Computing real roots of real polynomials
Computing real roots of real polynomials
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DOI：
10.1016/j.jsc.2015.03.004
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发表时间：
2016-03-01
影响因子：
0.7
通讯作者：
Mehlhorn, Kurt
中科院分区：
文献类型：
--
作者：
Sagraloff, Michael;Mehlhorn, Kurt
计算单变量多项式的根源是计算代数的基本和长期研究的问题，该问题在数学，工程，计算机科学和自然科学中的应用。为了隔离和近似所有复杂根，已知的最佳算法是基于PAN在2002年引入的几乎最佳多项式分解的几乎最佳方法。PAN的分解算法返回到1982年Schonhage的分裂圆圈方法。 PAN方法的缺点是它非常参与（2），并且所有根都必须同时计算。对于仅必须计算真正根源的重要特殊情况，实践中使用了更简单的方法。但是，它们在潘的复杂性方面的方法很大程度上落后于本文。在本文中，我们通过引入笛卡尔方法和牛顿迭代的混合来解决这种差异，表示anewDSC比Pan的方法更简单，但实现了运行时的可比性。对此。我们的算法计算任何无形多项式的真实根的隔离间隔，由甲骨文提供，该甲骨文提供了多项式系数的任意良好近似值。 ANEWDSC还可以在给定间隔中仅隔离根，并将隔离间隔优化为任意的小尺寸；对于后一个任务，它几乎达到了最佳的复杂性。 （c）2015 Elsevier Ltd.保留所有权利。
Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schonhage in 1982. The main drawbacks of Pan's method are that it is quite involved(2) and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity.In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted ANEwDsc, which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. ANEWDsc can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task. (C) 2015 Elsevier Ltd. All rights reserved.