Introduction to Programming
using FORTRAN 95

http://www.fortrantutorial.com/

These worksheets aim to provide an introduction toprogramming. The language chosen for this is FORTRAN 95. Thisis because FORTRAN is particularly suitable for science andengineering; it is also very widely available.
The skills you acquire working through these notes can beapplied to any computing language. The concepts you will learn
are shared in common with every other computing language.

This document and all the examples may be found online at:
http://www.fortrantutorial.com/© Janet A Nicholson 2011

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1

THE BASICS .............................................................................................................................................. 3
AIMS ............................................................................................................................................................ 3
INSTALL FTN95 PERSONAL EDITION .................................................................................................................... 3
YOUR FIRST PROGRAMMING SESSION .................................................................................................................... 3
PLATO ‐ A PROGRAMMING ENVIRONMENT ............................................................................................................. 3
RUNNING YOUR FIRST FORTRAN 95 PROGRAM .................................................................................................... 4
PROGRAM STRUCTURE...................................................................................................................................... 6
MORE ON INPUT AND OUTPUT ........................................................................................................................... 6
MORE DATA TYPES – INTEGER AND CHARACTER ...................................................................................................... 8
SAVING THE CONTENTS OF OUTPUT WINDOW ...................................................................................................... 10

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2

MAKING DECISIONS ............................................................................................................................... 11
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11

3

AIMS .......................................................................................................................................................... 11
ASSIGNMENT ................................................................................................................................................ 11
ARITHMETIC ................................................................................................................................................. 11
INTRINSIC FUNCTIONS ..................................................................................................................................... 12
MAKING DECISIONS ....................................................................................................................................... 13
PROGRAM STYLE ........................................................................................................................................... 14
MORE ON DECISION MAKING ............................................................................................................................ 14
OTHER LOGICAL OPERATORS ............................................................................................................................. 14
MULTIPLE CONDITIONS ................................................................................................................................... 15
THE SIMPLE IF STATEMENT .......................................................................................................................... 15
IMPORTANT NOTE – TESTING FOR ZERO.......................................................................................................... 16

LOOPS.................................................................................................................................................... 17
3.1
3.2
3.3
3.4
3.5

4

AIMS .......................................................................................................................................................... 17
MIXING VARIABLE TYPES .................................................................................................................................. 17
THE DO LOOP ................................................................................................................................................ 18
NESTED DO LOOPS ........................................................................................................................................ 19
USING LOOPS TO DO SUMMATION ..................................................................................................................... 20
USING FILES AND EXTENDING PRECISION............................................................................................... 22

4.1
4.2
4.3
4.4
4.5
4.6
5

AIMS .......................................................................................................................................................... 22
READING FROM FILES ...................................................................................................................................... 22
WRITING TO FILES .......................................................................................................................................... 23
EXTENDING THE PRECISION............................................................................................................................... 23
MAGNITUDE LIMITATIONS ............................................................................................................................... 25
CONVERGENCE – EXITING LOOPS ON A CONDITION ................................................................................................. 25
ARRAYS AND FORMATTED I/O ............................................................................................................... 27

5.1
5.2
5.3
5.4
5.5

AIMS .......................................................................................................................................................... 27
ARRAYS ....................................................................................................................................................... 27
ARRAY MAGIC ............................................................................................................................................... 29
MULTI DIMENSIONAL ARRAYS ........................................................................................................................... 30
FORMATTING YOUR OUTPUT............................................................................................................................. 31

5.5.1
5.5.2
5.5.3
5.5.4
5.6
6

Integer Specification ....................................................................................................... 32
Floating point Specification ............................................................................................ 32
Exponential Specification ................................................................................................ 32
Character Specification ................................................................................................... 33

IMPLIED DO LOOP TO WRITE ARRAYS .................................................................................................................. 33
SUBROUTINES AND FUNCTIONS............................................................................................................. 35

6.1
6.2
6.3
6.4
7

AIMS .......................................................................................................................................................... 35
RE‐USING CODE – THE SUBROUTINE ................................................................................................................... 35
ARGUMENTS TO SUBROUTINES.......................................................................................................................... 36
USER DEFINED FUNCTIONS .............................................................................................................................. 38
ADVANCED TOPICS ................................................................................................................................ 40

7.1
7.2
7.3

AIMS .......................................................................................................................................................... 40
ARRAY FUNCTIONS......................................................................................................................................... 40
WRITING REAL PROGRAMS ‐ FLOW CHARTS ....................................................................................................... 42

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1 The Basics
1.1 Aims
By the end of this worksheet, you will be able to: Create and run a FORTRAN 95 program Understand basic program structure Start to deal with programming errors
 Start to understand real, integer and character variable types.
 Save a copy of your output in Word.

1.2 Install FTN95 Personal Edition



1.3

Search for Silverfrost FTN5 personal edition or click this link
http://www.silverfrost.com/32/ftn95/ftn95_personal_edition.aspx.
Download and install the software accepting all the defaults.

Your first programming session

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icon
Locate and double click the Plato
Click File, New
Select Free Format Fortran FileClick File, Save As
Create a directory called fortranprograms and open it
Type first.f95

1.4 Plato ‐ a programming environment

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Plato is a "programming environment". Within Plato, you can create and edit programs and get themto run. Plato's editor is special – it understands the syntax of various programming languages. We tell
Plato which language we are using when we create our empty file and save it with a .f95 (FORTRAN
95) extension. Provided you have given your file the appropriate extension, Plato's editor will be able
to check the syntax of the program, highlighting the various keywords that it knows about using acolour code to distinguish between the various elements of the language.

Always ensure that your program files have a .f95 extension

1.5 Running your first FORTRAN 95 Program

Exercise 1.1
Type in the following exactly as shown:!My first program
program firstprint *,'This is my first program'end program first

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





Click the black ► , (the Execute button).
Plato will get FTN95 to check your program for errors. If it finds any problems, it will give you
the details. If you have typed in the program exactly as shown above, an executable file will
be generated (first.exe). Plato will then automatically get the program to start executing.
A banner will appear for a couple of seconds and will then disappear (that"s the price wehave to pay for using the free software)
A black console window will appear.
Press Return to close the window. Do not click the X at the top right of the window.

Plato can get upset if you do not press Return to close the window, try this…
Save your program first!
Run the program again (click ►)This time click the X at the top right of the window to close it.
Make up your own mind about which is the better way to close this window in future!

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1.6 Program Structure
Examine the following short program:!a: name of program!b: a comment!c: declarations!d: output!e: input!e: input!f :arithmetic!g: output!h: end of program

program sum!an example of program structure

real :: answer,x,y
print *, 'Enter two numbers'
read *, x
read *, yanswer=x+yprint *, 'The total is ', answer
end program sum

There are a number of general points here: The program is made up of a number of lines. Each line is called a statement.
 Each statement is made up of variable names e.g. answer, x, y
 operators e.g. +,- etc
 keywords e.g. read, print The statements are executed sequentially.
Let's break the program down, line by line:a) The name of the program. Keep it reasonably short and meaningful.
b) A comment explaining the purpose of the program. Comments are indicated by an
exclamation mark. All text to the right of an exclamation mark is ignored by the compiler.
Programmers use comments to help them remember how a program works. Use of
appropriate comments in programs aids understanding and is good practice.
c) Variables ‐ answer, x and y are used to store floating point numbers – we indicate
this by declaring them as real.
d) print *, outputs to the screen – the asterisk means use the default number of decimal
places when the number is written to the screen.
e) We read information from the keyboard and store the values in x and y.
f) Do some arithmetic and store the answer in answer.
g) Output the result to the screenh) Conclude the program

1.7 More on Input and Output
Exercise 1.2



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

Open a new file and call it io.f95.
Type in the following program:
program io
real :: x,y,z
print *, 'enter the values x,y and z'
read *, x,y,z
print *, 'the values you typed are for z,y,x are: ',z,y,x
end program io
Execute it by pressing ►You can enter the numbers one at a time and press the Enter key each time.
Execute the program again
This time type all three numbers on one line separated by commas.

