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1.1 Theﬁrstprogrammingencounter:aformula

Therearemanymorewaystospecifyformats.Forexample,ewrites

anumberinscientiﬁcnotation,i.e.,withanumberbetween1and10

followedbyapower of10,asin1.2432·10−3.Onacomputersucha

numberiswrittenintheform1.2432e-03.CapitalEintheexponentis

alsopossible,justreplaceebyE,withtheresult1.2432E-03.

For decimalnotation weusetheletter f,asin%f,andtheoutput

number thenappearswithdigitsbefore and/or after a comma, , e.g.,

0.0012432insteadof1.2432E-03.Withthegformat,theoutputwill

usescientiﬁcnotationforlargeorsmallnumbersanddecimalnotation

otherwise.Thisformatisnormallywhatgivesmostcompactoutputofa

realnumber.Alowercasegleadstolowercaseeinscientiﬁcnotation,

whileuppercaseGimpliesEinsteadof eintheexponent.

Onecanalsospecifytheformatas10.4for14.6E,meaninginthe

ﬁrst case that a ﬂoat is written in decimal notation with four decimals

in a ﬁeld of width equal to 10 characters, and in the second case a ﬂoat

written in scientiﬁc notation with six decimals in a ﬁeld of width 14

characters.

Here is a list of some important printf format speciﬁcations (the

program printf_demo.py exempliﬁes many of the constructions):

Format

Meaning

astring

an integer

%0xd

an integer padded with x leading zeros

decimal notation with six decimals

compact scientiﬁc notation, e in the exponent

compact scientiﬁc notation, E in the exponent

compact decimal or scientiﬁc notation (with e)

compact decimal or scientiﬁc notation (with E)

%xz

format z right-adjusted in a ﬁeld of width x

%-xz

format z left-adjusted in a ﬁeld of width x

%.yz

format z with ydecimals

%x.yz

format z with y decimals in a ﬁeld of width x

the percentage sign % itself

For a complete speciﬁcation of the possible printf-style format strings,

follow the link from the item printf-style formatting in the index2 of the

Python Standard Library online documentation.

We may try out some formats by writing more numbers to the screen

in our program (the corresponding ﬁle is ball_print2.py):

v0 = 5

g = 9.81

t = 0.6

y = v0*t - 0.5*g*t**2

print """

At t=%f s, a ball with

initial velocity v0=%.3E m/s

is located at the height %.2f m.

""" % (t, v0, y)

http://docs.python.org/2/genindex.html

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1 Computing with formulas

Observe here that we use a triple-quoted string, recognized by starting

and ending with three single or double quotes:”’ or""". Triple-quoted

strings are used for text that spans several lines.

In the e print statement above, we print t t in the f format, which

by default implies six decimals;v0 is written in the.3E format, which

implies three decimals and the number spans as narrow ﬁeld as possible;

andy is written with two decimals in decimal notation in as narrow ﬁeld

as possible. The output becomes

Terminal

Terminal> python ball_print2.py

At t=0.600000 s, a ball with

initial velocity v0=5.000E+00 m/s

is located at the height 1.23 m.

You should look at each number in the output and check the formatting

in detail.

Format string syntax. . Pythonoﬀersallthefunctionalityoftheprintf

format and much more through a diﬀerent syntax, often known as format

string syntax. Let us illustrate this syntax on the one-line output previ-

ously used to show the printf construction. The corresponding format

string syntax reads

print ’At t={t:g} s, the height of the ball is {y:.2f} m.’.format(

t=t, y=y)

The “slots” where variables are inserted are now recognized by curly

braces rather than a percentage sign. The name of the variable is listed

with an optional colon and format speciﬁer of the same kind as was

used for the printf format. The various variables and their values must

be listed at the end as shown. This time the “slots” have names so the

sequence of variables is not important.

The multi-line example is written as follows in this alternative format:

print """

At t={t:f} s, a ball with

initial velocity v0={v0:.3E} m/s

is located at the height {y:.2f} m.

