Structured matrices, continued fractions, and root localization of polynomials

TL;DR

This paper explores the deep connections between structured matrices, continued fractions, and polynomial root localization, offering both a comprehensive survey of classical results and new insights into these mathematical relationships.

Contribution

It introduces new results linking structured matrices and continued fractions to polynomial root localization, expanding understanding of these classical mathematical concepts.

Findings

New relations between Hankel, Hurwitz, Toeplitz, Vandermonde matrices and continued fractions.

Enhanced methods for root localization of univariate polynomials.

Comprehensive survey of classical facts with novel contributions.

Abstract

We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices, Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results.