We study the Lanczos method for approximating the action of a symmetric matrix function $f(\mathbf{A})$ on a vector $\mathbf{b}$ (Lanczos-FA). For the function $\mathbf{A}^{-1}$, it is known that the error of Lanczos-FA after $k$ iterations matches the error of the best approximation from the Krylov subspace of degree $k$ when $\vec{A}$ is positive definite. We prove that the same holds, up to a multiplicative approximation factor, when $f$ is a rational function with no poles in the interval containing $\mathbf{A}$'s eigenvalues. The approximation factor depends the degree of $f$'s denominator and the condition number of $\mathbf{A}$, but not on the number of iterations $k$. Experiments confirm that our bound accurately predicts the convergence of Lanczos-FA. Moreover, we believe that our result provides strong theoretical justification for the excellent practical performance that has long by observed of the Lanczos method, both for approximating rational functions and functions like $\mathbf{A}^{-1/2}\mathbf{b}$ that are well approximated by rationals.

翻译：我们研究用Lanczos方法逼近对称矩阵函数$f(\mathbf{A})$作用于向量$\mathbf{b}$（即Lanczos-FA）的效果。对于函数$\mathbf{A}^{-1}$，已知当$\vec{A}$正定时，经过$k$次迭代的Lanczos-FA误差与从$k$阶Krylov子空间得到的最佳逼近误差一致。我们证明，当$f$为在包含$\mathbf{A}$特征值的区间内无极点的有理函数时，上述结论仍然成立，仅需乘以一个乘法逼近因子。该逼近因子依赖于$f$分母的次数和$\mathbf{A}$的条件数，但与迭代次数$k$无关。实验证实我们的界限能准确预测Lanczos-FA的收敛性。此外，我们认为该结果为Lanczos方法长期以来在逼近有理函数以及可用有理函数良好逼近的函数（如$\mathbf{A}^{-1/2}\mathbf{b}$）时所表现出的优异实践性能提供了强有力的理论依据。