Which Circulant Preconditioner is Better?

The eigenvalue clustering of matrices $% S_n^{-1}A_n$ and $% C_n^{-1}A_n$ is experimentally studied, where $A_n$, $S_n$ and $C_n$ respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners generated by the Fourier expansion of a function $f(x)$. Some illustrations are given to show how the clustering depends on the smoothness of $f(x)$ and which preconditioner is preferable. An original technique for experimental exploration of the clustering rate is presented. This technique is based on the bisection idea and on the Toeplitz decomposition of a three-matrix product $CAC$, where $A$ is a Toeplitz matrix and $C$ is a circulant. In particular, it is proved that the Toeplitz (displacement) rank of $CAC$ is not greater than 4, provided that $C$ and $A$ are symmetric.