Numerical stability properties of a QR-based fast least squares algorithm

The numerical stability of a recent QR-based fast least squares algorithm is established from a backward stability perspective. The paper introduces a stability domain approach applicable to any least squares algorithm, so constructed from the set of reachable states in exact arithmetic. The error propagation question is shown to be subordinate to a backward consistency constraint, which requires that the set of numerically reachable variables be contained within the stability domain associated to the algorithm. This leads to a conceptually lucid approach to the numerical stability question which frees the analysis of stationarity assumptions on the filtered sequences, and obviates the tedious linearization methods of previous approaches. Moreover, initialization phenomena and considerations behind poorly exciting inputs admit clear interpretations from this perspective. The notion of minimality (in the system theory sense) is shown to be a crucial consideration from the consistency perspective, and the fast QR algorithm under study is, in contrast to many fast algorithms, proved to be minimal.