We obtain an expression for the error in the approximation of $f(A) \boldsymbol{b}$ and $\boldsymbol{b}^T f(A) \boldsymbol{b}$ with rational Krylov methods, where $A$ is a symmetric matrix, $\boldsymbol{b}$ is a vector and the function $f$ admits an integral representation. The error expression is obtained by linking the matrix function error with the error in the approximate solution of shifted linear systems using the same rational Krylov subspace, and it can be exploited to derive both a priori and a posteriori error bounds. The error bounds are a generalization of the ones given in [T. Chen, A. Greenbaum, C. Musco, C. Musco, SIAM J. Matrix Anal. Appl., 43 (2022), pp. 787--811] (arXiv:2106.09806) for the Lanczos method for matrix functions. A technique that we employ in the rational Krylov context can also be applied to refine the bounds for the Lanczos case.

翻译：我们得到了有理Krylov方法逼近$f(A) \boldsymbol{b}$和$\boldsymbol{b}^T f(A) \boldsymbol{b}$的误差表达式，其中$A$是对称矩阵，$\boldsymbol{b}$是向量，函数$f$允许积分表示。该误差表达式通过将矩阵函数误差与使用相同有理Krylov子空间求解移位线性系统的近似误差相关联而获得，并可加以利用来推导先验和后验误差界。这些误差界是对[T. Chen, A. Greenbaum, C. Musco, C. Musco, SIAM J. Matrix Anal. Appl., 43 (2022), pp. 787--811] (arXiv:2106.09806)中关于矩阵函数Lanczos方法所给误差界的推广。我们在有理Krylov语境中采用的一种技术也可应用于改进Lanczos情形下的误差界。