a very similar problem in Section “Looping with Indices” in Chapter 9.)

Test your function with the examples above, and with the examples you used for exercise 3(a).

Also test your function on an empty sequence (that is, a list or array with no elements). An empty list can be created with the expression [] (and an empty array with np.array([])). Does your function work? Does it work on a sequence with one element?

Exercise 3(c)#

The average (or mean) of a sequence of numbers is the sum of the numbers divided by the length of the sequence. You can calculate the average of a sequence of numbers using python’s built-in function sum (which works on any sequence type, as long as it contains numbers), using the NumPy function np.mean (if you convert the sequence to an array first), or writing your own function using a loop over the sequence (as was shown in the lecture).

Write a function most_average(numbers) which finds and returns the number in the input that is closest to the average of the numbers. (You can assume that the argument is a sequence of numbers.) By closest, we mean the one that has the smallest absolute difference from the average. You can use the built-in function abs to find the absolute value of a difference. For example, most_average([1, 2, 3, 4, 5]) should return 3 (the average of the numbers in the list is 3.0, and 3 is clearly closest to this). most_average([3, 4, 3, 1]) should also return 3 (the average is 2.75, and 3 is closer to 2.75 than is any other number in the list).

Programming problems#

Note: These are more substantial programming problems. We do not expect that everyone will finish them within the lab time. If you do not have time to finish them during the lab, you should continue working on them later (at home, in the CSIT labs after teaching hours, or on one of the computers available in the university libraries or other teaching spaces).

Closest matches#

(a) Write two functions, smallest_greater(seq, value) and greatest_smaller(seq, value), that take as argument a sequence and a value, and find the smallest element in the sequence that is greater than the given value, and the greatest element in the sequence that is smaller than the given value, respectively.

For example, if the sequence is [13, -3, 22, 14, 2, 18, 17, 6, 9] and the target value is 4, then the smallest greater element is 6 and the greatest smaller element is 2.

You can assume that all elements in the sequence are of the same type as the target value (that is, if the sequence is an array of numbers, then the target value is a number).
You can not assume that the elements of the sequence are in any particular order.
You should not assume that the sequence is of any particular type; it could be, for example, a NumPy array, a list, or some other sequence type. Use only operations on the sequence that are valid for all sequence types.
What happens in your functions if the target value is smaller or greater than all elements in the sequence?

(b) Same as above, but assume the elements in the sequence are sorted in increasing order; can you find an algorithm that is more efficient in this case?

Counting duplicates#

If the same value appears more than once in a sequence, we say that all copies of it except the first are duplicates. For example, in array(-1, 2, 4, 2, 0, 4), the second 2 and second 4 are duplicates; in the string “Immaterium”, the ‘m’ is duplicated twice (but the ‘i’ is not a duplicate, because ‘I’ and ‘i’ are different characters).

Write a function count_duplicates(seq) that takes as argument a sequence and returns the number of duplicate elements (for example, it should return 2 for both the sequences above). Your function should work on any sequence type (for example, both arrays and strings), so use only operations that are common to all sequence types. For the purpose of deciding if an element is a duplicate, use standard equality, that is, the == operator.

Putting stuff in bins#

A histogram is way of summarising (1-dimensional) data that is often used in descriptive statistics. Given a sequence of values, the range of values (from smallest to greatest) is divided into a number of sections (called “bins”) and the number of values that fall into each bin is counted. For example, if the sequence is array(2.09, 0.5, 3.48, 1.44, 5.2, 2.86, 2.62, 6.31), and we make three bins by placing the dividing lines at 2 and 4, the resulting counts (that is, the histogram) will be the sequence 2, 4, 2, because there are 2 elements less than 2, 4 elements between 2 and 4, and 2 elements > 4.

(a) Write a function count_in_bin(values, lower, upper) that takes as argument a sequence and two values that define the lower and upper sides of a bin, and counts the number of elements in the sequence that fall into this bin. You should treat the bin interval as open on the lower end and closed on the upper end; that is, use a strict comparison lower < element for the lower end and a non-strict comparison element <= upper for the upper end.

(b) Write a function histogram(values, dividers) that takes as argument a sequence of values and a sequence of bin dividers, and returns the histogram as a sequence of a suitable type (say, an array) with the counts in each bin. The number of bins is the number of dividers + 1; the first bin has no lower limit and the last bin has no upper limit. As in (a), elements that are equal to one of the dividers are counted in the bin below.

