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Parallel Homotopy Algorithms to Solve Polynomial Systems

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Mathematical Software - ICMS 2006 (ICMS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4151))

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Abstract

Homotopy continuation methods to compute numerical approximations to all isolated solutions of a polynomial system are known as “embarrassingly parallel”, i.e.: because of their low communication overhead, these methods scale very well for a large number of processors. Because so many important problems remain unsolved mainly due to their intrinsic computational complexity, it would be embarrassing not to develop parallel implementations of polynomial homotopy continuation methods. This paper concerns the development of “parallel PHCpack”, a project which started a couple of years ago in collaboration with Yusong Wang, and which currently continues with Anton Leykin (parallel irreducible decomposition) and Yan Zhuang (parallel polyhedral homotopies). We report on our efforts to make PHCpack ready to solve large polynomial systems which arise in applications.

2000 Mathematics Subject Classification. Primary 65H10. Secondary 14Q99, 68W30.

This material is based upon work supported by the National Science Foundation under Grant No. 0134611 and Grant No. 0410036. This work was partially supported by the National Center for Supercomputing Applications under DMS060008N and utilized the IBM pSeries 690 system copper. Date: 22 June 2006.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL, 60607-7045, USA

    Anton Leykin, Jan Verschelde & Yan Zhuang

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  1. Anton Leykin
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  2. Jan Verschelde
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  3. Yan Zhuang
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Editors and Affiliations

  1. Department of Applied Mathematics and Computational Sciences, University of Cantabria, Avda. de los Castros, s/n, E-39005, Santander, Spain

    Andrés Iglesias

  2. Faculty of Science Department of Mathematics, Kobe University, 1-1, Rokkodai, Nada-ku, 657-8501, Kobe, Japan

    Nobuki Takayama

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Leykin, A., Verschelde, J., Zhuang, Y. (2006). Parallel Homotopy Algorithms to Solve Polynomial Systems. In: Iglesias, A., Takayama, N. (eds) Mathematical Software - ICMS 2006. ICMS 2006. Lecture Notes in Computer Science, vol 4151. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11832225_22

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  • DOI: https://doi.org/10.1007/11832225_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38084-9

  • Online ISBN: 978-3-540-38086-3

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