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Introduction to Python for Science, Release 0.9.23
import
numpy
as
np
a
=
np
.
linspace(
0
,
32
,
1e7
)
(a)
a
=
a
*
a
(a)
Running this on my computer returns the results faster than I can discern, but certainly
much less than a second. This illustrates an important point:
for loops are slow
. Array
operations run much faster and are therefore to be preferred in any case where you have
a choice. Sometimes finding an array operation that is equivalent to a loop can be diffi-
cult, especially for a novice. Nevertheless, doing so pays rich rewards in execution time.
Moreover, the array notation is usually simpler and clearer, providing further reasons to
prefer array operations over loops.
6.3 List Comprehensions
List comprehensions are a special feature of core Python for processing and constructing
lists. We introduce them here because they use a looping process. They are used quite
commonly in Python coding and they often provide elegant compact solutions to some
common computing tasks.
Consider, for example the
3
×
3
matrix
In [1]:
x
=
[[
1
,
2
,
3
], [
4
,
5
,
6
], [
7
,
8
,
9
]]
Suppose we want to construct a vector from the diagonal elements of this matrix. We
could do so with a
for
loop with an accumulator as follows
In [2]:
diag
=
[]
...:
for
i
in
[
0
,
1
,
2
]:
...:
diag
.
append(x[i][i])
...:
In [3]:
diag
Out[3]:
[
1
,
5
,
9
]
List comprehensions provide a simpler, cleaner, and faster way of doing the same thing
In [4]:
diagLC
=
[x[i][i]
for
i
in
[
0
,
1
,
2
]]
In [5]:
diagLC
Out[5]:
[
1
,
5
,
9
]
110
Chapter 6. Conditionals and Loops
Introduction to Python for Science, Release 0.9.23
A one-line list comprehension replaces a three-line accumulator plus loop code.
Suppose we now want the square of this list:
In [6]:
[y
*
y
for
y
in
diagLC]
Out[6]:
[
1
,
25
,
81
]
Notice here how
y
serves as a dummy variable accessing the various elements of the list
diagLC
.
Extracting a column from a 2-dimensaional array such as
x
is quite easy. For example the
second row is obtained quite simply in the following fashion
In [7]:
x[
1
]
Out[7]:
[
4
,
5
,
6
]
Obtaining a column is not as simple, but a list comprehension makes it quite straightfor-
ward:
In [7]:
c1
=
[a[
1
]
for
a
in
x]
In [8]:
c1
Out[8]:
[
2
,
5
,
8
]
Another, slightly less elegant way to accomplish the same thing is
In [9]:
[x[i][
1
]
for
i
in
range
(
3
)]
Out[9]:
[
2
,
5
,
8
]
Suppose you have a list of numbers and you want to extract all the elements of the list that
are divisible by three. A slightly fancier list comprehension accomplishes the task quite
simply and demonstrates a new feature:
In [10]:
y
=
[
-
5
,
-
3
,
1
,
7
,
4
,
23
,
27
,
-
9
,
11
,
41
]
In [14]:
[a
for
a
in
y
if
a
%
3
==
0
]
Out[14]:
[
-
3
,
27
,
-
9
]
As we see in this example, a conditional statement can be added to a list comprehension.
Here it serves as a filter to select out only those elements that are divisible by three.
6.3. List Comprehensions
111
Introduction to Python for Science, Release 0.9.23
6.4 Exercises
1. Write a program to calculate the factorial of a positive integer input by the user.
Recall that the factorial function is given by
x
! =
x
(
x
−
1)(
x
−
2)
...
(2)(1)
so that
1! = 1
,
2! = 2
,
3! = 6
,
4! = 24
, ...
(a) Write the factorial function using a Python
while
loop.
(b) Write the factorial function using a Python
for
loop.
Check your programs to make sure they work for 1, 2, 3, 5, and beyond, but espe-
cially for the first 5 integers.
2. The following Python program finds the smallest non-trivial (not 1) prime factor of
a positive integer.
n
=
int
(
raw_input
(
"Input an integer > 1: "
))
i
=
2
while
(n
%
i)
!=
0
:
i
+=
1
(
"The smallest factor of n is:"
, i )
(a) Type this program into your computer and verify that it works as advertised.
Then briefly explain how it works and why the while loop always terminates.
(b) Modify the program so that it tells you if the integer input is a prime number
or not. If it is not a prime number, write your program so that it prints out the
smallest prime factor. Using your program verify that the following integers
are prime numbers: 101, 8191, 947431.
3. Consider the matrix list
x = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
.
Write a list comprehension to extract the last column of the matrix [3, 6, 9]. Write
another list comprehension to create a vector of twice the square of the middle
column
[8, 50, 128]
.
4. Write a program that calculates the value of an investment after some number of
years specified by the user if
(a) the principal is compounded annually
(b) the principle is compounded monthly
(c) the principle is compounded daily
112
Chapter 6. Conditionals and Loops
Introduction to Python for Science, Release 0.9.23
Your program should ask the user for the initial investment (principal), the interest
rate in percent, and the number of years the money will be invested (allow for
fractional years). For an initial investment of $1000 at an interest rate of 6%, after
10 years I get $1790.85 when compounded annually, $1819.40 when compounded
monthly, and $1822.03 when compounded daily, assuming 12 months in a year and
365.24 days in a year, where the monthly interest rate is the annual rate divided
by 12 and the daily rate is the annual rate divided by 365 (don’t worry about leap
years).
5. Write a program that determines the day of the week for any given calendar date
after January 1, 1900, which was a Monday. Your program will need to take into
account leap years, which occur in every year that is divisible by 4, except for years
that are divisible by 100 but are not divisible by 400. For example, 1900 was not
a leap year, but 2000 was a leap year. Test that your program gives the following
answers: Monday 1900 January 1, Tuesday 1933 December 5, Wednesday 1993
June 23, Thursday 1953 January 15, Friday 1963 November 22, Saturday 1919
June 28, Sunday 2005 August 28.
6.4. Exercises
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Introduction to Python for Science, Release 0.9.23
114
Chapter 6. Conditionals and Loops