Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers

Abstract

We consider the problem of tracking one solution path defined by a polynomial homotopy on a parallel shared memory computer. Our robust path tracker applies Newton's method on power series to locate the closest singular parameter value. On top of that, it computes singular values of the Hessians of the polynomials in the homotopy to estimate the distance to the nearest different path. Together, these estimates are used to compute an appropriate adaptive step size. For ndimensional problems, the cost overhead of our robust path tracker is O(n), compared to the commonly used predictor-corrector methods. This cost overhead can be reduced by a multithreaded program on a parallel shared memory computer.

Parallel Software to Offset the Cost of Higher Precision

“…A loss of 16 decimal places of accuracy in double double precision still leads to sufficiently accurate results. This argument is expressed formally in [13].…”

“…The focus of this paper is on recently developed code for new algorithms described in [3], [12,13], added to PHCpack [14]. PHCpack is a free and open source package to apply Polynomial Homotopy Continuation to solve systems of many polynomials in several variables.…”

“…The synchronization between those two stages is performed by terminating all tasks and launching a new set of tasks for the next stage. Details are described in [13].…”

“…At the end of [13], we reported an instance where quad double precision was insufficient for Newton's method to converge and compute the coefficients of the series past degree 15.…”

Accelerated Polynomial Evaluation and Differentiation at Power Series in Multiple Double Precision

“…The main motivation for this paper is to accelerate a new robust path tracker [15], added recently to PHCpack [17], which requires power series expansions of the solution series of polynomial systems. As shown in [16], double precision may no longer suffice to obtain accurate results, for larger systems, and for longer power series, truncated at higher degrees. The double precision can be extended with double doubles, triple doubles, quad doubles, etc., applying multiple double arithmetic [14].…”