多项式同伦的一条路径在共享存储并行计算机上的鲁棒数值跟踪
Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers
摘要：
我们考虑在一台并行共享内存计算机上跟踪由多项式同伦定义的一条解路径的问题。我们的鲁棒路径跟踪应用牛顿法的幂级数，以找到最接近的奇异参数值。在此基础上，计算同伦中多项式的Hessian函数的奇异值来估计到最近的不同路径的距离，并利用这些估计值来计算合适的自适应步长。对于n维问题，我们的鲁棒路径跟踪器的成本开销是O（n），相比常用的预估校正法。这种开销可以通过在并行共享内存计算机上的多线程程序来减少。
英文摘要：
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 stepsize. For n-dimensional 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.
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