Homotopy continuation methods to solve polynomial systems scale well on parallel computers, but their reliability and accuracy is often in doubt because of their reliance on floating-point arithmetic. In a recent joint work with Genady Yoffe we experimentally determined using QD-2.3.9 (a software library of Y. Hida, X.S. Li, and D.H. Bailey) that the overhead of double double arithmetic compared to working with ordinary doubles is of the same magnitude as computing with complex numbers. With multiple cores we can compensate for this overhead and thus achieve a so-called quality up. In particular, with 8 cores we can compute more accurately with double doubles in roughly the same time as in standard double precision. Similar factors apply to quadruple the working precision. This talk will address how the quad doubles of QD-2.3.9 are integrated and applied in the software package PHCpack to solve larger polynomial systems more accurately.
This is work with Genady Yoffe.
SIAM/NSF/MSRI Workshop on Hybrid Methodologies for Symbolic-Numeric Computation, MSRI, Berkeley, California, November 17-19, 2010.