Positive root isolation for poly-powers by exclusion and differentiation. (March 2018)

• Record Type: Journal Article
• Authors: Huang, Cheng-Chao; Li, Jing-Cao; Xu, Ming; Li, Zhi-Bin
• Is Part Of: Journal of symbolic computation. Volume 85(2018)
• Page Start: 148
• Page End: 169
• Publication Date: 2018-03
• Subjects: Real root isolation -- Generalized polynomial -- Transcendental number -- Interval arithmetic -- Systems biology; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.07.007
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We consider a class of univariate real functions—poly-power s—that extend integer exponents to real algebraic exponents for polynomials. Our purpose is to isolate positive roots of such a function into disjoint intervals, each contains exactly one positive root and together contain all, which can be easily refined to any desired precision. To this end, we first classifypoly-power s into simple and non-simple ones, depending on the number of linearly independent exponents. For the former, based on Gelfond–Schneider theorem, we present two complete isolation algorithms—exclusion and differentiation. For the latter, their completeness depends on Schanuel's conjecture. We implement the two methods and compare them in efficiency via a few examples. Finally the proposed methods are applied to the field of systems biology to show the practical usefulness.