Stability of Levinson-Durbin method for Toeplitz system solutions ?

How stable is Levinson-Durbin method for solution of systems of linear equations ?

I mean if condition number of matrix is $k$, does intermidiate steps involve matrixes with higher condition number ? For example QR is easy to see preserve condition number so it is stable, but Cholesky for example increase condition number from $k$ to $k^2$, so Cholesky is less stable.

The question is related with possible fix-point implementation of this method and we need to understand how many bits should be given, for matrices of small size.

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@Alexander The same paper shows some examples of PD matrices in which Levinson gives a much larger error than GKO and the conventional $O(n^3)$ Gaussian elimination methods. –  Federico Poloni Aug 16 '12 at 10:51