Numerical Investigation of the Spectral Distribution of Toeplitz-Function Sequences

Solving Toeplitz-related systems has been of interest for their ubiquitous applications, particularly in image science and the numerical treatment of differential equations. Extensive study has been carried out for Toeplitz matrices T_n∈C^n×n as well as Toeplitz-function matrices h(T_n)∈C^n×n, where h(z) is a certain function. Owing to its importance in developing effective preconditioning approaches, their spectral distribution associated with Lebesgue integrable generating functions f has been well investigated. While the spectral result concerning {h(T_n)}_n is largely known, such a study is not complete when considering {Y_nh(T_n)}_n with Y_n∈R^n×n being the anti-identity matrix. In this book chapter, we attempt to provide numerical evidence for showing that the eigenvalues of {Y_nh(T_n)}_n can be described by a spectral symbol which is precisely identified.