TWO EXAMPLES AS MOTIVATION FOR THE STUDY OF COMPUTER ERRORS Prof Jorge Lemagne Faculty of Science, Bindura University jlemagne@buse.ac.zw, jorgelemagneperez@gmail.com

Summary Introduction Example 1: The Patriot Missile Failure Example 2: An apparently contradictory result Exhortations

Introduction. To encourage students to study Mathematics For many people, Mathematics is a boring discipline. They ignore its fundamental role in Science, Technology and in general, in life. Showing the students these relationships would motivate and encourage them to study this discipline.

9 strategies for increasing student motivation in Math (Posamentier [2013]) (1) S1. Call attention to a void in student’s knowledge. S2. Show a sequential achievement. S3. Discovering a pattern S4. Present a challenge. S5. Entice the class with a “Gee-Whiz” mathematical result.

9 strategies for increasing student motivation in Math (Posamentier [2013]) (2) S6. Indicate the usefulness of a topic. S7. Use recreational mathematics. S8. Tell a pertinent story. S9. Get students actively involved in justifying mathematical curiosities.

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.”

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.

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.

Summary (1)  Introduction  Example 1: The Patriot Missile Failure Example 2: An apparently contradictory result Exhortations

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.

To produce the time in seconds  

But, what was actually stored?  

Error  

Total time error  

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.

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.

Cause of this disaster  

Summary (2)  Introduction  Example 1: The Patriot Missile Failure  Example 2: An apparently contradictory result Exhortations

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:

MATLAB Presentation

Simple experiment >> 0.3*4==1.2 ans = 1 >> 0.4*3==1.2 >> Why? (To be explained)

Traditionally  

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.

Scientific notation  

Significant digits  

Floating point number  

   

IEEE (Institute of Electrical and Electronics Engineers)  

For the sake of clarity  

The other data  

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.

Results  

Why unexpected results?   That is why we obtain unexpected results.

In general, laws of arithmetic do not hold on scientific computing  

Further information To deepen on floating point arithmetic and analysis of error: Conte and de Boor [1980] and Heath [2002] (for instance)

Summing up In general, on scientific computing the representation of numbers is not exact. Nor the result of arithmetic operations is exact. Hence, most laws of arithmetic do not hold on scientific computing.

Summary (3)  Introduction  Example 1: The Patriot Missile Failure  Example 2: An apparently contradictory result  Exhortations

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.

You are exhorted to Verify the previous results, and to experiment computationally. Learn conversion from decimal to binary notation. Study floating point arithmetic and analysis of error. These are ways for developing skills.

It is also recommended these examples to be used As motivation in Scientific Computing and Numerical Analysis courses, and a vocational guidance to O and A level students. To show the students (once more) the strong relationships between Mathematics and Computer Science. To illustrate some of the difficulties and challenges to be faced when new technologies are used.

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)

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-alfred-posamentier