1. Advances in the VAS CF method using better bounds Alkiviadis G. Akritas Department of Computer & Communication Engineering University of Thessaly Volos, Greece (joint work with Strzebonski and Vigklas)

2. CASC 2007, Bonn, Germany2 Outline of the talk Presentation of two methods derived from Vincent’s theorem. Better estimations of upper bounds on the positive roots of polynomials. Tables showing improvement of the VAS CF real root isolation method.

3. CASC 2007, Bonn, Germany3 The rule of signs var(p): exact only if var(p) = 0 or 1

4. CASC 2007, Bonn, Germany4 Vincent’s theorem (1836) (Continued Fractions Version)

5. CASC 2007, Bonn, Germany5 Real Root Isolation

6. CASC 2007, Bonn, Germany6 Most Important Observation

7. CASC 2007, Bonn, Germany7 VAS – continued fractions method (uses Descartes’ test)

8. CASC 2007, Bonn, Germany8 Vincent’s theorem (2000) (Alesina-Galuzzi: Bisection)

9. CASC 2007, Bonn, Germany9 Vincent’s Termination Test

10. CASC 2007, Bonn, Germany10 Uspensky’s Termination Test (special case of Vincent’s test)

11. CASC 2007, Bonn, Germany11 Termination test named after Uspensky because: Uspensky was the one to use it as a test, since he was not aware of Budan’s theorem.

12. CASC 2007, Bonn, Germany12 Budan’s theorem (from Vincent’s paper of 1836)

13. CASC 2007, Bonn, Germany13 Vincent vs Uspensky

14. CASC 2007, Bonn, Germany14 The VCA algorithm --- original version

15. CASC 2007, Bonn, Germany15 REL: Fastest implementation of VCA bisection method

16. CASC 2007, Bonn, Germany16 Comparison times using Cauchy’s rule in VAS CF

17. CASC 2007, Bonn, Germany17 Stefanescu’s theorem (2005)

18. CASC 2007, Bonn, Germany18 Matching coefficients plus breaking up coefficients Stefanescu introduced the concept of matching (or pairing) a positive coefficient with a negative one of lower degree. We introduced the concept of breaking up a positive coefficient --- into parts to be matched with negative coefficients. (for ANY number of sign variations!)

19. CASC 2007, Bonn, Germany19 Our theorem (1/2)

20. CASC 2007, Bonn, Germany20 Our theorem (2/2)

21. CASC 2007, Bonn, Germany21 Problems with a single method of computing bounds

22. CASC 2007, Bonn, Germany22 Use two methods to compute the bound; pick the minimum

23. CASC 2007, Bonn, Germany23

24. CASC 2007, Bonn, Germany24 Comparison times using new bounds in VAS CF

25. CASC 2007, Bonn, Germany25 Conclusions

26. CASC 2007, Bonn, Germany26 References I

27. CASC 2007, Bonn, Germany27 References II