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Certification Using Newton-Invariant Subspaces

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Mathematical Aspects of Computer and Information Sciences (MACIS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10693))

Abstract

For a square system of analytic equations, a Newton-invariant subspace is a set which contains the resulting point of a Newton iteration applied to each point in the subspace. For example, if the equations have real coefficients, then the set of real points form a Newton-invariant subspace. Starting with any point for which Newton’s method quadratically converges to a solution, this article uses Smale’s \(\alpha \)-theory to certifiably determine if the corresponding solution lies in a given Newton-invariant subspace or its complement. This approach generalizes the method developed in collaboration with F. Sottile for deciding the reality of the solution in the special case that the Newton iteration defines a real map. A description of the implementation in alphaCertified is presented along with examples.

J. D. Hauenstein—Supported in part by NSF ACI-1460032 and Sloan Research Fellowship BR2014-110 TR14.

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Acknowledgments

The author would like to thank Charles Wampler for helpful discussions related to using isotropic coordinates in kinematics.

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Authors and Affiliations

  1. Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Jonathan D. Hauenstein

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  1. Jonathan D. Hauenstein
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Correspondence to Jonathan D. Hauenstein .

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Editors and Affiliations

  1. University of Paderborn, Paderborn, Germany

    Johannes Blömer

  2. Wilfrid Laurier University, Waterloo, Ontario, Canada

    Ilias S. Kotsireas

  3. Johannes Kepler University Linz, Linz, Austria

    Temur Kutsia

  4. SBA Research, Vienna, Austria

    Dimitris E. Simos

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Hauenstein, J.D. (2017). Certification Using Newton-Invariant Subspaces. In: Blömer, J., Kotsireas, I., Kutsia, T., Simos, D. (eds) Mathematical Aspects of Computer and Information Sciences. MACIS 2017. Lecture Notes in Computer Science(), vol 10693. Springer, Cham. https://doi.org/10.1007/978-3-319-72453-9_3

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  • DOI: https://doi.org/10.1007/978-3-319-72453-9_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-72452-2

  • Online ISBN: 978-3-319-72453-9

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