On Bounds for Real Roots of Polynomials

The computation of the real roots of univariate polynomials with real coefficients is done using several algorithmic devices. Many of them are based on the isolation of the real roots, i.e. the computation of a finite number of intervals with the property that each of them contains exactly one root. For that one of the steps is that of computing bounds for the roots. This can be realized using classical bounds for the absolute values of complex roots, see [9]. However there exist bounds specific to real roots. We obtain a device for computing absolute positiveness bounds for the real roots. The method is based on an improvement of the results of D. Ştefănescu [12] on upper bounds for positive roots. It it useful for the isolation of these roots, i.e. for the computation of a finite number of intervals such that each interval contains exactly one root [1]. These methods allows us to compute an interval containing all the positive roots of the derivatives of the polynomial. Analytical properties of polynomials and the algorithmic methods for the computation of their roots are relevant for the study of many physical problems, see, for example [7], [10], [11].