On toeplitz matrices construction algorithm with a given number of connected components of the limitary spectrum complement

The simplest topological properties of the approximate spectrum, namely the connectivity of its complement in the complex plane, are studied. A numerical verification of the lower bounds for the maximum number of the connected components of the limitary spectrum complement of the band Toeplitz matrices whose symbol is Laurent polynomial of the specified degree, is carried out. The algorithm for computation of the Toeplitz matrix symbol parameters with its approximate spectrum dividing the complex plane into a given number of connected components is adduced. The examples of polynomials which are Toeplitz matrices symbols with the limitary spectrum dividing the complex plane into a given set of connected components are numerically investigated. Graphs of the limitary spectra of Toeplitz matrices illustrating the results obtained are given. The obtained limitary spectra are compared to the Toeplitz matrices spectra of large size with a given symbol.