A Self-sorting In-place Fast Fourier Transform Algorithm Suitable for Vector and Parallel Processing

We propose a new algorithm for fast Fourier transforms. This algorithm features uniformly long vector lengths and stride one data access. Thus it is well adapted to modern vector computers like the Fujitsu VP2200 having several oating point pipelines per CPU and very fast stride one data access. It also has favorable properties for distributed memory computers as all communication is gathered together in one step. The algorithm has been implemented on the Fujitsu VP2200 using the basic subroutines for fast Fourier transforms discussed elsewhere. We develop the theory of index digit permutations to some extent. With this theory we can derive the splitting formulas for almost all mixed-radix FFT algorithms known so far. This framework enables us to prove these algorithms but also to derive our new algorithm. The development and systematic use of this framework is new and allows us to simplify the proofs which are now reduced to the application of matrix recursions. 1. Introduction Fast Fourier transforms are of primary importance in computationally intensive applications. They are used for large-scale data analysis and to solve partial diier-ential equations. Applications of FFTs in data analysis include computer tomography, data ltering and structure or uid-structure interaction analysis. Other areas of interest here are spectral analysis of speech, sonar, radar, seismic and vibration detection. They are used for digital ltering, convolution evaluation and signal decomposition. A second eld of FFT applications are in the solution of partial diierential equations. An earlier example here is the fast Poisson solver which is closely related to FFTs. In recent years spectral methods featuring FFTs have been used successfully to solve computational uid dynamics (CFD) problems. They are routinely used in weather forecasting and for simulation in geophysics and quantum mechanics. All these applications use discrete Fourier transforms. They are nite-dimensional, linear, complex operators related to the transforms of classical Fourier analysis which