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 some fixed $\varepsilon>0$. 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方法依赖于对$n\times n$矩阵$A$和向量$u_1,\ldots,u_m$进行迭代矩阵-向量乘积$A^k u_j$。所有迭代向量$A^k u_j$张成的空间存在一组特殊基——\emph{极大Krylov基}——该基由第一个向量$u_1$的迭代序列$u_1, Au_1, A^2u_1,\ldots$构成，直至出现线性相关性，随后以相同方式迭代后续向量直至获得完整基。寻找极小多项式与Frobenius标准型与计算极大Krylov基密切相关。在本文之前，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$O(n^\omega \log(n))$来源于$A$的重复平方运算与对数次高斯消元。此处$\omega>2$是基域上矩阵乘法的可行指数。我们提出一种算法，当$m \in O(n)$时可在$O(n^\omega\log\log(n))$次域运算内计算极大Krylov基，若$m\in O(n/\log(n)^c)$（其中$c>0$为固定实数）甚至可达$O(n^\omega)$复杂度。由此我们证明，Frobenius标准型及其变换矩阵可在确定性复杂度$O(n^\omega (\log\log(n))^2)$内计算，因此当$\log(k) \in O(n^{\omega-1-\varepsilon})$（$\varepsilon>0$为固定值）时，矩阵幂运算~$A^k$可在后述复杂度内完成。这些改进的关键思想在于利用针对平均次数为$n/m$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计算。