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Look at the print statement
print *, 'the values you typed are for z,y,x are: ',z,y,xIn this statement, we are outputting four separate things, a literal string of characters,'the values you typed are for z,y,x are: 'and the variables z, y, and x. We may output several items at one time, provided they are separated
by commas.

Exercise 1.3
The following program has a number of errors.
 Create a new file called bug.f95 and then type in the following program exactly as shown.
 You can also download this file fromhttp://fortrantutorial.com/fortrantutorial‐example‐programs/index.php
program bug
this program is full of errors
real :: a,b,c
a = b + c
read *,cprint *,aend program simple
The compiler will report two error messages when it attempts to compile. Click on the details button.
Each error generates a message.

Double clicking on the message willtake you to the line in the program where the fault occurs.

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
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

Correct the two errors.
Click Execute
There is now one further error, Plato will provide a yellow warning alert. Watch the screen
carefully! The window will close and then the program will start to execute. Something is
not correct however… the program will "hang". It is actually waiting for you to input a value,
because of the line read *,c. To the user of the program, this is not at all obvious – they
may have thought that the program has crashed!
Type in a number then press enter
The program returns an strange value. This is an "execution time" error.
We need to find out what the warning message was. Click the "compile" button (to the rightof the binoculars). Then click the "details" button. Plato will tell you that the variable b has
not been given a value.
Correct the program to give b a value, and then execute the program again.
There is still a problem. This time, it is a problem with the program's logic.

Need a Hint?
The program statements are executed sequentially.
a=b+c
read *, c
print *, aThe statement a=b+c doesn't make sense, as at this stage of the program, we haven't yet given a
value to c.
Important points to note
 There are two types of errors associated with this program: compiler errors and run‐timeerrors.
 The program is also user‐unfriendly. The program waits for input without telling the user
what is needed.
Fix the run time error by:

read in a value for b
correct the order of the statements
make the program more user‐friendly,

then compare your program with the one called bugfixed.f95 at
http://fortrantutorial.com/fortrantutorial‐example‐programs/

1.8 More Data types – integer and character
So far, we have only used real (floating point numbers) in our programs. We can also specify that
numbers are integer and character. Program convert, below, demonstrates their use.
Within a given range, integers are always represented exactly whereas the precision of real numbers
is limited by the architecture of the machine. The real variable type gives us 6 figure decimal
precision. (If this doesn't seem enough – don't worry we'll come back later on when we examine how
to increase the number of digits of precision in Section 4).
Character variables hold strings of characters like'A happy day was had by all'
'Yes'
'N'
'3 + 4 equals 7'

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When the character variable is declared, we show the maximum length that the string can occupy byfollowing the name by a * then its maximum length. The example below has a maximum length of 10
characters allowed for a person's name – this might not always be enough! You have to make a
judgement here.
program convert!This example shows the use of integer and character variables.
implicit none
integer:: pounds,pence,total
character :: name*10
print *,'What is your name?'read *,nameprint *, 'Hi ',name,'! Enter number of pounds and pence'read *, pounds,pence
total =100 * pounds + pence
print *,'the total money in pence is ',total
end program convert

NOTE the inclusion of the line
implicit none

By including it in your program, FORTRAN will check that you have properly declared all your variabletypes. In the bad old days of programming, declaration of variables was thought to be unnecessary
and the old FORTRAN compilers used an implicit convention that integers have names starting withthe letters in the range i – n, all the others being real. FORTRAN still allows you to do this if we don't
include the line, implicit none. Time has shown that one of the commonest reasons for error in a
program is the incorrect use of variables.

Always use implicit none at the start of every program.

Exercise 1.4
With the program convert in section 1.5 as a guide, write a program to test out everything you've
learned so far. You might include different types of variables, for example real, integer, andcharacter. Include input and output using read and print. An example might be a program that asks
people questions, including things like their age and name and so on. It could, for example, print out
their year of birth with a suitable message. It's up to you, just use your imagination.

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1.9 Saving the contents of Output Window
Run your last program again. When the black output window opens right click on the
Plato icon in the top left corner

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


Click on edit
Click Select all
Click copy
Open a new document in Word or Notepad and click paste.

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2 Making Decisions
2.1 Aims
By the end of this worksheet, you will be able to: Do arithmetic
 Start to use FORTRAN intrinsic functions
 Begin to understand program flow and logic
 Know how to test for zero – important!
 Learn more about good programming style

2.2 Assignment
When we start programming, the similarity between mathematical equations and FORTRAN
statements can be confusing.
Consider the following FORTRAN statements:
Store the value 2 in memory location x
Store the value 3 in memory location y
Add the values stored in memory locationx and y and store the result in memory location z
In mathematics, “x = 2” means that the variable x is equal to 2. In FORTRAN it means “store the value
2 in the memory location that we have given the name x”.
x = 2
y = 3
z = x + y

The significance of this is made clearer by the following equation in mathematics:
x + y =zIn mathematics, this means that the left hand side of the equation is equal to the right hand side.
In FORTRAN, this expression is meaningless: there is no memory location "x+y" and so it would lead to
a compiler error.

Rule – there can only ever be ONE variable name on the left hand side of an equals sign

Exercise 2.1
Write a program which reads in two numbers a and b. Get the program to swap the values around sothat the value that was in a is now in b, and print out the result. Hint you need to declare a third
variable for intermediate storage of the data. (Check your program by examining program swap.f95 at
http://fortrantutorial.com/fortrantutorial‐example‐programs/

2.3 Arithmetic
The arithmetic operators are+,plus and minus*,/
multiply and divide**
exponentiation (raise to the power)()brackets
 The order of precedence in FORTRAN is identical to that of mathematics.
 Unlike algebra, the operator must always be present xy is not the same as x*y Where operations are of equal precedence they are evaluated left to right Consecutive exponentiations are evaluated right to left
 We can override the order of evaluation by use of brackets

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Exercise 2.2
The following program is an example of the use of arithmetic.
program calculate
implicit none
! a simple calculator
real :: x,y,z,answer
x=1.5
y=2.5
z=3.5
answer=x+y/z
print *,'result is ',answer
end program calculate
Explore the use of arithmetic operators by modifying program calculate. Use it to calculate the values:
1.

x y
xz

2.

xyz

3.

