LOCATING ROOTS OF A CERTAIN CLASS OF POLYNOMIALS

Abstract

Abstract. We introduce a special class of real recurrent polynomials f m (m  1) of degree m+1, with positive roots s m , which are decreasing asm increases. The rst root s 1 , as well as the last one denoted by s 1 areexpressed in closed form, and enclose all s m (m > 1).This technique is also used to nd weaker than before [6] sucientconvergence conditions for some popular iterative processes convergingto solutions of equations. 1. IntroductionWe introduce a special class of recurrent polynomials f m (m1) of degreem+ 1 with real coecients.Then, we nd sucient conditions under which each polynomial f m has apositive root s m , such that s m+1 s m (m1). The rst root s 1 , as well asthe last one denoted by s 1 are expressed in simple closed form.Applications are provided. In the rst one, we show how to use s 1 and s 1 to locate any s m belonging in (s 1 ;s 1 ] (m1).In the second one, using this technique on Newton’s method (16), we showthat the famous for its simplicity and clarity Newton{Kantorovich condition(46) for solving equations can always replaced by a weaker one (49).We also show how to use our results to generate majorizing sequences ap-pearing in connection for solving abstract equations in a Banach space settingusing Newton{type methods [3], [6].2. Enclosing roots of polynomialsWe introduce the main result of this section on enclosing roots of polynomi-als.Theorem 2.1. Assume:there exist constants K0, M0, 0, L0, ‘0, and 0;