Numerical Methods in Scientific Computing I

Math 2070 is our gateway course in computational
mathematics.
**
Syllabus for Fall 2008
Welcome to Math 2070!
Numerical Methods in Scientific Computing
**

This is the first part of the sequence Math 2070, 2071 and is introduction

_to numerical methods and the fundamentals of the analysis (stability,

consistency, _convergence, efficiency and accuracy) of numerical methods.

The course covers both _theoretical and practical aspects of the methods.

Math 2070 covers _numerical analysis except numerical linear algebra and

numerical Diff Eqs and Math 2071 covers _the remainder,

Math 2070: CRN 13384 _Lecture: MW 3-4:15Ty. 704 ,

Recitation: CRN 16072 MW 4:20-5:35 in GSCC 126

Lab Instructor is Dr. Mike Sussman .

Dr. Mike Sussman .

Labs are available online.

Labs are available online. _ MATLAB will be taught in the lab and used for the labs. _

Office: Ty. 603, (412) 624-8312 _E-mail: W J L at P I T T dot E D U _http://www.math.pitt.edu/~wjl _ _

the lectures in class. Attend and take good notes!

You might find some material presented simpler than in the text,

some differently than the

text, some small extensions and omissions from the text. The topics are

standard so variations will be minor rather than major. My lecture notes

have evolved through the search for the simplest mathematically rigorous

way to explain WHY and HOW and WHEN and WHEN NOT algorithms that are

important for practical computing work.

Textbooks are only useful when they give the same topics from a different

point of view than the lectures. (This will not be a course where the

teacher puts the book on the board.)

A Ralston and P Rabinowitz, A first course in Numerical Analysis, Dover reprint, 2001,

K. Atkinson, An Introduction to Numerical Analysis, 2nd Ed,

also an excellent book but more expensive.

Make Up Exams will NOT be given.

HW will be collected and graded.

You are not graded on what you can prove.

You are graded on what your explanation of a proof conveys

in a clear and direct manner to others.

Write in complete and correctly punctuated sentences.

Avoid intoxication with the symbols of symbolic logic.

(Write for people not for automatons!)

Negative partial credit is possible for extraneous and irrelevant nonsensical information.

No quibbling over "points"; Focus on learning!

For useful material related to this course go to **Kendall
Atkinson's class web page****
**

**Accessing MATLAB Remotely**

Dr. Sussman
has graciously provided instructions for accessing Matlab remotely.

The first answer
is: Yes, Matlab can be accessed remotely using your usual network access.
I am not familiar with

"PittNet" -- is
it a dialup connection (slow) or some

sort of fast
connection?
In either case, directions follow.

**ASSUMING YOU ARE
USING LINUX **or another type of Unix:

Use ssh to get to
unixs1 or unixs2, type "matlab": example

ssh
youraccountname@unixs2.cis.pitt.edu

password

matlab

Wait a long while
(1-2 minutes for a fast connection,

a looooong time
for dialup) and Matlab will come up.

**WARNING:** if
you have a (slow) dialup connection, you can

turn off the
graphical
stuff and run matlab in command mode:

(you get no
graphics
and no plotting)

ssh
YoUrAcCoUnTnAmE@unixs2.cis.pitt.edu

password

DISPLAY=

matlab

If you have a dialup
and you want plotting but no gui, do NOT

reset DISPLAY to
nothing. Just do:

matlab -nosplash
-nojvm

If you turn off the
gui, you will need some sort of editor.

Either open another
ssh session with your favorite editor

in it (pico, vi,
emacs, etc.) or use the exclamation point

to start an editor
from the Matlab command line (!pico ...)

**ASSUMING YOU ARE
USING WINDOWS**

You either have
an "X-server" program or you don't. I do

not know whether
one is included in the programs they supply

to students.

**ASSUMING YOU
HAVE A DIALUP CONNECTION OR YOU DO NOT HAVE AN X-server**

Use the Start->Run
menu and type

telnet
unixs2.cis.pitt.edu

(or unixs1 or
unixs)

It will ask you
to log in. Then just type

matlab

to get the LINE
version of Matlab (no graphics, no plotting). There

is no way to do
plotting without an X-server. You also will not have

the Matlab editor,
but you can edit a file FROM THE MATLAB COMMAND

LINE with the
command
"!pico filename.m" where you don't type the

quotation marks
and you put your own name where I have "filename.m".

When you exit pico,
you will return to Matlab.

