Multiply - Add Optimized Fft Kernels

Modern computer architecture provides a special instruction|the fused multiply-add (FMA) instruction|to perform both a multiplication and an addition operation at the same time. In this paper newly developed radix-2, radix-3, and radix-5 FFT kernels that e ciently take advantage of this powerful instruction are presented. If a processor is provided with FMA instructions, the radix-2 FFT algorithm introduced has the lowest complexity of all Cooley-Tukey radix-2 algorithms. All oating-point operations are executed as FMA instructions. Compared to conventional radix-3 and radix-5 kernels the new radix-3 and radix-5 kernels greatly improve the utilization of FMA instructions resulting in a signi cant complexity reduction. In general, the advantages of the FFT algorithms presented in this paper are their low complexity, their high e ciency, and their striking simplicity. Numerical experiments show that FFT programs using the new kernels clearly outperform conventional FFT routines, even the best available FFT programs.