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Polynomial homotopy continuation on GPUs. (English) Zbl 1365.65149

MSC:

65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
65Y10 Numerical algorithms for specific classes of architectures
65Y15 Packaged methods for numerical algorithms

Software:

CUDA; LBNL; PHCpack; phcpy; Python
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Full Text: DOI

References:

[1] N. Bliss, J. Sommars, J. Verschelde, and Xiangcheng Yu. Solving polynomial systems in the cloud with polynomial homotopy continuation. In V.P. Gerdt, W. Koepf, E.W. Mayr, and E.V. Vorozhtsov, editors,Computer Algebra in Scientific Computing, 17th International Workshop, CASC 2015, Aachen, Germany, volume 9301 ofLecture Notes in Computer Science, pages 87–100. Springer-Verlag, 2015. · Zbl 1439.13078
[2] A. Griewank and A. Walther.Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation.SIAM, second edition, 2008. · Zbl 1159.65026
[3] Y. Hida, X. S. Li, and D. H. Bailey. Algorithms for quad-double precision floating point arithmetic. In15th IEEE Symposium on Computer Arithmetic (Arith-15 2001), 11–17 June 2001, Vail, CO, USA, pages 155–162. IEEE Computer Society, 2001. Shortened version of Technical Report LBNL-46996, software at http://crd.lbl.gov/ dhbailey/mpdist.
[4] A. Leykin and J. Verschelde. Interfacing with the numerical homotopy algorithms in PHCpack. In N. Takayama and A. Iglesias, editors,Proceedings of ICMS 2006, volume 4151 ofLecture Notes in Computer Science, pages 354–360. Springer-Verlag, 2006. · Zbl 1230.65061
[5] M. Lu, B. He, and Q. Luo. Supporting extended precision on graphics processors. In A. Ailamaki and P.A. Boncz, editors,Proceedings of the Sixth International Workshop on Data Management on New Hardware (DaMoN 2010), June 7, 2010, Indianapolis, Indiana, pages 19–26, 2010. Software at http://code.google.com/p/gpuprec/.
[6] S. M. Rump. Verification methods: Rigorous results using floating-point arithmetic.Acta Numerica, 19:287449, 2010. · Zbl 1323.65046
[7] J. Verschelde. Algorithm 795: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation.ACM Trans. Math. Softw., 25(2):251–276, 1999. · Zbl 0961.65047
[8] J. Verschelde. Polynomial homotopy continuation with PHCpack.ACM Communications in Computer Algebra, 44(4):217–220, 2010. · Zbl 1308.68198
[9] J. Verschelde. Modernizing PHCpack through phcpy. In P. de Buyl and N. Varoquaux, editors,Proceedings of the 6th European Conference on Python in Science (EuroSciPy 2013), pages 71–76, 2014.
[10] J. Verschelde and G. Yoffe. Polynomial homotopies on multicore workstations. In M.M. Maza and J.-L. Roch, editors,Proceedings of the 4th International Workshop on Parallel Symbolic Computation (PASCO 2010), July 21-23 2010, Grenoble, France, pages 131–140. ACM, 2010.
[11] J. Verschelde and G. Yoffe. Evaluating polynomials in several variables and their derivatives on a GPU computing processor. InProceedings of the 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops (PDSEC 2012), pages 1391–1399. IEEE Computer Society, 2012.
[12] J. Verschelde and G. Yoffe. Orthogonalization on a general purpose graphics processing unit with double double and quad double arithmetic. InProceedings of the 2013 IEEE 27th International Parallel and Distributed Processing Symposium Workshops (PDSEC 2013), pages 1373–1380. IEEE Computer Society, 2013.
[13] J. Verschelde and X. Yu. GPU acceleration of Newton’s method for large systems of polynomial equations in double double and quad double arithmetic. InProceedings of the 16th IEEE International Conference on High Performance Computing and Communication (HPCC 2014), pages 161–164. IEEE Computer Society, 2014.
[14] J. Verschelde and X. Yu. Accelerating polynomial homotopy continuation on a graphics processing unit with double double and quad double arithmetic. In J.-G. Dumas and E.L. Kaltofen, editors,Proceedings of the 7th International Workshop on Parallel Symbolic Computation (PASCO 2015), July 10–11 2015, Bath, United Kingdom, pages 109–118. ACM, 2015.
[15] J. Verschelde and X. Yu. Tracking many solution paths of a polynomial homotopy on a graphics processing unit in double double and quad double arithmetic. InProceedings of the 17th IEEE International Conference on High Performance Computing and Communication (HPCC 2015), pages 371–376. IEEE Computer Society, 2015.
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