Characteristics for a new preconditioner of linear systems

We propose to develope a preconditioner HC for solving the toeplitz T linear system,. Since toeplitz matrices have a nice structure, it has been researched in toeplitz matrix properties and developed precoditioners for the linear toeplitz system. In this paper, we develope a hermitian circulant matrix as a preconditioner HC for the toeplitz linear system. We have two conditions such as (i) ∥HC - T∥F is minimized (ii) HC has also a nice structure such as given matrix T. This preconditioner has an advantage contrary to the existing preconditioners. That is, all eigenvalues of the given toeplitz matrix T are very close to all eigenvalues of our preconditioner HC. Also all eigenvalues of HC-1T are very close to 1. It supports it is a good choice as a preconditioner HC.