Quadratic bounding functions

In this paper we propose two algorithms based on branch and bound method and reduced interval techniques to compute all real zeros of a polynomial. Quadratic bounding functions are proposed which are better than the well known linear underestimator. Experimental result shows the efficiency of the two algorithms when facing ill-conditionned polynomials.

17p vinguyentuongdanh 19-12-2018 13 0   Download

Introduction In 1903 Voronoi [42] postulated the existence of explicit formulas for sums of the form (1.1) n≥1 an f (n) , for any “arithmetically interesting” sequence of coefficients (an )n≥1 and every f in a large class of test functions, including characteristic functions of bounded intervals. He actually established such a formula when an = d(n) is the number of positive divisors of n [43]. He also asserted a formula for (1.2) an = #{(a, b) ∈ Z2 | Q(a, b) = n} , where Q denotes a positive definite integral quadratic form [44]; ...

67p noel_noel 17-01-2013 52 8   Download

The Oppenheim conjecture, proved by Margulis [Mar1] (see also [Mar2]), asserts that for a nondegenerate indefinite irrational quadratic form Q in n ≥ 3 variables, the set Q(Zn ) is dense. In [EMM] (where we used the lower bounds established in [DM]) a quantitative version of the conjecture was established. Namely: Let ρ be a continuous positive function on the sphere {v ∈ Rn | v = 1}, and let Ω = {v ∈ Rn | v

48p noel_noel 17-01-2013 40 5   Download

Abstract In this paper we demonstrate how Lyapunov-Krasovskii functionals can be used to obtain exponential bounds for the solutions of time-invariant linear delay systems. Keywords: time delay system, exponential estimate, Lyapunov-Krasovskii functional. The objective of this note is to describe a systematic procedure of constructing quadratic Lyapunov functionals for (exponentially) stable linear delay systems in order to obtain exponential estimates for their solutions.

13p vietanh_90 30-11-2010 82 10   Download