Robust parallel computation in floating-point and SLI arithmetic

In this paper we consider the parallel computation of vector norms and inner products in floating-point and a proposed new form of computer arithmetic, the symmetric level-index system. The vector norms provide an illuminating example of the contrast between the two arithmetic systems under discussion in terms of the ability to program for (complete) robustness and parallelizability. The conflict between robustness of the computation—in the sense of the dual requirements of accuracy and freedom from overflow and underflow—and easy parallelization of the algorithms within a floating-point environment is made plain. It is seen that this conflict disappears if the symmetric level-index system of arithmetic is used. The freedom from overflow and underflow offered by this system allows the programming of the straightforward definitions in a way which is simple, robust and immediately parallelizable. Numerical results are given to illustrate the fact that the symmetric level-index system yields results of comparable accuracy to those of floating-point in cases where the latter system works and still yields results of high accuracy when the floating-point system fails altogether.