TWO EXAMPLES AS MOTIVATION FOR THE STUDY OF COMPUTER ERRORS

Prof Jorge Lemagne Faculty of Science, Bindura University

[email protected]

, [email protected]

Summary

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

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. 3

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. 4

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.

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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.”

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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. 7

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.

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Summary

(1)  Introduction  Example 1: The Patriot Missile Failure Example 2: An apparently contradictory result Exhortations 9

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.

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To produce the time in seconds

• • In Patriot computer, the system's internal clock measured the time in tenths of second.

1 So it was multiplied by 10 in seconds. to produce the time 11

But, what was actually stored?

1 In binary notation, the expansion of 10 terminating: 0.1

10 is non = 0.00011001100110011001100 … 2 1 However the multiplication by 10 was performed using a 24 bit fixed point register, so it stored instead 0.00011001100110011001100

2 ≈ 0.1

10 12

Error

Consequently, it is introduced an error 𝑒 = 0.11001100 … 2 ∙ 2 −23 ≈ 9.5 ∙ 10 −8 , called rounding error (in this case by chopping).

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Total time error

The Patriot battery was up around 100 hours.

Multiplying by the number of tenths of a second in 100 hours gives 𝐸 = 9.5 ∙ 10 −8 × 100 × 60 × 60 × 10 = 0.34

, the time error (in seconds).

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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.

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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.

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Cause of this disaster

• • The small rounding error 𝑒 , when multiplied by the large number giving the time in tenths of a second, led to a significant error 𝐸 , due to the magnified rounding error.

It turns out that the cause of this disaster was an inaccurate calculation of the time since boot due to computer arithmetic errors.

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Summary

(2)  Introduction  Example 1: The Patriot Missile Failure  Example 2: An apparently contradictory result Exhortations 18

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: 19

MATLAB Presentation

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>> 0.3*4==1.2

ans = 1 >> 0.4*3==1.2

ans = 0 >> Why? (To be explained)

Simple experiment

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Traditionally

• • • We are used to perform calculations using decimal notation.

Moreover, in some cases, they are done exactly. That happens when we perform 0.3 ∙ 4 = 0.4 ∙ 3 = 1.2

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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.

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Scientific notation

• • Essentially, as we know, its form has mantissa and exponent. For example: 3748 = 0.3748 ∙ 10 4 0.000001643 = 0.1643 ∙ 10 −5 −83.92 = −0.8392 ∙ 10 2 Since decimal notation is used, the base is 10, and each mantissa digit is between 0 and 9.

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Significant digits

3748 = 0.3748 ∙ 10 4 0.000001643 = 0.1643 ∙ 10 −5 −83.92 = −0.8392 ∙ 10 2 In the three examples the mantissa has 4 digits, or equivalently, each number has 4 significant digits.

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Floating point number

• • • 𝑥 = ± . 𝑑 1 𝑑 2 … 𝑑 𝑛 𝛽 ∙ 𝛽 𝑒 In general, the base is 𝛽 , and 𝑑 1 , 𝑑 2 are digits between 0 and 𝛽 − 1 , … , 𝑑 (normally 𝑛 𝑑 1 ≠ 0 ).

For most computers, 𝛽 = 2 .

The reason is mainly technological.

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Can we represent exactly 1.2 in floating point with 𝛽 = 2 ? • • • • Using the binary notation: 1.2

10 = 1.0011 0011 0011 … = 0.10011 0011 0011 … 2 ∙ 2 1 2 Observe that the mantissa has infinitely many binary digits or bits.

So, again, we need to approximate (round) the number. This means to take only 𝑛 digits. 27

IEEE (Institute of Electrical and Electronics Engineers) • • • Standard 754, published in 1985, defines a binary floating point arithmetic system.

Values of 𝑛 : 24 (simple precision), 53 (double precision) and 64 (extended precision).

Correspondingly, the precision in decimal system is about 7, 15 or 19, obtained multiplying by log 10 2 ≈ 0.3

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For the sake of clarity

We consider a computer with 𝑛 = 5 , so instead of 1.2

10 = 0.10011 0011 0011 … 2 ∙ 2 1 we have 1.2

10 ≈ 0.10011

Or equivalently: 1.2

10 ≈ 𝟏. 𝟎𝟎𝟏𝟏 Hence some error is committed.

2 ∙ 2 1 𝟐 Observe the number that is actually represented.

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The other data

Similarly, the other data represented in our computer are the following: 0.3

10 ≈ 0.010011

4 10 0.4

10 = 100 2 ≈ 0.011010

3 10 = 11 2 2 2 30

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. 31

Results

Thus, we have the following results: 1.2 ≈ 1.0011

0.3 ∙ 4 ≈ 1.0011

2 0.4 ∙ 3 ≈ 1.0100

2 2 Therefore, in our computer: 0.3 ∙ 4 = 1.2

But 0.4 ∙ 3 ≠ 1.2

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Why unexpected results?

In our example, when 0.3, 1.2

and 0.4

are represented in the computer, some error is committed.

Afterwards, when the multiplications 0.3 ∙ 4 3 and are performed, additional error is committed.

0.4 ∙ That is why we obtain unexpected results.

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In general, laws of arithmetic do not hold on scientific computing For some numbers 𝑎, 𝑏 and 𝑐 : 𝑎 + 𝑏 + 𝑐 ≠ 𝑎 + (𝑏 + 𝑐) 𝑎𝑏 𝑐 ≠ 𝑎 𝑏𝑐 𝑎 𝑏 + 𝑐 ≠ 𝑎𝑏 + 𝑎𝑐 𝑎 + 𝑏 = 𝑎 : 𝑏 ≠ 0 In arithmetic of 𝑛 = 3 decimal digits: 1 + 0.001 + 0.001 + ⋯ + 0.001

= 1 (1000 𝑡𝑖𝑚𝑒𝑠) 𝑜𝑛𝑤𝑎𝑟𝑑𝑠 2 𝑏𝑎𝑐𝑘𝑤𝑎𝑟𝑑 34

Further information

To deepen on floating point arithmetic and analysis of error: Conte and de Boor [1980] and Heath [2002] (for instance) 35

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.

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Summary

(3)  Introduction  Example 1: The Patriot Missile Failure  Example 2: An apparently contradictory result  Exhortations 37

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.

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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.

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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.

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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) 41

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 42