""".format(t=t, v0=v0, y=y)

The newline character. . Weoftenwantacomputerprogramtowrite

out text that spans several lines. In the last example we obtained such

output by triple-quoted strings. We could also use ordinary single-quoted

strings and a special character for indicating where line breaks should

occur. This special character reads\n, i.e., a backslash followed by the

letter n. The two print statements

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1.2 Computer science glossary

print """y(t) is

the position of

our ball."""

print ’y(t) is\nthe position of\nour ball’

result in identical output:

y(t) is

the position of

our ball.

1.2 Computer science glossary

It is now time to pick up some important words that programmers use

when they talk about programming: algorithm, application, assignment,

blanks (whitespace), bug, code, code segment, code snippet, debug, de-

bugging, execute, executable, implement, implementation, input, library,

operating system, output, statement, syntax, user, verify, and veriﬁcation.

These words are frequently used in English in lots of contexts, yet they

have a precise meaning in computer science.

Program and code are interchangeable terms. A code/program segment

is a collection of consecutive statements from a program. Another term

with similar meaning is code snippet. Many also use the word application

in the same meaning as program and code. A related term is source code,

which is the same as the text that constitutes the program. You ﬁnd the

source code of a program in one or more text ﬁles. (Note that text ﬁles

normally have the extension.txt, while program ﬁles have an extension

related to the programming language, e.g.,.py for Python programs.

The content of a .py ﬁle is, nevertheless, plain text as in a .txt ﬁle.)

We talk about running a program, or equivalently executing a program

or executing a ﬁle. The ﬁle we execute is the ﬁle in which the program

text is stored. This ﬁle is often called an executable or an application. The

program text may appear in many ﬁles, but the executable is just the

single ﬁle that starts the whole program when we run that ﬁle. Running

aﬁle can be done in several ways, for instance, by double-clicking the

ﬁle icon, by writing the ﬁlename in a terminal window, or by giving the

ﬁlename to some program. This latter technique is what we have used

so far in this book: we feed the ﬁlename to the programpython. That

is, we execute a Python program by executing another programpython,

which interprets the text in our Python program ﬁle.

The term library is widely used for a collection of generally useful

program pieces that can be applied in many diﬀerent contexts. Having

access to good libraries means that you do not need to program code

snippets that others have already programmed (most probable in a better

way!). There are huge numbers of Python libraries. In Python terminology,

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1 Computing with formulas

the libraries are composed of modules and packages. Section 1.4 gives

a ﬁrst glimpse of themath module, which contains a set of standard

mathematical functions forsinx,cosx,lnx ,ex,sinhx,sin−1x, etc. Later,

you will meet many other useful modules. Packages are just collections of

modules. The standard Python distribution comes with a large number

of modules and packages, but you can download many more from the

Internet, see in particularwww.python.org/pypi. Very often, when you

encounter a programming task that is likely to occur in many other

contexts, you can ﬁnd a Python module where the job is already done.

To mention just one example, say you need to compute how many days

there are between two dates. This is a non-trivial task that lots of other

programmers must have faced, so it is not a big surprise that Python

comes with a module datetime to do calculations with dates.

The recipe for what the computer is supposed to do in a program is

called algorithm. In the examples in the ﬁrst couple of chapters in this

book, the algorithms are so simple that we can hardly distinguish them

from the program text itself, but later in the book we will carefully set

up an algorithm before attempting to implement it in a program. This

is useful when the algorithm is much more compact than the resulting

program code. The algorithm in the current example consists of three

steps:

• initialize the variables v

,g, and t with numerical values,

• evaluate y according to the formula (1.1),

• print the y value to the screen.

The Python program is very close to this text, but some less experienced

programmers may want to write the tasks in English before translating

them to Python.

The implementation of an algorithm is the process of writing and

testing a program. The testing phase is also known as veriﬁcation: After

the program text is written we need to verify that the program works

correctly. This is a very important step that will receive substantial

attention in the present book. Mathematical software produce numbers,

and it is normally quite a challenging task to verify that the numbers

are correct.

