Pseudospectra of exponential matrix polynomials

This paper extends a result of Graillat, which was based on Rump’s Theorem, to show that pseudospectra of matrix polynomials expressed in other bases, and also some nonlinear matrix functions, are unaffected by drawing the matrix coefficients from certain structured families. We show by example that this behaviour is not universal. This result is of interest because computation of structured pseudospectra is much more expensive in general than computation of unstructured pseudospectra, and in the cases covered by Rump’s Theorem there is no point going to the extra effort.The paper also contains some examples that may be of interest to the community, including an analytical solution of a nested exponential equation that shows why exponential polynomial decomposition algorithms may be of interest.