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Robust Certified Numerical Homotopy Tracking

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Abstract

We describe, for the first time, a completely rigorous homotopy (path-following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial zero are rational our algorithm involves only rational computations, and if the homotopy is well posed an approximate zero with integer coordinates of the target system is obtained. The total bit complexity is linear in the length of the path in the condition metric, and polynomial in the logarithm of the maximum of the condition number along the path, and in the size of the input.

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Notes

  1. This property inspired its definition, see [33].

  2. If \(R\geq\sqrt{2}\) then values t i as described in Algorithm 1 exist. They also exist for smaller values of R>1 like R=1.0003 but taking \(R\geq\sqrt {2}\) will make our formulas look prettier. This assumption does not affect much to the running time of the algorithm. We will only use R 2 in the algorithm. Hence, we take \(R\in\sqrt{{\mathbb{Q}}}\).

  3. Note the slight abuse of notation: we just use ζ i the zero of \(g_{i}=f_{s_{i}}\), so we should actually denote it by \(\zeta_{s_{i}}\).

  4. Recall that \(d_{\mathrm{R}}(x,y)=\arccos\frac{\langle x,y\rangle}{\| x\|\|y\|}\) is the usual distance from x to y as projective points in ℙ(ℂn+1).

  5. By a.o. we mean an operation of the form +,−,×,/, or a comparison <,≤ or an assignment of a value to a variable, or computation of the integer part of a number.

  6. Exact linear algebra is a large research field, see http://linalg.org/people.html for a list of people working on the subject, as well as software and research articles.

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Acknowledgements

In earlier stages of the ideas behind this work, we maintained many related conversations with Clement Pernet; thanks go to him for helpful discussions and comments. We also thank Gregorio Malajovich, Luis Miguel Pardo and Michael Shub for their questions and answers. Our beloved friend and colleague Jean Pierre Dedieu also inspired us in many occasions. The second author thanks Institut Mittag-Leffler for hosting him in the Spring semester of 2011. A part of this work was done while we were participating in several workshops related to Foundations of Computational Mathematics in the Fields Institute. We thank this institution for its kind support.

C. Beltrán partially supported by MTM2010-16051, Spanish Ministry of Science (MICINN).

A. Leykin partially supported by NSF grant DMS-0914802.

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Authors and Affiliations

  1. Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Santander, Spain

    Carlos Beltrán

  2. School of Mathematics, Georgia Tech, Atlanta, GA, USA

    Anton Leykin

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  1. Carlos Beltrán
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Correspondence to Carlos Beltrán.

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Communicated by Peter Buergisser.

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Beltrán, C., Leykin, A. Robust Certified Numerical Homotopy Tracking. Found Comput Math 13, 253–295 (2013). https://doi.org/10.1007/s10208-013-9143-2

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