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GPU Accelerated Newton for Taylor Series Solutions of Polynomial Homotopies in Multiple Double Precision

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Abstract

A polynomial homotopy is a family of polynomial systems, typically in one parametert. Our problem is to compute power series expansions of the coordinates of the solutions in the parametert, accurately, using multiple double arithmetic. One application of this problem is the location of the nearest singular solution in a polynomial homotopy, via the theorem of Fabry. Power series serve as input to construct Padé approximations.

Exploiting the massive parallelism of Graphics Processing Units (GPUs) capable of performing several trillions floating-point operations per second, the objective is to compensate for the cost overhead caused by arithmetic with power series in multiple double precision. The application of Newton’s method for this problem requires the evaluation and differentiation of polynomials, followed by solving a blocked lower triangular linear system. Experimental results are obtained on NVIDIA GPUs, in particular the RTX 2080, RTX 4080, P100, V100, and A100.

Code generated by the CAMPARY software is used to obtain results in double double, quad double, and octo double precision. The programs in this study are self contained, available in a public github repository under the GPL-v3.0 License.

Supported by the National Science Foundation under grant DMS 1854513.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan St. (m/c 249), Chicago, IL, 60607-7045, USA

    Jan Verschelde

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  1. Jan Verschelde

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Correspondence toJan Verschelde.

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Editors and Affiliations

  1. Université de Lille, Villeneuve d’Ascq, France

    François Boulier

  2. School of Mathematical Sciences, Beihang University, Beijing, China

    Chenqi Mou

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Khristianovich Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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Verschelde, J. (2024). GPU Accelerated Newton for Taylor Series Solutions of Polynomial Homotopies in Multiple Double Precision. In: Boulier, F., Mou, C., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2024. Lecture Notes in Computer Science, vol 14938. Springer, Cham. https://doi.org/10.1007/978-3-031-69070-9_19

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