利用三次Hermite插值进行解析函数的根隔离

中文摘要 AI 辅助

我们提出了一种针对解析函数的根隔离算法。该方法将区间细分与基于局部三次Hermite插值的启发式测试相结合。当函数在少数采样点上与其三次Hermite插值足够接近时，我们假设三次插值的临界点能为原函数提供有效的根隔离。此类算法通常无法对所计算区间提供严格的认证。然而，大量数值实验表明，该算法效率极高，且几乎总能产生正确的结果。我们希望从这项工作中能够推导出概率性或精确的算法。

英文摘要

We present a root isolation algorithm for an analytic function. The method combines interval subdivision with a heuristic test based on local cubic Hermite interpolation. Whenever the function is sufficiently close to its cubic Hermite interpolant at a few sample points, we assume that the critical points of the cubic interpolant give a valid root isolation for the original function. Such an algorithm cannot, in general, provide a rigorous certification of the computed intervals. Nevertheless, extensive numerical experiments show that it is highly efficient and almost always produces the correct results. We hope that a probabilistic or exact algorithm can be derived from this work.

发表机构

• G.A.A.T.I. Université de la Polynésie Française（法属波利尼西亚大学 G.A.A.T.I.）

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