An error in a program is known as a bug, and the process of locating

and removing bugs is called debugging. Many look at debugging as the

most diﬃcult and challenging part of computer programming. We have in

fact devoted Appendix F to the art of debugging in this book. The origin

of the strange terms bug and debugging can be found in Wikipedia

Programs are built of statements. There are many types of statements:

http://en.wikipedia.org/wiki/Software_bug#Etymology

1.2 Computer science glossary

v0 = 3

is an assignment statement, while

print y

is a print statement. It is common to have one statement on each line, but

it is possible to write multiple statements on one line if the statements

are separated by semi-colon. Here is an example:

v0 = 3; g = 9.81; t = 0.6

y = v0*t - 0.5*g*t**2

print y

Although most newcomers to computer programming will think they

understand the meaning of the lines in the above program, it is important

to be aware of some major diﬀerences between notation in a computer

program and notation in mathematics. When you see the equality sign=

in mathematics, it has a certain interpretation as an equation (x+2 = 5)

or a deﬁnition (f(x) =x2 + 1). In a computer program, however, the

equality sign has a quite diﬀerent meaning, and it is called an assignment.

The right-hand side of an assignment contains an expression, which is a

combination of values, variables, and operators. When the expression is

evaluated, it results in a value that the variable on the left-hand side will

refer to. We often say that the right-hand side value is assigned to the

variable on the left-hand side. In the current context it means that we in

the ﬁrst line assign the number 3 to the variablev0, 9.81 tog, and 0.6 to

t.Inthenextline,theright-handsideexpression v0*t - 0.5*g*t**2

is ﬁrst evaluated, and the result is then assigned to the y variable.

Consider the assignment statement

y = y + 3

This statement is mathematically false, but in a program it just means

that we evaluate the right-hand side expression and assign its value to

the variabley. That is, we ﬁrst take the current value of f y and add 3.

The value of this operation is assigned toy. The old value of f y is then

lost.

You may think of the= as an arrow,y <- y+3, rather than an equality

sign, to illustrate that the value to the right of the arrow is stored in the

variable to the left of the arrow. In fact, the R programming language

for statistical computing actually applies an arrow, many old languages

(like Algol, Simula, and Pascal) used:= to explicitly state that we are

not dealing with a mathematical equality.

An example will illustrate the principle of assignment to a variable:

y = 3

print y

y = y + 4

1 Computing with formulas

print y

y = y*y

print y

Running this program results in three numbers:3,7,49. Go through the

program and convince yourself that you understand what the result of

each statement becomes.

Acomputer program must have correct syntax, meaning that the text

in the program must follow the strict rules of the computer language for

constructing statements. For example, the syntax of the print statement

is the word d print, followed by one or more spaces, followed by an

expression of what we want to print (a Python variable, text enclosed in

quotes, a number, for instance). Computers are very picky about syntax!

For instance, a human having read all the previous pages may easily

understand what this program does,

myvar = 5.2

prinnt Myvar

but the computer will ﬁnd two errors in the last line: prinnt is an

unknown instruction andMyvar is an undeﬁned variable. Only the ﬁrst

error is reported (a syntax error), because Python stops the program

once an error is found. All errors that Python ﬁnds are easy to remove.

The diﬃculty with programming is to remove the rest of the errors, such

as errors in formulas or the sequence of operations.

Blanks may or may not be important in Python programs. In Sec-

tion 2.1.2 you will see that blanks are in some occasions essential for a

correct program. Around = or arithmetic operators, however, blanks do

not matter. We could hence write our program from Section 1.1.7 as

v0=3;g=9.81;t=0.6;y=v0*t-0.5*g*t**2;print y

This is not a good idea because blanks are essential for easy reading

of a program code, and easy reading is essential for ﬁnding errors, and

ﬁnding errors is the diﬃcult part of programming. The recommended

layout in Python programs speciﬁes one blank around=,+, and-, and no

blanks around*,/, and**. Note that the blank afterprint is essential:

printisacommandinPythonand printyisnotrecognizedasanyvalid

command. (Python will complain thatprinty is an undeﬁned variable.)

Computer scientists often use the term whitespace when referring to a

blank. (To be more precise, blank is the character produced by the space

bar on the keyboard, while whitespace denotes any character(s) that, if

printed, do not print ink on the paper: a blank, a tabulator character

(produced by backslash followed by t), or a newline character (produced

by backslash followed by n). (The newline character is explained in

Section 1.1.11.)

