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A continuation method to solve polynomial systems and its complexity

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Abstract

In a recent work Shub (Found. Comput. Math. 9:171–178, 2009), Shub obtained a new upper bound for the number of steps needed to continue a known zero η 0 of a system f 0, to a zero η T of an input system f T , following the path of pairs (f t , η t ), where \({f_t,t\in[0,T]}\) is a polynomial system and f t (η t ) = 0. He proved that if one can choose the step-size in an optimal way, then the number of steps is essentially bounded by the length of the path of (f t , η t ) in the so-called condition metric. However, the proof of that result in Shub (Found. Comput. Math. 9:171–178, 2009) is not constructive. We give an explicit description of an algorithm which attains that complexity bound, including the choice of step-size.

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Authors and Affiliations

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

    Carlos Beltrán

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  1. Carlos Beltrán
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Correspondence to Carlos Beltrán.

Additional information

C. Beltrán was partially supported by MTM2007-62799, Spanish Ministry of Science.

The author wants to thank Mike Shub for many helpful conversations, and the referees for their many comments and suggestions.

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Beltrán, C. A continuation method to solve polynomial systems and its complexity. Numer. Math. 117, 89–113 (2011). https://doi.org/10.1007/s00211-010-0334-3

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