FPGA implementation of an exact dot product and its application in variable-precision floating-point arithmetic

Abstract

The current paper explores the capability and flexibility of field programmable gate-arrays (FPGAs) to implement variable-precision floating-point (VP) arithmetic. First, the VP exact dot product algorithm, which uses exact fixed-point operations to obtain an exact result, is presented. A VP multiplication and accumulation unit (VPMAC) on FPGA is then proposed. In the proposed design, the parallel multipliers generate the partial products of mantissa multiplication in parallel, which is the most time-consuming part in the VP multiplication and accumulation operation. This method fully utilizes DSP performance on FPGAs to enhance the performance of the VPMAC unit. Several other schemes, such as two-level RAM bank, carry-save accumulation, and partial summation, are used to achieve high frequency and pipeline throughput in the product accumulation stage. The typical algorithms in Basic Linear Algorithm Subprograms (i.e., vector dot product, general matrix vector product, and general matrix multiply product), LU decomposition, and Modified Gram–Schmidt QR decomposition, are used to evaluate the performance of the VPMAC unit. Two schemes, called the VPMAC coprocessor and matrix accelerator, are presented to implement these applications. Finally, prototypes of the VPMAC unit and the matrix accelerator based on the VPMAC unit are created on a Xilinx XC6VLX760 FPGA chip.

Compared with a parallel software implementation based on OpenMP running on an Intel Xeon Quad-core E5620 CPU, the VPMAC coprocessor, equipped with one VPMAC unit, achieves a maximum acceleration factor of 18X. Moreover, the matrix accelerator, which mainly consists of a linear array of eight processing elements, achieves 12X–65X better performance.