一般Toeplitz型和Hankel型矩阵的特征多项式的计算
Computing the Characteristic Polynomial of Generic Toeplitz-like and Hankel-like Matrices
摘要：
给出了计算域上Toeplitz、Hankel及更一般的Toeplitz+ Hankel类矩阵的零化多项式的新算法。我们的方法如下工程Coppermith的块Wiedemann方法与结构化的预测，最近已成功地应用于计算的二元结果。第一个婴儿步骤/巨大步骤的方法-直接使用已知的技术结构化矩阵-给出了一个随机蒙特卡罗算法的最小多项式的$n\times n$ Toeplitz或Hankel矩阵的位移秩$α$使用$\tilde O(n^{ω- c(ω)} α^{c(ω)})$算术运算，其中$ω$是矩阵乘法的指数和$c(2.373)\approx 0.523$的最佳已知值的$ω$。对于一般的Toeplitz+ Hankel类矩阵，第二种算法在位移秩被认为是常数时计算$\tilde O(n^{2-1/ω})$运算中的特征多项式。以前的算法需要$O(n^2)$操作，而这里给出的指数分别小于$ω$的最佳已知估计的$1.86$和$1.58$。
英文摘要：
New algorithms are presented for computing annihilating polynomials of Toeplitz, Hankel, and more generally Toeplitz+ Hankel-like matrices over a field. Our approach follows works on Coppersmith's block Wiedemann method with structured projections, which have been recently successfully applied for computing the bivariate resultant. A first baby-step/giant step approach -- directly derived using known techniques on structured matrices -- gives a randomized Monte Carlo algorithm for the minimal polynomial of an $n\times n$ Toeplitz or Hankel-like matrix of displacement rank $α$ using $\tilde O(n^{ω- c(ω)} α^{c(ω)})$ arithmetic operations, where $ω$ is the exponent of matrix multiplication and $c(2.373)\approx 0.523$ for the best known value of $ω$. For generic Toeplitz+Hankel-like matrices a second algorithm computes the characteristic polynomial in $\tilde O(n^{2-1/ω})$ operations when the displacement rank is considered constant. Previous algorithms required $O(n^2)$ operations while the exponents presented here are respectively less than $1.86$ and $1.58$ with the best known estimate for $ω$.
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