# high performance computing on graphics processing units: hgpu.org

## Accelerating Polynomial Homotopy Continuation on a Graphics Processing Unit with Double Double and Quad Double Arithmetic

Jan Verschelde, Xiangcheng Yu
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL 60607-7045, USA
arXiv:1501.06625 [cs.MS], (26 Jan 2015)

@article{verschelde2015accelerating,

title={Accelerating Polynomial Homotopy Continuation on a Graphics Processing Unit with Double Double and Quad Double Arithmetic},

author={Verschelde, Jan and Yu, Xiangcheng},

year={2015},

month={jan},

archivePrefix={"arXiv"},

primaryClass={cs.MS}

}

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Numerical continuation methods apply predictor-corrector algorithms to track a solution path defined by a family of systems, the so-called homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of the solution path. With double double and quad double arithmetic, we can solve larger problems that we could not solve with hardware double arithmetic, but at a higher computational cost. This cost overhead can be compensated by acceleration on a Graphics Processing Unit (GPU). We describe our implementation and report on computational results on two benchmark polynomial systems. In case the linear algebra dominates the total computational cost of a path tracker, the dimension needs to be of the order of at least several hundreds to offset the computational overhead caused by double double arithmetic. For general polynomials of higher degrees, when evaluating and differentiating the polynomials is computationally expensive, already in smaller dimensions, acceleration may offset the cost of higher precision arithmetic.
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