Structured Matrices and Displacement Operators

Abstract

In Chapter 1, we outlined the displacement rank approach (COMPRESS, OPERATE, DECOMPRESS) to computations with structured matrices and in more detail covered its OPERATE stage. In this chapter, we systematically cover the basic techniques required at the COMPRESS and DECOMPRESS stages of this approach. We present these techniques in a unified way but also detail the decompression techniques separately for each of the most popular classes of structured matrices. As an immediate result, we obtain superfast algorithms for multiplying these matrices by vectors and by each other. We accentuate the power of the approach based on the displacement transformations of two kinds that extend successful algorithms from one class of structured matrices to various other classes. We also briefly comment on parallel implementation of computations with structured matrices.