xy

z

2.4 Intrinsic Functions
FORTRAN is especially useful for mathematical computation because of its rich library of inbuilt
functions (intrinsic functions). We shall mention a few briefly here:
function namesin(x)
cos(x)
tan(x)
atan(x)abs(x)
sqrt(x)exp(x)
log(x)

type of argument
real
real
real
real
real/integer
real
real
real

type of result
real
real
real
real
real/integer
real
real
real

Definition
sine
cosinetangent
arctangent
absolute valuesquare root
ex
log x
10

Trigonometric functions are calculated in radians (1 radian = 180/Pi degrees).
There are, of course, many more, and this list doesn't cover all FORTRAN variable types. The following
example shows the use of some of the inbuilt functions.
program trig
implicit none
real :: a,pi
print *,'Enter an angle between 0 and 90'
read *, api=4.0*atan(1.0)
print *,'the sine of ',a,' is ',sin(a*pi/180)
end program trig

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2.5 Making Decisions
So far, our programs have worked as little more than basic calculators. The power of programmingcomes in when we have to make decisions. Copy the example program, test.f95, to your own filespace. See if you can understand what is going on.
program test
implicit none
!use of a simple menu
real :: x,y,answer
integer :: choice!set up the menu – the user may enter 1, 2 or 3
print *,'Choose an option'
print *,'1
Multiply'print *,'2
Divide'print *,'3
Add'read *,choice
x=3.4
y=2.9
!the following line has 2 consecutive!equals signs – (no spaces in between)
if (choice = = 1) thenanswer=x*y
print *,'result = ',answer
end ifif (choice = = 2) thenanswer=x/yprint *,'result = ',answer
end ifif (choice = = 3) thenanswer=x+yprint *,'result = ',answer
end ifend program test
The bolded lines in the above program are called if … end if statements. They work like this:if (condition is true) thenexecute this line
and thisand so on until we get to …
end if
It follows that if the condition is NOT true then the code 'jumps' to the next statement following the'end if'. The statements between the if and the end if are deliberately indented, this makes the
program easier to read.
We use two consecutive equals signs (no space in the middle) to test for equality. Compare
test
assignment

if (choice == 3) thenchoice = 3

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Exercise 2.3
Examine program test above. The line
print *,'result = ',answer
is repeated several times. Is this a good idea? Can you modify the program to make it more efficient?

2.6 Program Style
A good program:
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Uses comments appropriately to explain what is happening.
Uses indentation to make the program easier to read.
Uses meaningful variable names.
Uses sensible prompts to let the user know what is going on.
Uses implicit none at the start of every program.
Is efficient!

If you want to get maximum marks for your assignments keep the above points firmly in mind. It is
not enough just to get a program to work!

2.7 More on decision making
In our test.f95 above, there was a problem if the user entered a value that wasn't catered for by the
program.
What happens if the user doesn't enter one of the values 1, 2 or 3?
We are going to look at a new structure, called if, else, endif that handles this situation. Examine thefollowing code snippet:
if (choice = = 1) thendo somethingelse if (choice = =2) thendo something else
else
do this if nothing else satisfies the conditions
end if

2.8 Other logical operators
So far, all our tests have been for equality. There are several tests we can make:
==
/=
<
<=
>
>=

equal to (there is no space between the equals signs)
not equal toless thanless than or equal togreater thangreater than or equal to

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2.9 Multiple Conditions
Suppose we need to test if x is greater than y and y is greater than z. There are different ways of doing
this:if (x > y) thenif (y > z) thendo somethingend ifend ifThis can also be handled by the following:
if (x > y .and. y > z) thendo somethingend ifnotice the .and.

If we wanted to check whether a number were less than a given value or greater than a given value
we could write:
if (x < 10 .or. x > 20) thendo somethingend ifnotice the .or.

Exercise 2.4
Write a program that reads a number from the keyboard. Get the program to decide whether:

the value of the number is greater than 0 but less than 1
or is greater than 1 but less than 10
or is outside of both these ranges

Print out a suitable message to inform the user.

2.10 The simple if statement
There is a simpler, one line form of the if statement. Say we just wanted to print out a simple
message such as
print *, 'enter a positive number'read *, num
This snippet also
if (num <0)stop
introduces a useful,if (num < 10) print *, 'less than 10'
simple statement
stop – it simply
if (num > 10) print *, 'greater than 10'
stops the program.
print *,'It is a positive number'

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2.11 Important note – testing for zeroSuppose that you wish to test whether a real variable is zero. The test
Make sure you
understand this !

if (x = = 0) then ….
is not a satisfactory test. Although integer numbers are held exactly
by the computer, real numbers are not.

The way around this is to test if the absolute value of the variable is less than some small predefinedvalue. For example:
if (abs(x) < .000001) thenprint *,’No zero values! Please enter another number’read *, x
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3 Loops
3.1 Aims
By the end of this worksheet, you will be able to: Understand more about the use of real and integer variables and how to use a mixture ofdata types in expressions
 Understand how to re‐use code by looping Know how to control the number of times a section of code is executed by using a do loop

3.2 Mixing variable types
Exercise 3.1
Copy divide.f95
Make sure you understand thisthoroughly!

program divide
implicit none
integer :: xreal :: yx = 1
y = x/3
print *, y
end program divide
And run it. This program produces the following output:
0.00000
Something odd is happening. The problem is the line:
y=x/3

FORTRAN evaluates the right hand side of the assignment first using integer arithmetic, because
both x and 3 are integer. 1 divided by 3 cannot be stored as an integer, and so the value 0 is returned.
The result, 0, is then converted to a real number and the assigned to y.
Replace the line in program divide
x = 1
x = 10byYour output should now be:
3.00000
Can you see what is happening? FORTRAN is keeping the integer part of the answer and throwing therest away.

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To get over this problem, we have to signal to FORTRAN that we want it to calculate the right handside of the expression using real arithmetic. If we want to keep x as integer data type, we could re‐write our expression as follows:
y=x/3.0
The presence of a real number on the right hand side causes the right hand side of the expression to
be evaluated using floating point arithmetic.
Actually, the problem is even more complicated! Where we have an expression like
y=x * ((2**i)/3)
where x and y are real and i is integer, FORTRAN computes the result in stages:
First it calculates (2**i)/3 and evaluates it as an integer number, then multiplies the result by x andevaluates it as real.

Exercise 3.2
Copy check.f95 to your computer.

program check!Integer and real arithmetic
implicit none
real :: x,y
integer ix=2.0
i=2y=x*((2**i)/3)
print *,yy=x*((2.0**i)/3)
print *,yend program check… and examine its output. Make sure you understand why this is happening.

3.3

The do loop

Unless we are able to re‐execute code, we might as well use a calculator… Now we start to takeadvantage of the power of the computer.

Exercise 3.3
Copy program loop.f95
program loop
implicit none
integer :: ido i=0,20
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end program loop
Run the program. It prints out the numbers from 0 to 20 in steps of 1.
Note:
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


i is called a loop counter. In this example, it has a start value of zero.
All the statements within the do and end do are executed. In this example there is just theone statement, ie print.
Each time the statements are executed, the loop counter, i, is incremented by 1.
When the value of i is 20, the loop terminates, and the program resumes after the end do.

Change the do statement in program loop as follows:
do i = 50,70,2
Run the program. What happens?
The third argument in the do statement, is the increment step. If omitted, the value is taken as 1.
Loops can also decrement: try thisdo i = 5,-5,-2

Exercise 3.4
Using a do loop to generate integer values of x between –10 and 10 in steps of 1, write a programthat constructs a table of values of
y=1.0/xWhat happened when x had the value zero? Use an if, end if to test for the condition that gives theincorrect value, and print out an appropriate message. Compare your result with divbyzero.f95.

