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 vector. 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 a priori and 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 $f (\ mathbf{A}) 的迭代算法, 当$\ mathbf{A} 是一个 Hermitian 矩阵和 $\ mathbf{b} 美元是给定矢量时, 当$\ maczos 是一个 Hermitian 矩阵和 $\ mathb{b} 时, 我们用 Caccus 组合或孤立的 fegenvalual 值来计算, 我们给出了一个框架, 这个框架可以用来为 Lanczos- FA 计算前端值和 数级值的后序值错误 。 与 Lanczos- Fax 的多个误算法不同, 这些误算法是用于 $\ frif 的精度 。