High performance implementation of the TFT

Abstract

This paper reports on a high-performance implementation of the truncated Fourier transform (TFT). A general CooleyTukey like algorithm for the TFT is developed that allows the implementation to automatically adapt to the memory hierarchy. Then the algorithm introduces a small relaxation for larger transform sizes which trades off slightly higher arithmetic cost for improved data flow which allows full vectorization and parallelization. The implementation is automatically derived and tuned using the SPIRAL system for code generation and adaptation. The resulting arbitrarysize TFT library smooths out the staircase performance associated with power-of-two modular FFT implementations while retaining the performance associated with state-of-theart FFT libraries. This provides significant performance improvement over approaches that pad to the next power of two even when using high-performance FFT libraries.

High performance implementation of the inverse TFT

“…For obtaining high performance implementations, the complexity of the ITFT has limited implementations to fixed decomposition strategies and scalar codes. This paper extends the results from [18] to obtain general-radix and parallel algorithms for the ITFT, providing an automatically tuned high performance implementation. Combined with the TFT library in [18], a high performance implementation of fast integer and polynomial arithmetic can be obtained.…”

“…SPL expressions such as I ⊗ A and A ⊗ I can be automatically derived and optimized for vectorization and parallelization as illustrated in [18].…”

“…In [14,1], Mateer and Arnold devised and improved the in-place cyclotomic TFT, respectively. In [18], a parallel general-radix algorithm for the TFT was designed and implemented using the SPIRAL system to automatically derive and tune the implementation. The resulting general-size TFT library achieved significant speedup over a state-of-the-art parallel modular DFT library.…”

Fast Polynomial Multiplication over F260

“…In the next section we recall and slightly adapt classical results, and present our implementation framework. Our use of codelets for small and moderate sizes of DFT is customary in other high performance software, such as FFTW3 and SPIRAL [10,29].…”

A parallel implementation for polynomial multiplication modulo a prime

“…In [8] Harvey presents a new serial FFT algorithm which combines van der Hoeven's truncated FFT with David Bailey's cachefriendly FFT [1]. Meng and Johnson [13] improve on Harvey's work by incorporating vector instructions and multithreading into the truncated FFT. The implementation is derived and tuned using the Spiral system for code generation.…”