Sliding window adaptive fast QR and QR-lattice algorithms

Sliding window formulations of the fast QR and fast QR-lattice algorithms are presented. The derivations are based on the partial triangularization of raw data matrices. Three methods for window downdating are discussed: the method of plane hyperbolic rotations, the Chambers’ method, and the LINPACK algorithm. A numerically ill-conditioned stationary signal and a speech signal are used in finite wordlength simulations of the full QR (nonfast), fast QR, and QR-lattice algorithms. All algorithms are observed to be numerically stable over billions of iterations for double-precision mantissas (53 bits), but as the number of bits is decreased in the mantissa, the algorithms exhibit divergent behavior. Hence, practically, the algorithms can be regarded as numerically stable for long wordlengths.