Infinite Precision Analysis of the Fast QR Algorithm Based on a Posteriori Backward Prediction Errors

Abstract

The conventional QR Decomposition (QRD) method requires order ¢ ¤£ -O[¢ ¥£ ]-multiplications per output samples. However a number of fast QRD algorithms have been proposed with O[¢ ] of complexity. Particularly the Fast QRD algorithm based on a posteriori backward prediction errors is well known for its good numerical behavior and low complexity. This algorithm has two distinct versions and, in order to decide which one to choose for a given implementation, its infinite precision analysis of the mean square values of the internal variables would be required. In addition to this implementation issue, the finite-precision analysis requires the estimates of these mean square values. In this work, we first present an overview of the Fast QRD algorithms based on a posteriori backward prediction errors, followed by an infinite precision analysis of the steady state mean square values of the internal variables. Finally, the validity of the analytical results are verified by computer simulations carried out in a system identification setup. In the appendices, the detailed description of each implementation is listed.