Using Floating-Point Intervals for Non-Modular Computations in Residue Number System

Abstract

The residue number system (RNS) provides parallel, carry-free, and high-speed arithmetic and is therefore a good tool for high-performance computing. However, operations such as magnitude comparison, sign computation, overflow detection, scaling, and division are difficult to perform in RNS, since it is problematic to determine the magnitude of an RNS number. In order to resolve this problem, we propose to compute the interval evaluation of the fractional representation of an RNS number in floating-point arithmetic of limited precision. No matter what the size n of the moduli set and dynamic range, only small arithmetic operations are required, and most of the computations are performed in parallel with n threads, which allows for efficient implementation of our method on many general-purpose computing platforms. Using this method, we propose new algorithms for magnitude comparison and general division in RNS and implement them for GPUs using the CUDA platform. We evaluate the performance of our algorithms on an NVIDIA GTX 1080 GPU using sets of 4 to 256 RNS moduli that provide dynamic ranges from 64 to 4096 bits. Experimental results show that the proposed new algorithms are efficient for large moduli sets and clearly outperform the existing RNS magnitude comparison and division algorithms in terms of execution time. INDEX TERMS Residue number system, floating-point arithmetic, non-modular operations, magnitude comparison, division, high performance, parallel algorithms, graphics processing unit, CUDA.

Technique to Adjust Adaptive Digital Filter Coefficients in Residue Number System Based Filters

“…Given the advantages and disadvantages of the considered adaptation algorithms, we can conclude that RNS is suitable to improve performance when adjusting the coefficients and reduce the computational complexity of the tuning procedure [22]. These results can be used for building effective parallel computational systems [13] based on computers with parallel structure like FPGA [17,18] and GPU [23,24].…”

Classification of Moduli Sets for Residue Number System With Special Diagonal Functions

“…After determining these parameters from Appendices 1 and 2, the most promising RNS moduli sets are these with the lowest possible β. These results can be used for building effective parallel computational systems [15] based on computers with parallel structure like FPGA and GPU [16,17]. The basic idea of a hardware implementation is that an algorithm (division, sign detection, comparison of numbers, reverse conversion) based on a diagonal function requires division by SQ.…”

Efficient GPU Implementation of Multiple-Precision Addition based on Residue Arithmetic

“…An alternative method for implementing time-consuming operations in the RNS is based on computing the floatingpoint interval evaluation of the fractional representation of an RNS number [19]. This method is designed to be fast on modern general-purpose computing platforms that support efficient finite precision floating-point arithmetic operations such as IEEE 754 operations.…”

“…In these interval formulas, the following notation are used: Using interval evaluations, new algorithms have been proposed in [19] to efficiently implement several difficult RNS operations, such as number comparison and general division.…”

“…Both of our multiple-precision addition algorithms use an interval floating-point evaluation technique for efficient RNS sign determination [19]. However, the first algorithm relies on if-else statements to test the signs of the operands, while the second one uses the radix complement RNS notation for negative numbers.…”