Improvement on the bound of Hermite matrix polynomials (Q630546)

A new upper bound on the 2-norm of Hermite matrix polynomials \(H_n(x,A)\) is derived, which improves results known from the literature, [see, \textit{E.~Defez}, \textit{A.~Hervás}, \textit{L.~Jódar} and \textit{A.~Law}, ``Bounding Hermite matrix polynomials,'' Math. Comput. Modelling 40, No.~1-2, 117--125 (2004; Zbl 1061.33007)]. Hereby, \(A\) is a matrix whose eigenvalues lie in the open right-half plane. Using this new bound a matrix version of the Riemann-Lebesgue lemma is proved. Furthermore a new procedure for computing an approximation of the matrix exponential \(\text{e}^A\) with a prescribed accuracy is presented. Finally, a numerical example illustrates the computation of \(\text{e}^A\).