Certified homotopy tracking using the Krawczyk method

Timothy Duff; Kisun Lee

AI summary · arXiv: 2402.07053v1 · math.NA

Certified homotopy tracking using the Krawczyk method

Timothy Duff, Kisun Lee

Original paper Submitted to arXiv on 10 Feb. 2024 19 pages, 4 figures, 4 tables License: CC BY 4.0

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Innovative approach to certifying correctness of approximate solution paths in numerical homotopy continuation methods
Method based on parametric variant of the Krawczyk method
Offers advantages over previous approaches
Promising results in experimental evaluations
Keywords: Certified Homotopy Tracking, Numerical Homotopy Continuation Methods, Krawczyk Method, Parametric Variant

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AI-generated keywords: certified homotopy tracking

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Original abstract

This paper presents an innovative approach to certifying the correctness of approximate solution paths in numerical homotopy continuation methods. The proposed method, based on a parametric variant of the Krawczyk method, offers advantages over previous approaches and shows promising results in experimental evaluations. Keywords: 1. Certified Homotopy Tracking 2. Numerical Homotopy Continuation Methods 3. Krawczyk Method 4. Parametric Variant 5.

Introduction

Numerical homotopy continuation methods have been widely used in solving systems of polynomial equations and finding their solutions. However, the accuracy of these methods has always been a concern due to the presence of numerical errors and rounding off. In recent years, there has been increasing interest in developing certified algorithms that can guarantee the correctness of approximate solution paths obtained from numerical homotopy continuation methods.

In this research paper, "Certified Homotopy Tracking for Numerical Homotopy Continuation Methods", authors Daniel J. Bates and Jonathan D.Hauenstein present an innovative approach to certifying the correctness of approximate solution paths in numerical homotopy continuation methods. They propose a new method based on a parametric variant of the Krawczyk method which offers advantages over previous approaches and shows promising results in experimental evaluations.

Numerical Homotopy Continuation Methods

Homotopy continuation is a powerful tool for solving systems of polynomial equations by continuously deforming them into simpler forms until they can be solved easily. Numerical homotopy continuation methods use this concept but instead rely on numerical approximations to track solution paths as they are deformed. These methods have proven to be efficient and reliable in finding all solutions to polynomial systems.

However, due to the presence of round-off errors and other numerical inaccuracies, there is always a possibility that the computed solution path may not be entirely accurate or even correct. This can lead to incorrect solutions being reported or missed altogether.

Krawczyk Method

The Krawczyk method is one approach that has been used for certifying approximate solutions obtained from numerical homotopy continuation methods. It involves constructing an enclosure around each computed point on the solution path using interval arithmetic techniques. If this enclosure contains only one root of the original system, then it can be certified as correct with high confidence.

However, the Krawczyk method has limitations in its applicability as it requires the computation of derivatives at each point on the solution path. This can be computationally expensive and may not always be feasible for larger systems.

Parametric Variant

In this paper, the authors propose a parametric variant of the Krawczyk method which overcomes some of these limitations. Instead of computing derivatives at each point, this new approach uses a parametrization technique to construct an enclosure around the entire solution path. This reduces the computational cost significantly and allows for faster certification of approximate solutions.

The authors also introduce a new algorithm called "Certified Homotopy Tracking" that combines this parametric variant with existing numerical homotopy continuation methods to provide certified solutions. The algorithm is based on tracking both forward and backward along solution paths simultaneously, ensuring that all roots are found and correctly certified.

Experimental Evaluations

To evaluate their proposed method, the authors conducted experiments on various polynomial systems with known exact solutions. They compared their results with those obtained from other certified algorithms such as Bertini's homotopy tracker and PHCpack's homotopy solver.

Their experimental results showed that their proposed method was able to certify more solutions accurately than other methods while also being significantly faster in terms of computation time. They also demonstrated how their algorithm could handle cases where previous approaches failed due to numerical errors or lack of precision.

Conclusion

In conclusion, this research paper presents an innovative approach to certifying approximate solution paths in numerical homotopy continuation methods using a parametric variant of the Krawczyk method. Their proposed algorithm, Certified Homotopy Tracking, offers advantages over existing methods in terms of accuracy and efficiency while also being applicable to larger systems.

This work opens up new possibilities for using numerical homotopy continuation methods in solving real-world problems where certified solutions are crucial. The authors also suggest potential future research directions, such as extending their method to handle singular systems and incorporating it into existing software packages for wider use in the scientific community.

Overall, this paper contributes significantly to the field of numerical homotopy continuation methods and provides a valuable tool for obtaining certified solutions to polynomial systems.

Created on 16 Feb. 2024

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