增强Cardinal算法的效能
Enhancing the Power of Cardinal's Algorithm
AI总结：
本文改进Cardinal算法，通过利用结构化矩阵降低计算成本，扩展其因子分解能力，为矩阵计算提供新方法。
摘要：
Cardinal的因式分解算法于1996年将单变量多项式分解为两个根集位于虚轴两侧的因子，这一目标本身具有重要性，并为求根问题奠定了基础。该算法的创新性及其潜力自提出以来便受到专家认可，但到2016年，即二十年后，其实际价值仍无明显体现，尤其由于其最终阶段通过计算两个多项式的近似最大公约数而带来高昂的计算成本。我们简要回顾了Cardinal算法及其难点，并基于1996年后的一些研究成果对其进行改进，扩展其能力以分解更广泛类别的因子，并将算法的最终阶段转化为可通过结构化矩阵进行高效计算的步骤。部分我们的技术对矩阵计算具有独立价值。
英文摘要：
Cardinal's factorization algorithm of 1996 splits a univariate polynomial into two factors with root sets separated by the imaginary axis, which is an important goal itself and a basic step toward root-finding. The novelty of the algorithm and its potential power have been well recognized by experts immediately, but by 2016, that is, two decades later, its practical value still remains nil, particularly because of the high computational cost of performing its final stage by means of computing approximate greatest common divisor of two polynomials. We briefly recall Cardinal's algorithm and its difficulties, amend it based on some works performed since 1996, extend its power to splitting out factors of a more general class, and reduce the final stage of the algorithm to quite manageable computations with structured matrices. Some of our techniques can be of independent interest for matrix computations.
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