Construction of Bounds for the Positive Root of a General Class of Polynomials with Applications

Abstract

We consider classes or polynomials arising in numerical analysis when determining the order of convergence and in classical mathematics of finance when calculating the effective rate of interest. Those polynomials always have a unique positive root according to Descartes’ rule of signs. In the presence of parameters, for example, we want to give a priori bounds for this root with an accuracy satisfying the demands of the applications. Starting with a review of the cases of polynomials treated in the literature and the bounds given there, we propose a kind of pertubation method using the monotonicity principle first established in Herzberger [5] using some slightly improved bounds for these cases. Thus we are able to construct a priori bounds for classes of polynomials with more general coefficients compared with those treated in earlier papers. Some numerical examples demonstrate how good our method works out in practice.