**ASSUMING YOU HAVE
BOTH AN X-server AND A FAST CONNECTION**

Start the X-server
first. You will need to know the "display" name

It will be
something
like 192.160.1.1:0 (your IP address first, then

colon zero).

Use the Start->Run
menu and type

telnet
unixs2.cis.pitt.edu

username, password,
etc.

DISPLAY=192.160.1.1:0

EXPORT
DISPLAY

matlab

There might be a
telnet or rlogin or ssh command that comes with

the X-server. If
so, the DISPLAY variable is automatically set.

**REMARK:**

There is a free
Matlab clone called "octave" that runs on Windows

(If you run linux,
install it from your CDs). See

http://octave.sourceforge.net/Octave_Windows.htm

It is not so pretty
as Matlab, and it is mostly command-line

driven, but it will
do plotting and most simple stuff.

If you have experience programming in **FORTRAN** or **C**,
the

least painful way to migrate to MATLAB is to begin by using MATLAB
as a low

level programming language and then learn the higher level features
as you need them.

These notes below were prepared by Dr. M. Sussman for exactly that
purpose.

For that reason, they parallel **"A Synopsis of 5 statement
FORTRAN".**

**1. INTRODUCTION: Matlab Statements.**

You can use "%" anywhere on a line to make Matlab skip

everything you write past the "%". This is

called "commenting it out."

Ending a line without a semicolon (";") means its result will

be printed. Ending a line with a semicolon means it will not

be printed.

**Types of Variables.**

All simple variables in Matlab are actually complex matrices.

Matlab tries to "treat them the way you meant them." Variable

names are case sensitive, so iT is different from It is

different from IT is different from it.

Arithmetic Operations.

These are denoted by the
usual symbols

+, -, , / and *

Exponentiation is denoted by ^

The arithmetic equals (=) has a different meaning than in arithmetic.
For

example, the statement

X=X+1.0;

makes perfect sense in Matlab. It means, look at the left hand side,
find

the storage location set aside to store the variable called X and find
the

value of X. Add one to that value then put the result back in the X

storage location. This requires the computer to do something; hence,
it is

called an executable statement. (It's described informally by: "look
in

the box marked X. Get the number there; add one to it and put it back
in

the X box.")

**CONTROL STATEMENTS**

Simply typing a variable name on a line by itself without a semicolon
after

it will cause it to be printed. A sequence of variables
separated
by

commas will all be printed.

Anything put in quotation marks is printed directly as written.
Examples

of this are given later.

**2. The Heart of the Program.**

A computed basically can only do 5 things:

* Input data,

* perform arithmetic calculations,

* decisions based on logical or arithmetic tests,

* output data and

* loop (meaning perform al the above iteratively).

We've seen how to perform basic arithmetic calculations. Let's now
review

some of the remainder. The real power of the computer is unleashed
with

loops (doing a lot of calculations) and matrices and vectors (handling
a

lot of data).

**INPUT, OUTPUT and CONTROL**

A single variable can be read from the keyboard using the command

r=input('What is the value?')

and the string 'What is the value?' will be printed and then the user

will type a number and the value of that number will become the value

of r.

**DECISIONS.**

Decisions in Matlab are
performed by "IF tests." There are many

kinds. One basic kind takes the form

if (condition is true)

executable
statement;

else

executable
statement;

end

For example, the following adds 1 to x if x = 0 and subtracts 1
otherwise:

if x>=0

x=x+1.0;

else

x=x-1.0;

end

Equality is denoted "==" to distinguish it from assignment.

**LOOPS.**

These are "DO-Loops" in
FORTRAN and "FOR-Loops" in C and Matlab.

for i = beginning: stepsize : end

executable statements;

end

where "beginning", "end" and "stepsize" are all integers.For example,
this

computes the sum

1+1/2+1/4+1/8++ 1/(2**1000).

sum=0.0

for I=0,1000,1

addend=1.0/(2.0^i);

sum=sum+addend;

end

Suppose we want to exit a
loop early when a condition holds. Use

a "break" statement. For example, to exit the above when SUM>

1.5 we add an if-test

sum=0.0

for k=0:1:1000

addend=1.0/(2.0^k);

sum=sum+addend;

if sum>1.5)

break;

end

end

k

sum

**Matlab's BUILT IN FUNCTIONS**

Essentially all basic mathematical functitons are built-in. For
example,

exp(x) computes the exponential of x, sin(x) computes its sine, inv(M)

computes the inverse of a matrix, eig(M) computes the eigenvalues of
a

matrix, etc.

