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Next: 2.7.5 Algebraic Properties Up: 2.7 Interval Arithmetic Previous: 2.7.3 Inclusion Property

2.7.4 Interval Extension

For any total function tex2html_wrap_inline32577 , an implementation tex2html_wrap_inline32571 which always returns the universal interval tex2html_wrap_inline32593 is valid. Clearly implementation validity is not a sufficient indicator of quality. A good interval arithmetic implementation of a function returns small intervals.

The interval extension of a unary real function tex2html_wrap_inline32577 is an interval function tex2html_wrap_inline32571 defined by:

math6699

where l is an extended real number which bounds tex2html_wrap_inline32601 from below for all x in i; l is rounded down to determine the lower bound of tex2html_wrap_inline32609 . l may have the value tex2html_wrap_inline32377 if tex2html_wrap_inline32601 has no finite lower bound. u is similarly used to determine the upper bound. Although the interval extension is not a method to construct good interval operators, it can be used to show that a particular implementation returns optimal values.

The interval extension tex2html_wrap_inline32571 of an n-ary function tex2html_wrap_inline32577 is defined by:

math6715


next up previous notation contents
Next: 2.7.5 Algebraic Properties Up: 2.7 Interval Arithmetic Previous: 2.7.3 Inclusion Property
Jeff TupperMarch 1996