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Pré-Publication, Document De Travail Année : 2020

A bivariate extension of the Crouzeix-Palencia result with an application to Fréchet derivatives of matrix functions

Résumé

A result by Crouzeix and Palencia states that the spectral norm of a matrix function $f(A)$ is bounded by $K = 1+\sqrt{2}$ times the maximum of $f$ on $W(A)$, the numerical range of $A$. The purpose of this work is to point out that this result extends to a certain notion of bivariate matrix functions; the spectral norm of $f\{A,B\}$ is bounded by $K^2$ times the maximum of $f$ on $W(A)\times W(B)$. As a special case, it follows that the spectral norm of the Fréchet derivative of $f(A)$ is bounded by $K^2$ times the maximum of $f^\prime$ on $W(A)$. An application to the convergence analysis of certain Krylov subspace methods and the extension to functions in more than two variables are discussed.
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Dates et versions

hal-04307472 , version 1 (26-11-2023)

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Michel Crouzeix, Daniel Kressner. A bivariate extension of the Crouzeix-Palencia result with an application to Fréchet derivatives of matrix functions. 2023. ⟨hal-04307472⟩
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