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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 . An interval is specified by two floating point numbers, a lower and upper bound.

The interval 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, can be represented as the interval

since

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

Next: 2.7.1 Syntax Up: 2 Numbers Previous: 2.6.3 Infinity Unveiled
 Jeff Tupper March 1996