The extension of the concept of the generating function to a class of preconditioned Toeplitz matrices

It is well known that the generating function f L1[−π, π] of a class of Hermitian Toeplitz matrices An(f) describes very precisely the spectrum of each matrix of the class. Recently, this theory has been extended, in various senses, in three directions: the non-Hermitian case, the block case, and the case of preconditioned Toeplitz matrices. Here, by using some results of the first and the second extension, we deduce new properties of distribution of the spectrum of preconditioned matrices and we define the concept of generating function for these matrices. Moreover, we discuss the consequences of these results on the analysis of convergence of preconditioned conjugate-gradient methods previously defined in the literature. Finally, we perform some numerical experiments which confirm the theoretical idea.