Certifying solutions to square systems of polynomial-exponential equations. (March 2017)

• Record Type: Journal Article
• Authors: Hauenstein, Jonathan D.; Levandovskyy, Viktor
• Is Part Of: Journal of symbolic computation. Volume 79:Part 3(2017)
• Page Start: 575
• Page End: 593
• Publication Date: 2017-03
• Subjects: Certified solutions -- Alpha theory -- Polynomial system -- Polynomial-exponential systems -- Numerical algebraic geometry -- Alphacertified; 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.2016.07.020
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Smale's α -theory certifies that Newton iterations will converge quadratically to a solution of a square system of analytic functions based on the Newton residual and all higher order derivatives at the given point. Shub and Smale presented a bound for the higher order derivatives of a system of polynomial equations based in part on the degrees of the equations. For a given system of polynomial-exponential equations, we consider a related system of polynomial-exponential equations and provide a bound on the higher order derivatives of this related system. This bound yields a complete algorithm for certifying solutions to polynomial-exponential systems, which is implemented inalphaCertified . Examples are presented to demonstrate this certification algorithm.