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Probabilistic Condition Number Estimates for Real Polynomial Systems I: A Broader Family of Distributions

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Abstract

We consider the sensitivity of real roots of polynomial systems with respect to perturbations of the coefficients. In particular—for a version of the condition number defined by Cucker and used later by Cucker, Krick, Malajovich, and Wschebor—we establish new probabilistic estimates that allow a much broader family of measures than considered earlier. We also generalize further by allowing overdetermined systems. In Part II, we study smoothed complexity and how sparsity (in the sense of restricting which terms can appear) can help further improve earlier condition number estimates.

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Notes

  1. Here, “complexity” simply means the total number of field operations over \(\mathbb {C}\) needed to find a start point \(x_0\) for Newton’s iteration, such that the sequence of Newton iterates \((x_n)_{n\in \mathbb {N}}\) converges to a true root \(\zeta \) of P (see, e.g., [3, Ch. 8]) at the rate of \(|x_n-\zeta | \le (1/2)^{2^{n-1}}|x_0-\zeta |\) or faster.

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Acknowledgements

The Authors would like to thank the anonymous referees for detailed remarks that greatly helped clarify our paper.

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Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Berlin, Sekretariat MA 3-2, Strae des 17. Juni 136, 10623, Berlin, Germany

    Alperen A. Ergür

  2. Department of Mathematics, Texas A&M University TAMU 3368, College Station, TX, 77843-3368, USA

    Grigoris Paouris & J. Maurice Rojas

Authors
  1. Alperen A. Ergür
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  2. Grigoris Paouris
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  3. J. Maurice Rojas
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Correspondence to Alperen A. Ergür.

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Communicated by Felipe Cucker.

Alperen A. Ergür was partially supported by NSF Grant CCF-1409020, NSF CAREER Grant DMS-1151711 and Einstein Foundation, Berlin. Grigoris Paouris was partially supported by BSF Grant 2010288 and NSF CAREER Grant DMS-1151711. J. Maurice Rojas was partially supported by NSF Grants CCF-1409020 and DMS-1460766.

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Ergür, A.A., Paouris, G. & Rojas, J.M. Probabilistic Condition Number Estimates for Real Polynomial Systems I: A Broader Family of Distributions. Found Comput Math 19, 131–157 (2019). https://doi.org/10.1007/s10208-018-9380-5

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  • DOI: https://doi.org/10.1007/s10208-018-9380-5

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