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## 2.9.6 Multi-Dimensional Linear Intervals

A number may describe something with several parameters. Several parameters may be integrated into the number system. The simplest such system is , where the interval bounds are linear functions of and :

Each parameter and may independently vary from zero to one.

In general, is defined as a real linear interval number system with k parameters. Each parameter may vary from zero to one independently:

The term linear interval was chosen over affine interval due to familiarity. Although the bounds are techincally affine functions, an interval system which used linear functions would not see much use. As will be seen when interval arithmetic application algorithms are discussed, there will often be a mapping from an ``actual'' parameter to a system parameter to allow for more complex parameter domains. Forcing the upper and lower bounds to be zero when would severely restrict these mappings, and the applicability of interval methods.

Consider our example problem, of determining the range of a function over a given domain. The linear interval chosen to represent the domain [a,b] was . The upper and lower bounds are not always linear functions, since for .

Next: 2.9.7 Functional Intervals Up: 2.9 Generalized Interval Arithmetic Previous: 2.9.5 Quadratic Intervals
 Jeff Tupper March 1996