Generating symmetric DFTs and equivariant FFT algorithms

Authors:
Jeremy Johnson;Xu Xu
Affiliations:
Drexel University Philadelphia, PA;Drexel University Philadelphia, PA
Venue:
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Year:
2007

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Abstract

This paper presents a code generator which produces efficient implementations of multi-dimensional fast Fourier transform (FFT) algorithms which utilize symmetries in the input data to reduce memory usage and the number of arithmetic operations. The FFT algorithms are constructed using a group theoretic version of the divide and conquer step in the FFT that is compatible with the group of symmetries. The GAP compute algebra system is used to perform the necessary group computations and the generated algorithm is represented as a symbolic matrix factorization, which is translated into efficient code using the SPIRAL system. Performance data is given that shows that the resulting code is significantly faster than state-of-the-art FFT implementations that do not utilize the symmetries.