We study the widely used Lanczos method for approximating the action of a matrix function $f(\mathbf{A})$ on a vector $\mathbf{b}$ (Lanczos-FA). For the function $f(\mathbf{A})=\mathbf{A}^{-1}$, it is known that, when $\mathbf{A}$ is positive definite, the $\mathbf{A}$-norm error of Lanczos-FA after $k$ iterations matches the optimal approximation from the Krylov subspace of degree $k$ generated by $\mathbf A$ and $\mathbf b$. In this work, we ask whether Lanczos-FA also obtains similarly strong optimality guarantees for other functions $f$. We partially answer this question by showing that, up to a multiplicative approximation factor, Lanczos-FA also matches the optimal approximation from the Krylov subspace for any rational function with real poles outside the interval containing $\mathbf{A}$'s eigenvalues. The approximation factor depends on the degree of $f$'s denominator and the condition number of $\mathbf{A}$, but not on the number of iterations $k$. This result provides theoretical justification for the excellent performance of Lanczos-FA on functions that are well approximated by rationals. Additionally, using different techniques, we prove that Lanczos-FA achieves a weaker notion of optimality for the functions $\mathbf A^{1/2}$ and $\mathbf A^{-1/2}$. Experiments confirm that our new bounds more accurately predict the convergence of Lanczos-FA than existing bounds, in particular those based on uniform polynomial approximation.

翻译：我们研究了广泛使用的Lanczos方法（Lanczos-FA）用于近似矩阵函数$f(\mathbf{A})$作用在向量$\mathbf{b}$上的计算。对于函数$f(\mathbf{A})=\mathbf{A}^{-1}$，已知当$\mathbf{A}$正定时，经过$k$次迭代后Lanczos-FA的$\mathbf{A}$-范数误差与由$\mathbf A$和$\mathbf b$生成的$k$阶Krylov子空间的最优近似相匹配。本文探讨Lanczos-FA是否对其他函数$f$也能获得同样强的最优性保证。我们部分回答了该问题：对于分母极点均为实数且位于包含$\mathbf{A}$特征值的区间之外的任意有理函数，Lanczos-FA在乘法逼近因子意义下也与Krylov子空间的最优近似相匹配。该逼近因子取决于$f$分母的次数以及$\mathbf{A}$的条件数，但与迭代次数$k$无关。这一结果为Lanczos-FA在可由有理函数良好近似的函数上表现优异提供了理论依据。此外，我们采用不同技术证明了Lanczos-FA对函数$\mathbf A^{1/2}$和$\mathbf A^{-1/2}$满足较弱的近似最优性。实验证实，与现有界（尤其基于一致多项式近似的界）相比，我们的新界更准确地预测了Lanczos-FA的收敛行为。