**Inline functions**

Simple functions can be put directly in a Matlab statement:

area=inline('pi*r^2','r');

defines a function area(r) that computes the area of a circle given

its radius.

Here is an example:

area=inline('pi*r^2','r');

x=2.0;

a=area(x);

disp(['radius=',
num2str(x),
' area=' , num2str(a)])

note that Matlab knows the value of pi.

If a function uses more
calculations or a longer formula then use

a function m-file. Function m-files are very useful!!! Here's an

example:

function a=area(x,y)

a=y*x^2;

If you put those lines in a file named "area.m" then you can call them

from a program easily.

r=2.0

a=area(r,pi)

Here is an example of a simple MATLAB program which does numerical

differentiation:

% THIS IS
IN MATLAB!!!

%

% This program computes two finite
difference approximations

% to the derivative of a given
function.

% this is Matlab syntax to define a function "in line"

% you can also make up a separate m-file

F = inline('X*exp(X)','X');

FP = inline('X*exp(X)+exp(X)','X');

a=1.0;

h=0.5;

disp('Here we go-hold on!');

for i=0:13

h=1.0/(2.0^i);

d1hf= ( F(a+h)-F(a-h))/(2.0*h);

d2hf=(F(a+2.0*h)-F(a-2.0*h))/(4.0*h);

df=(4.0/3.0)*d1hf-(1.0/3.0)*d2hf;

e1=abs(FP(a)-df);

disp(['h=', num2str(h),' f''~',num2str(df),'
error=',num2str(e1)]);

end

disp('Whew-that was fast!')

**3. Arrays = Vectors and Matrices.**

Vectors and matrices are
the way to store and manipulate lots of

data.

For example, this computes the first 20 Fibonacci number:

fibo(1)=1.0;

fibo(2)=2.0;

for k=3:20

fibo(k)=fibo(k-1)+fibo(k-2);

end

fibo(20)

Remark: It's interesting to try running this changing "20" to, e.g., "50".

**4. Programming "Rules of Thumb".**

These days one is more
often
modifying an existing code or linking

pieces of codes than programming a new application from scratch.

Nevertheless, the following "rules of thumb" will, if followed, save
you

hours of headaches in debugging:

1. Be careful and correct: every hour you spend in checking, planning
and

double checking will save four hours of debugging.

2. Use pseudo code to organize and plan your program.

3. Check it twice before running it! Then run it on several test cases

with known true-solutions.

4. Use modules. Break the program into clear chunks (one way is to use

FUNCTION m-files). Check and debug each module independently of the

whole program before testing the whole program.

5. Use the right amount of generalization!

6. Print intermediate results.

7. Include warning messages.

8. Use variable names that are easy to understand and declare all

variables.

9. Put lots of comments in!

10. Keep your loops simple!

11. Think a lot about your data structures.

12. Use arrays to store data.

13. Use built in functions and program libraries.

14. Put clarity (for people) above compactness or even speed of
execution

**FORTRAN INFORMATION**

Some people might choose to use FORTRAN (or be forced
to because

they deal with legacy programms). I've inclluded some
information next for those lucky ones.

**HOW TO RUN A FORTRAN PROGRAM**

Here are some
basic , basic instructions on how to

(a) write a fortran
program [ie, create a file].

(b) how to run
it.

(a) To create
a file you need to use an editor. In unix, the pico editor

is probably easiest
because it's menu driven. Dont forget though that unix

is case sensitive:
large and small letters have different meanings , so

get the
capitalization
exactly right!

At the prompt type:

pico progname.for

(the "progname " you may choose but be sure to make it .for at the

end.)

Next paste in or type in your program. Editing it is easy-read the

commands at

the bottom of the
screen. When you are done type

'control'O

then type

'control'X

Now you've created a file progname.for.

(b) Here's how to run progname.for

At the
prompt type

setup sunpro

At the
next prompt type:

f77 -C progname.for

This compiles
your program. Now run it by typing at the next prompt

./ a.out (i.e., " dot slash a dot out" without any
spaces!)

This tells
the computer to dump your output to the screen.

Try
writing
a simple program to add 2 numbers and output it. If it doesnt

work, check
carefully
the cases [small letters or caps] and the spaces in

your
commands.

Another
comment:
If you discover something interesting, email me and I'll

send it to the
whole
class.

