Certifying solutions to square systems of polynomial-exponential equations

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• 作者：Jonathan D. Hauenstein^a ; ¹ ; ^{hauenstein@nd.edu" class="auth_mail" title="E-mail the corresponding author} ; Viktor Levandovskyy^b ; ^{Viktor.Levandovskyy@math.rwth-aachen.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Certified solutions ; Alpha theory ; Polynomial system ; Polynomial-exponential systems ; Numerical algebraic geometry ; Alphacertified
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P3
• 页码：575-593
• 全文大小：416 K

文摘

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 in alphaCertified. Examples are presented to demonstrate this certification algorithm.