Division by zero is one of the commonest reasons for a program to fail.

3.4 Nested Do Loops
We want to construct a table of values for z wherez = xyfor values of

x in the range 1 to 2 in steps of 0.5 andy in the range 1 to 2 in steps of 0.5

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Exercise 3.5
Copy program xytab.f95 to your filespace.

program xytab
implicit none!constructs a table of z=x/y for values of x from 1 to 2 and!y from 1 to 4 in steps of .5
real::x, y, zprint *, 'xyz'
do x = 1,2
do y = 1,4,0.5
z = x/yprint *, x,y,z
end doend doend program xytab

Examine its output. Notice the use of the first print to give a heading to the table.

3.5 Using loops to do summation
Earlier on, we discussed the idea of assignments.
x = 1.0
means store the value 1.0 in the memory location called x.
If we had the following code:
x = 1.0x = x + 1.0
print *, xCan you guess what value would be printed out for x?
The answer would be 2.0.

Bearing in mind the definition of an assignment, the statement

Really important!

x = x +1.0
means “add 1.0 to the value currently stored in memory location x and then store the result inmemory location x”.

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Exercise 3.6
Copy file increment.f95 to your file space and examine its output.
program increment
implicit none
integer :: ireal :: xx=1.0
do i=1,10x=x+1.0
print *, i,x
end doend program increment


Note carefully that we have set the initial value of x outside of the do loop. Why have wedone this? If you aren't sure – change the code to put the line x = 1.0 inside the loop – thenexamine the output.
It is important to understand that if we use constructions such as x = x + 1.0, then it is vital toinitialise x to some value. If we don't, it is possible that the value might be set to any random
number. Run the program, make a note of the final value of x then put an exclamation markin front of the x = 1.0 statement and run the program again.

Exercise 3.7
Edit the line x = x + 1.0 in program increment.f95, and change it to x = x * i. Re‐run the program andexamine the output. What is significant mathematically about the sequence of numbers that has been
generated?

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4 Using Files and Extending Precision
4.1 Aims
By the end of this worksheet, you will be able to: Read from and write to files
 Use extended precision

4.2 Reading from files
In the real world, most of the data we use for our programs will be kept in files. We just need amodification to the read statement that we are already familiar with to do this.
This program reads 3 numbers from a file called 'mydata.txt' into an array. Use Windows Notepad tocreate such a file for yourself, or copy the file from mydata.txt which is on the website.
program readdata
implicit none
!reads data from a file called mydata.txt
real :: x,y,z
open(10,file='mydata.txt')
read(10,*) x,y,z
print *,x,y,z
end program readdata
The new material here are the lines
open(10,file='mydata.txt')
read(10,*) x,y,z
The open statement links the file called 'mydata.txt' with an input device numbered 10 (it doesn't
have to be 10, it could be any positive integer). To read from device 10, we just use it as the firstargument in the read statement.

Exercise 4.1
Use Notepad to create a file called evenodd.txt. In the file type 10 numbers, one per line. Write a
program that reads data from evenodd.txt one line at a time. Check if each number is even or odd andprint out a suitable message. One way to check if a number is even or odd is to use the mod intrinsic
function, like this…
if (mod(num,2)>0) then……
mod returns the remainder of the first argument divided by the second. If the return value is greaterthan zero, then the number must be odd. Check program evenodd.f95 to see if you are correct.

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4.3 Writing to files
This is a similar idea to reading from files. We need a new statement, though, instead of print, we use
write.
program io2!illustrates writing arrays to files
implicit none
real :: num
integer :: iopen(12,file='myoutput')
do i = 1,100
num = i/3.0
write(12,*) nums
end doprint *, 'finished'
end program io2

Exercise 4.2
Write a program which reads in numbers from a file one at a time. If the number is positive, it should
store it in a file called 'positive.txt' and negative numbers in a file called 'negative.txt'.

4.4 Extending the precision
So far, we have used two types of variables, real and integer. The problem so far, as you will havenoticed on output, is that we are extremely limited by the number of significant digits that areavailable for computation. Clearly, when we are dealing with iterative processes, this will lead rapidly
to errors. We can, however, extend the precision available from the single precision default, which
gives us 6 figure decimal precision to 15 figures by using a new specification for real numbers.
program extended
implicit none
integer, parameter :: ikind=selected_real_kind(p=15)
real (kind=ikind) :: sum,x
integer :: isum=0.0
do i=1,100x=i
sum = sum + 1.0/(x**6)
end doprint *, sum
end program extended
produces the following output:
1.01734306196
Don't be put off by the odd looking code. In practice, the way of setting up this extended precision, ispretty much the same for every program.
We state the precision we want by the argument p
integer, parameter :: ikind=selected_real_kind(p=15)

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in this case, 15 decimal places. ikind is a new data type – a parameter. FORTRAN returns a value tothe parameter ikind that will be adequate to provide 15 digit precision. This code will work on any
machine irrespective of the architecture.
We declare that the variables are using extended precision byreal (kind=ikind) :: sum,xValid values for p are 6, 15 and 18. The default value for p is 6. If you ask for more precision than 18
digits, the compiler will complain with an error message. Try changing the values of p and see what
effect this has on the output.
The trouble with PRINT is that the programmer has no control over the number of digits output
irrespective of the selected precision .
Later on we'll come back to this when we learn about the WRITE statement, and output
formatting.
Note Unlike variables, parameters may not change once they are declared.
If we want to use constants in a program that uses extended precision, we have to tell FORTRAN thatthey are also extended precision explicitly. This leads to the rather strange syntax you can see in thefollowing program.
program extendedconstants!demonstrates use of extended precision
implicit none
integer, parameter :: ikind=selected_real_kind(p=18)
real (kind=ikind) :: val,x,yval=10/3
print*,val!10/3 calculated as integer - wrong!
x=10.0y=3.0
val=x/y!x/y assigned to extended precision - right!
print*,val
val=10.0_ikind/3!extend precision constant - right!
print*,val
val=10.0/3.0!real constants - wrong!
print*,val
val = .12345678901234567890!real constants - wrong!
print *, val
val = .12345678901234567890_ikind !ext precision consts - right!
print *, val
end program extendedconstants
You should run this program for yourself and think carefully about its implications. This program
demonstrates how easy it is to get calculations wrong. I’ll leave this to you to experiment to ensurethat you fully understand the importance of properly declaring variables and the use of constants in
FORTRAN programming. A systematic approach to your programming will reduce the risk of errors as
will running programs with test data that have known solutions so that you can confirm that yourprogram is error free.

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4.5 Magnitude limitations
We have already observed that there is a limitation of the accuracy with which we can do calculationsin FORTRAN (and indeed, any, computer language). There are also limitations on the magnitude of anumber. The various magnitude and precision limits are summarized in the following table:

Value of p
6
15
18

Decimal places
6 (default)
15
18

Range
 1038
 10307 104931

Exercise 5.3
To illustrate these limits copy file magnitude.f95 and run the program. Take a while to get a feel forwhat is going on. Try inputting various values for the variable maxpower (eg 400). Can you confirm
that the table above is correct?
One interesting construct isprint *,i,2.0_ikind**i
Here, we are telling the compiler that the real constant 2.0 is also using extended precision. Check
what happens if you select extended precision (option 3) and enter a value of maxpower of 400. See
what happens if you rewrite the line to beprint *,i,2.0**i
Run the program again and enter the same values. Can you explain what is going on?

