Structured matrices, continued fractions, and root localization of polynomials (Q2913267)

The paper gives an extensive and detailed account of the relations between different classes of objects and topics: structured matrices of Hankel-, Hurwitz-, Vandermonde- and other types, continued fraction representations of rational functions, and root localization of univariate polynomials.NEWLINENEWLINENEWLINEThe first section is concerned with complex rational functions. It is explained how Hankel- and Hurwitz-matrices associated with such functions can be used to localize and count poles and roots, to compute resultants and discriminants, and to find continued fraction representations. The second section considers real rational functions, Sturm's algorithm, Frobenius' rule of signs and Cauchy-indices. The third section is devoted to \(R\)-functions, functions that map the upper complex half-plane to itself or the lower complex half-plane. Again, connections to root counting and localization, continued fractions, and determinants of associated matrices are explained. The fourth and last section deals with real root counting for univariate polynomials.NEWLINENEWLINEThe contents of the paper range from classic results to more recent developments. The exhaustive presentation makes it a valuable source for finding and citing results.