The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n \times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the method's error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial uniformly approximating $f(x)$ on the range $[\lambda_{min}(A), \lambda_{max}(A)]$. However, despite decades of work, it has been unclear if this powerful guarantee holds in finite precision. We resolve this problem, proving that when $\max_{x \in [\lambda_{min}, \lambda_{max}]}|f(x)| \le C$, Lanczos essentially matches the exact arithmetic guarantee if computations use roughly $\log(nC\|A\|)$ bits of precision. Our proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating $f(x)$. We also study the special case of $f(A) = A^{-1}$, where stronger guarantees hold. In exact arithmetic Lanczos performs as well as the best polynomial approximating $1/x$ at each of $A$'s eigenvalues, rather than on the full eigenvalue range. In seminal work, Greenbaum gives an approach to extending this bound to finite precision: she proves that finite precision Lanczos and the related CG method match any polynomial approximating $1/x$ in a tiny range around each eigenvalue [Gre89]. For $A^{-1}$, this bound appears stronger than ours. However, we exhibit matrices with condition number $\kappa$ where exact arithmetic Lanczos converges in $polylog(\kappa)$ iterations, but Greenbaum's bound predicts $\Omega(\kappa^{1/5})$ iterations. It thus cannot offer significant improvement over the $O(\kappa^{1/2})$ bound achievable via our result. Our analysis raises the question of if convergence in less than $poly(\kappa)$ iterations can be expected in finite precision, even for matrices with clustered, skewed, or otherwise favorable eigenvalue distributions.

翻译：无处不在的Lanczos方法可以逼近任意对称$n \times n$矩阵$A$、向量$x$和函数$f$所对应的$f(A)x$。在精确算术中，该方法经过$k$次迭代后的误差受限于在$[\lambda_{min}(A), \lambda_{max}(A)]$区间上一致逼近$f(x)$的最佳$k$次多项式的误差。然而，尽管经过数十年的研究，这个强有力的保证在有限精度下是否成立仍不明确。我们解决了这个问题，证明了当$\max_{x \in [\lambda_{min}, \lambda_{max}]}|f(x)| \le C$时，若计算使用大约$\log(nC\|A\|)$比特的精度，Lanczos方法本质上能达到精确算术的保证。我们的证明扩展了Druskin和Knizhnerman的工作[DK91]，利用经典切比雪夫递推关系的稳定性来界定逼近$f(x)$的任意多项式的稳定性。我们还研究了$f(A) = A^{-1}$的特殊情况，此时存在更强的保证。在精确算术中，Lanczos方法的性能相当于在$A$的每个特征值处逼近$1/x$的最佳多项式，而非在整个特征值区间上。在开创性工作中，Greenbaum提出了一种将该界限推广到有限精度的方法：她证明了有限精度下的Lanczos及相关共轭梯度法能够匹配在每个特征值附近微小区间内逼近$1/x$的任何多项式[Gre89]。对于$A^{-1}$，这个界限看似比我们的更强。然而，我们构造了条件数$\kappa$的矩阵实例，其中精确算术Lanczos在$polylog(\kappa)$次迭代内收敛，但Greenbaum的界限预测需要$\Omega(\kappa^{1/5})$次迭代。因此，该界限无法显著优于通过我们的结果可实现的$O(\kappa^{1/2})$界限。我们的分析提出了一个问题：即使在特征值分布具有聚类性、偏斜性或其他有利特性的矩阵中，是否仍可预期在有限精度下以少于$poly(\kappa)$次迭代实现收敛。