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 $\alpha$ using $\tilde O(n^{\omega - c(\omega)} \alpha^{c(\omega)})$ arithmetic operations, where $\omega$ is the exponent of matrix multiplication and $c(2.373)\approx 0.523$ for the best known value of $\omega$. For generic Toeplitz+Hankel-like matrices a second algorithm computes the characteristic polynomial in $\tilde O(n^{2-1/\omega})$ 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 $\omega$.
翻译:为计算托普利茨、汉克勒、以及更普遍的托普利茨和汉克式汉克特等基质在一字段上的折叠多式矩阵,提出了新的算法。我们采用的方法是Copersmith的块 Wiedemann 方法的工程和结构化预测,这些方法最近成功地用于计算双变量结果体。第一个婴儿步/基级方法 -- -- 直接使用结构化基质的已知技术推算出 -- -- 给出一个随机化的蒙特卡洛算法,用于一个最低多式多式的托普利茨或类似汉克尔的流离失所矩阵。对于通用托普利茨和汉克尔类基母体,使用 $+alpha$(n ⁇ omega-c)}\alphetfreyfret Wiede Wiedeeedhighen 计算操作。$\\\ c(omega)\\\\ omga}levelopen exmocal exmology exmoal exdeal ex develop lishal developments) $n On_On\\\\\\\\\\ listrations dismagistrations dismismisal dedeal developmental developmental dismisal de lexn On On===