When we interact with computer programs, we usually provide some

information to the program and get some information out. It is common

1.2 Computer science glossary

to use the term input data, or just input, for the information that must

be known on beforehand. The result from a program is similarly referred

to as output data, or just output. In our example,v

,g, andt constitute

input, whiley is output. All input data must be assigned values in the

program before the output can be computed. Input data can be explicitly

initialized in the program, as we do in the present example, or the data

can be provided by the user through keyboard typing while the program

is running (see Chapter 4). Output data can be printed in the terminal

window, as in the current example, displayed as graphics on the screen,

as done in Section 5.3, or stored in a ﬁle for later access, as explained in

Section 4.6.

The word user usually has a special meaning in computer science:

It means a human interacting with a program. You are a user of a

text editor for writing Python programs, and you are a user of your

own programs. When you write programs, it is diﬃcult to imagine how

other users will interact with the program. Maybe they provide wrong

input or misinterpret the output. Making user-friendly programs is very

challenging and depends heavily on the target audience of users. The

author had the average reader of the book in mind as a typical user when

developing programs for this book.

Acentral part of a computer is the operating system. This is actually a

collection of programs that manages the hardware and software resources

on the computer. There are three dominating operating systems today

on computers: Windows, Mac OS X, and Linux. In addition, we have

Android and iOS for handheld devices. Several versions of Windows

have appeared since the 1990s: Windows 95, 98, 2000, ME, XP, Vista,

Windows 7, and Windows 8. Unix was invented already in 1970 and comes

in many diﬀerent versions. Nowadays, two open source implementations

of Unix, Linux and Free BSD Unix, are most common. The latter forms

the core of the Mac OS X operating system on Macintosh machines,

while Linux exists in slightly diﬀerent ﬂavors: Red Hat, Debian, Ubuntu,

and OpenSuse to mention the most important distributions. We will use

the term Unix in this book as a synonym for all the operating systems

that inherit from classical Unix, such as Solaris, Free BSD, Mac OS X,

and any Linux variant. As a computer user and reader of this book, you

should know exactly what operating system you have.

The user’s interaction with the operation system is through a set of

programs. The most widely used of these enable viewing the contents of

folders or starting other programs. To interact with the operating system,

as a user, you can either issue commands in a terminal window or use

graphical programs. For example, for viewing the ﬁle contents of a folder

you can run the commandls in a Unix terminal window ordir in a

DOS (Windows) terminal window. The graphical alternatives are many,

some of the most common are Windows Explorer on Windows, Nautilus

and Konqueror on Unix, and Finder on Mac. To start a program, it is

1 Computing with formulas

common to double-click on a ﬁle icon or write the program’s name in a

terminal window.

1.3 Another formula: Celsius-Fahrenheit conversion

Our next example involves the formula for converting temperature mea-

sured in Celsius degrees to the corresponding value in Fahrenheit degrees:

F =

C+32

(1.3)

In this formula,C is the amount of degrees in Celsius, andF is the

corresponding temperature measured in Fahrenheit. Our goal now is to

write a computer program that can computeF from (1.3) whenC is

known.

1.3.1 Potential error: integer division

Straightforward coding of the formula. . Astraightforwardattemptat

coding the formula (1.3) goes as follows:

C = 21

F = (9/5)*C + 32

print F

The parentheses around d 9/5 are not strictly needed, i.e., (9/5)*C is

computationally identical to9/5*C, but parentheses remove any doubt

that9/5*C could mean9/(5*C). Section 1.3.4 has more information on

this topic.

When run under Python version 2.x, the program prints the value 53.

You can ﬁnd the program in the ﬁlec2f_v1.py in thesrc/formulas

folder in the folder tree of example programs from this book (downloaded

fromhttp://hplgit.github.com/scipro-primer). Thev1 part of the

name stands for version 1. Throughout this book, we will often develop

several trial versions of a program, but remove the version number in

the ﬁnal version of the program.

Verifying the results. . Testingthecorrectnessiseasyinthiscasesince

we can evaluate the formula on a calculator:

·21+32is69.8,not53.

What is wrong? The formula in the program looks correct!

Float and integer division. . Theerrorinourprogramaboveisoneofthe

most common errors in mathematical software and is not at all obvious

for a newcomer to programming. In many computer languages, there are

two types of divisions: ﬂoat division and integer division. Float division is

what you know from mathematics: 9/5 becomes 1.8 in decimal notation.