More Remarks:

These instructions
are for our UNIXS system. There are a lot of free

FORTRAN compilers
available [FORTRAN 77] as well as a lot of free C

compilers too. I'll
email to you an addres on where to download one, if

you are interested
in working on a PC. I suggest though that you start on

UNIXS if you've
never programmed before.

Have fun!

FORTRAN Compilers-

Here are some FORTRAN resources including free and/or cheap FORTRAN compilers for a PC.

"The Fortran
Library,
http://www.fortranlib.com, is a comprehensive guide

to online Fortran
resources, including compiler vendors and

resellers,
benchmarks,
programming tools, books and articles on Fortran

and numerical
methods,
and commerical and public domain Fortran software.

The Fortran Market,
http://www.fortran.com/fortran/ , also provides

many links to
Fortran
resources and is a reseller for several Fortran 90

and 95 compilers,
benchmarking and test suites, and books on Fortran."

One former student installed one. Here below are his instruction on how to do this:

Subject: Fortran Compiler for PC's

First of all, the stuff has to be downloaded.

It can be gotten via anonymous ftp (using Netscape, for instance) from

ftp://ftp.bu.edu/pub/mirrors/simtelnet/gnu/djgpp/v2gnu/g770523b.zip

and you also
need

ftp://ftp.bu.edu/pub/mirrors/simtelnet/gnu/djgpp/unzip386.exe

There are other
things
like additional documentation, but this provides the

bare essentials
(I realized that it needed to be cut down for people to be

able to download
it).

Then this file
(g770523b.zip)
needs to be unzipped. I recommend making the

directory C:\DJGPP
by typing

mkdir \DJGPP

at an MS-DOS
prompt.
Then, go to the directory where you downloaded

g770523b.zip to,
and type

move g770523b.zip
\djgpp

move unzip386.exe
\djgpp

cd \djgpp

unzip386
g770523.zip

You'll also need
to make sure you can access the compiler, so add

C:\DJGPP\BIN to
your PATH statement in your CONFIG.SYS. If you don't know

how to do this,
just type the following from an MS-DOS prompt.

echo PATH=%PATH%;C:\DJGPP\BIN >> \CONFIG.SYS

After that, you'll
need to write the FORTRAN programs in something like

Notepad (do not
use a Word Processor for this!). To compile your program,

named (for
instance)
progname.for, type the following.

g77 -o progname progname.for

and to run the new program, type

progname

**A Synopsis of "5-Statement FORTRAN"**

**ABSTRACT**

This is prepared
solely for the benefit of my students. ItÕs intended to have 2
purposes.

1. To introduce
math students whoÕve never programmed before but are used
to the algorithmic

thought in rigorous
proofs to a language quickly and get them working, and

2. To do the same
for students who have experience programming in another language
like C, for example.

**1.
Introduction.**

FORTRAN is short for FORmula TRANslation language. It is a low

level language
designed
for doing scientific calculations and lots of

them. Although it
is out of favor in CS departments, it is extremely handy

to know a bit of
FORTRAN because:

(estimated) 95% of scientific software is in FORTRAN.

Thus, if you must
modify an existing program it will likely be a FORTRAN

one.

It is also useful
for us because it is simple ( = easy to learn) and its

logic parallels
the logic of mathematical formulas.
Generally, for

these and other
reasons, if you program in another language, it is a good

idea to know how
to call a FORTRAN subroutine from that language.

**2. FORTRAN
Statements.**

When you type a line of a FORTRAN program, different locations of

characters (i.e.,
which column it is types in) have different meanings.

Column
1
Column 2-5 Column
6
Column 7-72

Comment indicator
Line number Blank or continuation Executable FORTRAN
statement

Generally, try not
to continue a statement from one line to another: Keep

formulas short
enough
that you can glance at it and see what it is doing.

Thus, try to leave
column 6 blank. A character in column 1 tells the

compiler to ignore
the line so you use these lines to write notes to

yourself. You can
also use "C " in column 1 to de-activate a line. This is

called "commenting
it out."

**Types of
Variables.**

Variable names can have no more than 7 characctters.

You should declare
variables but in FORTRAN you don't need to

because there are
defaults.

INTEGERS:

Integers are stored
as such and added exactly. Any variable

beginning
with

I, J, K, L, M, N

is by default an
integer. Thus, old-style FORTRAN would use the

(undeclared)
variable
ICOUNT. Variables can also be declared as integers

initially, meaning
before the first executable statement. For example,

_ _ _ _ _ _ _ _ _ INTEGER COUNT

will make COUNT an
integer even though the undeclared default of a

variable COUNT is
real. In integer arithmetic

I =1/4

Returns I = 0 while
in real arithmetic

X = 1.0/4.0

Returns X = 0.25.
Don't use mixed mode arithmetic!

