Enclosing roots of polynomial equations and their applications to iterative processes

We introduce a special class of real recurrentpolynomials fn (n ≥ 1) of degree n,with unique positive roots sn, which are decreasingas n increases. The first root s1, as well asthe last one denoted by s∞ are expressed in closedform, and enclose all sn (n > 1). This technique is also used to find weaker than before[Kantorovich and Akilov, 1982] sufficient convergence conditions for some popular iterative processes converging to solutions of equations.