On the Description and Stability of Orthogonal Transformations of Rank Structured Matrices

Abstract

This paper develops a formal framework for the derivation and error analysis of algorithms for the orthogonal transformation of a rank structured matrix $A$. The formalism works with the Givens-weight representation and swapping procedures of Delvaux and Van Barel [SIAM J. Matrix Anal. Appl., 29 (2007), pp. 1147--1170], in which $A$ is parameterized in terms of plane rotations and a banded weight matrix $B$. In the Delvaux and Van Barel formulation, orthogonal transformations are applied to $A$ by swapping from one type of parameterization to another while simultaneously applying the desired transformations and computing the new parameterization so as to preserve bandedness of $B$. In this paper, swapping is formalized in terms of recurrences that rigorously justify the procedures for a relatively general class of two-sided transformations, while also providing the foundation for a rigorous proof of backward stability. The recurrences are used to describe a fast system solver based on $QR$ factorization. ...