For example, suppose the sequence of values is the numbers 1,..,10 and the bin dividers are array(2, 5, 7); the histogram should be array(2, 3, 2, 3).

To test your function, you can create arrays of random values using NumPy’s random module:

In [1]: import numpy.random as rnd
In [2]: values = rnd.normal(0, 1, 50)

This creates an array of 50 numbers drawn according to the normal distribution with mean 0 and standard deviation 1. The following creates 10 evenly sized bins covering the range of values:

In [1]: import numpy as np
In [2]: range = np.max(values) - np.min(values)
In [3]: dividers = (np.arange(1, 10) * (range / 10)) + np.min(values)

As you increase the size of the value array, you should find that the histogram becomes more symmetrical and more even.

You can also test your function by comparing it with the histogram function provided by NumPy (see help(numpy.histogram)).

(Advanced) Slicing and array operations#

Python’s built-in sequence types and NumPy’s array provide a mechanism, called slicing, to select parts of a sequence. It is done using the notation sequence[start:end]. There is also an extended form of slicing, which takes three arguments, written sequence[start:end:step].

(Punch & Enbody’s book has a detailed description of slicing, including its extended form, in Section 4.1.5 (page 183). Downey’s book discusses slicing in Section “String Slices” in Chapter 8; the extended form of slicing is only briefly mentioned in Exercise 8-3.) We will come back to slicing in future labs, so if you do not have time to cover it this week, that’s ok.

To understand what the arguments in a slicing expression mean, you can try the following examples:

In [1]: my_array = np.arange(0,20)

In [2]: L = len(my_array)

In [3]: my_array[1:L]
Out [3]: ...

In [4]: my_array[0:L - 1]
Out [4]: ...

In [5]: my_array[0:L:2]
Out [5]: ...

In [6]: my_array[L:0:-1]
Out [6]: ...

In [7]: my_array[6:6+6]
Out [7]: ...

In [8]: my_array[11:11-6:-1]
Out [8]: ...

In [9]: my_array[2*L]
Out [9]: ...

In [10]: my_array[0:2*L]
Out [10]: ...

All of the slice expressions above should work the same on a list with the same elements.

NumPy arrays also support two generalised forms of indexing:

If i is an array of integers, a[i] returns an array with the elements of a at the indices in i. All values in i must be valid indices for a, that is, between 0 and len(a) - 1.
If i is an array of Boolean values a[i] returns an array with the elements of a at positions where the value in i is True. The length of i must be equal to that of a.

Note that these forms of indexing do not work on other python sequence types (such as lists).

Arithmethic operators (+, -, *, **, /, //, %), comparisons (==, !=, <, >, <=, >=) and bit-wise logical operators (& for “and”, | for “or” and ~ for “not”) can all be applied to NumPy arrays. They all perform their operation element-wise. That is, if a and b are two arrays, c = a + b is another array such that c[i] = a[i] + b[i] for i in the range 0 to len(a) - 1. If a is an array and b is a non-array value (such as an int or a float), the operation is done between each element of a and b; again, the result is an array. For the operators that take two arguments (all except ~ and unary -) if they are applied to two arrays, these have to be of the same length. Comparison operators (‘==’, !=, <, >, <=, >=) applied to arrays return arrays of Boolean values.

Exercise 5(a) (advanced)#

Use the arange function, together with array operations, to write expresssions that construct the following arrays:

An array with integers 0, .., n, followed by n - 1, .., 0.
An array of n alternating 1’s and -1’s, starting with a 1.
An array of n alternating 1’s and -1’s, starting with a -1.

It may be helpful to know that:

When an arithmetic operation (such as + or *) is done on an array of Boolean values, these are converted to integers (False == 0 and True == 1).
If a and b are two arrays, the call np.concatenate((a, b)) returns an array that contains the elements of a followed by the elements of b (and thus its length is len(a) + len(b)). Note that there is an extra pair of parentheses around a,b in the function call; this is not a typo.
The function np.repeat can also be useful. Look it up using the help system!

Exercise 5(b) (advanced)#

As was demonstrated in last week’s lecture, with NumPy arrays it is sometimes possible to write complex operations very compactly. For example, counting negative values (as in Exercise 3(a) above) can be done with

def count_negative(sequence):
    return np.sum(np.array(sequence) < 0)

Can you use slicing, array operations and other NumPy functions to implement the is_increasing and most_average functions (on arrays) in a similar way?

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Semester 2, 2019: Lab 4