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Towards Soft Exact Computation (Invited Talk)

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Computer Algebra in Scientific Computing (CASC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11661))

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Abstract

Exact geometric computation (EGC) is a general approach for achieving robust numerical algorithms that satisfy geometric constraints. At the heart of EGC are various Zero Problems, some of which are not-known to be decidable and others have high computational complexity. Our current goal is to introduce notions of “soft- correctness” in order to avoid Zero Problems. We give a bird’s eye view of our recent work with collaborators in two principle areas: computing zero sets and robot path planning. They share a common Subdivision Framework. Such algorithms (a) have adaptive complexity, (b) are practical, and (c) are effective. Here, “effective algorithm” means it is easily and correctly implementable from standardized algorithmic components. Our goals are to outline these components and to suggest new components to be developed. We discuss a systematic pathway to go from the abstract algorithmic description to an effective algorithm in the subdivision framework.

This work is supported by NSF Grants CCF-1423228 and CCF-1564132.

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Notes

  1. 1.

    After Bernard Bolzano (1817). Bolzano’s Theorem states that if \(a<b\) and then there is some \(c\in (a,b)\) such that \(f(c)=0\). See also [3, 42] for this principle in real root isolation.

  2. 2.

    Monotone means for all i.

  3. 3.

    The factor \(\sigma >1\) was call the “effectivity factor” in [49]. In the present paper, we avoid this terminology since it conflicts with our notion of “effectivity” of this paper.

  4. 4.

    The -notation is like the O-notation except that logarithm factors in n and in L are ignored. In the subdivision setting, “near-optimality” may be taken to be .

  5. 5.

    Some authors introduce a new symbol, say F, to signal this change.

  6. 6.

    In the spirit of Knuth’s “Literate Programming”.

  7. 7.

    It is possible that such proofs contribute to the poor reputation of error analysis as a topic.

  8. 8.

    We use “precision” for the a priori user-specified bound. The algorithm delivers a value whose a posteriori error is at most this precision.

  9. 9.

    We write \(a = b\pm c\) to mean there exists a constant \(\theta \in [-1,1]\) such that \(a = b+\theta \cdot c\). Alternatively, \(|a-b|\le |c|\).

  10. 10.

    Despite the appearance of asymmetry, x and y are treated symmetrically by this definition.

  11. 11.

    There should no confusion with the notion support of a simplicial complex .

  12. 12.

    Strict inequality may arise in subsets of zero sets: if where \(F_i\) are polynomials, and where each \(G_i\) divides \(F_i\), then might exhibit this phenomenon.

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Acknowledgements

The author is deeply grateful for the feedback and bug reports from Michael Burr, Matthew England, Rémi Imbach, Juan Xu and Bo Huang.

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Authors and Affiliations

  1. Department of Computer Science, Courant Institute, NYU, New York, NY, 10012, USA

    Chee Yap

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  1. Chee Yap
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Correspondence to Chee Yap .

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Editors and Affiliations

  1. Coventry University, Coventry, UK

    Matthew England

  2. University of Kassel, Kassel, Germany

    Wolfram Koepf

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. University of Kassel, Kassel, Germany

    Werner M. Seiler

  5. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Yap, C. (2019). Towards Soft Exact Computation (Invited Talk). In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_2

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