Interval Arithmetic Training For Beginners

organized by

Sun Microsystems
and
The Danish Technical University

August 26-28, 2002

Lyngby
Denmark

Introduction

With great pleasure we announce this unique training on Interval Arithmetic. The event is a joint effort between the Danish Technical University (DTU) and Sun Microsystems. DTU is the Sun Center of Excellence (COE) for Interval Arithmetic and Dynamic Systems.

What Is An Interval ?

An interval is a range of numbers bounded by the interval's endpoints.

What Is Interval Arithmetic ?

Interval Arithmetic is the system for performing arithmetic on intervals.  The most significant characteristic of interval arithmetic is that resulting intervals are guaranteed  to contain the set of all possible results from any interval computation.  This is true, even when interval arithmetic is implemented on a computer using finite precision floating-point arithmetic.

As a consequence, interval arithmetic is a new computing paradigm.  Almost all the existing rules and assumptions for floating-point computations need to be reexamined.

Why Is Interval Arithmetic  Needed ?

There is a problem with floating-point numbers: Floating-point numbers are logically disconnected from the real worlds of science, engineering, and mathematics. The causes of this disconnect are:

• A single floating-point number contains no accuracy information.
• Most real numbers cannot be represented using finite precision floating-point numbers.
The consequences of this disconnect are profound, including:
• Computing speed produces unprecedented opportunities and risks. Unfortunately, without intervals, the opportunities cannot be exploited and the risks cannot be avoided.
• Intervals create a tight logical connection between computing and scientific, engineering, and mathematical reality.  As a consequence, it becomes possible to:
• Compute correct results the first time.
• Guarantee the accuracy of computed results.
• Validate computed problem solutions without benefit of independent verification.
• Compute answers to new questions that have never before been asked or answered.
• The interval paradigm has precisely the properties needed to connect computing to the real world.  Intervals are tightly connected to the real world in precisely the way that floating-point numbers, alone, are not:
• The width of an interval can be used to represent error in an approximation.
• Because an interval represents all the values it contains, a single pair of interval endpoints represents an infinite number of values, or a continuum. It is this fact that connects intervals to mathematics, so that nonlinear problems can be numerically solved.
A number is a single numeric value.  An interval is a range of numbers, bounded by the interval's endpoints.  The width of an interval can be zero, in which case it is a point, or it can be very large.  A somewhat more formal definition of  Intervals and Interval Arithmetic is the following:

 "An interval is the set of all real numbers between and including the interval's lower and upper bound. Interval arithmetic is used to evaluate arithmetic expressions over sets of numbers contained in intervals. Any interval arithmetic result is a new interval that is guaranteed to contain the set of all possible resulting values."

A simple example is the following. Define two intervals A and B. Assume A = [-1.0000, 2.0000] and B = [1.0000, 2.0000]. We can ask our selves the following questions:

• What do B+A, B-A, B*A, or A/B mean ?
• What does B/A mean ? Does the fact that A contains zero cause a problem ?
• How about testing A < B ?  If A and B overlap, the less-than relation is ambiguous.
• Perhaps an alternative interval relational test is: "Is A possibly less than B?"
Interval Arithmetic opens up a whole new world of possibilities. Uncertainty in input data can be dealt with, round-off behavior can be rigorously taken into account and even complete new algorithms can be designed that exploit the the properties of Interval Arithmetic to solve problems that have no other known solution.

The SUN Fortran 95 and C++ compilers support this new paradigm in a language native way.  Language support is provided for interval data types, basic arithmetic on intervals, interval extensions of standard intrinsic functions, relational operators, I/O, etc.  In addition to existing intrinsic functions, many interval-specific intrinsic operators and functions are included.

If you are interested in this exciting technology and want to learn more about it, sign up for the training ! Do it now, space is limited !

