On the conditioning of a Symmetric Toeplitz Matrix I have the following problem, which I hope is enough interesting for you to help me. I have a matrix $A$ which is **Toeplitz, Symmetric and Positive definite**. Such a matrix is an _autocorrelation matrix_ (< Maybe is also important to say that $dim(A)>250$. In my case of analysis, $A$ is extremely ill-conditioned with values $\kappa(A) >10^{13}$. I'm trying to find a suitable way of preconditioning or regularization of the problem because that matrix is involved in a linear system of equations $Ax=b$. Things that I have already tried, without success: * Optimal and Superoptimal preconditioners for Toeplitz matrices (< * Use a Cholesky decomposition. * Thikhonov Regularization (< Any other ideas or suggestions will be extremely appreciated! Thanks!!!

Your matrix $A$ is numerically singular, and the problem $Ax = b$ cannot be solved, by any algorithm. Any preprocessing, postprocessing, iterative refinement, etc. will be useless.

Possible solutions:

1. use high precision arithmetics,
2. reformulate the problem.



As the second method you may consider different regulations; performing SVD , removing small singular values, and then solve least square problems; or something else. But in this case you are solving completely different problem, which has nothing to do with solving $Ax=b$. Such obtained result may be usefull or not, it highly dependents on the problem.