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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 tex2html_wrap_inline33001 , where the interval bounds are linear functions of tex2html_wrap_inline32761 and tex2html_wrap_inline33005 :

math8149

Each parameter tex2html_wrap_inline32761 and tex2html_wrap_inline33005 may independently vary from zero to one.

In general, tex2html_wrap_inline33011 is defined as a real linear interval number system with k parameters. Each parameter may vary from zero to one independently:

math8155

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 tex2html_wrap_inline33015 to a system parameter tex2html_wrap_inline33017 to allow for more complex parameter domains. Forcing the upper and lower bounds to be zero when tex2html_wrap_inline33019 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 tex2html_wrap_inline32875 . The upper and lower bounds are not always linear functions, since tex2html_wrap_inline33025 for tex2html_wrap_inline33027 .


next up previous notation contents
Next: 2.9.7 Functional Intervals Up: 2.9 Generalized Interval Arithmetic Previous: 2.9.5 Quadratic Intervals
Jeff TupperMarch 1996