Krylov methods rely on iterated matrix-vector products $A^k u_j$ for an $n\times n$ matrix $A$ and vectors $u_1,\ldots,u_m$. The space spanned by all iterates $A^k u_j$ admits a particular basis -- the \emph{maximal Krylov basis} -- which consists of iterates of the first vector $u_1, Au_1, A^2u_1,\ldots$, until reaching linear dependency, then iterating similarly the subsequent vectors until a basis is obtained. Finding minimal polynomials and Frobenius normal forms is closely related to computing maximal Krylov bases. The fastest way to produce these bases was, until this paper, Keller-Gehrig's 1985 algorithm whose complexity bound $O(n^\omega \log(n))$ comes from repeated squarings of $A$ and logarithmically many Gaussian eliminations. Here $\omega>2$ is a feasible exponent for matrix multiplication over the base field. We present an algorithm computing the maximal Krylov basis in $O(n^\omega\log\log(n))$ field operations when $m \in O(n)$, and even $O(n^\omega)$ as soon as $m\in O(n/\log(n)^c)$ for some fixed real $c>0$. As a consequence, we show that the Frobenius normal form together with a transformation matrix can be computed deterministically in $O(n^\omega \log\log(n)^2)$, and therefore matrix exponentiation~$A^k$ can be performed in the latter complexity if $\log(k) \in O(n^{\omega-1-\varepsilon})$, for $\varepsilon>0$. A key idea for these improvements is to rely on fast algorithms for $m\times m$ polynomial matrices of average degree $n/m$, involving high-order lifting and minimal kernel bases.

翻译：Krylov方法依赖于矩阵-向量乘积的迭代计算 $A^k u_j$，其中 $A$ 为 $n\times n$ 矩阵，$u_1,\ldots,u_m$ 为向量。所有迭代向量 $A^k u_j$ 张成的空间存在一组特殊基——*最大Krylov基*——该基由首个向量 $u_1, Au_1, A^2u_1,\ldots$ 的迭代序列构成，直至出现线性相关，随后类似地对后续向量进行迭代，直至获得完整基。极小多项式与Frobenius标准型的求解与最大Krylov基的计算密切相关。在此之前，生成这些基的最快方法是Keller-Gehrig于1985年提出的算法，其复杂度上界 $O(n^\omega \log(n))$ 源于矩阵 $A$ 的重复平方运算及对数级别的Gauss消元。此处 $\omega>2$ 为基域上矩阵乘法的可行指数。我们提出一种新算法，当 $m \in O(n)$ 时，可在 $O(n^\omega\log\log(n))$ 次域运算内计算最大Krylov基；若存在常数 $c>0$ 使得 $m\in O(n/\log(n)^c)$，则复杂度可进一步降至 $O(n^\omega)$。作为推论，我们证明Frobenius标准型及变换矩阵可在 $O(n^\omega \log\log(n)^2)$ 内确定性地计算，且当 $\log(k) \in O(n^{\omega-1-\varepsilon})$（$\varepsilon>0$）时，矩阵幂 $A^k$ 的运算具有相同复杂度。这些改进的关键在于采用平均度数为 $n/m$ 的 $m\times m$ 多项式矩阵的快速算法，涉及高阶提升与极小核基技术。