4.6 Convergence – exiting loops on a condition
In the program extended.f95, we found the sum of
x 10

1

x 1

x



6

It is useful to determine at what point such sums converge to a steady value – otherwise we may
make arbitrary assumptions about the summation range. Clearly, a point will be reached, within the
precision range that can be handled on our computer, that the term

1
6xwill be too small to contribute to the sum. At this point we should exit the loop otherwise theprogram will do more computation than is required.
One way to do this is to compare the value of the variable sum with its previous value, and if thedifference between the two is very small, then exit the loop.

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program whileloop
implicit none
integer, parameter :: ikind=selected_real_kind(p=15)
real (kind=ikind) :: sum,previoussum,x,smallnumber,error
integer :: isum=0.0
previoussum=0.0
smallnumber = 10.0**(-15.0)
do i=1,1000
x=i
sum = sum + 1.0 /(x**6)
error=abs(sum-previoussum)
if (error<smallnumber) thenprint *,'sum ',sum,' number of loops ',i
exit
end ifprevioussum = sum
end doend program whileloop
IMPORTANT NOTE
In the real world, we have to make choices about the amount of precision we need to work to. It is
pointless asking for 15 digits of precision if, for example, we can only take a measurement to + or –1% accuracy!

It is not necessary to always use a loop counter in a do loop. If we don't actually specify a counter,the program will loop forever. Constructs like this are OK:

smallnumber = .0000001_ikind
doprint *, 'enter a positive number 'read *, number
if (number <= smallnumber) exit
end doThe disadvantage is that, if you get the code wrong, you run the risk of the program looping forever –generally it's safer to use a loop counter!

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5 Arrays and Formatted I/O
5.1 Aims
By the end of this worksheet you will be able to: Understand the use of arrays
 Improve the appearance of your output

5.2 Arrays
Let us imagine that we want to find the average of 10 numbers. One (crude) method is shown in thenext program.
program av
real :: x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,averageread *, x1,x2,x3,x4,x5,x6,x7,x8,x9,x10
average= (x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10)/10
print *, 'the average is ',average
print *, 'the numbers are:'print *, x1
print *, x2
print *, x3
print *, x4
print *, x5
print *, x6
print *, x7
print *, x8
print *, x9
print *, x10
end program av

This approach is messy, involves a lot of typing and is prone to error. Imagine if we had to deal withthousands of numbers!
The way around this is to use arrays. An array is a list that we can access through a subscript. To
indicate to FORTRAN that we are using an array, we just specify its size when we declare it.
real, dimension(100) ::x.
.
x(1) = 3
x(66) = 4This snippet of code allocates 100 memory locations to the array x. To access an individual location,
called an array element, we use a subscript – here we are assigning the number 4 to the 66th elementof array x and 3 to the 1st element.
Now let's return to program av at the start of this worksheet, we'll re‐write it using an array.

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program av2
implicit none
real ,dimension(10) :: xreal:: average,sum
integer:: iprint *, 'enter 10 numbers'
sum=0.0
do i=1,10
read *, x(i)sum=sum+x(i)end doaverage=sum/10
print *, 'the average is ',average
print *, 'the numbers are'print *,x
end program av2

Notice that if we type
print*, x
the program will print out the entire contents of the array.
The additional benefit of this program is that with very few changes, we could make it deal with any
number of items in our list. We can improve on this still further by making use the parameter data
type:
program av3!just change the value of the parameter to change the size of the!arrayimplicit none
integer, parameter:: imax = 10
real,dimension(imax) :: xreal:: average,sum
integer:: iprint *, 'enter’ ,imax, ’ numbers'
sum=0.0
do i=1,imax
read *, x(i)sum=sum+x(i)end doaverage=sum/imax
print *, 'the average is ',average
print *, 'the numbers are'print *,x
end program av3
Note this is an example of good programming. The code is easily maintainable – all we have to do tofind an average of a list of numbers of any size is just to change the size of the parameter imax. We
can also allocate the size of the array at run time by dynamically allocating memory.

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The following program demonstrates the use of arrays where we do not know the size of the array.

program alloc
implicit none
integer, allocatable,dimension(:):: vector!note syntax - dimension(:)
integer:: elements,iprint *,'enter the number of elements in the vector'
read *,elements
allocate(vector(elements))!allocates the correct amount of memory
print *,' your vector is of size ',elements,'. Now enter eachelement'
do i=1,elements
read *,vector(i)end doprint *,'This is your vector'
do i=1,elements
print *,vector(i)end dodeallocate(vector)!tidies up the memory
end program alloc

The program is called alloc.f95 and can be copied from the web page. Note in particular the boldedlines. The new way of declaring the array vector tells the compiler that it is allocatable – ie the size
will be determined at run time.
We shall look at this further in Section 7.

Exercise 5.1
Write a program that asks the user how many numbers they want to enter, call this value imax.
Allocate imax elements to two arrays, a and b. Read in imax numbers to a and do the same to b. Print
out the arrays a, b and print out the sum of a and b. Compare your attempt with sumalloc.f95.

5.3 Array magic
One of the benefits of arrays is that you can easily do operations on every element by using simple
arithmetic operators.
program ramagic
implicit none
real ,dimension(100) :: a,b,c,dopen(10,file='data.txt')
read(10,*) ab=a*10c=b-a

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d=1
print *, 'a= ',aprint *, 'b= ',bprint *, 'c= ',cprint *, 'd= ',dend program ramagic

Exercise 5.2
Copy program ramagic.f95 and file data.txt to your own filespace. Run the program and examine theoutput.

Exercise 5.3
Write a program that fills a 10 element array x with values between 0 and .9 in steps of .1. Print thevalues of sin(x) and cos(x) using the properties of arrays to simplify your program. Compare youranswer with ramagic2.f95.

5.4 Multi dimensional arrays
The arrays we have looked at so far have been one dimensional, that is a single list of numbers thatare accessed using a single subscript. In concept, 1 dimensional arrays work in a similar way tovectors. We can also use two dimensional arrays which conceptually are equivalent to matrices.
So, for example,
Integer, dimension(5,5) :: asets up a storage space with 25 integer locations.
The next program creates a 2 dimensional array with 2 rows and 3 columns. It fills all locations incolumn 1 with 1, columns 2 with 2, column 3 with 3 and so on.
program twodra
implicit none:: ainteger,dimension(2,3)
integer::row,col,count
count = 0
!creates an array with 3 cols and 2 rows
!sets col 1 to 1, col2 to 2 and so on
do row=1,2
count=0
do col =1,3
count=count+1a(row,col)=countend doend dodo row=1,2
do col =1,3
print *,a(row,col)end doend doend program twodra

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FORTRAN actually allows the use of arrays of up to 7 dimensions, a feature which is rarely needed. To
specify a extended precision 3 dimensional array b with subscripts ranging from 1 to 10, 1 to 20 and 1
to 30 we would write:
real (kind=ikind),dimension(10,20,30) :: b

Exercise 5.4
Using a 4*4 array create an identity matrix, that is, a matrix of the form:
1 0 0 00 1 0 00 0 1 00 0 0 1and output it. Wouldn't it be nice if we could actually output the matrix elements in rows andcolumns? At the end of this section we shall see exactly how to do this.

