Abstract
The eigenvalue clustering of matrices S n − 1 A n S_n^{-1}A_n and C n − 1 A n C_n^{-1}A_n is experimentally studied, where A n A_n , S n S_n and C n C_n respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners generated by the Fourier expansion of a function f ( x ) f(x) . Some illustrations are given to show how the clustering depends on the smoothness of f ( x ) 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 C A C CAC , where A A is a Toeplitz matrix and C C is a circulant. In particular, it is proved that the Toeplitz (displacement) rank of C A C CAC is not greater than 4, provided that C C and A A are symmetric.
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