Abstract
This article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step, it uses a newly developed Newton corrector algorithm which rejects an initial guess if it is not an approximate zero. The algorithm also uses an adaptive step size control that builds on a local understanding of the region of convergence of Newton’s method and the distance to the closest singularity following Telen, Van Barel, and Verschelde. To handle numerically challenging situations, the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrated in several numerical examples.
Highlights
Systems of polynomial equations arise in many applications across the sciences including computer vision [20, 36], chemistry [28], kinematics [41], and biology [32]
This article introduces a new path tracking algorithm, Algorithm 3, that does not require the choice of a path tracking tolerance ε or a maximal number N of corrector iterations allowed per step
The implementation of the proposed algorithm significantly outperforms the available implementation of the adaptive precision algorithm [5] in the the considered examples
Summary
Systems of polynomial equations arise in many applications across the sciences including computer vision [20, 36], chemistry [28], kinematics [41], and biology [32]. A numerical method for finding all isolated solutions of a system F of n polynomials in n unknowns is homotopy continuation [35]. This method constructs a homotopy H (x, t) : Cn × C → Cn such that H (x, t) is a polynomial system for all t, H (x, 1) = F (x) and H (x, 0) is a system whose isolated solutions are known.
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