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
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
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:
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