A stability analysis of predictor-based least squares algorithm

The numerical property of the recursive least squares (RLS) algorithm has been extensively studied. However, very few investigations are reported concerning the numerical behavior of the predictor-based least squares (PLS) algorithms which provide the same least squares solutions as the RLS algorithm. In Ref. [9], we gave a comparative study on the numerical performances of the RLS and the backward PLS (BPLS) algorithms. It was shown that the numerical property of the BPLS algorithm is much superior to that of the RLS algorithm under a finite-precision arithmetic because several main instability sources encountered in the RLS algorithm do not appear in the BPLS algorithm. This paper theoretically shows the stability of the BPLS algorithm by error propagation analysis. Since the time-variant nature of the BPLS algorithm, we prove the stability of the BPLS algorithm by using the method as shown in Ref. [6]. The expectation of the transition matrix in the BPLS algorithm is analyzed and its eigenvalues are shown to have values within the unit circle. Therefore we can say that the BPLS algorithm is numerically stable.