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首页> 外文会议>International Symposium on Symbolic and Algebraic Computation Aug 3-6, 2003 Philadelphia, Pennsylvania, USA >A Method Computing Multiple Roots of Inexact Polynomials
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A Method Computing Multiple Roots of Inexact Polynomials

机译：一种计算不精确多项式根的方法

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摘要

We present a combination of two novel algorithms that accurately calculate multiple roots of general polynomials. For a given multiplicity structure and initial root estimates, Algorithm I transforms the singular root-finding into a nonsin-gular least squares problem on a pejorative manifold, and calculates multiple roots simultaneously. To fulfill the input requirement of Algorithm Ⅰ, we design a numerical GCD-finder, including a partial singular value decomposition and an iterative refinement, as the main engine for Algorithm Ⅱ that calculates the multiplicity structure and the initial root approximation. The combined method calculates multiple roots with high forward accuracy without using multipreci-sion arithmetic, even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities. To measure the true sensitivity of the multiple roots, a pejorative condition number is proposed and error bounds are given. Extensive computational experiments are presented. The error analysis and numerical results confirm that a polynomial being ill-conditioned in conventional sense can be well conditioned pejoratively. In those cases, the multiple roots can be computed with remarkable accuracy.
机译：我们提出了两种新颖的算法的组合，它们可以准确地计算出多项式的多个根。对于给定的多重性结构和初始根估计，算法I将奇异的根寻找转换为有形流形上的非正弦最小二乘问题，并同时计算多个根。为了满足算法Ⅰ的输入要求，我们设计了一个数值GCD-finder，包括部分奇异值分解和迭代细化，作为算法Ⅱ的主机，计算了多重结构和初始根近似。即使系数不精确，该组合方法也可以不使用多重精度算法而以高前向精度计算多个根。这也许是第一个具有这种功能的黑盒型寻根器。为了测量多个根的真实敏感度，提出了一个有条件的条件数并给出了误差范围。进行了广泛的计算实验。误差分析和数值结果证实，在常规意义上处于不良状态的多项式可以具有贬义性。在那些情况下，可以以极高的精度计算多个根。

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