Vectorised/Semi-Parallel Interval Multiplication

Abstract

To date, two principle methods for the multiplication of two intervals have been proposed. Namely, the multiplication of all four bounds and finding their minima/maxima; or by pre-processing the bounds and determining which multiplicands to use based upon their signs. In either case, a minimum of four multiplications are required for complete coverage and special cases such as [0, 1] times [-infin, 1] can result in the less than enlightening [-infin, +infin]. This paper describes a new method of interval multiplication that never requires more than two multiplications, has no special cases and elegantly handles the above case. We continue by describing reformulations of the brute-force and 9-case methods which, through making use of SIMD technology, parallelise and vectorise their operation, ultimately allowing the complete removal of branching. We conclude with an analysis of the algorithms and their performance, compared with the two forementioned traditional techniques.