Some pointers to more information on Interval Arithmetic:

The Sun web site:

Other links to more information, which are also available as links in the second URL, above are:

Prerequisites

Other than basic programming skills in Fortran (or other languages, such as Pascal, C, C++, and/or Java) and an interest in this topic, no specific prerequisites are required. The training targets the absolute beginners in Interval Arithmetic. We will build everything up from scratch using Fortran. All aspects of intervals will be covered during the course. Last,  but certainly not least, you will also be able to ask any questions you ever wanted to ask about this topic.

Agenda

There is only topic: Interval Arithmetic.

We are delighted to have Bill Walster from Sun Microsystems to come over from the US and give this course. He is not only the Interval Arithmetic expert within Sun, but also a very enthusiastic and talented presenter.

But there is more. We plan to have some guest speakers from DTU on the last day. They will talk about their use of Interval Arithmetic. As part of this addition to the training, Bill will also share some of his research projects with us.

The best way to learn Interval Arithmetic is to try it yourself. Therefore we have allocated a substantial amount of time to the labs and exercises. You will be working on specific problems to explore the possibilities of this fascinating technology.

Monday August 26, 2002

 Topic Start time End time Welcome 9:00 9:30 Training - Part I 9:30 10:45 Break 10:45 11:00 Training - Part II 11:00 12:00 Lunch 12:00 13:00 Labs 13:00 14:15 Training - Part III 14:15 15:15 Break 15:15 15:30 Training - Part IV 15:30 17:00

Tuesday August 27, 2002

 Topic Start time End time Training - Part V 9:00 10:30 Break 10:30 10:45 Labs 10:45 12:00 Lunch 12:00 13:00 Training - Part VI 13:00 14:30 Break 14:30 14:45 Training - Part VII 14:45 15:45 Labs 15:45 17:00

Wednesday August 28, 2002

 Topic Start time End time Training - Part VIII 9:00 10:30 Break 10:30 10:45 Labs 10:45 12:00 Lunch 12:00 13:00 Guest presenter(s) 13:00 14:30 Break 14:30 14:45 Guest presenter(s) 14:45 16:15 Wrap-up/Q&A 16:15 16:30

Cost of the training

We will not charge for the training itself. You will be responsible for all other costs, like travel, hotel and meals.

Training Location

The Danish Technical University (DTU)
Lyngby (near Copenhagen)
Denmark

 Building 305 Lectures Room 053 Labs Rooms 117 and 225

The URL of the DTU web site (also in English) is: http://www.dtu.dk

On this web site you will find campus maps.

Registration

This training is open to all Sun customers, partners, employees and others interested.

Registration is very easy. Send an email to Anne Sorensen (anne.sorensen@sun.com) with the following information:

• Name of the participant(s)
• Company/Institute
• Full address
• Phone number and/or email address
• If you would like to stay in the preferred hotel (see below), please indicate so and give us your arrival/departure date and time
You will receive a confirmation of your registration request. If you indicated you would like to stay in the preferred hotel, we will make a room reservation for you.

The registration deadline is July 28, 2002. Seating is limited and we recommend to register as early as possible ! Allocation is on a first come, first serve basis. If demand exceeds our capacity, we may have to restrict attendance to 2-3 persons per company/institute.

Note: if you cancel your registration after August 19, we may have to charge you a cancellation fee.

Hotel

You are most welcome to stay wherever you like.

One site with hotel information is: http://www.copenhagen-hotels.dk/index.asp

For those interested to stay in a nearby location, we have reserved a block of rooms at the following hotel:

Hotel Marina
Vedbaek Strandvej 391
DK-2950 Vedbaek
Denmark

Phone +45 45891711
Fax +45 45891722

You can find more information on this hotel through the above given website. In the "Search by city" field select "Vedbaek"  and you'll get a link to the Marina Hotel.

The price per night is DKr. 875 (approximately 120 Euro). Breakfast is included.

Transportation

On the training days we will arrange a shuttle bus between the hotel and the training location. This bus will leave at a fixed time in the morning and at the end of the training day. All other transportation is at your own expense.

Dress code

There is no specific dress code. Please wear whatever you feel comfortable in.

Weather

Somewhat unpredictable at this time of the year, but typically the temperature is around 20-25 deg. C. Be prepared for cooler weather and/or rain though.