5.5 Formatting your output
You may now be wondering if there is any way to have better control over what your output looks
like. So far we have been using the default output option – that's what the *'s are for in the write andprint statements:
write(10,*) x,y,z
print *, 'program finished'

Exercise 5.5
Copy format.f95, and run it
program format
implicit none
!demonstrates use of the format statement
integer, parameter :: ikind=selected_real_kind(p=15)
real , dimension(4):: xinteger, dimension(4):: nums
integer:: ireal(kind=ikind),dimension(4):: computed!fill up the arrays with something
do i = 1,4nums(i)= i * 10computed(i)= cos(0.1*i)x(i)= computed(i)end doprint *,'nums - integer'
write(*,1) nums
1format(2i10)
print *, 'x - real'
write(*,2) x2format(f6.2)
print *, 'computed - double precision'
write(*,3) computed
3
format(f20.7)

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end program format
You can see that the write and format statements come in pairs.

labelwrite(output device,label) variable(s)format(specification)

We are using in this example a * as the output device – in other words, the screen.
The format statement can actually go anywhere in the program, but by convention we usually placethem just after the associated write or all together at the end of the program. It's just a matter oftaste.
The tricky part here is the specification. There are different specifications for integer, real, andcharacter variables.

5.5.1 Integer Specification
General form : nim

Right justified
m is the number of character spaces reserved for printing (including the sign if there is one)If the actual width is less than m, blanks are printed
n is the number of integers to output per line. If omitted, one number is output per line.

5.5.2 Floating point Specification
General form : nfm.d
 Right justified
 m is the number of character spaces reserved for printing (including the sign if there is one),and the decimal point.
 If the actual width is less than m, blanks are printed
 n is the number of real numbers to output per line. If omitted, one number is output perline.
 d is the number of spaces reserved for the fractional part of the number – filled with 0's if
fewer spaces are needed. If the fractional part is too wide it is rounded.
If the total width for output (m) is too small, FORTRAN will just output *'s.
Rule

m >=

width of the integer part plus
d plus1 (space for decimal point) plus
1 (space for sign – if negative)

Essentially, make m nice and wide and you won't have any trouble!

5.5.3 Exponential Specification
General form nEm.d
 Alternative specification for outputting real d is the number of decimal places
 m is the total width of the field including the sign (if any), the character E and its sign, thedecimal point and the number of places of decimals. Again make m nice and wide to ensure
the field is properly printed out.
 n is the number of exponential numbers to output per line. If omitted, one number is output
per line.

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Examplereal :: a,ba = sqrt(5.0)
b = -sqrt(a)write(*,10) a,b
10 format(2E14.5)
produces:
0.22361E+01-0.14953E+01

5.5.4 Character Specification
General form nAm
 n is the number of strings to print
 m is the maximum number of characters to output
Example:10

program chars
implicit none
character ::a*10,b*10
a='hello'b='goodbye'
write(*,10) a,b
format(2a10)
end program chars

Exercise 5.6
Using the format specifications in format.f95 as a guide, produce a table ofxexwhere 0  x  1 , for values of x in increments of 0.1. Write your output to a file called myoutput.
Ensure that your output lines up neatly in columns. An example program is neatoutput.f95 is available
on the website.

5.6 Implied Do Loop to write arrays
So far, the method we have used for input and output of arrays is:
integer :: col,row
real :: ra(10,10)!using do loop
do row = 1,10
do col = 1,10
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The trouble with this method is that the rows and columns are not preserved on output. An
alternative, and neater method is to use an implied do loop in the write statement.
real :: ra(10,10)integer :: row,col!use implied do
do row = 1,10
do col = 1,10
read *,ra(row,col)end doend dodo row=1,10
write(*,10) (ra(row,col),col=1,10)
end do10 format(10f5.1)

Exercise 5.7
In Exercise 5.4 you wrote a program to produce and identity matrix. Apply what you know about
formatting now to make a neatly formatted matrix onscreen. There is an example identity1.f95
available on the website.

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6 Subroutines and Functions
6.1 Aims
By the end of this worksheet you will be able to:

Understand the use of subroutines and functions to make your code more efficient andeasier to read.

6.2 Re‐using code – the subroutine
Examine the following program

program output
implicit none
real,dimension(3) :: a,b,c
character :: answer*1!initialise arraysa = 1.5
b = 2.5
c = 3.5
write(*,1) 'a',aprint *, 'type y to continue or any other key to finish'
read *, answer
if (answer /= 'y') stop
write(*,1) 'b',bprint *, 'type y to continue or any other key to finish'
read *, answer
if (answer /= 'y') stop
write(*,1) 'c',cprint *, 'type y to continue or any other key to finish'
read *, answer
if (answer /= 'y') stop
write(*,1) 'a*b*c',a * b * c
1 format(a,3f8.3)
end program output

The program sets up some arrays and then outputs them. At three stages in the program (bolded), it
asks whether it should continue; it stops if the answer is not 'y'. Notice that the three bolded parts ofthe code are identical.
Simple enough – but look at the amount of code! Most of it is the same – wouldn't it be nice to re‐use
the code and cut down on the typing? The answer is to use subroutines.

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program output1
implicit none
real,dimension(3) :: a,b,c!initialise arraysa = 1.5
b = 2.5
c = 3.5
write(*,1) 'a',acall prompt()
write(*,1) 'b',bcall prompt()
write(*,1) 'c',ccall prompt()
write(*,1) 'a*b*c',a * b * c
1format(a,3f8.3)
end program output1!++++++++++++++++++++++++++++++++++++++++++++++
subroutine prompt()!prompts for a keypress
implicit none
character answer*1
print *, 'type y to continue or any other key to finish'
read *, answer
if (answer /= 'y') stop
end subroutine prompt
Examine the code, each time we use typecall prompt()
the program jumps to the line
subroutine prompt()then executes each line of the code it finds in the subroutine until it reaches the line
end subroutine prompt
and then returns to the main program and carries on where it left off.
The program is much easier to understand now. All the code for prompting is in one place. If we everneed to change the code which prompts the user to continue, we will only ever need to change it
once. This makes the program more maintainable.

6.3 Arguments to subroutines
We have seen that subroutines are very useful where we need to execute the same bit of code
repeatedly.
The subroutine can be thought of as a separate program which we can call on whenever we wish to
do a specific task. It is independent of the main program – it knows nothing about the variables usedin the main program. Also, the main program knows nothing about the variables used in thesubroutine. This can be useful – we can write a subroutine using any variable names we wish and weknow that they will not interfere with anything we have already set up in the main program.