**REAL
NUMBERS:**

These can be
declared
as real before the first executable

statement by,
e.g.,

_ _ _ _ _ _ _ REAL X

By default any
variable
beginning with

A-H or O-Z

is also real. Reals
are numbers with decimal digits such as

PI = 3.141592736

XINDEX = 1.0

They are stored
in scientific notation and carry a certain number of

significant digits.
Carrying 7 digits base 10 is typical although not

universal.

**DOUBLE
PRECISION**

variables must
always
be declared n before the first

executable
statement,
e.g.,

_ _ _ _ _ _ _ DOUBLE PRECISION X

They work like reals
but carry more than 2x the number of significant

figures as reals
(also called single precision). Double precision is the

normal mode for
many scientific calculations. If you add a real to a

double precision
variable, you immediately lose al the extra accuracy of

the double
precision
variable so:

Avoid mixed mode arithmetic.

** Arithmetic
Operations.**

These are denoted by the usual symbols

+, -, , / and *

Exponentiation is
denoted by ** with differences between real and integer

exponents:

(Integer) X**2
means
X * X

(Real) X**2.0 means
EXP(2.0 LOG( X ) ).

The arithmetic
equals
(=) has a different meaning than in arithmetic. For

example, the
statement

X=X+1.0

makes perfect sense
in FORTRAN. It means, look at the left hand side, find

the storage
location
set aside to store the variable called X and find the

value of X. Add
one to that value then put the result back in the X

storage location.
This requires the computer to do something; hence, it is

called an
executable
statement. (It's described informally by: "look in

the box marked X.
Get the number there; add one to it and put it back in

the X
box.")

**CONTROL
STATEMENTS**

Programs always end with two statements that terminate the

program:

STOP

END

Another useful statement is the unformatted print statement which

just dumps the
output
to your computer screen. This is fine for moderate

amounts of output.
(For lots of it, read about formatted print statements

in a FORTRAN book.)
An unformatted print takes the form

PRINT *, variables to be printed separated by commas.

Anything put in
quotation
marks is printed directly as written. Examples

of this are given
later. Generally, for moderate amounts of data, an

unformatted print
is fine. On your terminal screen, highlight the output.

Next copy and paste
it into your favorite text editor and format it and

then print
it.

**3. The Heart of
the Program.**

A computed basically can only do 5 things:

¥ Input
data,

¥ perform
arithmetic
calculations,

¥ decisions
based on logical or arithmetic tests,

¥ output data
and

¥ loop (meaning
perform al the above iteratively).

We've seen how to
perform basic arithmetic calculations. Let's now review

some of the
remainder.
The real power of the computer is unleashed with

loops (doing a lot
of calculations) and matrices and vectors (handling a

lot of
data).

**INPUT, OUTPUT
and CONTROL**

Unformatted input is (usually from your terminal screen)

READ*, variables

While unformatted
output is

PRINT*, variables.

You can label the
variables by putting notes to be printed in quotes as in:

PRINT*, ÒNext we print the value of XÓ, X

This results in a printted line like:

Next we print the value of X .34567E2

when X has the value
34.567.

**
DECISIONS.**

Decisions in FORTRAN are performed by "IF tests." There are many

kinds. One basic
kind takes the form

IF (condition is true) THEN executable statement

ELSE executable statement

ENDIF

For example, the
following adds 1 to x if x = 0 and subtracts 1 otherwise:

IF(X.GE.0) THEN X=X+1.0

ELSE X=X-1.0

ENDIF

The conditions are arithmetic inequalities using the following.

Math
FORTRAN
Example

<
.LT.
X<Y is written as: (X.LT.Y)

>
.GT.
X>Y is written as: (X.GT.Y)

=
.EQ.
I=J is written as: (I.EQ.J)

> or
=
.GE.
is written as: (X.GE.Y)

< or
=
.LE.
is written as: (X.LE.Y)

not
equal
.NE.
is written as: (I.NE.J)

**LOOPS.**

These are "DO-Loops" in FORTRAN and "FOR-Loops" in C. If r is a

statement number,
they take the form

DO r I = beginning, end, stepsize

executable statements

r CONTINUE

where "beginning",
"end" and "stepsize" are all integers.For example, this

computes the
sum

1+1/2+1/4+1/8++
1/(2**1000).

