Open Access
2012 Certified Numerical Homotopy Tracking
Carlos Beltrán, Anton Leykin
Experiment. Math. 21(1): 69-83 (2012).


Given a homotopy connecting two polynomial systems, we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuation rooted in the alpha theory of Smale. Experimental results obtained with an implementation in the numerical algebraic geometry package Macaulay2 demonstrate the practicality of the algorithm. In particular, we confirm the theoretical results for random linear homotopies and illustrate the plausibility of a conjecture by Shub and Smale on a good initial pair.


Download Citation

Carlos Beltrán. Anton Leykin. "Certified Numerical Homotopy Tracking." Experiment. Math. 21 (1) 69 - 83, 2012.


Published: 2012
First available in Project Euclid: 31 May 2012

zbMATH: 1238.14048
MathSciNet: MR2904909

Primary: 14Q20 , 65H10 , 65H20

Keywords: approximate zero , certified algorithms , Complexity , Computational aspects of algebraic geometry , continual methods , homotopy methods , systems of equations

Rights: Copyright © 2012 A K Peters, Ltd.

Vol.21 • No. 1 • 2012
Back to Top