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This immediately poses a problem – what if we want the subroutine to do calculations for us that we
can use in the main program? The following program uses arguments to do just that.
Example: a program that calculates the difference in volume between 2 spheres.
program vols
!Calculates difference in volume of 2 spheres
implicit none
real :: rad1,rad2,vol1,vol2
character :: responsedoprint *, 'Please enter the two radii'
read *, rad1,rad2
call volume(rad1,vol1)
call volume(rad2,vol2)
write(*,10) 'The difference in volumes is, ',abs(vol1-vol2)
10
format(a,2f10.3)
print *, 'Any more? - hit Y for yes, otherwise hit any key'read *, response
if (response /= 'Y' .and. response /= 'y') stopend doend program vols!________________________________________________
subroutine volume(rad,vol)implicit none
real :: rad,vol,pi!calculates the volume of a spherepi=4.0*atan(1.0)
vol=4./3.*pi*rad*rad*rad!It's a little quicker in processing to do r*r*r than r**3!
end subroutine volume
When the program reaches the lines
call volume(rad1,vol1)
It jumps to the line
subroutine volume(rad,vol)The values, rad1 and vol1 are passed to the subroutine. The subroutine calculates a value for thevolume and when the line :end subroutine volume
is reached, the value of the volume is returned to the main program
Points to notice – these are very important – please read carefully You may have several subroutines in your program. Ideally, a subroutine should do a specifictask – reflected by its name.
 All the variables in subroutines, apart from the ones passed as arguments, are 'hidden' fromthe main program. That means that you can use the same names in your subroutine as in themain program and the values stored in each will be unaffected – unless the variable is passed
as an argument to the subroutine.
 It is very easy to forget to declare variables in subroutines. Always use implicit none in yoursubroutines to guard against this.

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




All the variables in the subroutine must be declared.
The positioning of the arguments (in this case, rad and vol) is important. The subroutine has
no knowledge of what the variables are called in the main program. It is vital that thearguments agree both in position and type. So, if an argument to the subroutine is real in
the main program, it must also be real in the subroutine.
If an argument to the subroutine is an array, it must also be declared as an array in thesubroutine.

Exercise 6.1
Write a program that calculates the difference in area between two triangles. Your program should
prompt the user for the information it needs to do the calculation. Use a subroutine to calculate theactual area. Pass information to the subroutine using arguments.

6.4 User Defined Functions
We have already met FORTRAN intrinsic functions like abs, cos, sqrt. We can also define our ownfunctions – they work in a similar way to subroutines.
As an example, let's write a program (func.f95) that does some trigonometry. As you know, the trig
routines in FORTRAN use radians, not degrees ‐ so it would be nice to write a function that does all
the conversion for us.
print *,'Enter a number'read *, api=4.0*atan(1.0)
print *,'the sine of ',a,' is ',sin(a*pi/180)
In this snippet, we are having to code the conversion from degrees to radians directly into the mainpart of the program. That's OK for a 'one‐off', but what if we needed to do the conversion several
times. Now look at this:program func
!demonstrates use of user defined functions
implicit none
integer, parameter :: ikind=selected_real_kind(p=15)
real (kind=ikind):: deg,rads
print *, 'Enter an angle in degrees'
read *, deg
write(*,10) 'sin = ',sin(rads(deg))
write(*,10) 'tan = ',tan(rads(deg))
write(*,10) 'cos = ',cos(rads(deg))
10
format(a,f10.8)
end program func!_____________________________________________
function rads(degrees)
implicit none
integer, parameter :: ikind=selected_real_kind(p=15)!
returns radians
real (kind=ikind) :: pi,degrees,rads
pi=4.0_ikind*atan(1.0_ikind)rads=(degrees*pi/180.0_ikind)end function rads
What we have done, in effect, is to create our own function rads, which is used in an identical way tothe intrinsic ones you have used already like sqrt, cos, and abs.
When the line

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write(*,10) 'sin = ',sin(rads(deg))is reached, the program jumps tofunction rads(degrees)
the value, degrees, is passed to the function. The function does some computation, then finally
returns the calculated value to the main program with the line
rads=(degrees*pi/180.0_ikind)
Note carefully that it doesn't return the value in the argument list (as does a subroutine) but actuallyassigns the value to its own name rads.
The function rads converts the value of the argument, degrees, to radians.
Notice that we must declare the data type of the function both in the main program, and in
the function itself as if it were a variable.
Functions return one value. This value, when calculated, is assigned to the name of thefunction as if it were a variable –rads=(degrees*pi/180.0_ikind)

Exercise 6.2
Write a program that includes a function called
real function average(n,list)
where n is integer and is the number of items in the list, and list is a real array.
Write suitable code for reading the numbers from a file (or keyboard), and output the average of thenumbers.

Exercise 6.3
Write a program that allows a user to enter the size of a square matrix. In the program write asubroutine to compute a finite difference matrix. Ensure your output is neatly formatted in rows andcolumns.
So, for a 10 by 10 matrix, we expect output to look like this2 ‐1 0 0 0 0 0 0 0 0‐1 2 ‐1 0 0 0 0 0 0 0
0 ‐1 2 ‐1 0 0 0 0 0 0
0 0 ‐1 2 ‐1 0 0 0 0 0
0 0 0 ‐1 2 ‐1 0 0 0 0
0 0 0 0 ‐1 2 ‐1 0 0 0
0 0 0 0 0 ‐1 2 ‐1 0 0
0 0 0 0 0 0 ‐1 2 ‐1 0
0 0 0 0 0 0 0 ‐1 2 ‐1
0 0 0 0 0 0 0 0 ‐1 2
Check your attempt with finite.diffs.f95 on the website.

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7 Advanced Topics
7.1 Aims
By the end of this worksheet you will be able to:

Use array functions
Create larger programs aided by "Flow Charts"

7.2 Array Functions
FORTRAN provides a number of intrinsic functions that are useful for working with arrays. Amongthese are some which are specifically aimed at working with matrices and vectors.

MATMUL

Matrix/vector

DOT_PRODUCTTRANSPOSEMAXVAL

VectorMatrixAny array

MINVALAny array

SUM

Any array

Matrix multiplication of two matrices or a matrix
and a vector.
Scalar (dot) product of two vectors
Transpose of a matrix
Maximum value of an array, or of all the elements
along a specified dimension of an array.
Minimum value of an array, or of all the elements
along a specified dimension of an array.
Sum of all the elements of an array, or of all theelements along a specified dimension of an array.

Program matrixmul.f95, demonstrates the use of these functions. Additionally, it includes two
subroutines that are likely to be useful when handling matrix/array manipulations: fill_array whichfills the array elements and outputra which prints the values of the array elements to the screen. This
program is also an example of dynamic memory allocation.
program matrixmul!demonstrates use of matmul array function and dynamic!allocation of array
real, allocatable, dimension(:,:) :: ra1,ra2,ra3
integer:: size!initialize the arrays
print*, 'Shows array manipulation using SQUARE arrays.'print*, 'Allocate the space for the array at run time.'
print*, 'Enter the size of your array'
read *, size
allocate(ra1(size,size),ra2(size,size),ra3(size,size))print*, 'enter matrix elements for ra1 row by row'call fill_array(size,ra1)
print*, 'enter matrix elements for ra2 row by row'call fill_array(size,ra2)!echo the arrays
print *,'ra1'
call outputra(size,ra1)

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print *,'ra2'
call outputra(size,ra2)!demonstrate the use of matmul and transpose intrinsic!functions
ra3=matmul(ra1,ra2)
print *,'matmul of ra1 and ra2'
call outputra(size,ra3)
ra3=transpose(ra1)
print *,'transpose of ra1'
call outputra(size,ra3)
deallocate(ra1,ra2,ra3)
end program matrixmul!--------------------------------------------------------subroutine outputra(size,ra)
implicit none
!will output a real square array nicely
integer:: size,row,colreal,dimension(size,size):: racharacter:: reply*1
do row =1,size
write(*,10) (ra(row,col),col=1,size)
10
format(100f10.2)!as we don't know how many numbers are to be output, specify!more than we need - the rest are ignored
end doprint*,'__________________________________________________'
print*,'Hit a key and press enter to continue'
read *,reply
end subroutine outputra!--------------------------------------------------------subroutine fill_array(size,ra)
implicit none
!fills the array by prompting from keyboard
integer:: row,col,size
real:: numreal, dimension(size,size):: ra
do row=1,size
do col=1,size
print *, row,colread *,numra(row,col)=num
end doend doend subroutine fill_array

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Exercise 7.1
Write a program to read in 2 square matrices (of any size). Confirm that the matrices obey the rule

where

(AB)T

=

B T ATAT

is the transpose of matrix A.