PROGRAM SUM.for

SUM=0.0

DO 10 I=0,1000,1

ADDEND=1.0/(2.0**I)

SUM=SUM+ADDEND

10
CONTINUE

STOP

END

Suppose we want to exit a loop early when a condition holds. In

this case, in
FORTRAN
the only way is to use a "GO TO" statement. (This is

the only valid use
of GO TO's.) For example, to exit the above when SUM>

1.5 we add an
if-test
with a GO TO:

PROGRAM SUM.for

SUM=0.0

DO 10 I=0,1000,1

ADDEND=1.0/(2.0**I)

SUM=SUM+ADDEND

IF(SUM.GT.1.5) THEN GO TO 20

ENDIF

10
CONTINUE

20
PRINT*, "I=",I, "and sum=", SUM

STOP

END

** FORTRAN's BUILT
IN FUNCTIONS**

There are many built in functions in FORTRAN. If you put a "D" in

front the function
returns a double precision answer. Some common ones

include:

ABS(X)=|x| ,

TAN(X) =
tangent(x),

ATAN(X)=
arctangent(x),

COS(X),
SIN(X),TAN(X),...
,

EXP(X) =
exponential
function of x= e**x ,

LOG(X)= natural
logarithm of x,

LOG10(X)= logarithm
base 10 of x,

SQRT(X)=square root
of x,

FLOAT(I) and
REAL(I)
convert integer I into a real to avoid mixed mode arithmetic,

DFLOAT(I) makes
I a double precision real,

and so
on!

FUNCTION
STATEMENTS

Function statements are very handy to code a simple formula.

IMPORTANT: MUST
BE BEFORE THE FIRST EXECUTABLE STATEMENT!

Here is an example:

PROGRAM AREA.for

AREA( R)=4.0*ATAN(1.0)*R**2

X=2.0

A=AREA(X)

PRINT*, "Radius=", X, "AREA=" , A

STOP

END

Note that
4.0*ATAN(1.0)
is a standard way to code PI without typing in all

its digits!

If a function uses more calculations or a longer formula then use

a function
subroutine.
Function subroutines are very useful!!! Here's an

example:

PROGRAM AREA.for

PI=4.0*ATAN(1.0)

R=2.0

A=AREA(R,PI)

STOP

END

FUNCTION AREA(X,Y)

AREA=Y*X**2

RETURN

END

Notes:

You must have "AREA
= " somewhere in function subroutine AREA

And you must always
have "RETURN, END" at the end.

** 4. Arrays =
Vectors
and Matrices.**

Vectors and matrices are the way to store and manipulate lots of

data. They must
be set up with various dimensioning statements

BEFORE THE FIRST
EXECUTABLE STATEMENT!!!

Examples of these dimensioning statements are the following:

REAL X(50), A(50, 50)

INTEGER INDEX(50)

which means the same as:

REAL X, A

INTEGER INDEX

DIMENSION X(50), A(50, 50), INDEX(50)

For example, this computes the first 20 Fibonacci number:

PROGRAM Fibonacci.for

REAL FIBO(20)

FIBO(1)=1.0

FIBO(2)=2.0

DO 10 I=3,20,1

FIBO(I)=FIBO(I-1)+FIBO(I-2)

10
CONTINUE

PRINT*,FIBO(20)

STOP

END

Remark: It's
interesting
to try running this changing "20" to, e.g., "50".

You will soon see
why FIBO(.) is stored as a real number rather than (at

first and second
consideration) the more natural integer.

**5. Programming
"Rules of Thumb".**

These days one is more often modifying an existing code or linking

pieces of codes
than programming a new application from scratch.

Nevertheless, the
following "rules of thumb" will, if followed, save you

hours of headaches
in debugging:

1. Be careful and
correct: every hour you spend in checking, planning and

double checking
will save four hours of debugging.

2. Use pseudo code to organize and plan your program.

3. Check it twice
before running it! Then run it on several test cases

with known
true-solutions.

4. Use modules.
Break
the program into clear chunks (one way is to use

FUNCTION
subroutines).
Check and debug each module independently of the

whole program
before
testing the whole program.

5. Use the right amount of generalization!

6. Print intermediate results.

7. Include warning messages.

8. Use variable
names
that are easy to understand and declare all

variables.

9. Put lots of comments in!

10. Keep your loops simple!

11. Think a lot about your data structures.

12. Use arrays to store data.

13. Use built in functions and program libraries.

14. Put clarity (for
people) above compactness or even speed of execution