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Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials
 

Summary: Linear and Quadratic Complexity Bounds
on the Values of the Positive Roots of Polynomials
Alkiviadis G. Akritas
(Department of Computer and Communication Engineering
University of Thessaly
Volos, Greece
akritas@uth.gr)
Dedicated to Professor Doru S¸tefanescu1
Abstract: In this paper we review the existing linear and quadratic complexity (upper)
bounds on the values of the positive roots of polynomials and their impact on the per-
formance of the Vincent-Akritas-Strzebo´nski (VAS) continued fractions method for the
isolation of real roots of polynomials. We first present the following four linear complex-
ity bounds (two "old" and two "new" ones, respectively): Cauchy's, (C), Kioustelidis',
(K), First-Lambda, (FL) and Local-M ax, (LM ); we then state the quadratic complex-
ity extensions of these four bounds, namely: CQ, KQ, FLQ, and LMQ -- the second,
(KQ), having being presented by Hong back in 1998. All eight bounds are derived from
Theorem 5 below. The estimates computed by the quadratic complexity bounds are
less than or equal to those computed by their linear complexity counterparts. Moreover,
it turns out that VAS(lmq) -- the VAS method implementing LMQ -- is 40% faster
than the original version VAS(cauchy).

  

Source: Akritas, Alkiviadis G. - Department of Computer and Communication Engineering, University of Thessaly

 

Collections: Computer Technologies and Information Sciences