Integer division a/b with integers (whole numbers) a and b results in

an integer that is truncated (or mathematically, rounded down). More

1.3 Another formula: Celsius-Fahrenheit conversion

precisely, the result is the largest integerc such thatbc≤a. This implies

that 9/5 becomes 1 since 1· 5 = 5≤ 9 while 2·5 = 10> 9. Another

example is 1/5, which becomes 0 since 0·5≤ 1 (and 1·5> 1). Yet another

example is 16/6, which results in 2 (try 2·6 and 3·6 to convince yourself).

Many computer languages, including Fortran, C, C++, Java, and Python

version 2, interpret a division operationa/b as integer division if both

operandsa andb are integers. If eithera orb is a real (ﬂoating-point)

number,a/b implies the standard mathematical ﬂoat division. Other

languages, such as MATLAB and Python version 3, interpretsa/b as

ﬂoat division even if both operands are integers, or complex division if

one of the operands is a complex number.

The problem with our program is the coding of the formula(9/5)*C

+ 32.Thisformulaisevaluatedasfollows.First, 9/5iscalculated.Since

9and 5areinterpretedbyPythonasintegers(wholenumbers),9/5isa

division between two integers, and Python version 2 chooses by default

integer division, which results in 1. Then 1 is multiplied byC, which

equals 21, resulting in 21. Finally, 21 and 32 are added with 53 as result.

We shall very soon present a correct version of the temperature conver-

sion program, but ﬁrst it may be advantageous to introduce a frequently

used term in Python programming: object.

1.3.2 Objects in Python

When we write

C = 21

Python interprets the number21 on the right-hand side of the assignment

as an integer and creates anint (for integer) object holding the value 21.

The variableC acts as a name for thisint object. Similarly, if we writeC

= 21.0,Pythonrecognizes 21.0asarealnumberandthereforecreates

afloat (for ﬂoating-point) object holding the value 21.0 and letsC be a

name for this object. In fact, any assignment statement has the form of

avariable name on the left-hand side and an object on the right-hand

side. One may say that Python programming is about solving a problem

by deﬁning and changing objects.

At this stage, you do not need to know what an object really is,

just think of an int object as a collection, say a storage box, with

some information about an integer number. This information is stored

somewhere in the computer’s memory, and with the nameC the program

gets access to this information. The fundamental issue right now is that

21 and 21.0 are identical numbers in mathematics, while in a Python

program 21 gives rise to an int object and 21.0 to a float object.

There are lots of diﬀerent object types in Python, and you will later

learn how to create your own customized objects. Some objects contain

1 Computing with formulas

alot of data, not just an integer or a real number. For example, when

we write

print ’A text with an integer %d and a float %f’ % (2, 2.0)

astr (string) object, without a name, is ﬁrst made of the text between

the quotes and then thisstr object is printed. We can alternatively do

this in two steps:

s = ’A text with an integer %d and a float %f’ % (2, 2.0)

print s

1.3.3 Avoiding integer division

As a quite general rule of thumb, one should be careful to avoid integer

division when programming mathematical formulas. In the rare cases

when a mathematical algorithm does make use of integer division, one

should use a double forward slash,//, as division operator, because this

is Python’s way of explicitly indicating integer division.

Python version 3 has no problem with unintended integer division,

so the problem only arises with Python version 2 (and many other

common languages for scientiﬁc computing). There are several ways to

avoid integer division with the plain/ operator. The simplest remedy in

Python version 2 is to write

from __future__ import division

This import statement must be present in the beginning of every ﬁle

where the e / operator always shall imply ﬂoat division. Alternatively,

one can run a Python programsomeprogram.py from the command line

with the argument -Qnew to the Python interpreter:

Terminal

Terminal> python -Qnew someprogram.py

Amore widely applicablemethod, also in other programminglanguages

than Python version 2, is to enforce one of the operands to be afloat

object. In the current example, there are several ways to do this:

F = (9.0/5)*C + 32

F = (9/5.0)*C + 32

F = float(C)*9/5 + 32

In the ﬁrst two lines, one of the operands is written as a decimal num-

ber, implying afloat object and hence ﬂoat division. In the last line,

float(C)*9meansfloattimes int,whichresultsina floatobject,

and ﬂoat division is guaranteed.

Arelated construction,

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