Exercise 7.2
Write a program that will read a 3 X 3 matrix from a data file. In the program, include a subroutinethat will generate any cofactor cof of the matrix mat. Call the subroutine cofactor and use thesearguments:
subroutine cofactor(i,j,mat,cof)
implicit none
real :: mat(3,3),minor(2,2),cof
integer :: elrow,elcol! cof – the cofactor of matrix mat for element i,j.
.

Exercise 7.3
Use the program you developed Exercise 7.2 to calculate the determinant of a 3 X 3 matrix.

7.3 Writing REAL programs ‐ Flow Charts
Now that you know all the main elements of FORTRAN 95, you are in a position to apply your skills to
writing REAL programs. Unlike most of the exercises in these worksheets, REAL programs tend to berather large. In large programs, the underlying logic can often be difficult to follow.
It helps, therefore, both in the devising of a program and later in its maintenance, to have a plan ofwhat you intend the program to do. Let’s take, as an example, a program that works like a calculator.
The flowchart is shown on the next page. The logic of the program, as a whole, is clear. Details likewhat will happen in the subroutines is glossed over at this stage.
In commercial programming, flowcharts are usually formalized, with specific shapes for boxes that dodifferent things. That need not concern us here. Essentially, we use flowcharts to provide a ‘map’ of
the underlying logic flow in the program – what connects with what.

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StartDisplay menu of choices
A,B,C,D
User choosesA

Subroutine Subtract

B
Subroutine multiplyC

Subroutine power

Subroutine trig
D

Sorry, bad input

Have you finished?
No
Yes

Stop

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De‐bugging Tips
7.4 Symptoms and probable causes
Have you got rounding errors?
 Don’t do floating point calculations using integers. Make sure your precision is consistent andadequate. Make sure the precision of the right hand side of an equation matches the type ofvariable you are assigning to.
Are your calculations completely wrong?
 Initialise all your variables – don’t forget arrays!
 Make sure your arrays are big enough to hold all the data.
 Check that arguments passed to subroutines agree exactly in size, type and position Is the program’s logic working the way it should?

7.5 Wise precautions and time saving tips
You must not test floating point numbers for equality. Example:
if (x == 1) then...

does not work.
 Should you be using the absolute value of a calculation? Example:if(abs(x-y)<.00001) then
Don’t have overly elaborate logical tests. It’s probably better to test one or two things at atime rather than this sort of thing…
if (((x.AND.y).OR.z > 10).OR.(.NOT. xx < 0)) then …
you might think you understood it when you wrote it, but imagine trying to figure out what’s
happening if the program breaks!






Don't try and write a complicated program all at once. Write it a piece at a time and checkthat each piece is working correctly before doing the next bit.
Use ‘implicit none’ at the start of all programs and subroutines.
If your program needs data to be entered by the user, you will save hours of time by taking
the trouble to read in the data from a file while the program is in development.
Always do a test for ‘division by zero’ when dividing.
BE NEAT! Good programming is like plain speaking – there’s no mileage in tricky, difficult toread code.

7.6 How to find the bugs
Use a print statement to print out the values within the program – take a look at this code…
x = x + 1
z = x * yprint *, ‘debug statement 1 value of x,y,z‘, x,y,z
do ii =1,10z = x * ii
if (ii == 5) thenprint *, ‘debug do loop value of z when ii = 5’,z
end ifend doif (z>2000) thenprint *, ‘debug statement – z>2000 value of z ‘,zstop
end if
Notice how we can print out values of specific variables, stopping the program if necessary.

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IndexArithmetic ..................................................... 11
arithmetic operators .................................... 11
Arrays ........................................................... 27
Assignment ................................................... 11
character ........................................................ 8
Character Specification................................. 33
Convergence ................................................. 25
Debugging Tips ............................................. 44
De‐bugging Tips............................................ 44
dot product ................................................... 40
dynamic memory allocation ......................... 40
dynamically allocating memory .................... 28
errors .............................................................. 8
Exercise 1.1 ..................................................... 4
Exercise 1.2 ..................................................... 6
Exercise 1.3 ..................................................... 7
Exercise 1.4 ..................................................... 9
Exercise 2.1 ................................................... 11
Exercise 2.2 ................................................... 12
Exercise 2.3 ................................................... 14
Exercise 2.4 ................................................... 15
Exercise 3.1 ................................................... 17
Exercise 3.2 ................................................... 18
Exercise 3.3 ................................................... 18
Exercise 3.4 ................................................... 19
Exercise 3.5 ................................................... 20
Exercise 3.6 ................................................... 21
Exercise 3.7 ................................................... 21
Exercise 4.1 ................................................... 22
Exercise 4.2 ................................................... 23
Exercise 5.1 ................................................... 29
Exercise 5.2 ................................................... 30
Exercise 5.3 ............................................. 25, 30
Exercise 5.4 ................................................... 31
Exercise 5.5 ................................................... 31
Exercise 5.6 ................................................... 33
Exercise 5.7 ................................................... 34
Exercise 6.1 ................................................... 38
Exercise 6.2 ................................................... 39
Exercise 6.3 ................................................... 39
Exercise 7.1 ................................................... 42
Exercise 7.2 ................................................... 42
Exercise 7.3 ................................................... 42

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exiting loops .................................................. 25
Exponential Specification .............................. 32
exponentiations ............................................ 11
finite difference ............................................ 39
floating point numbers ................................... 8
Floating point Specification .......................... 32
Flow Charts ............................................. 40, 42
Formatting your output ................................ 31
function ......................................................... 39
identity .......................................................... 31
ikind .............................................................. 26
Implied Do Loop to write arrays ................... 33
integer ............................................................. 8
Integer Specification ..................................... 32
intrinsic functions ............................. 11, 12, 40
logical operators ........................................... 14
loop ............................................................... 18
loop counter.................................................. 19
Magnitude..................................................... 25
maintainable ................................................. 36
matrix ............................................................ 39
Matrix multiplication .................................... 40
Mixing variable types .................................... 17
Multi dimensional arrays .............................. 30
Nested Do Loops ........................................... 19
operators ........................................................ 6
parameter ..................................................... 24
Plato ................................................................ 3
precedence ................................................... 11
precision.................................................... 8, 23
programming style ........................................ 11
Reading from files ......................................... 22
simple if statement ....................................... 15
statement........................................................ 6
stop ............................................................... 15
Style .............................................................. 14
subroutine ..................................................... 35
summation .................................................... 20
Transpose of a matrix ................................... 40
User Defined Functions................................. 38
variable ........................................................... 6
Writing to files .............................................. 23

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