Computing Krylov iterates in the time of matrix multiplication
| Autor: | Neiger, Vincent, Pernet, Clément, Villard, Gilles |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1145/3666000.3669715 |
| Popis: | 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. Comment: 10 pages, 5 algorithms |
| Databáze: | arXiv |
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