Implementations of a New Theorem for Computing Bounds for Positive Roots of Polynomials

Akritas, Alkiviadis G.; Strzeboński, Adam; Vigklas, Panagiotis S.

doi:10.1007/s00607-006-0186-y

Cited by 24 publications

(33 citation statements)

References 5 publications

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“…So it is still an open question whether the maximum computing time of the bisection method dominates n 5 . Experimental evidence (Johnson, 1991(Johnson, , 1998Rouillier and Zimmermann, 2004;Akritas et al, 2006) suggests that Mignotte polynomials with a suitable choice of a perhaps require a computing time that dominates n 5 .In Section 2 we present the CF-method and characterize the intervals that can appear in its output. We also define the universal CF-tree, a notion we use in Section 3, ''A Road Map'', to construct difficult input polynomials.…”

mentioning

confidence: 98%

On the computing time of the continued fractions method

Collins¹,

Krandick

2012

Journal of Symbolic Computation

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a b s t r a c tThe maximum computing time of the continued fractions method for polynomial real root isolation is at least quintic in the degree of the input polynomial. This computing time is realized for an infinite sequence of polynomials of increasing degrees, each having the same coefficients. The recursion trees for those polynomials do not depend on the use of root bounds in the continued fractions method. The trees are completely described. The height of each tree is more than half the degree. When the degree exceeds one hundred, more than one third of the nodes along the longest path are associated with primitive polynomials whose low-order and high-order coefficients are large negative integers. The length of the forty-five percent highest order coefficients and of the ten percent lowest order coefficients is at least linear in the degree of the input polynomial multiplied by the level of the node. Hence the time required to compute one node from the previous node using classical methods is at least proportional to the cube of the degree of the input polynomial multiplied by the level of the node. The intervals that the continued fractions method returns are characterized using a matrix factorization algorithm. (W. Krandick). 1 Tel.: +1 919 602 4231.2 As the result of a tragic accident Werner Krandick passed away on March 26, 2012, after this article was accepted for publication. He was a major contributor to real root isolation of univariate polynomials with rational coefficients. He worked on both theoretical complexity and practically efficient algorithms and implementations. This article, reporting joint work with George Collins, provides a lower bound, conclusively resolving some long discusssions on comparisons between the continued fractions method and bisection-based methods. The article exemplifies Werner's work: precise, dedicated and cogent, giving generous credit to related results. He will be greatly missed by the JSC community. 0747-7171/$ -see front matterMignotte (1981, 1982), have a pair of close real roots, one on either side of 1/a. The distance between either root and 1/a is less than a −n/2−1 . Johnson concludes that the number of bisections required to separate the two roots dominates n, a fallacy in case 1/a is a bisection point. The polynomials S n (−x+1) have a pair of close real roots, one on either side of 1−1/a. This solves a perceived numbertheoretical problem: If the binary expansion of 1/a does not contain sufficiently many digits 1, then the expansion of 1 − 1/a will. Johnson asserts without proof that the bisection method will call itself for polynomials that are dense and have long coefficients. How dense? How many of the coefficients are long? Do their signs matter? There is no proof here, just speculation. Some statements do not make any sense, for instance the assertion that the length of an arbitrary level number in the recursion tree of the bisection method is codominant with the height of the tree. A few more errors can be discerned but much of the presentation is...

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mentioning

confidence: 98%

On the computing time of the continued fractions method

Collins¹,

Krandick

2012

Journal of Symbolic Computation

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“…Ştefȃnescu's theorem, , (Akritas, Strzeboński, and Vigklas, 2006), we obtain a general theorem, which includes the above three methods as special cases, and from which new, sharper, bounds can be derived. …”

Section: Kioustelidis' Methodsmentioning

confidence: 99%

“…In Sage reference manual, (SAGE , 2004(SAGE , -2010 implementing our linear complexity bounds "first-λ", "local-max " and min{"first-λ", "local-max "}, described earlier, (Akritas, Strzeboński, and Vigklas, 2006). Given a polynomial represented by a list of its coefficients, (cl) (as RealIntervalFieldElements, RIF ), an upper bound on its largest real root is being computed.…”

Section: Sage Session Demonstration Of New Boundsmentioning

confidence: 99%

“…The two linear complexity bounds are: Cauchy's, ub C and min(ub F L , ub LM ), the minimum of "first-λ" and "local-max " bounds, (Akritas, Strzeboński, and Vigklas, 2006), whereas the two quadratic complexity ones are: ub KQ , the Quadratic complexity variant of K ioustelidis' bound, studied by Hong, (Hong, 1998), and ub LM Q , the Quadratic complexity version of the "local-max " bound, (Akritas, Strzeboński, and Vigklas, 2008a).…”

Section: Benchmarking Vas With New Boundsmentioning

confidence: 99%

“…(a) Cauchy's bound, ub C , to be used as a point of reference, since it has been in use for the past 30 years, and (Akritas, Strzeboński, and Vigklas, 2006), which is the best among the linear complexity bounds, in order to see when it's implementation will outperform that of the two quadratic complexity bounds.…”

Section: Benchmarking Vas With New Boundsmentioning

confidence: 99%

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