Exact analysis of the finite precision error generation and propagation in the FAEST and the fast transversal algorithms. A general methodology for developing stable a posteriori RLS computational schemes

摘要

An accurate analysis of the finite precision error in the FAEST-5p and the fast transversal algorithm (FTF) is undertaken here, on the basis of an entirely new methodology and practice. The cause of the numerical instability and divergence of these two schemes is clearly pointed out and demonstrated. In particular, it is proved that, out of all the formulas for these two algorithms, only four (4) specific formulas generate an amount of finite precision error that consistently makes the algorithms fail after a certain number of iterations. Moreover, it is shown that there is a limited number of specific formula that transmit this generated finite precision error. Finally, a general methodology is introduced that allows for the development of new a posteriori algorithms that are intrinsically free of the corresponding finite precision numerical problems and that therefore are, in practice, fully stable.