Tridiagonal factorizations of Fourier matrices and applications to parallel computations of discrete Fourier transforms

Abstract

Let ω n =e −2πi n , and F n be the n × n matrix defined by F n = 1 n (ω ij n ) , where i and j run from 0 to n – 1. Two different methods are developed for factoring F n into products of tridiagonal and permutation matrices. One method is based on matrix identities associated with FFTs and the Rader prime algorithm. The other method is based on a numerical technique, never before applied to Fourier matrices, called minimal-variable oblique elimination. New results established in this paper include the establishment of necessary and sufficient conditions for which minimal-variable oblique elimination can be used to compute tridiagonal decompositions of arbitrary square matrices and explicit descriptions of minimal-variable solutions in the case that the descriptions are satisfied, proof that minimal-variable oblique elimination can be applied successfully to Fourier matrices of all orders, and explicit descriptions of various tridiagonal decompositions of PF n for any n × n permutation matrix P . Complexity estimates are derived for both the parallel algorithms resulting from the decompositions and computation of the decompositions, and timings are estimated for processor arrays constructed using GAPP chips.