2020/07/19 by Michel Crouzeix, Crouzeix, Michel, Daniel Kreßner +1 · 1 citation
Computer Science · Mathematics · #Matrix Theory and Algorithms #Mathematical functions and polynomials #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2007.09784
A result by Crouzeix and Palencia states that the spectral norm of a matrix function f(A) is bounded by K = 1+√(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 K2 times the maximum of f on W(A)× W(B). As a special case, it follows that the spectral norm of the Fréchet derivative of f(A) is bounded by K2 times the maximum of f^′ 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.