Tracking Many Solution Paths of a Polynomial Homotopy on a Graphics Processing Unit in Double Double and Quad Double Arithmetic

Polynomial systems occur in many areas of science and engineering. Unlike general nonlinear systems, the algebraic structure enables to compute all solutions of a polynomial system. We describe our massively parallel predictor-corrector algorithms to track many solution paths of a polynomial homotopy. The data parallelism that provides the speedups stems from the evaluation and differentiation of the monomials in the same polynomial system at different data points, which are the points on the solution paths. Polynomial homotopies that have tens of thousands of solution paths can keep a sufficiently large amount of threads occupied. Our accelerated code combines the reverse mode of algorithmic differentiation with double double and quad double arithmetic to compute more accurate results faster.

[1]  David Defour,et al.  Interval Arithmetic in CUDA , 2011 .

[2]  Ferenc SzollHosi,et al.  Construction, classification and parametrization of complex Hadamard matrices , 2011, 1110.5590.

[3]  Jonathan D. Hauenstein,et al.  Software for numerical algebraic geometry: a paradigm and progress towards its implementation , 2008 .

[4]  Ruchira S. Datta.  Finding all Nash equilibria of a finite game using polynomial algebra , 2006 .

[5]  Jan Verschelde,et al.  Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation , 1999, TOMS.

[6]  Jan Verschelde,et al.  Evaluating Polynomials in Several Variables and their Derivatives on a GPU Computing Processor , 2012, 2012 IEEE 26th International Parallel and Distributed Processing Symposium Workshops & PhD Forum.

[7]  Bingsheng He,et al.  Supporting extended precision on graphics processors , 2010, DaMoN '10.

[8]  Layne T. Watson,et al.  Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms , 1987, TOMS.

[9]  Akiko Takeda,et al.  PHoM – a Polyhedral Homotopy Continuation Method for Polynomial Systems , 2004, Computing.

[10]  Siegfried M. Rump,et al.  Verification methods: rigorous results using floating-point arithmetic , 2010, Acta Numerica.

[11]  Jon G. Rokne,et al.  Interval Arithmetic , 1992, Graphics Gems III.

[12]  Tsung-Lin Lee,et al.  HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method , 2008, Computing.

[13]  Tien-Yien Li.  Numerical Solution of Polynomial Systems by Homotopy Continuation Methods , 2003 .

[14]  Jonathan D. Hauenstein,et al.  What is numerical algebraic geometry , 2017 .

[15]  Jan Verschelde,et al.  Accelerating polynomial homotopy continuation on a graphics processing unit with double double and quad double arithmetic , 2015, PASCO.

[16]  Marc Moreno Maza,et al.  Solving Bivariate Polynomial Systems on a GPU , 2011, ACCA.

[17]  Jan Verschelde,et al.  Modernizing PHCpack through phcpy , 2013, ArXiv.

[18]  Nathan Bliss,et al.  Solving Polynomial Systems in the Cloud with Polynomial Homotopy Continuation , 2015, CASC.

[19]  Joachim Rosenthal,et al.  Pole Placement and Matrix Extension Problems: A Common Point of View , 2004, SIAM J. Control. Optim..

[20]  Andrew McLennan,et al.  The Maximal Number of Regular Totally Mixed Nash Equilibria , 1997 .

[21]  Jan Verschelde,et al.  Orthogonalization on a General Purpose Graphics Processing Unit with Double Double and Quad Double Arithmetic , 2012, 2013 IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd Forum.

[22]  Jan Verschelde,et al.  GPU Acceleration of Newton's Method for Large Systems of Polynomial Equations in Double Double and Quad Double Arithmetic , 2014, 2014 IEEE Intl Conf on High Performance Computing and Communications, 2014 IEEE 6th Intl Symp on Cyberspace Safety and Security, 2014 IEEE 11th Intl Conf on Embedded Software and Syst (HPCC,CSS,ICESS).

[23]  Xiaoye S. Li,et al.  Algorithms for quad-double precision floating point arithmetic , 2000, Proceedings 15th IEEE Symposium on Computer Arithmetic. ARITH-15 2001.

[24]  Tsung-Lin Lee,et al.  Hom4PS-3: A Parallel Numerical Solver for Systems of Polynomial Equations Based on Polyhedral Homotopy Continuation Methods , 2014, ICMS.

[25]  Jan Verschelde,et al.  Pieri Homotopies for Problems in Enumerative Geometry Applied to Pole Placement in Linear Systems Control , 2000, SIAM J. Control. Optim..

[26]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[27]  Masha Sosonkina,et al.  Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms , 1997, TOMS.

[28]  Marc Moreno Maza,et al.  Dense Arithmetic over Finite Fields with the CUMODP Library , 2014, ICMS.

[29]  Shubhabrata Sengupta,et al.  Efficient Parallel Scan Algorithms for GPUs , 2011 .

[30]  BERND STURMFELS,et al.  Numerical Schubert Calculus , 1998, J. Symb. Comput..

[31]  Sosonkina Maria,et al.  HOMPACK90: A Suite of FORTRAN 90 Codes for Globally Convergent Homotopy Algorithms , 1996 .