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Improved Computation of Bounds for Positive Roots of Polynomials

A new lower bound for computing positive roots of polynomial equations is proposed. We discuss a twostage algorithm for computing positive roots of polynomial equations. In the first stage, we use the continued fraction method based on Vincent’s theorem, which employs the lower bound of real roots, for isolating the positive roots into intervals. In the second stage, we apply a bisection method to each interval. In order to compute the proposed lower bound, we follow three steps. First, we compute a candidate for the lower bound generated by Newton’s method. Second, by using Laguerre’s theorem, we check whether the candidate for the lower bound is suitable. Third, we compare the localmax bound and the proposed lower bound. Then, we employ the larger bound to accelerate the continued fraction method based on Vincent’s theorem. Finally, we conduct experiments to evaluate the effectiveness of the proposed lower bound. (This paper is submitted to PDPTA’13.)