We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(\mathbf{A}) \mathbf{b}$ when $\mathbf{A}$ is a Hermitian matrix and $\mathbf{b}$ is a given mathbftor. Assuming that $f : \mathbb{C} \rightarrow \mathbb{C}$ is piecewise analytic, we give a framework, based on the Cauchy integral formula, which can be used to derive {\em a priori} and \emph{a posteriori} error bounds for Lanczos-FA in terms of the error of Lanczos used to solve linear systems. Unlike many error bounds for Lanczos-FA, these bounds account for fine-grained properties of the spectrum of $\mathbf{A}$, such as clustered or isolated eigenvalues. Our results are derived assuming exact arithmetic, but we show that they are easily extended to finite precision computations using existing theory about the Lanczos algorithm in finite precision. We also provide generalized bounds for the Lanczos method used to approximate quadratic forms $\mathbf{b}^\textsf{H} f(\mathbf{A}) \mathbf{b}$, and demonstrate the effectiveness of our bounds with numerical experiments.

翻译：我们分析矩阵函数近似( Lanczos) 的 Lanczos 方法( Lanczos 方法 ) (Lanczos- FA), 这是计算 $f (\ mathbf{A})\ mathbf{A} 的迭代算法, 当$\ mathbf{A} 是一个 Hermitian 的矩阵和$\ mathb{b} 是一个给定的数学ftor 。 假设$f:\ mathb{C} \ rightrow \ mathb{C} 是一个小分析器, 我们给出了一个框架, 这个框架可以用来计算 $( mathb{A} ) 和\ mapha postioria} 的元值, 当 以 兰czos 用于解决线性体系的错误来计算 时, 计算 。 兰茨- falfb} 错误是用来计算精度的精度 。 我们用精确度 的精确度计算方法来得出 精确度, 我们的精确度的精确度的计算方法也提供了我们现有的精确度。