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2.7 Interval Arithmetic

Although floating point computations are simple and efficient, rounding can cause a stream of floating point computations to quietly diverge from the envisioned stream of real computations. Interval arithmetic guarantees rigorous results yet is built from floating point arithmetic. Interval arithmetic will not prevent a series of computations from wandering but it will inform the user how much the computed result could deviate from the real result (the result using real numbers for the computations). The presentation given here differs somewhat from conventional introductions [4, 56, 57], due to the impending generalizations.

The set of   intervals is denoted by tex2html_wrap_inline32517 . An interval is specified by two floating point numbers, a lower and upper bound.

figure6517

The interval tex2html_wrap_inline32017 represents any particular real number between a and b.   Rather than returning a single floating point number each operation will return a range of numbers which the real result is guaranteed to be in.

For example, tex2html_wrap_inline32525 can be represented as the interval

math6539

since

math6547

Operations involving tex2html_wrap_inline32527 are not ``aware'' that tex2html_wrap_inline32527 represents tex2html_wrap_inline32525 . The operations only assume that tex2html_wrap_inline32527 represents some fixed real number between tex2html_wrap_inline32535 and tex2html_wrap_inline32537 .


next up previous notation contents
Next: 2.7.1 Syntax Up: 2 Numbers Previous: 2.6.3 Infinity Unveiled
Jeff TupperMarch 1996