Bounding hermite matrix polynomials

For the Hermite matrix polynomials Hn(χ, A) in which the parameter A is a matrix whose eigenvalues lie in the open right half plane, a bound on ∥Hn(χ, A)∥2 is developed. It has the form of a simple expression in the integer n, multiplied by exp ((5/2)∥A∥2χ2), for any real χ. The bound is exploited to construct an algorithm for computing any matrix exponential to within an arbitrary tolerance which may be prescribed a priori. Numerical effectiveness of the computational process is illustrated.