TWO EXAMPLES AS MOTIVATION FOR THE STUDY OF COMPUTER ERRORS

1
TWO EXAMPLES AS MOTIVATION FOR THE STUDY OF
COMPUTER ERRORS

• Prof Jorge Lemagne
• Faculty of Science, Bindura University
• jlemagne_at_buse.ac.zw, jorgelemagneperez_at_gmail.com

2
Summary
3
Introduction. To encourage students to study
Mathematics
4
9 strategies for increasing student motivation in
Math (Posamentier 2013) (1)
5
9 strategies for increasing student motivation in
Math (Posamentier 2013) (2)
6
Preceding strategies might be applied

• In this talk Two examples as motivation for the
  study of computer errors.
• In both, all S1 to S9 might be applied.
• Especially S6. Indicate the usefulness of a
  topic can be carried out by introducing a
  practical application of genuine interest to the
  class at the beginning of a lesson.

7
Inevitable presence of error

• Scientific computing Discipline concerned with
  the development and study of numerical algorithms
  for solving mathematical problems that arise in
  science and engineering.
• The most fundamental feature of numerical
  computing is the inevitable presence of error.

8
Consequences of careless numerical computing

• Scientists and engineers often wish to believe
  that the numerical results of a computer
  calculation, especially those obtained as output
  of a software package, contain no error at least
  not a significant or intolerable one.
• But careless numerical computing does
  occasionally lead to disasters.
• Among them one of the most spectacular was the
  Patriot missile failure.

9
Summary (1)
10
Example 1 The Patriot Missile Failure

• February 25, 1991 (Gulf War), Dharan, Saudi
  Arabia
• An American Patriot Missile was supposed to
  track and intercept an incoming Iraqi Scud
  missile.

11
To produce the time in seconds

•  

12
But, what was actually stored?

•  

13
Error

•  

14
Total time error

•  

15
The distance travelled

• A Scud travels at about 1676 meters per second.
• So travels more than half a kilometre in this
  time (0.34 sec).
• This was far enough that the incoming Scud was
  outside the "range gate" that the Patriot
  tracked.

16
Consequence

• As a consequence, the Patriot failed to track and
  intercept the incoming Iraqi Scud missile.
• The Scud struck an American Army barracks,
  killing 28 soldiers and injuring around 100 other
  people.

17
Cause of this disaster

•  

18
Summary (2)
19
Example 2 An apparently contradictory result

• The second example of this talk is far less
  tragic than the preceding one.
• We initially propose you to make a simple
  experiment.
• It will be used an environment that is suitable
  for technical computing MATLAB (MathWorks
  2013). We open the application

20
MATLAB Presentation
21
Simple experiment
gtgt 0.341.2 ans 1 gtgt 0.431.2 ans
0 gtgt Why? (To be
explained)
22
Traditionally

•  

23
Scientific calculations

• However, scientific calculations are not exact or
  use decimal notation.
• Why?
• Scientific calculations are usually carried out
  in floating point arithmetic.
• Actually, this is just a generalization of what
  is called scientific notation.

24
Scientific notation

•  

25
Significant digits

•  

26
Floating point number

•  

27
 

•  

28
IEEE (Institute of Electrical and Electronics
Engineers)

•  

29
For the sake of clarity

•  

30
The other data

•  

31
Rounding again

• Now, we perform the multiplications with these
  rounded numbers.
• Each multiplication gives a number with 10
  significant digits.
• Hence, it must be rounded again.

32
Results

•  

33
Why unexpected results?
34
In general, laws of arithmetic do not hold on
scientific computing

•  

35
Further information

• To deepen on floating point arithmetic and
  analysis of error
• Conte and de Boor 1980 and
• Heath 2002 (for instance)

36
Summing up
37
Summary (3)
38
So, what have we seen?

• Two examples as motivation for the study of
  computer errors.
• These may be startling to readers who are not
  familiarized with computer arithmetic.

39
You are exhorted to
40
It is also recommended these examples to be used
41
Bibliography (1)

• 1 Arnold, D. N. (2000) The Patriot Missile
  Failure, http//www.ima.umn.edu/arnold/disasters/
  disasters.html
• 2 Conte, S. D. and de Boor, C. (1980)
  Elementary Numerical Analysis, an Algorithmic
  Approach, Third Edition, McGraw-Hill Book
  Company, ISBN 0-07-012447-7
• 3 Heath, M. T. (2002) Scientific Computing An
  introductory survey, Second edition, The
  McGraw-Hill Companies, Inc.,
  ISBN 0-07-239910-4, ISBN 0-07-112229-X (ISE)

42
Bibliography (2)

• 4 Higham, N. J. (1996) Accuracy and stability
  of numerical algorithms, SIAM, Philadelphia,
  ISBN O-8987 l-355-2 (pbk.)
• 5 MathWorks, Inc., The (2013) MATLAB R2013a
• 6 Posamentier A. (2013) 9 Strategies for
  Motivating Students in Mathematics, EDUTOPIA, The
  George Lucas Educational Foundation,
  http//www.edutopia.org/blog/
  9-strategies-motivating-students-mathematics-alfre
  d-posamentier