Exact and first-order error analysis of the Schur and split Schur algorithms: theory and practice

Abstract

A new analytical methodology is introduced here for fixed-point error analysis of various Toeplitz solving algorithms. The method is applied to the very useful Schur algorithm and the lately introduced split Schur (1918, 1986) algorithm. Both exact and first order error analysis are provided in this paper. The theoretical results obtained are consistent with experimentation. Besides the intrinsic symmetry of the error propagation recursive formulae, the technique presented here is capable of explaining many practical situations. For signals having a small eigenvalue spread the Schur algorithm behaves better than the split Schur in the fixed-point environment. The intermediate coefficients of the split Schur algorithm leading to the PARCOR's cannot serve as alternatives to the reflection coefficients in error sensitive applications. It is demonstrated that the error-weight vectors of the Schur propagation mechanism follow Levinson-like (second order) recursions, while the same vectors of the split Schur propagation mechanism follow split Levinson-like (third-order) recursions.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>