On the generation of algorithms for linear transforms and systems with symmetry

Abstract

For many applications, especially from digital signal processing, linear maps must be applied frequently. This amounts to computing matrix-vector multiplications. To achieve real-time performance, it is essential to have fast algorithms. Many of the matrices that come up in the applications have a lot of structure that can be exploited to reduce the number of scalar operations to be performed in order to multiply with these matrices. This article is concerned with the case when this structure can be characterized by a special kind of symmetry. Its main objective is to describe how such a symmetry can be used to obtain a factorization of the matrix with all factors being sparse. Such a factorization yields an algorithm for multiplying with the matrix. In the literature various ways of applying one linear transform through another one by doing some minor extra calculations have been published. A well known example of this is the application of the Fourier transform via the Hartely transform. A further results presented here is a theoretical explanation for this on the basis of a relation on the symmetries of the involved transforms. The methods have been used to automatically generate fast algorithms for many of the transforms and filters from digital signal processing. The resulting algorithms compare well with manually derived ones from the literature. (orig.)SIGLEAvailable from TIB Hannover: RR 631(1997,3) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman