How stable is [Levinson-Durbin](http://en.wikipedia.org/wiki/Levinson_recursion) 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.



Stability of Levinson-Durbin method for Toeplitz system solutions ?
It can be unstable, and the condition number is not an adequate measure to tell when it fails. You may want to check the numerical experiments in <http://www.jstor.org/stable/2153371> for specific examples and discussion.


If I remember correctly, you can prove stability only for symmetric matrices whose principal minors are all positive (totally positive matrices), otherwise things can go wrong. More precisely, there are numbers called *reflection coefficients* that can be defined along the algorithm; you get stability only if they are all positive, as for instance in the totally positive case (sorry but I do not remember the details exactly, it's some years since I worked on that).


The GKO algorithm, developed in the article I linked above, is generally considered to be a more stable alternative. However, you have to pay for that with larger computational times and memory overhead (you can save the memory overhead though).


