Stability of QR-based fast system solvers for a subclass of quasiseparable rank one matrices

The development of fast algorithms to perform computations with quasiseparable matrices has received a lot of attention in the last decade. Many different algorithms have been presented by several research groups all over the world. Despite this intense activity, to the best of our knowledge, there is no rounding error analysis published for these fast algorithms. In this paper, we present error analyses for two fast solvers of quasiseparable linear systems when they are applied on order one quasiseparable matrices that include the diagonal in the lower triangular rank structure. Both solvers are based on computing first the QR factorization of the coefficient matrix, and their error analyses require novel structured techniques for proving rigorously that only one of the considered algorithms is backward stable, while the other one is not. Two fundamental consequences of this work are: (i) users should employ with caution fast algorithms for quasiseparable matrices since they may be unstable; and (ii) a lot of work has to be done to identify which fast algorithms for quasiseparable matrices are backward stable among the large family available in the literature.