Advances in the VAS CF method using better bounds

Alkiviadis G. AkritasDepartment of Computer & Communication Engineering

University of ThessalyVolos, Greece

(joint work with Strzebonski and Vigklas)

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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.

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The rule of signs var(p): exact only if var(p) = 0 or 1

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Vincent’s theorem (1836)(Continued Fractions Version)

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Real Root Isolation

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Most Important Observation

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VAS – continued fractionsmethod (uses Descartes’ test)

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Vincent’s theorem (2000)(Alesina-Galuzzi: Bisection)

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Vincent’s Termination Test

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Uspensky’s Termination Test(special case of Vincent’s test)

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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.

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Budan’s theorem(from Vincent’s paper of 1836)

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Vincent vs Uspensky

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The VCA algorithm ---original version

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REL: Fastest implementation of VCA bisection method

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Comparison times using Cauchy’s rule in VAS CF

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Stefanescu’s theorem (2005)

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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!)

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Our theorem (1/2)

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Our theorem(2/2)

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Problems with a single method of computing bounds

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Use two methods to compute the bound; pick the minimum

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Comparison times using new bounds in VAS CF

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Conclusions

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References I

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References II