Numerical properties of digit on-line arithmetic (pipeline)
Richard J. Zaccone · 1984
In digit on-line arithmetic, operands are introduced a digit at a time. After the first few operand digits have been introduced, the result begins to appear a digit at a time. This feature of digit on-line arithmetic allows a significant amount of overlapping of arithmetic operations. This work develops an error bound for digit on-line arithmetic operations. The bound is good and it indicates that accurate results can be expected from using this type of arithmetic. Digit on-line arithmetic can sometimes produce unnormalized results. This can present a problem for the divide and square root algorithms. If the divisor and radicand are highly unnormalized, these algorithms will not produce the correct results. Two advances in overcoming this problem are presented. First, several techniques for producing results that are closer to being normalized are developed. Second, it is shown that normalized results are not necessary for divide and square root to work properly. Combining these advances yields algorithms that will always give the correct results. Some problems, by their nature, do not produce highly unnormalized results. It is possible to solve these problems without any of the new operand conditioning techniques presented. This leads to architectures for solving these problems that are simpler and easier to build. A large class of problems for which a triangular decomposition can be formed without any normalization problems is defined. A simulator for performing digit on-line arithmetic has also been developed. It allows the coding of complicated problems involving structures such as vectors and matrices with very little difficulty. A description of the simulator and some experimental results are included. Finally, some preliminary results on using digit on-line arithmetic to solve ordinary differential equations are presented.