The article on the ForteTM Tools Software Web page, "Interval Arithmetic for Guaranteed Results," by Dr. Bill Walster, the Sun architect of interval arithmetic functionality in ForteTM Developer language products, provides an overview of interval arithmetic and why it is an important technology for certain types of software projects.

Walster defines interval arithmetic and notes that the technology for using interval arithmetic in computer science projects is now included in the Forte Developer language products.

There are two basic reasons why interval arithmetic is important, according to Walster:

1. Unlike floating-point numbers, intervals can be used to represent and bound numerical errors from all sources. The resulting interval error bounds are mathematically rigorous in that they are guaranteed to contain the set of all possible results.

2. Because intervals represent continuous sets of numbers, intervals contain numbers that are not representable on a machine. This gives intervals a logically rigorous connection to mathematics that does not exist for floating-point numbers. The result is that interval algorithms can be used to create numerical "proofs." Nothing like that is possible with floating-point numbers alone.

The topics Walster covers in this article include:

• Interval Widths
• The Inaccuracy of Floating-Point Numbers
• Intervals Have No Missing Numbers
• "The Fundamental Theorem of Interval Arithmetic"

As a direct result of this theorem, it is possible to develop interval algorithms for solving important nonlinear problems that many mathematicians continue to believe are impossible to solve numerically. Examples include:

• Computing bounds on parameters from observations where the model relating the parameters and observations is nonlinear
• Solving design optimization problems where the criterion for a good design is a nonlinear function of design parameters.

Also note that because intervals bound errors, when performing parameter estimation or solving for the best possible design, errors from all sources can be rigorously and automatically bounded.

There are additional details and equations on the Sun Web page:

http://www.sun.com/forte/info/features/intervals2.html

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