特征多项式的有效计算
Efficient Computation of the Characteristic Polynomial
摘要：
本文讨论了小有限域上和整数上稠密矩阵的特征多项式的计算。首先给出了有限域上的两种算法：一种是基于Krylov迭代和高斯消去法的算法。我们比较它的Keller-Gehrig的第二个算法的改进。然后，我们证明了Keller-Gehrig的第三个算法的推广可以提高复杂度和计算时间。我们使用这些结果作为计算整数矩阵的特征多项式的基础。我们首先对稠密矩阵使用提前终止和中国式排序。然后，基于整数最小多项式和Hensel分解的概率方法，特别适合于稀疏和/或结构化矩阵。
英文摘要：
This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices.
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