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• W3175126076 abstract "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." @default.
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• W3175126076 date "2022-05-12" @default.
• W3175126076 modified "2023-10-16" @default.
• W3175126076 title "Error Bounds for Lanczos-Based Matrix Function Approximation" @default.
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• W3175126076 doi "https://doi.org/10.1137/21m1427784" @default.
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