Error Bounds for Lanczos-Based Matrix Function Approximation

نویسندگان

چکیده

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

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ژورنال

عنوان ژورنال: SIAM Journal on Matrix Analysis and Applications

سال: 2022

ISSN: ['1095-7162', '0895-4798']

DOI: https://doi.org/10.1137/21m1427784