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

07/19/2020
by   Michel Crouzeix, et al.
0

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 K^2 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 K^2 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.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset